Microwave and RF Engineering

Microwave and RF Engineering Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & S...

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Microwave and RF Engineering

Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-75862-5

Wiley Series in Microwave and Optical Engineering KAI CHANG, Editor Texas A&M University FIBER-OPTIC COMMUNICATION SYSTEMS, Third Edition, Govind P. Agrawal ASYMMETRIC PASSIVE COMPONENTS IN MICROWAVE INTEGRATED CIRCUITS, Hee-Ran Ahn COHERENT OPTICAL COMMUNICATIONS SYSTEMS, Silvello Betti, Giancarlo De Marchis, and Eugenio Iannone PHASED ARRAY ANTENNAS: FLOQUET ANALYSIS, SYNTHESIS, BFNs, AND ACTIVE ARRAY SYSTEMS, Arun K. Bhattacharyya HIGH-FREQUENCY ELECTROMAGNETIC TECHNIQUES: RECENT ADVANCES AND APPLICATIONS, Asoke K. Bhattacharyya RADIO PROPAGATION AND ADAPTIVE ANTENNAS FOR WIRELESS COMMUNICATION LINKS: TERRESTRIAL, ATMOSPHERIC, AND IONOSPHERIC, Nathan Blaunstein and Christos G. Christodoulou COMPUTATIONAL METHODS FOR ELECTROMAGNETICS AND MICROWAVES, Richard C. Booton, Jr. ELECTROMAGNETIC SHIELDING, Salvatore Celozzi, Rodolfo Araneo, and Giampiero Lovat MICROWAVE RING CIRCUITS AND ANTENNAS, Kai Chang MICROWAVE SOLID-STATE CIRCUITS AND APPLICATIONS, Kai Chang RF AND MICROWAVE WIRELESS SYSTEMS, Kai Chang RF AND MICROWAVE CIRCUIT AND COMPONENT DESIGN FOR WIRELESS SYSTEMS, Kai Chang, Inder Bahl, and Vijay Nair MICROWAVE RING CIRCUITS AND RELATED STRUCTURES, Second Edition, Kai Chang and Lung-Hwa Hsieh MULTIRESOLUTION TIME DOMAIN SCHEME FOR ELECTROMAGNETIC ENGINEERING, Yinchao Chen, Qunsheng Cao, and Raj Mittra HIGH EFFICIENCY RF AND MICROWAVE SOLID STATE POWER AMPLIFIERS, Paolo Colantonio, Franco Giannini and Ernesto Limiti DIODE LASERS AND PHOTONIC INTEGRATED CIRCUITS, Larry Coldren and Scott Corzine RADIO FREQUENCY CIRCUIT DESIGN, W. Alan Davis and Krishna Agarwal MULTICONDUCTOR TRANSMISSION-LINE STRUCTURES: MODAL ANALYSIS TECHNIQUES, J. A. Brand~ ao Faria PHASED ARRAY-BASED SYSTEMS AND APPLICATIONS, Nick Fourikis FUNDAMENTALS OF MICROWAVE TRANSMISSION LINES, Jon C. Freeman OPTICAL SEMICONDUCTOR DEVICES, Mitsuo Fukuda MICROSTRIP CIRCUITS, Fred Gardiol HIGH-SPEED VLSI INTERCONNECTIONS, Second Edition, Ashok K. Goel FUNDAMENTALS OF WAVELETS: THEORY, ALGORITHMS, AND APPLICATIONS, Jaideva C. Goswami and Andrew K. Chan HIGH-FREQUENCY ANALOG INTEGRATED CIRCUIT DESIGN, Ravender Goyal (ed.) ANALYSIS AND DESIGN OF INTEGRATED CIRCUIT ANTENNA MODULES, K. C. Gupta and Peter S. Hall PHASED ARRAY ANTENNAS, R. C. Hansen STRIPLINE CIRCULATORS, Joseph Helszajn THE STRIPLINE CIRCULATOR: THEORY AND PRACTICE, Joseph Helszajn LOCALIZED WAVES, Hugo E. Hern andez-Figueroa, Michel Zamboni-Rached, and Erasmo Recami (eds.) MICROSTRIP FILTERS FOR RF/MICROWAVE APPLICATIONS, Jia-Sheng Hong and M. J. Lancaster MICROWAVE APPROACH TO HIGHLY IRREGULAR FIBER OPTICS, Huang Hung-Chia NONLINEAR OPTICAL COMMUNICATION NETWORKS, Eugenio Iannone, Francesco Matera, Antonio Mecozzi, and Marina Settembre FINITE ELEMENT SOFTWARE FOR MICROWAVE ENGINEERING, Tatsuo Itoh, Giuseppe Pelosi, and Peter P. Silvester (eds.)

INFRARED TECHNOLOGY: APPLICATIONS TO ELECTROOPTICS, PHOTONIC DEVICES, AND SENSORS, A. R. Jha SUPERCONDUCTOR TECHNOLOGY: APPLICATIONS TO MICROWAVE, ELECTRO-OPTICS, ELECTRICAL MACHINES, AND PROPULSION SYSTEMS, A. R. Jha OPTICAL COMPUTING: AN INTRODUCTION, M. A. Karim and A. S. S. Awwal INTRODUCTION TO ELECTROMAGNETIC AND MICROWAVE ENGINEERING, Paul R. Karmel, Gabriel D. Colef, and Raymond L. Camisa MILLIMETER WAVE OPTICAL DIELECTRIC INTEGRATED GUIDES AND CIRCUITS, Shiban K. Koul ADVANCED INTEGRATED COMMUNICATION MICROSYSTEMS, Joy Laskar, Sudipto Chakraborty, Manos Tentzeris, Franklin Bien, and Anh-Vu Pham MICROWAVE DEVICES, CIRCUITS AND THEIR INTERACTION, Charles A. Lee and G. Conrad Dalman ADVANCES IN MICROSTRIP AND PRINTED ANTENNAS, Kai-Fong Lee and Wei Chen (eds.) SPHEROIDAL WAVE FUNCTIONS IN ELECTROMAGNETIC THEORY, Le-Wei Li, Xiao-Kang Kang, and Mook-Seng Leong ARITHMETIC AND LOGIC IN COMPUTER SYSTEMS, Mi Lu OPTICAL FILTER DESIGN AND ANALYSIS: A SIGNAL PROCESSING APPROACH, Christi K. Madsen and Jian H. Zhao THEORY AND PRACTICE OF INFRARED TECHNOLOGY FOR NONDESTRUCTIVE TESTING, Xavier P. V. Maldague METAMATERIALS WITH NEGATIVE PARAMETERS: THEORY, DESIGN, AND MICROWAVE APPLICATIONS, Ricardo Marqu es, Ferran Martıń, and Mario Sorolla OPTOELECTRONIC PACKAGING, A. R. Mickelson, N. R. Basavanhally, and Y. C. Lee (eds.) OPTICAL CHARACTER RECOGNITION, Shunji Mori, Hirobumi Nishida, and Hiromitsu Yamada ANTENNAS FOR RADAR AND COMMUNICATIONS: A POLARIMETRIC APPROACH, Harold Mott INTEGRATED ACTIVE ANTENNAS AND SPATIAL POWER COMBINING, Julio A. Navarro and Kai Chang ANALYSIS METHODS FOR RF, MICROWAVE, AND MILLIMETER-WAVE PLANAR TRANSMISSION LINE STRUCTURES, Cam Nguyen FREQUENCY CONTROL OF SEMICONDUCTOR LASERS, Motoichi Ohtsu (ed.) WAVELETS IN ELECTROMAGNETICS AND DEVICE MODELING, George W. Pan OPTICAL SWITCHING, Georgios Papadimitriou, Chrisoula Papazoglou, and Andreas S. Pomportsis SOLAR CELLS AND THEIR APPLICATIONS, Larry D. Partain (ed.) A complete list of the titles in this series appears at the end of this volume. ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES, Clayton R. Paul INTRODUCTION TO ELECTROMAGNETIC COMPATIBILITY, Second Edition, Clayton R. Paul ADAPTIVE OPTICS FOR VISION SCIENCE: PRINCIPLES, PRACTICES, DESIGN AND APPLICATIONS, Jason Porter, Hope Queener, Julianna Lin, Karen Thorn, and Abdul Awwal (eds.) ELECTROMAGNETIC OPTIMIZATION BY GENETIC ALGORITHMS, Yahya Rahmat-Samii and Eric Michielssen (eds.) INTRODUCTION TO HIGH-SPEED ELECTRONICS AND OPTOELECTRONICS, Leonard M. Riaziat NEW FRONTIERS IN MEDICAL DEVICE TECHNOLOGY, Arye Rosen and Harel Rosen (eds.) ELECTROMAGNETIC PROPAGATION IN MULTI-MODE RANDOM MEDIA, Harrison E. Rowe ELECTROMAGNETIC PROPAGATION IN ONE-DIMENSIONAL RANDOM MEDIA, Harrison E. Rowe HISTORY OF WIRELESS, Tapan K. Sarkar, Robert J. Mailloux, Arthur A. Oliner, Magdalena Salazar-Palma, and Dipak L. Sengupta PHYSICS OF MULTIANTENNA SYSTEMS AND BROADBAND PROCESSING, Tapan K. Sarkar, Magdalena Salazar-Palma, and Eric L. Mokole SMART ANTENNAS, Tapan K. Sarkar, Michael C. Wicks, Magdalena Salazar-Palma, and Robert J. Bonneau NONLINEAR OPTICS, E. G. Sauter

APPLIED ELECTROMAGNETICS AND ELECTROMAGNETIC COMPATIBILITY, Dipak L. Sengupta and Valdis V. Liepa COPLANAR WAVEGUIDE CIRCUITS, COMPONENTS, AND SYSTEMS, Rainee N. Simons ELECTROMAGNETIC FIELDS IN UNCONVENTIONAL MATERIALS AND STRUCTURES, Onkar N. Singh and Akhlesh Lakhtakia (eds.) MICROWAVE AND RF ENGINEERING, Roberto Sorrentino and Giovanni Bianchi ANALYSIS AND DESIGN OF AUTONOMOUS MICROWAVE CIRCUITS, Almudena Su arez ELECTRON BEAMS AND MICROWAVE VACUUM ELECTRONICS, Shulim E. Tsimring FUNDAMENTALS OF GLOBAL POSITIONING SYSTEM RECEIVERS: A SOFTWARE APPROACH, Second Edition, James Bao-yen Tsui RF/MICROWAVE INTERACTION WITH BIOLOGICAL TISSUES, Andr e Vander Vorst, Arye Rosen, and Youji Kotsuka InP-BASED MATERIALS AND DEVICES: PHYSICS AND TECHNOLOGY, Osamu Wada and Hideki Hasegawa (eds.) COMPACT AND BROADBAND MICROSTRIP ANTENNAS, Kin-Lu Wong DESIGN OF NONPLANAR MICROSTRIP ANTENNAS AND TRANSMISSION LINES, Kin-Lu Wong PLANAR ANTENNAS FOR WIRELESS COMMUNICATIONS, Kin-Lu Wong FREQUENCY SELECTIVE SURFACE AND GRID ARRAY, T. K. Wu (ed.) ACTIVE AND QUASI-OPTICAL ARRAYS FOR SOLID-STATE POWER COMBINING, Robert A. York and Zoya B. Popovic’ (eds.) OPTICAL SIGNAL PROCESSING, COMPUTING AND NEURAL NETWORKS, Francis T. S. Yu and Suganda Jutamulia SiGe, GaAs, AND InP HETEROJUNCTION BIPOLAR TRANSISTORS, Jiann Yuan ELECTRODYNAMICS OF SOLIDS AND MICROWAVE SUPERCONDUCTIVITY, Shu-Ang Zhou

Microwave and RF Engineering Roberto Sorrentino University of Perugia, Italy

Giovanni Bianchi Verigy Ltd, Bo¨blingen, Germany

This edition first published 2010 Ó 2010, John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloguing-in-Publication Data Sorrentino, Roberto. Microwave and RF engineering / R. Sorrentino, G. Bianchi. p. cm. Includes bibliographical references and index. ISBN 978-0-470-75862-5 (cloth) 1. Microwave devices. 2. Radio–Transmitters and transmission. I. Bianchi, Giovanni. II. Title. TK7876.S666 2010 621.381’3–dc22 2009051049 A catalogue record for this book is available from the British Library. ISBN: 978-0-470-75862-5 (Hbk) Set in 9/11pt, Times Roman by Thomson Digital, Noida, India Printed in Singapore by Markono Print Media Pte Ltd

Contents About the Authors Preface 1 Introduction 1.1 Microwaves and radio frequencies 1.2 Frequency bands 1.3 Applications Bibliography

xv xvii 1 1 4 6 8

2 Basic 2.1 2.2 2.3 2.4 2.5 2.6 2.7

electromagnetic theory Introduction Maxwell’s equations Time-harmonic EM fields; polarization of a vector Maxwell’s equations in the harmonic regime Boundary conditions Energy and power of the EM field; Poynting’s theorem Some fundamental theorems 2.7.1 Uniqueness theorem 2.7.2 Lorentz’s reciprocity theorem 2.7.3 Love’s equivalence theorem 2.8 Plane waves 2.9 Solution of the wave equation in rectangular coordinates 2.9.1 Plane waves: an alternative derivation 2.9.2 TEM waves 2.9.3 TE and TM waves 2.10 Reflection and transmission of plane waves; Snel’s laws 2.10.1 Snel’s laws; total reflection 2.10.2 Reflection and transmission (Fresnel’s) coefficients 2.10.3 Reflection from a conducting plane 2.11 Electrodynamic potentials Bibliography

9 9 9 12 14 15 17 19 19 19 20 21 22 24 25 26 27 28 31 34 36 38

3 Guided EM propagation 3.1 Introduction 3.2 Cylindrical structures; solution of Maxwell’s equations as TE, TM and TEM modes 3.3 Modes of propagation as transmission lines 3.4 Transmission lines as 1-D circuits 3.5 Phase velocity, group velocity and energy velocity 3.6 Properties of the transverse modal vectors et, ht; field expansion in a waveguide 3.7 Loss, attenuation and power handling in real waveguides

39 39 41 48 52 55 57 59

viii

CONTENTS 3.8 3.9 3.10 3.11 3.12 3.13 3.14

The rectangular waveguide The ridge waveguide The circular waveguide The coaxial cable The parallel-plate waveguide The stripline The microstrip line 3.14.1 The planar waveguide model 3.15 The coplanar waveguide 3.16 Coupled lines 3.16.1 Basic principles for EM analysis 3.16.2 Equivalent circuit modelling Bibliography

61 67 68 72 74 76 78 82 82 84 85 86 88

4 Microwave circuits 4.1 Introduction 4.2 Microwave circuit formulation 4.3 Terminated transmission lines 4.4 The Smith chart 4.5 Power flow 4.6 Matrix representations 4.6.1 The impedance matrix 4.6.2 The admittance matrix 4.6.3 The ABCD or chain matrix 4.6.4 The scattering matrix 4.7 Circuit model of a transmission line section 4.8 Shifting the reference planes 4.9 Loaded two-port network 4.10 Matrix description of coupled lines 4.11 Matching of coupled lines 4.12 Two-port networks using coupled-line sections Bibliography

91 91 91 94 97 105 109 109 110 111 112 119 123 124 125 126 127 129

5 Resonators and cavities 5.1 Introduction 5.2 The resonant condition 5.3 Quality factor or Q 5.4 Transmission line resonators 5.5 Planar resonators 5.6 Cavity resonators 5.7 Computation of the Q factor of a cavity resonator 5.8 Dielectric resonators 5.9 Expansion of EM fields 5.9.1 Helmholtz’s theorem 5.9.2 Electric and magnetic eigenvectors 5.9.3 General solution of Maxwell’s equations in a cavity 5.9.4 Resonances in ideal closed cavities 5.9.5 The cavity with one or two outputs 5.9.6 Excitation of cavity resonators Bibliography

131 131 131 134 136 139 142 144 146 147 148 148 153 154 155 157 161

CONTENTS

ix

6 Impedance matching 6.1 Introduction 6.2 Fano’s bound 6.3 Quarter-wavelength transformer 6.4 Multi-section quarter-wavelength transformers 6.4.1 The binomial transformer 6.4.2 Chebyshev polynomials; the Chebyshev transformer 6.5 Line and stub transformers; stub tuners 6.6 Lumped L networks Bibliography Simulation files

163 163 163 165 167 171 172 178 180 185 185

7 Passive microwave components 7.1 Introduction 7.2 Matched loads 7.3 Movable short circuit 7.4 Attenuators 7.5 Fixed phase shifters 7.5.1 Loaded-line phase shifters 7.5.2 Reflection-type phase shifters 7.6 Junctions and interconnections 7.6.1 Guide-to-coaxial cable transition 7.6.2 Coaxial-to-microstrip transition 7.7 Dividers and combiners 7.7.1 The Wilkinson divider 7.7.2 Hybrid junctions 7.7.3 Directional couplers 7.8 Lumped element realizations 7.9 Multi-beam forming networks 7.9.1 The Butler matrix 7.9.2 The Blass matrix 7.9.3 The Rotman lens 7.10 Non-reciprocal components 7.10.1 Isolator 7.10.2 Circulator Bibliography Simulation files

187 187 187 188 190 193 193 194 195 198 203 204 205 209 211 221 223 224 225 227 230 232 232 234 235

8 Microwave filters 8.1 Introduction 8.2 Definitions 8.3 Lowdpass prototype 8.3.1 Butterworth filters 8.3.2 Chebyshev filters 8.3.3 Cauer filters 8.3.4 Synthesis of the lowdpass prototype 8.4 Semi-lumped lowdpass filters 8.5 Frequency transformations 8.5.1 Lowdpass to highpass transformation 8.5.2 Lowdpass to bandpass transformation

237 237 237 239 240 240 244 245 250 254 255 257

x

9

CONTENTS 8.5.3 Lowdpass to bandstop transformation 8.5.4 Richards transformation 8.6 Kuroda identities 8.7 Immittance inverters 8.7.1 Filters with line-coupled short-circuit stubs 8.7.2 Parallel-coupled filters 8.7.3 Comb-line filters Bibliography Simulation files

260 261 264 267 273 277 281 286 286

Basic 9.1 9.2 9.3 9.4

289 289 289 302 303 303 305 306 313 314 316 317 317 319 321 326 328 334 334 336 338 339 340 342 342 348 360 360

concepts for microwave component design Introduction Cascaded linear two-port networks Signal flow graphs Noise in two-port networks 9.4.1 Noise sources 9.4.2 Representation of noisy two-port networks 9.4.3 Noise figure and noise factor 9.4.4 Noise factor of cascaded networks 9.4.5 Noise bandwidth 9.5 Nonlinear two-port networks 9.5.1 Harmonic and intermodulation products 9.5.2 Harmonic distortion 9.5.3 Intermodulation distortion 9.5.4 Gain compression 9.5.5 Intercept points 9.5.6 Saturation and intercept point of cascaded two-port networks 9.6 Semiconductors devices 9.6.1 Basic semiconductor physics 9.6.2 Junction diode 9.6.3 Bipolar transistor 9.6.4 Junction field effect transistor 9.6.5 Metal oxide field effect transistor 9.7 Electrical models of high-frequency semiconductor devices 9.7.1 Linear models 9.7.2 Nonlinear semiconductor models Bibliography Related Files

10 Microwave control components 10.1 Introduction 10.2 Switches 10.2.1 PIN diode switches 10.2.2 FET switches 10.2.3 MEMS switches 10.2.4 Alternative multi-port switch structures 10.3 Variable attenuators 10.4 Phase shifters

363 363 363 368 375 379 385 389 400

CONTENTS 10.4.1 10.4.2 10.4.3 10.4.4 10.4.5 Bibliography Related files

True-delay and slow-wave phase shifters Reflection phase shifters Stepped phase shifters Binary phase shifters Final considerations on phase shifters

xi 402 404 407 408 412 412 413

11 Amplifiers 11.1 Introduction 11.2 Small-signal amplifiers 11.2.1 Gain definitions 11.2.2 Stability 11.2.3 Matching networks 11.2.4 Maximum gain impedance matching 11.3 Low-noise amplifiers 11.4 Design of trial amplifier 11.5 Power amplifiers 11.5.1 Output power optimization with negligible transistor parasitics 11.5.2 Output power optimization in presence of transistor parasitics 11.5.3 Load pull 11.5.4 Balanced amplifiers 11.5.5 PA classes 11.5.6 Amplifier linearization 11.5.7 Additional PA issues 11.6 Other amplifier configurations 11.6.1 Feedback amplifiers 11.6.2 Distributed amplifiers 11.6.3 Differential pairs 11.6.4 Active loads 11.6.5 Cascode configuration 11.7 Some examples of microwave amplifiers 11.7.1 Two-stage millimetre-wave amplifier 11.7.2 Low-noise amplifier Bibliography Related files

415 415 415 416 420 424 425 429 432 440 440 444 451 454 459 473 481 482 483 485 489 494 495 497 497 499 501 501

12 Oscillators 12.1 Introduction 12.2 General principles 12.3 Negative resistance oscillators 12.4 Positive feedback oscillators 12.5 Standard oscillator configuration 12.5.1 Inductively coupled oscillator 12.5.2 Inductive gate feedback oscillator 12.5.3 Hartley oscillator 12.5.4 Colpitts oscillator

503 503 503 508 512 518 521 523 525 526

xii

CONTENTS 12.5.5 Clapp oscillator 12.5.6 Differential oscillator 12.6 Design of a trial oscillator 12.7 Oscillator specifications 12.8 Special oscillators 12.8.1 Lumped element and transmission line oscillators 12.8.2 Cavity oscillators and dielectric resonator oscillators 12.8.3 Voltage-controlled oscillators 12.8.4 Push–push oscillators 12.8.5 Amplitude-stabilized oscillators 12.9 Design of a push –push microwave VCO Bibliography Related files

527 528 530 534 543 543 547 549 553 555 557 559 559

13 Frequency converters 13.1 Introduction 13.2 Detectors 13.2.1 Quadratic diode detector 13.2.2 Envelope detectors 13.2.3 FET detectors 13.3 Mixers 13.3.1 Product detector 13.3.2 Single-ended diode mixers 13.3.3 Singly balanced diode mixers 13.3.4 Doubly balanced diode mixers 13.3.5 Subharmonically pumped mixers 13.3.6 Image reject mixers 13.3.7 Suppression in presence of amplitude and phase imbalance 13.3.8 FET mixers 13.3.9 Mixers based on differential pairs 13.3.10 Mixer nonlinearities 13.4 Frequency multipliers Bibliography Related files

561 561 561 563 570 573 577 579 581 584 590 594 597 600 602 606 617 625 630 630

14 Microwave circuit technology 14.1 Introduction 14.2 Hybrid and monolithic integrated circuits 14.2.1 High-frequency PCB 14.2.2 Hybrid MICs 14.2.3 MMICs 14.2.4 Advanced hybrid MICs 14.2.5 Parasitic elements associated to physical devices 14.3 Basic MMIC elements 14.3.1 Transmission lines 14.3.2 Via holes 14.3.3 Resistors 14.3.4 Inductors

633 633 633 634 635 636 637 637 639 640 640 641 643

CONTENTS 14.3.5 Capacitors 14.3.6 Semiconductor devices 14.4 Simulation models and layout libraries 14.4.1 Single element models 14.4.2 Scalable models 14.4.3 Nonlinear models 14.4.4 MMIC statistical models 14.4.5 Temperature-dependent models 14.5 MMIC production technique 14.5.1 Lithography 14.5.2 On-wafer testing 14.5.3 Cut and selection 14.6 RFIC Bibliography

xiii 645 646 649 650 650 651 651 652 652 653 655 655 656 657

15 RF and microwave architectures 15.1 Introduction 15.2 Review of modulation theory 15.2.1 Amplitude modulation 15.2.2 Angular modulation 15.3 Transmitters 15.3.1 Direct modulation transmitters 15.3.2 Polar modulator 15.3.3 Cartesian modulator 15.3.4 Transmitters with frequency translation 15.4 Receivers 15.4.1 RF tuned receivers 15.4.2 Superetherodyne receivers 15.4.3 Zero-IF and low-IF receivers 15.4.4 Walking IF receivers 15.4.5 One practical IC-based receiver 15.4.6 Digital receivers 15.5 Further concepts on RF transmitters and receivers 15.5.1 Transceivers 15.5.2 CAD analysis of a radar transmitting subassembly 15.5.3 Receiver performance analysis 15.6 Special radio functional blocks 15.6.1 Quadrature signal generation 15.6.2 PLL 15.6.3 ALC and AGC 15.6.4 SDLVA Bibliography Related files

659 659 659 660 663 665 665 675 677 681 682 682 692 696 699 701 703 710 710 719 725 731 731 735 744 749 753 754

16 Numerical methods and CAD 16.1 Introduction 16.2 EM analysis 16.2.1 The method of moments

757 757 760 761

xiv

CONTENTS 16.2.2 16.2.3 16.2.4 16.2.5 16.3 Circuit 16.3.1

The finite difference method The FDTD method The finite element method The mode matching method analysis Linear analysis: the signal flow graph and the admittance matrix methods 16.3.2 Time domain nonlinear analysis 16.3.3 Frequency domain nonlinear analysis 16.4 Optimization 16.4.1 Definitions and basic concepts 16.4.2 Objective function 16.4.3 Constraints 16.4.4 Optimization methods Bibliography

763 766 770 771 780 780 785 786 788 789 790 791 791 792

17 Measurement instrumentation and techniques 17.1 Introduction 17.2 Power meters 17.3 Frequency meters 17.3.1 RF digital frequency meter 17.3.2 Microwave digital frequency meter 17.3.3 Frequency conversion frequency meters 17.3.4 Frequency conversion frequency meter without preselector 17.4 Spectrum analyzers 17.4.1 Panoramic receiver 17.4.2 Superheterodyne spectrum analyzer 17.5 Wide-band sampling oscilloscopes 17.6 Network analyzers 17.6.1 Scalar analyzers 17.6.2 Vector analyzers 17.6.3 Noise figure meters 17.7 Special test instruments 17.7.1 IFM 17.7.2 Complex test benches 17.7.3 Test instruments for non-electrical quantities Bibliography Related files Appendix A Useful relations from vector analysis and trigonometric function identities Appendix B Fourier transform Appendix C Orthogonality of the eigenvectors in ideal waveguides Appendix D Standard rectangular waveguides and coaxial cables Appendix E Symbols for electric diagrams Appendix F List of acronyms

795 795 795 798 798 799 800 802 803 803 806 809 816 817 821 833 837 837 843 846 849 849

Index

883

851 861 865 869 873 877

About the Authors Roberto Sorrentino received the Laurea degree in Electronic Engineering from the University of Rome ‘‘La Sapienza’’, Rome, Italy, in 1971, where he was an Associate Professor until 1986. From 1986 to 1990 he was a Professor at the University of Rome ‘‘Tor Vergata’’. Since 1990 he has been a Professor at the University of Perugia, Perugia, Italy. He has authored and co-authored over 100 technical papers in international journals, 300 refereed conference papers and three books in the area of the analysis and design of microwave passive circuits and antennas. He is an IEEE Fellow (1990), a recipient of the IEEE Third Millennium Medal (2000) and of the Distinguished Educator Award from IEEE MTT-S (2004). He was the President of the European Microwave Association from 1998 to 2009. Giovanni Bianchi received the Laurea degree in Electronic Engineering from the University of Rome ‘‘La Sapienza’’, Rome, Italy, in 1987. In 1988, he joined the microwave department of Elettronica S.p.A. where he was involved in microwave components (including GaAs MMICs) and subassembly design. He joined Motorola PCS in 2000, where he worked on GSM and WCDMA mobile phone design, and in 2004 joined SDS S.r.L where he was responsible for microwave designs. Since January 2008 he has worked as a R&D Engineer in the hardware/RF division at Verigy, and is an expert of high frequency theory and techniques. In his 23 years of design experience he has covered both passive and active microwave components, including filters, amplifiers, oscillators, and synthesizers. He is the author of four books (including the present one) as well as 12 papers.

Preface This book deals with a rather complex discipline that involves many different techniques and approaches: the result is a difficult and alluring subject at the same time. Most academic tradition focuses on the electromagnetic-related aspects of microwaves, i.e. on the science of the solution of Maxwell’s equations, which is quite difficult to divulgate because of the remarkable difficulties it involves. The electromagnetic theory is the basis of the high-frequency techniques, from radio frequency (RF) up to millimetre waves; therefore it must be well understood, in order to comprehend and dominate a variety of phenomena utilized in many applications, mainly – but not exclusively – in telecommunications. Microwaves, however, do not reduce to the electromagnetic theory. The microwave engineer, i.e. the designer operating with frequencies that are so high as to need a specific methodological approach, must have a basic knowledge of the electromagnetic theory, but must also be familiar with network theory, signal theory, linear and nonlinear circuits, and electronic technology – particularly microwave integrated circuits, CAD techniques and test instruments. Such considerations have motivated us to write this book. It differs from traditional microwave books because it includes several topics, such as semiconductor device modelling or test instruments, that are commonly considered at the boundary with other electronic disciplines, but are equally important for practising microwave engineers. This book is intended for both students and professionals. Therefore the topics are presented – wherever possible – at different levels of depth. The reader will find some topics, discussed in specific sections printed in line boxes, that can be ignored without lack of continuity of the discussion. In this way, we tried to circumvent the difficulties related to the conventional approach to microwaves, without losing a detailed and rigorous exposition. We are aware that many important topics have not been included in the book, particularly propagation in dielectric waveguides and optical fibres. The study of propagation in microstrips and printed circuits has been limited to a brief qualitative description. The same treatment applies to the analysis of many components and devices, as well as many other specific topics. The size limitation for the book to be manageable has imposed some exclusion. We hope nevertheless to have provided a useful tool for a first approach and for subsequent in-depth study aswell. Based on own different experience, we had to realize a book combining formal academic rigour with a practical approach useful for the designer. The conventional research-oriented list of subjects has thus been extended to cover a number of application-oriented aspects. Examples from actual engineering practice are included in all chapters. After the introduction to the field of microwaves and radio frequencies, the basic electromagnetic theory is concisely recalled in Chapter 2, with more emphasis on the propagation of plane waves. The reader is assumed already to have a background in the electromagnetic theory, so that this chapter serves as a reference and as a reminder. Chapter 3 is then devoted to the study of guided electromagnetic propagation along unlimited transmission lines. In contrast with most textbooks, the telegrapher’s equations are introduced here as a special case of the mode propagation in cylindrical waveguides. The conventional derivation from a lumped circuit model is presented in a subsequent section. Some of the most common guiding structures are discussed, including coupled transmission lines. The concept of a microwave circuit, a powerful model for the characterization of microwave structures, is introduced in Chapter 4. The chapter concerns the modelling of microwave structures and transmission lines of finite length, including the Smith chart, N-port circuits and terminated coupled lines. Chapters 5 to 8 are devoted to the study of various classes of passive microwave components. Chapter 5 deals with microwave cavities and resonators. The theory of resonant mode expansion is also introduced

xviii

PREFACE

as a significant theoretical approach to the solution of Maxwell’s equations in a volume. The matching of microwave circuits is treated in Chapter 6, a significant part being devoted to quarter-wave transformers. A number of microwave passive components are then presented in Chapter 7. The term passive is interpreted here as linear. Switches and tuning elements are not included in such components, but are discussed instead in Chapter 10. Chapter 7 is devoted to interconnections, the various types of directional couplers, dividers and combiners, in various technologies, including a brief description of microwave multi-beam forming networks. Non-reciprocal components are described briefly in the last sections of the chapter. Because of their importance in the design activity of the practising microwave engineer, microwave filters are treated in some detail in Chapter 8. The basic concepts needed for the study of control and active components in the subsequent chapters are introduced in Chapter 9. Chapter 10 is devoted to microwave control components: these are passive components using control devices, such as diodes and transistors, to operate on the microwave signal without increasing its associated energy. Microwave amplifiers based on solid state devices are dealt with in Chapter 11. Due to space limitations, the discussion is necessarily limited to the main concepts and to the most common configurations, specifically to the one-transistor amplifier with one input and one output matching network, including small-signal, low-noise and power amplifiers. The generation of microwaves is discussed in Chapter 12, which is devoted to oscillators, presenting the most common configurations and illustrating some analytical techniques. Frequency conversion, discussed in Chapter 13, is a fundamental technique in the nonlinear processing of microwave and RF signals, from detection to mixing to frequency multiplying. The technologies for the fabrication of microwave circuits are the subject of Chapter 14. Here attention, for obvious reasons, is confined to integrated circuits (ICs), spanning microwave integrated circuits (MICs) and the most sophisticated monolithic microwave integrated circuits (MMICs), including an overview of silicon radio frequency integrated circuits (RFICs). The system perspective is taken into consideration in Chapter 15 on RF and microwave architectures for transmitters and receivers, including a summary of basic modulation theory. Chapter 16 introduces the reader to the foundations of numerical methods and CAD techniques for microwave circuit design. Although much more space could gave been devoted to this important subject, we feel that we have provided enough fundamental information for the reader prior to consulting specialized books on the subject. Measurement and instrumentation are the subjects of the concluding chapter. We felt, indeed, that the microwave engineer should be conscious of the basic aspects related to such essential steps which are the final verification of the results of his or her work. In order to illustrate and extend the material presented in text form in Chapters 5 to 17 (except 14), 98simulation files have been developed and collected in a separate CD-ROM. These simulation files will allow the reader to ‘play with the numbers’ to see what happens to the circuit response when some parameters are changed, and to generate different examples from those already presented in the book, and so on. The CD also includes two setup programs to install the required application tools: namely, Ansoft Designer SV and SIMetrix. The simulation files are grouped in folders, one for each chapter. Each folder is further divided into subfolders, one per file type, i.e. Ansoft Designer SV, Mathcad and SIMetrix, which are the commercial programs we have employed. Although they have much wider capabilities, these programs have been adopted here for the following use: .

Ansoft Designer SV A functional subset of Ansoft Designer, the commercially distributed designmanagement environment and circuit simulator for RF and microwave hardware development. We used this program for linear S-parameter and noise analyses.

.

Mathcad A computer mathematical manipulation program to implement and simultaneously document mathematical calculations. The Mathcad visual format and user interface integrate the familiar standard mathematical notation with text and graphs in a single worksheet. Our use of Mathcad is in the analysis and synthesis of microwave/RF structures.

PREFACE .

xix

SIMetrix This comprises a SPICE simulator with a schematic editor, and a waveform viewer in a unified environment. We have used it here to provide the reader with examples of nonlinear circuit and subsystem analysis.

For detailed descriptions and/or to download updated versions, the interested reader can visit the respective websites: http://www.ansoft.com/, http://www.ptc.com and http://www.simetrix.co.uk/. We gratefully acknowledge the help received from Elisa Fratticcioli and Cristiano Tomassoni for text and figure editing and revision. Luca Pelliccia is gratefully acknowledged for carefully and patiently revising the whole text and removing many typos and mistakes. Michele Ancis, Simone Bastioli, Loris Caporali, Federico Casini, Paola Farinelli, Elisa Sbarra and Roberto Vincenti Gatti have kindly contributed to some parts of the book. Their contribution is acknowledged in the specific sections. We are aware that, in spite of many revisions, the book may need some improvement and that some mistakes may unavoidably still be present. We welcome input from anyone who has read this material and wishes to point out mistakes, make suggestions or ask questions to clarify any issue. Suggestions can be sent to us at out respective email addresses.

September 2009

Roberto Sorrentino ([email protected]) Giovanni Bianchi ([email protected])

1

Introduction 1.1 Microwaves and radio frequencies Technical terms sometimes have odd histories. The term microwaves, after being confined for decades to a restricted circle of specialists in radar, telecommunications and electromagnetism, has become, with the introduction of microwave ovens about 30 years ago, a popular term associated with cooking. This might be the reason why its use among specialists has declined somewhat, being often replaced by the ampler and more generic term radio frequencies (RFs) and, above 30 GHz, millimetre waves. At about the same time the term radio is being replaced by the term wireless, an expression coined and adopted by Marconi which has become fashionable in recent years. Microwaves were first introduced in the technical literature in 1932 by Nello Carrara, to designate those electromagnetic (EM) waves whose wavelength was smaller than 30 cm, i.e. the electromagnetic spectrum above 1 GHz [1]. In those years, the use of such high frequencies was motivated by the research on radar, for which many studies were launched and enormous resources spent worldwide.1 The simplest definition of microwaves is the one based on a precise interval of frequencies. Figure 1.1 depicts the full electromagnetic spectrum from long waves up to ultraviolet. According to the majority of textbooks, microwaves correspond to frequencies between 300 MHz and 300 GHz, corresponding to wavelengths between 1 millimetre and 1 metre. According to some sources, the lower limit is raised to 500 MHz or 1 GHz. The frequency range between 30 and 300 GHz is also referred to as millimetre waves, since the wavelengths are between 1 and 10 mm. This terminological uncertainty reflects the fact that there is no specific physical phenomenon identifying a precise frequency boundary. Also, RF does not correspond to a precise frequency range but indicates all frequencies employed in the radio technique, usually below the microwave range. Rather than a frequency range, microwaves actually identify a methodology, i.e. a specific approach to the study of electromagnetic phenomena. Such an approach is intermediate between the two other methodologies derived from Maxwell’s equations, namely circuit theory and optics. To be more precise, the discriminating element of the three methodologies is not really the frequency but rather the wavelength, or, better, the ratio between the wavelength and the dimensions of the circuits or objects where the EM field manifests itself.

1

The development of radar absorbed more funds than the atomic bomb (see [2], p. 22).


2

MICROWAVE AND RF ENGINEERING

Figure 1.1

The electromagnetic spectrum.

In the case of low-frequency EM fields, for which the wavelength is much larger than the dimensions of the circuits and the EM field propagation times from one point to another of the circuit are a small fraction of the period, one applies the lumped circuit theory which represents a simplification of Maxwell’s equations. In the opposite case, i.e. when the objects and the circuit elements are much larger than the wavelength, one can apply the optical laws which are another type of simplification of Maxwell’s equations. The microwave regime corresponds to those cases when the wavelengths are of the same order (roughly, from one-tenth to 10 times) as the circuit dimensions, so that neither one nor the other approximation is permissible: Maxwell’s equations must be solved in their entirety. Peculiar and often non-intuitive effects arise when the wavelength is comparable with the dimensions of the objects involved in the EM field. This confers a special difficulty on this discipline: circuit elements, such as the capacitors and inductors that are familiar in low-frequency ranges, not only assume totally unconventional shapes but do not actually exist as distinct regions of space that store only electric or magnetic energy. Because of their peculiarity, the intuitive perception of EM phenomena may sometimes be misleading. Special attention and specific expertise are therefore often required in the study of devices, circuits and systems operating at microwave frequencies. A typical phenomenon is wave diffraction from obstacles. As already mentioned, such a phenomenon is strictly related to the ratio between the wavelength l and the obstacle dimension d. The following example illustrates this point. While an acoustic signal whose wavelength is of the order of tenths of centimetres can easily reach a listener sitting behind a wall 2 or 3 metres high, the same is not possible for an optical signal, whose wavelength is of the order of fractions of a micrometre. While two people sitting on opposite sides of the wall can hear each other because the acoustic wave is diffracted by the edge of the wall, they cannot see each other because the optical wave is not significantly diffracted. The diffraction of waves by an aperture of width d created in a wall is illustrated in Figure 1.2 under different conditions when d l or d l. As can be seen, in the former case the wave is diffracted by the aperture edges so as to propagate beyond the wall in all directions. On the contrary, in the latter case, the wave propagates through the aperture in a straight fashion reaching only the points located in the direction of the incident wave.

INTRODUCTION

Figure 1.2

3

EM wave diffraction by an aperture for (a) d l and (b) d l.

Other phenomena difficult to perceive on the basis of simple intuition are due to evanescent waves, i.e. waves that are attenuated in a lossless medium.2 Indeed, such waves can produce interactions between distant objects and circuit elements producing energy exchanges between them. This is a phenomenon that is totally unexpected on the basis of normal experience of mechanical phenomena. One of these effects is the optical tunnel effect, which consists of the transmission of power through space regions where the EM wave does not propagate but is evanescent.3 The interaction through evanescent fields is also responsible for the altered responses of circuit elements when put in relatively close proximity, so that the circuit models of the isolated elements can no longer be employed. The distributed character of microwave circuits is responsible for other phenomena that do not occur at low frequencies when all circuit elements can be considered as lumped. A microwave filter never behaves as an ideal lumped element filter, which normally possesses one passband and one stopband, but contains a virtually unlimited number of spurious passbands. A microwave amplifier is not merely a diagram block with an associated gain (and possibly a noise figure) but has frequency-dependent gain and mismatch, distorts the signal, adds noise, and, in the worst case, self-oscillates. When, in the design of microwave circuits, we are confronted with the practical implementation of the theory, surprises may occur if we blindly trust the design tools at our disposal. It must be borne in mind that the simulators, though indispensable tools in the analysis and design of our circuits, are based on 2 3

See for example the phenomenon of total reflection described in Section 2.10.1. This has recently been proposed as a possible way to provide a means to recharge the batteries of portable devices.

4


models, and thus can predict the actual responses of our circuits as long as such models accurately represent the corresponding structures or circuit elements. In the same way, the numerical values that appear on the display of a measuring instrument are not the quantity under measurement, but just another physical quantity related to it in a more or less accurate way. In spite of such difficulties, or perhaps just because of them, microwaves are a technology and a discipline that are at the same time both stimulating and fascinating. This book attempts to attenuate some theoretical difficulties by presenting the discipline in as simple a way as we could, but we need to stress that the problems one has to face when dealing with microwaves never run out, even after years of study and professional practice.

1.2 Frequency bands Although the term microwaves should concern a methodology rather than a frequency range, a conventional subdivision into frequency bands is clearly needed for practical reasons. This does not eliminate some confusion since different conventions are in use. Table 1.1 lists the frequency band designations according to the CCIR (Consultative Committee on International Radio) over the full 30 Hz to 300 GHz spectrum. The microwave spectrum actually occupies the bands 9 (UHF), 10 (SHF) and 11(EHF). The most common designation of microwave bands is that quoted in Table 1.2, where a letter is used to designate the various bands. Such a denomination dates back to the Second World War, when random letters were chosen in order to confuse the enemy, but some confusion is still present [2] (e.g. in some books the Q band is used for the 33–50 GHz band). From a practical point of view, the selection of a frequency band is based on the specific application and on the characteristics of the EM wave propagating in the atmosphere. The ratio d/l between the circuit dimension and the wavelength is of paramount importance in determining the ability of an antenna to radiate the EM field into space. To this end, the circuit dimensions and the wavelength should be of the same order of magnitude, so that the higher the frequency, the smaller the antenna size. Similarly, the speed of data transmission depends on the frequency band employed. The use of higher frequencies allows one to increase the channel capacity and thus to increase the data transmission rate. The propagation through the atmosphere, however, produces attenuations that depend not only on the distance, as in free space, but also on the physical and chemical properties of the medium, as illustrated by Figure 1.3, where atmospheric attenuation is plotted at sea level and at 4000 m altitude.

Table 1.1 Denomination of radio bands. Band

Denomination

Frequency range

Wavelengths

1 2 3 4 5 6 7 8 9 10 11 12

ELF SLF ULF VLF LF MF HF VHF UHF SHF EHF LHF

< 30 Hz 30–300 Hz 300 Hz–3 kHz 3–30 kHz 30–300 kHz 300 kHz–3 MHz 3–30 MHz 30–300 MHz 300 MHz–3 GHz 3–30 GHz 30–300 GHz >300 GHz

>10 000 km 10 000–1000 km 1000–100 km 100–10 km 10–1 km 1 km–100 m 100–10 m 10–1 m 1 m–10 cm 10–1 cm 10–1 mm < 1 mm

INTRODUCTION

5

Table 1.2 IEEE denomination of microwave frequency bands. Denomination

Frequency range (GHz)

UHF L S C X Ku K Ka V W Millimetre waves Submillimetre waves

0.3–1 1–2 2–4 4–8 8–12 12–18 18–27 27–40 40–75 75–110 30–300 300–3000

As can be seen in the figure, the attenuation rapidly increases over 10 GHz with a non-monotonic behaviour, reaching the peaks due to water vapour absorption at 22 GHz and oxygen absorption at 63 GHz, the minima of attenuation being located at 24 and 94 GHz. In general, as observed above, the use of ever higher frequencies is spurred by a number of advantages such as the reduced dimensions of the components (antennas, line sections, circuit elements), wider bandwidths, high signal processing and data transmission speeds, higher radar resolution, higher antenna directivities and thus reduced interference. By contrast, the use of higher frequencies involves a number of practical problems, such as higher atmospheric attenuation (although not necessarily), more stringent fabrication tolerances (because of the reduced dimensions), higher fabrication costs, higher circuit loss and reduced available power from the solid state devices, and lower or insufficient maturity of the semiconductor technology. For such contrasting reasons, most civil RF systems (such as television, cellular communications, GPS, microwave ovens) employ frequencies located between 500 MHz and 5 GHz, corresponding to Wavelength, mm 100

30

1

3

millimetre waves

10 Attenuation, dB/km

10

sea level

4000 MSL

1

0.1 O2

O2 0.01

H2O

H2O

H2O

1E-3 10

30

100 Frequency, GHz

Figure 1.3 Atmospheric attenuation.

300

400

6


wavelengths between 6 and 60 cm. As pointed out in [2], this is due, on the one hand, to the antenna size, which needs to be small enough, and, on the other hand, to the increase of atmospheric attenuation at higher frequencies. The latter would require higher radiated powers, also involving a potential risk to the population. For example, if for practical reasons a maximum antenna size of 10 cm is chosen, the condition d/l > 0.1 implies a frequency no lower than 300 MHz.

1.3 Applications RF and microwave technology, originally finalized for military applications (radar), is nowadays spurred by a number of civil applications, especially cellular telephony and the so-called personal communication systems (PCS). Communications remain the most important application area where, besides cellular telephony and satellite communications, we may include radio and television broadcasting, wireless local area networks (WLANs) and point–multipoint broadcasting systems, namely LMDS (Local Multipoint Distribution Systems) and MMDS (Multipoint Multichannel Distribution Systems). RF and microwave technology also concerns several application sectors including, among others: .

Navigation and localization systems, such as GPS, based on 24 orbiting satellites and providing the user with geographical coordinates and height, or the corresponding European system Galileo, and aircraft landing systems such as MLS (Microwave Landing System).

.

Electromagnetic sensors for the measurement and characterization of physical quantities and the properties of materials for industrial applications.

.

Weather forecasting and remote sensing of environmental parameters (e.g. temperature, wind speed, water content) and monitoring of natural resources.

.

Automotive, road traffic aids and control.

.

Civil and military surveillance systems.

.

Healthcare and medicine, for investigation, diagnosis and treatment, such as microwave hyperthermia for treating cancer.

.

Radio astronomy and space exploration.

.

Microwave imaging, for civil and military applications.

.

RF identification (RFID), a technique which is rapidly replacing the bar code system to identify and track products, animals or persons using RFs.

.

Food processing, industrial treatment (drying, curing, heating, etc.) of materials and goods (e.g. for killing pests).

We should not fail to mention scientific research, which is the basis for future developments in various and newsworthy directions (e.g. lower energy consumption) to support humanity. The reader should consult the website of EURAMIG, the European Radio and Microwave Interest Group (http://www.euramig.org/), a non-profit European initiative for the promotion of microwaves and RF. It is not necessary to recall that the pervasiveness of microwave technology in everyday life is such as to induce fears about possible biological risks to humans and living organisms. Avast area of investigation has been developed concerning the interaction of EM fields (not only within the RF spectrum) with biological systems. The reader may deepen his or her knowledge on what is called, in general terms, electromagnetic compatibility (EMC), by consulting the research literature on this subject (e.g. [3, 4]). Figure 1.4 displays the main applications of microwaves and RF. Actually, the list could be extended much further, but it might shortly become obsolete. One can indeed expect that the use of EM fields

MMDS LMDS

WLAN

Terrestrial

PCS

Cellular Telephony

Civilian

Surveillance

Guidance of weapons

Ship traffic control

Space vehicles

Radar

Military

Electronic warfare

Navigation

Curing

Sensing and monitoring

Waste treatment

Drying

Process control

Industrial

Figure 1.4 Applications of microwaves and radio frequencies.

Remote sensing

Car traffic control

Air traffic control

Satellite

Communications

Applications of Microwaves and RF

Hyperthermia

Imaging

Biomedical

INTRODUCTION 7

8


is destined to spread further to the most diverse applications – some useful, others less so, or possibly useless – like many everyday items that the market imposes on us for solving problems that nobody perceives today but that tomorrow we might not be able to do without. A vast literature of treatises, textbooks and handbooks has been published on the subject of microwaves, RF and applications since the first decades of the twentieth century. In the bibliography at the end of this chapter the reader will find a very short and somewhat arbitrary list of those books, chosen from the most popular ones.

Bibliography 1. N. Carrara, ‘The detection of microwaves’, Proceedings of the Institute of Radio Engineers (IRE), Vol. 20, No. 10, pp. 1615–1625, 1932. 2. T. H. Lee, Planar Microwave Engineering, Cambridge University Press, Cambridge, 2004. 3. C. R. Paul, Introduction to Electromagnetic Compatibility, John Wiley & Sons, Ltd, Chichester, 2006. 4. H. Ott, Electromagnetic Compatibility Engineering, John Wiley & Sons, Ltd, Chichester, 2009. 5. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. 6. N. Marcuvitz, Waveguide Handbook, McGraw-Hill, New York, 1951. 7. R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, New York, 1960. 8. C. G. Montgomery, R. H. Dicke and E. M. Purcell, Principles of Microwave Circuits, McGraw-Hill, New York, 1948 and Peter Peregrinus, Stevenage, 1987. 9. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961. 10. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., New York, 1962. 11. S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley & Sons, Inc., New York, 1984. 12. J. D. Kraus, Electromagnetics, McGraw-Hill, New York, 1984. 13. C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, Inc., New York, 1989. 14. D. M. Pozar, Microwave Engineering, John Wiley & Sons, Ltd, Chichester, 2004.

2

Basic electromagnetic theory 2.1 Introduction This chapter summarizes the basic elements of the electromagnetic (EM) theory, from Maxwell’s equations to the propagation of EM waves. Since this subject is normally taught in basic courses on EM theory, only the fundamental concepts are provided as they are needed for subsequent discussions in the rest of the book. The scope here is essentially to recall the foundations of EM theory and to establish the basic formulae and symbolism that will be employed throughout the book. The experienced reader may skip this chapter.

2.2 Maxwell’s equations EM fields can be established and propagate both in a vacuum and in natural media. EM phenomena are governed by Maxwell’s equations combined with the constitutive equations of the material. Using modern notation and adopting Giorgi’s or the MKS system [1], Maxwell’s equations in differential form can be written as rE ¼ rH ¼

@B @t

@D þJ @t

ð2:1Þ ð2:2Þ

rD ¼ r

ð2:3Þ

rB ¼ 0

ð2:4Þ

In Equations (2.1)–(2.4) vector quantities are indicated in bold and: E (V/m) is the electric field D (C/m2) is the electric displacement or induction


10

MICROWAVE AND RF ENGINEERING H B J r

(A/m) is the magnetic field (Wb/m2) is the magnetic induction (A/m2) is the electric current density (C/m3) is the electric charge density.

The above quantities are, in general, functions of both space and time. If we take the divergence of (2.2) (remember that the divergence of a curl is zero) and combine the result with (2.3), we obtain the continuity equation for the electric current rJ ¼

@r @t

ð2:5Þ

The continuity equation can be interpreted as expressing the principle of conservation of the electric charge in local or differential form. The decrease with time of the charge density in an infinitesimal volume equals the current flowing outward from the same volume. Equations (2.1)–(2.4) are the differential form of Maxwell’s equations. Their integral form is sometimes more useful for computational purposes or in the derivation of their numerical counterparts (see Chapter 16). By applying Stokes’ theorem (A.61) to (2.1) and (2.2) and the divergence theorem (A.60) to (2.3) and (2.4) one obtains þ ð @ E dl ¼ B dS ð2:6Þ @t S C þ

@ H dl ¼ @t C

ð

ð D dS þ S

J dS

ð2:7Þ

S

ð

þ D dS ¼ S

r dV

ð2:8Þ

V

þ B dS ¼ 0

ð2:9Þ

S

Similarly, the continuity equation can be converted into its integral counterpart using the divergence theorem (A.60) þ J dS ¼ S

@ @t

ð r dV

ð2:10Þ

V

Maxwell’s equations (2.1)–(2.4) need to be supplemented by the constitutive relations which characterize the EM properties of the material (or medium) where the EM is established. In the case of linear isotropic media, including the vacuum, the constitutive relations are simply scalar and linear relationships between the electric displacement and the electric field and between the magnetic induction and the magnetic field D ¼ eE

ð2:11Þ

B ¼ mH

ð2:12Þ

In (2.11) e is the dielectric permittivity, usually expressed in terms of the dielectric constant (or relative permittivity) er and the permittivity of the vacuum e0 e ¼ e0 er

ð2:13Þ

BASIC ELECTROMAGNETIC THEORY

11

where e0 ¼ 8:854 10 12 ffi

10 9 F=m 36p

ð2:14Þ

Similarly, in (2.12), m is the magnetic permeability, usually expressed as the product of the relative permeability mr and the vacuum permeability m0. The latter has the value m0 ¼ 4p 10 7 H=m

ð2:15Þ

Permittivity and permeability are the constitutive parameters of the medium. If this is inhomogeneous, i.e. its physical properties vary in space, the constitutive parameters are functions of the space coordinates. When, as we will always assume in this book, the constitutive parameters are independent of the electromagnetic field, the medium is linear. In conducting materials where an electric current is induced by the electric field, Ohm’s law holds, stating the proportionality between the conduction current density J c and the electric field E through the conductivity s (S/m) of the medium: J

c

¼ sE

ð2:16Þ

The constitutive relations (2.11) and (2.12) can alternatively be expressed using another pair of vectors that characterize the electric and magnetic properties of the medium, i.e. the electric polarization P and the magnetic polarization M . Such relations are as follows: D ¼ e0 E þ P

ð2:17Þ

B ¼ m0 ðH þ M Þ

ð2:18Þ

For linear and isotropic media, P and M

are proportional to the electric and magnetic fields P ¼ e0 we E

ð2:19Þ

¼ wm H

ð2:20Þ

M

where we and wm are the electric and magnetic susceptibilities, respectively. It is easily found that we ¼ er 1

ð2:21Þ

wm ¼ mr 1

ð2:22Þ

In many practical cases, the material where the EM exists is such that a more complex relation than the simple proportionality (2.19) holds between the polarization and the electric field. Such a relationship may also depend on the time variation of the field. As an example, the molecules of many materials, including water, have their own oscillation frequencies. As a consequence the dielectric properties of such materials are frequency dependent and the material is said to be dispersive. Its constitutive relation can be rather complex in the time domain, but in the frequency domain (see next section) it simply implies a frequency dependence of the dielectric constant. For a discussion on the permittivity dispersion see for instance [2]. In the case of a material with crystal structure, the orientation of the field with respect to the crystal lattice may affect its electrical or magnetic properties. In particular, the electric or magnetic polarizations may have directions that do not coincide with those of the originating electric or magnetic field, respectively. In such a case the material is said to be anisotropic. The relationships (2.19) and (2.20), although linear, become of tensor type:

12

MICROWAVE AND RF ENGINEERING P ¼ e0 v E

ð2:23Þ

M

ð2:24Þ

e

¼ v H m

a tensor or dyadic being represented by a matrix. Correspondingly, (2.11) and (2.12) become D ¼ eE

ð2:25Þ

B ¼ m H

ð2:26Þ

where e and m are referred to as tensor permittivity and tensor permeability. The tensor permittivity has the matrix form 3 2 exx exy exz 7 6 ð2:27Þ e ¼ 4 eyx eyy eyz 5 ezx

ezy

ezz

and similarly for the tensor permeability. In scalar form (2.25) is thus written as Dx ¼ exx E x þ exy E y þ exz E z Dy ¼ eyx E x þ eyy E y þ eyz E z

ð2:28Þ

Dz ¼ ezx E x þ ezy E y þ ezz E z and similarly for (2.26).

2.3 Time-harmonic EM fields; polarization of a vector EM field quantities depend on the three spatial coordinates and time. The complexity of linear EM problems may be considerably reduced by referring to a time-harmonic regime. In such a case, all timevarying quantities (due to the linearity of Maxwell’s equations) are assumed to have a sine or cosine behaviour with the same radian frequency o ¼ 2pf ¼ 2p=T, where f is the frequency and T the corresponding period. This approach allows us to eliminate time from Maxwell’s equations and, as shown below, to replace the time derivatives with simple algebraic products. Under the time-harmonic regime, in fact, a time-varying quantity, aðtÞ ¼ jAjcosðot þ fÞ

ð2:29Þ

where |A| is a real positive constant, can be expressed in the form aðtÞ ¼ Re½Ae jot

ð2:30Þ

A ¼ jAje jf ¼ jAjðcos f þ j sin fÞ

ð2:31Þ

where A is a complex quantity given by

The complex number A, which is called the phasor of a(t), can thus be used to represent the corresponding time variable. It is straightforward to verify that the time derivative of a(t) has joA as its phasor, thus the time derivative is equivalent to multiplication by jo.


13

When dealing with quantities that are squares or products of time-harmonic variables, such as energy or power, time variations at twice the radian frequency o are produced along with steady state components. In such cases, we normally refer to the average over a time period T. Using the phasor notation (2.30), consider the product cðtÞ ¼ aðtÞbðtÞ. It can easily be seen that the time-averaged value of this quantity is ð 1 T 1 c ¼ cðtÞ dt ¼ Re AB* ð2:32Þ T 0 2 Let us now consider a vector quantity A ðtÞ: each component behaves as (2.29) so that the vector can be expressed in the form A ðtÞ ¼ Re½Ae jot

ð2:33Þ

where A is a complex vector consisting of a real and an imaginary part A ¼ Ar þ jAj

ð2:34Þ

A ðtÞ ¼ Ar cosðotÞ Aj sinðotÞ

ð2:35Þ

From (2.33) and (2.34) we obtain

Equation (2.35) shows that the vector A ðtÞ lies in the plane of the vectors Ar, Aj, its tip describing an ellipse (see Figure 2.1a). The vector is therefore said to be elliptically polarized. In special cases the ellipse may degenerate into a circle or a segment: in such cases we have circular polarization or linear polarization, respectively. The circular polarization occurs when Ar and Aj are orthogonal to each other and have equal magnitudes depending on their relative directions, the resulting vector can be right or left circularly polarized, as shown in Figures 2.1b,c. The linear polarization occurs when Ar and Aj are parallel or one of them is null (which can be considered as a special case of parallelism). Thus we have Circular polarization:

Ar Aj ¼ 0 and

Linear polarization:

Ar Aj ¼ 0

jAr j ¼ Aj

As already pointed out, a substantial simplification is obtained by assuming a time-harmonic regime. In the next section this will become more explicit by writing the corresponding expressions of Maxwell’s equations. Although at first sight this may appear as a totally arbitrary assumption, it must be recalled that Fourier series and Fourier integrals allow us to expand in harmonic components any time-periodic or time-integrable function.

ωt =

−Aj

ωt = π

30º

a(t)

ωt =

3 π 2

Ar

π 2

ωt =

ωt = 0, 2π ...

ωt = π

−Aj

3 π 2

Ar a(t)

(a)

ωt =

ωt = 0, 2π ... ωt = π

−Aj

π 2 a(t)

Ar

(b) ωt =

π 2

ωt = 0, 2π ...

(c) ωt =

3 π 2

Figure 2.1 Vector polarization: (a) elliptic; (b) right circular; (c) left circular.

14


From now on, unless otherwise stated, we will refer to the time-harmonic regime so that all timevarying quantities will be represented by the corresponding phasors. The time variable disappears from our equations, since the time derivative is replaced by the product jo. The commonly adopted terminology is that we operate in the frequency domain in contrast to the normal time domain.

2.4 Maxwell’s equations in the harmonic regime As shown in the previous section, time-varying quantities, both scalar and vectorial, can be represented in terms of the corresponding phasors. Since the time derivative corresponds to a multiplication by jo, the time dependence can be totally omitted from Maxwell’s equations (2.1)–(2.4), which therefore become r E ¼ joB

ð2:36Þ

r H ¼ joD þ J

ð2:37Þ

rD ¼ r

ð2:38Þ

rB ¼ 0

ð2:39Þ

We say that Equations (2.36)–(2.39) are the frequency domain Maxwell’s equations, while (2.1)–(2.4) are their time domain counterparts. In integral form (2.36)–(2.39) are written as þ ð E dl ¼ jo B dS ð2:40Þ C

S

þ

ð

ð

H dl ¼ jo D dS þ C

S

þ

J dS

ð2:41Þ

S

ð D dS ¼

S

r dV

ð2:42Þ

V

þ B dS ¼ 0

ð2:43Þ

S

The continuity equation (2.5) becomes r J ¼ jor

ð2:44Þ

The constitutive relations (2.11) and (2.12) are also expressed in the frequency domain as D ¼ eE

ð2:45Þ

B ¼ mH

ð2:46Þ

For lossy dielectrics, the permittivity e is a complex quantity, whose imaginary part (if not zero) is negative:1 e ¼ e0 je00 1

A positive imaginary part would violate the principle of energy conservation.

ð2:47Þ


15

Similar considerations hold for the permeability of magnetic materials. It should be stressed that the complex dielectric permittivity does not correspond to a time-varying quantity. The presence of the imaginary part of e corresponds, in time domain, to the fact that the polarization P, thus the displacement D, is phase-shifted with respect to the electric field. This has the effect of producing within the medium an energy dissipation which depends on the imaginary part of the complex permittivity (see Section 2.6). Indeed, the presence of an imaginary part is equivalent to the presence of a conduction current. According to Ohm’s law (2.16),2 the latter is proportional to the electric field. In (2.37) this current can be lumped into the displacement current in just one term proportional to E: r H ¼ joeE þ sE ¼ joðe j s=oÞE ¼ joðe0 je00 ÞE

ð2:48Þ

e00 ¼ s=o

ð2:49Þ

with

Maxwell’s equation (2.37) can thus be written as r H ¼ joec E

ð2:50Þ

where the subscript ‘c’ indicates that the permittivity is complex, its imaginary part (2.49) incorporating the ohmic current density. The complex permittivity (2.47) is often put in the form ec ¼ e0 ð1 j tan dÞ

ð2:51Þ

where tan d ¼

e00 e0

ð2:52Þ

is a parameter (called tangent delta) commonly used to characterize the quality of the dielectric material. Its values range from a few 10 4 for excellent dielectrics such as quartz (SiO2) or alumina (Al2O3) to 5 10 4 for glass and up to 10 or 100 10 4 for silicon. The permittivity is also, in general, a function of frequency. In such a case, the medium is said to be dispersive. Dispersion is indeed a feature of the material, or a transmission line, consisting of properties that are frequency dependent. A non-dispersive medium, on the contrary, does not change its properties with the frequency.

2.5 Boundary conditions The differential form of Maxwell’s equations (2.36)–(2.39) can be applied provided that the constants of the medium, such as the permittivity, permeability and, possibly, the conductivity, are continuous functions of the spatial coordinates. In practical cases we have clearly to deal with a variety of different media, so that such constants vary abruptly at the interfaces between different materials.3 The differential form of Maxwell’s equations can thus be solved only within the spatial regions where the medium’s constants are continuous. In order to obtain a unique and physically meaningful solution, suitable boundary conditions must then be imposed at the boundaries between regions occupied by different materials. Such conditions can be derived using the integral form of Maxwell’s equations and applying a 2 Since (2.16) is an instantaneous relation between J c and E , it is unchanged between the corresponding phasors: Jc ¼ sE. 3 This is true from a macroscopic point of view, where we assume a geometrical surface represents the boundary between two materials.

16


Figure 2.2 Continuity of B and E components at the boundary between two media. The same conditions apply to D and H, respectively, unless there are surface charges and currents.

limit procedure where a continuous variation of the medium’s constants is gradually transformed into an abrupt change. Only the results of such a procedure are reported here, along with simple hints on how it can be developed. For a complete derivation, the reader is directed to the classical textbooks in the bibliography, e.g. [3, 4]. We consider the surface S to represent the boundary between two media with constants e1, m1 and ^ be the unit vector normal to S directed from medium 1 towards medium 2 (see Figure 2.2). e2, m2. Let n From (2.42) and (2.43) we obtain4 ^ ðD2 D1 Þ ¼ 0 n

ð2:53Þ

^ ðB2 B1 Þ ¼ 0 n

ð2:54Þ

The former condition holds as long as there is no surface charge (i.e. a finite charge in an infinitesimal volume) on the boundary, which may happen only if one of the media is a perfect conductor. In all other cases, the above equations state the continuity of the normal components of both the electric displacement D and the magnetic induction B at the boundary between different materials. As far as the other EM field components are concerned, with reference to Figure 2.2, from (2.40) and (2.41) we obtain5 ^ ðE2 E1 Þ ¼ 0 n

ð2:55Þ

^ ðH2 H1 Þ ¼ 0 n

ð2:56Þ

The former condition holds as long as there is no surface current density (i.e. a finite current through an infinitesimal cross-section) on the boundary, which may happen only if one of the media is a perfect conductor. In all other cases, the above conditions state the continuity of the tangential components of both the electric and magnetic fields at the boundary between two media. When medium 1 is a perfect conductor (Figure 2.3), both a surface charge rs (C/m2) and a surface current density Js (A/m) are associated with the EM field on its boundary. In reality, these are charges and currents concentrated within a very thin layer ds (the skin depth) under the surface of a good conductor. As we will see later on in Section 2.10.3, the skin depth is given by ds ¼ ðoms=2Þ 1=2

ð2:57Þ

4 Apply (2.42) and (2.43) to a small cylinder of height h crossing the boundary, and compute the limit for vanishing h. 5 Apply (2.40) and (2.41) to a small rectangular loop with side h crossing the boundary in the limit as h ! 0.


17

Figure 2.3 Boundary conditions at the surface of a perfect electric conductor (pec).

For good conductors such as silver, copper or gold,6 the skin depth at a frequency of 10 GHz is a fraction of a micrometre. From a mathematical point of view, however, it is more convenient to refer to the limit case of infinite conductivity (perfect electric conductor, or pec); in this case the EM field in medium 1 is exactly zero and the skin depth is zero as well. Omitting the subscript 2, Equations (2.53)–(2.54) become in this case ^ D ¼ rs n

ð2:58Þ

^B ¼ 0 n

ð2:59Þ

On the surface of a perfect conductor, the normal component of the electric induction thus equals the surface current density, while that of the magnetic induction is zero. Concerning (2.55) and (2.56), for a perfect conductor we have Ê¼0 n

ð2:60Þ

^ H ¼ Js n

ð2:61Þ

The former equation states that on the surface of a perfect conductor the tangential component of the electric field is zero, while that of the magnetic field, after a 90 clockwise rotation, equals the surface current density. In the solution of some EM problems it may be convenient to introduce fictitious magnetic charges rsm and currents Jsm .7 Consequently, the boundary conditions (2.59) and (2.60) assume the dual form of (2.58) and (2.61) respectively: ^ B ¼ rms n

ð2:62Þ

^ E ¼ Jsm n

ð2:63Þ

2.6 Energy and power of the EM field; Poynting’s theorem The creation of an EM field requires energy to be consumed. If not transformed into other forms (e.g. thermal energy), by virtue of the conservation principle such energy is then to be found stored in the electric and magnetic fields.

6 7

Silver, copper and gold conductivities are 63 106 , 60 106 and 45 106 S/m respectively. Correspondingly, magnetic charges rm and currents Jm are introduced in the r.h.s. of (2.1) and (2.4).

18

MICROWAVE AND RF ENGINEERING The electric energy stored in an electric field in a volume V at time t is given by ð ð 1 e E ðtÞ DðtÞ dV ¼ E 2 ðtÞ dV W e ðtÞ ¼ 2 V 2 V

ð2:64Þ

using the constitutive relation (2.11). In a dual fashion, using (2.12), the instantaneous magnetic energy in V is ð ð 1 m H ðtÞ B ðtÞ dV ¼ H 2 ðtÞ dV ð2:65Þ W m ðtÞ ¼ 2 V 2 V In the case of conducting media, the electric field produces a conduction current according to Ohm’s law (2.16). The electric energy dissipates in the medium and is transformed into heat by the Joule effect. The power dissipated in a volume V is given by ð

ð E ðtÞ J ðtÞ dV ¼ s E 2 ðtÞ dV

P d ðtÞ ¼ V

ð2:66Þ

V

For time-harmonic fields, one refers to the corresponding quantities averaged over a time period T according to the formula (2.32). We therefore have, for the average electric and magnetic energies stored in one period, 1 We ¼ Re 4

ð

E D dV ¼

e0 4

ð E E* dV

ð2:67Þ

ð ð 1 m H B* dV ¼ H H* dV Wm ¼ Re 4 4 V V

ð2:68Þ

*

V

V

while the time-averaged power dissipated is ð ð 1 s E J* dV ¼ E E* dV Pd ¼ Re 2 2 V V

ð2:69Þ

the last equality holding for ohmic currents. From Maxwell’s equations (2.36)–(2.39), and using expansion (A.25) for the divergence of the product of two vectors, one obtains the following identity: r ðE H* Þ ¼ joðE D* H* BÞ E J*

ð2:70Þ

Integrating over a volume V bounded by a closed surface S, and applying the divergence theorem to the l.h.s. (A.60), one obtains the expression for Poynting’s theorem: þ ð ð 1 jo P dS ¼ E J* dV þ ðH* B E D* Þ dV ð2:71Þ 2 V 2 V S The l.h.s. of (2.71) is the flow of the complex Poynting vector defined as 1 P ¼ E H* 2 entering into the volume V. The real part of (2.71) þ ð ð 1 s E E* dV ¼ Pd Re P dS ¼ Re E J* dV ¼ 2 2 V S V

ð2:72Þ

ð2:73Þ


19

is interpreted as the real power entering into the volume V through S; this quantity equals the power dissipated by the conduction current by Joule effect.8 The imaginary part of (2.71) ð þ 2o ðmH H* e0 E E* Þ dV ¼ 2oðWm We Þ ð2:74Þ Im P dS ¼ 4 V S is the flow of reactive power entering in V and is equal to 2o times the difference of the average values of the magnetic and electric energy densities.

2.7 Some fundamental theorems In this section we describe some fundamental theorems of electromagnetism; the proofs can be found in many textbooks like those cited at the end of this chapter.

2.7.1

Uniqueness theorem

In the solution of EM problems it is of paramount importance to know under what conditions a solution exists and is unique. The uniqueness theorem, which we consider here only for time-harmonic fields, states that, in a volume V,9 the solution to Maxwell’s equations exists and is unique once the following quantities are known: 1. The sources of the EM fields, i.e. the electric current densities J (and possibly the magnetic currents Jm in case they are assumed to be non-zero). Here J and Jm are so-called impressed currents, corresponding to generators in lumped circuit theory. ^ E or of the magnetic field n ^ H on the 2. The tangential component of either the electric field n boundary S of V. In a source-free region the EM field is thus uniquely determined by the tangential electric or magnetic distributions on its boundary.

2.7.2

Lorentz’s reciprocity theorem

This theorem is the basis for a number of fundamental properties of both microwave and lumped circuit elements. In a volume V filled with a linear and isotropic medium with constants e, m, let J1 and J2 be two current distributions that produce the EM field distributions E1H1 and E2H2 respectively. Making use of Maxwell’s equations (2.36) and (2.37) along with the constitutive relations (2.45)–(2.46) and the expression (A.25) we obtain the following identity: r ðE1 H2 E2 H1 Þ ¼ . . . ¼ J2 E1 þ J1 E2 Integrating over the volume V and applying the divergence theorem we obtain ð þ ðE1 H2 E2 H1 Þ dS ¼ ð J2 E1 þ J1 E2 Þ dV S

8

ð2:75Þ

ð2:76Þ

V

Power loss due to polarization currents may formally be lumped into the conductivity term, recalling (2.49). For the theorem to hold, the volume is required to be filled with a lossy material, although the loss can be arbitrarily small. In a totally lossless region, such as an ideal cavity resonator, there are infinite solutions to Maxwell’s equations, called the resonant modes of the cavity (see Chapter 5). 9

20


S being the boundary of V. This is the basic form of Lorentz’s reciprocity theorem. In many practical cases, the l.h.s. of (2.76) vanishes (the proof can be found in [3]) so that the theorem assumes a much simpler form. This occurs when all sources of the EM field are within the volume V, for instance when V coincides with the whole space and no sources are located at infinity, or when V is bounded by a perfect electric conductor or by a surface exhibiting a surface impedance Zs such that ^H Et ¼ Zs n

ð2:77Þ

^ is the outward-directed unit normal vector. As shown later on in Section 2.10.3, a special case of where n Equation (2.77) is the so-called Leontovic boundary condition on the surface of a good conductor: Zs ¼ ð1 þ jÞ

rffiffiffiffiffiffiffi om 2s

ð2:78Þ

In all the above cases, Equation (2.76) reduces to ð

ð J2 E1 dV ¼ V

J1 E2 dV

ð2:79Þ

V

Equation (2.79) constitutes the reciprocity theorem. In the case of point sources we may put Ji ¼ Ji dðr ri Þti ;

i ¼ 1; 2

ð2:80Þ

d being the Dirac delta function and ti the unit vector directed along Ji. Equation (2.79) therefore reduces to E1 ðr2 Þ t2 J2 ¼ E2 ðr1 Þ t1 J1

ð2:81Þ

The above result implies that the component of the electric field produced by the current J1 in r2 in the direction of J2 is equal to the electric field component produced by J2 in r1 in the direction of J1. As we will see in Chapter 4, such a theorem implies the symmetry properties of the impedance, admittance or scattering matrices used to characterize the electric circuits. It should be noted that the validity of the theorem is subject to (2.11)–(2.12) being applicable. It does not therefore apply to nonlinear or anisotropic materials.

2.7.3

Love’s equivalence theorem

This theorem is a consequence of the uniqueness theorem. It allows one to replace a source outside the region V with a suitable distribution of electric and magnetic currents that produce the same tangential E or H components on the boundary S of V. Consider (see Figure 2.4) a set of known sources J1 located outside the region V. Let E1, H1 be the EM field generated by J1 within V. By virtue of the uniqueness theorem, the field in V will remain unaltered if we modify the external sources, provided that the same tangential component of either the ^ E and n ^ H on S is conserved. In particular, we may put a perfect electric or magnetic field n conductor on the outer surface of V impressing at the same time on S a surface magnetic current ^ E. In this way the tangential component of the electric field on S remains unaltered. In a Jms ¼ n similar fashion we may put a perfect magnetic conductor on S by impressing at the same time on S an ^ H. The theorem can clearly be applied to the region outside V when the electric current density Js ¼ n sources are inside V.


21

Figure 2.4 Illustrating Love’s equivalence theorem.

2.8 Plane waves In this section we cover the basic concepts concerning wave functions and show that these are solutions to Maxwell’s equations. Let us consider a space region filled with a lossless material deprived of any sources of an EM field, thus J ¼ 0, s ¼ 0. The time-harmonic EM fields are governed by the homogeneous Maxwell’s equations r E ¼ jomH

ð2:82Þ

r H ¼ joeE

ð2:83Þ

Taking the curl of the second equation and substituting in the first one, we obtain an equation with only the E field, i.e. r2 E þ k2 E ¼ 0

ð2:84Þ

pffiffiffiffiffi k ¼ o me

ð2:85Þ

where we have put

In the derivation of (2.84) we have used the expansion (A.31) and that r E ¼ 0. Equation (2.79) is Helmholtz’s equation or the wave equation in the frequency domain. For the sake of simplicity and as an introduction, let us consider the one-dimensional case, which corresponds to the well-known harmonic motion equation d 2 f ðxÞ þ k2 f ðxÞ ¼ 0 dx2

ð2:86Þ

The general solution can be expressed as the superposition of two exponentials: f ðxÞ ¼ C þ e jkx þ C e þ jkx C

þ

and C

being arbitrary complex constants.

ð2:87Þ

22


As we will see later on, expression (2.87) represents voltages and currents along a transmission line. The phasor f(x) corresponds to the time function f ðx; tÞ ¼ Re½ f ðxÞe jot ¼ Re½ðC þ e jkx þ C e þ jkx Þe jot ð2:88Þ ¼ jC þ jcosðot kx þ f þ Þ þ jC jcosðot þ kx þ f Þ jj putting C ¼ C e . Either term in (2.88) is a harmonic function of the variable c ¼ ot kx or c ¼ ot þ kx. The time period is related to the radian frequency o by T ¼ 2p=o

ð2:89Þ

while the spatial period or wavelength is related to the phase constant k by l ¼ 2p=k

ð2:90Þ

In the space–time domain (x, t), the former function in (2.88) is a cosine travelling in the positive x direction with a velocity vp ¼ o=k

ð2:91Þ

This is called the phase velocity: it is the velocity of a hypothetical observer that sees a constant phase: c ¼ ot kx ¼ const:

ð2:92Þ

Differentiating (2.92), one indeed obtains dx=dt ¼ o=k. The second term in (2.88) is a cosine travelling with opposite velocity in the negative x direction. In contrast to the 1-D case, in the 3-D case no general solution exists for the wave equation (2.84). The solution is to be found on a case-by-case basis depending on the geometry of the problem and on the boundary conditions. It is nevertheless possible to identify some solutions of (2.84) that, although they do not constitute the general integral, are nevertheless meaningful solutions from a physical point of view and, due to linearity, useful for building up more complex solutions of a number of practical problems. This is specifically the case for plane waves that are obtained as special solutions of Maxwell’s equation in rectangular coordinates.

2.9 Solution of the wave equation in rectangular coordinates Using a rectangular coordinate system, Equation (2.84) can be broken down into three scalar equations, one for each scalar component: r2 Ei þ k2 Ei ¼ 0;

i ¼ x; y; z

ð2:93Þ

Solutions to the above scalar wave equations can be obtained by the method of separation of variables.

The method consists of factorizing the unknown function Ei(x,y,z) into the product of three functions, each depending on a single coordinate: Ei (x,y,z) ¼ X(x)Y(y)Z(z). In such a way, with simple algebra, Equation (2.93) becomes 1 d 2X 1 d2Y 1 d 2Z þ þ ¼ k2 ¼ ðkx2 þ ky2 þ kz2 Þ 2 2 X dx Y dy Z dz2

ð2:94Þ

where the constant k2 has been expressed as the sum of three constants kx2 , ky2 , kz2 . The above equality can hold at any point (x, y, z) only if each term on the r.h.s. is a constant. This yields three ordinary differential equations (harmonic motion equations) whose solutions are of the type XðxÞ ¼ Cx e jkx x etc., whence we easily get (2.95).


23

In a rectangular coordinate system the solutions are of the type Eðx; y; zÞ ¼ E0 e jk r

ð2:95Þ

where E0 is a (complex) constant vector, r is the position vector r ¼ x^x þ y^y þ z^z

ð2:96Þ

and k is the propagation vector which has only to satisfy the separability condition k k ¼ kx2 þ ky2 þ kz2 ¼ o2 me

ð2:97Þ

The magnetic field is obtained from the E field (2.95) using (2.82): Hðx; y; zÞ ¼ H0 e jk r

ð2:98Þ

where H0 ¼

1 k E0 om

ð2:99Þ

Note that (2.97) is equally satisfied by reversing the sign of k. Therefore, if (2.95) and (2.98) are a solution to Maxwell’s equations, then another solution is obtained by reversing the sign of k. Inserting (2.98) and (2.95) into (2.83), we obtain also E0 ¼

1 k H0 oe

ð2:100Þ

In the general case, k is a complex vector which has to satisfy the condition (2.97). Let us split k into its real and imaginary parts by putting k ¼ b ja

ð2:101Þ

The reason for the minus sign in the imaginary part will become clear later on. Equation (2.97) then implies b2 a2 ¼ o2 me ab ¼ 0

ð2:102Þ

The second of the above equations shows that a and b are orthogonal one to each other, or a is zero, which is a particular case of orthogonality. Inserting expression (2.101) into (2.95), we obtain for the E field Eðx; y; zÞ ¼ E0 e jb r e a r Hðx; y; zÞ ¼ H0 e jb r e a r

ð2:103Þ

The electric field has a constant phase when b r ¼ const:, thus on any plane perpendicular to the vector b, which is therefore called the phase vector. The EM field represented by (2.103) is called a plane wave because the constant-phase surfaces are planes. The field amplitude is constant on the planes a r ¼ const: which are orthogonal to a. The field decays exponentially in the direction of the vector a, which is therefore called the attenuation vector, and has constant amplitudes on the planes orthogonal to a. It is important to point out that the orthogonality between a and b, thus of the constant-phase and constant-amplitude planes, depends on the medium where the wave propagates being without loss. Otherwise, the dielectric constant would be complex and the r.h.s. of the second equation of (2.102) would be non-zero.

24

2.9.1


Plane waves: an alternative derivation

As we have just seen, plane waves are a solution to homogeneous (i.e. source-free) Maxwell’s equations in rectangular coordinates. This result has been obtained by solving the wave equation for the E field using the method of separation of variables. An alternative and speedier way to reach the same result consists of searching directly for a solution of homogeneous Maxwell’s equations (2.82) and (2.83) by assuming Eðx; y; zÞ ¼ E0 e jk r

ð2:104Þ

Hðx; y; zÞ ¼ H0 e jk r

ð2:105Þ

with E0 and H0 constant vectors and k a (so far) arbitrary vector. E0 and H0 are not independent of each other since the EM field must satisfy Maxwell’s equations (2.82)–(2.83). It is easy to realize that, because of (2.104)–(2.105), the divergence and curl operators become simple algebraic vector operators: r ¼ jk r ¼ jk

ð2:106Þ

Using the above equalities, after inserting (2.104) and (2.105) into (2.82) and (2.83), we obtain k E0 ¼ omH0

ð2:107Þ

k H0 ¼ oeE0

ð2:108Þ

which are clearly equivalent to (2.100) and (2.99) respectively. From (2.107), and (2.108) in particular, we deduce that k E0 ¼ 0 k H0 ¼ 0

ð2:109Þ

The above expressions imply the orthogonality of k, E and H;10 they are a consequence of E and H having zero divergence (see the first equation of (2.106)). Equations (2.107)–(2.108) constitute six homogeneous scalar equations in six unknown EM field components. In order for non-trivial solutions to exist, the determinant of the coefficient matrix must be zero. It can easily be verified that such a condition coincides with the separability condition (2.97); moreover, when such a condition is satisfied, the rank of the coefficient matrix reduces to 4. This means that only four out of six equations are linearly independent and two arbitrary quantities must be given. For example, we may arbitrarily fix the Ez and Hz components and express all other components in terms of them. In this fashion, provided that kx and ky are not both zero (i.e. the wave does not propagate in the z direction), we obtain E0x ¼

1 ðkx kz E0z þ omky H0z Þ kx2 þ ky2

E0y ¼

1 ðky kz E0z omkx H0z Þ kx2 þ ky2

H0x ¼

1 ðoeky E0z kx kz H0z Þ kx2 þ ky2

H0y ¼

ð2:110Þ

1 ðoekx E0z þ ky kz H0z Þ þ ky2

kx2

10 The term orthogonality is used here in a generalized sense to mean that the scalar product of two possibly complex vectors is zero. If such vectors have non-zero real and imaginary parts, this type of orthogonality clearly does not imply geometrical orthogonality, as is the case when the vectors are real.


25

When E0z and H0z are given, all remaining EM field components are thus determined. Other solutions can be obtained by simply circularly interchanging the indexes (x ! y, y ! z, z ! x).

2.9.2

TEM waves

Let us now suppose that kx ¼ ky ¼ 0, thus k2 ¼ kz2 . Since the propagation vector has only the z component, because of (2.102) a must be zero and the propagation vector is real: k¼b

ð2:111Þ

Because of (2.111), Equations (2.104) and (2.105) may be rewritten as Eðx; y; zÞ ¼ E0 e jbz Hðx; y; zÞ ¼ H0 e jbz

ð2:112Þ

x0 . Let us assume for simplicity that E0 is real and directed along the x axis, thus E0 ¼ E0r ¼ E0x ^ Using (2.33), the corresponding time-harmonic quantity is E ðx; y; z; tÞ ¼ Re½E0 e jbz e jot ¼ E0r cosðot bzÞ

ð2:113Þ

This is a linearly polarized vector wave function propagating in the positive z direction with phase velocity, as already defined in (2.91), vp ¼

o 1 ¼ pffiffiffiffiffi b me

ð2:114Þ

Expressions (2.110) cannot clearly be applied as all denominators are zero. From (2.107)–(2.108) we easily obtain instead b H0y oe b E0y ¼ H0x oe E0z ¼ 0 E0x ¼

ð2:115Þ

H0z ¼ 0 where pffiffiffiffiffi b ¼ kz ¼ o me

ð2:116Þ

Equations (2.115) are two distinct solutions of (2.107)–(2.108), each possessing only two non-zero components of the EM field: either Ex, Hy or Ey, Hx. Both are therefore linearly polarized. One can be obtained from the other by a simple 90 rotation around the z axis (Ex ! Ey, Hy ! Hx). Such solutions represent TEM (or Transverse Electromagnetic) waves, since the electric and magnetic fields are both perpendicular to the direction of propagation z. Since the attenuation is zero, the EM field has the same amplitude in the whole space. For this reason they are also called uniform plane waves. The ratio between the transverse components of the electric and magnetic fields is given in both cases by rffiffiffi Ex Ey m ð2:117Þ Z¼ ¼ ¼ e Hy Hx where we have used (2.116). The above quantity is called the wave impedance. Observe that the E- and H-field components and the propagation direction z must be taken according to a right-handed orthogonal

26


triplet (e.g. [x, y, z], [y, x, z], [y, x, z], etc.). This is the reason for the minus sign in front of Hx in (2.117). Equation (2.99) can be written in the form 1 H0 ¼ b^ E0 Z

ð2:118Þ

where ^b is the unit vector in the direction of b. Using (2.112) and (2.118), the Poynting vector (2.72) is 1 1 P ¼ E H* ¼ . . . ¼ E02 ^ b 2 2Z

ð2:119Þ

The wave impedance of TEM waves depends only on the parameters of the medium. For a generic material we define its intrinsic impedance as pffiffiffiffiffiffiffiffiffi ð2:120Þ Zi ¼ m=ec For a lossless material ec ¼ e, thus this quantity clearly coincides with the wave impedance of the TEM waves. This result does not hold in general, as the wave impedance may differ from the intrinsic impedance of the medium. For the vacuum we have rffiffiffiffiffi m0 Z0 ¼ ffi 120p ¼ 377O ð2:121Þ e0 It is also important to observe that, because of linearity, any linear superposition of the two solutions (2.115) is still a solution of (2.107)–(2.108).

2.9.3

TE and TM waves

Expressions (2.110) show that, apart from the case of TEM waves propagating in the z direction already discussed above, plane wave solutions in the general case when a „ 0 can be expressed as the combination of TE (Transverse Electric) waves with E0z ¼ 0 and TM (Transverse Magnetic) waves with H0z ¼ 0. Let us look at those solutions. Because of (2.102), the phase vector and the attenuation vector must be orthogonal. For simplicity, suppose that the waves propagate in the z direction and attenuate in the x direction. Therefore k ¼ b^z ja^ x

ð2:122Þ

Eðx; y; zÞ ¼ E0 e jbz e ax

ð2:123Þ

Hðx; y; zÞ ¼ H0 e jbz e ax

ð2:124Þ

Equations (2.104) and (2.105) become

Let us consider TE waves first. From (2.110), putting E0z ¼ 0 we obtain jom H0z a jb H0z ¼ a

E0y ¼ H0x

ð2:125Þ

while E0x ¼ 0 and H0y ¼ 0. The wave impedance of TE waves is ZTE ¼

E0y om ¼ H0x b

ð2:126Þ


27

The Poynting vector is

1 1 2ax 2 * 1 1 a 2 2ax ^z þ j ^ E0y k ¼ e P ¼ E H* ¼ . . . ¼ e x E0y 2 2om 2 ZTE om

ð2:127Þ

In contrast with the TEM case, the Poynting vector is complex and the power density decays exponentially with attenuation 2a in the x direction. The real part of the Poynting vector is directed along the direction z of the phase vector, while the imaginary part is directed along the direction of the attenuation. In a dual way, for TM waves (H0z ¼ 0) one obtains jb E0z a joe H0y ¼ E0z a

E0x ¼

ð2:128Þ

while E0y ¼ 0 and H0x ¼ 0. The wave impedance of TM waves is ZTM ¼

E0x b ¼ H0y oe

ð2:129Þ

The Poynting vector is

1 1 2ax 2 1 a 2 2ax ^ e ZTM ^z j x H0y H0y k ¼ e P ¼ E H* ¼ . . . ¼ 2 2oe 2 om

ð2:130Þ

Similar considerations hold as for the TE waves.

2.10 Reflection and transmission of plane waves; Snel’s laws The solutions to Maxwell’s equations discussed in the previous section are valid in unbounded media, but practical problems involve discontinuities between media with different EM properties. To take a step further towards reality, and still using an oversimplified model, let us consider two half spaces, each filled with a different dielectric material with parameters e1, m1 for z < 0 and e2, m2 for z > 0. Both media are supposed to be lossless. The geometry of our problem is depicted in Figure 2.5, the z axis being perpendicular to the boundary z ¼ 0 between the two media. Suppose there exists a uniform plane wave, produced by an infinitely remote source, with phase vector b1, forming an angle yi, the angle of incidence, to the normal to the surface of separation bi ¼ b1 ð^x sin yi þ ^z cos yi Þ

Figure 2.5

Reflection and transmission of plane waves.

ð2:131Þ

28


pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where b1 ¼ o me0 er1 ¼ b0 er1 and b0 is the phase constant of vacuum.The EM field of the wave can be expressed as Ei ¼ E0i e jbi r ¼ E0i e jb1 ðx sin yi þ z cos yi Þ Hi ¼ bi Ei =ðomÞ

ð2:132Þ

E0i is orthogonal to bi and in general may be linearly or elliptically polarized. Expressions (2.132) for the EM field would be a valid solution if all space were filled with dielectric 1. However, they do not satisfy Maxwell’s equations in medium 2 for z > 0. The solution to the problem must be found by summing in (2.132) the ‘reaction’ of the structure to the wave (2.132) that we have impressed on it, and which we call the incident wave. Such a reaction must be sought among the solutions to Maxwell’s equations in such a way that, after summing to the incident wave, the boundary conditions at z ¼ 0 are satisfied. To this end it is sufficient for z < 0 to add to the incident wave (2.132) another wave, which we call the reflected wave, and, for z > 0, to consider another wave in medium 2, which we call the transmitted wave. The overall field for z < 0 that results from the superposition of the incident plus the reflect wave, and the field of the transmitted wave for z > 0, must satisfy the boundary conditions at z ¼ 0, i.e. the continuity of the tangential components of the electric and magnetic fields. With the notation of Figure 2.5, the reflected wave for z < 0 is expressed as Er ¼ E0r e jbr r ¼ E0r e jb1 ðx sin yr z cos yr Þ Hr ¼ br Er =ðomÞ

ð2:133Þ

where br ¼ b1 ð^x sin yr ^z cos yr Þ

ð2:134Þ

while the transmitted wave in z > 0 can be written as Et ¼ E0t e jbt r ¼ E0t e jb2 ðx sin yt þ z cos yt Þ Ht ¼ bt Et =ðomÞ

ð2:135Þ

with bt ¼ b2 ð^x sin yt þ ^z cos yt Þ pffiffiffiffiffiffi b2 ¼ b0 er2

2.10.1

ð2:136Þ ð2:137Þ

Snel’s laws; total reflection

Expressions (2.132), (2.133) and (2.135) are all solutions of Maxwell’s equations in the respective media 1 and 2. The solution to our problem, as already stated, must satisfy the boundary conditions, i.e. the continuity conditions at z ¼ 0. Such conditions first imply that the law of variation along the common x direction for z ¼ 0 must be the same for all waves. Therefore bix ¼ brx ¼ btx

ð2:138Þ

Using (2.131), (2.134) and (2.136) we easily obtain yr ¼ yi pffiffiffiffiffiffi pffiffiffiffiffiffi er1 sin yi ¼ er2 sin yt

ð2:139Þ


29

The above equations constitute Snel’s laws. The former states that the angle of reflection is equal to the angle of incidence; according to the second law, the transmission angle (or angle of refraction) can be expressed as sin yt ¼ where ni ¼

rffiffiffiffiffiffi er1 n1 sin yi ¼ sin yi er2 n2

ð2:140Þ

pffiffiffiffiffi eri is the refractive index of the ith medium.

Willebrord Snel van Royen (1580–1626) was a Dutch mathematician, whose Latinized name was Snellius. Snel observed that a ray of light incident on a rectangular tank of water, as shown in Figure 2.6, refracted at the point O, is such that the distance of the actual refracted ray C from the point of refraction O is in a fixed ratio to the distance B from O of the extrapolated ray, regardless of the angle of incidence. The reader can easily prove that OC=OB ¼ n2 , n2 being the refractive index of water.

From (2.140) it follows that cosyt ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 2 1 sin2 yt ¼ 1 sin yi er2

ð2:141Þ

Note that when er1 > er2 , i.e. the wave comes from a denser to a less dense medium, e.g. from glass to air, the angle of refraction is greater than the angle of incidence: yt > yi . Clearly there exists a critical angle yc such that yt ¼ p/2. Using (2.140) we obtain yc ¼ sin 1

er2 er1

ð2:142Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi For angles of incidence greater than yc, er1 =er2 sin yi > 1 and Equation (2.140) does not yield real solutions for yt.11 Such a seeming inconsistency occurs because we have implicitly assumed that the 11

In fact, solving for yt , (2.140) yields

yt ¼ p2 þ j cosh 1

qffiffiffiffi er1 er2

sin yi

ð2:143Þ

Putting yt ¼ ytr þ jytj , and using well-known trigonometric formulae, we have

sin yt ¼ sin ytr cosð jytj Þ þ cos ytr sinð jytj Þ ¼ sin ytr cosh ytj þ j cos ytr sinh ytj

ð2:144Þ

In deriving the last equality we used the following properties:

cosð jjÞ ¼ cosh j sinð jjÞ ¼ j sinh j

ð2:145Þ

Inserting (2.139) into (2.135) and equating the real and imaginary parts of both sides, we easily find

ytr ¼

p 2

cosh ytj ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 sin yi er2

ð2:146Þ

30


Figure 2.6

Snel’s experiment.

transmitted wave is a uniform plane wave, i.e. that at ¼ 0. In reality, for angles of incidence greater than the critical angle, the transmitted wave is attenuated along the z direction, the propagation vector being kt ¼ b2 ^x ja2^z . In fact, (2.141) can be written as cos yt ¼ j

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 2 sin yi 1 er2

ð2:147Þ

The minus sign in front of the square root has been used for a reason that will soon become apparent. Inserting (2.140) and (2.147) into the expression for the phase vector of the transmitted wave (2.136) and using kt instead of b t, we obtain rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 er kt ¼ bt jat ¼ b2 x^ sin yi j^z sin2 yi 1 er2 er2

ð2:148Þ

The complex propagation vector represents a TE or TM wave propagating in the x direction and attenuated in the z direction, with rffiffiffiffiffiffi er1 sin yi ¼ ^ xb1 sin yi er2

ð2:149Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er1 2 er2 sin yi 1 ¼ ^zb1 sin2 yi er2 er1

ð2:150Þ

bt ¼ ^xb2

at ¼ ^zb2

As we saw in the previous section, such a wave does not carry any real power in the z direction, and thus inside the second medium. This is called a surface wave since it propagates along the boundary between the two media and decays exponentially towards the second medium. The real power carried by the incident wave is conveyed into the reflected wave. For this reason we say that for angles of incidence greater than the critical angle yc, the total reflection occurs. It is worth specifying that, although the incident power is totally reflected by the discontinuity between the two media, the field inside the second medium is not zero, but it decays exponentially as it enters from its surface.


2.10.2

31

Reflection and transmission (Fresnel’s) coefficients

Conditions (2.138), and thus Snel’s laws (2.139), are necessary but not sufficient to ensure the continuity of the tangential components of the electric and magnetic fields across the boundary z ¼ 0. Let us assume that the incident field is linearly polarized; by virtue of the linearity of Maxwell’s equations, the case of elliptical polarization can be studied by superposition. We intend now to derive the expressions for the reflection and transmission coefficients at the boundary between the two media. To do this in a speedy way we will follow a procedure based on the equivalence between waves and transmission lines. More precisely, an equivalence can be stated between the E and H components tangential to the discontinuity between the media and voltages and currents along transmission lines. The reader already familiar with transmission lines will easily recognize the similarity of the procedure that we are going to develop with that used in the simple problem of computing the reflection coefficient of a load (medium 2) seen by a transmission line of different impedance (medium 1). Using the subscript ‘t’ to indicate components parallel to the z ¼ 0 plane, let E0it ; E0rt ; E0tt H0it ; H0rt ; H0tt be the E-field and H-field components of the incident, reflected and transmitted waves, respectively, tangential to the boundary z ¼ 0. The continuity of the fields across the plane z ¼ 0 requires E0it þ E0rt ¼ E0tt H0it þ H0rt ¼ H0tt

ð2:151Þ

Let us now introduce the concept of tangential impedance Zt as the ratio of the tangential E- and H-field components. Using the field equations (2.132), (2.133) and (2.135), we may write E0it E0rt ¼ H0it H0rt E0tt Zt2 ¼ H0tt

Zt1 ¼

ð2:152Þ

As we will see in detail later on, the tangential impedances Zt1, Zt2 depend in practice on the polarization and angle of incidence of the incoming wave. The reflection coefficient and the transmission coefficient are defined as E0rt E0it E0tt T¼ E0it

G¼

ð2:153Þ

respectively. Using (2.152) and (2.153), the system (2.151) can be written as 1þG ¼ T 1 1 ð1 GÞ ¼ T Zt1 Zt2

ð2:154Þ

The above system of equations is easily solved to get G¼

Zt2 Zt1 Zt2 þ Zt1

T¼

2Zt2 Zt2 þ Zt1

ð2:155Þ

32


Figure 2.7

Perpendicular polarization.

In order to specify the values of the tangential impedances, we need to distinguish two cases, depending on whether the electric field is perpendicular or parallel to the plane of incidence, i.e. the plane (xz) containing the phase vector b 1 of the incident wave and the direction (z) normal to the boundary (see Figure 2.5).

Perpendicular polarization This case is illustrated in Figure 2.7. The electric field vectors of the incident, reflected and transmitted fields in (2.132), (2.133) and (2.135) can be written as follows: E0i ¼ E0i ^y E0r ¼ E0r y^

ð2:156Þ

E0r ¼ E0t ^y Correspondingly, the magnetic field vectors are H0i ¼

E0i ð ^x cos yi þ ^z sin yi Þ Z1

H0r ¼

E0r ð^x cos yi þ ^z sin yi Þ Z1

H0t ¼

E0t ð ^x cos yt þ ^z sin yt Þ Z2

ð2:157Þ

recalling that yr ¼ yi . The tangential impedances are therefore Zt1 ¼

E0i E0r Z ¼ ¼ 1 H0ix H0rx cos yi

Zt2 ¼

E0t Z ¼ 2 H0tx cos yt

ð2:158Þ

Inserting (2.158) into (2.155) and using (2.141), we obtain after some simple algebra rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er2 sin2 yi er1 Z2 cos yi Z1 cos yt . . . rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G? ¼ ¼ ¼ er2 Z2 cos yi þ Z1 cos yt sin2 yi cos yi þ er1 cos yi

T? ¼

2Z2 cos yi ¼ ... ¼ Z2 cos yi þ Z1 cos yt

2 cos yi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er2 cos yi þ sin2 yi er1

ð2:159Þ


Figure 2.8

33

Reflection coefficients vs. angle of incidence.

Typical behaviour of the reflection coefficient versus the angle of incidence for the case of perpendicular polarization is shown in Figure 2.8. Note that the reflection coefficient is an increasing function of yi.

Parallel polarization

This is a dual case of the perpendicular polarization. As sketched in Figure 2.9, the magnetic field is directed along y while the electric field lies is in the xz plane: Ei ¼ E0i ð^x cos yi ^z sin yi Þ

H0i ¼ ^ yE0i =Z1

Er ¼ E0r ð ^x cos yi ^z sin yi Þ

H0r ¼ ^ yE0r =Z1

Et ¼ E0t ð^x cos yt ^z sin yt Þ

H0t ¼ ^ yE0t =Z2

ð2:160Þ

The tangential impedances are Zt1 ¼

E0ix E0rx ¼ ¼ Z1 cos yi H0i H0r

Zt2 ¼

E0tx ¼ Z2 cos yt H0t

Figure 2.9

Parallel polarization.

ð2:161Þ

34


Using (2.161) in (2.155) we get E0rx Gjj ¼ E0ix

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er2 sin2 yi er1 Z2 cos yt Z1 cos yi ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ er2 Z2 cos yt þ Z1 cos y sin2 yi þ er1

er2 cos yi er1 er2 cos yi er1

ð2:162Þ

As far as the transmission coefficient is concerned, notice that the definition (2.153) corresponds to the ratio of the tangential x components to the transmitted and incident fields. Rather, one commonly refers to the transmission coefficient as the ratio of the full amplitudes of the fields.12 Therefore Tjj ¼

E0t E0tx =cos yt 2Z2 cos yi 2cos yi ¼ ¼ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er2 er2 E0i E0ix =cos yi Z2 cos yt þ Z1 cos yi sin2 yi þ cos y er1 er1

ð2:163Þ

Typical behaviour of the reflection coefficient (2.162) as a function of the angle of incidence is also shown in Figure 2.8. A special feature is apparent, since for a particular incident angle yB, called the Brewster angle, the reflection coefficient becomes zero. Putting Gjj ¼ 0 in (2.162), one obtains13 yt ¼ p=2 yi Therefore, inserting this into Snell’s law (2.139) we obtain rffiffiffiffiffiffi er2 tan yB ¼ er1

ð2:164Þ

ð2:165Þ

The Brewster angle is also called the polarization angle. Indeed, when the incident wave is elliptically polarized, it can be seen as the superposition of two linearly polarized waves, one parallel and one perpendicularly polarized. Since the parallel polarized wave is totally absorbed by the second medium, the reflected wave will be linearly polarized.

2.10.3

Reflection from a conducting plane

So far we have assumed that both media where the EM field propagates are perfect dielectrics. While we will not consider the case of lossy dielectrics, an important case is to be considered, i.e. when medium 2 is a good conductor. In fact, this case is representative of a number of practical applications and is the basis for the evaluation of the loss in a conductor. As shown in Figure 2.10, if medium 2 is a good conductor, then s oe0 er2 so that its complex dielectric constant can be written as ec2 ¼ e0 er2 js=o ffi js=o

ð2:166Þ

In contrast with the case of a perfect dielectric considered so far, the propagation vector kt of the transmitted wave in medium 2 has both a real and an imaginary part; the condition (2.97) becomes kt kt ¼ b2t a2t 2jbt at ¼ o2 mec ffi joms

ð2:167Þ

Equating the real and imaginary parts of the above complex equation we obtain b2t ¼ a2t bt at ffi

oms 2

ð2:168Þ ð2:169Þ

12 In the case of the reflection coefficient the two definitions coincide since the angle of reflection equals the angle of incidence. pffiffiffiffi 13 pffiffiffiffi e1 cos yt ¼ e2 cos yi . Combining with (2.139) we get sin yt cos yt ¼ sin yi cos yi , thus sin 2yt ¼ sin 2yi . This equality implies that yt ¼ p=2 yi , since the case yt ¼ yi may not be considered as it would imply e2 ¼ e1.


Figure 2.10

35

Reflection from a conducting plane.

To ensure the continuity of the fields along the x axis for z ¼ 0, we must have ktx ¼ kix, thus atx ¼ 0

ð2:170Þ

btx ¼ bt sin yt ¼ bix ¼ b1 sin yi

ð2:171Þ

and pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where b1 ¼ o me0 er1 ¼ b0 er1 . Using the above equations, (2.168) and (2.169) become b21 sin2 yi þ b2tz ¼ a2tz btz atz ¼

oms 2

ð2:172Þ ð2:173Þ

In the hypothesis of very high conductivity of medium 2, the x component (2.171) of bt becomes pffiffiffi negligible with respect to the z component since, according to (2.173), both btz and atz increase with s. 2 2 Neglecting the term b1 sin yi in (2.172) we obtain rffiffiffiffiffiffiffiffiffi oms 1 ð2:174Þ bt ¼ at ¼ ¼ 2 ds where

sffiffiffiffiffiffiffiffiffi 2 ds ¼ oms

ð2:175Þ

is the skin depth of the conductor (see below).14 As the EM field penetrates into the conductor, its amplitude decays exponentially with attenuation at given by (2.174); at the distance ds ¼ 1=at the field

14

If the reader is interested in the exact solution of (2.167) here it is:

1 ¼ 2 ds

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 d b b2 1 þ s ix ix 2 2

1 a2tz ¼ 2 ds

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 d b b2 1 þ s ix þ ix 2 2

b2tz

Since the quantity

2 2 ds bix oe1 2

¼ sin yi 2 s is negligible with respect to unity, the above expressions for btz and atz reduce to (2.174).

ð2:176Þ

ð2:177Þ

36


amplitude is reduced to e 1 ffi 0:137. From (2.171) we get for the angle of refraction rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 sin yi 2oe0 er1 sin yt ¼ ¼ ds b1 sin yi ¼ sin yi 1 bt s

ð2:178Þ

Consider for example a plane wave at a frequency f ¼ 10 MHz impinging from air on a metal surface of iron at an angle of 45 . Since the conductivity of iron is s ¼ 1 107 S=m, we obtain in this case that the skin depth is ds ¼ 0:159 mm. Moreover, b1 ¼ 0:210 per metre and the angle of refraction is yt ¼ 0:33 10 6 rad. Using (2.174), the propagation vector of the transmitted wave is therefore rffiffiffiffiffiffiffiffiffi oms ð1 jÞ^z ð2:179Þ kt ffi 2 This result tells us that the transmitted wave is a uniform plane wave in practice, since the equal-phase planes are also equal-amplitude planes; indeed the x component of the phase vector is much smaller that the z component, so that the angle of refraction can be very well approximated with zero whatever the angle of incidence. In such conditions, the electric and magnetic fields of the transmitted wave are tangential to the surface z ¼ 0 of the conductor. More specifically, suppose that the incident field is perpendicularly polarized: Ei ¼ ^yE0i e jb1 ðx sin yi þ z cos yi Þ Hi ¼

E0i ð ^x cos yi þ ^z sin yi Þe jb1 ðx sin yi þ z cos yi Þ Z1

ð2:180Þ

The electric field transmitted in the conductor is Et ¼ ^yE0t e jbt r ¼ ^yE0t e jb1 x sin yi jbt z e at z

ð2:181Þ

while the magnetic field can be written as Ht ¼

1 1 kt Et ¼ ^z Et om Zc

where Zc is the intrinsic impedance of the conductor rffiffiffiffiffiffiffi rffiffiffiffiffiffi m om ffi ð1 þ jÞ Zc ¼ ec2 2s

ð2:182Þ

ð2:183Þ

Such a quantity plays the role of a surface impedance as it determines the ratio of the electric and magnetic fields tangential to the surface of the conductor. Equation (2.182) can be put in the form ^ Ht Et ¼ Zc n

ð2:184Þ

^ being the normal unit vector directed outside the metal. Equation (2.184) is called the Leontovic n condition: it allows us to determine the electric field at the surface of a conductor in terms of the magnetic field. The same result is obtained for a parallel polarized incident wave.

2.11 Electrodynamic potentials Maxwell’s equations can be solved formally in terms of auxiliary variables, called electrodynamic potentials, which represent the generalization to the dynamic case of the static scalar potential V(r) from which the static electric field can be obtained as its gradient E ¼ rV.


37

Consider Maxwell’s equations in a medium with constants m and e. r E ¼ jomH

ð2:185Þ

r H ¼ joeE þ J

ð2:186Þ

If the magnetic permeability is the same in the whole space, from (2.185) it follows that r H ¼ 0. Since a zero divergence vector can be expressed as the curl of another vector we may write H¼rA

ð2:187Þ

A is called the magnetic vector potential.15 Inserting (2.187) into (2.185), we find r ðE þ jomAÞ ¼ 0 ) E þ jomA ¼ rF

ð2:188Þ

where the last equality is a consequence of the fact that a zero curl vector, i.e. a conservative field, can be expressed as the gradient of a scalar potential F. The electric field can thus be expressed in terms of the vector potential A and the scalar potential F: E ¼ jomA rF

ð2:189Þ

Inserting (2.189) and (2.185) into (2.186), we obtain r r A ¼ rr A r2 A ¼ joeð jomA rFÞ þ J

ð2:190Þ

rr A r2 A ¼ k2 A joerF þ J

ð2:191Þ

whence

where k2 ¼ o2me. So far, we have specified only the curl of the vector potential A; without affecting its curl we may arbitrarily fix its divergence in such a way as to simplify Equation (2.190). Let us therefore put r A ¼ joeF

ð2:192Þ

This is called the Lorentz gauge. As a consequence, (2.191) becomes r2 A þ k2 A ¼ J

ð2:193Þ

Suppose we are able to solve (2.193) with appropriate boundary conditions: the magnetic field can then be calculated using (2.187), while the electric field can be expressed in terms of only A using (2.189) and (2.192). The resulting expression is E ¼ jomA þ

rr A jom

ð2:194Þ

A totally dual procedure can be developed assuming that there are magnetic current sources Jsm instead of electric sources Js. In such a case Maxwell’s equations are r E ¼ jomH Jm

ð2:195Þ

r H ¼ joeE

ð2:196Þ

15 Usually (see for instance [3] and [4]) the magnetic vector potential is defined through the magnetic induction, i.e. we put B ¼ r A. Assuming that the medium is homogeneous, and thus m is constant, the choice of (2.187) is preferred here as it yields dual expressions for E and H.

38


It is now the turn of the E field to be solenoidal. By introducing an electric vector potential F and continuing in much the same way, we find that it must satisfy Helmholtz’s equation r2 F þ k 2 F ¼ J m

ð2:197Þ

E ¼ r F

ð2:198Þ

while the EM field is obtained from

H ¼ joeF þ

rr F jom

ð2:199Þ

In the absence of both electric (J ¼ 0) and magnetic (Jm ¼ 0) sources, Equations (2.193) and (2.197) become homogeneous. The EM field can be derived either in terms of A through (2.187) and (2.194) or in terms of F through (2.198) and (2.199).

Bibliography 1. 2. 3. 4.

J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., New York, 1962. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, 1992. C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, Inc., New York, 1989.

3

Guided EM propagation 3.1 Introduction Transmission lines are fundamental components of any RF apparatus or system. They are necessary to connect devices and components in order to transmit the RF signal from one point to the other. At RFs the passive components themselves behave as distributed elements made of transmission line or waveguide sections. Because of the high operating frequencies, in fact, the dimensions of the circuit elements are normally of the same order of magnitude as the wavelength. There is an ample variety of transmission lines or, more generally, of guiding structures or waveguides. The choice of one rather than another type is based on a number of practical considerations, such as loss, weight and size, cost, compatibility with active devices. The use of high frequencies requires on the one hand more sophisticated models capable of accounting for propagation delay effects along the circuit elements and within the electronic devices themselves; on the other hand, the frequency increase is normally associated with an increase in the losses. These are mainly due to electric currents flowing in the conductors and to spurious radiation from discontinuities and interconnections. For such reasons, the structures employed to guide the EM signal must be chosen properly depending on the operating frequency, in order to minimize the power loss. As an example, the twin line commonly employed at industrial or low frequencies cannot be used at RF because of its high radiation loss. It is therefore replaced by the coaxial cable, which, in turn, becomes quite lossy above a few tens of gigahertz and must be replaced by the waveguide. Figure 3.1 shows the most common transmission lines and guiding structures employed at RF and microwaves. On the one side, except for the circular dielectric waveguide where the EM field is confined by the dielectric discontinuity, we have conventional closed structures such as the coaxial cable and metal waveguides of various shapes, where the field is confined by metal walls. On the other side, we have planar, printed or integrated circuits fabricated on top of a thin dielectric substrate, such as the stripline or the microstrip line; because of their advantages in terms of reduced size, weight and cost, they are the most frequently employed transmission lines, except when the requirements of low loss or high power can only be matched by the use of waveguides. Multi-conductor transmission lines such as the coaxial cable support the propagation of a TEM mode starting from DC. Waveguides that are made of only one conductor, on the contrary, are characterized by a cut-off frequency, below which EM propagation cannot take place. Intermediate between the printed circuit and waveguide is the fin line, consisting of a slotline


40


Figure 3.1

Some common transmission lines for RF and microwaves.

located in the so-called E plane of the rectangular waveguide. The image line consists of a rectangular dielectric guide laid on a metal plane. With the exception of the stripline, planar circuits are characterized by the inhomogeneity between the dielectric substrate and the air over it. As a consequence they exhibit a dispersive behaviour, i.e. a frequency dependence of the propagation characteristics. A comparison of the properties of the various structures according to application is presented in Table 3.1. Under the term waveguide we consider circular, rectangular and ridge waveguides, in the order of increasing loss (thus decreasing quality factor). Similarly, the useful frequency range is narrowest for the circular and broadest for the ridge waveguide – and just the opposite with the transverse size and power handling. The quality factors are of the order of a few thousands for rectangular guides, some hundreds for coaxial cables, a few hundreds for the suspended stripline and some tenths for striplines and microstrip lines. Another important characteristic of a transmission line is its integrability with active devices. From this point of view, printed circuit lines, particularly the microstrip and the coplanar waveguide, offer the best advantages as they allow the use of multi-terminal devices. The coplanar waveguide has the further advantage over the microstrip of having all conductors on the same side of the substrate, so allowing easy mounting of devices in shunt configuration. On the contrary, the other technologies, such as the stripline, the suspended stripline, the fin line, the coaxial cable and the waveguide, do not allow anything but two-terminal devices (diodes) to be employed.

GUIDED EM PROPAGATION

41

Table 3.1 Comparison between various types of transmission lines. Transmission line Waveguide Coaxial cable Stripline Microstrip Coplanar line Slotline Suspended stripline Image line

Z0 ðOÞ

Q

Small size

Handled power

Compatibility with active devices

Low cost

1–300 > : jom

for TM modes for TE modes

ð3:52Þ

for TEM modes

8 joe > > > 2 < k Y¼ j z > om > > : joe

for TM modes for TE modes

ð3:53Þ

for TEM modes

Equations (3.51) constitute a set of linear homogeneous differential equations known as the telegrapher’s equations, whose general solution can be put in the following form: VðzÞ ¼ V þ egz þ V e þ gz 1 IðzÞ ¼ ðV þ egz V e þ gz Þ Z0

ð3:54Þ

where Z0 ¼ g¼

pffiffiffiffiffiffiffiffiffi Z=Y

ð3:55Þ

pffiffiffiffiffiffi ZY

ð3:56Þ

Equations (3.54) provide expressions for the equivalent voltages and currents along a generalized transmission line, corresponding to the complex amplitudes of the E and H fields propagating in the guiding structure. As shown in the next section, the telegrapher’s equations can also be derived from conventional circuit theory by considering an infinitesimal length of a two-wire transmission line. This approach has the advantage of associating the per-unit-length impedance and admittance with the familiar lumped element concept. The quantity Z0 is the characteristic impedance and, as can easily be verified, it is coincident with the wave impedance (3.44), (3.45) and (3.46) of the respective mode. The quantity g is the propagation constant; as can be found using (3.52) and (3.53), it coincides with jkz . Both Z0 and g are, in the general case, complex quantities. In the case of lossless structures both Z and Y in (3.52)–(3.53) are imaginary, so that the characteristic impedance is real while the propagation constant is imaginary. An exception is represented by the modes below cut-off, or evanescent modes, for which the propagation constant g becomes real and the characteristic impedance is imaginary. From (3.55) and (3.40) in fact we obtain 8 a z > > < joe for TM modes ð3:57Þ Z0 ¼ jom > > : for TE modes az Note that, as solutions of a homogeneous problem, the functions T representing the transverse potential are determined apart from an arbitrary factor. This implies a corresponding uncertainty in the transverse modal vectors et, ht in Table 3.3. Such an uncertainty can be eliminated by imposing that the modal vectors satisfy the normalization condition: ð et ht ^z dS ¼ 1 ð3:58Þ S


51

Such a condition implies that V(z) and I(z) may be interpreted as equivalent voltages and currents in such a way that the power associated with the voltage and current waves equals that transported by the EM field. Using (3.47), the latter is given by P¼

1 2

ð

ð 1 1 E H* ^z dS ¼ VI * et ht ^z dS ¼ VI * 2 2 S S

ð3:59Þ

It is worth noting that only the transverse components of the EM field contribute to the power flow, since, as can easily be verified, ðE H* Þ ^z ¼ ðEt H*t Þ ^z. The normalization condition (3.57) therefore ensures that the power travelling along the equivalent transmission lines is equal to that transported by the mode. It should also be noted that the equivalent voltages and currents V(z) and I(z) in (3.47) can be chosen in different ways, still leaving the electric and magnetic fields unaltered. In fact, we may factorize the fields as Et ¼ VðzÞet ðx; yÞ ¼ ½CVðzÞ½et ðx; yÞ=C ¼ V 0 ðzÞe0t ðx; yÞ Ht ¼ IðzÞht ðx; yÞ ¼ ½IðzÞ=C½Cht ðx; yÞ ¼ I 0 ðzÞh0t ðx; yÞ

ð3:60Þ

where C is an arbitrary real constant. An alternative definition of equivalent voltages and currents is therefore V 0 ðzÞ ¼ CVðzÞ I 0 ðzÞ ¼ IðzÞ=C e0t ðx; yÞ ¼ et ðx; yÞ=C

ð3:61Þ

h0t ðx; yÞ ¼ Cht ðx; yÞ It can easily be verified that condition (3.59) on the power transported is also satisfied by the new voltages and currents V 0 I 0 : ð ð 1 1 1 * * E H* ^z dS ¼ V 0 I 0 e0t h0t ^z dS ¼ V 0 I 0 ð3:62Þ P¼ 2 S 2 2 S The transverse field vectors e0t and h0t are orthogonal, but instead of (3.48) the following relation holds: ht ðx; yÞ ¼ C 2^z et ðx; yÞ

ð3:63Þ

The characteristic impedance of the new equivalent transmission line defined in terms of V 0 I 0 is Z00 ¼

V 0 ðzÞ VðzÞ ¼ C2 ¼ C 2 Z0 I 0 ðzÞ IðzÞ

ð3:64Þ

In the general case we can write ht ðx; yÞ ¼

Z0 ^z et ðx; yÞ Zz

ð3:65Þ

where Zz is the wave impedance of the mode [20]. As we will see in the particular case of the coaxial line, the voltage V(z) for TEM modes, rather than using the definition in Table 3.3, is defined in terms of the line integral of the electric field between the two conductors. This can be done because rt Et ðx; yÞ ¼ 0, thus the electric field of a TEM mode is conservative in the cross-section and therefore its line integral between two points is independent of the integration path. As a consequence, the characteristic impedance of the TEM line differs from the wave impedance Zz given by (3.46).

52

MICROWAVE AND RF ENGINEERING I(z,t) Z0, β0

I(z+dz,t)

R dz L dz

V(z,t)

C dz

G dz

z

V(z+dz,t) z+dz

dz

Figure 3.4

An infinitesimal length of transmission line and its equivalent lumped element circuit.

3.4 Transmission lines as 1-D circuits The telegrapher’s equations (3.51) can alternatively be obtained by considering an infinitesimal length dz of a two-wire transmission line (twin conductor line, coaxial line, etc.). Since dz is negligible with respect to the wavelength, the conventional lumped element circuit theory may be employed. With reference to Figure 3.4, the infinitesimal length is characterized by a series resistance R dz and inductance L dz associated with the electric current flowing along the conductors, and by a conductance G dz and capacitance C dz, associated with the voltage difference between the conductors. The quantities R, L, G and C are the resistance, inductance, conductance and capacitance per unit length, respectively, of the transmission line, and are therefore expressed in O/m, H/m, S/m and F/m. The series resistance depends on the conductivity of the conductor(s) of which the line is made; it is zero in the ideal case of perfect conductors. Similarly, the conductance G depends on the dielectric interposed between the conductors: it is zero in the ideal case of a perfect dielectric. An ideally lossless transmission line has, therefore, both R ¼ 0 and G ¼ 0. By applying Kirchhoff’s voltage law (i.e. the total voltage drop along a closed path must be zero) to the circuit of Figure 3.3b we obtain Vðz þ dzÞVðzÞ þ ðR dz þ joL dzÞI ¼ 0

ð3:66Þ

From Kirchhoff’s current law (the total current entering a node must be zero) we get Iðz þ dzÞIðzÞ þ ðG dz þ joC dzÞV ¼ 0

ð3:67Þ

From simple algebra (writing Vðz þ dzÞVðzÞ ¼ dV etc.), we get the differential equations governing voltages and currents along the line: dV ¼ ðR þ joLÞI dz

ð3:68Þ

dI ¼ ðG þ joCÞV dz

ð3:69Þ

These equations are nothing but the telegrapher’s equations (3.51) with R þ joL ¼ Z

ð3:70Þ

G þ joC ¼ Y

ð3:71Þ

being the line per-unit-length impedance and admittance, respectively. The solution of (3.68)–(3.69) is given by (3.54). Correspondingly, the characteristic impedance and the propagation constant are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi R þ joL Z0 ¼ Z=Y ¼ ð3:72Þ G þ joC

GUIDED EM PROPAGATION g ¼ a þ jb ¼

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ZY ¼ ðR þ joLÞðG þ joCÞ

53 ð3:73Þ þ

The solutions (3.54) represent two voltage and current waves propagating in the positive (V ) and negative (V ) directions of the z axis. Indeed, the time domain voltage is expressed as Vðz; tÞ ¼ Re½Ve jot ¼ V þ eaz cosðotbzÞ þ V e þ az cosðot þ bzÞ

ð3:74Þ

The first term in (3.74) represents an incident wave propagating in the positive z direction with phase velocity (see Equation (2.91)) vph ¼

o b

ð3:75Þ

and attenuation a (nepers per metre). Similarly, the second term in (3.74) represents a reflected wave travelling in the negative z direction with the same phase velocity and attenuation as the incident wave. V þ and V are arbitrary constants that can be determined only when additional conditions are provided, namely the boundary conditions at the line ends, the line having so far been supposed to have an indefinite length. In general, the voltage to current ratio, i.e. the impedance along the line, is a quantity varying with z. Using Equations (3.54) we obtain ZðzÞ ¼

VðzÞ V þ egz þ V e þ gz ¼ Z0 þ gz V e V e þ gz IðzÞ

ð3:76Þ

When only the incident wave is present, then V ¼ 0 and the impedance equals everywhere the characteristic impedance Z0. Similarly, if only the reflected wave is present, the impedance is independent of z and equal to the opposite of the characteristic impedance (Z0).3 The characteristic impedance thus represents the ratio of the voltage and current along the line when only the incident (or, apart from the sign, the reflected) wave is present. In the general case the impedance varies along the line according to (3.76). The reflection coefficient, in general varying with z as well, is defined as the ratio of the reflected and incident voltages.4 From the first equation of (3.54) GðzÞ ¼

V e þ gz V þ 2gz ¼ e V þ egz V þ

ð3:77Þ

From some easy algebra we obtain the relations relating the impedance and the reflection coefficient: ZðzÞ ¼ Z0

GðzÞ ¼

1 þ GðzÞ 1GðzÞ

ZðzÞZ0 ZðzÞ þ Z0

ð3:78Þ

ð3:79Þ

When the admittance YðzÞ ¼ 1=ZðzÞ is employed instead, the following formulae hold between the admittance and reflection coefficient: Y ¼ Y0

1G 1þG

ð3:80Þ

3 The minus sign corresponds to the convention for the sign of the current, which is positive when flowing in the positive z direction. For the reflected wave the current flows in the negative direction. 4 A current reflection coefficient can be defined as well, but it is seldom used. When no specification is given, the reflection coefficient is invariably the voltage reflection coefficient.

54

MICROWAVE AND RF ENGINEERING Y0 Y Y0 þ Y

G¼

ð3:81Þ

where, to simplify the notation, the z dependence has been omitted. For lossless transmission lines R ¼ 0 and G ¼ 0, and Equations (3.72)–(3.73) become rffiffiffiffi L Z0 ¼ C pffiffiffiffiffiffi g ¼ jb ¼ jo LC

ð3:82Þ ð3:83Þ

Therefore, the characteristic impedance is real, while the propagation constant is imaginary. Equations (3.54) hold with g being replaced by jb. Correspondingly, the phase velocity (3.75) is expressed in terms of the inductance and capacitance per unit length of the line: pffiffiffiffiffiffi vph ¼ 1= LC ð3:84Þ Equations (3.76) and (3.77) become ZðzÞ ¼ Z0

V þ ejbz þ V e þ jbz V þ ejbz V e þ jbz

GðzÞ ¼

V þ 2jbz e Vþ

ð3:85Þ ð3:86Þ

respectively. The reflection coefficient thus varies only by its phase, while its amplitude remains constant along the line. If we put Gð0Þ ¼

V Vþ

and

Zð0Þ ¼

V þ þ V V þ V

the previous formulae can be written in the form ZðzÞ ¼ ¼ Z0

Zð0Þcos bzjZ0 sin bz Zð0ÞjZ0 tan bz ¼ Z0 Z0 cos bzjZð0Þsin bz Z0 jZð0Þtan bz GðzÞ ¼ Gð0Þe þ 2jbz

ð3:87Þ

ð3:88Þ

In most practical cases, microwave and RF transmission lines have small losses, i.e. R oL and G oC. For low-loss transmission lines, (3.73) can be greatly simplified. By neglecting the product RG pffiffiffiffiffiffiffiffiffiffiffi and by approximating 1 þ x ffi 1 þ x=2 one obtains pffiffiffiffiffiffi 1 pffiffiffiffiffiffi R G g ¼ a þ jb ffi LC þ þ jo LC ð3:89Þ 2 L C To the same degree of approximation, the characteristic impedance is still given by (3.82). The phase constant is therefore the same as in the lossless case, while the attenuation due to small losses is expressed as 1 pffiffiffiffiffiffi R G 1 ð3:90Þ LC þ ¼ ðRY0 þ GZ0 Þ a¼ 2 L C 2 pffiffiffiffiffiffiffiffiffi where Z0 ¼ 1=Y0 ¼ L=C . In the following sections, for the sake of simplicity we will normally refer to lossless transmission lines.


55

3.5 Phase velocity, group velocity and energy velocity In Section 2.8 we have seen that a function such as eðx; y; z; tÞ ¼ e0 cosðotbzÞ

ð3:91Þ

represents a wave propagating with phase velocity vp ¼

o b

ð3:92Þ

The phase velocity is indeed the velocity at which a hypothetical observer should travel in order to ‘see’ the wave phase as constant.5 Equation (3.91) represents a monochromatic wave, whose spectrum consists of just a single radian frequency o. This is therefore a more mathematical than physical solution, since it represents a signal of infinite time duration. Any practical signal used for transmitting information, on the contrary, has a spectrum which occupies a finite frequency band. This is not without important consequences for the quality of the transmission. In fact, except for the case when the phase constant b is proportional to the frequency, as for plane waves or TEM modes, the phase velocity varies with the frequency. In the former case, each spectral component of the signal propagates with the same phase velocity and the signal propagates without distortion. In the latter case, each spectral component travels with a different phase velocity so that the signal, which is nothing but the superposition of its spectral components, undergoes a progressive distortion as it travels. Such a phenomenon is called dispersion: it causes a waveform to be distorted at the output of a transmission line. This inevitably happens not only in any non-TEM lines but also in any real line when some loss is present. The practical relevance of this phenomenon depends, as might be expected, also on the signal’s bandwidth and on the line length. For narrow-band signals and not-too-long lines it can often be neglected. For wide-band or ultra wide-band (UWB) applications, which are recently being widely exploited, the concept of phase velocity becomes almost meaningless and cannot be adopted to quantify the propagation velocity of the signal. In such cases one has to resort to the concept of group velocity. Instead of a monochromatic signal such as (3.91), let us consider a signal consisting of two monochromatic waves with angular frequencies close to each other, say o0 Do. Correspondingly, the waves will propagate with phase constants b0 Db. After travelling a distance z the resulting signal is aðz; tÞ ¼ sin½ðo0 DoÞtðb0 DbÞz þ sin½ðo0 þ DoÞtðb0 þ DbÞz ¼ 2cosðDotDbzÞsinðo0 tbzÞ

ð3:93Þ

where we have made use of known trigonometric formulae. The above expression can be seen as a waveform at frequency o0, like (3.91), its amplitude being modulated by the function cosðDotDbzÞ which has an angular frequency Do and a phase constant Db. Figure 3.5 shows the waveform (3.93) at a given time instant t. The group velocity is the velocity with which the modulating function propagates, in the limit of infinitesimal frequency shift Do. It is therefore given by the expression 1 do db vg ¼ ð3:94Þ ¼ db do Although obtained with an oversimplified procedure based on just two frequencies, the above expression (3.94) can be demonstrated in the general case of a continuous spectrum.6 5 It should be stressed that this is a mathematical rather than a physical velocity, as it does not imply any motion of particles. The reader should not be astonished when ascertaining that in many cases the phase velocity is greater than the velocity of light, seemingly contradicting Einstein’s principle of relativity. 6 See [3], pp. 200–204.

56

MICROWAVE AND RF ENGINEERING 1 0.8 signal amplitude

0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 0

10

20

30

40 50 z (m)

60

70

80

90

Figure 3.5 Waveform resulting from the superposition of two waves at frequencies f0 Df, with f0 ¼ 100 MHz and Df ¼ 4 MHz. Equation (3.94) holds as long as the dispersion of the transmission line is limited or, in other terms, if the phase velocities at the band limits are close enough so that b can be approximated by a linear function of frequency. Otherwise, the signal distortion is such that a velocity of the signal cannot be defined. In the case of a waveguide, using (3.37) with b ¼ kz , we obtain for the phase and group velocities c vp ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:95Þ o 2 c 1 o rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 2 c o

ð3:96Þ

P Wm þ W e

ð3:97Þ

vg ¼ c 1

pffiffiffiffiffi where c ¼ 1= me is the velocity of light in the medium. Note that the product of the two velocities is 2 equal to c ; their frequency behaviours are shown in Figure 3.6. While the phase velocity is always greater than c, the group velocity is always less than the velocity of light in the medium. At the cut-off frequency, the phase velocity becomes infinite. This corresponds to the phase being everywhere the same at any distance (b ¼ 0), thus the wave has simultaneously the same phase at points infinitely distant from one another. On the contrary, at cut-off the group velocity is zero because there is actually no field propagation. Both velocities approach c asymptotically at very high frequencies. This implies that dispersion becomes negligible at frequencies well above cut-off, as reflected by the dispersion diagram of Figure 3.3. Since a power flow is associated with the EM field travelling along a transmission line, it seems reasonable to define an energy velocity vE. This can be computed on the basis of the following reasoning. The power transmitted through the waveguide cross-section is nothing but the energy flowing through it in 1 second. Such energy is to be found in a cylindrical volume whose length equals the distance d covered by the energy in 1 second, thus d ¼ vE 1. By equating the real power P through the cross-section S to the total energy stored in the cylinder, we find the following expression for the energy velocity: vE ¼

where P is computed as the flow of the real part of the Poynting vector through S, and Wm ; We are the magnetic and electric energy densities, respectively, per unit length of guide, or, equivalently, the energy densities over the guide cross-section:


10

57

x 108

9 8 7 6

vp

m 5 s 4 3

vg

2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω ωc Figure 3.6

Phase and group velocities of a waveguide mode. ð m jHj2 dS 4 S ð e We ¼ jEj2 dS 4 S

Wm ¼

ð3:98Þ

Using the field expressions quoted in Table 3.2, it can be shown that the energy velocity (3.97) has exactly the same value as the group velocity (3.94).7

3.6 Properties of the transverse modal vectors et, ht; field expansion in a waveguide The functions T(x, y) introduced in Section 3.2 in the factorization of the potentials Az or Fz show a number of important mathematical properties that derive from Helmholtz’s equation (3.13) and the relevant boundary conditions for TM, TE or TEM modes. As an immediate consequence, also the transverse vectors et and ht (3.47) that determine the field dependence on the transverse coordinates possess specific mathematical properties referred to, as we will see in a moment, as orthogonality properties. To simplify our discussion, we are not going to demonstrate these properties, although it is an instructive procedure – the interested reader will find the proofs in Appendix C. Let us consider an ideal waveguide, i.e. one made of a perfect conductor and filled with a perfect dielectric, and having two different modes, identified by the superscripts ‘p’ and ‘q’. The following equation holds between the corresponding electric field transverse vectors: ð ðpÞ ðqÞ et et dS ¼ 0 ð3:99Þ S

The above equation states the orthogonality of the transverse modal vectors of an ideal waveguide. This is a consequence of the boundary conditions where the tangential electric field must be zero on the metal wall. Therefore, (3.99) does not hold true in the case of a lossy conductor. 7

A complete treatment of the various velocities can be found in the classical book by J. A. Stratton [2], pp. 330–340.

58


If p and q refer to the same mode, the r.h.s. of (3.99) is not zero. Using the normalization condition (3.58) along with relation (3.48) to express ht in terms of et, we find ð ðpÞ ðpÞ et et dS ¼ 1 ð3:100Þ S

Equations (3.99) and (3.100) can be combined into the orthonormalization of the modal vectors ð 1 for p ¼ q ðpÞ ðqÞ et et dS ¼ dpq ¼ ð3:101Þ 0 for p 6¼ q S where dpq is called the Kronecker delta. Because of (3.48), Equation (3.101) also implies8 ð ðpÞ ðqÞ ht ht dS ¼ dpq

ð3:102Þ

S

ð

ðpÞ

ðqÞ

et ht ^z dS ¼ dpq

ð3:103Þ

S

It can be proved that the orthogonality (3.99) holds without any exceptions if p and q refer to different types of modes, one TM and the other TE. If both p and q refer to TM or TE modes, the orthogonality holds provided that the modes have different eigenvalues, i.e. if 2 2 6¼ ktq ktp

ð3:104Þ

Different modes having the same eigenvalue are called degenerate modes. Although they have different field distributions, they have the same cut-off frequency, wavelength and phase velocity.9 As we will see later on, degenerate modes are systematically present in circular and square waveguides. A very important consequence of the modal orthogonality, from the physical point of view, is that the power carried by a combination of modes equals the sum of the powers individually carried by each mode. In other words, each mode carries power independently of the others. Such a result is not obvious at all, since the power is quadratic with respect to the EM field, and thus, in general, the power of the superposition of various field distributions differs from the sum of the powers associated to each of those. The above property is easily proved by computing the power carried by the superposition of the p and q modes: ð 1 ðpÞ ðqÞ ðpÞ ðqÞ P¼ ðE þ Et Þ ðHt þ Ht Þ* ^z dS 2 S t ð ð 1 1 ðpÞ ðpÞ ðqÞ ðqÞ ¼ V ðpÞ I ðpÞ* et ht ^z dS þ V ðqÞ I ðqÞ* et ht ^z dS 2 2 ð3:105Þ S ð S ð 1 ðpÞ ðqÞ* ðpÞ 1 ðqÞ ðpÞ* ðqÞ ðqÞ ðpÞ et ht ^z dS þ V I et ht ^z dS þ V I 2 2 S S ¼ PðpÞ þ PðqÞ 8

If we adopt (3.63) instead of (3.48), then (3.101) and (3.102) must be replaced by ð ðpÞ ðqÞ C 2 et et dS ¼ dpq S

1 C2

ð

ðpÞ

ðqÞ

ht ht dS ¼ dpq S

respectively. 9 Although the orthogonality (3.99) cannot be proved for a pair of degenerate modes, it is nevertheless always possible to resort to mode orthogonality by the so-named Gram-Schmidt orthogonalization procedure consisting of ðpÞ ðqÞ 0ðpÞ 0ðqÞ substituting the pair et ; et with two linear combinations et ; et such that (3.101) holds.


59

where P(p) and P(q) are the powers carried by the p and q modes respectively. As already observed in connection with (3.59), it can be seen that only the transverse components of the EM fields contribute to the power carried by the modes. From the mathematical point of view, the mode orthogonality is a basic property which allows the expansion of any EM field in the waveguide in terms of a summation (or a series) of the waveguide modes. It can be proved, in fact, that the three groups of TM, TE and TEM waveguide modes form a complete set of orthonormal functions, which means that an arbitrary piecewise-continuous square-integrable vector function defined in the waveguide cross-section can be expanded in terms of the waveguide modal vectors [1]. The most general EM field in a waveguide can thus be expressed by the summation of all its modes; using (3.47)–(3.50) with an index n to designate the generic mode we can therefore write Et ¼

1 X Vn ðzÞetn ðx; yÞ n¼0

1 X In ðzÞhtn ðx; yÞ Ht ¼

ð3:106Þ

n¼0

Ez ¼

Hz ¼

1 1 X k2 Tn ðx; yÞIn ðzÞ joe n¼0 tn 1 1 X k2 Tn ðx; yÞVn ðzÞ jom n¼0 tn

for TM modes

for TE modes

The above expansions are the basis for the so-called mode matching method, discussed in Chapter 16. One application is the calculation of the field excited in a waveguide by a current source, presented in Section 7.6.1.

3.7 Loss, attenuation and power handling in real waveguides Power loss, and thus signal attenuation, specifically distinguishes real from ideal waveguides. Moreover, the amount of power that can be handled practically is limited because the filling dielectric breaks down when subject to very high electric fields. Losses in a real waveguide are mainly due to the finite conductivity of the metal and, to a lesser extent, to the conduction and polarization currents in the filling dielectric. It can therefore be concluded that, by virtue of energy conservation, the EM field cannot propagate in a real waveguide without attenuation. In most cases, however, the losses are usually so small that a simplified approach can be employed in order to evaluate the attenuation in a waveguide. Since the EM field propagates along the waveguide axis z under an exponential law, the attenuation can be expressed in terms of the power dissipated per unit length Pd (this is thus a power density per unit length, to be measured in W/m) and the power travelling in the guide. The latter in fact can be expressed as PðzÞ ¼ P0 e2az

ð3:107Þ

where P0 is the power transmitted through the section in z ¼ 0 and a is the attenuation of the EM field. To ensure energy conservation, the rate of power decrease along an infinitesimal length dz must be equal to the power dissipated per unit length; from (3.107) we easily get Pd ¼

dP ¼ 2aP0 e2az ¼ 2aP dz

ð3:108Þ

60


The attenuation is therefore found as the ratio of the power density dissipated per unit length to the propagating power a¼

Pd 2P

ð3:109Þ

By simple extension of the above reasoning it can be seen that when more than one type of loss occurs, producing the power losses Pd1, Pd2. . . , the total attenuation will then be the sum of the individual attenuations. A rigorous treatment of conductor loss in a waveguide would be very difficult. This is because the finite conductivity of the metal implies different boundary conditions than those we have employed to derive the waveguide modes. Indeed, the tangential electric field is not exactly zero at the metal walls. Moreover, the metal’s conductivity is strongly affected by the status and roughness of its surface, and thus by the manufacturing process of the guide. For such reasons, the approach universally adopted to evaluate conductor loss in a transmission line is based on the following points: 1. The boundary condition of zero tangential electric field on the perfect electric conductor is replaced by the Leontovic condition (2.184) between the tangential components of the E and H fields on the metal surface ^ Et ¼ Zc Ht n ^ is the inner-directed normal unit vector at the metal surface and where n rffiffiffiffiffiffiffi om Zc ¼ ð1 þ jÞ 2s

ð3:110Þ

ð3:111Þ

is the intrinsic impedance of the conductor, s its conductivity and m its magnetic permeability. 2. The magnetic field at the surface of the metal is assumed to be the same as in the ideal case. Equation (3.110) is thus used to compute the tangential electric field at the metal surface. Note that this is the most significant difference with respect to the ideal case. Indeed, because of Ohm’s law, the conduction current flowing on the surface of a real conductor is associated with a non-zero tangential electric field parallel to the current. 3. Once the tangential E field at the metal surface is known, the power density Pd dissipated in the metal can be computed. This can be done using two alternative but fully equivalent approaches. The first consists of computing, via the Leontovic condition, the Poynting vector flow into the metal. The second approach consists of computing the power dissipated in the metal due to the Joule effect. Poynting’s theorem ensures the equality of both results. Following the former approach, the power dissipated in a waveguide section of unit length is expressed as the flow of the Poynting vector into the metal: rffiffiffiffiffiffiffi þ þ þ 1 1 1 om ^ dl ¼ Re Zc Ht H*t dl ¼ Pd ¼ Re Et H*t n Ht H*t dl ð3:112Þ 2 2 2 2s C C C ^ is the unit inward-directed normal to the metal wall and the index t indicates the where n ^. C is the linear contour of the crosscomponent tangential to the metal and thus orthogonal to n section. By virtue of the boundary condition (2.61) at the surface of a perfect conductor, the tangential magnetic field Ht is equal (in amplitude) to the surface current density Js. The previous expression can then be put in the form rffiffiffiffiffiffiffi þ 1 om Js J*s dl ð3:113Þ Pd ¼ 2 2s C


61

The same expression would have been obtained by computing the power lost in the metal by the Joule effect (2.69), using (3.110) and (3.111). For TM modes the H field has only one tangential component transverse to the axial direction and associated with the longitudinal current density. For TE modes there is also an H-field longitudinal component associated with a transverse current density. The axial and transverse currents give rise to different attenuation behaviours. The former decreases, while the latter increases with frequency. Dielectric loss can be computed much more easily than conductor loss. In fact, if we know the conductivity or the complex permittivity of the filling dielectric (in most cases air), then, recalling (2.69), the power dissipated is ð ð ð 1 s oe00 Pd ¼ E J* dS ¼ E E* dS ¼ E E* dS ð3:114Þ 2 S 2 S 2 S The power transported is

ð ð 1 1 * ^ P ¼ Re E H z dS ¼ ReY0 E E* dS 2 2 S S

ð3:115Þ

where Y0 ¼ 1=Z0 is the characteristic admittance of the mode, having implicitly supposed that only one mode is present. From (3.109), (3.114) and (3.115) we obtain ad ¼

1 oe00 2 ReðY0 Þ

ð3:116Þ

As an alternative, the attenuation due to dielectric loss can be computed by inserting the complex permittivity in expression (3.35) for the propagation constant: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kz ¼ o2 mðer js=oÞkt2 ¼ o2 mer kt2 je00 and computing the imaginary part. The result is exactly the same. Let us now discuss the maximum power that can be handled by a practical waveguide. As for the attenuation, the power limitation may be due to the dielectric (most often) or to the conductor. Any real dielectric is characterized by a maximum electric field value, called the dielectric rigidity or dielectric strength, that can be withstood, above which breakdown occurs. In the case of air at normal atmospheric pressure, the dielectric rigidity is 29 kV/cm. The maximum power that can be transmitted by the waveguide can be evaluated from the dielectric rigidity by expressing the power carried by the fundamental mode in terms of the maximum amplitude of the electric field and equating the latter to the dielectric rigidity. This can easily be done using the expressions for the electric field and for the power associated with the fundamental mode propagating in the waveguide. It might be worth specifying that any mismatch can only reduce the maximum power handling, not only because of the power reflected, but also because the reflected wave increases the maximum electric field amplitude. Another potential limitation on the maximum power is due to the excessive heating of the metal walls due to the current flowing in them. In this case, rather than the maximum instantaneous power, the limitation is put on the average power. For the applications considered in this book, power handling is of no concern.

3.8 The rectangular waveguide The rectangular waveguide is the most common type of waveguide, which is used in an ample frequency range from 320 MHz to 333 GHz, with decreasing inner dimensions from 1483.87 741.93 mm for guide WR-2300 down to 0.086 0.043 mm for guide WR-3. Appendix D reports the data on standard rectangular waveguides.

62


Figure 3.7 Geometry of the rectangular waveguide. The geometry of the rectangular waveguide is shown in Figure 3.7. As we have seen, waveguides cannot support TEM modes, but only TE and TM ones. Such modes can be derived as illustrated in Section 3.2 by first solving the eigenvalue equation (3.13) for the transverse potential T(x, y) with boundary conditions (3.21) or (3.28) for TM and TE modes, respectively. The solutions can be obtained without difficulty using the separation of variables method (see Section 2.9), which consists of assuming the unknown function as the product of two functions of a single variable x or y: Tðx; yÞ ¼ XðxÞYðyÞ

ð3:117Þ

mpx npy sin ; a b

m; n ¼ 1; 2 . . .

ð3:118Þ

m; n ¼ 0; 1; 2 . . .

ð3:119Þ

In this manner we obtain TTM ðx; yÞ ¼ Csin TTE ðx; yÞ ¼ Ccos

mpx npy cos ; a b

In both cases the eigenvalue is given by kt2 ¼

mp2 a

þ

np2 b

ð3:120Þ

Note that for TE modes the indexes m and n should not be simultaneously zero since this would lead to a trivial solution.

The proof of (3.118)–(3.119) is as follows. Inserting (3.117) into (3.13) we obtain two ordinary differential equations d 2X þ kx2 X ¼ 0 dx2 d 2Y þ ky2 Y ¼ 0 dy2

ð3:121Þ

where the constants kx and ky are linked by the separability condition kx2 þ ky2 ¼ kt2

ð3:122Þ


63

Equations (3.121) are the very well-known equations of harmonic motion, whose solutions can be put in various equivalent forms. In this instance it is convenient to use trigonometric functions: XðxÞ ¼ C1 sinðkx xÞ þ C2 cosðkx xÞ YðyÞ ¼ D1 sinðky yÞ þ D2 cosðky yÞ

ð3:123Þ

with C1, C2, D1, D2 arbitrary coefficients to be determined using the relevant boundary conditions. Imposing the boundary conditions (Table 3.2) for TM modes, X(x) must vanish for x ¼ 0 and x ¼ a. The former condition yields Xð0Þ ¼ C2 ¼ 0

ð3:124Þ

XðaÞ ¼ C1 sinðkx aÞ ¼ 0

ð3:125Þ

thus the latter implies

Since C1 ¼ C2 ¼ 0 would yield a zero EM field everywhere, we must have kxa ¼ mp, where m is a non-zero integer number. Therefore, in order for a non-trivial solution to exist, kx may only assume a discrete set of values given by kx ¼ mp=a;

m ¼ 1; 2 . . .

ð3:126Þ

Correspondingly, there are an infinite number of solutions for X(x): mpx a

ð3:127Þ

npy b

ð3:128Þ

m ¼ 1; 2 . . .

ð3:129Þ

XðxÞ ¼ C1 sin Using an identical procedure we obtain YðyÞ ¼ D1 sin ky ¼ np=b;

For TM modes we thus obtain (3.118), where C ¼ C1D1; using (3.122) the eigenvalue kt2 is found to be given by (3.120). For TE modes an identical procedure can be applied, except that the boundary conditions require the normal derivative rather than the function itself to vanish. For example, in x ¼ 0 we must have dX ð0Þ ¼ kx C1 ¼ 0 dx

ð3:130Þ

The procedure easily yields (3.119).

The cut-off frequency associated with the eigenvalue (3.120) is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 n2 1 fc ¼ pffiffiffiffiffi þ 2 me a b

ð3:131Þ

The expressions for the EM fields of the TM and TE modes are finally obtained by inserting (3.118) and (3.119) into (3.18) or (3.26), respectively, and are given in Table 3.4. The modes are labelled as TM or TE and with indexes m and n, which indicate the number of half swings of the EM fields in the x and y directions, respectively: we then speak of TMmn or TEmn modes. One can immediately realize that the fundamental mode of the rectangular waveguide (i.e. the mode with

64


Table 3.4 TM and TE modes in a rectangular waveguide. TM modes Tðx; yÞ

Cmn cos

mpx npy cos a b

Cmn sin

Et ¼ Zz;mn et ðx; yÞejkz;mn z Ht ¼ ht ðx; yÞejkz;mn z kz;mn Zz;mn ¼ oe Ez ðx; yÞ ¼

TE modes

kt2 Tmn ðx; yÞ joe

mpx npy sin a b

Et ¼ et ðx; yÞejkz;mn z ht ðx; yÞ jkz;mn z Ht ¼ e Zz;mn om Zz;mn ¼ kz;mn Hz ðx; yÞ ¼

kt2 Tmn ðx; yÞ jom

ex ðx; yÞ

Cmn

mp mpx npy cos sin a a b

Cmn

ey ðx; yÞ

Cmn

np mpx npy sin cos b a b

Cmn

Ez ðx; yÞ

Cmn

kt2 mpx npy sin sin a b joe

0

hx ðx; yÞ

Cmn

np mpx npy sin cos b a b

Cmn

mp mpx npy sin cos a a b

hy ðx; yÞ

Cmn

Cmn

np mpx npy cos sin b a b

Hz ðx; yÞ

0

mp mpx npy cos sin a a b

Cmn

np mpx npy cos sin b a b mp mpx npy sin cos a a b

kt 2 mpx npy cos cos a b jom

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 þ a b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 o2 mekt;mn

kt;mn kz;mn

rffiffiffiffiffiffiffiffiffiffi dm dn 1 ab kt;mn

Cmn

lowest cut-off frequency) is the TE10 mode. By putting m ¼ 1 and n ¼ 0 in the equations of Table 3.4 we obtain the expressions for the EM field of the TE10 mode: Ex ¼ 0 Ey ¼ j

px oma C sin ejkz;10 z p a

Ez ¼ 0 px kz;10 a Ey ejkz;10 z ¼ Hx ¼ j C sin Z10 p a Hy ¼ 0 px ejkz;10 z Hz ¼ C cos a

ð3:132Þ


65

with kt10 ¼ p=a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi kz10 ¼ o m0 e 1ð fc10 = f Þ2 1 c pffiffiffiffiffiffiffi ¼ pffiffiffiffi 2a m0 e 2a er om0 Z10 ¼ kz10 pffiffiffiffiffiffiffiffiffi c ¼ 1= m0 e0

fc10 ¼

ð3:133Þ

As can be seen, the EM field is independent of y (because n ¼ 0); the electric field has only the y component with a sinusoidal variation along the x axis, being zero, as it must, at the side walls in x ¼ 0 and x ¼ a. Identical behaviour occurs for the transverse magnetic field Hx. On the waveguide crosssection, there is only the Ey component of the electric field and the Hx component of the magnetic field. The plane containing the waveguide axis and the transverse E field is called the E plane, while that containing the waveguide axis and the transverse H field is called the H plane. With the reference system used in Figure 3.7, the E plane and H plane coincide with the yz and xz planes, respectively. The third non-zero component of the EM field, Hz, has a cosinusoidal behaviour with maxima at the side walls. The surface current density flowing on the metal walls can be obtained from the tangential magnetic field using the boundary condition (2.61), which we repeat here: ^H Js ¼ n

ð3:134Þ

The EM field and electric current lines of the TE10 mode are shown in Figure 3.8. In practical cases we are interested in evaluating the attenuation of the dominant mode. The procedure is that of Section 3.7. Longitudinal and transverse currents are found to produce two different contributions to the total attenuation due to conductor loss: ac ¼ acl þ act

ð3:135Þ

pffiffiffiffipffiffiffi 1:90 104 er f acl ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1ð fc = f Þ2

ð3:136Þ

where

is the attenuation due to the longitudinal currents, and 2 2b fc act ¼ acl f a

ð3:137Þ

is the attenuation due to the transverse currents. The above expressions have been obtained by assuming the wall to be made of copper (s ¼ 5:8 107 S=m). Note also that at cut-off both attenuations become exceedingly high, while atpfrequencies well above cut-off the longitudinal currents give rise to an ffiffiffi attenuation increasing with f , and the attenuation associated with the transverse currents tends to zero. The latter phenomenon, however, occurs at frequencies when also higher order modes propagate and is therefore hard to exploit to get attenuation-free EM wave propagation. The attenuation due to the filling dielectric is found to be ad ¼ 27:3

er Zc tan d dB=m f eo

ð3:138Þ

Figure 3.9 shows the total resulting attenuation along with its various contributing terms versus the frequency normalized to the cut-off frequency, in the useful band. Below cut-off the mode cannot propagate, while above twice the cut-off frequency higher order modes can propagate, which is an

66


Signal direction

λg/4

λg

Signal direction

H Js Figure 3.8 Electric and magnetic field lines and current density lines of the TE10 mode. Reprinted with permission from Ingegneria delle microonde e radiofrequenze by R. Sorrentino and G. Bianchi, Copyright (2006) McGraw-Hill.


67

0.50

Rectangular Waveguide WR-75 ( a=19.05 mm; b=a/2 )

0.45

Attenuation (dB/m)

0.40

b

0.35

a

0.30 0.25

α cl

0.20

α c = α ct + α cl

0.15 0.10

α ct

0.05 0.00 1

1.1

1.2

1.3

1.4

1.5 1.6 f/fc

1.7

1.8

1.9

2

Figure 3.9 Attenuation of the TE10 mode due to conductor loss. undesirable situation since it produces a signal distortion because of the different propagation behaviours of the various modes.

Useful bandwidth The frequency band of unimodality of the rectangular waveguide depends on the aspect ratio b/a, but in any case cannot be more than one octave.10 Equation (3.131) in fact shows that the cut-off of the TE20 mode is twice that of the TE10 mode: c fc20 ¼ pffiffiffiffi ¼ 2 fc10 a er If b > a=2, however, the cut-off of the TE01 mode is lower than that of the TE20 mode: c c fc01 ¼ pffiffiffiffi < pffiffiffiffi ¼ fc20 2b er a er In order not to reduce the useful bandwidth of the guide, the smaller size of the guide should not exceed a/2. On the other hand, since the greater the value of b, the smaller the attenuation, the optimum b is exactly a/2: smaller values would not increase the useful bandwidth but would increase conductor loss.

3.9 The ridge waveguide The ridge waveguide is derived from the rectangular waveguide by inserting one or two symmetrical rectangular ridges on one or both broader sizes, as sketched in Figure 3.10. The ridges have the effect of confining the EM field of the fundamental mode within the central gap. As a result, the cut-off of the fundamental mode of the rectangular waveguide is lowered while the cut-off of the TE20 mode, whose E field is zero at the centre of the broad walls, is almost unaffected. As a consequence, the ridge waveguide has a wider bandwidth than the corresponding rectangular guide. The price for this improvement is paid in terms of higher attenuation due to higher current density in the metal walls. The ridge guide is thus employed in those applications where a useful bandwidth greater than one octave is required. More details on this type of waveguide can be found in [4, 5].

10

Taken from music terminology, this term indicates a frequency interval from f0 to 2f0.

68


Figure 3.10

Single and double ridge waveguide.

3.10 The circular waveguide The circular waveguide, whose geometry is sketched in Figure 3.11, is employed in all applications where for mechanical reasons a circular geometry is necessary, as in rotary joints or for particular applications where the rotation symmetry can be usefully exploited, as in devices like the rotary phase shifter or the Faraday isolator. The analytical procedure for deriving the TM and TE modes of the circular waveguide is the same as for the rectangular waveguide, with the obvious difference that the coordinate system must be cylindrical with variables (r, f). As a consequence, the separation of variables must be performed by factorizing the potential T(r, f) as the product of a function of the radius r with a function of the azimuth angle f: Tðr; fÞ ¼ RðrÞ FðfÞ

ð3:139Þ

The transverse potential for the circular waveguide is found to be Tðr; jÞ ¼ CJn ðkt rÞcosðnj þ j0 Þ

ð3:140Þ

Tðr; fÞ ¼ Jn ðkt rÞ½C1 cosðnfÞ þ C2 sinðnfÞ

ð3:141Þ

or, equivalently,

with C1 ¼ C cos f0 and C2 ¼ C sin f0. Jn is the nth-order Bessel function of the first kind. As shown in Figure 3.12, such functions have a pseudo-periodical behaviour with decreasing maximum amplitudes.

Figure 3.11 The circular waveguide.


Bessel functions Jn(x)

1

0.5

69

← J (x) 0

← J (x) 1 ←J

2

(x) ←J

10

(x)

0

-0.5 0

5

10

15

20

25

30

x

Figure 3.12 Bessel functions of the first kind.

The proof of (3.140) is as follows. In polar coordinates the eigenvalue equation (3.13) is written as (see A.42): 1@ @T 1 @2T ð3:142Þ r þ 2 2 þ kt2 ¼ 0 r @r @r r @f Inserting (3.139) into (3.142) and separating the variables, we obtain two ordinary differential equations in the unknown functions R(r) and F(f): r2

d 2R dR þr þ ðkt2 r2 n2 ÞR ¼ 0 dr2 dr d2F þ n2 F ¼ 0 df2

ð3:143Þ ð3:144Þ

where n2 is the separation constant of the two equations. The solution of (3.144) is easily found and can be put in the form FðfÞ ¼ P cosðnf þ f0 Þ

ð3:145Þ

with P and f0 arbitrary constants. Equation (3.145) shows that n must be an integer number (including 0) in order for the function F(f) to be periodic with period 2p. The solution of (3.143) is RðrÞ ¼ C1 Jn ðkt rÞ þ C2 Yn ðkt rÞ ð3:146Þ where C1 and C2 are integration constants and Jn and Yn are Bessel functions of the first and second kind of order n, respectively. The behaviours of the Bessel functions Jn(x) and Yn(x) (n ¼ 0,1,2) are shown in Figures 3.12 and 3.13, respectively. The reader can observe that Bessel functions of the second kind diverge when x ! 0. As a consequence, because the EM field must remain finite, the corresponding integration constant must be zero, C2 ¼ 0, so that (3.146) reduces to ð3:147Þ RðrÞ ¼ C1 Jn ðkt rÞ Inserting (3.147) and (3.145) into (3.139), we obtain the final expression (3.140). Equations (3.140) and (3.141) show that when n ¼ 0 the field is independent of f. In all other cases, T(r, f) results from the superposition of two distinct solutions.

70


Bessel functions Yn(x)

1

0.5

← Y (x) 0

← Y (x) 1 ← Y (x) 2

←Y

10

(x)

0

-0.5

-1

0

5

10

15

20

25

30

x

Figure 3.13 Bessel functions of the second kind.

TM modes

According to the boundary condition (3.21) for TM fields, (3.141) must vanish on the guide boundary, and thus for r ¼ a. Therefore we must impose Jn ðkt aÞ ¼ 0

ð3:148Þ

As can be seen from Figure 3.13, such an equation has infinite solutions corresponding to the curve Jn crossing the abscissa. Let pnm be the mth zero of Jn(x). We then have the following expression for kt: kt ¼

pnm a

ð3:149Þ

The first zeros of Jn are given in Table 3.5. Equation (3.148) identifies the eigenvalues of the TM modes of the circular waveguide. They are labelled by a pair of indexes (n, m), m representing the number of half swings in the radial direction and n those in the azimuth direction. The EM field components of TMnm modes are given in Table 3.6. Except when n ¼ 0, each pair (n, m) corresponds to two distinct field distributions sharing the same eigenvalue kt2 . They are therefore degenerate modes (see Section 3.6).

TE modes

Instead of (3.148), for TE modes we need to impose (3.28), i.e. the vanishing of the normal derivative of the potential at the boundary. Since the normal direction coincides with the radial direction, we must have dJn ðkt rÞ ¼0 ð3:150Þ dr r¼a Let p0nm be the mth zero of (3.150), i.e. the mth maximum or minimum of the Bessel function Jn(x); the TEnm eigenvalues are obtained as p0 ð3:151Þ kt ¼ nm a Table 3.5 First zeros of Jn ðxÞ: n

pn1

pn2

pn3

0 1 2

2.405 3.832 5.135

5.520 7.016 8.417

8.654 10.174 11.620


71

Table 3.6 EM fields of TM and TE modes of the circular waveguide. Jn0 is the derivative of Jn : TM modes

TE modes

(

jnoe pnm r sin nj Jn rkt2 a cos nj ( jpnm oe 0 pnm r cos nj ¼ Jn akt2 a sin nj

Hr ¼

Hr ¼

Hj

Hj ¼

Hz ¼ 0 Er

Ej

Ez

( jkz pnm 0 pnm r cos nj ¼ Jn a akt2 sin nj ( jnkz pnm r sin nj ¼ Jn rkt2 a cos nj ( p r cos nj nm ¼ Jn a sin nj

Hz ¼ Er ¼ Ej ¼

( cos nj jkz p0nm 0 p0nm r J n akt2 a sin nj 0 ( sin nj jnkz p r Jn nm 2 a rkt cos nj 0 ( cos nj p r Jn nm a sin nj 0 ( sin nj jnom p r Jn nm rkt2 a cos nj 0 ( 0 cos nj jpnm om 0 pnm r J akt2 n a sin nj

Ez ¼ 0

The first p0nm values are given in Table 3.7. By comparison with Table 3.5 we note that p00m ¼ p1m . This is not by chance but is a consequence of the following property of Bessel functions:11 dJ0 ðxÞ ¼ J1 ðxÞ dx

ð3:152Þ

The field components of TEnm modes are given in Table 3.6. Also, in this case, whenever n 6¼ 0, modes are degenerate in pairs. Inspection of Tables 3.5 and 3.7 indicates that the fundamental mode is the TE11 mode corresponding to the eigenvalue kt11 ¼ 1:841=a. The field lines are sketched in Figure 3.14. The degenerate mode has its field lines rotated by 90 . The normalized frequency behaviour of the attenuation associated with the axial and transverse currents is shown in Figure 3.15, where, as for the case of the rectangular waveguide in (3.136)–(3.137), acl ¼

pffiffiffiffipffiffiffi 0:420 3:80 104 er f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a 1ð fc =f Þ2 Þ

ð3:153Þ

is the attenuation due to the longitudinal currents, and act ¼

2 1 fc acl 0:420 f

ð3:154Þ

Table 3.7 Zeros of the derivatives of Jn ðxÞ.

11

n

p0n1

p0n2

p0n3

0 1 2

3.832 1.841 3.054

7.016 5.331 6.706

10.174 8.536 9.970

More generally, xQ0n ðxÞ ¼ nQn ðxÞxQn þ 1 ðxÞ where Qn denotes any Bessel function of order n.

72


Figure 3.14

Electric (solid line) and magnetic (dashed line) field lines of the TE11 mode.

0.50 0.45

circular waveguide

Attenuation (dB/m)

0.40 0.35

20 mm

0.30 0.25

α ct

0.20

α c = α ct + α cl

0.15

α cl

0.10 0.05 0.00 1

1.1

1.2

1.3

1.4

1.5 f/fc

1.6

1.7

1.8

1.9

2

Figure 3.15 Attenuation due to conductor loss for the TE11 mode.

is the attenuation due to the transverse currents, while the total attenuation due to conductor losses is the sum of (3.153) and (3.154) as given in (3.135). The above expressions have been obtained by assuming the wall to be made of copper (s ¼ 5:8 107 S=m).

3.11 The coaxial cable The coaxial cable consists of two concentric cylindrical conductors separated by a dielectric material where the EM field propagates (Figure 3.16). Due to the presence of two distinct conductors and a homogeneous dielectric, the fundamental mode is a TEM mode, which has a zero cut-off frequency.


Figure 3.16

73

Geometry of the coaxial cable.

Higher TM and TE modes do exist and start propagating at frequencies approximately above fc ¼

1 1 pffiffiffiffiffi ffi pffiffiffiffiffi lc me pða þ bÞ me

ð3:155Þ

Up to this frequency, the cable is unimodal. As in the case of the circular waveguide, the coordinate system is the cylindrical one. As we have seen in Section 3.2, the EM field of the fundamental TEM mode is obtained by solving Laplace’s equation for the potential T(r, f) along with the boundary condition that T must be constant on the contour. The cylindrical symmetry allows us to search for solutions independent of f. In such a case, if @=@f ¼ 0 Laplace’s equation simplifies to 1d dT r ¼0 ð3:156Þ r dr dr After two integrations the general solution is obtained as Tðr; fÞ ¼ A ln r þ B

ð3:157Þ

with A and B integration constants. Assuming T ¼ 0 on the outer conductor (r ¼ b) and T ¼ T0 on the inner one (r ¼ a), we get lnðr=bÞ Tðr; fÞ ¼ T0 ð3:158Þ lnða=bÞ T0 is an arbitrary constant that determines the amplitude of the EM field. The latter is obtained by inserting (3.158) into expressions (3.32) for the TEM mode and recalling that in the present case rt T ¼ ð@T=@rÞ^r. For the progressive (or incident) wave we obtain Et ¼

k jkz z @T T0 Z 1 jkz z ^r ¼ ^r e e oe @r lnðb=aÞ r

Ht ¼ ejkz z

@T T0 1 jkz z ^ ^¼ u u e @r lnðb=aÞ r

ð3:159Þ

ð3:160Þ

pffiffiffiffiffiffiffi where Z ¼ m=e is the wave impedance. Note that the electric and magnetic fields are orthogonal at any point and form a right triplet with the direction of propagation. The E field lines are purely radial, while the H lines are purely circumferential (Figure 3.17).

74


Figure 3.17

Field lines of the TEM mode of the coaxial cable.

The current density flows in the axial direction with opposite signs on the outer and inner conductors: ^ Hjr¼a ¼ ^r Hjr¼a ¼ Js;int ¼ n

T0 ejkz z ^z lnðb=aÞ a

^ Hjr¼b ¼ ð^rÞ Hjr¼a ¼ Js;ext ¼ n

T0 ejkz z ^z lnðb=aÞ b

ð3:161Þ ð3:162Þ

The total current is obtained by integrating the current density along the circumference of the inner conductor þ ð 2p T0 ejkz z 2pT0 jkz z a dj ¼ e IðzÞ ¼ Js;int ^z dl ¼ ð3:163Þ lnðb=aÞ C 0 lnðb=aÞ a The same result is obtained by performing the integral along any line (in particular a circumference) encompassing the inner conductor. As already observed in Section 3.3, in the case of the TEM mode the electric field is conservative in the cross-sectional plane; we may therefore uniquely define a potential difference between the two conductors. It therefore appears natural to define the voltage along the cable in terms of the line integral of the electric field: ðb ð ða T0 Z jkz z b 1 ð3:164Þ e dr ¼ T0 Zejkz z VðzÞ ¼ E dl ¼ Er dr ¼ lnðb=aÞ b a a r Equations (3.163) and (3.164) define the characteristic impedance of the coaxial cable as Z Z0 ¼ lnðb=aÞ 2p

ð3:165Þ

Note that (3.165) differs from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the wave impedance (3.46) for TEM modes. The present choice corresponds to putting C ¼ lnðb=aÞ=2p in (3.60)–(3.64). The attenuation due to conductor loss can be computed using the method described in Section 3.7. In the case of the TEM mode, only the axial current is present. Since both the attenuation and the characteristic impedance of the coaxial cable depend on lnðb=aÞ, it is possible to express the former as a function of the latter, as shown in Figure 3.18. Note that the minimum attenuation is achieved for characteristic impedance close to 75 O. Data on standard coaxial cables are reported in Appendix D.

3.12 The parallel-plate waveguide Rather than an actual waveguide, the parallel-plate waveguide (PPW) is an EM model of the propagation in practical TEM lines; it can be used for example as a simplified model of the microstrip line. The PPW


75

–3

x 10

1.4

Coaxial cable

εr

1.3

2b

1.2 1.1 1

Attenuation

.

2b (dB/ GHz ) ε r fGHz

1.5

0.9 0.8 0.7 0.6

20

40

60

80

100

120

140

160

180

εr Z0

Figure 3.18

Conductor loss attenuation of the coaxial cable vs. characteristic impedance.

consists of two infinite metal plates, supposedly made of a perfect conductor, set at a distance h from one another, as sketched in Figure 3.19. The two plates are supposed to be of infinite extent in both z and x directions. The z axis is assumed, as usual, to be the direction of propagation. As long as the filling dielectric is homogeneous, as we assume here, the presence of two different conductors assures us that the fundamental mode is the TEM mode. To determine the EM field we can follow the procedure described in Section 3.2 and solve Laplace’s equation (3.31) for the transverse potential T(x, y) with the boundary condition (3.34) that T is constant on the metal boundaries, i.e. for y ¼ 0 and y ¼ h. The EM field can then be obtained from T(x, y) using Table 3.2. In implementing this procedure it is convenient to look for solutions that are independent of the x coordinate. The 2-D Laplace’s equation thus reduces to an ordinary differential equation in T(y) that can immediately be integrated, yielding TðyÞ ¼ T0 y=h

ð3:166Þ

where T0 is an arbitrary constant representing the potential for y ¼ h, while T has been assumed to be zero for y ¼ 0. Using the formulae in Table 3.2, the corresponding EM field is easily found to be

Figure 3.19

The parallel-plate waveguide.

76

MICROWAVE AND RF ENGINEERING Et ¼ Z Ht ¼

T0 jbz ^ye h

T0 jbz ^xe h

ð3:167Þ

pffiffiffiffiffiffiffi pffiffiffiffiffi with b ¼ o me and Z ¼ m=e. We immediately recognize that the above field is nothing but a uniform plane wave propagating in the z direction. By putting E0 ¼ ZT0 =h we can indeed write Et ¼ E0 ^yejbz E0 jbz ^xe Ht ¼ Z

ð3:168Þ

The PPW, as mentioned above, is representative of two-conductor TEM lines of finite lateral extent. Indeed, we may take a slice of width w by inserting two side walls made of a perfect magnetic conductor (pmc). Since the magnetic field is directed along the x axis, such walls do not affect the EM field because they do not modify the boundary conditions.12 The PPW therefore becomes a closed waveguide with upper and lower walls made of pec and side walls made of pmc. We are now interested in evaluating the characteristic impedance of such a transmission line, in much the same way as we did for the coaxial cable in Section 3.11. Let us compute the voltage and current along the PPW. The voltage is obtained as the line integral of the E field between the two conductors: ðh ð3:169Þ V ¼ E0 ejbz dy ¼ hE0 ejbz 0

The current flowing on the plates can be obtained by integrating the axial current density over the width w. Because of the boundary condition on a metal surface (2.61), the z-directed current density is equal to the x-directed magnetic field. The total current is thus given by ðw E0 Hx ejbz dx ¼ w ejbz ð3:170Þ I¼ Z 0 By taking the ratio of (3.169) to (3.170), we finally get the characteristic impedance of the line: rffiffiffi mh Z0 ¼ ð3:171Þ ew This equation shows that the characteristic impedance of the PPW with magnetic side walls is equal to the wave impedance times the aspect ratio h/w.

3.13 The stripline The stripline belongs to the category of planar circuits that can be fabricated with printed circuit technology. As shown in Figure 3.20, it consists of a metal strip embedded in a dielectric substrate symmetrically sandwiched between two metal ground planes. The stripline can be considered as a printed version of the coaxial cable. As for the coaxial cable, because of the presence of the two distinct conductors13 and a homogeneous dielectric, the fundamental pffiffiffiffiffi mode is a TEM mode, with zero cut-off frequency and propagation constant kz ¼ o me. The EM analysis is thus based on the solution of Laplace’s equation for the transverse potential T(x, y) with boundary conditions T ¼ 0 on the ground planes and T ¼ T0 on the metal strip. Assuming to 12

It is worth specifying that this reasoning is valid only for the TEM mode propagating in the z direction. The ground planes are to be considered as the same conductor since in practice they are electrically connected at some distance from the central strip. From a mathematical viewpoint they are connected at infinity. 13


77

Figure 3.20 Geometry of the stripline (a) and electric field lines of the TEM mode (b). a first approximation that the metal strip has zero thickness, Laplace’s equation can be solved by the method of conformal mapping. This is a mathematical method for solving Laplace’s equation in two dimensions, based on the properties of complex analytical functions. In practice, the conformal mapping transforms the original problem into another simpler configuration for which the solution to Laplace’s equation is known. The reader is directed to [3] for details of the method. Figure 3.20b shows the electric field lines of the TEM mode, while the following expression for the characteristic impedance of the stripline with an infinitely thin strip is obtained [3]: 8 pffiffiffi! > 1 1 þ k > > pffiffiffi ln 2 0:707 k < 1 > > 1 4 er Kðk0 Þ 4 er > 0 1 þ k > > pffiffiffiffi 0 < k 0:707 > : p ln 2 1 k0 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi where k0 ¼ tanhðpw=2bÞ, k ¼ 1k0 2 , Z0 ¼ m0 =e0 ¼ 377 O. KðkÞ is a complete elliptic integral of the first kind; the ratio KðkÞ=Kðk0 Þ can be expressed to an excellent approximation as indicated in (3.172). In the case of finite thickness t of the strip, some corrections to the above formulae can be implemented. Figure 3.21 shows the characteristic impedance of the stripline for different values of the

200 180 160 140

εZ

r 0

120 100

t /b

=0

0.05 0.10 0.20 0.15 0.25

80 60 40 20 0 0.1

1 w/b

Figure 3.21 Characteristic impedance of the stripline.

4

78


normalized metal thickness t/b, based on Cohn’s formulae [6]. Note that Zc decreases with w/b and with t/ b as a consequence of the increasing capacitance per unit length.

3.14 The microstrip line The microstrip line, or simply microstrip, is the most common transmission line employed in microwave and RF engineering. This is because of a number of advantages such as its simplicity of manufacturing, low cost, low weight and excellent integrability with active devices. As shown in Figure 3.22, it is a planar transmission line like the stripline, but, in contrast, it is an open structure; it is not fully embedded in a dielectric but printed on top of a dielectric substrate metallized on the opposite side. The EM field guided by the metal strip propagates in a inhomogeneous medium consisting of the dielectric substrate and the air on top of it. In spite of the presence of two separate conductors so that Laplace’s equation has a non-trivial solution, the dielectric inhomogeneity prevents a TEM from propagating. Indeed, for a TEM mode we pffiffiffiffiffi know that kt2 ¼ 0, and the propagation constant is given by kz ¼ o me according to (3.42). In the presence of two different dielectric materials we would have two different propagation constants, one for the substrate and the other for air: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi kz0 ¼ o m0 e0 ; kzd ¼ o m0 e0 er ð3:173Þ This result is in contrast with the continuity condition of the fields at the separation between the two media. Note, nevertheless, that the two propagation constants (3.173) tend to coincide in the limit of zero frequency. Only in the static case is it indeed possible to get a TEM solution. Under such a condition, however, the electric and magnetic fields are independent solutions of the static Maxwell’s equations (@=@t ¼ 0) representing the electrostatic and magnetostatic conditions, respectively. The former is associated with a static potential between the strip and the ground plane, while the latter is associated with a direct current flowing along the line. When a harmonic regime is established at a finite frequency, the electric and magnetic fields are linked by the full Maxwell’s equations. However, as long as the frequency is small enough so that the wavelength is much greater than the cross-section of the microstrip, the EM field distributions in the cross-section can be approximated by the static ones. We are thus led to the characterization of the microstrip in terms of an approximate model based on the static solution, called the quasi-static model. In practice, the quasi-static approach consists of modelling the microstrip as a TEM line with a perunit-length capacitance C and a per-unit-length inductance L, evaluated by solving Laplace’s equation. Such quantities allow us to compute the characteristic impedance and phase constant of the transmission line using (3.82) and (3.83). The fundamental mode is thus approximated by a TEM mode and for this reason is named the quasi-TEM mode. Figure 3.23 shows a typical electric field line distribution based

Figure 3.22

The microstrip line.


Figure 3.23

79

Electric field lines of the quasi-TEM mode (static approximation).

on the static approximation. As we will see better in a moment, this approximation cannot account for the frequency dispersion of the line parameters, so some proper modifications need to be implemented. Before going any further, let us for the moment show that the static model of the microstrip, instead of using C and L, can be expressed in terms of the capacitances C and C0, the latter being the per-unit-length capacitance when the dielectric is hypothetically removed or, equivalently, when the substrate has a unit relative permittivity er ¼ 1. Under static conditions the change in the substrate permittivity affects only the electric field distribution, thus the capacitance, but not the magnetic field nor the inductance L. The last can therefore be expressed in terms of the phase velocity and capacitance of the ‘empty’ microstrip L¼

1 v20 C0

ð3:174Þ

so that the phase velocity of the ‘full’ microstrip can be written as 1 v0 vp ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi LC C=C0 where v0 ¼ ðm0 e0 Þ1=2 . The phase constant is therefore given by pffiffiffiffiffiffiffiffiffiffiffi o o pffiffiffiffiffiffiffiffiffiffiffi C=C0 ¼ k0 C=C0 kz ¼ ¼ vp v0 Concerning the characteristic impedance, we may write rffiffiffiffi rffiffiffiffiffiffiffiffiffi L LC0 1 Z0 ¼ ¼ pffiffiffiffiffiffiffiffiffi ¼ CC0 v0 CC0 C

ð3:175Þ

ð3:176Þ

ð3:177Þ

Equations (3.176) and (3.177) show that both the characteristic impedance and the phase constant can be expressed in terms of C and C0. The capacitance ratio ee f f ;stat ¼

C C0

ð3:178Þ

is the static microstrip effective permittivity. Besides being the ratio of the microstrip capacitances with and without the substrate material, this quantity allows us to express the phase constant (3.176) as pffiffiffiffiffiffiffiffiffiffiffiffiffi kz ¼ k0 ee f f ;stat ð3:179Þ

80


Figure 3.24

Derivation of the quasi-static model of the microstrip.

Similarly, the characteristic impedance can be expressed in terms of the characteristic impedance of the empty microstrip Z00 and the effective permittivity. In fact rffiffiffiffiffiffirffiffiffiffiffiffi L C0 Z00 ð3:180Þ Z0 ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ee f f ;stat C0 C With reference to Figure 3.24, the effective permittivity can be interpreted as the permittivity of a hypothetical homogeneous material filling the whole space in such a way as to produce the same per-unitlength capacitance C and phase constant kz as the original microstrip line. Needless to say, the homogeneous transmission line of Figure 3.24c supports a true TEM mode, which is not the case in Figure 3.24a. From a computational point of view, C and C0 can be computed by solving Laplace’s equation with various techniques, in particular, as for the stripline, using the conformal mapping technique, originally used by Wheeler [8]. For practical applications it is more convenient to use simple formulae; though approximate, they easily provide the capacitance values required for evaluating the microstrip parameters. Among the most popular formulae, those due to Hammerstad and Jensen [9] are quite accurate and simple enough, and thus worth reporting here:14 u ¼ w=h "

1 u4 þ ðu=52Þ2 A ¼ 1 þ ln 49 u4 þ 0:432

!

# #" u 3 1 er 0:9 0:053 ln 1 þ þ 0:564 18:7 18:1 er þ 3

ee f f ;stat ¼

er þ 1 er 1 10 A þ 1þ 2 2 u

2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 0:7528 60 6 2p6 2 5 Z0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ln4 þ þ 1þ eð30:666=uÞ ee f f ;stat u u u

ð3:181Þ

ð3:182Þ

ð3:183Þ

The effective permittivity and the characteristic impedance are plotted in Figure 3.25 for various substrate permittivities er and aspect ratios u. Observe that the effective permittivity is a sort of average between the substrate and air permittivities, and it is closer to one or the other depending on whether the field is more or less confined to the dielectric substrate. Expression (3.182) shows that, depending on the aspect ratio u ¼ w=h, the effective permittivity varies between the limits er þ 1 ee f f er 2

ð3:184Þ

14 The interested reader can open the Mathcad file 05_Microstrip_Analysis_Synthesis.MCD (associated with Chapter 12) which enables the analysis and synthesis of arbitrary isolated microstrips, based on Hammerstad–Jensen formulae.


Figure 3.25

81

Characteristic impedance and effective permittivity of the microstrip.

which correspond to the limit cases of extremely narrow (w=h ! 0) or extremely wide (w=h ! 1) strips, respectively. The effective permittivity defined so far as a capacitance ratio (3.178) corresponds to a static, thus frequency-independent, microstrip model where the dominant quasi-TEM mode is approximated with a TEM mode. Figure 3.26 shows a typical dispersion diagram of the microstrip, i.e. the normalized phase constant kz =k0 versus the frequency. While at lower frequencies kz =k0 is almost constant, as the pffiffiffiffi frequency increases it progressively departs from (3.177) and increases asymptotically towards er . If the dominant mode were a TEM, its normalized phase constant would have been frequency independent. On the contrary, the microstrip is dispersive as the phase velocity varies with the frequency. From a physical point of view this causes the EM field to be progressively more confined, as the frequency increases, within the substrate. A frequency fstat can be introduced that represents the boundary between the quasi-static and the dynamic regimes. Above fstat we need to resort to a more accurate model to

εr

3.0 2.9

w/h = 5

2.8

w/h = 2

kz /k0

2.7

w/h = 1

2.6

w/h = 0.5

2.5 2.4 2.3 0

10

20

30

Frequency, GHz

Figure 3.26

Microstrip dispersion diagram for er ¼ 9.

40

82


characterize the propagation and the field distribution of the fundamental mode. The very definition (3.178) of effective permittivity loses its meaning. A more general definition of the effective permittivity stems from (3.179), where it appears that ee f f ¼ ðkz =k0 Þ2

ð3:185Þ

This can be seen as a dynamic effective permittivity since it implies that the effective permittivity must be calculated using a dynamic rather than a static model, the latter being a particular case of the former in the limit of low frequencies. Various analytical and numerical methods have been developed in past years for the rigorous modelling of planar circuits such as microstrip circuits. A partial and very limited account is given in Chapter 16. The interested reader is referred to the vast literature on this subject, see for instance [10–12]. Here it suffices to say that rigorous analyses can be performed to compute the EM field distribution in a microstrip line and the corresponding values of the effective permittivity (thus the phase constant) and characteristic impedance. Such analyses involve a full solution of Maxwell’s equations without neglecting the axial field components Ez, Hz that are actually non-zero though smaller that the others. The full wave solution constitute a hybrid mode, i.e. a mode that is neither TM nor TE but a combination of both. For practical purposes, however, what interests us most is the knowledge of kz =k0 and Zc rather than a complete description of the EM field. From this perspective, we may just use the same model as in the quasi-static case, but introduce a frequency dependence in the expression for the effective permittivity so as to account for the dispersion. Among the dispersion formulae proposed in the literature we report here those by Yamashita et al. [13]: pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi er ee f f ;stat pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð f Þ ¼ þ e ee f f e f f ;stat 1 þ 4F11:5 F1 ¼

h 4h f pffiffiffiffiffiffiffiffiffiffi w i2 er 1 0:5 þ 1 þ 2log 1 þ c0 h

ð3:186Þ

where ee f f ;stat is the static value given by (3.178). As far as the characteristic impedance is concerned, the dispersion is less pronounced, so it can usually be neglected.

3.14.1

The planar waveguide model

A simple and popular approach to the approximate modelling of microstrip lines and circuits is the planar waveguide model. Accordingly, the microstrip line is modelled as a parallel-plate waveguide with magnetic side walls (see Section 3.12), filled with a homogeneous dielectric of relative permittivity ee f f , and having the same height h and the same characteristic impedance Z0 as the microstrip line. From Equation (3.171) it follows that the last condition is fulfilled by assuming for the parallel-plate guide an effective width rffiffiffiffiffiffiffiffiffiffiffi h m0 h 120p we f f ¼ ffi pffiffiffiffiffiffiffi Z0 e0 ee f f Z0 ee f f In practice, we f f > w, i.e. the effective width is larger than the microstrip line as it accounts for the presence of fringing fields at the strip edges. For example, for h ¼ 100 mm, er ¼ 12:5 and Z0 ¼ 50 O, the inversion of formulae (3.181)–(3.183) gives w ¼ 76 mm and ee f f ¼ 8:102, then we f f ¼ 265 mm.

3.15 The coplanar waveguide The coplanar waveguide or CPW is the second most common printed circuit configuration after the microstrip line. As its name suggests, both strip and ground are made on the same side of the dielectric


Figure 3.27

83

The coplanar waveguide.

substrate. As sketched in Figure 3.27, the ground consists of two half planes symmetrically located on both sides of the metal strip. This geometry offers the great advantage of allowing the shunt connection of solid state devices without requiring perforation of the substrate, as required by the microstrip. The CPW can be seen as the planar version of the coaxial cable, but unlike the latter it is inhomogeneous so that the dominant mode is, as for the microstrip, a quasi-TEM mode. The characteristic impedance depends on both the width w of the strip and the gap g: essentially, it depends on their ratio, similar to the coaxial cable. With respect to the microstrip, this confers on the CPW an additional degree of freedom in the realization of a given characteristic impedance. The main drawback of the CPW is due to the existence of two different ground planes that must be electrically connected to each other. Otherwise, another spurious quasi-TEM mode would appear. This can be easily seen by recognizing that, referring for simplicity to the static case, two independent TEM modes can be supported by three distinct conductors. To suppress the spurious mode, the ground planes must be maintained at the same potential. As sketched in Figure 3.28, metal bridges located at various critical points of the CPW, such as bends, junctions and discontinuities, are fabricated to connect the ground planes electrically.

Figure 3.28

Metal bridges for spurious mode suppression in CPW.

84


Figure 3.29

Static EM field lines in the coplanar waveguide.

Figure 3.29 shows the typical electric field lines of the dominant mode of the CPW. As for the microstrip, a quasi-static approach based on the solution of Laplace’s equation can be adopted. If the substrate thickness is large enough (g=h 1) and the metal thickness t can be assumed to be zero, the solution can be found with relative ease by conformal mapping as originally proposed by Wen [14]. More accurate formulae were then developed by Ghione and Naldi [15] and are reported here. With the notation of Figure 3.27, the characteristic impedance Z0 and the effective permittivity ee of the CPW are expressed as ee ¼ 1 þ

er 1 K 0 ðkÞ Kðk1 Þ 2 KðkÞ K 0 ðk1 Þ

ð3:187Þ

Z00 Z K 0 ðkÞ Z0 ¼ pffiffiffiffi ¼ p0ffiffiffiffi ee p ee KðkÞ where

8 pffiffiffi! > 1 1þ k > > > > ln 2 1pffiffiffi k Z0 KðkÞ Z0 4 er Kðk Þ 4 er > 1 1 þ k0 > > > pffiffiffiffi : p ln 2 1 k0

0:707 k < 1 ð3:188Þ 0 < k 0:707

KðkÞ is a complete elliptic integral of the first kind. Fortunately, the ratio K 0 ðkÞ=KðkÞ can be very well approximated, as indicated by the last equality in (3.188).

3.16 Coupled lines So far we have considered various microwave guiding structures and studied their modes of propagation, each line being considered as isolated. In many practical cases, on the contrary, two or more transmission lines are located, intentionally or not, close to each other so that the field guided by each of them interacts with the field of the other. Actually the field is simultaneously guided by the whole set of coupled lines, so that one must refer to the modes of the entire set rather than to the modes of each transmission line. The coupling between different lines is used in the realization of directional couplers and coupledline filters. The characterization of coupled lines, however, is required not only for the design of such devices, but also to model cross-coupling and interference phenomena occurring within the same circuit. In the following discussion we will limit ourselves to the simplest case, though the most common one, of a symmetrical pair of identical lines. The symmetry indeed allows a substantial simplification of the


85

Figure 3.30 Cross-sections of some coupled lines: (a) stripline; (b) microstrip line; (c) coaxial cable; (d) square coaxial cable. analysis. The generalization to an arbitrary number of non-symmetric coupled lines is beyond the scope of this discussion. A thorough discussion can be found in [16, 17].

3.16.1

Basic principles for EM analysis

We consider here coupled lines made of three conductors with a longitudinal symmetry plane: two identical conductors with uniform cross-section along the z axis plus a third one acting as the common ground. The three conductors are separated by one or more dielectrics. Figure 3.30 shows the crosssections of four types of coupled lines: (a) the stripline; (b) the microstrip line; (c) the coaxial cable; (d) the rectangular coaxial cable. Without going into the EM analysis of coupled lines, we will just make some general considerations from the EM point of view in order to get an insight into their behaviour. In cases (a), (c) and (d) the dielectric involved by the EM field propagation is homogeneous. The structures can therefore support the propagation of TEM modes. Let us recall that, as we saw in Section 3.2, a TEM mode is obtained by solving Laplace’s equation (3.31) for the potential T(x, y) along with the boundary condition (3.34) of constant T on the conductor boundaries. We can immediately observe that, because of the presence of three conductors, there are two distinct TEM modes. We can in fact arbitrarily choose two values of T on the first two conductors, while the third one is taken as the reference (T ¼ 0). Given the symmetry of the structure, the two TEM modes are determined by ascribing equal (T1 ¼ T2 ¼ 1) or opposite potentials (T1 ¼ T2 ¼ 1) to the two conductors.15 The former case gives rise to the even mode: it corresponds to putting a magnetic wall on the symmetry plane (thus the normal electric field is zero); the latter gives rise to the odd mode: it corresponds to putting an electric wall on the symmetry plane (thus the tangential electric field is zero). As an example, Figure 3.31 shows the electric and magnetic field flux lines of the even and odd modes of a pair of coupled striplines. In the case (b) relevant to the coupled microstrip lines, the filling dielectric is inhomogeneous. As we saw in Section 3.14 for the single microstrip line, true TEM modes cannot exist, but two quasi-TEM modes can be supported, one even and one odd. The basic difference with the former cases where the 15 It should be observed that any linear combination of the two modes is still a solution of the boundary value problem, so it can be considered as another mode of the structure. There are, however, two and only two independent solutions.

86


Figure 3.31

Odd and even mode field lines in a pair of coupled striplines.

dielectric is homogeneous is that the even and odd modes have different effective permittivities and thus exhibit different phase velocities. In all cases, in spite of the limitations involved in the case of the microstrip, we will refer to a static analysis based on the solution of Laplace’s equation. Such an equation allows us to compute the per-unitlength capacitances and inductances of the coupled lines. Each conductor, in fact, exhibits a capacitance C towards the ground and a mutual capacitance Cm towards the other conductor. Similarly, there will be a mutual inductance M between the two conductors, in addition to the self-inductance L. Once such parameters have been computed by solving Laplace’s equation, a simple circuit model can be employed to characterize the coupled lines.

3.16.2

Equivalent circuit modelling

Figure 3.32 shows the equivalent circuit of an infinitesimal length of lossless symmetrically coupled lines. The circuit is the generalization of the elementary equivalent circuit of Figure 3.4 for the isolated

Figure 3.32

Equivalent network of an infinitesimal length of coupled lines.


87

transmission line, by adding the mutual capacitance CM and inductance M. For reasons to become clear soon, CM has been replaced by the series of two capacitances 2CM. By applying Kirchhoff’s laws to the nodes and loops of the circuit one obtains four rather than two of the telegrapher’s equations. Rather than using this procedure, which is somewhat lengthy, we can greatly simplify the analytical treatment by exploiting the structure’s symmetry. Any solution, in fact, can be obtained by superimposing an even and an odd solution, which are equivalent to putting on the symmetry plane a magnetic wall (or open circuit) and an electric wall (short circuit). In this manner, the analysis is reduced to that of two isolated transmission lines, each having its characteristic impedance and phase constant. Let us therefore separately compute the even and odd mode characteristics.

Even mode In the even mode, equal voltages and currents are present on both lines. The symmetry plane is an open circuit that prevents the current flow. In such conditions, the mutual capacitance does not play any role, while the mutual inductance produces an equal voltage drop M I2 ¼ M I1 on both lines. The even mode thus behaves as a transmission line with a capacitance per unit length equal to C and an inductance per unit length equal to L þ M. Its characteristic impedance and phase velocity are thus given by rffiffiffiffiffiffiffiffiffiffiffiffiffi LþM 1 Z0e ¼ ; ve ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:189Þ C ðL þ MÞC Odd mode

The symmetry plane in this case is a grounded (V ¼ 0) electric wall. Voltages and currents on the two lines have the same amplitudes but opposite signs. Concerning the mutual capacitance CM, we may replace it with the series of two identical capacitors 2CM: the symmetry plane will thus connect the node between the two capacitors to ground. The mutual inductance, on the contrary, because of the opposite directions of the two currents, produces opposite voltage drops M I2 ¼ M I1 on the two lines. One infers that the odd mode is equivalent to a transmission line with per-unit-length capacitance C þ 2CM and inductance L M. The characteristic impedance and phase velocity of the odd mode are therefore given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LM 1 ; vo ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:190Þ Z0o ¼ C þ 2CM ðLMÞðC þ 2CM Þ The four parameters C, CM, L and M are actually not independent of each other. This is found by using the EM analysis. Consider the particular case when the dielectric is homogeneous (excluding the microstrip). We know that both even and odd modes are TEM modes for which the phase constant is given by (3.33) pffiffiffiffiffi (kz ¼ o me) and depends only on the dielectric permittivity of the filling dielectric. Both modes thus have equal phase velocities: c v ¼ ve ¼ vo ¼ pffiffiffiffi er

ð3:191Þ

As a consequence ðL þ MÞC ¼ ðLMÞðC þ 2CM Þ ¼

er c2

ð3:192Þ

where c 3108 m/s is the phase velocity in vacuum and er is the dielectric permittivity. Equation (3.192) establishes two links among the four parameters C, CM, L, M of the coupled lines. In the case of a single line, this equation corresponds to the condition where the phase velocity of the transmission line must be equal to that of the TEM mode: pffiffiffiffiffiffi pffiffiffiffi v ¼ c= er ¼ 1= LC ) LC ¼ er =c2

88


In the general case, voltages and currents propagating along the coupled lines result from the linear superposition of the corresponding even and odd mode quantities. In the formulae, the voltages and currents on the two lines are given by V1 ðzÞ ¼ Ve ðzÞ þ Vo ðzÞ ¼ Veþ ejbe z þ Ve e þ jbe z þ Voþ ejbo z þ Vo e þ jbo z 1 1 I1 ðzÞ ¼ Ie ðzÞ þ Io ðzÞ ¼ ðV þ ejbe z Ve e þ jbe z Þ þ ðV þ ejbo z Vo e þ jbo z Þ Z0e e Z0o o V2 ðzÞ ¼ Ve ðzÞVo ðzÞ ¼ Veþ ejbe z þ Ve e þ jbe z Voþ ejbo z Vo e þ jbo z 1 1 I2 ðzÞ ¼ Ie ðzÞIo ðzÞ ¼ ðV þ ejbe z Ve e þ jbe z Þ ðV þ ejbo z Vo e þ jbo z Þ Z0e e Z0o o

ð3:193Þ

where expressions (3.54) for the voltages and currents along a transmission line have been used, g being replaced by jb, and with the subscripts ‘e’ and ‘o’ used to designate the even and odd modes respectively. In the case of a TEM line for which be ¼ bo ¼ b, the above formulae simplify to V1 ðzÞ ¼ Ve ðzÞ þ Vo ðzÞ ¼ ðVeþ þ Voþ Þejbz þ ðVe þ Vo Þe þ jbz þ Ve Vþ V V I1 ðzÞ ¼ Ie ðzÞ þ Io ðzÞ ¼ þ o ejbz e þ o e þ jbz Z0e Z0o Z0e Z0o V2 ðzÞ ¼ Ve ðzÞVo ðzÞ ¼ ðVeþ Voþ Þejbz ðVe þ Vo Þe þ jbz þ Ve Voþ jbz Ve Vo þ jbz e e I2 ðzÞ ¼ Ie ðzÞIo ðzÞ ¼ Z0e Z0o Z0e Z0o

ð3:194Þ

Notice that in the case of coupled lines, we cannot define a unique characteristic impedance as the ratio of progressive voltage to current wave amplitudes. Such a ratio depends on the respective amplitudes of the even and odd modes: it equals the even (or odd) characteristic impedance only when only one mode is present, either Voþ ¼ 0 or Veþ ¼ 0. When the lines are located far enough one from each other so that the coupling becomes negligible, then M ¼ 0 and CM ¼ 0 and both even and odd impedances (3.189) and (3.190) become equal to the characteristic impedance of the isolated line. The four equations (3.193) become equivalent to two pairs of the unrelated telegrapher’s equations. The modelling of coupled lines of finite length is discussed in Sections 4.10–4.12.

Bibliography 1. K. Kurokawa, An Introduction to the Theory of Microwave Circuits, Academic Press, New York, 1969. 2. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. 3. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, 1992. 4. C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, Ltd, Chichester, 1989. 5. S. Hopfer, ‘The design of ridged waveguides’, IRE Transactions on Microwave Theory and Techniques, Vol. MTT-3, pp. 20–29, 1955. 6. S. B. Cohn, ‘Problems in strip transmission lines’, IRE Transactions, Vol. PGMTT-3, No. 2, pp. 119–126, 1955. 7. G. Matthaei, L.Young and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1964. 8. H. A. Wheeler, ‘Transmission-line properties of a strip on a dielectric sheet on a plane’, IEEE Transactions on Microwave Theory and Techniques, Vol. 25, No. 8, pp. 631–647, 1977.


89

9. E. Hammerstad and O. Jensen, ‘Accurate models for microstrip computer-aided design’, IEEE Microwave Symposium, pp. 407–409, 1980. 10. T. Itoh (ed.), Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, John Wiley & Sons, Inc., New York, 1989. 11. J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design, Artech House, Norwood, NJ, 1991. 12. R. C. Booton Jr, Computational Methods for Electromagnetics and Microwaves, John Wiley & Sons, Inc., New York, 1992. 13. E. Yamashita, K. Atsuki and T. Hirahata, ‘Microstrip dispersion in a wide frequency range’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-29, No. 6, pp. 610–611, 1981. 14. C. P. Wen, ‘Coplanar waveguide: a surface strip transmission line suitable for nonreciprocal gyromagnetic device application’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-17, pp. 1087–1090, 1969. 15. G. Ghione and C. Naldi, ‘Analytical formulas for coplanar lines in hybrid and monolithic MICs’, Electronics Letters, Vol. 20, pp. 179–181, 1984. 16. J. A. G. Malherbe, Microwave Transmission Line Couplers, Artech House, Norwood, NJ, 1988. 17. C. L. Paul, Analysis of Multiconductor Transmission Lines, John Wiley & Sons, Inc., Hoboken, NJ, 2007. 18. K. C. Gupta, R. Garg, F. Bahl and P. Barthia, Microstrip Lines and Slotlines, 2nd edition, Artech House, Norwood, NJ, 1996. 19. R. K. Hoffmann, Handbook of Microwave Integrated Circuits, Artech House, Norwood, NJ, 1987. 20. G. Conciauro, M. Guglielmi and R. Sorrentino, Advanced Modal Analysis, John Wiley & Sons, Ltd, Chichester, 1999.

4

Microwave circuits 4.1 Introduction In the previous chapter we showed that each propagating mode of a generalized transmission line can be modelled in terms of equivalent voltages and currents or, in other words, a transmission line. The transmission line model not only allows for a notable simplification of the electromagnetic problem by breaking it down into the longitudinal and transverse problems, but also allows us to employ established models and tools taken from the transmission line theory and circuit theory. As we will show in this chapter, by adopting the concept of microwave circuit it is possible to represent electromagnetic problems relevant to three-dimensional structures in terms of quantities typical of lumped element circuits. In this manner we can get intuitive vision and a physical insight into otherwise extremely complex electromagnetic models.

4.2 Microwave circuit formulation The analysis carried out in the previous chapter dealt with EM propagation in cylindrical guiding structures (generalized transmission lines) of infinite length. We have thus determined the modes of propagation as solutions to homogeneous Maxwell’s equations, i.e. free ‘oscillations’ of the system. In reality, of course, there are no transmission lines or guides of infinite length, but sections and stubs of finite lengths connecting circuit elements and components such as bends, bifurcations and junctions between different waveguides etc., or devices for the processing of the RF signal (filters, attenuators, etc.). In general terms, any deviation from the axial uniformity of the guide represents a discontinuity. Discontinuities are unavoidably present in any RF circuit either by the constraints imposed on the circuit (e.g. space occupation) or in order to perform specific functions on the RF signal (e.g. an attenuator). For the sake of illustration, Figure 4.1 shows some typical discontinuities in rectangular waveguide and microstrip lines. As an example, let us consider a metal iris or diaphragm (ideally made of a perfect conductor) inserted in the cross-section of a rectangular waveguide, as depicted in Figure 4.2. Suppose that an incident wave, represented by the dominant TE10 mode coming from an infinite distance z ¼ 1, impinges onto the iris. As we saw in the previous chapter, waveguide modes are homogeneous solutions


92


Figure 4.1 Examples of discontinuities: inductive iris (a), T junction (b) and bend (c) in a rectangular waveguide; double step (d), bend (e), gap (f) and T junction (g) in microstrip.

to Maxwell’s equations in an infinite waveguide; as such they satisfy the boundary conditions at the waveguide walls. From a mathematical point of view, the presence of an obstacle represents additional boundary conditions for the EM field. In our case, the tangential electric field must vanish on the iris surface. The TE10 mode (whose field lines are shown in Figure 3.8) clearly does not satisfy such a condition, since it has a non-zero electric field component Ey ¼ sinðpx=aÞ tangential to the iris. In order to satisfy the boundary conditions on the iris, we may add other waveguide modes to the TE10 mode. If the superposition of such modes satisfies the boundary conditions we are assured that this is the solution to our problem. In fact, since each mode satisfies Maxwell’s equation, so does their superposition by virtue of linearity. In other words, Maxwell’s equations and the boundary conditions on the waveguide wall are individually satisfied by each mode, while the boundary conditions on the obstacle are collectively satisfied by the mode superposition. As mentioned in Section 3.6, waveguide modes represent a complete set of vector functions. Similar to the well-known Fourier series for scalar functions in a linear interval, any vector function defined on the cross-section of the waveguide can be expanded in terms of waveguide modes. Such a property tells us that there exists a suitable superposition of waveguide modes such that, together with the incident TE10 mode, the boundary conditions on the iris are satisfied. Needless to say, knowing that the combination exists does not mean that we know it. We will come back to this point in the discussion of the mode matching method in Chapter 16.

Figure 4.2

(a) Symmetrical inductive iris in rectangular waveguide and (b) its equivalent circuit.

MICROWAVE CIRCUITS

93

At the plane z ¼ 0 where the iris is located, the EM field can be represented as the superposition of the TE10 mode plus a certain number (possibly infinite) of higher order modes. Under the normal assumption that the working frequency is such that only the fundamental mode can propagate while all other modes are below cut-off, the higher modes excited by the obstacle will die out as soon as we move away from it. On the contrary, the TE10 mode will propagate after the obstacle and will also be reflected by it towards the negative z direction. While the reader is referred to the rigorous treatment based on the mode matching method presented in Chapter 16, here we simply conclude that at a sufficient distance from the discontinuity, where we put our reference planes T1 and T2, the amplitude of the higher order modes becomes negligible so that the field distribution is that of the dominant mode. The latter consists of the incident plus the reflected waves for z < 0, and of the transmitted wave for z > 0. If we adopt the microwave circuit point of view by representing the mode as a transmission line, i.e. using equivalent voltages and currents, the iris can be modelled by a lumped element equivalent circuit as shown in Figure 4.2b. The reactance X that, as can be proved, is inductive represents the EM energy stored through the higher modes excited in the proximity of the obstacle. It should be stressed that since we have put the reference planes T1 and T2 far enough from the discontinuity, the EM field distribution on them is that of the dominant mode and is thus known apart from its amplitude coefficient. In conclusion, the region between the reference planes T1 and T2 can be represented by a simple two-port circuit where the voltages and currents represent the electric and magnetic fields of the dominant mode at the reference planes. Generalizing what we have just discussed, we define a microwave circuit as a region of space confined by perfectly conducting walls, enclosing an EM field and connected to the external space by N generalized transmission lines, as schematically depicted in Figure 4.3. Normally, though not always, we assume that the operating frequency is such that only the dominant mode propagates along each transmission line, while all other modes are below cut-off. At the ith (i ¼ 1. . . N) line we put a reference plane Ti at sufficient distance from the circuit so that higher modes can be neglected; from Ti on, the electric and magnetic fields are those of the dominant mode and can be represented in terms of equivalent voltages and currents, respectively, as in the previous chapter. The volume V, together with the line sections up to the reference planes, can thus be represented as an N-port network, with the notation shown in Figure 4.3b. The voltage and current at port i represent the electric and magnetic fields at the reference plane Ti. The reader might observe that the latter has a somewhat arbitrary position. Indeed, such an arbitrariness affects the values that voltages and currents may assume at the ports of the network of Figure 4.3b. We will see in Section 4.8 how the effect of a possible reference plane shift can be quantified. For the moment we may say that, because of the linearity of Maxwell’s equations, there will be a linear relationship between the set of currents and the set of voltages at the

Figure 4.3

A microwave circuit and its N-port network representation.

94


network ports. Seen from its terminals, the N-port network can be described in terms of 2N scalar quantities (i.e. N voltages þ N currents at the ports) instead of 2N vector functions (N electric þ N magnetic fields at the waveguide cross-sections). The simplification is apparent, although one has still to determine the relationship between the voltages and currents. In practice this indeed implies solving Maxwell’s equation in the volume V along with the proper boundary conditions. In this chapter we are not dealing with the field-theoretical problem of solving the EM field equations, but rather with the various ways of how the EM fields can be efficiently represented using circuit models, as discussed in the next section.

4.3 Terminated transmission lines Microwave circuits, as defined in the previous section, are connected together via lengths of generalized transmission lines, where we normally assume that the frequency is such that only the fundamental mode can propagate. EM propagation along transmission lines is governed by the telegrapher’s equations (3.51), the solution for voltage and current (or electric and magnetic field) waves being represented by Equation (3.54). After the study of infinite transmission lines presented in Chapter 3, in this and the following sections we study the behaviour of transmission lines of finite length from the circuit point of view, i.e. in terms of the usual parameters, such as impedance and admittance, employed in circuit theory. Consider a transmission line of finite length l terminated in z ¼ 0 with a load ZL (Figure 4.4). From its input terminals located in z ¼ l the line is seen as a dipole whose impedance we now wish to compute. The termination with ZL implies that the voltage to current ratio in z ¼ 0 is equal to ZL. Using (3.76), Zð0Þ ¼

Vð0Þ Vþ þV ¼ Z0 þ ¼ ZL Ið0Þ V V

ð4:1Þ

From (4.1), expressing V in terms of V þ , we have V ¼ GL V þ

ð4:2Þ

ZL Z0 ZL Z0

ð4:3Þ

where GL ¼

is the reflection coefficient at the load, i.e. in z ¼ 0. Such a quantity can be utilized instead of the impedance ZL to characterize the load. The reflection coefficient of the load is therefore the reflection

I(0)

+ Z0

0

V (0)

Zin

Z=

Z=0

Figure 4.4

Terminated transmission line.

ZL

MICROWAVE CIRCUITS

95

coefficient at the end of a transmission line of characteristic impedance Z0 terminated by ZL. It is important to stress that the reflection coefficient depends not only on the load but also on the transmission line to which the load is connected. On another line with a different Z0, the same load ZL has a different reflection coefficient. Once the reflection coefficient (4.3) at the load has been found, Equations (3.87)–(3.88) provide the reflection coefficient and the impedance at any point on the line, in particular at its input z ¼ l: Gin ¼ Gð lÞ ¼ GL e þ 2jbl Zin ¼ Z0

ZL cos bl þ jZ0 sinbl ZL þ jZ0 tan bl ¼ Z0 Z0 cos bl þ jZL sinbl Z0 þ jZL tan bl

ð4:4Þ ð4:5Þ

Equation (4.5) is an important formula that allows the line input impedance to be expressed in terms of the load impedance and the line characteristic impedance. In terms of admittance rather than impedance, with a similar procedure one obtains Yin ¼ Y0

YL cos bl þ jY0 sinbl YL þ jY0 tan bl ¼ Y0 Y0 cos bl þ jYL sinbl Y0 þ jYL tan bl

ð4:6Þ

Note that the impedance along the line, and the reflection coefficient as well, are periodic functions of the electrical length bl ¼ 2pl=l and take equal values every l ¼ l=2. In many cases it is preferable to refer to impedances normalized to the characteristic impedance Z0 of the line. This is equivalent to assuming a unitary characteristic impedance of the line. In such cases, using the symbol ^ to indicate normalized quantities, the load reflection coefficient (4.3) becomes Z^ L 1 Z^ L 1

ð4:7Þ

1 þ GL Z^ L ¼ 1 GL

ð4:8Þ

Z^ L cos bl þ j sinbl Z^ L þ j tan bl ¼ Z^ in ¼ cos bl þ j Z^ L sinbl 1 þ j Z^ L tan bl

ð4:9Þ

GL ¼ Conversely,

The input impedance (4.5) becomes

The voltage amplitude along the line can be expressed as a function of the amplitude of the incident wave and of the reflection coefficient: ð4:10Þ jVðzÞj ¼ jV þ e jbz þ V e þ jbz j ¼ jV þ jj1 þ Ge þ j2bz j ¼ jV þ j1 þ jGje þ jð2bz þ fÞ This equation shows that the voltage amplitude varies along the line, swinging between a maximum jVmax j ¼ jV þ jj1 þ jGjj

for 2bz þ f ¼ 0; 2p; 4p . . .

ð4:11Þ

jVmin j ¼ jV þ jj1 jGjj

for 2bz þ f ¼ p; 3p; 5p . . .

ð4:12Þ

and a minimum

The ratio between the maximum and minimum voltage amplitudes along the line is called the voltage standing wave ratio(VSWR): VSWR ¼

Vmax 1 þ jGj ¼ Vmin 1 jGj

ð4:13Þ

96


VSWR=

1.8

8

2

1.6 1.4

Volt

1.2

VSWR=3

1

VSWR=1

0.8 0.6 0.4 0.2 0 0

π

π

2

βz

3 2

π

2π

Figure 4.5 Voltage along a loaded line: |G| ¼ 0, VSWR ¼ 1; |G| ¼ 0.5, VSWR ¼ 3; |G| ¼ 1, VSWR ¼ 1. The VSWR clearly takes a value between 1 (when G ¼ 0, and thus the line is matched) and 1 (when |G| ¼ 1, and thus the line is totally mismatched). Figure 4.5 shows the voltage amplitude along the line for three different values of VSWR. Let us consider some noteworthy cases. In such a case ZL ¼ 0, YL ¼ 1, the reflection coefficient is unitary with phase p. GL ¼ 1, and VSWR ¼ 1. From (4.4)–(4.6) we obtain

Short-circuited line

Zin ¼ jZ0 tanðblÞ; Yin ¼ jY0 cotðblÞ Gin ¼ e 2jbl

ð4:14Þ

Open-circuited line In such a case ZL ¼ 1, YL ¼ 0, the reflection coefficient is unitary with phase 0: GL ¼ 1. Moreover, VSWR ¼ 1. From (4.4)–(4.6) we obtain

Zin ¼ jZ0 cotðblÞ; Yin ¼ jY0 tanðblÞ Gin ¼ e 2jbl

Matched line

ð4:15Þ

Since ZL ¼ Z0, YL ¼ Y0, then GL ¼ 0, VSWR ¼ 1. We obtain Zin ¼ Z0 Yin ¼ Y0 Gin ¼ 0

The input impedance equals the characteristic impedance of the line, whatever its length.

Half-wavelength line When the line length is equal to an integer multiple n of l/2, then bl ¼ bnl=2 ¼ np. Equation (4.5) gives Zin ¼ ZL: the input impedance equals the load impedance. Correspondingly, Gin ¼ GL.

MICROWAVE CIRCUITS

Quarter-wavelength transformer bl ¼ ð2n þ 1Þp=2. From (4.5) we get

97

When the line length is an odd multiple 2n þ 1 of l/4, then Z02 ZL

ð4:16Þ

1 Z^ in ¼ Z^ L

ð4:17Þ

Zin ¼ or, using normalized impedances,

A quarter-wavelength section of transmission line has therefore the property of ‘inverting’ the load impedance. With a suitable choice of its characteristic impedance Z0, such a circuit can be used to match the load ZL to a generator of internal impedance Zg. By imposing in (4.16) the condition Zin ¼ Zg we obtain pffiffiffiffiffiffiffiffiffiffi Z0 ¼ Zg ZL ð4:18Þ When (4.18) is satisfied, the input reflection coefficient is zero at the frequency corresponding to the line length being l/4 (or an odd multiple of it), thus the frequency f0 ¼ vph =ð4lÞ (or an odd multiple). The matching property of the quarter-wave transformer is limited to a narrow frequency band centred on f0. Using multiple quarter-wave sections, a wide-band impedance transformer can be designed, as discussed in Section 6.4.

4.4 The Smith chart The Smith chart, shown in Figure 4.6, is an extremely useful graphical tool for the representation and solution of a variety of transmission line problems. It is nothing but the representation of the normalized ^ þ jX ^ in the complex plane of the reflection coefficient. It thus allows for an immediate impedance Z^ ¼ R conversion of the impedance into the reflection coefficient, and vice versa. In mathematical terms, the chart represents the Z^ to G mapping expressed by Equation (4.7) that we repeat here for the reader’s convenience: GðzÞ ¼

^ 1 ZðzÞ ^ ZðzÞ þ 1

ð4:19Þ

This expression represents a so-called bilinear transform between the plane of the complex quantity Z^ and that of the complex quantity G. The transform is such that a circle in one plane is transformed into a circle in the other, the term circle being meant to include the limit case of infinite radius when the circle degenerates into a straight line. In addition, the angle formed by two intersecting curves is conserved in the transformation. As a consequence, two orthogonal lines in the Z^ plane transform into two ^ orthogonally crossing circles in the G plane. The Smith chart simply maps in the G plane the constant R ^ ^ and constant X lines of the Z plane (Figure 4.8). As stated above, such orthogonal line families are transformed into orthogonal circle families. Let us now present some examples for the sake of illustration. Consider first a purely reactive load for ^ From (4.19) we obtain which Z^ ¼ j X. GðzÞ ¼

^ 1 ^ jX 1 jX ¼ ^ þ1 ^ jX 1 þ jX

ð4:20Þ

This equation shows that the amplitude of G, being the ratio of complex conjugate numbers, is unitary and ^ varies from 1 to þ 1, the phase of G varies from 0 to 2p, being equal to p for X ^ ¼ 0. that, as X Equation (4.20) is thus the equation of a circle of unit radius in the G plane, corresponding to the outer ^ ¼ 0. Since for any circle of the Smith chart. Such a circle is indeed the mapping of the straight line R

98

MICROWAVE AND RF ENGINEERING 90º

0.3 λ

0.2 λ 60º

8.0

1.0

120º

2.0

9.0

0.4 λ 150º

0. 4

3.0

0.1 λ 30º 4.0 5.0

0.2 10.0

4.0 5.0

3.0

2.0

1.0

0.8

0.6

180º 0.5 λ

0.4

0

0.2

10.0

0º 0λ –10.0

–0.2

0

–5.

.0

–4

.0

.4

–0

0.9 λ

–1.0

–0.8

–0.

6

–2.

0

0.6 λ

–30º

–3

–150º

–120º

–60º 0.8 λ

0.7 λ –90º

1

8.0

∞

9

0 ∞

0.6 5 3

5

3

2

0.4 3

2

6

10 1

0.2

0

1.5

1.2 1

0.2 1.2

1.5

15 20

∞

20 15

0.5 0.2 0.05

0

0.05 0.2

0.4

0.6

2

3

10

6

0.5

1

0.8 5

∞ SWR

9

0 Return Loss, dB

3 2

3

1 Γ

5

∞ Reflection Loss, dB

Figure 4.6 The Smith chart.

^ 0, then jGj 1: any point corresponding to a passive load is contained in the unit circle passive load R ^ 0 in the complex Z^ plane is transformed into the area internal to the of the Smith chart. The half plane R circle jGj 1. ^ straight lines. It can easily be verified that they are Let us now consider the other constant R ^ ¼ 0, having their centres on the real axis, all transformed into circles internal to the outer circle R passing through the point G ¼ 1. As shown in Figure 4.6, each of such circles is labelled with the ^ value. Note that, in particular, the circle R ^ ¼ 1 intersects the origin G ¼ 0. This is the corresponding R ^ ¼ 1, X ^ ¼ 0) and the reflection matching condition for which the normalized impedance is unitary (R ^ circles intersect G ¼ 1 is explained as follows. As shown coefficient is zero. The fact that all constant R ^ is, the load behaves as an open circuit, by (4.20), when the load reactance is infinite, whatever R corresponding to G ¼ 1.

MICROWAVE CIRCUITS

jX

99

X=2 X=const

R=const

R=3

R=1

X=1

R

X=-1 X=-2

Figure 4.7

The constant R and constant X lines transform into the Smith chart circles of Figure 4.6.

^ lines in the Z^ plane transform into the G plane. Let us We now wish to examine how the constant X ^ ^ ^ Equation (4.19) gives start with a purely resistive load X ¼ 0, thus Z ¼ R. G¼

^ 1 R ^ þ1 R

ð4:21Þ

^ varies from 0 (short circuit) to 1 (open In such a case G is purely real and varies from 1 to þ 1 as R ^ ^ circuit). The half line X ¼ 0, R 0 of Figure 4.7 thus maps into the real axis of the G plane. ^ lines. It can be proved that they map in G plane circles all Let us finally consider the other constant X passing through the point G ¼ 1 (open circuit) with centres lying on the straight line Re(G) ¼ 1. The ^ > 0) have their centres on the upper half line (Im(G) > 0), circles corresponding to inductive loads (X ^ < 0) have their centres on the lower half line while those corresponding to capacitive loads (X (Im(G) < 0). As shown by Figure 4.6, the two circle families are separated by the straight line (degenerate ^ ¼ 0. circle) X The usefulness and simplicity of the Smith chart are apparent if one considers that when moving along a lossless transmission line the normalized impedance varies according to a somewhat complicated law expressed by (3.87), while the reflection coefficient varies only by its phase, as shown by (3.88). As we move from the load in z ¼ 0 towards the generator (negative z), the phase of the reflection coefficient decreases by the angle z ð4:22Þ f ¼ 2bz ¼ 2p l=2 and the corresponding point in the G plane of the Smith chart moves clockwise along an arc f. Note that a half-wave displacement towards the generator corresponds to a full 2p rotation of the corresponding point in the Smith chart. Vice versa, moving towards the load (increasing z) corresponds to counterclockwise rotations. On the outer circle, an angular scale is attached where angles are measured in terms of fractions of wavelength. In such a way one can immediately identify the reflection coefficient at any location on ^ and constant X ^ circles, the corresponding impedance values. the line as well as, through the constant R The Smith chart is also usually equipped with various additional rulers to evaluate graphically the amplitude of the reflection coefficient, the corresponding VSWR, etc.

100


It is worth observing that the mapping (4.19) between the impedance and the reflection coefficient depends on the reference impedance Z0 used for normalization. When transmission lines of different characteristic impedances are cascaded together, the problem arises of denormalizing the impedances when we move from one line to the next one. This typically happens when dealing with multi-section impedance transformers (Section 6.4). Let us go into more detail by considering again the simple problem in Figure 4.4 of a transmission line terminated at one end with a load ZL. We want to determine how the reflection coefficient changes as the result of a change of reference impedance. Let the transmission line with characteristic impedance Z0 be connected at its left end to another line of impedance Z1. The reflection coefficient Gin;1 seen from Z1 is different from Gin;0 seen, although at the same point, from Z0. In order to determine the relation between the two reflection coefficients, we use Equations (3.78) and (3.79) to write

Gin;1

Zin Z1 ¼ ¼ Zin þ Z1

1 þ Gin;0 Z1 Gin;0 þ G01 1 Gin;0 ¼ ... ¼ 1 þ Gin;0 Gin;0 G01 þ 1 Z0 þ Z1 1 Gin;0 Z0

ð4:23Þ

where G01 ¼

Z0 Z1 Z0 þ Z1

ð4:24Þ

As expected, if Z0 ¼ Z1 then the above formula gives Gin;1 ¼ Gin;0 . Equation (4.23) is a bilinear transform between the Gin;1 and Gin;0 planes, mapping circles into circles. As we move along the line Z0 from the load towards the generator, the representative point in the Gin;0 plane describes a circle centred at the origin, as expressed by (3.88). The corresponding point in the plane Gin;1 still describes a circle; in order to find its centre and radius, let us put Gin;0 ¼ re jy . Since G01 in (4.24) is real, when y1 is varied the circle described by (4.23) has its centre on the real axis and intersects it when y1 is 0 or p, i.e. when Gin;0 ¼ r. Inserting such values into (4.23), we easily obtain that the centre x0 (on the real axis) and the radius r are given by 1 G01 þ r G01 r 1 r2 þ ¼ G01 x0 ¼ 2 1 þ G01 r 1 G01 r 1 G201 r2 ð4:25Þ 1 G01 þ r G01 r 1 G201 r ¼ ¼ r1 2 1 þ G01 r 1 G01 r 1 G201 r2

EXAMPLE 4.1 A 10 mm long section of coaxial cable with 50 O characteristic impedance, filled with a dielectric with er ¼ 4, is terminated with a load ZL ¼ 100 þ j70. We want to determine the input impedance, input reflection coefficient and VSWR at the frequency f ¼ 3 GHz. We first need to determine the normalized load impedance: 100 þ j70 Z^ L ¼ ¼ 2 þ j1:4 50 The corresponding point in the Smith chart is shown in Figure 4.8. Using the angular ruler attached to the outer circle, we read off the phase of the reflection coefficient of the load GL (about 30 , or p/6). Its magnitude is found by projecting the end of the radius onto the scale provided below the chart. One thus finds jGj ¼ 0:52

ð4:26Þ

MICROWAVE CIRCUITS

101

90º 60º

8.0

1.0

120º

2.0

9.0

30º

3.0

0. 4

150º

4.0 5.0

0.2

2 + j1.4

10.0

10.0

4.0 5.0

2.0

3.0

0.8

1.0

0.6

180º

0.4

0

0.2

ρ = 0.52

0º

–10.0

–0.2

–5. 0

.0

0.56 + j0.72

–4

–30º

–3 .0

–150º

.4

–1.0

–0.8

–0.

6

–2.

0

–0

–120º

–60º

144º

–90º

1

8.0

0.6

9 0 ∞

5 3

5

3

2

0.4

0.2

0

1.5

1.2 1

3

2

6

10

15 20

0.5

0.2 0.05

1

0

0.2 1.2

1.5

0.52 0.6

0.4 2

3 3.2

20 15

10

6 5.7

0.05 0.2

0.5

1

0.8

5

1.4 2

∞ SWR

9

0 Return Loss, dB

3 3

1 Γ

5

∞ Reflection Loss, dB

Figure 4.8 Using the Smith chart.

In order to determine the corresponding point at the line input we first need to compute the electrical length of the line. Using the data provided and assuming the velocity of light in a vacuum to be 3108 m/s, the wavelength is 3 108 ¼ 50 mm l ¼ pffiffiffi 4 3 109

ð4:27Þ

The line section is 0.2l long and corresponds to an angular rotation j ¼ 2p

0:01 ¼ 2p 0:4 radians 0:05=2

ð4:28Þ

corresponding to 144 . After moving 144 clockwise, the input reflection coefficient is found at 114 and the following input normalized impedance can be read on the chart: Z^ in ¼ 0:56 j0:72

ð4:29Þ

102


Denormalizing, Zin ¼ 50 Z^ in ¼ 28 j36 Using the VSWR scale we also obtain VSWR ¼ 3:2

ð4:30Þ

The Smith chart can also be used for admittance rather than impedance computation. Indeed, as already seen, the normalized impedance of a quarter-wave section is equal to its inverse, thus to the normalized admittance. Since a l/4 shift corresponds to a 180 rotation, we conclude that symmetrical points with respect to the origin correspond to impedances that are the inverse of each other. The numerical value associated with one point in the Smith chart is thus the admittance (or impedance) corresponding to the impedance (admittance) of the symmetrical point. As an alternative, the Smith chart can simply be rotated by 180 in order to represent the admittances rather than the impedances. Constant Q curves In some applications it is useful to consider the constant quality factor curves because they can give information at a glance about the performance of a matching network.1 The quality factor or the Q factor or simply the Q of an impedance Z ¼ R þ jX corresponding to the admittance Y¼

1 1 R X j 2 ¼ G þ jB ¼ ¼ Z R þ jX R2 þ X 2 R þ X2

is defined as2 Q¼

jX j jBj ¼ R G

Note that the Q factor is invariant with respect to any impedance normalization with respect to a resistance. The reflection coefficient corresponding to Z or Y can be expressed in terms of Q: R0 ð1 jQÞ R0 Z R0 R jX R0 ð1 jRQÞR G¼ ¼ ¼ ¼ R R0 þ Z R0 þ R þ jX R0 þ ð1 jRQÞR R0 þ ð1 jQÞ R By taking the real and imaginary parts G ¼ GR þ jGI we obtain 2 R0 R0 1 Q2 2Q R R GR ¼ ; GI ¼ 2 2 R0 R0 2 þ1 þQ þ 1 þ Q2 R R

ð4:31Þ

As Q is fixed and R is varied, the above are the parametric equations of two curves in the complex G plane. Such curves are symmetrical with respect to the real axis, since Q does not change when an impedance Z ¼ R þ jX is replaced by its conjugate Z ¼ R jX. Moreover, by recalling that by a 180 rotation in the G plane the normalized impedance transforms into the normalized admittance

1 2

Matching networks are discussed in Chapter 6. See Section 5.3.

MICROWAVE CIRCUITS

103

and that the Q factor computed from the impedance is the same as that computed from the admittance, we find that the two curves (4.31) are also symmetrical with respect to the imaginary axis. After some manipulation, R can be eliminated from (4.31) so as to obtain G2R þ G2I

2 GI 1 ¼ 0 Q

ð4:32Þ

It is readily seen that the above are the equations of two circles with centres 1 c 0; j Q and radius

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 r ¼ 1þ 2 Q

All circles pass through the points ð1; j0Þ. The two circles, when Q becomes infinite, degenerate into the unitary circle of the Smith chart. The pair of circles for Q ¼ 0 degenerate in the real axis. Figure 4.9 shows some of the constant Q circles (Q ¼ 0, 0.5, 1, 2, 4, 1) in the Smith chart. Constant Q circles can be used to identify the Q factor of a matching network. Higher Q factors generally correspond to narrower bandwidths and higher sensitivities with respect to component tolerances. As an example consider the quarter-wavelength impedance transformer. Suppose we have to match a resistive 200 O load to a 50 O source. This can be done using either one of the matching networks in Figure 4.10a,b representing two impedance transformers made of quarterwavelength line sections at the frequency f0. The simplest circuit is that of Figure 4.10a, consisting of just a line section whosepcharacteristic impedance is the geometrical average of the load and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi source impedances: Z01 ¼ 50 200 ¼ 100. At f0 the 200 O load is perfectly matched to 50 O, but the matching degrades as the frequency departs from f0. 10.0

0

6.0

4

2.0

8.0

∞

0.4

3.

2

4.0 5.0

1

0.2

10.0

4.0 5.0

Q=0

3.0

2.0

1.0

0.8

0.6

0.4

10.0

–10 .0

0.2

Q=0.5

Q=0.5 –0 .2

–4 .0 –5. 0

1 .0

2

–2

–0.10

–0.8

–0 .6

4

.0

–3

.4

–0

Figure 4.9 Constant Q circles.

104


Zin

Zin

Z01

Z01 ′

Z02 ′

RL = 200 Ω (a)

RL = 200 Ω (b)

ZB

Figure 4.10

Impedance matching transmission line circuits: (a) single section; (b) two sections.

As we move from the load towards the source, the trajectory of the reflection coefficient describes the semicircle A–C in Figure 4.11. Note that while both A and C correspond to resistive (zero-reactance) loads, the impedance is complex in all intermediate points along the line. The Q factor is zero in A, increases up to a maximum of about 0.75, and then goes back to zero in C. The maximum Q can be reduced using a two-section transformer like that shown in Figure 4.10b. The impedance matching is achieved in two steps. The first section transforms 200 Offi load into pthe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 the intermediate value of 100 O. The characteristic impedance is Z02 ¼ 100 200 ffi 141 O. The reflection coefficient from the load to the junction between the two sections describes the semicircle A–B in Figure 4.11. The second l/4 section transforms 100 O into 50 O as required. Its

Figure 4.11

Reflection coefficient trajectories of the circuits in Figure 4.10.

MICROWAVE CIRCUITS

105

4

VSWR

3

1 section (∆f/f0≈0.625)

2

2 sections (∆f/f0≈0.964)

1 0.0

0.5

1.0

1.5

2.0

f/f0

Figure 4.12

VSWR of the circuits in Figure 4.10.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 characteristic impedance is Z01 ¼ 50 100 ffi 70:7 O. The reflection coefficient describes the semicircle B–C in Figure 4.11. As in the single section transformer, the impedance is real at the ends but complex along the lines. The maximum Q factor is half that of the single section transformer. Although the result is already well known, simple inspection of Figure 4.11 allows us to conclude that the two-section transformer has a wider bandwidth that the single section transformer. A quantitative comparison of the performances of the two circuits in Figure 4.10 can be obtained, for example, in terms of the VSWR vs. frequency, as shown in Figure 4.12. Observe first that both responses are periodical with period 2f0. Only the first period is shown in the figure. For both circuits the mismatch is maximum at the band ends (VSWR ¼ 4) and zero at mid-band. Elsewhere, the VSWR of circuit (b) is better than that of circuit (a). Finally, for a given maximum VSWR, the frequency band of the two-section transformer is wider than the single section one. For example, for VSWR < 2, the former has a 96% bandwidth, while the latter has only 60% bandwidth. In conclusion, monitoring the Q factors of the reflection coefficients in the design of a matching network can provide some useful information; minimizing the Q generally leads to less critical networks and wider bandwidths.

4.5 Power flow In this section, we are concerned with the power flow associated with EM waves travelling along a generalized transmission line. Let us suppose, for the moment, that only the incident wave is present; this implies that the line is matched and the reflected wave amplitude is zero (V ¼ 0). We also assume that the line is lossless. Using (3.54) for the voltage and current along the line, the real power3 travelling along the line in the positive z direction is 1 V þ* þ jbz 1 2 ¼ e ð4:33Þ P þ ¼ Re V þ e jbz jV þ j 2 2Z0 Z0 3

Using the phasor representation, the real power is the average power in one period T ¼ 2p/o, see (2.32).

106

MICROWAVE AND RF ENGINEERING Zg Vg +

Iin

+ Vin –

Z0 , β0

ΓL

ZL

Zin Z= 0

Z= –

Figure 4.13

ΓL =

ZL – Z0 ZL + Z0

Γin =

Zin – Zg Zin + Zg

z

Power transfer from a source to a load through a transmission line.

When, on the contrary, only the reflected wave is present, thus V þ ¼ 0, the power travelling in the negative z direction is P ¼

1 jV j2 2Z0

ð4:34Þ

In the general case, when both incident and reflected waves are present on the line, 1 Re ðV þ e jbz þ V e þ jbz ÞðV þ* e þ jbz V * e jbz Þ 2Z0 i 1 h þ 2 ¼ jV j jV j2 ¼ P þ P 2Z0

P¼

ð4:35Þ

This equation shows that, as might be anticipated, the flow of real power is constant along the lossless line and equals the difference between the incident and the reflected power. This is clearly because the line is lossless and by virtue of the principle of energy conservation. Note, however, that in addition to the real or active power there is an imaginary or reactive power Pj ¼

1 1 Im ðV þ e jbz þ V e þ jbz ÞðV þ* e þ jbz V * e jbz Þ ¼ ImðV þ* V ej2bz Þ 2Z0 Z0

ð4:36Þ

that varies along the line and depends on the simultaneous presence of both waves. Indeed, if either V þ or V is zero, the reactive power is also zero. Let us now study the power transfer from a source Vg to a load ZL through a transmission line section of characteristic impedance Z0 and phase constant b0, as depicted in Figure 4.13. Zg is the internal impedance of the source, located in z ¼ ‘. We want to determine the power delivered to the load and the conditions for maximum power transfer. Since the line is lossless, the power delivered to the load equals the input power entering into the line: " # " # Vg Zin 2 1 1 1 jVin j2 Pin ¼ Re Vin Iin* ¼ Re Re ¼ Zg þ Zin 2 Z * 2 2 2 Zin* in 2 Vg 1 1 Rin 2 ¼ Re½Zin ¼ Vg 2 Zg þ Zin 2 2 ðRin þ Rg Þ2 þ ðXin þ Xg Þ2 having put Zin ¼ Rin þ jXin and Zg ¼ Rg þ jXg. We now consider some particular cases.

ð4:37Þ

MICROWAVE CIRCUITS

107

Matched load ZL ¼ Z0. In such a case no reflection occurs at the load, thus GL ¼ 0 and Zin ¼ Z0. Equation (4.37) becomes 1 2 Z0 Pin ¼ Vg 2 ðZ0 þ Rg Þ2 þ Xg2

ð4:38Þ

The power delivered is just the same as if the load is connected directly to the source. Zin ¼ Zg. In this case the input reflection coefficient is zero. If, however, Zin 6¼ Z0 the line is generally matched neither to the load nor to the source, so that there will be a standing wave. The power transferred to the load is

Matched source

1 2 Rg Pin ¼ Vg 2 4ðR2g þ Xg2 Þ

ð4:39Þ

It is interesting to note that the power (4.39) delivered when the source is matched may be less than that when the load is matched to the line (4.38).

Maximum power transfer; available power

The condition for maximum power transfer for a given source impedance can be established by simply equating to zero the derivatives of (4.37) with respect to Rin and Xin. From easy algebra one obtains the well-known condition for conjugate matching: Zin ¼ Zg*

ð4:40Þ

When the impedance seen by the source equals the conjugate of the internal impedance of the generator, the power delivered to the load is a maximum and equal to the power available from the source 1 2 1 Pav ¼ Vg 2 4Rg

ð4:41Þ

Note that (4.40) does not imply the matching of the generator to the line (Zg ¼ Z0), nor of the load (Z0 ¼ ZL), nor of the generator (Zg ¼ Zin) except, in the last case, when Xg ¼ Xin ¼ 0. Let us finally observe that, as far as the input terminals of the line are concerned, the voltage source Vg with internal impedance Zg is equivalent to an incident wave of amplitude Vgþ ¼ Vg =2 travelling along a line with characteristic impedance Zg. Suppose for the moment, as illustrated in Figure 4.14, that the line Z0 is matched and thus is terminated with Z0. Let us compute the voltage Vin at the line input in both cases. In the former case, when the line is fed by the source Vg, we have Vin ¼ Vg

Z0 Zg þ Z0

ð4:42Þ

while in the latter case, when the line Z0 is connected to the transmission line Zg with an incident voltage Vgþ , the input voltage at z ¼ l is Vin ¼ Vgþ ð1 þ Gin Þ ¼ Vgþ

Figure 4.14

ð4:43Þ

+

V g = Vg /2

Zg Vg

2Z0 Zg þ Z0

+

Z0

Zg

Z0

Equivalence between a voltage source and an incident voltage wave.

108


Both circuits produce the same voltages and currents on the line if 2Vgþ ¼ Vg . The power carried by the incident wave Vgþ is þ 2 V 1 g ð4:44Þ Pav ¼ 2 Rg and is thus coincident with the available power from the generator (4.41). Consider Figure 4.13 again, where the line Z0 is terminated with ZL. Let V þ and V be the wave amplitudes on the line Z0, with V ¼ GL V þ . The voltage and current at z ¼ l are given by Vð lÞ ¼ V þ ðebl þ GL e bl Þ V þ bl ðe GL e bl Þ Z0

Ið lÞ ¼

ð4:45Þ

The above quantities are related to the source voltage Vg by Vg ¼ Vð lÞ þ Zg Ið lÞ

ð4:46Þ

þ

Inserting (4.45) into (4.46) and solving for V , we find that the incident wave voltage on the line is Vþ ¼

Vg 1 1 þ Z^ g e jbl GL Gg e jbl

ð4:47Þ

with Zg Z^ g ¼ ; Z0

Gg ¼

Z^ g 1 Z^ g þ 1

ð4:48Þ

Let us now replace the voltage source Vg with internal impedance Zg by a transmission line of characteristic impedance Zg with an incident voltage wave Vgþ . The voltage and current at z ¼ l can be expressed in terms of the incident and reflected voltages with the usual formulae: Vgþ þ Vg ¼ Vð lÞ Vgþ Vg ¼ Zg Ið lÞ

ð4:49Þ

Inserting (4.45) into (4.49) and solving for V þ , we obtain V þ¼

2Vgþ 1 1 þ Z^ g e jbl GL Gg e jbl

ð4:50Þ

Again we see the equivalence between the two models of Figure 4.14, when 2Vgþ ¼ Vg . Note that from (4.49) and (4.45) we also get Vg ¼ Vgþ

Gg þ GL e 2jbl 1 GL Gg e 2jbl

ð4:51Þ

Therefore, the input reflection coefficient seen from the Zg line is Gin;g ¼

Gg þ GL e 2jbl 1 GL Gg e 2jbl

ð4:52Þ

while the input reflection coefficient seen from the Z0 line is Gin;0 ¼ GL e 2jbl

ð4:53Þ

It can easily be shown that (4.52) can also be obtained by (4.23) with Gin;1 ¼ Gin;g and G01 ¼ Gg .

MICROWAVE CIRCUITS

109

4.6 Matrix representations As we have seen above, a microwave circuit with N apertures can be represented as an N-port network. Different representations may be adopted and are in use to characterize such a circuit by expressing the relationships between the 2N voltages and currents at its ports or between other variables related to such quantities. Here we will limit our attention to the most common and practical representations.

4.6.1

The impedance matrix

The impedance matrix relates the voltages V1, V2, . . . , VN at the circuit ports to the currents I1, I2, . . . , IN entering into those ports. Because of linearity, we can write V1 ¼ z11 I1 þ z12 I2 þ þ z1N IN V2 ¼ z21 I1 þ z22 I2 þ þ z2N IN .. . VN ¼ zN1 I1 þ zN2 I2 þ þ zNN IN

ð4:54Þ

½V ¼ ½Z ½I

ð4:55Þ

In matrix form,

where [V] and [I] are N 1 column vectors and [Z] is an N N matrix called the impedance matrix of the network. The physical meaning of the impedance matrix becomes apparent by considering how, at least in principle, we can measure its elements, also called the impedance parameters or Z parameters of the network. Suppose the current Ij is applied to the jth port while all other ports are open-circuited (thus Ii ¼ 0 for any i 6¼ j). From (4.54) we obtain Vi ¼ zij Ij , thus Vi ð4:56Þ zij ¼ Ij I1 ¼I2 ¼¼IN ¼0;Ij 6¼ 0 The impedance parameters are thus evaluated by feeding one port with a given current while all remaining ports are open-circuited. In particular, zii represents the input impedance at the ith port when all remaining ports are open-circuited, while the quantity zij (with i 6¼ j) is the transimpedance between ports i and j. For lossless networks, the Z parameters, thus the Z matrix, are purely imaginary. For reciprocal networks, i.e. when Lorentz’s theorem holds, it can be proved that zij ¼ zji

ð4:57Þ

This implies that the impedance matrix is symmetrical with respect to its main diagonal. Such a property does not hold true for networks containing nonlinear devices (such as diodes and transistors) or anisotropic materials (such as ferrites). Let us now consider the particular case of a two-port network, as depicted in Figure 4.15a. Figure 4.15b shows a particular circuit topology, called a T network because of its shape.

I1 V1

I2

[Z]

V2 (a)

Figure 4.15

I1

I2 ZA

V1

ZB ZC

V2

(b)

A two-port network and its T network representation.

110


The T network lends itself to represent a generic two-port network in terms of its impedance parameters. To prove this statement, let us determine its Z parameters by adopting their definitions. To compute z11 and z12 we apply a unit current at port 1 and compute V1 and V2. By examining Figure 4.15b we easily obtain z11 ¼ ZA þ ZC z21 ¼ ZC

ð4:58Þ

In much the same way, we obtain z22 ¼ ZB þ ZC z12 ¼ ZC

ð4:59Þ

Conversely, given the Z matrix of a two-port network, by reversing (4.58)–(4.59) we can compute the impedances ZA, ZB and ZC of the T network: ZA ¼ z11 z12 ZB ¼ z22 z12 ZC ¼ z12 ¼ z21

ð4:60Þ

From the external terminals, the original network and its T network representation are fully equivalent as they have the same Z matrix. Note that a reciprocal network is characterized by three complex parameters (ZA, ZB and ZC), or, equivalently, by six real parameters. They reduce to three real parameters (XA, XB, XC) in the lossless case.

4.6.2

The admittance matrix

Dual to the impedance matrix, the admittance matrix expresses the currents entering into the network in terms of the voltages applied to the ports. Using the matrix notation already used for the impedance matrix, we can write ½I ¼ ½Y ½V

ð4:61Þ

where [Y] is the admittance matrix of the network. Its physical interpretation is analogous to that of the impedance matrix, except that now we have to set the voltages to zero rather than the currents. Instead of (4.56) we now have Ii yij ¼ ð4:62Þ Vj V1 ¼V2 ¼¼VN ¼0;Vj 6¼ 0 Therefore, yij equals the current Ii entering into the ith port when a unit voltage (Vj ¼ 1) is applied to the jth port, while all remaining ports are short-circuited. By comparing (4.55) with (4.61) we can deduce that the Z and Y matrices are inverses of each other, provided that the inverse matrices exist (which is not always true, as we will soon see). Therefore ½Z ¼ ½Y 1

ð4:63Þ

We deduce that, like the Z matrix, the Y matrix is also purely imaginary for lossless networks and symmetrical for reciprocal networks: yij ¼ yji

ð4:64Þ

In the case of two-port networks, the network topology suited to the Y matrix representation is the P network shown in Figure 4.16.

MICROWAVE CIRCUITS

I1

111

I2 YC

V1

YB

YA

Figure 4.16

V2

A P network.

By a similar reasoning as for the Z parameters, the parameters y11 and y12 are obtained by computing the currents I1 and I2 when voltage V1 ¼ 1 V is applied at port 1 with port 2 short-circuited (V2 ¼ 0). With reference to Figure 4.16, we easily find y11 ¼ YA þ YC y21 ¼ YC

ð4:65Þ

y22 ¼ YB þ YC y12 ¼ YC

ð4:66Þ

Similarly,

By inverting the above formulae we can express the admittances of the P network in terms of the Y parameters: YA ¼ y11 þ y12 YB ¼ y22 þ y12 YC ¼ y12 ¼ y21

ð4:67Þ

These formulae allow us to identify the P equivalent of a generic two-port network of which the Y matrix is known. It is hardly necessary to specify that the admittance parameters are not the inverse of the impedance parameters, i.e. that yij 6¼ 1/zij. In particular, the input admittance y11 is not the inverse of the input impedance z11, as it clearly appears since the former is obtained when port 2 is short-circuited and the latter when port 2 is open-circuited.

4.6.3

The ABCD or chain matrix

The ABCD matrix, also called the chain matrix, is particularly useful for characterizing two-port networks when such networks are cascaded together, as shown in Figure 4.17. The ABCD matrix belongs to the category of transmission matrices, for which the quantities at port 1 are expressed in terms of those at port 2. The ABCD parameters in fact express the voltage and current at the first port in terms of the corresponding quantities at the second port, as follows: V1 A B V2 ¼ ð4:68Þ I1 I2 C D The reader will notice that the current I2 appears with a minus sign; this is because I2 corresponds to the current flowing out of port 2, and thus is equal to the current entering into the next network. With reference to Figure 4.17, since the cascade connection implies that the voltages and currents at the output of the first network are equal to the corresponding quantities at the input of the second network, we can write

112


Cascade connection of two-port networks.

Figure 4.17

"

ð1Þ

V1

ð1Þ

I1

#

A1 ¼ C1

B1 D1

"

#

ð1Þ

V2

ð1Þ

I2

A1 ¼ C1

B1 D1

"

ð2Þ

V1

ð2Þ

I1

#

A1 ¼ C1

B1 D1

A2 C2

B2 D2

"

ð2Þ

V2

ð2Þ

I2

# ð4:69Þ

The ABCD matrix of the whole network is thus simply the product, taken in the proper order, of the ABCD matrices of the constituent networks:

B A1 ¼ D C1

A C

B1 D1

A2 C2

B2 D2

ð4:70Þ

This result can be immediately generalized to the cascade of any number of cascaded two-port networks. In a lossless network, it can be proved that A and D are real while B and C are imaginary. The reciprocity of the network implies that the matrix determinant is unitary: AD CD ¼ 1

ð4:71Þ

From simple algebra we can obtain the relationship between the chain matrix and the impedance matrix. In fact, solving V1 ¼ z11 I1 þ z12 I2 V2 ¼ z12 I1 þ z22 I2 for V1 and I1 we obtain

A

B

C

D

¼

Z11 =Z21

ðZ11 Z22 Z12 Z21 Þ=Z21

1=Z21

Z22 =Z21

ð4:72Þ

The relationships between the ABCD parameters and the other network parameters are quoted in Table 4.1.

4.6.4

The scattering matrix

In contrast with the matrix representations discussed so far, the scattering matrix, or S matrix, does not deal with voltages and currents but with incoming and outgoing waves at the ports of the network. From the physical point of view, the scattering matrix is the matrix that best represents the behaviour of a microwave circuit, where voltages and currents are somewhat fictitious quantities that cannot be measured directly. On the contrary, we can measure amplitudes and phases of the waves travelling along the transmission lines connected to the circuit ports.

MICROWAVE CIRCUITS

113

In order to introduce the scattering parameters, however, it is first necessary to clarify what we mean as incoming and outgoing waves, since the commonly adopted terminology for such waves is somewhat improper and confusing. Indeed, the waves travelling towards the network are usually called incident waves, while those travelling from the network are called reflected waves. The latter term does not account for the fact that the wave coming out of a circuit can result not only from the reflection of a wave incident at that port, but also from the transmission of other wave(s) incident at other port(s). In spite of this, we will occasionally use the term reflected wave to indicate a wave coming out of a network. Let us now refer to the generic microwave circuit of Figure 4.3. Along the ith feeding line with characteristic impedance Z0i there are voltage and current waves travelling towards (Viþ ; Iiþ ) and away from (Vi ; Ii ) the circuit. We define the following quantities as incoming and outgoing waves at the reference plane Ti : ai ¼

Vi þ Z0i Ii Vþ pffiffiffiffiffiffi ¼ pi ffiffiffiffiffiffi 2 Z0i 2 Z0i

ð4:73Þ

bi ¼

Vi Z0i Ii V pffiffiffiffiffiffi ¼ pi ffiffiffiffiffiffi 2 Z0i 2 Z0i

ð4:74Þ

Conversely, the voltage and current are expressed in terms of the wave amplitudes ai, bi as follows: pffiffiffiffiffiffi Vi ¼ ðai þ bÞi Z0i ð4:75Þ pffiffiffiffiffiffi Ii ¼ ðai bi Þ= Z0i In these expressions we have assumed that the characteristic impedance of the line is real; moreover, we have considered that, according to the expressions for voltage and current along a transmission line, we have Vi ¼ Viþ þ Vi

Ii ¼ ðViþ Vi Þ=Z0i

ð4:76Þ

pffiffiffiffiffiffi Apart from the factor 1=ð2 Z0i Þ, therefore, ai and bi represent the incident ( þ ) and reflected ( ) voltage waves, respectively, along the ith line at the reference plane Ti. Note that the power entering into the ith port can be expressed in terms of the (complex) wave amplitudes (4.74) 1 Pi ¼ Vi I *i ¼ jai j2 jbi j2 ¼ Piþ Pi 2

ð4:77Þ

where 1 Piþ ¼ Viþ Iiþ * ¼ jai j2 2 1 Pi ¼ Vi Ii * ¼ jbi j2 2

ð4:78Þ

are the incoming (incident) and outgoing (reflected) power from the ith port, respectively. The scattering matrix expresses the outgoing waves bi as linear functions of the incoming waves ai: b1 ¼ s11 a1 þ s12 a2 þ þ s1N aN b2 ¼ s21 a1 þ s22 a2 þ þ s2N aN .. . bN ¼ sN1 a1 þ sN2 a2 þ þ sNN aN

ð4:79Þ

114


or, in matrix form, ½b ¼ ½S ½a

ð4:80Þ

The elements of the scattering matrix are the scattering parameters or simply S parameters. It is important to note that they can be defined only when the set of impedances Z0i (i ¼ 1, 2, . . ., N) of the feeding lines has been specified. Such impedances are also called the reference impedances. By changing the reference impedances, the scattering parameters change too. This should not be surprising, since the scattering parameters in the case of multi-port circuits play the same role as the reflection coefficient of a dipole. The latter (see Equation (3.79)) indeed depends on the characteristic impedance of the transmission line connected to it. In most practical cases, however, the reference impedances are chosen to be the same and equal to 50 O. To appreciate the physical meaning of the S parameters, let us proceed as for the impedance or admittance parameters. From (4.79) we obtain bi sij ¼ ð4:81Þ aj a1 ¼a2 ¼¼aN ¼0;aj 6¼ 0 The scattering parameter sij is thus the ratio of the wave amplitude bi coming out of port i to the wave amplitude ai incident on port i when no other wave is incident on the remaining ports. Considering that all ports are assumed to be terminated with transmission lines with characteristic impedances Z0i (i ¼ 1. . .N), the last statement implies that all such lines are matched, i.e. are of infinite length or terminated with their own characteristic impedance. In other words, all ports must be terminated with the respective reference impedance Z0i. Such a condition ensures that no wave is reflected back from the load towards the network (thus ai ¼ 0). From (4.81) for j ¼ i we deduce that sii is the reflection coefficient at the ith port when all ports are matched (i.e. when they are terminated with their reference impedance Z0i). The parameters sij are directly related to the power travelling through the network. Indeed, using (4.78) and (4.81) we find that jsij j2 ¼ Pi =Pjþ

ð4:82Þ

The relationship between the S matrix and the Z matrix can be written as follows: ^ ½U Þ ^ þ ½U Þ 1 ð½Z

½S ¼ ð½Z

ð4:83Þ

pffiffiffiffiffiffi pffiffiffiffiffiffi ^ ¼ diag 1= Z0i ½Z diag 1= Z0i ½Z

ð4:84Þ

zij ^zij ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Z0i Z0j

ð4:85Þ

where [U] is the unit matrix and

or, equivalently,

^ as defined by (4.84) or (4.85) is the normalized impedance matrix of the network. It The matrix ½Z

coincides with the impedance matrix when the reference impedances Z0i are unitary. In order to prove (4.83), let us write (4.76) in matrix form: ½V ¼ ½V þ þ ½V

½I ¼ diag½Z0i ð½V þ ½V Þ Equations (4.73)–(4.74) can be also written in matrix form as follows: pffiffiffiffiffiffi ½a ¼ diag½1=ð2 Z0i Þ ½V þ

pffiffiffiffiffiffi ½b ¼ diag½1=ð2 Z0i Þ ½V

ð4:86Þ

ð4:87Þ

MICROWAVE CIRCUITS The above equations can immediately be reversed to get pffiffiffiffiffiffi ½V þ ¼ diag½2 Z0i ½a

pffiffiffiffiffiffi ½V ¼ diag½2 Z0i ½b

115

ð4:88Þ

Inserting (4.86) into (4.55) and reordering, we obtain ½V ¼ ð½Z diag½1=Z0i þ ½U Þ 1 ð½Z diag½1=Z0i ½U Þ½V þ

Using the second equation of (4.87) and the first equation of (4.88), after some algebra we find pffiffiffiffiffiffi pffiffiffiffiffiffi ½b ¼ diag½1= Z0i ð½Z diag½Z0i þ ½U 1 Þð½Z diag½Z0i ½U Þdiag½ Z0i ½a

ð4:89Þ pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ diag½ Z0i ð½Z þ diag½Z0i Þ 1 ð½Z diag½Z0i Þdiag½1= Z0i ½a

Therefore we have pffiffiffiffiffiffi pffiffiffiffiffiffi ½S ¼ diag½ Z0i ð½Z þ diag½Z0i Þ 1 ð½Z diag½Z0i Þdiag½1= Z0i

ð4:90Þ

The previous expression reduces to (4.83) using the position (4.84). Note the similarity of (4.83) to the reflection coefficient of a dipole with normalized impedance Z^ L : _

GL ¼

ZL 1 _ ZL þ 1

In the common case of reciprocal circuits, the impedance matrix is symmetrical and therefore the S matrix is symmetrical too: sij ¼ sji

ð4:91Þ

^ ½U Þð½Z

^ þ ½U Þ 1 ½S ¼ ð½Z

ð4:92Þ

Equation (4.83) can then also be written as

For lossless networks, the scattering matrix satisfies the so-called unitary property ½S T ½S* ¼ ½U

ð4:93Þ

Equation (4.93) is obtained by imposing that the power entering into the network be equal to the power going out of it. Because of (4.78), we must then impose that X X jb j2 ¼ ja j2 ð4:94Þ i i i i The sum of the squared amplitudes can be put in the following form: X ja j2 ¼ ½a T ½a*

i i

ð4:95Þ

Recalling that the transpose of the product of matrices is the product of the transposed matrices in reverse order, we therefore have that, in order for the power to be conserved, ½b T ½b* ¼ ½a T ½S T ½S* ½a* ¼ ½a T ½a*

Since the excitation is totally arbitrary, the last equality holds only if (4.93) holds.

116


Equation (4.93) implies, in particular, that N X i¼1

sin s*in ¼

N X

jsin j2 ¼ 1 for n ¼ 1; 2; . . . ; N

ð4:96Þ

i¼1

N X i¼1

sin s*im ¼ 0

for n 6¼ m

ð4:97Þ

According to (4.96), the sum of the squared amplitudes of any column must be equal to one, while, according to (4.97), the product of any column with the conjugate of another column must be zero. Let us now consider the particular case of a lossless two-port network. The unitary condition (4.93) becomes " #" # " # " # 1 0 s11 s21 js11 j2 þ js21 j2 s11 s*12 þ s21 s*22 s*11 s*12 T * ¼ ¼ ð4:98Þ ½S ½S ¼ 0 1 s12 s22 s*21 s*22 s12 s*11 þ s22 s*21 js12 j2 þ js22 j2 Equating the single elements of the matrix to those of the unit matrix we obtain js11 j2 þ js21 j2 ¼ 1 js12 j2 þ js22 j2 ¼ 1

ð4:99Þ

s11 s*12 þ s21 s*22 ¼ 0 s12 s*11 þ s22 s*21 ¼ 0

ð4:100Þ

If the network is reciprocal, then s12 ¼ s21, so that (4.99) implies js11 j ¼ js22 j; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi js12 j ¼ 1 js11 j2

ð4:101Þ

Therefore, the reflection coefficients at both ports have the same amplitude, but the phases are not necessarily the same, unless the network is symmetrical.4 In addition, if the network is reciprocal, Equations (4.100) become equivalent, since one is the complex conjugate of the other. By writing the S parameters in polar form, i.e. sij ¼ jsij je jjij we obtain from (4.100) s11 s*12 þ s21 s*22 ¼ js11 jjs12 jðe jðj11 j12 Þ þ e jðj12 j22 Þ Þ ¼ 0

ð4:102Þ

having used (4.101). Equation (4.102) is satisfied when the two terms within brackets are opposite, i.e. if the phases differ by odd multiples of p: Thus, j12 ¼

j11 þ j22 p þ þ np 2 2

ð4:103Þ

Apart from multiples of p, the phase of s21 therefore differs by p/2 from the average phase of s11 and s22. In summary, the three complex S parameters of a reciprocal lossless two-port are related by the three real equations (4.101)–(4.102) so that the degrees of freedom of the network are reduced to three. This is

4 One should not confuse the symmetry of the network with that of the S matrix. The former implies that the network looks identical as seen from the first or second port, thus s11 ¼ s22; the latter depends on reciprocity, it implies s12 ¼ s21 and it holds true independently of the symmetry of the network.

MICROWAVE CIRCUITS

I1

117

I2 Yc

V1

Figure 4.18

V2

A two-port network which does not have a Z matrix representation.

consistent with the representation in terms of impedance parameters that, for a lossless reciprocal network, reduce to three reactances X11, X12 ¼ X21, X22. In contrast to the impedance and admittance matrices that may not exist for particular network topologies, the scattering matrix does exist for whatever real network. Let us take as an example the network made by the series element shown in Figure 4.18. The network equations are V1 ¼ V2 ZC I2 I1 ¼ I2

ð4:104Þ

where we have put Zc ¼ 1/Yc. By comparison of (4.104) with (4.68) we find that the ABCD matrix is A B 1 ZC ¼ ð4:105Þ C D 0 1 The network may also be considered as a special case of the P network with YA ¼ YB ¼ 0. From (4.65)–(4.66) we find that the admittance matrix is 1 1 ½Y ¼ YC 1 1 Such a network does not have an impedance matrix since the Y matrix has no inverse (its determinant is zero). This reflects the fact that the impedance parameters are open-circuit parameters. Because of the network topology I1¼ I2: it is therefore impossible to excite one port with a non-zero current while the other port is open-circuited. Or, in other terms, the input impedance at one port is infinite when the other port is open-circuited. In order to compute the S matrix of Figure 4.18, we assume that the network is connected at both ports with transmission lines with Z0 ¼ 1/Y0 characteristic impedance. This will be our reference impedance at both ports. We can use the dual expression of (4.83) to express [S] in terms of [Y]: ½S ¼ ð½U þ ½Y^ Þ 1 ð½U ½Y^ Þ " # 1" # " # 1 1 þ Y^ C Y^ C 1 Y^ C Y^ C 1 2Y^ C ¼ ¼ ... ¼ 1 þ 2Y^ C 2Y^ C 1 Y^ C 1 Y^ C Y^ C 1 þ Y^ C

ð4:106Þ

^ C , and the S parameters become where Y^ C ¼ YC =Y0 . If the network is lossless, Y^ C ¼ j B 1 ^C 1 þ j2B ^C j2B ¼ ^C 1 þ j2B

s11 ¼ s22 ¼ s12 ¼ s21

ð4:107Þ

118


The reader can easily verify that the lossless conditions (4.101)–(4.102) are satisfied. Note that (4.107) can also be calculated without resorting to the admittance matrix, but using the scattering parameter definition. With both ports terminated on transmission lines with Z0 characteristic impedance, we consider a wave a1 incident from line 1 to port 1, while the second line is matched (a2 ¼ 0). In such conditions, the input reflection coefficient is s11 and is given by s11 ¼

ðZC þ Z0 Þ Z0 ZC 1 1 ¼ ¼ ¼ ðZC þ Z0 Þ þ Z0 ZC þ 2Z0 1 þ 2Z0 =ZC 1 þ 2Y^ C

ð4:108Þ

To compute s21, since a2 ¼ 0, from the second equation of (4.75) we have I1 ¼ a1 b1 ¼ a1 ð1 s11 Þ I2 ¼ b2 where, for simplicity but without loss of generality, we have put Z0 ¼ 1. Because I2 ¼ I1 from previous equations, we obtain I2 b2 s21 ¼ ¼ ¼1 I1 a1 ð1 s11 Þ 1 s11 thus s21 ¼ 1 s11 ¼

2Y^ C 1 þ 2Y^ C

ð4:109Þ

This is the same expression we already found. Let us now consider the dual case of the network made of shunt impedance ZC, as shown in Figure 4.19. The relevant network equations are V1 ¼ V2 I1 ¼ YC V2 I2

ð4:110Þ

From the above, the ABCD matrix is immediately obtained: A B 1 0 ¼ C D YC 1

ð4:111Þ

The impedance matrix is obtained from (4.110) or from (4.58)–(4.59) by putting ZA ¼ ZB ¼ 0: 1 1 ð4:112Þ ½Z ¼ ZC 1 1 while an admittance matrix cannot be defined. Indeed, since V1 ¼ V2 one port cannot be set to a non-zero voltage when the other is short-circuited. In other words, the input admittance at one port is infinite when the other port is short-circuited.

I1

V1

Figure 4.19

I2

Zc

V2

A two-port network which does not have a Y matrix representation.

MICROWAVE CIRCUITS

119

Let us compute the scattering parameters using Z0 as the reference impedance at both ports. Consider a wave a1 incident to port 1, while the second line is matched (a2 ¼ 0); the reflection coefficient at port 1 is s11 ¼

ZC jjZ0 Z0 Y0 ðY0 þ YC Þ Y^ C 1 ¼ ¼ ¼ ZC jjZ0 þ Z0 Y0 þ Y0 þ YC 2 þ Y^ C 2Z^ C þ 1

ð4:113Þ

Since V1 ¼ a1 þ b1 ¼ a1 ð1 þ s11 Þ V2 ¼ b2 we obtain from the equality of the voltages V2 b2 s21 ¼ ¼1 ¼ V1 a1 ð1 þ s11 Þ 1 þ s11 Therefore s21 ¼ 1 þ s11 ¼

2Z^ C 1 þ 2Z^ C

ð4:114Þ

The same results could obviously be obtained by inserting (4.112) in (4.83). If the network is lossless, then ZC ¼ jXC. It can be easily verified that (4.101)–(4.102) are satisfied. Table 4.1 provides the relationships among the Z, Y, ABCD and S parameters of a two-port network.

4.7 Circuit model of a transmission line section The matrix representations introduced in Section 4.6 can be applied to represent a section of transmission line as a two-port network. With reference to Figure 4.20, let us consider a transmission line section of characteristic impedance Zc located between z ¼ 0 and z ¼ l, with electrical length y ¼ bl. From the transmission line equations VðzÞ ¼ V þ e jbz þ V e þ jbz 1 IðzÞ ¼ ðV þ e jbz V e þ jbz Þ Zc

ð4:115Þ

expressing the voltages and currents at the two ports as the voltages and currents along the transmission line at the electrical lengths 0 and y, i.e. putting V1 ¼ Vð0Þ V2 ¼ VðyÞ

I1 ¼ Ið0Þ I2 ¼ IðyÞ

ð4:116Þ

we obtain V1 ¼ V þ þ V Vþ V I1 ¼ Zc V2 ¼ V þ e jy þ V e jy

1 þ jy I2 ¼ V e V e þ jy Zc

ð4:117Þ

The first two equations can be used to express the quantities V þ and V in terms of V1 and I1: 1 V ¼ ðV1 Zc I1 Þ 2

ð4:118Þ

y12 y21 y22

z22

z22 jZj

z12 jZj

z21 jZj

z11 jZj

z11 z21

jZj z21

1 z21

z22 z21

z22

y11

y12

y21

y22

A

B

C

D

D

C

jYj y21 y11 y21

B

A

A B

1 B

BD AC B

B D

1 y21

y22 y21

y11

z21

z21 D C

y11 jYj

z12

z12

A C AD BC C 1 C

y22 jYj y12 jYj y21 jYj

z11

[ABCD]

[Y]

z11

[Z]

Table 4.1 Relations among the Z, Y, ABCD and S parameters of a two-port network

ð1 þ s11 Þð1 s22 Þ þ s12 s21 ð1 þ s11 Þð1 þ s22 Þ s12 s21

2s21 ð1 þ s11 Þð1 þ s22 Þ s12 s21

s12 ð1 þ s11 Þð1 þ s22 Þ s12 s21

ð1 s11 Þð1 þ s22 Þ þ s12 s21 ð1 þ s11 Þð1 þ s22 Þ s12 s21

ð1 s11 Þð1 þ s22 Þ þ s12 s21 ð1 s11 Þð1 s22 Þ s12 s21

ð1 s11 Þð1 s22 Þ s12 s21 2s21

ð1 þ s11 Þð1 þ s22 Þ s12 s21 2s21

ð1 s11 Þð1 þ s22 Þ þ s12 s21 2s21

Z0

Z0

ð1 þ s11 Þð1 s22 Þ þ s12 s21 2s21

Y0

Y0

Y0

Y0

Z0

ð1 þ s11 Þð1 s22 Þ þ s12 s21 ð1 s11 Þð1 s22 Þ s12 s21 2s12 Z0 ð1 s11 Þð1 s22 Þ s12 s21 2s21 Z0 ð1 s11 Þð1 s22 Þ s12 s21 Z0

[S]

120 MICROWAVE AND RF ENGINEERING

2y12 Y0 DY 2y21 Y0 DY ðY0 þ y11 ÞðY0 y22 Þ þ y12 y21 DY

2z12 Z0 DZ

2z21 Z0 DZ

ðz11 þ Z0 Þðz22 Z0 Þ z12 z21 DZ

s12

s21

s22

A þ B=Z0 CZ0 D A þ B=Z0 þ CZ0 þ D

2 A þ B=Z0 þ CZ0 þ D

2ðAD BCÞ A þ B=Z0 þ CZ0 þ D

A þ B=Z0 CZ0 D A þ B=Z0 þ CZ0 þ D

DZ ¼ ðz11 þ Z0 Þðz22 þ Z0 Þ z12 z21 ; DY ¼ ðy11 þ Y0 Þðy22 þ Y0 Þ y12 y21 ; Y0 ¼ 1=Z0

jZj ¼ z11 z22 z12 z21 ; jYj ¼ y11 y22 y12 y21 ;

ðy0 y11 ÞðY0 þ y22 Þ þ y12 y21 DY

ðz11 Z0 Þðz22 þ Z0 Þ z12 z21 DZ

s11

s22

s21

s12

s11

MICROWAVE CIRCUITS 121

122


Figure 4.20

A transmission line section as a two-port network.

Substituting into the remaining equations and reordering, we obtain V1 ¼ jZc cotyI1 j V2 ¼ j

Zc I2 sin y

ð4:119Þ

Zc I1 jZc cot yI2 sin y

The impedance matrix of the line sections is therefore " # 1 jZc cos y coty ðsinyÞ 1 ¼ ½Z ¼ jZc sin y 1 cos y ðsinyÞ 1 coty

ð4:120Þ

Notice that, consistent with the definition of the impedance matrix, Z11 ¼ Z22 is nothing but the input impedance of an open-circuited stub. Reordering the equations so as to express the currents in terms of the voltages, we obtain the admittance matrix of the line section: " # jYc cos y 1 coty ðsinyÞ 1 ð4:121Þ ½Y ¼ jYc ¼ siny 1 cos y ðsinyÞ 1 coty where Yc ¼ 1/Zc. Note that y11 ¼ y22 is the input admittance of a short-circuited stub. Finally, for the ABCD matrix we obtain from simple manipulations A B cos y jZc siny ¼ ð4:122Þ C D cos y jYc sin y To compute the scattering matrix, let us choose the same reference impedance Z0 at both ports. Using (4.83), " ½S ¼ 1 ¼ D

"

j Z^ c cot y þ 1 j Z^ c ðsin yÞ 1

j Z^ c ðsin yÞ 1 j Z^ c cot y þ 1

#! 1 "

2 Z^ c 1

2j Z^ c ðsin yÞ 1

2j Z^ c ðsin yÞ 1

2 Z^ c 1

#

j Z^ c cot y 1 j Z^ c ðsin yÞ 1

j Z^ c ðsin yÞ 1 j Z^ c cot y 1

#! ð4:123Þ

where Z^ c ¼ Zc =Z0 D ¼ 1 2j Z^ c cot y þ Z^ c

2

ð4:124Þ

MICROWAVE CIRCUITS

123

If the reference impedance Z0 is the same as the characteristic impedance Zc of the line, then Z^ c ¼ 1. From simple algebra (4.123) reduces to " # 0 e jy ½S ¼ ð4:125Þ e jy 0 As might be expected, the reflection coefficients s11 ¼ s22 are zero, while the transmission coefficients s21 ¼ s12 consist of just a phase shift equal to the electrical length of the line. It should be remembered that this result holds true only when Zc ¼ Z0.

4.8 Shifting the reference planes In deriving the circuit model of an N-port microwave circuit, on each feeding line we have introduced a reference plane where we evaluate voltages and currents relative to that port. The location of such reference planes is to a large extent arbitrary, as long as they are sufficiently far from the microwave circuit that all higher modes have died out. In many practical cases it is useful to shift the reference planes, e.g. in order to obtain simpler circuit representations. In this section we want to determine how the shift of the reference planes affects the scattering matrix of a two-port network. The extension to the general case of N-port networks is straightforward. Let us refer to Figure 4.21, where a two-port network is characterized in terms of the scattering matrix [S] with respect to the reference planes T1, T2. Shifting such planes to T01 ; T02 is equivalent to adding two line sections of electrical lengths y1, y2, respectively. By simple inspection of Figure 4.21 we can write a1 ¼ a01 e jy1 a2 ¼ a02 e jy2 thus

Finally,

b0 1 ¼ b1 e jy1 ¼ ðs11 a1 þ s12 a2 Þe jy1 ¼ s11 a01 e jy1 þ s12 a02 e jy2 e jy1

b0 2 ¼ b2 e jy2 ¼ ðs21 a1 þ s22 a2 Þe jy2 ¼ s21 a01 e jy1 þ s22 a02 e jy2 e jy2 "

# " # # " s11 s12 e jy1 0 e jy1 0 s11 e j2y1 s12 e jðy1 þ y2 Þ ½S ¼ ¼ 0 e jy2 s21 s22 0 e jy2 s21 e jðy1 þ y2 Þ s22 e j2y2 jy jy ¼ diag e i ½S diag e i 0

T′1

T1

θ1

T2

θ2

a′1

ð4:126Þ

T′2 a′2

Z0

[S]

Z0 b′2

b′1 a1

a2

b1

b2

Figure 4.21

Shifting the reference planes.

124


More explicitly, s0ij ¼ sij e jðyi þ yj Þ

ð4:127Þ

A much harder effort would be required to obtain the corresponding relation between the impedance or admittance matrix.

4.9 Loaded two-port network We want here to determine the behaviour of a two-port network at port 1 when port 2 is terminated with a load ZL, as sketched in Figure 4.22. The latter condition is expressed by V2 ¼ ZL I2

ð4:128Þ

Combining (4.128) with the network description in terms of the impedance matrix V1 ¼ z11 I1 þ z12 I2 V2 ¼ z21 I1 þ z22 I2 we easily obtain the input impedance at port 1: Zin ¼

V1 z12 z21 ¼ z11 I1 z22 þ ZL

ð4:129Þ

Note that Zin reduces to z11 when the load is an open circuit (ZL ¼ 1). In much the same way the input admittance is found to be Yin ¼ y11

y12 y21 y22 þ YL

ð4:130Þ

which coincides with y11 when the load is a short circuit (YL ¼ 1). If the load is characterized by its reflection coefficient GL with respect to a given reference Z0, we have a2 ¼ GL b2

ð4:131Þ

Utilizing the scattering matrix (using the same reference impedance), we obtain with a similar procedure the input reflection coefficient Gin ¼ s11

s12 s21 s12 s21 GL ¼ s11 þ s22 1=GL 1 s22 GL

ð4:132Þ

Two port network

Γin

1

[Z]

2

[Y] [S]

Zin Yin Figure 4.22

A terminated two-port network.

ZL

MICROWAVE CIRCUITS

125

4.10 Matrix description of coupled lines In Section 3.16.2 we derived the voltages and currents along a set of coupled lines in terms of their even and odd modes. In this section we will derive the Z matrix description of a length of coupled lines. We will also provide an expression for the scattering matrix, without any proof, since the derivation procedure is the same as for the Z matrix. Let us therefore consider a length l of TEM coupled transmission lines. For simplicity of notation, rather than physical length l we will refer to the electrical length y¼o

l v

ð4:133Þ

representing the phase shift of the signal through the distance l. For example, when the lines are l/4 long, the electrical length is y ¼ p/2. Figure 4.23 shows the schematic for a four-port network consisting of a section of coupled lines with characteristic impedances Z0e, Z0o and electrical length y. For simplicity, the ground terminals on the l.h.s. and r.h.s. are not indicated. Line 1 connects terminals 1 and 2, while line 2 connects terminals 3 and 4. As mentioned above, we can express the electrical quantities at the four ports as a linear superposition of the even and odd mode quantities. Taking into account that the even mode corresponds to equal quantities on both lines, while the odd mode corresponds to opposite quantities, we obtain V1 ¼ V1e þ V1o

I1 ¼ I1e þ I1o

V2 ¼ V2e þ V2o V3 ¼ V1e V1o V4 ¼ V2e V2o

I2 ¼ I2e þ I2o I3 ¼ I1e I1o I4 ¼ I2e I2o

ð4:134Þ

As we have seen, each mode can be modelled as a transmission line. Rather than manipulating (4.134), the Z matrix can be easily obtained by applying expression (4.120) for the impedance matrix of a line section to relate the voltages and currents of the even and odd modes:

V1e;o V1e;o

¼ jZ0e;o

1 cosy sin y 1

1 cosy

I1e;o I1e;o

ð4:135Þ

where for brevity we have used subscripts ‘e’ or ‘ o’ to refer to the even or odd mode. Substituting (4.135) into (4.134), we obtain the relationships between the voltages and currents at the four ports of the network:

3

4

1

2

Figure 4.23 Four-port network made of a section of coupled lines.

126

MICROWAVE AND RF ENGINEERING 3 2 3 V1 I1 6V 7 6I 7 6 27 6 27 6 7 ¼ ½Z 44 6 7 4 V3 5 4 I3 5 V4 I4 2

ð4:136Þ

where the elements of the matrix [Z] are Z0e þ Z0o cot y 2 Z0e þ Z0o ðsinyÞ 1 ¼ j 2 Z0e Z0o ¼ j cot y 2 Z0e Z0o ¼ j ðsinyÞ 1 2

z11 ¼ z22 ¼ z33 ¼ z44 ¼ j z12 ¼ z21 ¼ z34 ¼ z43 z13 ¼ z31 ¼ z24 ¼ z42 z14 ¼ z41 ¼ z32 ¼ z23

ð4:137Þ

As can be seen, because of the symmetry and reciprocity of the structure, the network is characterized by only four independent parameters. The same procedure can also be applied to compute other matrix representations, such as the Y matrix or the scattering matrix S. In the former case, instead of (4.137) by duality, one obtains Y0e þ Y0o cot y 2 Y0e þ Y0o ¼ j ðsinyÞ 1 2 Y0e Y0o cot y ¼j 2 Y0e Y0o ðsinyÞ 1 ¼j 2

y11 ¼ y22 ¼ y33 ¼ y44 ¼ j y12 ¼ y21 ¼ y34 ¼ y43 y13 ¼ y31 ¼ y24 ¼ y42 y14 ¼ y41 ¼ y32 ¼ y23

ð4:138Þ

where Y0e ¼ 1=Z0e , Y0o ¼ 1=Z0o . The scattering matrix elements are found to be s11 s13

! 2 2 1 Z^ 0e 1 Z^ 0o 1 ¼ þ 2 De Do ! 2 2 1 Z^ 0e 1 Z^ 0o 1 ¼ 2 De Do

s12 ¼ s14

j Z^ 0e Z^ 0o þ Do siny De

j Z^ 0e Z^ 0o ¼ siny De Do

ð4:139Þ

where De;o ¼ 1 2j Z^ 0e;o cot y þ Z^ 0e;o . The other elements of the matrix are simply obtained by permutations of the indexes, just like the Z matrix. In (4.139) the impedances are normalized with respect to the reference impedance Z0. 2

4.11 Matching of coupled lines As already observed, a set of coupled lines does not possess a unique characteristic impedance. We are therefore interested in investigating whether and how the four-port network constituted by a section of coupled lines can be matched to an external reference impedance Z0. To answer this question it is

MICROWAVE CIRCUITS

127

sufficient to consider the scattering parameters (4.139) of the network and to impose the matching conditions at all ports; that is, the reflection coefficient s11 is zero (the reflection coefficients at the other ports are equal to s11). After some algebra, the condition s11 ¼ 0 in the first equation of (4.139) yields Z^ 0e Z^ 0o ¼ 1

ð4:140Þ

Z0e Z0o ¼ Z02

ð4:141Þ

or, using denormalized impedances,

The above equations imply that the coupled-line section is matched at any frequency to an impedance equal to the geometric mean of the even and odd impedances. Let us assume that the coupled-line length is matched. Inserting (4.141) into (4.139), we obtain s11 ¼ s22 ¼ s33 ¼ s44 ¼ 0

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C2 s12 ¼ s21 ¼ s34 ¼ s43 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C2 cos y þ j sin y jC sin y s13 ¼ s31 ¼ s24 ¼ s42 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C2 cos y þ j sin y s14 ¼ s41 ¼ s32 ¼ s23 ¼ 0

ð4:142Þ

where C¼

Z0e Z0o Z0e =Z0o 1 ¼ Z0e þ Z0o Z0e =Z0o þ 1

ð4:143Þ

is the coupling coefficient. When y ¼ p/2, i.e. at the frequency when the coupled-line section is l/4 long, C is equal to the magnitude of s31, whose square is proportional to the power transfer from the input port 1 to the coupled port 3. Notice that when the matching condition is verified, ports 1 and 4 are mutually isolated; similarly, ports 2 and 3 are isolated as well. The power entering into port 1 is thus split between ports 2 and 3, while port 4 is uncoupled. This is the behaviour of a directional coupler, a device discussed in Section 7.7.3. Ports 2, 3 and 4 are called the direct, coupled and isolated ports, respectively. The behaviour of the scattering parameters is periodic with period y ¼ 2p. Considering their magnitude, the period is p. The magnitudes of the transmission parameters s21 and s31 of the direct and coupled branches are shown in Figure 7.33. Notice that the maximum coupling occurs at y ¼ p/2 (thus the length is l/4), for which |s31| ¼ C.

4.12 Two-port networks using coupled-line sections Sections of coupled transmission lines can also be employed as two-port networks by short- or opencircuiting two out of four ports in Figure 4.23 and using the remaining ports as the input/output ports. Let us first observe that the four-port network of Figure 4.23 is equivalent to a network made of three transmission lines, as shown in Figure 4.24. The transmission lines A and C have characteristic impedance ZA ¼ ZC ¼ Z0e ¼ ðY0e Þ 1 , while the central line has characteristic impedance ZB ¼ ðYB Þ 1 ¼

Y0o Y0e 1 2

128


1

1

3

2

3

1

3

4

V1A

I1A

+ V1B I1B –

4

3

V1C

I2A

ZA

I2B

ZB

I1C

I2C

ZC

V2A

2

+ V2B – 4

V2C

ZA = ZC = Z0e –1 –1 ZB= 2 (Y0o –Y0e)–1=2/(Z0o – Z0e )

Figure 4.24

Equivalent circuit (right) of a pair of coupled lines (left).

The equivalence is readily proved by computing the admittance matrix of the three-line network of Figure 4.24. Because of the symmetry of the structure, it suffices to compute the first row of the admittance matrix, i.e. y1j ; j ¼ 1 . . . 4, the remaining elements being obtained by simple permutation of the indexes. According to (4.121) and with the notation of Figure 4.24 we can write

(Z0e + Z0o)/2

1

3

4

2

1

2

open

1

2 (a) 2 Z0e Z0o (Z0e + Z0o) (Z0e – Z0o)2

Z0o

1

3

4

open

1

2

1

3

4

1

2

open

Z0o

1

2

2 (b)

(Z0e – Z0o)/2

–1

–1

2/(Z0e – Z0e )

1 2 (c)

2 Z0e

Z0e

–1

–1

2/(Z0o – Z0e )

1

3

4

2

1

2

1

2 (d) Z0e

Figure 4.25

Z0e

Circuits using coupled lines.

MICROWAVE CIRCUITS A I1A A V1 ½

¼ Y IA VA 2B 2B V I1 ¼ ½Y B 1B I2B V C 2C V I1 ¼ ½Y C 1C I2C V2

129

where

# coty ðsin yÞ 1 ½Y ¼ ½Y ¼ jY0e ðsinyÞ 1 cot y " # Y0o Y0e cot y ðsin yÞ 1 B ½Y ¼ j 2 ðsinyÞ 1 cot y

ð4:144Þ

"

A

C

With reference to Figure 4.24, we find that the input current at port 1 is given by

I1 ¼ I1A þ I1B ¼ yA11 V1A þ yA12 V2A þ yB11 V1B þ yB12 V2B

¼ yA11 V1 þ yA12 V2 þ yB11 ðV1 V3 Þ þ yB12 V2 yB12

¼ yA11 þ yB11 V1 þ yA12 þ yB12 V2 yB11 V3 yB12 V4

ð4:145Þ

ð4:146Þ

In conclusion, from (4.144)–(4.146) we find Y0e þ Y0o cot y 2 Y0e þ Y0o ðsin yÞ 1 y12 ¼ yA12 þ yB12 ¼ j 2 ð4:147Þ Y0e Y0o coty y13 ¼ yB11 ¼ j 2 Y0e Y0o B y14 ¼ y12 ¼ j ðsinyÞ 1 2 which are clearly coincident with (4.138). Of special interest are the cases illustrated in Figure 4.25, where coupled-line sections are used as two-port networks, the remaining two ports being open- or short-circuited depending on the case. Such circuits are used in the realization of bandstop and bandpass filters (see Sections 8.7.2 and 8.7.3). The equivalences illustrated in the figure can easily be proved after some algebra by computing the matrix representation of each pair of equivalent networks. Other equivalences involving coupled line two-port networks can be found, for example, in [5, 6]. y11 ¼ yA11 þ yB11 ¼ j

Bibliography 1. N. Marcuvitz, Waveguide Handbook, McGraw-Hill, New York, 1951. 2. C. G. Montgomery, R. H. Dicke and E. M. Purcell, Principles of Microwave Circuits, McGraw-Hill, New York, 1948. 3. J. C.Slater, Microwave Electronics, Van Nostrand, Princeton, NJ, 1950. 4. K. Kurokawa, An Introduction to the Theory of Microwave Circuits, Academic Press, New York, 1969. 5. G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1964. 6. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007.

5

Resonators and cavities 5.1 Introduction Due to their property of sustaining, and thus stabilizing, EM field oscillations at given frequencies, resonators play a special role in electronic techniques at both low and high frequencies. They are employed in a large variety of circuits and devices, including filters, oscillators, amplifiers and frequency meters. Obviously, depending on the frequency band of the application, different technologies and configurations are employed to fabricate resonators, but the basic features are common to all of them. After introducing the general features of resonators in this chapter, we examine lumped element, then one-dimensional (or transmission line) resonators, two-dimensional (or planar) resonators and finally three-dimensional or cavity resonators. The last case is of special interest from the theoretical point of view, since it offers the opportunity to present a very general and powerful theory concerning EM expansion in resonant modes.

5.2 The resonant condition A generic one-port circuit like that depicted in Figure 5.1 can exchange energy with the outer space through a terminal pair or through the cross-section of a generalized transmission line (as we saw in Chapter 4, this can also be represented as a terminal pair) depending on whether we adopt the point of view of circuit theory or the microwave network. We can therefore enclose the circuit within a region V such that the EM energy exchange occurs only through the terminal pair (or the transmission line crosssection), while the EM field is zero on the remaining part of the surface. In the absence of external sources, non-zero voltages or currents can occur in a passive circuit only at specific frequencies, called resonant frequencies, of the circuit. The condition for such free oscillations to occur is the resonant condition. It is apparent that only an ideal lossless circuit can sustain such oscillations indefinitely, while a lossy circuit can only sustain damped oscillations. A resonant circuit, or resonator, usually exhibits low loss so that its oscillations are maintained over a prolonged period of time that includes several oscillation cycles. In the ideal case of no losses, free oscillations of a two-terminal network occur at real frequencies such that either the input impedance Zin or admittance Yin is zero. In the former case, in fact, a non-zero


132


One port circuit

Figure 5.1

V

A generic one-port circuit.

current may flow in the network, although with zero input voltage (V ¼ ZinI ¼ 0); in the latter case, conversely, a non-zero voltage may be present at the input port without any current flow (I ¼ YinV ¼ 0). For a lossless network, such conditions therefore imply that the input reactance or susceptance, respectively, is zero: Xin ¼ 0

ð5:1aÞ

Bin ¼ 0

ð5:1bÞ

or

These are the resonant conditions for a lossless one-port network. In practice, since for a lossless circuit Xin ¼ 1=Bin the resonant frequencies correspond to the zeros and poles of the input reactance or susceptance. As illustrated below, the two conditions above correspond to series and parallel resonances, respectively. When losses are present in the circuit, the resonant conditions (5.1) still occur, but in this case, as we will see below, at resonance the resonant circuit exhibits a pure real input impedance or reactance. The resonant condition can be related to the EM energy stored in the circuit by making use of Poynting’s theorem (2.71). When applied to the volume V, the theorem can be put in the form Pin ¼ Pd þ 2joðWm We Þ

ð5:2Þ

where Pin is the complex power entering into the network, Pd is the real power dissipated by the Joule effect, Wm and We are the magnetic and electric energies, respectively, stored in V in a period T ¼ 2p/o. Expressing the input power in terms of voltage and current, Pin ¼ VI * =2, we obtain from (5.2) the energybased definitions of such quantities: Zin;par ¼

Pd þ 2joðWm We Þ II * =2

ð5:3Þ

Yin;ser ¼

Pd 2joðWm We Þ VV * =2

ð5:4Þ

The above expressions show that the input impedance and admittance are real when Wm ¼ We

ð5:5Þ

RESONATORS AND CAVITIES

133

Equation (5.5) can be considered as the network resonant condition: when the average magnetic energy equals the average electric energy, the input impedance and admittance become real. If the network is lossless, thus Pd ¼ 0, the resonant condition implies zero input impedance or admittance, as stated above in (5.1). In such conditions, the network can sustain oscillations indefinitely without an external source. For this reason, the resonant condition can be seen as the condition such that the network, in the absence of loss, can oscillate freely. The resonant condition therefore corresponds to a free oscillation of the network. As already mentioned, when the input impedance (or, in the presence of losses, the reactance) is zero we will speak of series resonance, and when the input admittance (or the susceptance) is zero we will speak of a parallel resonance or anti-resonance. To illustrate this point, let us consider for example the parallel and series circuits of Figures 5.2a,b. The average magnetic energy stored in an inductor carrying the current IL and the average electric energy in a capacitor subject to the voltage VC are 1 Wm ¼ LjIL j2 4 1 We ¼ CjVC j2 4

ð5:6Þ

respectively. For the parallel circuit of Figure 5.2, both the magnetic and electric energies can be expressed in terms of the common voltage V: 1 1 jV j2 1 jV j2 ¼ Wm ¼ LjIL j2 ¼ L 2 4 4 ðoLÞ 4 o2 L 1 We ¼ C j V j 2 4

ð5:7Þ

Equating and solving for o, we obtain the resonant radian frequency 1 o0 ¼ pffiffiffiffiffiffi LC

ð5:8Þ

In a dual fashion, for the series resonant circuit of Figure 5.2b we express the energies in terms of the common current I: 1 Wm ¼ LjI j2 4 1 1 jI j2 W e ¼ C j VC j 2 ¼ 4 4 o2 C

ð5:9Þ

Equating both terms, we obtain for the resonant frequency the same expression (5.8) as for the parallel resonant circuit.

Figure 5.2

Parallel (a) and series (b) resonant circuits.

134


5.3 Quality factor or Q In the lossless case, since the input impedance or admittance is zero, the resonator is able to sustain free oscillations at the resonant frequency (or frequencies). The unavoidable presence of dissipative elements in the circuit is such that the free oscillations are damped in time. The quality of a resonator is measured by a figure of merit called the quality factor or Q factor, or simply the Q of the resonator defined by the following general expression: Q¼

o0 W Pd

ð5:10Þ

where o0 is the radian resonant frequency, W ¼ Wm þ We is the total average energy stored in the resonator in one period T ¼ 2p/o0, and Pd is the average power dissipated in the resonator in one period. For the parallel and series circuits of Figure 5.2 we have Qpar ¼

2 o0 W o0 2We o0 2 14 CjV j o0 C ¼ ¼ 1 ¼ 2 Pd G Pd 2 GjV j

ð5:11Þ

Qser ¼

2 o0 W o0 2Wm o0 2 14 LjI j o0 L ¼ ¼ 1 ¼ 2 Pd Pd R R I j j 2

ð5:12Þ

An ideal resonator with no loss (G ¼ 0, or R ¼ 0) therefore has an infinite Q factor. In deriving (5.11)–(5.12) we have taken into account the equality of the electric and magnetic energies at resonance, so that W ¼ 2Wm ¼ 2We. Note that in both cases Q is the ratio of the imaginary and real parts of the admittance or impedance, respectively. This result can be generalized to the case of non-resonant oneport circuits. Let us now consider the behaviour of the input impedance of a parallel resonator in the proximity of the resonant frequency. In the lossless case (G ¼ 0) the input impedance goes to infinity producing an open circuit. In the general lossy case when G 6¼ 0, from inspection of Figure 5.2a we can write 1 1 1 1 o2 joC o2 Zin;par ¼ G þ joC þ ¼ G þ joC 1 02 ¼ R 1þ ð5:13Þ 1 02 joL G o o where R ¼ 1/G. At frequencies close to o0, we may put o ¼ o0 þ Do, with jDo=oj 1, so that in (5.13) the following approximation holds: o2 oDo 2 2Do ffi 1 02 ¼ 1 ð5:14Þ o o o neglecting the second power of Do=o. Using (5.11) and (5.14), then (5.13) becomes 2Do 1 Zin;par ffi R 1 þ jQ o0

ð5:15Þ

The magnitude of the impedance attains its maximum 1/G ¼ R at the resonant frequency. In much the same way it can be seen that at the resonant frequency the magnitude of the input admittance of the series resonator is maximum, 1/R ¼ G. Figure 5.3 shows the behaviour of the magnitude and the imaginary part of (5.15) as functions of Do/o0. The magnitude has a bell-shaped pffiffiffibehaviour. The 3.01 dB fractional bandwidth is the interval between the points where jZin j ¼ R= 2; from (5.15) it can be seen that such points occur when Q2Do=o0 ¼ 1. The fractional bandwidth is therefore B¼

2Do 1 ¼ o0 Q

ð5:16Þ


135

1 0.9 0.8 0.7

1/Q

Zin 0.6 R

0.5 0.4 0.3 0.2 0.1

(a)

–0.5 –0.4 –0.3 –0.2 –0.1

0

0.1

0.2 0.3

0.4

0.5

0.2 0.3

0.4

0.5

∆ω /ω0 80

Phase of Zin (deg)

60 40 20 0 –20 –40 –60 –80

(b)

–0.5 –0.4 –0.3 –0.2 –0.1

0

0.1

∆ω /ω0

Figure 5.3

Input impedance of a parallel resonator.

The quality factor can thus be evaluated as the reciprocal of the fractional bandwidth of the resonator. In a dual fashion we obtain the admittance of the series resonator in the proximity of the resonant frequency: 2Do 1 Yin;ser ffi G 1 þ jQ ð5:17Þ o0 while Equation (5.16) still holds. The quality factor is also related to the damping of the free oscillations of the circuit. Indeed, the amplitude of the field oscillations decreases in time as ext, x being the damping factor. The instantaneous energy stored in the circuit will thus decay as WðtÞ ¼ W0 ej2xt where W0 is the energy stored at t ¼ 0. Because of energy conservation, the rate of energy decrease in time dW=dt must be equal to the power loss: Pd ¼

dW ¼ 2xW0 ej2xt ¼ 2xWðtÞ dt

ð5:18Þ

The damping factor can therefore be expressed as x¼

Pd o0 ¼ 2W 2Q

ð5:19Þ

136


using (5.10).1 This equation shows that the higher the value of Q, the slower the damping of the oscillations. The lossy elements in the parallel and series resonators of Figure 5.2 can represent either the unwanted power loss Pd within the circuit or the power loss Pe due to an external load coupled to the resonator. In order to distinguish between the two effects, we define an external quality factor Qe due to the external load only and a loaded quality factor QL resulting from both types of loss. Based on the definition (5.10), we thus have QL ¼

o0 W 1 1 ¼ ¼ Pd þ Pe Pd =o0 W þ Pe =o0 W 1=Q þ 1=Qe

ð5:20Þ

where Qe ¼

o0 W Pe

Q¼

o0 W Pd

is the external Q and

is the resonator’s intrinsic or unloaded Q resulting only from the internal power loss. As an example, consider the parallel resonator and assume there is an external load represented by a resistance Re parallel to the internal resistance R ¼ 1/G. The loaded Q is o0 C 1 1 1 ¼ ð5:21Þ þ QL ¼ 1=R þ 1=Re Q Qe with Q ¼ o0 RC and Qe ¼ o0 Re C. In the most general case when various loss sources are present in the circuit (including, for example, radiation loss), each one absorbing a power Pdi (i ¼ 1, 2, . . ., N), the resulting loaded Q is given by !1 N X o0 W 1 1 ¼ N ð5:22Þ ¼ QL ¼ N X X Qi i¼1 Pdi Pdi =ðo0 WÞ i¼1

i¼1

where Qi is the Q associated with the ith loss source.

5.4 Transmission line resonators At radio and microwave frequencies electronic circuits are normally distributed and so are resonant circuits. The latter can be realized with transmission line sections like those shown in Figure 5.4. In contrast with lumped elements that resonate at a finite number of frequencies, distributed resonators have an infinite number of resonances as they are described by trigonometric, and thus periodic, rather than rational functions of frequency. Let us first consider the lossless case. As discussed in the previous sections, when no losses are present the resonant frequencies of the distributed circuits of Figure 5.4 are obtained when the input impedance or admittance is zero. Recalling the input impedance of a short-circuited stub (3.85a), we thus have for Figure 5.4a: Zin ¼ jZ0 tanðblÞ

ð5:23Þ

1 Strictly speaking, (5.10) refers to average values while (5.18) refers to instantaneous values. The equality (5.19) can be assumed to be valid provided that the damping factor is small with respect to the period T of the oscillations.


Z0, α, β

Z0, α, β Zin

Zin

(a)

Figure 5.4

137

(b)

Resonant circuits realized with a short- (a) or open-circuited (b) stub.

where b is the phase constant of the line and Z0 its characteristic impedance. Series resonances are obtained by setting Zin ¼ 0, which implies bl ¼ np

n ¼ 0; 1; 2 . . .

ð5:24Þ

Recalling the relation (3.75) between radian frequency, phase constant and phase velocity vph , the series resonant frequencies are npvph ð5:25Þ on ¼ l Similarly, there are infinite parallel resonances when Yin ¼ 0, thus Zin ¼ 1: bl ¼ ð2n þ 1Þp=2

n ¼ 0; 1; 2 . . .

ð5:26Þ

Thus at the radian frequencies on ¼

ð2n þ 1Þpvph l

ð5:27Þ

The resonant conditions can also be expressed in terms of line lengths. Recalling that b ¼ 2p=l, l being the wavelength, it can be immediately seen that series resonances occur when the stub length l is an integer multiple of l/2, while parallel resonances are obtained when l is an odd multiple of l/4. This is easily understood by recalling the properties of half- and quarter-wave line sections. When the line is half a wavelength long, the input impedance is the same as the load impedance, thus a short circuit is seen at the input. For the quarter-wavelength line, instead, the input impedance is the reciprocal of the load impedance, thus an open circuit is seen at the input. For the open stub of Figure 5.4b, we have a dual condition so that parallel resonances are obtained when the line is a multiple of half wavelengths, while series resonances occur when the length is an odd multiple of l/4. In fact the input impedance is Zin ¼ jZ0 cotðblÞ so that the condition (5.24) corresponds to parallel resonances while (5.26) corresponds to series resonances. Let us now consider a more realistic case, assuming a non-zero per-unit-length resistance R of the line conductor, while the dielectric loss can still be neglected so that G ¼ 0. From the complex propagation constant (3.89) of a low-loss transmission line, assuming G ¼ 0, we find rffiffiffiffi pffiffiffiffiffiffi R C g ¼ a þ jb ¼ þ jo LC ð5:28Þ 2 L so that the attenuation is R a¼ 2

rffiffiffiffi C R ¼ L 2Z0

ð5:29Þ

138


while the phase constant b is the same as in the lossless case. Using the above results, the input impedance (5.23) of the short-circuited stub becomes2 Zin ¼ jZ0 tanhðglÞ ¼ Z0

tanh al þ jtan bl 1 þ jtanh al tan bl

ð5:30Þ

In the proximity of the series resonance, e.g. for bl ffi p, let us put b ¼ p=l þ Db o ¼ pvph =l þ Do

ð5:31Þ

with both Db=b 1 and Do=o 1. Under the assumption of low loss, al 1, thus tanhðalÞ ffi al. Moreover, by virtue of (5.31), tanðblÞ ¼ tanðp þ lDbÞ ¼ tanðlDbÞ ffi lDb so that (5.30) becomes Zin ffi Z0 ðal þ jlDbÞ

ð5:32Þ

In deriving (5.32) we have neglected in the denominator of Zin the product (al)(Dbl) with respect to unity. Using (5.29) and writing Db ¼ DðoL=Z0 Þ ¼ DoL=Z0 , the input impedance (5.32) of the lossy line resonator becomes Zin ¼

R l þ jlLDo 2

ð5:33Þ

By comparison, let us now consider the input impedance of a series R0L0C0 resonator in the proximity of the resonant frequency: 1 o2 Zin ¼ R0 þ joL0 þ ¼ R0 þ joL0 1 0 ffi R0 þ j2L0 Do ð5:34Þ joC0 o where we have used the approximation (5.14). Comparing (5.33) and (5.34), we conclude that the resonant line is equivalent to a series resonator having a resistance R0 ¼ Rl=2 and an inductance L0 ¼ Ll=2, so that the Q factor can be computed from that of its equivalent circuit: Q¼

o0 L0 o0 L 1 b ¼ ¼ R 2a R0

ð5:35Þ

The Q of a low-loss resonant line with negligible dielectric loss is therefore one-half the ratio of the phase and attenuation constants. Note that the line resonator behaves as if it has half overall resistance Rl and half overall inductance Ll. The factor 1/2 derives from the current having a sinusoidal behaviour along the line, so that its effective value is one-half of its maximum magnitude. In fact, the magnetic energy and the power loss associated with the current on the l/2 line, assuming sinusoidal behaviour, are given by ð 1 l=2 l Wm ¼ LjI0 j2 sin2 bz dz ¼ LjI0 j2 4 0 16 ð5:36Þ ð 1 l=2 l Pd ¼ RjI0 j2 sin2 bz dz ¼ RjI0 j2 2 0 8 I0 being the current amplitude. Using (5.10) with W ¼ 2Wm, we again find (5.35).

2

Remember that tanhða þ bÞ ¼

tanh a þ tanh b 1 þ tanh a tanh b

and tanhðjbÞ ¼ jtan b


139

Figure 5.5 Calculating the resonant frequencies of a shorted stub loaded with a capacitor at the input port.

EXAMPLE 5.1 We want to determine the resonances of a line section terminated with a capacitor C on one end and with a short circuit on the other. Compared with Figure 5.4a, we have a shunt capacitive load located at the input of the resonator. Seen from the junction of the line with the capacitor, the circuit looks like that in Figure 5.5b, where the admittances at the left and right of the junction are 1 joC Z2 ¼ jZ0 tanðblÞ

Z1 ¼

ð5:37Þ

The input impedance is Zin ¼ jZ0

tanbl 1oCZ0 tanbl

ð5:38Þ

The condition for series resonance Zin ¼ 0 is (5.24), the same as for the unloaded line. Indeed, the line resonates when it is a multiple of half a wavelength, so that the capacitor is short-circuited. Parallel resonances occur at the poles of Zin or zeros of Yin, thus when Y0 cotðblÞ ¼ oC

ð5:39Þ

Figure 5.6 shows the graphical solution (dots) of the above equation for two different values of C. Notice that in the limit for C ! 0 (open circuit) the solutions tend to (5.26) (quarter-wavelength line); conversely, if C ! 1 (short circuit) the solutions tend to (5.24) (half-wavelength line). Compared with the unloaded line, the capacitor, as we can see, has the effect of lowering the parallel resonant frequencies and is thus equivalent to an extension of the line. Such an extension, however, is not the same for all resonant modes. It is interesting to note that if the stub is open circuited, as the reader can easily verify, the resonant condition is (5.39), while the anti-resonances are obtained when Y0 cotðblÞ ¼ oC. The graphical solutions are shown with triangles in Figure 5.6.

5.5 Planar resonators Modern microwave and RF technology commonly employs printed circuits realized by depositing a metal film on a thin dielectric substrate metallized on the opposite side. Since the substrate’s thickness is normally very small (of the order of a fraction of a millimetre) with respect to the wavelength, the EM field can be considered to a first approximation to be constant along the direction normal to the substrate, so as to neglect the presence of a fringing field at the metal strip edges. This is equivalent to assuming that

140

MICROWAVE AND RF ENGINEERING Y0cot(βl) = Y0cot(πω/2ω0)

ωC2

C1 > for TM modes þ > < d ffi a c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s fr;nmq ¼ pffiffiffiffi ð5:53Þ 0 2 2p er > p nm qp 2 > > þ for TE modes : a d The same procedure can be applied to cylindrical cavities of other shapes, the problem being reduced to that of determining the cut-off frequencies of the corresponding waveguides.

5.7 Computation of the Q factor of a cavity resonator As for waveguides, in a cavity resonator the EM power dissipates on the metal walls and in the filling dielectric. In this section we show how to compute the quality factor of a cylindrical cavity like that shown in Figure 5.9, by separately considering the two loss factors. Correspondingly, a quality factor Qd associated with the dielectric loss and a quality factor Qc associated with the conductor loss can be introduced. The overall Q factor of the cavity, according to (5.22), will be Qtot ¼ ð1=Qd þ 1=Qc Þ1

ð5:54Þ

Dielectric loss

The loss in the dielectric is represented by the imaginary part of the complex dielectric constant (see (2.47)) ec ¼ ejs=o ¼ e0 je00

ð5:55Þ

where s is the finite conductivity of the dielectric material. The power dissipated in the dielectric is, according to (2.69), ð ð 1 s E J* dV ¼ jEj2 dV ð5:56Þ Pd ¼ 2 V 2 V while the energy stored in the electric field is given by (2.67): ð e0 We ¼ jEj2 dV 4 V

ð5:57Þ

From the definition (5.10) of the Q factor, and recalling that at resonance the total energy stored is twice the electric energy, thus W ¼ 2We, we obtain Qd ¼

2o0 We o0 e0 e0 1 ¼ 00 ¼ ¼ Pd s e tan d

ð5:58Þ

Notice that Qd depends only on the dielectric, not on the resonant mode. It is interesting to note also that one arrives at the same result (5.58) by computing the damping factor due to the imaginary part of ec. In fact, since the l.h.s. is real, Equation (5.50) can be satisfied only for complex values of o. Replacing jo with s ¼ x þ jo and solving (5.50), we obtain the complex resonant frequency sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s kt2 þ kz2 s 2 s kt2 þ kz2 ð5:59Þ s ¼ x þ jo0 ¼ 0 j 0 ffi 0j me0 me0 2e 2e 2e The damping factor is therefore x ¼ s=2e0 . Using Equation (5.19) we obtain again (5.58).


145

Conductor loss

The computation of the loss due to the finite conductivity of the metal walls can be carried out using the same procedure as for the waveguides (Section 3.7); that is, by expressing the tangential electric field on the metal through the Leontovic condition (2.184). For the power dissipated in the metal, similar to (3.112) and (3.113) we thus obtain rffiffiffiffiffiffiffiþ þ 1 om Rc Pd ¼ Ht H*t dS ¼ Js J*s dS ð5:60Þ 2 Sc 2 2s Sc

where Ht is the tangential magnetic field on the metal surface Sc and rffiffiffiffiffiffiffi om Rc ¼ ReðZc Þ ¼ 2s is the surface resistance of the conductor. The metal surface Sc is composed of the lateral wall and the end sections, so that the Q factor can be expressed as ð ð ð * 2o0 m 14 H H* dV H H dV H H* dV pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 V V Qc ¼ qffiffiffiffiffiffiþ ¼ 2o0 ms þ ¼ þ ð5:61Þ ds o0 m * * * 1 H H dS H H dS H H dS t t t t t t 2 2s Sc

Sc

Sc

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The quantity ds ¼ o0 ms=2 is the skin depth (2.175) in the metal at the resonant frequency. In an equivalent manner, expressing the Q in terms of electric rather than magnetic energy, instead of (5.61) we obtain ð E E* dV 2e0 V þ Qc ¼ ds m Ht H*t dS

ð5:62Þ

Sc

The above expressions for Q must be evaluated depending on the resonant mode of the resonator. In the case of a rectangular resonator with sides a, b, d, if d > a > b the first resonant mode is the TE101 mode, which corresponds to the first resonance of the dominant mode TE10 of the rectangular waveguide of sides a, b. Using (5.52), its resonant frequency is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r c p2 p2 þ fr;101 ¼ pffiffiffiffi ð5:63Þ 2p er a d The expressions for the EM field of the TE101 resonant mode are obtained from those of the TE10 mode of the rectangular waveguide (Table 3.4) by adding to the incident wave (kz ¼ kz10 ¼ p/d) a reflected wave (kz ¼ kz10 ¼ p/d) of the same amplitude and opposite phase, so that the Ey field is zero in z ¼ 0, d. We therefore obtain the three components of the EM field of the mode: oma px oma px pz Ey ¼ j Csin ðejkz;10 z ejkz;10 z Þ ¼ sin sin p a p a d a px a px pz Hx ¼ j Csin ðkz;10 ejkz;10 z þ kz;10 ejkz;10 z Þ ¼ j sin cos ð5:64Þ p a d a d px px pz Hz ¼ Ccos ðejkz;10 z ejkz;10 z Þ ¼ jcos sin a a d In order to simplify the expressions, we have put C ¼ 1/2 in the last terms of (5.64). The computation of the Q can be more easily done using (5.62). The stored electric energy is ð ð ðb ðd 2 e0 m2 a3 bd e0 o2 e0 m2 a2 a 2 px o2 e0 m2 a3 bd fr;101 2 pz We ¼ Ey2 dx dy dz ¼ sin dy sin ¼ dx dz ¼ 2 4 V 4p2 a d 8p2 0 0 0 ð5:65Þ

146


The power loss is computed by integrating (5.60) over the six cavity walls: ð Rc Pd ¼ jHt j2 dS 2 Sc ðb ðd ð ð Rc a Rc a dx ðHx2 ðx; y; 0Þ þ Hx2 ðx; y; dÞÞ dy þ dx ð2Hx2 ðx; 0; zÞ þ 2Hz2 ðx; 0; zÞÞ dz ¼ 2 0 2 0 0 ð 0 ðd Rc b Rc 2 2 dy ðHz ð0; y; zÞ þ Hz ða; y; zÞÞ dz ¼ . . . ¼ 2 ð2a3 b þ a3 d þ ad 3 þ 2d 3 bÞ þ 2 0 4d 0 ð5:66Þ Finally, we obtain o3 m2 e0 a3 bd 3 4pb fr;101 ad 3 Zd ¼ c 2p2 Rc ð2a3 b þ a3 d þ ad 3 þ 2d 3 bÞ ð2a3 b þ a3 d þ ad 3 þ 2d 3 bÞ Rc pffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi p where Zd ¼ m=e0 is the intrinsic impedance of the dielectric and c ¼ 1= me0 . Qc ¼

ð5:67Þ

5.8 Dielectric resonators As in the case of waveguides, the metal walls in cavity resonators are used to confine the EM field within the cavity. At the same time, however, because of the finite conductivity, the current flowing in the metal produces a power loss that reduces the Q factor of the resonator. Metal loss can be avoided using resonators made of high-permittivity dielectrics. Typical values are of the order of 30–40 or higher. The strong discontinuity between the resonator and the outside space (usually air) has the effect of confining the EM field within the dielectric. The phenomenon is analogous to the total reflection discussed in Section 2.10.1. Of course, as for that case, the EM field is not zero outside the dielectric resonator, but decays exponentially from its surface. As a consequence, the resonator can be coupled to a closely spaced external circuit element, such as a microstrip or another dielectric resonator. Dielectric resonators are usually, though not necessarily, disc shaped. The dominant resonant mode is designated as TE01d (see below). Because of the high permittivity, the dimensions of a dielectric resonator are considerably smaller than those of an empty cavity resonator for the same resonant frequency. The Q factor is usually of the order of 5000–10 000. Computation of the resonant frequencies of a dielectric resonator is a rather difficult procedure. In a first approximation, if the dielectric constant is very high, the air–dielectric boundary can be modelled as a perfect magnetic conductor.4 Although not extremely accurate, this model can be used to compute the resonant modes of the dielectric resonator, in much the same way as for the cavity resonators of Section 5.6, except that the boundaries are now made of a perfect magnetic conductor rather than a perfect electric conductor. For a thin dielectric disc, the lowest mode is a TE010 mode. The three indexes correspond to the cylindrical coordinates j, r, z and indicate a constant field with respect to the azimuth (j), one half-wave variation in the radial direction (r) and constant field along the thickness (z). In practice, since the field is not fully confined within the dielectric, along the disc thickness the field varies

4

The reflection coefficient of a plane wave incident from the dielectric to the air is pffiffiffiffi er 1 Z Z ¼ pffiffiffiffi G¼ 0 er þ 1 Z0 þ Z

This value approaches unity (as for an open circuit) when er becomes very large.


Figure 5.10

147

Microstrip line coupled to a dielectric resonator.

but by less than one half wave. As a consequence this mode is designated as TE01d, where d < 1. More details can be found in [5–7]. Figure 5.10a shows the layout of a dielectric resonator coupled to a microstrip. The coupling mainly occurs through the magnetic field of the TE01d mode and is such that the signal travelling along the line is reflected back at the disc resonances. From a circuit point of view, each resonance is seen as an RLC resonant circuit inductively coupled to the line. This is equivalent to putting a parallel resonant circuit in series with the line, as shown in Figure 5.10b. Figure 5.11 shows the measured and simulated frequency responses of the circuit compared with that of the isolated microstrip. As can be observed, the RF signal is transmitted through the line with small attenuation but at the disc resonances.

5.9 Expansion of EM fields In the previous sections we saw that lossless cavity resonators, fully enclosed by perfect electric conductors, possess an infinite number of resonant modes, each corresponding to a homogeneous solution of Maxwell’s equations. In practice, the cavity is not isolated but must be connected to the external world in order to exchange energy with it. The connection often occurs through apertures produced in the cavity walls, so that the boundary conditions are modified with respect to the isolated cavity. As a consequence, the EM field distribution in a cavity with openings results from the superposition of several resonant modes, the boundary conditions on the openings being satisfied by the collective contribution of all the modes. The theory of resonant modes is a powerful and very general approach to the expansion of the EM field in an arbitrary region of space. This theory is succinctly described here, the various proofs being omitted for brevity and simplicity. The reader is referred to [2, 8, 9] for a complete and exhaustive treatment.

148

MICROWAVE AND RF ENGINEERING 0.0

–0.2

20 log10(Is21I)

–0.4 measure model –0.6

line

–0.8

–1.0

–1.2 2.0

2.2

2.4

2.6

2.8

3.0

Frequency, GHz

Figure 5.11

5.9.1

Responses of the circuit of Figure 5.10.

Helmholtz’s theorem

Let us first introduce a fundamental theorem of field theory, namely Helmholtz’s theorem [8]. The theorem states that any vector A(x, y, z) defined in a space region V bounded by the surface S can be expressed as the sum of an irrotational term plus a solenoidal term according to the following formula: ð þ ^ 0 r0 Aðr0 Þ 0 Aðr0 Þ n AðrÞ ¼ r dV þ dS 4pR 4pR V S ð þ ^ 0 r0 Aðr0 Þ 0 Aðr0 Þ n dV þ dS þr 4pR 4pR V S

ð5:68Þ

where R ¼ jrr0 j is the distance between the observation point and the integration point. Equation (5.68) shows that the vector originates from volume sources r A and r A and ^ and A n ^ . The vector A is irrotational, and thus can be sources located on the boundary S, i.e. A n ^ ¼ 0 on S; derived from the gradient of a scalar potential, if and only if r A ¼ 0 in V and A n conversely, A is solenoidal, and thus can be derived from the curl of a vector potential, if and only ^ ¼ 0 on S. if r A ¼ 0 in V and A n

5.9.2

Electric and magnetic eigenvectors

In order to create a base of solutions that can be used to expand any EM field inside the volume V, we need to specify suitable sets of boundary conditions, in such a way that the modal base is unique and complete. The term complete means that any vector function which is square integrable and piecewise continuous in V can be expressed as a combination of the modes. Even if not explicitly


149

stated, we will implicitly consider only square-integrable piecewise-continuous functions. Complete modal bases can be obtained from the solutions of eigenvalue problems that, depending on the boundary conditions, are of the electric or magnetic type. Electrical eigenvectors are solutions of5 r2 E þ k2 E ¼ 0 rE ¼ 0 Ê¼0 n

ð5:69aÞ

in V

on S

ð5:69bÞ

Magnetic eigenvectors are solutions of r2 H þ k2 H ¼ 0

in V

ð5:70aÞ

^rH¼0 n ^H ¼ 0 n

on S

ð5:70bÞ

It is worth specifying that, although we have used the same symbols for the electric and magnetic fields which are solutions to Maxwell’s equations, the above quantities are not the electric and magnetic fields, but should be regarded as merely mathematical quantities. Similarly, k2 is not the wavenumber but simply denotes the generic eigenvalue of (5.69) or (5.70). The two problems differ in the boundary conditions, which are of the electric or magnetic type respectively. However, both lead to solutions having the following properties: 1. There is a countable infinity of real non-negative eigenvalues and real eigenvectors. 2. Eigenvectors are orthogonal and can be normalized in such a way that ð Am An ¼ dmn V

ð5:71Þ

where Am ; An are two independent eigenvectors. Each set of solutions consists of three different groups: I. Solenoidal electric eigenvectors: r Em 6¼ 0; r Em ¼ 0. As a consequence, (5.69) becomes 2 E¼0 r r Em km

in V

ð5:72aÞ

^ Em ¼ 0 n

on S

ð5:72bÞ

6 0. These are obtained from II. Irrotational electric eigenvectors: r f m ¼ 0; r f m ¼ scalar potentials f m ¼ rvm =lm , where the potential is obtained by solving the scalar eigenvalue equation

5

r2 vm þ l2m vm ¼ 0

in V

ð5:73aÞ

vm ¼ 0

on S

ð5:73bÞ

Recall that r2 A ¼ r r A þ rr A.

150

MICROWAVE AND RF ENGINEERING III. Harmonic electric eigenvectors: r E0 ¼ 0; r E0 ¼ 0. Such eigenvectors correspond to the zero eigenvalue of (5.69), if it exists. They are non-trivial solutions of r2 E0 ¼ 0 r E0 ¼ 0 ^ E0 ¼ 0 n

ð5:74aÞ

in V on S

ð5:74bÞ

Such eigenvectors satisfy the same equation and boundary conditions as the electrostatic field. Non-trivial solutions can exist only if the volume is multiply connected, so that the boundary S is made of different parts. There will be as many independent solutions as the number of parts minus one. Let the boundary consist of P portions; then, there will be P 1 independent harmonic eigenvectors E01 ; E02 ;. . . ; E0P1 . Due to the completeness of the above base, any piecewise-continuous vector A in V can be expanded as follows: A¼

1 X

am Em þ

m¼1

1 X

bm f m þ

m¼1

P1 X m¼1

cm E0m

ð5:75Þ

where the expansion coefficients are Ð am ¼ V Em A dV Ð bm ¼ V f m A dV Ð cm ¼ V E0m A dV

ð5:76Þ

The last term in (5.75) using harmonic eigenvectors is present only if the volume is multiply connected. If A is solenoidal it is easily seen that6 ð A f m dV ¼ 0 ð5:77Þ V

In much the same way, we obtain a complete base of magnetic eigenvectors that can be divided into three groups: I. Solenoidal magnetic eigenvectors: r Hm 6¼ 0; r Hm ¼ 0. As a consequence, (5.70) become 2 r r Hm km Hm ¼ 0

^ r Hm ¼ 0 n

on S

in V

ð5:78aÞ ð5:78bÞ

In fact, taking into account the properties of the irrotational eigenvectors II and that, since r A ¼ 0, A rvm ¼ r ðAvm Þ, one obtains ð ð þ ð rvm 1 1 ^vm dS ¼ 0 A f m dV ¼ A dV ¼ r ðAvm Þ dV ¼ An lm lm lm S 6

V

V

V


151

II. Irrotational magnetic eigenvectors: r gm ¼ 0; r gm 6¼ 0. These are obtained from scalar potentials, gm ¼ rwm =lm , where the potential wm is obtained by solving the scalar eigenvalue equation r2 wm þ l2m wm ¼ 0 @wm ¼0 @n

in V

ð5:79aÞ ð5:79bÞ

on S

III. Harmonic magnetic eigenvectors: r H0 ¼ 0; r H0 ¼ 0. Such eigenvectors correspond to the zero eigenvalue of (5.71), if it exists. They are non-trivial solutions of r2 H0 ¼ 0 ^ r H0 ¼ 0 n ^ H0 ¼ 0 n

ð5:80aÞ

in V on S

ð5:80bÞ

Such eigenvectors satisfy the same equation and boundary conditions as the magnetostatic field. Non-trivial solutions can exist only if the volume is multiply connected, like a torus. There will be as many independent solutions as the number of parts minus one. Let the boundary consist of P portions; then, there will be P 1 independent harmonic eigenvectors H01 ; H02 ; . . . ; H0P1 . An alternative expansion of a piecewise-continuous vector A in V is therefore the following one, based on the use of the magnetic eigenvectors: A¼

1 X

Am Hm þ

m¼1

1 X m¼1

Bm gm þ

P1 X m¼1

Cm H0m

ð5:81Þ

The expansion coefficients are given by Am ¼ Bm ¼ Cm ¼

Ð ÐV Ð

Hm A dV

V gm A dV 0 V Hm A dV

ð5:82Þ

The last term using harmonic eigenvectors is present only if the volume is multiply connected. If A is solenoidal and tangential to the boundary, it is easily seen that7 ð gm A dV ¼ 0 ð5:83Þ V

The solenoidal eigenvectors of electric and magnetic type are not independent of each other. Indeed, they are related by r Hm ¼ km Em r Em ¼ km Hm

ð5:84Þ

Therefore, every solenoidal electric eigenvector has a corresponding solenoidal magnetic eigenvector and vice versa.

7

The proof is just the same as for (5.77). See footnote 6.

152

MICROWAVE AND RF ENGINEERING EXAMPLE 5.2 Compute the electric and magnetic eigenvectors of a lossless rectangular cavity of size a b c. Let us start with the solenoidal electric eigenvectors of (5.72). Notice that (5.72a) is 2 equivalent to solving r2 Em km Em ¼ 0 with the additional condition that r Em ¼ 0. Using the method of separation of variables we obtain mpx npy ppz sin sin a b c mpx npy ppz ¼ Cy;mnp sin cos sin a b c mpx npy ppz ¼ Cz;mnp sin sin cos a b c

Ex;mnp ¼ Cx;mnp cos Ey;mnp Ez;mnp

ð5:85Þ

where the index m has been replaced by the triplet mnp. The corresponding eigenvalues are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 pp2 ð5:86Þ þ þ kmnp ¼ a b c In a similar manner, the solution of (5.78) is found to be mpx npy ppz cos cos a b c mpx npy ppz ¼ Dy;mnp cos sin cos a b c mpx npy ppz ¼ Dz;mnp cos cos sin a b c

Hx;mnp ¼ Dx;mnp sin Hy;mnp Hz;mnp

ð5:87Þ

It should be remembered that the class of solutions of (5.85) is related to (5.87) through (5.84), so that the coefficients Cx,mnp determine Dx,mnp and vice versa. Moreover, Cx, mnp must be such as to comply with the condition r Em ¼ 0 and the normalization condition (5.71). The same conditions must be applied to Dx,mnp. Such conditions, however, do not completely determine the three constants – indeed we have one degree of freedom left in the choice of the coefficients. In much the same way as with the resonant modes discussed in Section 5.6, we may further distinguish the solutions in TE and TM eigenvectors, the former being obtained by putting Cz,mnp ¼ 0 in (5.85), the latter by putting Dz,mnp ¼ 0 in (5.87). In this way we obtain the expressions quoted in Table 5.1. Concerning the irrotational eigenvectors, we first need to solve (5.73) and (5.79). The solutions are readily found using the separation of variables method: mpx npy ppz sin sin a b c mpx npy ppz wm ¼ Dmnp cos cos cos a b c vmnp ¼ Cmnp sin

with the same eigenvalues as (5.86): r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 pp2 þ þ lmnp ¼ a b c

ð5:88Þ ð5:89Þ

ð5:90Þ

By taking rvmnp and rwmnp we obtain the same expressions as (5.85) and (5.87), replacing Es and Hs with fs and gs, respectively, and with the expressions for the C and D coefficients quoted in Table 5.1.


153

Table 5.1 Eigenvectors of a rectangular cavity. The electric field components vary according to (5.85), while the magnetic field components vary according to (5.87).

TE eigenvectors

TM eigenvectors

where

Solenoidal electric eigenvectors rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2dm dn np Cx;mnp ¼ kmn abc b rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2dm dn mp Cy;mnp ¼ kmn abc a Cz;mnp ¼ 0 rffiffiffiffiffiffiffi dp mpp2 abc ac rffiffiffiffiffiffiffi dp npp2 2 Cy;mnp ¼ kmn kmnp abc bc rffiffiffiffiffiffiffi dp kmn Cz;mnp ¼ kmnp abc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 þ kmn ¼ a b 2 Cx;mnp ¼ kmn kmnp

Irrotational electric eigenvectors rffiffiffiffiffiffiffi 8 mp kmnp abc a rffiffiffiffiffiffiffi 1 8 np Cy;mnp ¼ kmnp abc b rffiffiffiffiffiffiffi 1 8 pp Cz;mnp ¼ kmnp abc c 1 for m ¼ 0 dm ¼ 2 for m 6¼ 0 Cx;mnp ¼

5.9.3

1

Solenoidal magnetic eigenvectors rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2dm dn mpp2 Dx;mnp ¼ kmn kmnp abc ac rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2dm dn npp2 Dy;mnp ¼ abc bc kmn kmnp rffiffiffiffiffiffiffiffiffiffiffiffiffi kmn 2dm dn Dz;mnp ¼ kmnp abc rffiffiffiffiffiffiffi dp np 2 Dx;mnp ¼ kmn abc b rffiffiffiffiffiffiffi dp mp 2 Dy;mnp ¼ kmn abc a Dz;mnp ¼ 0

kmnq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 pp2 ¼ þ þ a b c

Irrotational magnetic eigenvectors rffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm dn dp mp kmnp abc a rffiffiffiffiffiffiffi 1 8 np ¼ kmnp abc b rffiffiffiffiffiffiffi 1 8 pp ¼ kmnp abc c

Dx;mnp ¼ Dy;mnp Dz;mnp

1

General solution of Maxwell’s equations in a cavity

We are now in a position to solve Maxwell’s equations in a cavity bounded by metal walls by using the field expansions developed in the previous section. It seems reasonable to use the electric and magnetic eigenvectors for expanding the electric and magnetic fields, respectively, since they satisfy the same type of boundary conditions at the metal surface. In this way we expect that the convergence of the respective expansions will be much faster. We are not going into details of the derivation, but simply report the final results. Consider first a cavity V fully enclosed by perfect conducting walls except for the presence of some apertures, where the tangential electric field is non-zero. Suppose also that an electric current density distribution J and a magnetic current distribution JM are present in the cavity. Maxwell’s equations are r H ¼ joeE þ J r E ¼ jomHJM

ð5:91Þ

154


Expanding E and r H with the electric eigenvectors expansions (5.75) and H and r E in terms of the magnetic eigenvectors expansion (5.81), we obtain E¼

1 X

am Em þ

i¼1

H¼

1 X

bm f m þ

m¼1

Am Hm þ

m¼1

with

1 X

1 X m¼1

P1 X m¼1

Bm gm þ

P1 X m¼1

cm E0m

ð5:92Þ

Cm H0m

ð5:93Þ

þ ð ð 1 k n E H dS þ k H J dV þ jom E J dV m m m m M m 2 k2 km S V V ð 1 bm ¼ f m J dV joe V ð 1 E0 J dV cm ¼ joe V m þ ð ð 1 Am ¼ 2 2 joe n E Hm dS þ joe Hm JM dVkm Em J dV k km S V V ð þ 1 Bm ¼ gm JM dV þ gm n E dS jom V S ð þ 1 H0m JM dV þ H0m n E dS Cm ¼ jom V S

am ¼

ð5:94aÞ ð5:94bÞ ð5:94cÞ ð5:95aÞ ð5:95bÞ ð5:95cÞ

pffiffiffiffiffi where k ¼ o me. Notice that the surface integrals in the above expressions are zero over those portions of S where there is a perfect conductor (thus n E ¼ 0). These expressions for the expansion coefficients show that the EM field is created by the electric and magnetic current densities Ji and Jm, and by the presence of a non-zero tangential electric field on the boundary, i.e. on the apertures where the cavity is connected to the external circuits. The general expressions above can be greatly simplified in most practical cases, as illustrated in the following examples.

5.9.4

Resonances in ideal closed cavities

Consider an ideal simply connected cavity (thus E0 ¼ 0), fully enclosed by a perfect electric conductor (thus n E ¼ 0), the only source being electric currents (JM ¼ 0). Under such hypotheses, the general expansions (5.92)–(5.93) reduce to E¼

1 X

bm f m þ

m¼1

H¼

1 X

am Em

ð5:96Þ

m¼1 1 X

Am Hm

ð5:97Þ

m¼1

where

ð 1 f m J dV joe V ð jom am ¼ 2 2 Em J dV k km V km am Am ¼ jom bm ¼

ð5:98Þ


155

The general EM field in the cavity results from the contribution of infinite terms (resonant modes); the ith term becomes infinitely large as the frequency approaches km om ¼ pffiffiffiffiffi me

ð5:99Þ

which represents the resonant frequency of the mth mode. The first term in (5.96) ‘resonates’ at zero frequency, and therefore represents an electrostatic field.

5.9.5

The cavity with one or two outputs

Consider now an ideal simply connected cavity (thus E0 ¼ 0), with no sources inside, fully enclosed by a perfect electric conductor (thus n E ¼ 0) except for two apertures connected to two waveguides. The case of one single output can be derived very easily by simply short-circuiting the second aperture. As sketched in Figure 5.12, we can extend the boundary of the volume so as to include part of the waveguides, in such a way that on the surfaces S1 and S2 we can assume that the EM field is that of the dominant modes of the waveguides.8 This assumption is not strictly necessary but simplifies the analytical treatment by simply neglecting higher order modes on the waveguides. The integrals over the boundary surface S of the cavity reduce to the only portions S1 and S2 where the tangential electric field is not zero. Under the above hypotheses, the general expansions (5.92)–(5.95) reduce to E¼

1 X

am Em

ð5:100Þ

m¼1

H¼

1 X m¼1

with

Bm gm þ

1 X

Am Hm

ð5:101Þ

m¼1

ð km ^ E Hm dS n 2 k2 km S þS ð 1 2 1 ^ E gm dS Bm ¼ n jom S1 þ S2 joe am Am ¼ km am ¼

ð5:102Þ

The tangential electric field on Si (i ¼ 1, 2) in (5.102) can be expressed in terms of the ith waveguide modes. Under the assumption that only the fundamental mode is present on the cross-section Si, we can simply write9 ðiÞ

Et ¼ Vi ei ðiÞ

Ht ¼ Ii hi ¼ Ii ð^ n i Þ ei

ð5:103Þ

where ð^ ni Þ is the unit vector in the direction of the ith waveguide axis, as shown in Figure 5.12, and ei and hi are the electric and magnetic field vectors corresponding to the dominant mode of the waveguide, and Vi and Ii the corresponding equivalent voltage and current. Using the first equation 8 If the reference planes are taken on the waveguide wall where higher order waveguide modes are present, the procedure is more complicated, as it yields an N N matrix representation of the cavity, N being the total number of waveguide modes taken into consideration. Correspondingly one obtains the so-called generalized scattering matrix (GSM), generalized admittance matrix (GAM), and so on. 9 In the most general case, the EM field can be expressed, rather than in terms of only the dominant mode, as a summation over all the waveguide modes. For details, see [9].

156


Figure 5.12 of (5.103) we obtain

ð S1 þ S2

where we have put

^ E Hm dS ¼ V1 x1m þ V2 x2m n

ð xim ¼

Similarly,

A cavity with two outputs.

ð5:104Þ

ð Si

^ ei Hm dS ¼ n

Si

hi Hm dS

ð5:105Þ

ð S1 þ S2

with

^ E gm dS ¼ V1 B1m þ V2 B2m n

ð Bim ¼

ð5:106Þ

ð Si

^ ei gm dS ¼ n

Si

hi gm dS

ð5:107Þ

The coefficients Ii of the magnetic field over the ith waveguide cross-section can be expressed in terms of the magnetic field itself: ð ð ni Þ ei dS ð5:108Þ Ii ¼ H hi dS ¼ H ð^ Si

Si

Substituting (5.101) into (5.108) and using (5.104)–(5.107), we finally obtain the relationship between the equivalent currents and voltages at the cavity outputs in the form of a 2 2 admittance matrix ½I ¼ ½Y½V ð5:109Þ the matrix elements being expressed as Yij ¼

1 1 X xim xjm 1 X Bim Bjm joe 2 k2 k jom m¼1 m m¼1

ð5:110Þ

The admittance matrix is expressed in the form of a partial fraction expansion, where each term corresponds to a resonant mode (including the static term resonating at DC). Although only one index m has been employed, in practice (5.110) involves a triple summation over three indexes corresponding to the three coordinates. The convergence of such an expansion is rather slow, so alternative expressions should be adopted for practical computations (see [9]).


157

The case of a cavity with one input is obtained in a straightforward manner by simply shortcircuiting one opening. The cavity can then be represented as a two-terminal circuit, whose admittance is given by Y11 (or Y22) of (5.110). By some straightforward algebra we find that the input admittance of the cavity is Ycavity ¼

1 X 1 joCm þ joL0 m¼1 1ðo=om Þ2

ð5:111Þ

where m L0 ¼ X ; 1 B2m

Cm ¼ e

x2m 2 km

ð5:112Þ

m¼1

km om ¼ pffiffiffiffiffi me In (5.112) we have simply written zm and xm , instead of z1m and x1m . The above formulae show that the cavity behaves as the parallel connection of the inductor L0 to an infinite number of series resonant LC cells representing the modes of the cavity, as shown in Figure 5.13a. In a dual fashion we can analyze a cavity enclosed by a perfect magnetic conductor. This can be considered as an approximation of dielectric resonators or planar microstrip resonators [10]. In this case, the inherent representation is that of the impedance rather than the admittance matrix. The dual series expansion of (5.111) leads to the equivalent circuit of Figure 5.13b, consisting of a capacitor C0 in series with infinite parallel resonant LC cells representing the mode of the cavity.

5.9.6

Excitation of cavity resonators

As an alternative to apertures in the cavity walls, resonators can be excited by electric or magnetic probes consisting of a small linear current or loop. In both cases the current filament can represent

L0

L1

L2

C1

C2 (a)

C1

C2

L1

L2

C0

(b)

Figure 5.13 Equivalent circuits of a cavity with one aperture: (a) cavity with perfect electric conductor walls; (b) cavity with perfect magnetic conductor walls.

158


h (a)

S

Figure 5.14

(b)

Excitation of a cavity with an electric (a) or a magnetic probe (loop) (b).

the inner conductor of a coaxial cable connected to the resonator. Consider first a short linear current I0 impressed along a short segment located near the inner wall of an ideal cavity, as sketched in Figure 5.14a. The cavity is assumed to be lossless and simply connected. Let I be the current flowing in the short filament with cross-section S and length h. If S is very small, we can assume the current density to be constant in the volume v ¼ hS:10 J ¼ J0^t

ð5:113Þ

in v

where J0 ¼ I=S; ^t is the unit vector in the direction of the current. Inserting (5.113) into (5.98), we obtain ð 1 1 bm ¼ f m J dv ¼ hI fmt joe v joe ð jom jom am ¼ 2 2 Em J dv ¼ 2 2 hIEmt k km v k km km Am ¼ am jom

ð5:114Þ

where fmt ¼ Emt ¼

10

1 Sh 1 Sh

ð ð

1 f m ^t dv ffi h v v

Em ^t dv ffi

ðh

1 h

0 ðh

f m ^t dl ð5:115Þ Em ^t dl

0

The volume is indicated here by a small v to avoid confusion with the voltage V.


159

where we have assumed fm and Em to be constant in the cross-section S.11 Using (5.96) and (5.114), the voltage across the probe is found to be ! ðh 1 1 X X ^ bm fmt þ am Emt V ¼ E t dl ¼ h 0 m¼1 m¼1 ! ð5:116Þ 1 1 X 1 X 1 2 2 2 f jom E ¼h I 2 mt joe m¼1 mt k2 km m¼1 The resulting impedance is therefore Zin ¼

1 X 1 Lm þ jo 2 joC0 1ðo=o mÞ m¼1

ð5:117Þ

with C0 ¼

2 fmt m¼1 mh2 2 Emt 2 km

h2 Lm ¼

e 1 X

ð5:118Þ

Equation (5.117) represents the series of the capacitor C0 with infinite parallel resonators, as in Figure 5.13b. Consider now the excitation of the cavity through a small current loop of area S, as in Figure 5.14b. In this case the volume integrals appearing in (5.114) can be transformed as follows: ð þ ð ð Em J dV ¼ I Em dl ¼ I r Em dS ¼ Ikm Hm dS ð5:119Þ V

S

S

where I ¼ SJ is the current flowing in the loop. In deriving the above equation we have used Stokes’ theorem and (5.84). Putting ð 1 Hm dS ð5:120Þ Hmn ¼ S S from (5.114) we obtain ð jom jom Em J dV ¼ 2 2 ISkm Hmn 2 k2 km k km V km Am ¼ 2 2 ISkm Hmn k km

am ¼

ð5:121Þ

With the same procedure, we find that bm ¼ 0 since r f m ¼ 0. From (5.93) we can determine the magnetic field in the cavity: H¼

1 X m¼1

Am Hm ¼ IS

1 X

2 km 2 2 k km m¼1

Hmn Hm ðrÞ

ð5:122Þ

11 Notice that such an approximation is valid as long as S is very small compared with the spatial variation of fm and Em, thus for modes up to a certain order m. For very large orders, however, the contribution of the modes tends to become negligible.

160


Figure 5.15

Equivalent circuit of the loop-coupled cavity of Figure 5.12b.

The voltage induced by the magnetic field flux through the loop is þ ð ð 1 1 X X V ¼ E dl ¼ jom H dS ¼ jom Am Hm dS ¼ jomIS2 m¼1

S

2 km 2 2 k km m¼1

S

2 Hmn

ð5:123Þ

so we obtain the expression for the input impedance: Zin ¼ jomS2

1 X

2 Hmn

m¼1 1ðo=om Þ

2

¼

1 X

joLm

m¼1 1ðo=om Þ

2

ð5:124Þ

with 2 Lm ¼ mS2 Hmn

ð5:125Þ

The equivalent circuit therefore consists of a series of an infinite number of anti-resonant circuits, as shown in Figure 5.15. An alternative simpler approach12 is based on the use of the energy definition (5.4) of the input admittance Yin ¼

2joðWm We Þ 4joðWm We Þ ¼ VV * =2 ðomSHn Þ2

ð5:126Þ

where Hn is the magnetic field in the loop, orthogonal to its plane. The above expression can be specialized to each specific resonant mode of the cavity as long as the frequency is close to its resonant frequency. For example, suppose the loop is located at half the height of the side wall of a circular cylindrical cavity. If the loop is orthogonal to the f direction, it will excite the TM010 mode. Using the expressions relevant to this mode, we obtain pl p201 2 Wm We ¼ ð5:127Þ E02 J12 ðp01 Þ ea 4 o2 m jHn j ¼

1 p01 E0 J1 ðp01 Þ om a

ð5:128Þ

where l is the height of the cavity, a its radius, p01 the first zero of the Bessel function J0 and E0 the (arbitrary) amplitude of the electric field. Inserting the above expressions into (5.126), we obtain Yin ¼

12

See [5], p. 49ff.

1 þ joCp joL0

ð5:129Þ


161

with L0 ¼

mS2 ; pla2

Cp ¼

pla4 e S2 p201

ð5:130Þ

In the proximity of the TM01 resonant mode, we again find that the cavity behaves as a parallel resonant circuit. In wider frequency ranges additional resonant modes should be included in the equivalent circuit of the cavity, leading to the same equivalent circuit of Figure 5.15.

Bibliography 1. J. C. Slater, Microwave Electronics, Van Nostrand, Princeton, NJ, 1950. 2. K. Kurokawa, An Introduction to the Theory of Microwave Circuits, Academic Press, New York, 1969. 3. T. Okoshi, Planar Circuits for Microwave and Lightwaves, Springer Series in Electrophysics, Vol. 18, 1984. 4. K. C. Gupta and M. Abouzhara(eds), Analysis and Design of Planar Microwave Components, IEEE Press, Piscataway, NJ, 1994. 5. D. Kajfez and P. Guillon, Dielectric Resonators, Artech House, Dedham, MA, 1986. 6. Kai Chang(ed.), Handbook of Microwave and Optical Components, Vol. 1, Ch. 4. John Wiley & Sons, Inc., New York, 1989. 7. C. A. Balanis, Advanced Electromagnetic Engineering, John Wiley & Sons, Ltd, Chichester, 1989, Ch. 9. 8. R.E. Collin, Field Theory of Guided Waves, IEEE Press, Piscataway, NJ, 1991. 9. G. Conciauro, M. Guglielmi and R. Sorrentino, Advanced Modal Analysis, John Wiley & Sons, Ltd, Chichester, 1999. 10. G. D’Inzeo, F. Giannini, C.M. Sodi and R. Sorrentino, ‘Method of analysis and filtering properties of microwave planar networks’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT26, pp. 462–471, 1978. 11. I. Wolff and N. Knoppik, ‘Rectangular and circular microstrip disk capacitors and resonators’, IEEE Transactions on Microwave Theory and Techniques, Vol. 22, No. 10, pp. 857–864, 1974. 12. R. Sorrentino, ‘Planar circuits, waveguide models, and segmentation method’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-33, pp. 1348–1353, 1985. 13. J. K. Kraus, Antennas, McGraw-Hill, New York, 1988.

6

Impedance matching 6.1 Introduction The power transfer between different parts of a microwave circuit, typically from a source to a transmission line or from the latter to an antenna, is limited by unavoidable impedance mismatchings. It is apparent how critical the problem is of designing and building appropriate matching and impedance transforming circuits capable of maximizing the power transfer through the microwave apparatus and optimizing its behaviour from an energy point of view. It must also be considered that the presence of standing waves reduces the power-handling capability of the system and that multiple internal reflections within the circuit deteriorate the quality of the RF signals. An impedance matching network, or more generally an impedance transformer, is used to optimize the power transfer from a source to a load in a prescribed frequency band with a prescribed return loss (or reflection coefficient s11). Equivalently, we may say that such a network ‘transforms’ (or, better, aims to transform) the load impedance into that corresponding to the maximum power transfer. The design of matching networks can be achieved by using various methods and circuit configurations. A first distinction can be obtained between purely reactive, thus lossless, and resistive (lossy) networks. A second classification is between lumped and distributed networks. Finally, the matching network depends on the frequency range of interest; it clearly appears that the complexity increases with the required passband width. Lossless matching networks are the only ones considered in this chapter, as they are the most commonly adopted networks. They can be reduced in practice to three categories: quarter-wavelength transformers, stub networks and lumped L networks. Before going into detail, however, it is important to present a fundamental limitation on the impedance matching.

6.2 Fano’s bound It is worth noting at the outset that there are specific theoretical bounds on wide-band impedance matching. Indeed, when a load is not purely resistive, Fano [1] theoretically determined an upper limit to the product of return loss and bandwidth.


164

MICROWAVE AND RF ENGINEERING Z Lossless matching network

Γ

Figure 6.1

C

R

Matching a complex load to a lossless two-port network.

To illustrate this point, let us consider a load consisting of an RC parallel cell, the matching network being a lossless two-port network as shown in Figure 6.1. Let Gð f Þ be the reflection coefficient at the input of the matching network when the second port is terminated with the RC cell. Fano’s bound states that 1 ð

0

1 ð 1 d f 1 ) RLð f Þd f < 4:343 ln Gð f Þ 2RC RC

ð6:1Þ

0

where RL ¼ 20 log10 ðjGjÞ is the return loss at the input of the matching network.1 If the RC cell is replaced with a generic reactive bipole, the r.h.s. of (6.1) takes a different form, but there always exists an upper limit a on the integral of RL.2 Suppose we want to match a bipole in the frequency band between f1 and f2, using a lossless network. The integral on the l.h.s. of (6.1) can be broken down into the sum of three integrals: ðf1

1 ð

RLðf Þd f ¼

ðf2 RLð f Þd f þ

RLð f Þd f þ f1

0

0

1 ð

RLð f Þd f < a

ð6:2Þ

f2

It follows that ðf2

1 ðf1 ð RLð f Þd f < a RLð f Þd f RLð f Þd f

f1

0

ð6:3Þ

f2

Since the return loss of a passive network is a non-negative quantity, the following inequality holds even more so: ðf2 RLð f Þd f < a

ð6:4Þ

f1

Let RLMIN be the minimum value of the return loss in the frequency interval [f1 f2]. Using (6.4) we may write ðf2 RLMIN ð f2 f1 Þ
0 (respectively, B < 0), it can be realized as a lumped capacitor (inductor) or as an open-circuited (short-circuited) stub.3 In a similar manner, a reactive admittance can be reduced to a conductance by adding a suitable series impedance. Once transformed into a resistance or a conductance, the load can be matched to the source using, for example, a quarter-wavelength transformer, as described in the next section.

6.3 Quarter-wavelength transformer The simplest distributed matching network is the l/4 transformer that we considered in Section 4.3. The load impedance ZL is connected to the reference impedance Z0 through a line section of impedance Z1 , as shown in Figure 6.2. At the frequency when the length is l/4, and thus the electrical length y is p/2, the input impedance seen at terminal 1 is Z0 ¼

Z12 ZL

ð6:9Þ

The circuit of Figure 6.2 can thus be used to match a real load ZL to a source Z0 (real) by choosing Z1 to be the geometric mean of ZL and Z0 . Though very simple, this solution is narrow band as the matching is realized only in the proximity of the central frequency f0 corresponding to the line length l being equal to l/4: vp ð6:10Þ f0 ¼ 4l where vp is the phase velocity along the line. 3 The stub is usually 50 O, since this is a value easy to realize. In general the stub length is chosen to be less than l/4, and it is open-circuited (or short-circuited) in order to have B > 0 (B < 0). (See Equations (4.14) and (4.15).)

166


In order to evaluate the frequency behaviour of the l/4 transformer, we compute the input reflection coefficient. Using well-known formulae (e.g. (4.5)), we obtain G¼

Zin Z0 ZL Z0 pffiffiffiffiffiffiffiffiffiffi ¼ ... ¼ Zin þ Z0 ZL þ Z0 þ 2j Z0 ZL tan y

ð6:11Þ

where y ¼ bl ¼

2pl 2pl f ¼ l vp

ð6:12Þ

is the line electrical length. The magnitude of G is r¼h

jZL Z 0 j

i1=2 ¼ " ðZL þ Z 0 Þ2 þ 4Z 0 ZL tan2 y

1 pffiffiffiffiffiffiffiffiffiffi 2 #1=2 2 Z0 ZL 1 1þ ZL Z0 cos y

ð6:13Þ

Its frequency behaviour is shown in Figure 6.3. The magnitude of G vanishes when the line length is an odd multiple of l/4 corresponding to electrical lengths y ¼ ð2n þ 1Þp=2 with n ¼ 0, 1, 2. . .. On the contrary, when the line length is a multiple of l/2 (thus y ¼ np) the line simply transfers the load impedance ZL to the input, so that the reflection coefficient is given by Gð0Þ ¼

ZL Z0 ZL þ Z0

ð6:14Þ

and is independent of Z1 . In the design of the impedance transformer we need to specify the maximum allowed reflection coefficient rM and the corresponding frequency range. The two quantities are clearly not independent of each other as can be seen from Figure 6.3, which shows that the reflection coefficient stays below rM in the interval p Dy ¼ 2 yM ¼ p2yM ð6:15Þ 2 corresponding to a relative bandwidth w¼

Df Dy 4 ¼ ¼ 2 yM f0 p=2 p

zL - z0 zL + z0

Figure 6.3 Reflection coefficient of a single section l/4 transformer.

ð6:16Þ

IMPEDANCE MATCHING

167

In (6.16) we have assumed that the line is non-dispersive, i.e. that the phase velocity is frequency independent. From (6.13), putting r ¼ rM we find for yM ffi ffi 2r pffiffiffiffiffiffiffiffiffi 2r pffiffiffiffiffiffiffiffiffi p M Z0 ZL M Z0 ZL 1 pffiffiffiffiffiffiffiffiffiffiffiffi ¼ sin pffiffiffiffiffiffiffiffiffiffiffiffi yM ¼ cos ðZL Z0 Þ 1r2M 2 ðZL Z0 Þ 1r2M 1

ð6:17Þ

using the formula cosðaÞ ¼ sinðp=2aÞ. For example, if we need to match a 200 O load to 50 O with a 20 dB maximum reflection coefficient, from the previous equations we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 50 200 ¼ 100 O rm ¼ 101 ¼ 0:1 0:2 100 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1:44 rad ¼ 82:5 ym ¼ cos1 150 10:01 Z1 ¼

ð6:18Þ

Df ¼ 16:7% f0 If the load is ZL ¼ 400 O, a similar computation would show that ym ¼ 1:51 rad ¼ 86:7 , implying a relative bandwidth of 6.6%.

6.4 Multi-section quarter-wavelength transformers A better performance can be obtained by using a multi-section transformer, consisting of a cascade of several l/4 line sections; the additional degrees of freedom available can be used to obtain impedance transformers with improved characteristics in terms of both bandwidth and return loss. Figure 6.4 depicts an impedance transformer consisting of N l/4 line sections of characteristic impedances Zi (i ¼ 1, 2, . . ., N), interposed between the load ZN+1 ¼ ZL and the source Z0. Our objective is to determine the values of Zi (i ¼ 1, 2, . . ., N) such that some prescribed requirements are satisfied in terms of input reflection coefficient, or, equivalently, return loss. The procedure for determining the unknown circuit elements from a prescribed response is called synthesis, in contrast to the opposite procedure, the analysis, consisting of determining the unknown response of the circuit from its element values. Though somewhat simplified, the procedure we will illustrate here is quite similar to the filter synthesis described in the next chapter and is thus also intended as an introduction to the concepts that will be illustrated in greater detail in Chapter 8. Without loss of generality, we can assume that the load impedance ZL is larger than the reference impedance Z0. In other words, the impedance ratio is greater than unity: R¼

θ Γ

Γ0

Z1

ZL > 1 Z0

θ Γ1

Z2

ð6:19Þ

θ Γ2

Z3

θ Γ3

Z0

Figure 6.4

Multi-section quarter-wavelength transformer.

ZN

ΓN

ZL

168


If the opposite is true, the synthesis can be reversed taking the load as the reference impedance and Z0 as the load. Equivalently, the synthesis can be performed using the admittances rather than the impedances. As a consequence of condition (6.19), the characteristic impedance of each line is greater than the previous one, Ziþ1 > Zi , and the reflection coefficients at the various junctions Gi ¼

Ziþ1 Zi Ziþ1 þ Zi

ð6:20Þ

are real and positive.4 With reference to Figure 6.4, where Z0 and ZL are supposed to be given, the synthesis procedure can be broken down into the following steps: 1. The analytical expression of the input reflection coefficient of Figure 6.4 is determined. 2. The type of response required is chosen. Two options are normally offered to the designer, namely the maximally flat and the equiripple, or Chebyshev, response. With the former, the input reflection coefficient at the centre frequency f0 is chosen to be zero along with all its derivatives up to the (N 1)th one. These conditions, as we will see, are sufficient to determine the N unknown impedances of the transformer. This type of transformer is called, for reasons that will become clear later on, the binomial transformer. As an alternative, rather than a maximally flat response we may prefer an equiripple response: the input reflection coefficient swings between zero and a maximum value rM in a prescribed frequency band centred at f0. This is the Chebyshev transformer, whose name derives from the fact that Chebyshev polynomials are employed. 3. By equating the expression in step 1 to the analytical expression of the required response in step 2, the N unknown impedances Zi are finally determined. Step 1 requires a considerable amount of algebraic effort. This can be avoided under the assumption that the impedance ratio (6.19) is not too large, so that the impedance ratios between consecutive line sections Zi =Ziþ1 are close to unity. As a consequence, the reflection coefficients (6.20) are so small that we can assume that jGi j2 1

ði ¼ 0; 1; . . . ; NÞ

ð6:21Þ

Under the above hypotheses, the expression for the input reflection coefficient Gin in the impedance transformer of Figure 6.4 can easily be computed using the theory of small reflections. Accordingly, Gin results from the sum of the reflection coefficients (6.20) of the individual discontinuities, each multiplied by the phase term ej2iy corresponding to the distance travelled by the signal from the input to the discontinuity and back to the input, i.e. Gin ¼ G0 þ G1 ej2y þ G2 ej4y þ G3 ej6y þ þ GN ej2Ny ¼

N X

Gi ej2iy

ð6:22Þ

i¼0

4 Note that the quantity (6.20) is not the reflection coefficient when looking to the right from the junction, but the reflection coefficient of the (i + 1)th line with respect to the ith line. The former, in fact, depends on all subsequent discontinuities, while the latter is the ‘local’ refelction coefficient which only depends on the ith and (i + 1)th lines.

IMPEDANCE MATCHING

169

The theory of small reflections is developed by assuming that the reflected signal at the input port results from an infinite number of subsequent local reflections from the various discontinuities. To illustrate the point, consider the simple case of Figure 6.2. Assume a unit voltage impinging on the input port. The sequence of events is schematically illustrated in Table 6.1: 1. A first reflection occurs at port 1, producing a reflected wave of amplitude G0 ¼

Z1 Z0 Z1 þ Z0

while the signal transmitted to line 1 is T01 ¼ G0 þ 1

ð6:23Þ

2. This signal travels along line 1 for an electrical length y, undergoing a phase shift ejy . 3. The signal reaches the second discontinuity Z2 ¼ ZL where it is partly reflected back with reflection coefficient G1 ¼

Z2 Z1 Z2 þ Z1

ð6:24Þ

4. The reflected signal travels back on line 1 undergoing another phase shift ejy . 5. The signal reaches the first discontinuity from the right, where it is partly reflected back to the load. The reflection coefficient is Z0 Z1 ¼ G0 Z0 þ Z1 while the transmission coefficient is T10 ¼ G0 þ 1

ð6:25Þ

The transmitted signal emerges from port 1 towards the source, contributing to an additional reflection term. 6. The signal reflected in step 5 travels towards the load, undergoing another phase shift. The process then repeats from step 4 to step 5 an infinite number of times. As can be seen from Table 6.1, the total signal emerging from port 1, thus the reflection coefficient at port 1, is Gin ¼ G0 þ T10 T01 G1 ej2y þ T10 T01 ðG0 ÞG21 ej4y þ T10 T01 G20 G31 ej6y þ . . . 1 X ¼ G0 þ T10 T01 G1 ej2y ðG0 G1 ej2y Þi

ð6:26Þ

i¼0

The summation in the above formula is the geometrical series 1 X i¼0

xi ¼

1 1x

Therefore Gin ¼ G0 þ

T10 T01 G1 ej2y G0 þ G1 ej2y ¼ j2y 1G0 G1 e 1 þ G0 G1 ej2y

ð6:27Þ

170


Table 6.1 Derivation of the input reflection coefficient of Figure 6.2. Step

Signal emerging Transmitted/ from port 1 reflected at port 1 G0

1

Signal incident to port 1

2

Signal travels along line 1

3

Signal reflected from load

4


5

Transmission and reflection at port 1

6


7

Reflection from load

8


9

Transmission and reflection at port 1

Signal along line 1

Reflected from port 2

T01

T01 ejy G1 T01 ejy

G1 T01 ej2y T10 G1 T01 ej2y G0 G1 T01 ej2y

G0 G1 T01 ej3y G0 G21 T01 ej3y

G0 G21 T01 ej4y T10 G0 G21 T01 ej4y G20 G21 T01 ej4y G20 G21 T01 ej5y

...

...

using (6.23) and (6.25). As the reader can verify, the above expression for Gin is fully equivalent to (6.11); however, (6.27) lends itself to the approximation for small reflections. If (6.21) holds for both G0 and G1 , then (6.27) can be approximated as Gin ¼

G0 þ G1 ej2y ffi G0 þ G1 ej2y 1 þ G0 G1 ej2y

ð6:28Þ

This formula can be immediately extended to the general case of N line sections so as to obtain (6.22). As already observed, because of assumption (6.19), the coefficients (6.20) in (6.22) are real and positive quantities and coincide with their respective magnitudes ri: Gin ¼

N X i¼0

ri ej2iy

ð6:29Þ

IMPEDANCE MATCHING

171

This expression can be further simplified by the assumption of a symmetrical transformer, i.e. a transformer where r0 ¼ rN ; r1 ¼ rN1 ; ri ¼ rNi Using (6.30), (6.29) becomes

ð6:30Þ

5

Gin ¼ 2ejNy ½r0 cos Ny þ r1 cosðN2Þy þ r2 cosðN4Þy þ . . . 8 ðN1Þ=2 > X > > > ri cosðN2iÞy for N odd > < i¼0 ¼ 2ejNy N=2 > X > 1 > > ri cosðN2iÞy þ rN=2 for N even > : 2 i¼0

ð6:32Þ

Once expression (6.32) for the input coefficient has been obtained (step 1 of the synthesis procedure), we can choose between the maximally flat and the Chebyshev response.

6.4.1

The binomial transformer

The input reflection coefficient is imposed to get a maximally flat behaviour at the centre frequency f ¼ f0, i.e. for y ¼ p=2. We put N 1 þ e2jy GBin ðyÞ ¼ Gð0Þ ¼ Gð0Þ cosN ðyÞejNy ð6:33Þ 2 where G(0) is the reflection coefficient at DC and at all frequencies for which the lines are l/2 long, and given by (6.14). As can easily be verified, for y ¼ p/2, GBin ðyÞ is zero along with its derivatives up to (N 1)th order. In other words, the input reflection coefficient has an Nth- order zero at the centre frequency. It is important to stress that, as we will show below, (6.33) is consistent with the general expression (6.32) obtained under the hypothesis of small reflections. In those cases when such an approximation cannot be applied, (6.33) may give rise to unphysical results. In particular, in order to determine the transformer bandwidth for any transformer ratio R, it is necessary to resort to a rigorous theory, which is beyond our scope. While the interested reader is referred to [3], we limit ourselves to providing the final result. By requiring that the squared magnitude of the reflection coefficient be less than 0.5 (or 3.01 dB), i.e. jGBin ðyM Þj2 ¼ 0:5, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4R p 4R 2N 1 ¼ sin yM ¼ cos1 2N 2 ðR1Þ2 ðR1Þ2 ð6:34Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 1 2N 4R w ¼ sin p ðR1Þ2 where we have used (6.16). Figure 6.5 shows the relative 3 dB bandwidth of binomial transformers as a function of the impedance ratio R and for a number of sections varying from N ¼ 1 to N ¼ 15. 5

To derive (6.32) we combine the first and last terms and make use of the following property:

r0 þ rN ej2Ny ¼ r0 ð1 þ ej2Ny Þ ¼ 2r0 ejNy cos Ny

ð6:1Þ

We proceed similarly with the other pairs. When N is even, the central term rN=2 ejNy is not combined with another term.

172

MICROWAVE AND RF ENGINEERING R = 5.828 (Γ= 0.707)

2 1

∆f/ f0

15

7

8 6

0.1 5 n=1

0

2

4

6

4

3

2

8

10

12

14

log10(R) = log10 [(1 +Γ) / (1 − Γ )]

Figure 6.5

The 3 dB relative bandwidth of binomial transformers.

The characteristic impedances Zi (i ¼ 1 . . . N) of the line sections can be obtained by equating (6.32) and (6.33) and using (6.20). In order to do that, we use the well-known expression of the Nth power of a binomial: N X N i Ni ða þ bÞN ¼ ð6:35Þ ab i i¼0 where

N i

¼

N Ni

¼

N! i!ðNiÞ!

ð6:36Þ

is the binomial coefficient. Using (6.35) and taking into account the symmetry property of the binomial coefficients (6.36), expression (6.33) becomes GBin ðyÞ ¼ Gð0Þ2N ð1 þ e2jy ÞN 8 ðN1Þ=2 > X N > > > cosðN2iÞy > X > N N 1 > > cosðN2iÞy þ > : 2 N=2 i i¼0 Equating (6.32) to (6.37) we obtain ri ¼ 2N

ZL Z0 ZL þ Z0

N i

for N odd

ð6:37Þ

for N even

¼ rNi

ði ¼ 1 . . . NÞ

ð6:38Þ

As already mentioned, these are approximate formulae that can be applied under the assumption of small reflections. In practice, a good approximation is obtained when ZL < 2Z0 [3]. The design of the maximally flat transformer can be carried out as illustrated in Example 6.1 at the end of the next section.

6.4.2

Chebyshev polynomials; the Chebyshev transformer

Instead of (6.33), an alternative expression for the input reflection coefficient can be adopted in order to obtain an equiripple behaviour in the operating band. To this end, we need to introduce the Chebyshev

IMPEDANCE MATCHING

173

polynomials: they are based on the property that the Nth power of the cosine of an angle y can be expressed as an Nth-degree polynomial of the cosine itself, i.e. cosðNyÞ ¼ TN ½cosðyÞ

ð6:39Þ

where TN is the Nth-order Chebyshev polynomial of the variable x ¼ cosðyÞ. One can easily recognize that T0 ðxÞ ¼ 1;

T1 ðxÞ ¼ x;

T2 ðxÞ ¼ 2x2 1

ð6:40Þ

Moreover, it can be proved that the following recursive relation holds: TN ðxÞ ¼ 2xTN1 ðxÞTN2 ðxÞ

for N > 2

ð6:41Þ

The computation of the Chebyshev polynomial of any order can thus be performed starting from (6.40) and then applying the recursive relation (6.41). In order to understand how we can utilize Chebyshev polynomials to get an equiripple response, we present the following interesting properties: (a) jTN ðxÞj 1 for

jxj 1

(b) jTN ð 1Þj ¼ 1 1 ðodd NÞ (c) jTN ð0Þj ¼ 0 ðeven NÞ (d) limx ! 1 TN ðxÞ ¼ limx ! 1 2N1 xN : In words, when |x| is less than unity, the amplitude of any Chebyshev polynomial swings between the values 1, touching N 1 times the values 1 or 1. Because of (b), the Chebyshev polynomials are 1 or 1 at the boundaries of the interval x ¼ 1. When the argument |x| is greater than 1, the polynomials increase monotonically towards infinity, the faster, the higher the degree N. Chebyshev polynomials of degrees 1 to 4 are shown in Figure 6.6. N=2

N=4

TN (x)

4 3 2

1 1

−1

1

x

−1 N=1

N=3

Figure 6.6 Chebyshev polynomials of degree N ¼ 1, 2, 3, 4.

174


Because of the above properties, it is easy to see that an equiripple behaviour is obtained by choosing the following expression for the input reflection coefficient: GCheb ðyÞ ¼ Gð0ÞejNy

TN ðcos y=cos ym Þ TN ð1=cos ym Þ

ð6:42Þ

Indeed, when jðcos yÞ=ðcos ym Þj 1, thus for ym y pym, the magnitude of GCheb is between 0 and Gð0Þ rM ¼ ð6:43Þ TN ð1=cos yM Þ For any given N, (6.43) provides the relationship between the transformer bandwidth and the maximum ripple in the band. It also shows that the reflection coefficient is reduced from Gð0Þ ¼ ðR1Þ=ðR þ 1Þ to rM by the factor Gð0Þ ¼ TN ð1=cos yM Þ ð6:44Þ rM As an example, Figure 6.7 shows the frequency behaviours of Chebyshev transformers for 1–4 sections, with impedance ratio R ¼ 20 and relative bandwidth w ¼ 100%. The frequency is normalized with respect to the centre frequency f0. It can be seen that, in the same band 0.5–1.5, the maximum reflection coefficient rM is progressively reduced from6 0.65 to 0.053 when the number of sections is increased from 1 to 4. The ratio (6.44) can be used to determine, for given impedance ratio R, ripple rM and relative bandwidth w, the degree N of the Chebyshev polynomial, thus the number of sections of the transformer. This is shown in Figure 6.8, where the square of (6.44)7 is plotted as a function of the relative bandwidth w, for N ¼ 1. . .15. This diagram is equivalent to Table 6.2 in [4]. 1.0

0.8 N =1

ρ

0.6 2 0.4

3

4

0.2

0.0

0

1

2

2 θ/π

Figure 6.7 Reflection coefficients of Chebyshev transformers up to fourth order vs. the normalized frequency y=ðp=2Þ ¼ f = f0. 6 More precisely, the maximum reflection coefficient is equal to 0.64, 0.302, 0.128 and 0.053 for N = 1, 2, 3 and 4, respectively. 7 2 TN ð1=cos yM Þ represents the improvement in excess loss due to the presence of the transformer, the excess loss being defined as EX ¼ Pav =PL 1 ¼ jGj2 =ð1jGj2 Þ where Pav is the power available and PL the power delivered to the load [4]. The rigorous theory of quarter-wave transformers is indeed based on this quantity rather than on the reflection coefficient. Instead of Equation (6.44) one obtains EXa =EXr ¼ TN2 ð1=cos yM Þ, where EXa is the maximum excess loss and EXr the maximum excess loss in the passband.

IMPEDANCE MATCHING

175

1040

N=15

TN2 (1/cos θM)

1030 11 9

1020 7

1010 5 3 1

100

0

1

2

w = 2 − (4/π)θM

Figure 6.8

Diagram showing TN2 ð1=cos yM Þ in terms of relative bandwidth w ¼ 2ð4=pÞyM .

Once the number of sections has been determined, the characteristic impedances of the various sections can be computed by a procedure similar to that used for the binomial transformer, i.e. by equating expression (6.42) to (6.32). In doing this the nth power of cos y must be expressed as a polynomial in cos y using the following expression: n X n cos½ðn2kÞy cosn ðyÞ ¼ 2n k k¼0

In this way, the Chebyshev polynomials in (6.42) can be expressed as follows: cos y cos y T1 ¼ cos yM cos yM cos y 1 þ cos 2y T2 1 ¼ cos yM cos2 yM cos y cos 3y þ 3 cos y cos y 3 ¼ T3 cos yM cos3 yM cos yM cos y cos 4y þ 4 cos 2y þ 3 1 þ cos 2y T4 4 ¼ cos yM cos4 yM cos2 yM

ð6:45Þ

These expressions can be used to compute the reflection coefficients of the various sections of the transformer up to N ¼ 4. For higher numbers of sections, the reader is referred to [3]. An exact design procedure, not resorting to the theory of small reflections, can also be developed [5].

Table 6.2 Impedances of the two-section transformer in Example 6.1.

Binomial Chebyshev

Z0

Z1

Z2

Z3

1 1

1.414 214 1.420 798

2.828 427 2.815 319

4 4

176

MICROWAVE AND RF ENGINEERING |Γ(θ)|

ZL – Z0 ZL + Z0

maximally flat Chebyshev

ρm,maximally flat = 0.33 ρm,Chebyshev = 0.145 O

θ

π/2

θm,Chebyshev =0.159 π

θm,maximally flat = 0.202 π

Figure 6.9 Frequency responses of binomial and Chebyshev l/4 transformers of fifth degree.

Figure 6.9 shows a comparison of the reflection coefficients of a Chebyshev and a maximally flat transformer of degree N ¼ 5 as a function of the electrical length y. Note that the former provides a better matching over a wider bandwidth.

EXAMPLE 6.1 An impedance ZL ¼ 200 O is to be matched to the reference impedance Z0 ¼ 50 O in the frequency band 900–1100 MHz with a minimum return loss of 20 dB. The relative bandwidth and the impedance ratio are w¼

D f 1100900 ¼ 0:2; ¼ f0 1000

R¼

Znþ1 200 ¼4 ¼ 50 Z0

The first step is to determine the required number N of sections. For N ¼ 1,8 and thus for the simple l/4 transformer since for one section there is no difference between binomial and Chebyshev transformers, the minimum in-band return loss is 20.55 dB ðr ffi 0:093 86Þ, which is too close to the specification. We will then choose N ¼ 2. For a binomial transformer, N ¼ 2 yields a maximum return loss of 36.664 dB ðr 0:014 68Þ. This is much better than the requirement and provides some margin against non-idealities of the actual circuit (such as loss, discontinuity effects and fabrication tolerances). In comparison, a Chebyshev transformer with the same number of sections has a minimum in-band return loss of 42.577 dB ðr 0:00743Þ. The second step consists of computing the impedances of the various sections. The physical length of all line sections is l/4 at the centre frequency. Using formulae (6.38) we obtain the characteristic impedances quoted in Table 6.2. Note the closeness of the values obtained for both transformers. Figure 6.10 shows the responses of the ideal binomial (curve 1) and Chebyshev (curve 2) transformers. The Chebyshev version looks more attractive for practical applications. Therefore, 8

See the Mathcad file 01_Multisection_TL_Transformer.MCD.

IMPEDANCE MATCHING

177

consider a microstrip circuit with substrate permittivity er ¼ 4:4 and thickness h ¼ 3 mm. By numerically inverting the analysis formulae (3.181)–(3.183)9 we obtain the widths and lengths of the two transformer sections given in Table 6.3. The response of the microstrip Chebyshev transformer is shown by curve 3 in Figure 6.10. Although this response has been computed10 without including the effect of the step discontinuity between lines 2 and 3, some degradation due to dispersion and loss can nonetheless be observed. The degradation becomes even more serious when the effect of the step discontinuity from w ¼ 3:004 to w ¼ 0:432 is taken into account, as shown by curve 4 in Figure 6.10. The effect of the reactance due to the discontinuities is somehow equivalent to a local alteration of the distributed capacitance and inductance of the microstrips. This means that the discontinuity modifies the effective impedance and electrical length of the two microstrips. The response can partially be realigned by adjusting widths and lengths. By circuit tuning, it was found that it is sufficient to shorten the lengths of the microstrip sections. The optimized lengths are given in the last row of Table 6.3, while the corresponding response of the transformer is shown by curve 5 in Figure 6.10. Although obtained by simulation, these results can be considered to be quite realistic.

0.06 1) Binomial, ideal 2) Chebyshev, ideal 3) Chebyshev, microstrip (no disc.) 4) Chebyshev, microstrip (disc.) 5) Chebyshev, microstrip (disc., opt.)

0.05

ρ

0.04 0.03 0.02 0.01 0.00 0.8

0.9

1.0

1.1

1.2

Frequency, GHz

Figure 6.10

Comparison of two-section transformers in Example 6.1.

Table 6.3 Values of two-section Chebyshev transformer discussed in Example 6.1. Normalized impedances Zk Denormalized impedances Zk wk (mm) c 1 lk ¼ pffiffiffiffiffiffiffiffi (mm) ee f f 4f0 c 1 lk;opt ¼ pffiffiffiffiffiffiffiffi (mm) ee f f 4 f0

9

Z0

Z1

Z2

Z3

1 50

1.420 798 71.0399 3.004

2.815 319 140.7659 0.432

4 200

42.137

43.784

41

43.7

See the Mathcad file 02_Microstrip_Analysis_Synthesis.MCD in Chapter 12. See the Ansoft file 03_Multisection_TL_Transformer.adsn.

10

178


6.5 Line and stub transformers; stub tuners A simple distributed network able to match complex impedances is the line and single stub circuit shown in Figure 6.11. The matching procedure can be illustrated by the example shown in Figure 6.12, where the load impedance is ZL ¼ ZA ¼ Z0 ð0:5j1:5Þ

ð6:46Þ

In the Smith chart this is represented by the point P corresponding to a reflection coefficient of magnitude 0.745. The transmission line of electrical length y0 connected to the load transforms ZP into ZQ, through a clockwise rotation of an angle 2y0. The line length must be such that the real part of the transformed ^ ¼ 1. In the present admittance YQ ¼ 1=ZQ is equal to Y0 ¼ 1=Z0 , so that the point Q lies on the circle G example, 2y0 ffi 37:3 and the resulting admittance at Q is YQ ¼ ð1 þ j2:236ÞZ01 . The admittance YQ can now be easily matched by parallel connecting a reactive bipole having an opposite susceptance B2. This susceptance can be realized as a lumped capacitor or inductor depending on whether B2 is positive or negative. In this example B2 ¼ ImðYQ Þ ¼ 2:236=Z0

ð6:47Þ

The solution indicated in Figure 6.12 makes use of a stub of impedance Z02 , short- or open-circuited at the other end (a broken line is used to indicate a short or an open circuit). The choice between the two options depends on different practical factors. From a technology point of view, for instance when microstrip technology is employed, short-circuited stubs are generally to be avoided since they require a hole to be drilled in the substrate. From the point of view of circuit sensitivity to fabrication tolerances, the stub should be shorter than l/4. Therefore if B2 < 0 (B2 > 0) one would employ a short-circuited (opencircuited) stub. In the latter case, for example, recalling the input admittance of an open stub (4.15), the length of the stub should be such that Y02 tanðblÞ ¼ B2 ¼ 2:236Y0

ð6:48Þ

Needless to say, the dual circuits of Figure 6.11 can be employed, using series rather than parallel stubs. The design procedure is quite similar and is thus not described here. If the stub is realized in the form of a sliding short circuit, the circuit can be used as a variable impedance tuner commonly employed in laboratory practice. This type of circuit, however, has some serious limitations. Not only does it not allow any complex admittance or impedance to be matched, but also it requires the stub to be located in a position which depends on the load and on the frequency.

θ1 Z 01

P2

P1

ZL

Z02 Open or short-circuit

θ2

Figure 6.11 Line and stub transformer.

IMPEDANCE MATCHING

Figure 6.12

179

Load matching with the circuit of Figure 6.10.

A double-stub tuner, shown in Figure 6.13, partially removes the requirement for variable distance from the load: the matching is obtained only by adjusting the lengths of the stubs, which are placed at a fixed distance d from each other and have the same characteristic impedance Z0 as the source. For simplicity we assume Z0 ¼ 1, and thus all impedances and admittances are considered to be normalized. Suppose the two stubs have lengths l1 ; l2 respectively and the second one is located at a distance l from the load ZL ¼ 1=YL . Figure 6.13b shows the reduced equivalent circuit of the tuner, where the stubs have been replaced by their susceptances: Bi ¼ cotðbli Þ

l

d Y0

Z0=1

Z0=1

Z02

Z01

l2

ði ¼ 1; 2Þ

ð6:49Þ

d ZL

Y0

B2 Z0=1

B1

l1

Open or short-circuit (a)

Figure 6.13

(b)

The double-stub tuner (a) and its reduced equivalent circuit (b).

Y′L

180


and the load has been moved to the first stub and transformed into the admittance YL0 ¼

YL þ j tanðblÞ 1 þ jYL tanðblÞ

ð6:50Þ

The matching of the circuit in Figure 6.13b requires the input admittance to be equal to unity: YL0 þ jðB1 þ tanðbdÞÞ þ jB2 ¼ 1 1 þ jðYL0 þ jB1 Þ tanðbdÞ

ð6:51Þ

This complex equation can be solved to give the two stub lengths l1 ; l2 . If we require the real part of the l.h.s. of (6.51) to be equal to unity, then we obtain the condition 2 2 0 G02 L ½1 þ cot ðbdÞGL þ ½cotðbdÞB1 B2 ¼ 0

ð6:52Þ

Since G0L must be real and non-negative, by solving the above equation the following condition is found: 0 G0L 1 þ cot2 ðbdÞ ¼

1 sin2 ðbdÞ

ð6:53Þ

In the Smith chart, Eqn. (6.53) represents the area external to the circle G ¼ 1=sin2 ðbdÞ. Note that the smaller the value of d, the wider the range of loads that can be matched. The complete solution for stub susceptances is found to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B1 ¼ cotðbdÞ 1B0L tanðbdÞ G0L sec2 ðbdÞ½G0L tanðbdÞ2 ð6:54Þ G02 tanðbdÞ B0L þ B1 þ tanðbdÞ 1ðB0L þ B1 Þ tanðbdÞ B2 ¼ L 2 2 0 0 ½GL tanðbdÞ þ ½1ðBL þ B1 Þ tanðbdÞ From (6.54) the stub lengths are computed by inverting (6.49). The limitation expressed by (6.53) can be removed using a triple-stub tuner. The additional stub enables any complex load to be matched. Moreover, the additional degree of freedom can be used to increase the bandwidth. Electronically driven tuners can be realized by loading the stubs with variable loads. A typical implementation consists of using RF MEMS switches to vary the lengths of the stubs [6].

6.6 Lumped L networks The distributed solutions illustrated in previous sections correspond approximately to a simple lumped L network. Depending upon the specific frequency characteristics of the arbitrary impedance to be matched, one of the four circuits shown in Figure 6.14 can be adopted to provide the widest bandwidth. By possibly interchanging the positions of the load and the source, the four networks can be reduced to just two: a series C/shunt L and a series L/shunt C as shown in Figure 6.15a,b, having highpass and lowdpass responses, respectively. Let us consider first the network of Figure 6.15a. By imposing that the input impedance be equal to Z1;a , the matching condition is easily found to be Z1a ¼

Z2a ð1o2 La Ca Þ þ joLa joCa Z2a o2 La Ca

Let us assume Z2a to be real, so that Z2a ¼ R2a and Z1a ¼ R1a . Reordering (6.55), we obtain L o2 La Ca ðR2a R1a ÞR2a þ joCa R1a R2a ¼0 C

ð6:55Þ

ð6:56Þ

IMPEDANCE MATCHING L 1

C

C

2

1

L

(a)

L 2

1

C

(b)

181

C 2

1

L

(c)

2

(d)

Figure 6.14 Lumped L networks: (a) shunt C, series L; (b) shunt L, series C; (c) series L, shunt C; (d) series C, shunt L. This complex equation is satisfied if the network elements are chosen in such a way that, at the working radian frequency o, one has o2 La Ca ¼

R2a R2a R1a

ð6:57Þ

La Ca

ð6:58Þ

R1a R2a ¼

Equation (6.57) can be satisfied only if R1a < R2a so that the network of Figure 6.15a can be used to match a load R2a larger than the source resistance. Moreover, since the r.h.s. of (6.57) is larger than unity, the equation is satisfied only if the working frequency is higher than the resonant frequency: pffiffiffiffiffiffiffiffiffiffi ð6:59Þ o > oa ¼ 1= La Ca pffiffiffiffiffiffiffiffiffiffiffiffiffi The network therefore has a highpass response. Note that, by interpreting La =Ca as the characteristic impedance of a transmission line, Equation (6.58) is just the same as the matching condition of a quarterwave transformer. Using an identical procedure, for the circuit of Figure 6.15b instead of (6.55) we obtain the matching condition Z2b ð1o2 Lb Cb Þ þ joLb joCb Z2b þ 1

Z1b ¼

ð6:60Þ

which implies that o2 Lb Cb ¼ 1 R1b R2b ¼

R1b R2b

ð6:61Þ

Lb Cb

ð6:62Þ

Also, the network of Figure 6.15b transforms higher into lower resistances (since R1b < R2b ) but, in contrast to Figure 6.14a, it has a lowdpass pffiffiffiffiffiffiffiffiffiffibehaviour, since the working frequency must be lower than the resonant frequency: o < ob ¼ 1= Lb Cb . The design of a matching L network can also easily be performed by using a Smith chart, where both impedance and admittance circles are shown simultaneously, as in Figure 6.16. As an example, consider a

1 Z1a

Ca La

(a)

Figure 6.15

2

1 Z2a

Z1b

Lb Cb

2 Z2b

(b)

Lumped L networks: (a) high pass; (b) low pass.

182


A

+X

–B

Load

G=1

R=1

–X

+B

Z-Chart Y-Chart

Figure 6.16

Matching a load with a lumped L network using the Smith chart.

load RL þ jXL with RL < Z0 , where Z0 is the source impedance. The example applies equally well to any complex load lying on the same constant-R circle. Let us use the network of Figure 6.15b. For simplicity we consider normalized impedances, so that Z0 ¼ 1. Since the load resistance is smaller than the source, we need to connect the load to port 1 and the source to port 2. The series inductor Lb will move P along the constant-R circle R ¼ RL ; its value must be such that the point A lies on the G ¼ 1 circle, i.e. RL þ jXL þ joLb ¼

1 1 þ jBA

Equating the real and imaginary parts of the above equation, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RL ð1 RL Þ XL Lb ¼ o rffiffiffiffiffiffiffiffiffiffiffiffi 1 1 BA ¼ RL

ð6:63Þ

ð6:64Þ

The parallel inductor Lb must be chosen in such a way as to move the representative point from A to the centre of the chart ðG ¼ 0Þ, i.e. to compensate for the susceptance BA . Using the second equation of (6.64), we obtain Cb ¼

BA XL þ oLb ¼ o oRL

ð6:65Þ

in agreement with (6.62) when XL ¼ 0. The reader can easily verify that the first equation of (6.64) coincides with (6.61) by putting XL ¼ 0, R1a ¼ RL and R2a ¼ 1. In the general case when the impedance to be matched is complex, as typically happens for active circuits, the networks of Figure 6.15 can be used in combination with proper reactive components capable of compensating for the imaginary part of the load.

IMPEDANCE MATCHING

183

Let ZL ¼ RL þ jXL be the load impedance and Z0 the reference or source impedance. There are four possible cases: 1. RL > Z0 , XL > 0, i.e. high resistance with excess inductance. The latter can be compensated for by series connecting to the load, at port 2 of Figure 6.15a or Figure 6.15b, a capacitor with opposite reactance XL. 2. RL > Z0 , XL < 0, i.e. high resistance with excess capacitance. Same as case 1, but using a series inductor with reactance XL . 3. RL < Z0 , XL > 0, i.e. low resistance with excess inductance. Since RL < Z0 we must use the networks of Figure 6.15 in reversed position, i.e. with the load at port 1 and the source at port 2. Since XL > 0, by using Figure 6.15b we can absorb the reactance into Lb . The inductor is thus replaced by L0 ¼ LXL =o. This is clearly possible under the condition Lb XL =o. 4. RL < Z0 , XL < 0, i.e. low resistance with excess capacitance. Proceeding as in case 3, we can use Figure 6.15a in reversed position, increasing the capacitance to Ca0 ¼ ð1=Ca þ oXL Þ1 , provided that Ca ðoXL Þ1 . Similar to the distributed networks, the lumped L networks described above perfectly match the load at a given frequency. When a broad band is required, multiple cascaded networks, even of different types, can be designed. Fano’s bound expressed by (6.1) is in any case insurmountable. EXAMPLE 6.2 Consider the matching of a real load using a lumped L network with the following design specifications:11 f0 ¼ 1 GHz Z1b ¼ R1b ¼ 50 O;

Z2b ¼ R2b ¼ 200 O

ðR ¼ Z2b =Z1b ¼ 4Þ

From Equations (6.61) and (6.62), it follows that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1b R2b R1b R1b R2b R1b ¼ Lb ¼ ffi 13:783 nH o ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pf0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 1 R2b 1 1 R2b 1 ¼ 1 ffi 1:378 pF Cb ¼ o R2b R1b 2pf0 R2b R1b pffiffiffiffiffiffiffiffiffiffi The resulting cut-off frequency is fb ¼ ob =2p ¼ 1= 2p Lb Cb ffi 1:155 GHz, which exceeds f by a margin of 155 MHz. The network is perfectly matched at 1 GHz, as expected, and shows a return loss better than 10 dB in the range 0:745 < f < 1:202 GHz. Figure 6.17 shows the return loss with the 10 dB limits marked with squares. Looking at the curve on the Smith chart12 shown in Figure 6.18, we can see that: 1. At DC the input impedance coincides with Z2b ¼ R2b . 2. At 1 GHz the input impedance coincides with Z1b ¼ R1b , as desired and in agreement with the scalar plot. 3. When the frequency approaches infinity, the impedance at port 1 of the network tends to the open circuit because of the presence of the series inductor Lb.

11

See the Ansoft file 02_Lumped_LC_Matching.adsn. Screenshot from the Ansoft program Ansoft Designer Student Version, running the file 02_Lumped_LC_Matching.adsn. 12

MICROWAVE AND RF ENGINEERING 0

−10 20 log10(ρ)

184

−20

−30

−40 0.0

Figure 6.17

0.5

1.0 Frequency, GHz

1.5

2.0

Reflection coefficient of the lumped L matching network of Example 6.2.

110

90

100

80

70

1.00

120

60 50

130 2.00

0.50

140

40 30

150 160

5.00

0.20

20 10

170

0 Hz 0.00 0.0

180

0.20

0.50

1.00

2.00

5.00

0

∞ Hz

1 GHz

−170

−10

−160

−5.00 −20

0.20

−150

−30 −40

−140 −130

−0.50

−120

1.0

Figure 6.18

−110

−2.00 −1.00 −100

−90 0.0

−80

−70

−50

−60 1.0 ρ

Reflection coefficient of the lumped L matching network of Example 6.2.

IMPEDANCE MATCHING

185

Bibliography 1. R. M. Fano, ‘Theoretical limitations on the broadband matching of arbitrary impedances’, Research Lab. Electronics, MIT Technical Report 41, January 1948. 2. D. M. Pozar, Microwave Engineering, 2nd edition, John Wiley & Sons, Inc., New York 1998, pp. 295–297. 3. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York. 1992. 4. G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1964. 5. R. E. Collin, ‘Theory and design of wide-band multisection quarter-wave transformers’, Proceedings of the IRE, pp. 179–185,1955. 6. M. Unluet al., ‘A reconfigurable RF MEMS triple stub impedance matching network’, Proceedings of the 36th European Microwave Conference, Manchester, pp. 1370–1373, 2006.

Simulation files Ansoft files 01_Multisection_TL_Transformer.adsn. Analyzes the two-section transmission line transformer of Example 6.1. 02_Lumped_LC_Matching.adsn. Analyzes the lumped LC matching network considered in Example 6.2.

7

Passive microwave components 7.1 Introduction In contrast with lumped element circuits which comprise a reduced number of elements (resistors, capacitors, inductors, etc.), there exists a large variety of microwave and RF components. They differ from each other depending on the function performed and the technology employed, whether waveguide, coaxial cable, microstrip line or others. Far from providing an exhaustive discussion of the subject (the interested reader is referred to the vast literature on this subject, see for instance [1–3]), we intend here to present a description of some of the most common or most important components from the application or educational points of view. Filters, matching networks and resonators are not considered here; because of their distinct importance they are treated in specific chapters. A possible classification of passive microwave components is based on the number of physical ports and is shown in Table 7.1. The presence of internal losses and the reciprocity of the components are important parameters that affect the performance of microwave components. In the ideal case, most components (except, for example, attenuators) are assumed to be lossless, since power loss is generally an undesirable, though to some extent unavoidable, phenomenon. Both losslessness and reciprocity, as we saw in Chapter 4, imply certain specific properties of the matrix representations (impedance, admittance or scattering) of such components. Special consideration is given to non-reciprocal components, for which Lorentz’s theorem (discussed in Section 2.7.2) does not apply, so that their representative matrices are not symmetrical (Section 4.6). Typical parameters used to characterize the performance of passive components are the insertion loss, the return loss and the isolation between ports of multi-port devices. In addition, the maximum power that the component can stand is an important parameter for high-power applications. It is worth stressing that, in practice, any parameter is frequency dependent and must therefore be specified in connection with a given frequency range.

7.2 Matched loads A matched load or matched termination is employed to absorb all the incident power in such a way as to realize the matching condition. It is thus a one-port device, which ideally exhibits over the operational Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-75862-5

188

MICROWAVE AND RF ENGINEERING Table 7.1 Classification of passive microwave components. One-port Two-port

Two-port non- reciprocal Three-port Three-port non- reciprocal Four-port (1 þ N)-port (M þ N)-port

Matched load Movable short Interconnection, bend Attenuator Phase shifter Isolator T, Y junction Wilkinson divider/combiner Circulator Hybrid junction Directional coupler Divider/combiner Multiplexer Multi-beam forming networks

frequency range a null reflection coefficient (G ¼ 0), and thus an input impedance equal to the reference impedance (Zc ¼ Z0 ). Matched loads are extensively and typically used when measuring the scattering parameters of microwave components, in order to realize the matching condition of those ports that are not connected to the source or to the measuring instrument. Matched terminations can be realized in lumped or distributed forms. An example of the latter category in rectangular waveguide technology is shown in Figure 7.1. The load is made of a slab of lossy material located at the centre of the rectangular waveguide (where the electric field of the dominant TE10 mode is maximum) and parallel to it. The slab is tapered in such a way as to create a discontinuity that is as smooth as possible within the maximum allowed length. Normally, the length of the slab is of the order of one guided wavelength. Distributed terminations have the advantage of producing very small reflections but are bulky. Lumped loads are simply realized with a layer of conducting material capable of absorbing the incident power. To minimize reflections, the conductivity of the material is chosen is such a way that the surface impedance is close to the wave impedance of the incident field. Figure 7.2 shows a lumped matched load in coaxial cable technology.

7.3 Movable short circuit A movable short circuit allows an arbitrary reactive load to be realized at the input of a waveguide section. The device consists of a metal sliding block terminating a waveguide stub, so realizing a short-circuited

Figure 7.1

Matched load in rectangular waveguide.

PASSIVE MICROWAVE COMPONENTS

Figure 7.2

189

Lumped matched load in coaxial cable.

stub whose length can be changed when operating with a micrometer plug. In rectangular waveguide technology, the sliding short can be realized through a sliding contact like that shown in Figure 7.3. The problem in the realization of such a device is that it should ensure the perfect closure of the currents flowing on the upper wall with those on the lower wall of the rectangular waveguide. This requires an excellent contact between the mobile leaf and the waveguide walls, but this would produce rapid wearing of the metal surfaces. To overcome such a difficulty, one has to resort to contactless plungers, such as that depicted in Figure 7.4a. The inner section of the plunger is in loose contact with the waveguide wall so as to realize, rather than a short circuit, a non-zero impedance ZS given by ZS ¼

RS 1 þ joCS RS

ð7:1Þ

Such an impedance results from the parallel contact resistance RS between the movable and fixed walls and the capacitance CS of the gap between them. In order to reduce the impedance seen from the waveguide end, the plunger is shaped in such a way as to realize two l/4 sections with impedances Za and Zb with Za Zb . Because of the well-known properties of l/4 sections, the input impedance is easily

Figure 7.3

Sliding short circuit.

190

MICROWAVE AND RF ENGINEERING λ/4

λ/4 micrometric screw

outer conductor inner conductor

(a)

Zs

Zin

Zin

sliding contact

λ/4

λ/4

Za

Zb

Zs

(b)

Figure 7.4

Two-section sliding plunger (a) and its equivalent circuit (b).

found to be Zin ¼

2 Za ZS Zb

ð7:2Þ

By choosing Zb ffi 10Za , a reduction of two orders of magnitudes in the impedance ZS can easily be realized.

7.4 Attenuators An attenuator is a two-port device used to reduce the amplitude of the input signal, ideally without reflections, thus with s11 ¼ 0. Attenuators may be fixed or variable and differ substantially one from each other depending on the technology adopted. Rectangular waveguide attenuators are realized by placing thin slabs of a lossy material along the E plane, thus parallel to the electric field of the dominant mode. The electric currents induced by virtue of Ohm’s law dissipate the EM energy into heat by the Joule effect, so reducing the power travelling along the guide. The variable attenuator in a rectangular waveguide is the rotary attenuator. It consists of a circular waveguide section, interposed within two rectangular waveguides and loaded with a thin slab of resistive material. The latter can be rotated around the longitudinal axis at different angles with respect to the electric field, so as to vary the quantity of dissipated power. It is apparent, in fact, that the power dissipated is maximum when the lossy slab is parallel to the electric field, and minimum when the slab is orthogonal to it. A detailed description of the rotary attenuator can be found in [4]. In microwave integrated circuits (MICs) attenuators may be realized in the form of T or P symmetrical resistive networks, as shown in Figure 7.5. Variable attenuation can be obtained using solid state devices such as PIN diodes or FETs whose resistance can be varied electronically. Let us consider the T attenuator. Its impedance matrix can be readily computed (see Section 4.6.1) as z11 z12 RT2 RT1 þ RT2 ¼ ð7:3Þ z21 z22 RT2 RT1 þ RT2


(a)

Figure 7.5

191

(b)

T (a) and P (b) resistive attenuators.

Using Table 4.1, we compute the scattering matrix:

s11 s21

s12 s22

^ T1 þ 2R ^ T2 1 ^ T2 ^ R 1 1 2R R T1 ¼ ^ T1 R ^ T2 ^ T1 þ 2R ^ T2 1 ð7:4Þ ^ T1 þ 1 R ^ T1 þ 2R ^ T2 þ 1 2R R R

where the symbol ^ indicates normalization with respect to the reference impedance Z0. Imposing the condition that the device must be matched at both ports, thus s11 ¼ s22 ¼ 0, we obtain the condition ^2 ^ T2 ¼ 1RT1 R ^ T1 2R

ð7:5Þ

Since both resistances must be non-negative, one can infer that ^ T1 1 or RT1 R0 R

ð7:6Þ

Using condition (7.5) in (7.4), we obtain for s21 ^ T1 1R ^ T1 1þR

ð7:7Þ

L ¼ 20 log10 ðs21 Þ

ð7:8Þ

s12 ¼ s21 ¼ From this equation we can evaluate the attenuation

as a function of the resistance RT1 . From the design point of view, we are interested in evaluating the resistance values for a prescribed attenuation. From (7.5), (7.7) and (7.8) we obtain s21 ¼ 10L=20 1s21 RT1 ¼ Z0 1 þ s21 2s21 RT2 ¼ Z0 1s221

ð7:9Þ

In a similar manner, using the admittance matrix we can obtain the synthesis formulae for the P attenuator: 1 þ s21 1s21 1s221 ¼ Z0 2s21

RP1 ¼ Z0 RP2

ð7:10Þ

192

MICROWAVE AND RF ENGINEERING substrate edge

microstrip RΠ1 (71.15 Ω)

1

2

(96.25 Ω) RΠ1

RΠ1

ground connection

3.7 mm

Figure 7.6 Layout of a microstrip attenuator in P configuration. The substrate is h ¼ 0.508 mm thick with dielectric constant er ¼ 9:9. -8

0

20 log10(|s21|)

-10

(8.7 GHz, -9.5 dB)

s11

-12

-20

-14

-30

-16

-40

-18

0

5

10

15

20 log10(|s11|)

s21

-10

-50 20

Frequency, GHz

Figure 7.7

Scattering parameters of the attenuator in Figure 7.6.

As an example, Figure 7.6 shows the layout of a P attenuator in microstrip line, designed to provide an attenuation of 10 dB. The resistors are realized with resistive films similar to the monolithic ones.1 The two resistors RP1 have been connected to ground by wrapping the back metallization around the substrate edge. The circuit response,2 shown in Figure 7.7, is affected by the parasitics associated with the resistors and the ground connection.3 The attenuation (nominally 10 dB) actually decreases with frequency, with a 1

See Section 14.3.3. See the Ansoft file 01_Attenuator.adsn. 3 See again Section 14.3.3. 2


a1 Figure 7.8

[S]

b2 = a1e-jθ

S

0 e

e

jθ

193

jθ

0

Two-port representation of an ideal phase shifter.

maximum deviation from the nominal 10 dB attenuation of slightly more than 2 dB at 20 GHz, while the deviation from the nominal value is less than 0.5 dB below 8.7 GHz. As the frequency increases, the matching also deteriorates: nominally infinite, the return loss is only 12 dB at 20 GHz. By replacing the resistors with variable-resistance diodes one can realize a continuously (or analogue) variable attenuation. As an alternative, digital attenuators can be made using various attenuating cells cascaded one after the other and connected via switched lines.4

7.5 Fixed phase shifters The phase shifter is a two-port device used to produce either a fixed or controllable phase shift to the input signal. Phase shifters are used in a variety of communication and radar systems. They are essential components in antenna systems and phased arrays, where the radiating elements must be fed with appropriate phases so as to create the required radiated beam. In general, fixed phase shifters can potentially become variable phase shifters by replacing some fixed reactances with electronically tunable ones. In this section we give only a brief account of fixed phase shift networks, the variable phase shifters being discussed in Section 10.4. The reader is referred to it for more detailed information. In the ideal case, a phase shifter can be defined as a two-port device perfectly matched (s11 ¼ s22 ¼ 0) and without loss (thus js21 j ¼ js12 j ¼ 1), where the output signal undergoes a prescribed phase shift Df with respect to a given reference. The two-port schematic and the scattering matrix of an ideal phase shifter are shown in Figure 7.8. The phase shift of the output signal is usually evaluated with reference to the phase shift of a line section of equal length as the device. Let l be the length of such a line section and b its phase constant; the phase shift is thus evaluated as Df ¼ ybl

ð7:11Þ

Depending on the frequency behaviour of Dfð f Þ, there are two fundamental types of phase shifters: .

The true-phase phase shifter, whose Dfðf Þ is constant with frequency.

.

The true-time-delay or TTD phase shifter, whose Dfðf Þ varies linearly with frequency, so that the device behaves as a delay line.

Depending on the circuit typology, we can have transmission or reflection phase shifters. In the former the phase shift is produced by the transmission of the signal through the device, while in the latter one exploits the phase shift produced by the reflection from a reactive load. We now consider the various typologies.

7.5.1

Loaded-line phase shifters

As shown in Figure 7.9a, the loaded-line architecture consists of a transmission line with characteristic impedance Zc and electrical length y loaded at both ends with identical parallel admittances that can switched between two different states. Let the admittances be Yi ¼ Gi þ jBi where i ¼ 1, 2 denotes the two possible load states. Switching between the two states produces a phase shift in the RF signal travelling 4

See Section 10.3.

194


θ

Yi

1

Zc

Yi

L

C/2

(a)

2 C/2

(b)

Figure 7.9

Loaded-line phase shifter (a) and a lumped element realization (b).

along the line. This phase shifter is attractive because of its simplicity of fabrication, but it gives rise to a mismatch that increases with the phase shift required. As a consequence, this configuration is usually limited to applications that require small phase shifts. Figure 7.9b shows the lumped element version of a loaded-line phase shifter. This circuit can be considered asffi a lumped element approximation pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi of a transmission line length of characteristic impedance ZC ¼ L=C and propagation delay t ¼ LC. This architecture therefore realizes a TTD phase shifter.

7.5.2

Reflection-type phase shifters

To illustrate the behaviour of a reflection-type phase shifter, we use a specific example. Consider a grounded variable capacitor C seen from a reference line of characteristic impedance Z0. The reflection coefficient G¼

^ 1joC ¼ e jf ; ^ 1 þ joC

^ j ¼ 2 tan1 ðoCÞ

ð7:12Þ

^ ¼ CZ0 . The phase is unitary in amplitude and has a phase j depending on the normalized capacitance C difference between the reflected and the incident waves can therefore vary from 0 for zero capacitance (thus when the load is an open circuit) to 180 in the limit when the capacitance becomes infinite (so realizing a short circuit). A phase shifter can thus be obtained using a 90 hybrid junction (see Section 7.7.2) for separating the incident from the reflected signal as shown in the schematic of Figure 7.10. The direct and coupled ports of the coupler are terminated with two identical reactances. If such reactances can be varied simultaneously by a control signal, a tunable phase shifter is obtained. As shown in Section 7.7.2, the signal incident on port 1 is split between port 2 (direct port) and port 3 (coupled port) with equal amplitudes and 90 phase shift. The same behaviour is shown, apart from the relevant index changes, for signals entering the other ports. As a consequence, the signals reflected back into the hybrid junction by the two equal loads at ports 2 and 3 will cancel at port 1 (since they arrive there 180 out of phase) while they sum at port 4 (where they arrive in phase). Depending on the values of the reactances, 1

2 1

4

0°

0° 90° hybrid junction

90° 90° 2

3

jX

Figure 7.10

jX

Schematic of the reflection-type phase shifter.


195

thus on their reflection coefficients G, the output signal (port 4 of the hybrid junction) undergoes a phase shift j, according to Equation (7.12).

7.6 Junctions and interconnections At low frequencies the interconnections between the components of an electric circuit are normally so small compared with the wavelength that they can be considered as infinitesimal points with no dimensions. At microwave frequencies, on the contrary, the interconnections between physical elements have finite sizes that involve time delay (in the time domain) and phase shift (in the frequency domain) phenomena. In addition, any discontinuity produces local distortions of the EM field distribution. Such distortions can be seen as due to the excitation of higher order modes; they are associated with the storage of reactive energy and, from a circuit point of view, can be modelled as lumped reactances. For the above reasons, even the simplest junction between different transmission lines should properly and accurately be designed in order to minimize unavoidable mismatch and reflections. In all cases one needs to be able to model such reactive phenomena in order to possibly compensate for them. In other cases, as we have mentioned in Chapter 4, reactive phenomena associated with discontinuities are intentionally used to realize specific circuit elements such as inductances or capacitances that, at high frequencies, cannot be realized using conventional geometries. Let us first consider conventional guiding structures such as the coaxial cable or the waveguide. The simplest junction is that between lines with equal geometries but filled with different dielectric materials. The abrupt change of dielectric filling does not alter the cross-sectional distribution of the electric and magnetic fields but only the wave impedance, thus the characteristic impedance, and the phase constant, thus the wavelength and the phase velocity5. Consider a coaxial cable filled with a material of dielectric constant er connected to an identical but empty cable. This type of discontinuity is encountered in coaxial connectors. Since both the characteristic pffiffiffiffi impedance and the wavelength are inversely proportional to er , removal of the dielectric produces an increase of both quantities by the same factor. Assuming that the cable is matched or infinitely long, the discontinuity will produce a reflection coefficient pffiffiffiffi pffiffiffiffi er 1 11= er G¼ ð7:13Þ pffiffiffiffi ¼ pffiffiffiffi 1 þ 1= er er þ 1 The same problem may arise in the case of a waveguide, e.g. a rectangular waveguide. In such a case, pffiffiffiffi however, the phase constant is not simply proportional to er but depends on the dielectric constant according to the following expression which is valid for the dominant TE10 mode (see (3.133)): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi kz ¼ o m0 e0 er 1ð fc10 = f Þ2 where fc10 ¼

1 c pffiffiffiffiffiffiffi ¼ pffiffiffiffi 2a m0 e 2a er

Removal of the dielectric thus also produces an increase in the cut-off frequency. For suitable values of the frequency and of the waveguide width a it may be that in the empty waveguide the mode is below cut-off, thus evanescent. In such a case, total reflection is produced (jGj ¼ 1). In many practical cases one needs to join cables or waveguides with different cross-sections, as shown by the examples in Figure 7.11. In contrast to the simple dielectric discontinuity that does not alter 5 The two-dimensional boundary value problem associated with computation of the waveguide modes depends on the geometry of the metal boundaries and does not include the dielectric material, as long as the latter uniformly fills the waveguide cross-section. The dielectric constant arises when determining the phase constant kz . See Section 3.2.

196


Figure 7.11 Junctions between cables and guides of different cross-sections: (a) step in the inner conductor of a coaxial cable; (b) E plane step in a rectangular waveguide; (c) symmetrical H plane step in a rectangular waveguide. the cross-section, thus the EM field distribution, the step discontinuity gives rise to excitation of higher order modes, which are normally below cut-off. This produces the storage of reactive energy in the proximity of metal edges. Such energy is mainly of electric (or magnetic) type when the electric (or magnetic) field is perpendicular to the edge. Indeed the normal components of the EM field become exceedingly large at very close distances from the metal edge.6 This is why the equivalent circuit contains a capacitor in the first two cases of Figure 7.11 and an inductor in the case of the H plane step – recall the field distributions of the TEM mode in the coaxial cable and that of the TE10 mode in the rectangular waveguide. In the case of the H plane step, one has also to consider that the change in the waveguide width implies a change in the cut-off frequency, with the same possible consequences as for the dielectric step, i.e. that the field is evanescent in the narrower section. One has also to keep in mind that the any step 6

For a detailed description of the phenomena occurring at metal edges see [5, 6].


197

Figure 7.12 Rectangular waveguide H plane (a) and E plane (b) junctions and their respective equivalent circuits. discontinuity implies a change in the characteristic impedance, as highlighted in the equivalent circuits of Figure 7.11. In many microwave circuits one needs to join together two or more transmission lines. There are clearly many different configurations. For a rectangular waveguide, typically there are the T junctions, where one branch waveguide is inserted orthogonally to a main waveguide, either in the E or H plane. Correspondingly, we have the E and H plane T junctions shown in Figure 7.12 along with their equivalent circuits. Note that the E plane junction corresponds to the series connection of the branch line, while the H plane junction corresponds to the shunt connection. This is easily understood by focusing attention on the waveguide longitudinal currents, which flow at the centre of the wider waveguide walls. In order to reduce the mismatch, and thus the reflections that a simple junction produces, suitable metal elements (posts, diaphragms, wedges, etc.) can be inserted in the junction region to compensate for the reactances associated with the discontinuity. In a similar fashion, also for printed circuits, such as striplines, mictrostrips, coplanar waveguides, etc., many different junctions and discontinuities are present. Two examples in microstrip line are shown in Figure 7.13.

Figure 7.13

Symmetrical step discontinuity (a) and T junction in mictrostrip line (b) [7].

198


Besides the junctions between guiding structures of different sizes, great importance has been attached to the so-called transitions, i.e. the junctions between transmission lines of different technologies, such as the interconnections of waveguide to coaxial cable, waveguide to microstrip, microstrip to coaxial cable, microstrip to CPW, etc. Such interconnections (coax connectors can be numbered among them) are necessary in order to join portions of circuits made with different technologies, as typically occurs in the measurement or testing of components. The transitions between lines with totally different configurations, such as a rectangular waveguide and a microstrip line, generally require complicated analyses in order to identify suitable transition geometries capable of transferring the signal from one line to the other with minimum power loss in the given frequency band. Because of the practically unlimited variety of possible solutions, no general criteria can be provided here to help identify such geometries, except that the transition has gradually to convert the field configuration of the dominant mode of one line into that of the other line. Once again we will limit our attention to a few examples that can at least provide an idea of the problems involved.

7.6.1

Guide-to-coaxial cable transition

The transition from a rectangular waveguide to a coaxial cable (sometimes called adapter) is necessary in many practical applications, specifically for measuring waveguide components in a coaxial cable measuring environment. The coupling between the waveguide TE10 mode and the coax TEM mode occurs by extending the cable’s inner conductor into the waveguide. The former can be placed parallel to the E field of the waveguide in order to maximize the interaction between the current flowing in the conductor and the E field of the waveguide. This is the voltage probe depicted in Figure 7.14a. As an alternative, the central conductor can be folded into a ring shape so as to couple with the magnetic field, creating a current probe as depicted in Figure 7.14b. The length of the conductor and its distance from the waveguide wall must be determined in such a way as to optimize the matching, thus maximizing the power transfer from one to the other guiding structure. Accurate design of the transition requires the use of numerical techniques to account for the thickness of the probe, the

(a) voltage probe

(b) current probe

y

y

z

O

x

x

y

O z

Figure 7.14

z

O

y

x

O

x

z

Geometries of the waveguide-to-coax transitions: (a) voltage probe; (b) current probe.

PASSIVE MICROWAVE COMPONENTS 0

199

15

-1

10

-2

5

-3

0

-4

-5

-5

-10

-6

-15

s11

-7 -8 15

10 log10(|s11|)

10 log10(|s21|)

s21

-20 20

25

30

35

-25 40

Frequency, GHz

Figure 7.15 Measured scattering parameters of a pair of WR-28-to-coax transitions (both input and output are in coaxial cable).

possible presence of dielectric coating, and other tiny details. A simplified theoretical approach, though not extremely accurate, is presented later in this section. For a more accurate numerical analysis the reader is referred to [8]. As an example of a practical result, Figure 7.15 shows the frequency response of a pair of transitions of the voltage probe type between a coaxial cable and a WR-28 rectangular waveguide. The latter operates in the Ka band, 26.4–40 GHz, has a 7.112 mm 3.556 mm cross-section and fc ¼ 21.08 GHz cut-off frequency. As can be seen, the transition shows a broad-band behaviour with an insertion loss of the order of 0.1–0.2 dB reaching a maximum of about 0.4 dB at the upper frequency edge (the measured values refer to a pair of transitions). The reader should note that at frequencies below the operating frequencies of the waveguide, the return loss is not exactly zero but swings around the zero level assuming positive values also. This of course does not mean that this is an active device, it is simply due to the directivity error of the scalar network analyzer, as discussed in Section 17.6.

From a modelling point of view, the interconnection between a coaxial cable and a waveguide can be treated in a general way as the excitation of a waveguide by a current source.7 To focus the problem, let us assume that a known (or ‘impressed’) current density Ji is present in the waveguide portion between z1 and z2 as sketched in Figure 7.16. The current will excite waveguide modes propagating in the positive z direction for z > z2 and modes propagating in the negative z direction for z < z1. To simplify the formulation and reduce the number of equations, we will use the superscript ‘ þ ’ for the fields in z z2 and ‘’ for the fields in z z1. Using the expansions (3.106) with Vn ðzÞ ¼ Vnþ ejkzn z for z z2 or Vn ðzÞ ¼ Vn ejkzn z for z z1, the EM field can be expressed as

7

The theory presented here follows that in [4] with some small changes.

200


Figure 7.16

E t ¼

Excitation of a waveguide by a current source.

1 1 X X Vn ðzÞetn ðx; yÞ ¼ Vn etn ðx; yÞe jkzn z n¼0

n¼0

1 1 X X Vn ^z etn ðx; yÞe jkzn z H In ðzÞhtn ðx; yÞ ¼ t ¼ Zzn n¼0 n¼0

Ez

1 1 2 X 1 X ktn 2 ¼ ktn In ðzÞTnTM ðx; yÞ ¼ j V þ T TM ðx; yÞe jkzn z j oe n¼0 k n n n¼0 zn

Hz ¼

ð7:14Þ

1 1 1 X 1 X 2 ktn Vn ðzÞTnTE ðx; yÞ ¼ k2 V þ T TE ðx; yÞe jkzn z j om n¼0 j om n¼0 tn n n

We have used the superscripts ‘TM’ and ‘TE’ to indicate that the corresponding potential is relative to TM or TE modes, respectively. The above expressions can be put in the simpler form E ¼

1 X ðetn ezn ÞVn e jkzn z n¼0

H ¼

1 X 1 ^z etn þ hzn Vn e jkzn z Zzn n¼0

ð7:15Þ

with ezn ¼ ^z

2 jktn T TM ðx; yÞ kzn n

k2 hzn ¼ ^z tn TnTE ðx; yÞ j om

ð7:16Þ

Notice that the series in (7.15), as far as the z components are concerned, involve only TM or TE terms, respectively. Let us now apply Lorentz’s reciprocity theorem (2.76) to the volume bounded by the crosssections at z ¼ z1 and z ¼ z2 and by the waveguide side walls. We use as the field E1, H1 that radiated


201

by Ji, i.e. the field (7.15), and as the field E2, H2 that of the nth waveguide mode for z z1: E2 ¼ ðetn ezn Þe jkzn z 1 ^z etn þ hzn e jkzn z H2 ¼ Zzn

ð7:17Þ

Since no current is associated with the waveguide mode (J2 ¼ 0), the expression of Lorentz’s theorem (2.76) in this case becomes þ ð ðE H2 E2 H Þ dS ¼ Ji E2 dV ð7:18Þ S

V

The surface integral on the l.h.s. of (7.18) vanishes on the waveguide side walls (since the tangential electric field is zero). Moreover, since the modes are orthogonal, only the nth terms in the expansions (7.15) of E and H give a non-zero contribution. Therefore, the surface integral reduces to the two cross-sections at z ¼ z1 and z ¼ z2. Using (7.14) and (7.17) in (7.18), it is immediately found that the integral over the cross-section at z ¼ z1 is also zero, so that the only remaining term of the surface integral is ð ð 2 þ 2 þ ðE H2 E2 H Þ dS ¼ Vn etn ð^z etn Þ ^z dS ¼ V ð7:19Þ Zzn Zzn n S2 S2 where we have taken into account that etn ð^z etn Þ ^z ¼ etn etn and used the orthonormalization property of the vectors etn . We can now compute the amplitude Vnþ of the nth mode for z z2. Inserting (7.19) into (7.18) and using (7.17), we obtain ð Zzn jkzn z2 Ji ðetn ezn Þ dV ð7:20Þ e Vnþ ¼ 2 V In much the same way, if we choose for E2, H2 the forward propagating mode in the region z > z2, we obtain ð Zzn jkzn z1 Ji ðetn þ ezn Þ dV ð7:21Þ e Vn ¼ 2 V Let us now apply the above formulae to characterize the voltage probe, sketched in Figure 7.14a. Under the assumption that the probe radius is very small compared with the wavelength, we can approximate the current density as a straight linear filament of zero thickness, parallel to y, extending into the waveguide by a length h, as schematically shown in Figure 7.17. That is, Ji ¼ ^y

I0 sin½b0 ðhyÞdðxa=2ÞdðzlÞ A

ð7:22Þ

where I0 is the total current flowing through the cross-sectional area A of the wire, b0 is the free space phase constant and d is the Dirac delta function. The probe is assumed to have a sinusoidal current distribution, the current obviously being zero at the end y ¼ h of the probe. Since the current is directed along y, (7.20) shows that only the y component of etn is involved. We observe, however, that in contrast to Figure 7.16, the waveguide is terminated at z ¼ 0 with a conducting wall. To take into account the presence of the metal wall, whose effect is to create a zero tangential electric field in z ¼ 0, we simply add a second current source in z ¼ l having the opposite sign. This additional source is the image of the real source and has the effect of creating the same boundary condition in z ¼ 0 as the electric wall. To include the effect of the electric wall we simply replace (7.22) with Ji ¼ ^y

I0 sin½b0 ðhyÞdðxa=2Þ½dðzlÞdðz þ lÞ A

ð7:23Þ

202


Figure 7.17 Modelling the voltage probe of Figure 7.14a. Using the expressions in Table 3.4 for rectangular waveguide modes, to be denoted by the pair of indexes m, n instead of just n, we obtain ð Zzmn I0 mp mp jkzmn l jkzmn l h npy þ Vmn ¼ e sin½b0 ðhyÞ cos Cmn sin e dy 2 a 2 b 0 ð7:24Þ ðh mp mp npy sin sinðkzmn lÞ sin½b0 ðhyÞ cos dy ¼ jZzmn I0 Cmn a 2 b 0 For the TE10 mode, the integral in (7.24) is ðh

sin½b0 ðhyÞdy ¼

0

1cosðb0 hÞ b0

ð7:25Þ

and (7.25) becomes p 1cosðb0 hÞ þ V10 ¼ jZz10 I0 C10 sinðkz10 lÞ a b0

ð7:26Þ

The electric field of the dominant mode is then found to be Eyþ ¼ V10þ ey10 ejkz10 z ¼ 2j

Zz10 I0 1cosðb0 hÞ px jkz10 z sin e sinðkz10 lÞ ab b0 a

ð7:27Þ

The higher order mode amplitudes can also be computed using (7.23). For n 6¼ 0 the integral in (7.24) is ðh

sin½b0 ðhyÞcos

0

npy np=b n ph cos hÞ dy ¼ 2 cosðb 0 b b b0 ðnp=bÞ2

ð7:28Þ

and the voltage expansion coefficients become þ ¼ j Zzmn I0 Cmn Vmn

mp np=b n ph cos hÞ sinðkzmn lÞ 2 cosðb 0 a b b0 ðnp=bÞ2

ð7:29Þ


203

In the computation of the EM field of the higher order modes, we may assume that the frequency is such that they are all below cut-off, thus evanescent. In this case, the propagation constants kzmn are imaginary and so are the wave impedances Zzmn. We are going no further into that, since, for simplicity, we are interested here in the dominant mode; this means that we are going to consider only the real power radiated into the waveguide, while the reactive power is associated with the evanescent modes. The real power transported by the fundamental mode is simply computed as follows: P¼

ð ð ð 1 1 a b

þ

2 Z I2 2 Eyþ Hxþ * dS ¼

Ey dx dy ¼ z10 20 ð1cosðb0 hÞÞ sin2 ðkz10 lÞ 2 2Zz10 0 0 abb0

ð7:30Þ

S

The current at the base of the probe (y ¼ 0) is I0 sinb0 h. The real part of the input impedance seen from the coaxial cable can be computed8 as Rin ¼

2P jI0 sin b0 hj2

¼

2Zz10 2 b0 h 2 tan sin ðkz10 lÞ 2 abb20

ð7:31Þ

It is worth noting that the input resistance depends on the various geometrical parameters, in particular the length h of the probe and its distance l from the end wall. Such parameters must be chosen so as to optimize the matching of the waveguide with the cable, i.e. to make the input resistance equal to the characteristic impedance of the cable and to minimize the input reactance. The latter can be evaluated, as mentioned earlier, by computing the reactive energy associated with higher order modes. Such a computation is omitted for brevity.

7.6.2

Coaxial-to-microstrip transition

This is also a transition of practical importance, necessary for the measurement of microstrip circuits. The quasi-TEM nature of the field in the microstrip makes the geometry of the transition almost natural, since it consists of connecting the cable’s central conductor to the microstrip conductor and the cable’s outer conductor to the microstrip ground plane. The in-line configuration is the most common one, though in some applications other geometries, for instance with the cable perpendicular to the microstrip, may be necessary. A pair of in-line transitions in back-to-back position is shown in Figure 7.18. The inner conductor is overlapped and soldered to a portion of the microstrip and suitably shaped in order to reduce the discontinuity. Note the metal flanges supporting the coaxial cable. Figure 7.19 shows the measured frequency behaviour of such a double transition. The performance looks good in a quite broad band, but starts deteriorating above 10 GHz, in agreement with the electromagnetic simulations. The ripple in s11 is due to the internal reflections within the microstrip section comprised between the transitions. The periodicity of the ripple would thus change with the length of the microstrip section.

8

See Equation (5.3). In computing expression (7.31) we have used the following equality: ð1cosaÞ2 ð1cosaÞ2 1cosa a ¼ ¼ ¼ tan2 1cos2 a 1 þ cosa 2 sin2 a

204


Figure 7.18 A pair of coax-to-microstrip transitions.

0

0.0

s21

-5

measured simulated

10 log10(|s21|)

-1.0

-10 -15

-1.5

-20

-2.0

s11

-2.5

-25

-3.0

-30

-3.5

-35

-4.0

1

2

3

4

5

6

7

8

9

10

11

10 log10(|s11|)

-0.5

-40

Frequency, GHz

Figure 7.19 Response of a pair of back-to-back coax-to-microstrip transitions.

7.7 Dividers and combiners The linear combination of different signals or, conversely, the subdivision of a signal into different components is normal practice in analogue signal processing at RFs. It should be stressed that division and combining are linear operations, not to be confused with mixing and frequency conversion, which are treated in Chapter 13. Due to the reciprocity, a combiner behaves also as a divider by interchanging outputs with input. They are therefore treated as a unique subject. There is an ample variety of circuits and devices that can be used as dividers and combiners. They also have a variety of names, such as combiner, splitter (divider), hybrid, hybrid junction, directional coupler, with some terminological uncertainties. Here we consider the Wilkinson divider, the hybrid junction (or simply the hybrid) and the directional coupler, but with a word of warning that there is some overlap between the last two categories.


7.7.1

205

The Wilkinson divider

Due to Wilkinson’s ingenuity, the device he invented [9] is universally known as the Wilkinson power divider or Wilkinson splitter/combiner. In its simplest version it is a three-port circuit capable of splitting the power incident form port 1 into two equal output signals from ports 2 and 3. Conversely, due to the use of a resistor placed between ports 2 and 3, those ports are uncoupled (s23 ¼ 0) so that a signal possibly reflected from an unmatched load at one of those ports does not affect the other port. Figure 7.20 shows the circuit schematic and the layout of the microstrip realization of a two-way Wilkinson divider.9 It consists of two l=4 sections of impedance Zc , connected at one end at port 1; at the other ends (ports 2 and 3), they are connected together through the resistor R. By cascading N levels of Wilkinson dividers, 2N -way dividers can be realized, as in the example of Figure 7.21, which shows an eight-way divider. Inspection of Figure 7.20 reveals that the device is matched at port 1 when pffiffiffi Zc ¼ Z0 2

substrate edge

Zc = /2, /4

1

(a)

ð7:32Þ

2

2 Z0 2

R = 2 Z0

7 mm

1

3

3

2 microstrip transmission lines 1 (b)

1 3

(c)

RF ports film resistors

Figure 7.20 Wilkinson divider: (a) schematic; (b) electrical symbol; (c) layout of a microstrip realization (h ¼ 0.58 mm, er ¼ 9:9).

9 More complicated versions have been developed in order to achieve unequal divisions, or to widen the operational bandwidth by the use of multi-section outputs.

206


3

4

5 1 6

7

8

9

Figure 7.21

An eight-way power divider using two-way Wilkinson dividers.

In fact, the input impedance at port 1 when ports 2 and 3 are matched (i.e. terminated on the reference impedance) is the parallel of two l=4 Zc sections terminated with Z0 (for symmetry reasons the resistor plays no role), i.e. Zin ¼

1 Zc2 2 Z0

ð7:33Þ

Equation (7.32) follows from the matching condition Zin ¼ Z0 . It can also be shown that the condition for matching at ports 2 and 3 (s22 ¼ s33 ¼ 0) is R ¼ 2Z0

ð7:34Þ

When the matching conditions are satisfied at the frequency when the line sections are l/4 long (thus the electrical lengths are p=2), the Wilkinson divider has the following scattering matrix: 2 0 j 4 ½S ¼ pffiffiffi 1 2 1

3 1 1 0 05 0 0

ð7:35Þ

This expression shows that the device is matched at all ports (s11 ¼ s22 ¼ s33 ¼ 0) and that ports 2 and 3 are uncoupled (s23 ¼ 0). The signal entering in port 1 is equally split between ports 2 and 3, undergoing equal phase shifts of10p=2. Figure 7.22 shows the frequency response11 of the Wilkinson divider of Figure 7.20. Notice that the loss has the effect of increasing the insertion loss above the 3 dB predicted theoretically. The device nevertheless ensures a matching and isolation better than 14 dB over a 40% bandwidth. 10 11

Remember that ejp=2 ¼ j. See the Ansoft file 02_Wilkinson_Divider.adsn.


-3

20

s21 = s31

10

-4

0

-5

-10 s11

s22=s33

-6

-20

s32 -7 8

Figure 7.22

9

10 Frequency, GHz

11

-30 12

20 log10(|s11|), 20 log10(|s32|), 20 log10(|s22|)

20 log10(|s21|) = 20 log10(|s31|)

-2

207

Response of the Wilkinson divider of Figure 7.20.

Computation of the scattering matrix As a result of symmetry, the scattering matrix (7.35) can be computed by reducing the analysis to two two-port circuits. With reference to Figure 7.23, a symmetry plane can be identified cutting the input line 1 and the resistor R into two halves. The input line can be decomposed as the parallel of two lines of impedance 2Z0 , and the resistor as the series of two resistors of value R=2. Any excitation of the circuit can then be decomposed into an even and an odd mode; correspondingly, the signals at ports 2 and 3 have the same amplitudes and equal or opposite phases respectively. Consider the even excitation first. The symmetry plane behaves as a magnetic wall, i.e. an open circuit. The device reduces to that of Figure 7.23b. The resistance R does not appear as no current flows through it. In the odd mode, the symmetry plane is equivalent to a zero-voltage plane, so that port 1 and the centre of the resistor R are grounded.12 The relevant circuit is shown in Figure 7.23c. Let us now consider both circuits. For simplicity we limit the analysis at the centre frequency, when the line is l=4 long. Even mode Because of the property of quarter-wavelength lines, the matching condition occurs when the line impedance is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Zc ¼ 2Z0 Z0 ¼ Z0 2 This condition is the same as (7.32). The scattering matrix of a matched line section of electrical length y ¼ p=2 is simply h i 0 j SðeÞ ¼ ð7:36Þ j 0 where the superscript ‘e’ stands for even. Odd mode With reference to Figure 7.23c, the short circuit at port 1 is transformed by the quarter-wavelength line into an open circuit at port 2. The latter is matched when R ¼ 2Z0 , hence Equation (7.34). The 12

The centre of the resistor R may be considered as an unused fourth port of the device.

208

MICROWAVE AND RF ENGINEERING 2 Z0 2

1

symmetry plan

R 2 Z0

(a) 3

2 Z0

Z0

2 Z0 2

2

2 Z0

2 Z0

1

R/2

1 (b) even

Figure 7.23

Z0

(c) odd

The symmetric Wilkinson divider (a) can be split into its even (b) and odd (c) parts.

scattering matrix can easily be computed by observing that port 1 is short-circuited, thus s11 ¼ 1, and the two ports are decoupled, thus s12 ¼ s21 ¼ 0, while port 2 is matched, thus s22 ¼ 0. In conclusion, the scattering matrix of the odd mode is h i 1 0 ð7:37Þ SðoÞ ¼ 0 0 where the subscript ‘o’ stands for odd. Note that [S(o)] is not unitary (see (4.93)); indeed the circuit is lossy because of the presence of the resistor R/2. We now have all the necessary information to reconstruct the scattering matrix in the general case of arbitrary excitation by superimposing the even and odd mode excitations, i.e. by expressing the wave amplitudes at the three ports in terms of their even and odd components. Recalling that only the even mode is present on input line 1 and that this line is split into the parallel of two lines of impedance 2Z0 , each carrying one-half of the total power, the incident and reflected waves a1 and b1 are related to the even mode wave amplitudes by pffiffiffi pffiffiffi a1 ¼ a1e 2; b1 ¼ b1e 2 ð7:38Þ For ports 2 and 3 we simply have a2 ¼ a2e þ a2o ;

a3 ¼ a2e a2o

b2 ¼ b2e þ b2o ;

b3 ¼ b2e b2o

ð7:39Þ

Using (7.38) and (7.39), we obtain the outgoing wave amplitudes of the device: b a a þ a3 p1ffiffiffi ¼ b1e ¼ s11e a1e þ s12e a2e ¼ s11e p1ffiffiffi þ s12e 2 2 2 2 a1 a2 þ a3 a2 a3 þ s22o b2 ¼ b2e þ b2o ¼ s21e a1e þ s22e a2e þ s22o a2o ¼ s21e pffiffiffi þ s22e 2 2 2 a1 a2 þ a3 a2 a3 s22o b3 ¼ b2e b2o ¼ s21e a1e þ s22e a2e s22o a2o ¼ s21e pffiffiffi þ s22e 2 2 2 Using (7.36) and (7.37), we then obtain the scattering matrix (7.35).

ð7:40Þ


209

Figure 7.24 The hybrid ring (or rat-race) (a) and its electrical scheme (b). Reproduced with kind permission of Verigy Germany GmbH.

7.7.2

Hybrid junctions

A hybrid junction or simply a hybrid is a four-port passive component originally conceived in telephony applications in order to employ just one pair of wires in both transmit and receive modes [10]. One port can be internally terminated on a matched load so that the device looks like a three-port rather than a fourport circuit. As a hybrid junction we define (the technical terminology is not consistent throughout the literature) a four-port component with the following properties: 1. All ports are matched (sii ¼ 0, i ¼ 1. . .4). 2. At least one port is isolated from the input port 1 (s14 ¼ 0, if port 4 is the isolated port). pffiffiffi 3. The input power is equally split between the two remaining outputs (js12 j ¼ js13 j ¼ 1= 2). For lossless symmetrical devices, the phase difference between output signals at ports 2 and 3, normally 90 or 180 , characterizes the hybrid junction, so that one usually speaks of 90 or 180 hybrid junctions.13 A typical 180 hybrid is the hybrid ring or rat-race shown in Figure 7.24 along with its electrical scheme. It consists of a closed loop (a circle or other shapes are allowed) 3l=2 long; the ports are located pffiffiffi l/4 from each other. To ensure matching, the characteristic impedance of the ring must be equal to Z0 2, Z0 being the reference impedance. The input signal from port 1 is equally divided between the two half branches of the ring.14 Both portions of the signal reach the other three ports along different paths, thus different phases: they sum at port 2 (both paths are 3l=4 long) and port 4 (the paths are l=4 and l þ l=4 long) but they cancel out at port 3 (the two paths are l=2 and l long, respectively) so that port 3 is isolated from port 1. Moreover, since ports 2 and 4 are 180 out of phase (their distance is l=2) the rat-race is a 180 hybrid. Similarly, it can be seen that ports 2 and 4 are isolated from each other and are in phase when the device is fed from port 3. As a consequence, when two signals enter the rat-race at ports 1 and 3, their sum will appear at port 4 and the difference will appear at port 2. Ports 4 and 2 are therefore indicated as the sum (S) and difference (D) ports, respectively. The Wilkinson divider can be considered as a 0 hybrid, port 4 being the centre of the resistor R. See footnote 12. This qualitative discussion assumes, to a first approximation, that the power absorbed by each port is so small that it negligibly affects the amplitudes of the waves travelling along the ring. 13 14

210

MICROWAVE AND RF ENGINEERING 12 mm

2

4

20 log10(|s41|) = 20 log10(|s21|)

1

-2

10

s21

-3

s41

0

-4 -10 -5 -20

s11 s31

3

-6

microstrip transmission lines 4

20

8

9

10

11

20 log10(|s11|), 20 log10(|s31|)

substrate edge

-30 12

Frequency, GHz

RF ports

Figure 7.25 Layout and frequency response of a microstrip rat-race. The dielectric substrate is alumina, h ¼ 0:508 mm thick, with er ¼ 9:9. Reproduced with kind permission of Verigy Germany GmbH.

Notice that the device is symmetrical with respect to a plane passing through the middle of ports 2 and 3. As a consequence, the roles of ports 1 and 2, and 4 and 3, can be interchanged. A full analysis of the device can be carried out by exploiting this symmetry. At the centre frequency when the ring length is 3l=2, the scattering matrix can easily be computed as 3 2 0 1 0 1 7 j 6 6 1 0 1 0 7 ð7:41Þ ½S ¼ pffiffiffi 6 7 1 0 15 24 0 1 0 1 0 The usual operating bandwidth of the rat-race is around 20%. Figure 7.25 shows the microstrip layout of a rat-race along with its frequency response.15 Although the rat-race configuration can also be used in rectangular waveguide technology, another type of hybrid junction is usually employed, the so-called hybrid T or magic T. Its geometry is shown in Figure 7.26. Seen from port 1, it consists of an H plane T junction plus an additional E plane T junction. The device’s symmetry with respect to the middle of the guides 1 and 4 allows one to infer its fundamental features. Recalling the field distribution of the dominant TE10 mode of the rectangular waveguide, it is apparent that ports 1 and 4 are isolated (s14 ¼ s41 ¼ 0). The former port indeed has the electric field parallel to the symmetry plane, which thus behaves as a magnetic wall (even symmetry), while port 4 has the field perpendicular to the symmetry plane, which thus behaves as an electric wall (odd symmetry). Less evident is the fact that ports 2 and 3 also are isolated. This condition occurs, in fact, as we will see below, at the condition when ports 1 and 4 are matched. For symmetry reasons, it also appears that s12 ¼ s13 and s22 ¼ s33 ; moreover, the electric field of waveguide 4 is seen from ports 2 and 3 with opposite phases, thus s24 ¼ s34 . Such considerations do not ensure that the device is a hybrid junction, since we also need to impose that the ports are matched. This can be realized in practice by inserting properly shaped metal posts or sheets within the junction in such a way to compensate for the unwanted reactances and ensure the matching of ports 1 and 4, so that s11 ¼ s44 ¼ 0. 15

See the Ansoft file 03_Rat_Race.adsn.


Figure 7.26

211

The magic T. Reproduced with kind permission of Verigy Germany GmbH.

From these considerations, the scattering matrix of the 2 0 s12 s12 6s s 6 12 22 s23 ½S ¼ 6 4 s12 s23 s22 0 s24 s24

magic T is therefore 3 0 s24 7 7 7 s24 5

ð7:42Þ

0

taking into account the reciprocity of the device (sij ¼ sji ). If we finally assume that the device is lossless, then the scattering matrix

2 be unitary (see Equation (4.93)). In particular, the elements of each row P

must of (7.42) must satisfy sij ¼ 1. Therefore: pffiffiffi 1. For the first and fourth rows we deduce that js12 j ¼ js24 j ¼ 1= 2. 2. Using the previous result, for the second row we obtain js22 j2 þ js23 j2 ¼ 0, thus s22 ¼ s23 ¼ 0. We conclude that ports 2 and 3 also are isolated and matched. For simplicity, we may choose the reference planes at ports 1 and 4 in such a way that s12 and s14 are real.16 The scattering matrix of the magic T can thus be written in the form 3 2 0 1 1 0 7 6 1 7 1 61 0 0 7 ½S ¼ pffiffiffi 6 ð7:43Þ 1 7 26 5 41 0 0 0 1 1 0

7.7.3

Directional couplers

A close relative of the hybrid junction is the directional coupler, one of the most useful devices in microwave circuits and applications. As a reciprocal ideally matched and lossless four-port network with the ports isolated in pairs, it is used to divide or combine RF signals travelling along different lines. It is 16 In Section 4.8 we have shown that a shift in the reference plane produces a phase shift in the scattering parameters.

212

MICROWAVE AND RF ENGINEERING Input 1

Isolated 4

Figure 7.27

P1

P2

P4

P3

2 Direct

3 Coupled

Schematic of a directional coupler.

essential in the measurement of the scattering parameters of devices as it allows separation of incident from reflected waves travelling along the transmission line, i.e. the waves travelling in opposite directions. This property gives the device its name. Figure 7.27 shows the schematic representation of a directional coupler. The signal entering into port 1 is divided between port 2 (direct port) and port 3 (coupled port), while port 4 is ideally isolated. The following parameters are used to characterize the behaviour: 1. Coupling. Normally, but not always, expressed in dB, the directional coupler measures the ratio between the input and the output power at the coupled port: CdB ¼ 10 log10

P1 P3

¼ js31 jdB

ð7:44Þ

Typical values of CdB are from 3 to 20 dB. The former value corresponds to an equal split of the input signal between ports 2 and 3. The reader may observe that such a coupler is nothing but a hybrid junction. However, a word of warning: higher CdB values correspond to lower coupling levels. A 0 dB coupler would transfer the whole input power to the coupled port (P3 ¼ P1 ). Such a device is used as a crossover. The reader can easily prove that a 0 dB coupler is obtained by cascading two 3 dB couplers. 2. Directivity. The directional coupler measures the device capability to discriminate between incident and reflected waves in the coupled port. It is defined as the ratio, expressed in dB, of the power from port 3 to that from port D ¼ 10 log10

P4 P3 P1 ¼ 10 log 10 log10 ¼ js31 jdB þ js41 jdB P3 P1 P4

ð7:45Þ

In ideal conditions P4 ¼ 0 and the directivity is infinite. Typical values vary from 10 dB for microstrip couplers to 30 dB for waveguide couplers. 3. Isolation. This is a measure of the isolation between the input port and the isolated port: I ¼ 10 log10

P1 P4

¼ js41 jdB

ð7:46Þ

In ideal conditions the isolation is infinite, like the directivity. The previous formulae (7.44)–(7.46) show that coupling, directivity and isolation are not independent of each other, but are related by the following equation: D ¼ CdB I

ð7:47Þ


213

The ideal directional coupler has the following properties: 1. All ports are matched to the reference impedance Z0 (usually 50 O). This means that if three ports are terminated with Z0 , the input impedance at the fourth port is Z0 as well. In terms of scattering parameters, this implies17 that sii ¼ 0 (i ¼ 1. . .4). 2. The device is lossless. Its scattering matrix is therefore unitary: ½S½S* T ¼ ½U. 3. The ports are isolated in pairs (1–4 and 2–3), thus s14 ¼ s41 ¼ 0 and s23 ¼ s32 ¼ 0. This implies infinite directivity and isolation. As can be expected, such properties are only approximately verified by actual directional couplers. If we impose properties 1 and 3 along with reciprocity and symmetry, the scattering matrix of an ideal directional coupler is found to have a very simple expression, containing only four complex independent parameters: 3 2 0 s12 s13 0 7 6 6 s12 0 0 s24 7 7 6 ð7:48Þ ½S ¼ 6 0 s34 7 5 4 s13 0 0 s24 s34 0 By further imposing the lossless condition 2, we obtain the following conditions for the scattering parameters: js12 j2 þ js13 j2 ¼ 1 js12 j2 þ js24 j2 ¼ 1 js13 j2 þ js34 j2 ¼ 1

ð7:49Þ

js24 j2 þ js34 j2 ¼ 1 s12 s*24 þ s13 s*34 ¼ 0 s12 s*13 þ s24 s*34 ¼ 0 After simple manipulations, and putting s13 ¼ Cejy13 , we obtain 2

0 6 pffiffiffiffiffiffiffiffiffiffiffi2ffi 6 1C ½S ¼ 6 6 jy13 4 Ce 0

pffiffiffiffiffiffiffiffiffiffiffiffi 1C 2 Ce jy13 0

0

0

0 pffiffiffiffiffiffiffiffiffiffiffiffi 1C2 e jy34

Ce jy24

0

3

7 7 Ce jy24 7 pffiffiffiffiffiffiffiffiffiffiffiffi jy34 7 2 1C e 5

ð7:50Þ

0

with the additional condition on the phases y12 y13 y24 þ y34 ¼ p

ð7:51Þ

For simplicity we have set the phase of s12 to zero. This is not a limitation since what matters are the phase differences between the scattering parameters rather than their absolute values. A particularly important and useful category of directional couplers is that of fully symmetrical couplers, for which all ports are indistinguishable from one another. In such cases it is apparent that the transmission from port 1 to 2 is identical to that from 3 to 4; similarly, that from port 1 to port 3 is the same 17

Actually, it can easily be proved that any matched and lossless four-port circuit is a directional coupler [4].

214


Figure 7.28

The two-hole coupler.

as that from 2 to 4. As a consequence, y12 ¼ y34 and y13 ¼ y24, and Equation (7.51) becomes p y12 ¼ y13 ð7:52Þ 2 This equation shows that in a fully symmetrical, ideally matched and isolated coupler, ports 2 and 3 are 90 out of phase. A directional coupler can be realized in various technologies by a variety of different configurations. For space limitations, here we just briefly describe the most common ones.

7.7.3.1 Two-hole coupler This is probably the simplest type of coupler. It is instructive to illustrate the principle behind its behaviour. The device consists of two rectangular waveguides set side by side by the broad walls and coupled with a pair of small holes at l/4 distance from one another. See Figure 7.28. Let us consider a wave travelling in the upper waveguide from port 1 towards port 2. At each hole a small portion of the incident power is transmitted into the second waveguide generating two waves travelling in opposite directions, one in the forward direction (towards port 3) and the other in the backward direction (towards port 4). Since the electrical distance between the two holes is p=2, the two forward waves add in phase at port 3, while the backward waves are 180 out of phase, and thus cancel at port 4, which is therefore isolated from port 1. This reasoning implicitly assumes that the amplitude of the wave produced in the second waveguide by the second hole is equal to that generated by the first hole: this is true only to a first approximation by omitting the fact that the first hole actually reduces the power carried by the wave travelling in the first waveguide. The smaller the holes, the better the approximation. In practice the maximum coupling obtainable is about 20 dB. The device is inherently narrow band, the band being centred around the frequency when the hole distance is l=4. Such limitations can be reduced or overcome by using a higher number of holes, which can also be located in different positions along the waveguide walls (both wide and narrow). Multi-hole directional couplers are discussed in [11].

7.7.3.2 Branch-line coupler The branch-line coupler consists of two transmission lines linked by two or more transverse l/4 branches located at l=4 distance one from each other. It can be realized in both waveguide and printed circuit (i.e. microstrip, stripline, CPW, etc.) technologies. The distributed circuit representation and the layout of a microstrip realization are shown in Figures 7.29a,b.18 Note that the circuit is fully symmetric, so, as we have proved before, its scattering matrix is of type (7.48) or, in the lossless case, (7.50). This symmetry also allows us to simplify the analysis. In fact, we may excite the structure at the four ports with signals of 18 The sharp junction discontinuities may alter the performance of the actual device with respect to the ideal behaviour of the circuit in Figure 7.27a. Such effects can be minimized by suitably rounding off the corners of the layout.


215

7 mm

1

2

4

3

ZS=Z0/ 2, /4 1

2

Z0, /4

Z0, /4

4

3

ZS=Z0/ 2, /4

substrate edge microstrip transmission lines 4

RF ports

(a)

Figure 7.29

(b)

Branch-line coupler: (a) schematic; (b) layout of a microstrip implementation.

the same amplitudes and phases, either equal or opposite, in such a way that on the two symmetry planes an open circuit (even symmetry) or a short circuit (odd symmetry) is realized. Correspondingly, the structure of Figure 7.29 reduces to a bipole loaded with two parallel stubs terminated with an open or a short circuit, as shown in Figure 7.30. Here the indexes ‘e’ and ‘o’ stand for even and odd, respectively. By

1

2 YS

Y0

Y0

YS

4

1

open Y0 open

3

1

YS

Y0 open

ee

open

1 Y0

symmetry planes

YS oe

YS eo

1 Y0

YS oo

Figure 7.30 Reduction of Figure 7.29a to four bipoles by replacing the symmetry planes with magnetic (even symmetry ‘e’ or open circuit) or electric (odd symmetry ‘o’ or short circuit) planes.


20 log10(|s21|) = 20 log10(|s31|)

-2

20

-3

10 s31

-4

0 s21

-5

-10

-6

-20

s41 s11

-7

20 log10(|s11|), 20 log10(|s41|)

216

-30 8

9

10

11

12

Frequency, GHz

Figure 7.31

Response of the branch-line coupler of Figure 7.29b.

pffiffiffi choosing the series and shunt branch impedances as Zs ¼ Z0 = 2 and Zp ¼ Z0 we obtain a 3 dB coupler whose scattering matrix is 3 0 j 1 0 7 1 6 6 j 0 0 17 ½S ¼ pffiffiffi 6 7 241 0 0 j 5 0 1 j 0 2

ð7:53Þ

The device constitutes a 90 hybrid junction which splits the power at port 1 into two signals of equal amplitudes and 90 out of phase at ports 2 and 3. We do not need to stress that (7.53) is only valid at the centre frequency when all branches are l=4 long. The operational bandwidth of the device depends on the specifications, but in general we can say that the bandwidth is of the order of 10–15%. As an example, Figure 7.31 shows the computed19 scattering parameters of a microstrip hybrid. Assuming a maximum return loss of 15 dB, the bandwidth is approximately 15%. When wider bandwidths are required, devices with a higher number of branches can be employed.

Computation of the scattering parameters To compute the scattering parameters of the branch-line coupler, we make use of Table 7.2, where the four cases of Figure 7.30 are considered. The second row of the table indicates the relations between the incident and reflected waves at the four ports, depending on the respective symmetry conditions specified in the first row. The third row gives the corresponding input admittances normalized with respect to Y0 . Such values are obtained by taking into account that each stub is l=8 long, so that tanðblÞ ¼ tanðp=4Þ ¼ 1. The fourth row gives the reflection coefficients relating the reflected to the incident waves.

19

See the Ansoft file 04_Branch_Line.adsn.


217

The scattering matrix of the branch-line coupler is computed by superimposing the four cases, i.e. by exciting the circuit with the sum of the four signals. In this way the outgoing waves are found to be b1 ¼ Gee aee þ Geo aeo þ Goe aoe þ Goo aoo b2 ¼ Gee aee Geo aeo Goe aoe þ Goo aoo

ð7:54Þ

b3 ¼ Gee aee þ Geo aeo Goe aoe Goo aoo b4 ¼ Gee aee Geo aeo þ Goe aoe Goo aoo Taking into account that (as can be easily verified) aee ¼ ða1 þ a2 þ a3 þ a4 Þ=4; aoe ¼ ða1 þ a2 a3 a4 Þ=4;

aeo ¼ ða1 a2 a3 þ a4 Þ=4 aoo ¼ ða1 a2 a3 a4 Þ=4

ð7:55Þ

s12 ¼ ðGee Geo þ Goe Goo Þ=4 s14 ¼ ðGee þ Geo Goe Goo Þ=4

ð7:56Þ

we obtain the first row of the scattering matrix: s11 ¼ ðGee þ Geo þ Goe þ Goo Þ=4; s13 ¼ ðGee Geo þ Goe Goo Þ=4;

The remaining rows are immediately obtained by exploiting the reciprocity and symmetry of the circuit: s11 ¼ s22 ¼ s33 ¼ s44 ; s13 ¼ s31 ¼ s24 ¼ s42 ;

s12 ¼ s21 ¼ s34 ¼ s43 s14 ¼ s41 ¼ s23 ¼ s32

ð7:57Þ

Let us now impose the matching condition s11 ¼ 0 and the condition that ports 1 and 4 are isolated, i.e. s14 ¼ 0. Using (7.57) along with the expressions in Table 7.2, we find that both conditions are satisfied for Ys2 ¼ 1 þ Yp2

ð7:58Þ

Table 7.2 Computing the parameters of the circuits of Figure 7.30 Even–even

Even–odd

Odd–even

Odd–odd

aee ¼ a1 ¼ a2 ¼ a3 ¼ a4

aeo ¼ a1 ¼ a2 ¼ a3 ¼ a4

aoe ¼ a1 ¼ a2 ¼ a3 ¼ a4

aoo ¼ a1 ¼ a2 ¼ a3 ¼ a4

Gee aee ¼ b1 ¼ b2 ¼ b3 ¼ b4

Geo aeo ¼ b1 ¼ b2 ¼ b3 ¼ b4

Goe aoe ¼ b1 ¼ b2 ¼ b3 ¼ b4

Goo aoo ¼ b1 ¼ b2 ¼ b3 ¼ b4

Yin ¼ Yee ¼ j Ys þ Yp

Yin ¼ Yeo ¼ j Ys þ Yp

Yin ¼ Yeo ¼ j Ys þ Yp

Yin ¼ Yoo ¼ j Ys þ Yp

Gee ¼

1jðYs þ Yp Þ 1 þ jðYs þ Yp Þ

Geo ¼

1jðYs þ Yp Þ 1 þ jðYs þ Yp Þ

Goe ¼

1 þ jðYs Yp Þ 1jðYs Yp Þ

¼ 1=Geo

Goo ¼

1 þ jðYs þ Yp Þ 1jðYs þ Yp Þ

¼ 1=Gee

218


Under such a condition we obtain from the last row of Table 7.2 Gee ¼ ðY2 þ jÞ=Y1 ;

Geo ¼ ðY2 þ jÞ=Y1 ¼ Gee

Goe ¼ ðY2 jÞ=Y1 ¼ Goo ;

Goo ¼ ðY2 þ jÞ=Y1

ð7:59Þ

Finally, from (7.56), s11 ¼ 0; s13 ¼

s12 ¼

Yp Gee þ Goo ¼ ; 2 Ys

Gee Goo 1 ¼ jYs 2

ð7:60Þ

s14 ¼ 0

Note that, as previously demonstrated in the general case, s12 and s31 are 90 out of phase. Comparing (7.60) with (7.50) we conclude that C ¼ Zs =Zp . To obtain a 3 dB coupler, i.e. a coupler such that the input power at port 1 is equally pffiffiffi divided between ports 2 and 3, it suffices to choose of (7.58), Y ¼ 2. The denormalized impedance of the series branch is Yp ¼ 1; thus, because s pffiffiffi therefore Zs ¼ Z0 = 2 and that of the shunt branches is Zp ¼ Z0 . Expression (7.53) for the scattering matrix is finally obtained from (7.59) and (7.60).

7.7.3.3 Coupled-line couplers The distributed coupling between closely spaced printed lines detailed in Section 3.16 enables the realization of another class of fully symmetrical directional couplers, whose scattering matrix is of type (7.50). The basic geometry is shown in Figure 7.32. It consists of a l=4 section of coupled lines connected to four lines with characteristic impedance Z0 . The width of the metal strips and their distance are determined so as to ensure the matching of the device, according to Equation (4.140), and to realize the prescribed coupling C, given by (4.143). Using these equations one can easily obtain the even and odd mode impedances of the coupled lines: rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi Z0e 1þC Z0o 1C ^ ^ ð7:61Þ Z 0e ¼ ¼ ¼ ; Z 0o ¼ 1C 1þC Z0 Z0 These values can be used, along with synthesis formulae, to compute the width and spacing of the lines, while their length is determined by the condition that it must equal l=4 at the centre frequency.

3

Z0

Z0 4 d=λ/4

1

Figure 7.32

Z0

Z0 2

Layout of a microstrip coupled-line directional coupler.


219

s21

0

2

20 log10(|s21|), 20 log10(|s31|)

10 log10(1-C ) -5 20 log10(C) -10

-15

s31

-20 0

1 =

Figure 7.33

2 = f/f0 0

Coupling and transmission coefficient of a 6 dB microstrip coupled-line coupler.

Contrary to what one might expect, the pairs of isolated ports are 1–4 and 2–3; rather than being on the same side of the device, they are located at diagonal positions. In some applications this may create topological difficulties. Normally, the difficulty of realizing metal strips that are too closely spaced (say, below 0.1 mm) does not allow coupling higher than 6 dB. Figure 7.33 shows the behaviour of the coupling C and the transmission coefficient s21 of a coupledline coupler versus the normalized frequency.20 Note that the coupling is maximum at the centre frequency when the coupled section is l=4 long. The drawbacks of limited bandwidth and small coupling, together with the difficulty of the isolated ports being located on opposite sides of the device, are overcome in the Lange coupler [12] shown in Figure 7.34. This employs an interdigitated configuration consisting of interleaved line segments like the fingers of two hands.21 The line between ports 3 and 4 is divided into two segments connected by bonding wires in such a way that ports 1 and 2, which are electrically connected, are on opposite sides of the device. As can be expected, the interdigitated geometry leads to higher couplings than the mere coupled lines. A Lange coupler can indeed realize a hybrid junction (i.e. with 3 dB coupling). There is, however, a price to pay, which is the need for bonding wires (thus additional fabrication cost) and, more importantly, the strict fabrication tolerances: the width and spacing of the interdigitated lines are very narrow (easily less than 0.1 mm) and thus require sophisticated fabrication technologies. Figure 7.35 shows the frequency response22 of the coupler of Figure 7.34. The typical behaviour of the coupling js31 j of such a device is observed: maximum at the centre frequency, then decreasing towards both higher and lower frequencies. Correspondingly, the direct signal amplitude js21 j is minimum at the centre frequency and increases towards the band edges. We also note that the nominal 3 dB coupling is not achieved because of the device loss, without which the js21 j and js31 j curves would touch each other at the centre frequency.

20 See the Ansoft file 05_Directional_Coupler.adsn for the analysis of both an ideal and a microstrip 10 dB directional coupler. 21 The term comes from the Latin digitus, meaning finger. 22 See the Ansoft file 06_Lange_Coupler.adsn.


1

4

2 mm

substrate edge 3

2

microstrip transmission lines RF ports

1

bond wires

Figure 7.34 Lange coupler in microstrip technology.

20

-2

10

-3

-4

0

s21

-10

-5

-20

-6 s41

s11 -7

4

8

12

16

-30 20

Frequency, GHz

Figure 7.35

Response of the Lange coupler of Figure 7.34.

20 log10(|s11|), 20 log10(|s41|)

s31 20 log10(|s21|) = 20 log10(|s31|)

220


221

7.8 Lumped element realizations The dividers, hybrid junctions and couplers considered so far consist of distributed circuit elements, mostly single or multiple l=4 sections. The hybrid ring, for instance, consists of six l=4 sections, so that its diameter is of the order of l=2. At low frequencies, the size of such components may become significant and often intolerable. Consider for instance the microstrip circuits of Figures 7.20c, 7.25a and 7.29b. They have been designed for a 10 GHz centre frequency and realized on a substratewith a permittivity er ¼ 9:9. The physical length of pffiffiffiffiffiffiffiffi a l=4 section is lmm ¼ 75=ðf0;GHz ee f f Þ, which is of the order of 2.7 mm. If the centre frequency were 1 GHz, each section would be around 27 mm long and the ring diameter would be over 50 mm. In many practical applications where space occupancy is of critical importance, one has to look for alternative configurations to reduce the circuit size substantially. One possibility consists of employing lumped elements, replacing each distributed line length with a lumped LC section. The latter is chosen in such a way as to approximate the corresponding line length and, in particular, to be equivalent to it at the centre frequency. Substitution of a transmission line length by a lumped LC cell can be implemented in various ways, but, due to space limitations, here we consider just one example. Other cases can be treated in a similar manner. In Figure 7.36, the two-port network consisting of a l=4 transmission line length is replaced by an LC lowdpass P cell. The following discussion can easily be extended to other similar cases (e.g. a T cell). The values of the LC elements of the cell can be determined by requiring that, at the central frequency f0 when the line is l=4 long, both circuits are equivalent, i.e. they cannot be distinguished when measured at the external terminal pairs, hence they have the same admittance matrix. Let us recall the admittance matrix of a line length (4.121): ½Yline ¼

jY0 cos y 1 sin y 1 cos y

ð7:62Þ

where y ¼ bl ¼ ðp=2Þ ðo=o0 Þ and o0 ¼ 2pf0 . The admittance matrix of a P cell is obtained from (4.65)–(4.66), where YA ¼ YB ¼ joC and YC ¼ 1=ðjoLÞ: 1 1o2 LC 1 ½YP ¼ ð7:63Þ 1 1o2 LC joL Equating both matrices (7.62) and (7.63) for y ¼ p=2, i.e. o ¼ o0 , we obtain L and C as functions of the characteristic impedance Z0 and the centre frequency o0 : Z0 o0 1 C¼ Z 0 o0 L ¼

Figure 7.36

Substituting a transmission line length (a) by an LC lowdpass P cell (b).

ð7:64Þ

222


C

C L35.35

1

2

L50

L50 L35.35

4

3

C

C

Figure 7.37 Lumped element realization of a branch-line coupler corresponding to Figure 7.29a, with a centre frequency of 500 MHz. In a similar way we can approximate the l=4 section with an LC lowdpass T cell or a l/2 section with a highpass T or P cell. Let us now consider specifically the branch-line coupler of Figure 7.29a consisting of four l=4 sections, two with 50 O characteristic impedance (between the node pairs 1–4 and 2–3) and two with 35.35 O characteristic impedance (between the node pairs 1–2 and 3–4). Let us suppose that we need to realize the coupler at a centre frequency of 500 MHz. Applying (7.64) we get 8 > > < L50O ¼

8 > > > < L35:35O ¼

> > : C50O

> > > : C35:35O

50 ffi 15:92 nH 2p 500 106 1 ¼ ffi 6:37 pF 50 2p 500 106

pffiffiffi 50= 2 ffi 11:25 nH 2p 500 106 1 pffiffiffi ¼ ffi 9 pF 50= 2 2p 500 106

By replacing the line sections with the LC P cells, the circuit of Figure 7.29a reduces to that shown in Figure 7.37, where each capacitor has been obtained as the parallel of two capacitors pertaining to the transmission lines connected to each node: C ¼ C50O þ C35:35O ffi 15:37 pF Figure 7.38 shows the response23 of Figure 7.37. For comparison, the black lines show the response of the ideal distributed branch line of Figure 7.29a with the same centre frequency f0 ¼ 500 MHz. Notice that the lumped circuit exhibits a slightly lower performance (worse matching, higher insertion loss and worse balance between the two outputs 2 and 3 as the frequency departs from f0 ). On the other hand, the lumped version is much smaller than the distributed one. If realized in microstrip with the same substrate, the distributed circuit of Figure 7.29b would have line sections about 54 mm long, while the four capacitors and four inductors can be realized with physical dimensions of the order of 1 mm using SMD technology with a 0402 package.24 In the present case, the lumped circuit has a size of order one-tenth that 23 24

See the Ansoft file 07_Lumped_Branch_Line.adsn. See Section 14.2.1.


s21

-4

0

ideal distributed lumped

s31 -5

-10 s11

-6

20 log10(|s11|)

20 log10(|s21|), 20 log10(|s31|)

-3

223

-20

-7 400

450

500

-30 600

550

Frequency, MHz

Figure 7.38 Response of the lumped branch line of Figure 7.37 (grey lines) compared with that of the ideal distributed branch line of Figure 7.29b (black lines). Broadside Wavefront

2 p

d sin

p

d 1

Figure 7.39

2

….

N-1

N

A linear array radiating a wavefront at an angle yp < 0.

of the distributed one.25 Two additional parameters to consider are the insertion loss (not included in any of the simulations of Figure 7.38) and the maximum power that can be carried by the circuit. The distributed realization has a superior performance from both points of view.

7.9 Multi-beam forming networks Modern advances in communication technology, such as satellite communications, radar systems, pointto-point communication links and imaging, have spurred the development of antennas made of one- or two-dimensional arrays of radiating elements. By feeding the radiating elements with suitable phases and amplitudes one can achieve a variety of radiation patterns to fit the most sophisticated requirements, such as multi-beam or scanning capabilities, and to be reconfigured according to time-varying requirements. An example is the possible variation of the area covered by satellite antennas. In order to provide such 25 Actually, in the case of the lumped circuit, we should consider not only the size of the components, but also the area occupied by the bonding pads. A rough estimate suggests that the overall size of the circuit in Figure 7.37 is about 5 5 mm.

224


capabilities, a microwave feeding network must be designed capable of varying the phases and/or the amplitudes of the signals feeding the array elements. This can be achieved by feeding networks consisting of the devices illustrated in previous sections, such as dividers, hybrids and phase shifters, possibly with control elements (as described in Chapter 10) to achieve the required reconfigurability of the radiated beam. This approach is extremely powerful and flexible, but, in the case of 2-D scanning, also quite expensive: consider that phased array antennas can easily contain several thousand elements. In the case of 1-D scanning, a cheaper approach consists of employing an M N multi-beam forming network (MBFN), where the N outputs are connected to the elements of the linear array. When fed at the ith port (i ¼ 1. . .M), the phase distribution at the N output ports creates a beam directed towards the direction of the angle yi. In this manner, M independent beams are simultaneously created; moreover, by switching among the input ports, a switched-beam rather than continuous-beam scanning is obtained. This solution can be implemented using basically two classes of MBFNs: matrices and lenses. We briefly consider here the most common ones, namely the Butler matrix, the Blass matrix and the Rotman lens.

7.9.1

The Butler matrix26

Consider a linear array of N radiating elements, located at a constant distance d from one another. If each element is fed with a constant phase difference Df with respect to the previous one, the radiated beam will be tilted by an angle yp with respect to the broadside direction, i.e. the direction perpendicular to the array. A Butler matrix can provide a linear phase distribution when the number of elements is a power of 2. The Butler matrix is in fact an N N MBFN where the number N of input and output ports is required to be N ¼ 2n where n is an integer. The Butler matrix can be realized in various technologies from waveguide to printed circuit. Constituting elements are directional couplers, either 90 or 180 hybrids, crossovers and phase shifters. They must be properly connected to get the desired phase distribution at the output ports. Designs exist using either 90 or 180 hybrids. Butler matrices with 90 hybrids do not produce a beam at broadside, i.e. orthogonal to the linear array, whereas those with 180 hybrids produce a broadside beam. The number of couplers employed is the same for both solutions, but the number and position of phase shifters depend on the type of hybrid employed. The phase progression at the output ports is related to the order N of the matrix. A detailed design procedure for both approaches can be found in [13] and [14] for the cases of 90 and 180 hybrids, respectively. Regardless of the kind of coupler, the operating principle is the same for both solutions. Concerning the topology of the device, if N ¼ 2n is the order of the matrix, the number of hybrids used in the network is nN=2. The hybrids are arranged in n columns with N/2 hybrids per column. The number of columns of phase shifter is (n 1) and the number of phase shifters per column is N/2, yielding a total of ðn1ÞN=2 phase shifters. The value of each phase shifter can be calculated by using the formulae given in [13]. By feeding the pth input port (p ¼ 1. . .N) a linear phase distribution at the output ports is obtained so that the main beam is tilted by the angle N þ 12p l p ¼ 1; 2; . . . ; N ð7:65Þ yp ¼ sin1 2N d l is the free space wavelength and d is the distance between adjacent radiating elements. As an example, consider the 4 4 matrix employing 90 hybrids shown in Figure 7.40. A total number of four couplers, arranged in two columns, and two phase shifters are employed. For each input port, the resulting phase distributions at the output ports of the device and the corresponding main beam direction, assuming d ¼ l/2, are given in Table 7.3.

26

Elisa Sbarra and Roberto Vincenti Gatti are gratefully acknowledged for their contribution to this section.


Figure 7.40

225

Schematic of a 4 4 Butler matrix with 90 hybrid junctions.

Table 7.3 Phase distributions at the output ports of Figure 7.40 and main beam direction yp for d ¼ l/2 Port p

OUT1

OUT2

OUT3

OUT4

45 135 90 180

2 4 1 3

7.9.2

90 0 225 135

135 225 0 90

180 90 135 45

yp 14.47 48.59 48.59 14.47

The Blass matrix27

In contrast to the Butler matrix, the Blass matrix [15] can in principle have any number of input and output ports. It consists of M rows corresponding to the number of beams to be generated and N columns connected to the radiating elements, as sketched in Figure 7.41. The two sets of transmission lines, usually referred to as rows and columns of the matrix, are interconnected at each crossover by a directional coupler. The signal applied at any input port (l.h.s. of Figure 7.41) propagates along the corresponding feed line. At each crossover a small portion of the signal is coupled to each column, ultimately exciting the corresponding radiating element. Each line is terminated with a matched load to absorb the residual signal. By feeding each input port, a different phase distribution is produced along the array, thus a different radiated beam orientation, depending on the various electrical paths. The network is therefore able to generate M different radiated beams whose orientations depend on the selected input port. The coupling coefficients of the directional couplers affect the power distribution among the array elements, thus the beam shape (beamwidth, sidelobe level, etc.). Compared with the Butler matrix, the Blass matrix is much more flexible, since it can generate multiple beams placed in arbitrary directions, covering a wide scanning range. However, part of the input power is lost as it is absorbed by the matched loads at the end of each line. The synthesis procedure can be illustrated with the aid of the schematic of Figure 7.42, where we consider an M N Blass matrix made of ideal transmission lines and directional couplers. For simplicity, we assume the phase shift to be the same for all branches and equal to y. The Blass matrix is identified by the M N complex quantities Xmn ¼ Cmn ejfmn , Cmn being the coupling coefficient of the mnth coupler and fmn the phase of the mnth line length. For the ith (i ¼ 1. . .M) beam to be generated, the N radiating elements must be excited with an appropriate complex signal distribution Oij (j ¼ 1, 2, . . ., N). This 27

Federico Casini is gratefully acknowledged for his contribution to this section.

226


Figure 7.41

Schematic of the Blass matrix [16] (copyright EuMA).

identifies the excitation matrix [O]. The synthesis of the Blass matrix consists of computing the coefficients Xmn (m ¼ 1. . .M, n ¼ 1. . .N) for a given excitation matrix Oij. This can be done by a recursive procedure starting with the first element of the array and then proceeding with the following ones, the computation becoming progressively heavier. Consider the signal coming from input 1 that has to generate beam number 1: the only path connecting input 1 to the first radiating element is through the coupler C11 and the phase shifter f11.

Figure 7.42

Schematic for the design procedure of a Blass matrix [16] (copyright EuMA).


Figure 7.43 EuMA).

227

Photographs of a Blass matrix in rectangular waveguide technology [16] (copyright

Because of the ideally infinite directivity of the couplers, a simple equation can be written to compute the excitation for node (1,1): jO11 je jffO11 ¼ C11 ejf11

ð7:66Þ

where O11 is the desired excitation for the first radiating element when the first beam is selected. Similarly we compute all C1j and f1j (j ¼ 1. . .N) and in the same manner we can immediately compute C21 and f21. Now consider the path connecting input 2 and output 2 of the network. As can be seen from Figure 7.42, there are two different paths: the primary path crossing node 22 (still to be computed), and a secondary path crossing three nodes of the matrix that have already been computed. Once again, the complex equation associated to the considered node has only one complex unknown, but this time the equation is made up of two different terms, each associated to a different path: jO22 je jffO22 ¼ ejy ½C12 C21 C11 ejðf12 þ f21 Þ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C ejðf12 þ f22 Þ 1C21 1C12 22

ð7:67Þ

Following the above procedure that accounts for all secondary paths, the synthesis of the ideal multi-beam Blass matrix is obtained. In this way, the excitations produced by each input port are not distorted by the presence of the secondary paths, and so produce almost ideal radiation patterns. Figure 7.43 shows the photograph of a 5 32 Blass matrix realized in rectangular waveguide technology operating at Ku band. The matrix was designed to have the beam oriented at 30 , 35 , 40 , 45 and 50 with respect to the broadside. The matrix was realized as a three-layer machine-drilled aluminium structure. Figure 7.44 shows the measured radiating patterns corresponding to inputs 1, 3 and 5.

7.9.3

The Rotman lens28

A Rotman lens is a planar M N device employed as an MBFN for switched-beam/multi-beam antennas [17]. Scanning is achieved without phase shifters or power dividers, thus strongly reducing the fabrication costs. Normal realizations are either in waveguide or printed circuit technology, the main advantages of the former being the lower loss and higher power-handling capability, while reduced 28

Elisa Sbarra and Roberto Vincenti Gatti are gratefully acknowledged for their contribution to this section.

228


Figure 7.44 Measured normalized radiation patterns generated by the Blass matrix of Figure 7.43 (copyright EuMA) [16].

dimensions and lower fabrication costs are typical of the latter. Specific features of the Rotman lens are the wide scanning angle range and the wide operating frequency band. The design procedure for Rotman lenses in microstrip technology is described here. Figure 7.45 shows the layout of the Rotman lens. The main part is a parallel-plate region bounded on the left by the circular profile C1, with radius R and centre Op, and on the right by the profile C2 which has to be determined. The M input ports lie on C1, whereas the N radiating elements are located on the straight profile CA and identified by their coordinates Yn (n ¼ 1. . .N), where the linear array elements are located. N transmission lines of length Wn (n ¼ 1. . .N) connect the points Pn(x, y) on C2

y

Parallel plate region F1 Input profile C1

Wn

•

•

F

•

G

• •

C

F•2

• • P (x,y)• • • W • Wave front •O • x • • • • •C • n

R

F0•

Op

•

0

A

Output profile C2

Figure 7.45

Yn

Radiating elements

Schematic layout of Rotman lens.


229

with the corresponding radiating elements Yn on CA. Beam steering is achieved by switching among the M input ports. The first step in the design procedure consists of choosing three points, F0, F1 and F2, symmetrically located on the input profile C1, as shown in Figure 7.45. We then need to introduce the following geometrical quantities, shown in the figure: .

G is the distance of F0 from the origin O of the reference system.

.

F is the length of F1O and F2O.

.

a and a are the angular positions of F1 and F2, respectively. The corresponding input ports have coordinates (G, 0), (F cos a, F sin a), (F cos a, F sin a).

.

c is the desired steering angle obtained by feeding the lens from F1.

F1, F2 and F0 are points of perfect focus: by feeding the lens from such points, a phase-error-free excitation is generated at the antenna radiating elements, with steering angles þ c, c and 0 , respectively. The design equations simply consist of imposing that the three rays from F1, F2 and F0 to the radiating elements Yn have the same electrical lengths, apart from the contribution required for the beam tilting in the (0, þ c or c) direction, respectively. As can be easily seen, such conditions are expressed as follows [18]: pffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi er ðF1 Pn Þ þ ee f f Wn þ Yn sin c ¼ er F þ ee f f W0 pffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi er ðF2 Pn Þ þ ee f f Wn Yn sin c ¼ er F þ ee f f W0 pffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi er ðF0 Pn Þ þ ee f f Wn ¼ er G þ ee f f W0

ð7:68Þ

where er is the substrate dielectric constant and eeff is the effective dielectric constant of the output microstrip lines. F1Pn, F2Pn and F0Pn are the path lengths from the points F1, F2 and F0, respectively, to the point Pn(x, y) on the profile C2. In the above equations, the unknowns are the coordinates of Pn on C2 and the length Wn. By solving (7.68) for n ¼ 1. . .N, one determines the array profile C2 and the line length Wn for Rotman lenses in microstrip technology. This procedure yields a lens with three perfect focal points F1, F2 and F0 corresponding to the steering angles c and 0 . By feeding the lens from a generic input point Cg with angular position g on the profile C1, the main lobe is tilted at the angle y, which is determined from the following formula:

sin y ¼ sin g

sin c sin a

ð7:69Þ

The last design step concerns the plate region side walls. They have to be designed in order to minimize multiple reflections in the parallel-plate region: common solutions consist of the use of properly shaped absorbing material for waveguide lenses or dummy ports terminated on matched loads for microstrip devices. A photograph of a Ku band microstrip Rotman lens operating in the 11.325–11.875 GHz frequency range is shown in Figure 7.46. This is a 7 32 beamforming network realized in microstrip technology. Dummy ports terminated with matched loads are located along the side walls of the parallel-plate regions. The measured radiation patterns of a slotted waveguide antenna fed by the proposed device at centre frequency are shown in Figure 7.47; the optimized output delay lines Wn are designed to produce a 25 –55 scanning angle range.

230


Figure 7.46

Ku band microstrip Rotman lens prototype: total dimensions 43 46 cm.

Figure 7.47 Rotman lens radiation patterns measured at centre frequency f0 ¼ 11.575 GHz [19].

7.10 Non-reciprocal components All the components considered so far are made of linear and isotropic materials for which Lorentz’s reciprocity theorem of Section 2.7.2 holds. There is an important class of microwave passive devices that make use of ferrites, ceramic materials with particular magnetic properties and, when subject to a static magnetic field, anisotropic properties as well. Even a concise treatment of EM wave propagation in


231

ferrites is beyond the scope of this book, so, due to space limitations, we will present just a sketchy description. The interested reader is directed to the rich literature available on the subject [20–22]. Ferrites are ferrimagnetic materials with magnetic permeabilities of the order of several thousands. They exhibit very low losses (typical resistivity is of the order of 1014 O m) with a dielectric constant between 10 and 15. They are made of oxides of iron (Fe), manganese (Mn), nickel (Ni) and zinc (Zn), or a mixture of them. Their peculiar magnetic properties derive from the presence of a magnetic dipole moment produced by the spins of the electrons. By applying a static magnetic field B0 producing a saturated magnetization Ms of the ferrite, anisotropic behaviour is exhibited concerning the RF magnetic field. The constitutive relation between the RF components of B and H is of the form B ¼ mH

ð7:70Þ

If the static field B0 is parallel to the z axis, the permeability tensor m is expressed as follows: 3 2 mxx mxy 0 7 6 ð7:71Þ m ¼ m0 4 myx myy 0 5 0

0

1

Therefore, the Cartesian components of (7.51) are expressed as Bx ¼ m0 ðmxx Hx þ mxy Hy Þ By ¼ m0 ðmyx Hx þ myy Hy Þ

ð7:72Þ

Bz ¼ m0 Hz The anisotropic behaviour is apparent because the vector B, even though its components are linearly related to those of the magnetic field, has different orientations and amplitudes depending on the direction of H. In particular, if the RF magnetic field H is parallel to the static magnetic field (thus is directed along z), the ferrite behaves as an ordinary non-magnetic material (B ¼ m0 H). If, on the contrary, the magnetic field is directed in another direction, e.g. along the x axis, the magnetic induction has both x and y components. Under some simplifying hypotheses, in particular by neglecting material losses, the elements of the permeability tensor can be written as mxx ¼ myy ¼ 1 þ mxy ¼ myx ¼

oM o0 o20 o2

jooM o20 o2

ð7:73Þ

where oM ¼ jgjMs =m0 o0 ¼ jgjB0 =m0

ð7:74Þ

g is the gyromagnetic ratio given by the ratio between the magnetic dipole moment and the angular momentum of the electron; Ms is the saturated magnetization of the ferrite. The frequency o0 is the Larmor frequency or cyclotron frequency. When the frequency approaches o0 , the elements of the permeability tensor become infinite as the result of a strong interaction between the RF magnetic field and the magnetic dipole moments associated with the electron spins. A specific consequence of the material anisotropy is that Lorentz’s reciprocity theorem does not hold, so that the microwave components containing such materials are not reciprocal: the [Z], [Y] or [S] matrices are thus expected not to be symmetrical.

232


Figure 7.48

7.10.1

Resonance isolator.

Isolator

The non-reciprocal character typically manifests itself in the isolator. An ideal isolator is a two-port device that allows the RF signal to be transmitted without attenuation from port 1 to port 2, while it inhibits transmission from port 2 to 1 by totally absorbing the incoming RF power. Its scattering matrix is thus 0 0 ½S ¼ ð7:75Þ 1 0 The [S] matrix is clearly non-symmetrical (s12 6¼ s21 ) and non-unitary (½S*T 6¼ ½S1 ); indeed the isolator is a lossy component since it absorbs the power entering from port 2. Put in cascade to a generator, the isolator ensures its matching while protecting it from possible reflections from the load. The isolator can be realized in a rectangular waveguide by exploiting the resonance behaviour of the ferrite at the cyclotron frequency o0. In the resonance isolator sketched in Figure 7.48, one or two ferrite slabs magnetized in the y direction are located at waveguide positions where the RF magnetic field is circularly polarized in the xz plane.29 The forward attenuation of such a device is of the order of 0.02 dB/ mm at a frequency of 10 GHz, while the corresponding backward attenuation is of the order of 0.25–0.40 dB/mm [4]. The isolator can also be realized by employing a circulator, as shown in the next section.

7.10.2

Circulator

The circulator is an n-port device (n usually being 3), ideally without loss and perfectly matched, used to transmit the signal from port i to the next port i þ 1 (or i 1), according to the scheme of Figure 7.49. As a consequence of the above properties, the scattering matrix of an ideal three-port circulator is 2 3 0 0 1 ½S ¼ 4 1 0 0 5 ð7:76Þ 0 1 0

29 Equations (3.132) show that the x and z components of the magnetic field of the dominant TE10 mode are in quadrature. Therefore, the magnetic field is circularly polarized in the xz plane at the x coordinate where the x and z components have the same amplitude.


233

2 3

Figure 7.49 Symbol of the three-port circulator. antenna

transmitter

receiver

Figure 7.50

Use of an antenna in receive and transmit modes using a circulator. 1

2 3

Figure 7.51

A circulator used to realize an isolator.

This expression implies that the signal entering at port 1 is transferred without attenuation to port 2, that from port 2 to port 3 and that from port 3 to port 1: b1 ¼ a3 b2 ¼ a1 b3 ¼ a2 The scattering matrix (7.76) is clearly unitary. The circulator can have very wide bandwidths, larger than one octave. It can typically be used to allow a transmitter and a receiver to share the same antenna,30 as sketched in Figure 7.50. It can also be used as a wide-band isolator by terminating one of the ports with a matched load, as shown in Figure 7.51. It can easily be proved that any three-port lossless, matched and non-reciprocal network is a circulator.31 Therefore, a circulator can in principle be realized by putting a ferrite sample into a threeport junction and ensuring that the device is matched. The latter is indeed the most critical condition, which is realized using suitably designed matching elements. 30 31

See also Section 15.5.1. It suffices to impose the unitary property on a 3 3 scattering matrix having s11 ¼ s22 ¼ s33 ¼ 0.

234


Bibliography 1. K. Chang, Handbook of Microwave and Optical Components, Vol. I, Microwave Passive and Antenna Components, pp. 62, 80, John Wiley & Sons, Hoboken, NJ, 1989. 2. C. G. Montgomery, R. H. Dicke and E. M. Purcell, Principles of Microwave Circuits, McGraw-Hill, New York, 1948. 3. N. Marcuvitz, Waveguide Handbook, McGraw-Hill, New York, 1951. 4. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, 1992. 5. R. E. Collin, Field Theory of Guided Waves, IEEE Press Series on Electromagnetic Wave Theory, 2001. 6. J. Van Bladel, Singular Electromagnetic Fields and Sources, Oxford Science Publications, 1991. 7. K. C. Gupta, R. Garg and R. Chadha, Computer-Aided Design of Microwave Circuits, Artech House, Norwood, MA, 1981. 8. M. E. Bialkowski, ‘Analysis of a coaxial-to-waveguide adaptor incorporating a dielectric coated probe’, IEEE Microwave and Guided Wave Letters, Vol. 1, No. 8, pp. 211–214, 1991. 9. E. J. Wilkinson, ‘N-way hybrid power combiner’, IRE Transactions on Microwave Theory and Techniques, Vol. 13, pp. 116–118, 1960. 10. T. H. Lee, Planar Microwave Engineering, Cambridge: Cambridge University Press, 2004. 11. R. Levy, ‘Directional couplers’, in L. Young, Advances in Microwaves, Vol. 1, Academic Press, New York, 1966. 12. J. Lange, ‘Interdigitated stripline quadrature hybrid’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-17, pp. 1150–1151, 1969. 13. H. J. Moody, ‘The systematic design of the Butler matrix’, IEEE Transactions on Antennas and Propagation, Vol. 11, pp. 786–788, 1964. 14. T. M. Macnamara, ‘Simplified design procedures for Butler matrices incorporating 90 hybrids or 180 hybrids’, IEEE Proceedings on Microwaves, Antennas and Propagation, Vol. 134, No. 1, pp. 50–54, 1987. 15. P. S. Hall and S. J. Vetterlein, ‘Review of radio frequency beamforming techniques for scanned and multiple beam antennas’, IEEE Proceedings on Microwaves, Antennas and Propagation, Vol. 137, pp. 293–303, 1990. 16. F. Casini, R. Vincenti Gatti, L. Marcaccioli and R. Sorrentino, ‘A novel design method for Blass matrix beam-forming networks’, Proceedings of the European Microwaves Conference, 9–12 October, pp. 1511–1514, 2007. 17. W. Rotman and R. F. Turner, ‘Wide-angle microwave lens for line source applications’, IEEE Transactions on Antennas and Propagation, Vol. 11, pp. 623–632, 1963. 18. P. K. Singhal, P. C. Sharma and R. D. Gupta, ‘Rotman lens with equal height of array and feed contours’, IEEE Transactions on Antennas and Propagation, Vol. 51, pp. 2048–2056, 2003. 19. R. Vincenti Gatti, L. Marcaccioli, E. Sbarra and R. Sorrentino, ‘Flat Array Antennas for Ku-Band Mobile Satellite Terminals’, International Journal of Antennas and Propagation, Sp. Issue on Active Antennas for Space Applications, Hyndawi Publishing Corporation, Volume 2009 (2009), Article ID 836074, 5 pages. 20. B. Lax and K. J. Button, Microwave Ferrites and Ferromagnetics, McGraw-Hill, New York, 1962. 21. J. Helszajn, Principles of Microwave Ferrite Engineering, John Wiley & Sons, Inc., New York, 1969. 22. A. J. Baden Fuller, Ferrites at Microwave Frequencies, Peter Peregrinus, London, 1987.


235

Simulation files 01_Attenuator.adsn. Describes the microstrip attenuator of Figure 7.6, producing the plot in Figure 7.7. 02_Wilkinson_Divider.adsn. Analyzes the power dividers in Figure 7.20 as per the result shown in Figure 7.22. 03_Rat_Race.adsn. Simulation of the rat-race of Figure 7.25, with the respective response on the same figure. 04_Branch_Line.adsn. Analysis of the microstrip branch-line couplers (ideal and microstrip) in Figure 7.29b. The result of the simulation is plotted in Figure 7.31. 05_Directional_Coupler.adsn. Analyzes an ideal and a microstrip 10 dB coupler working at 10 GHz. 06_Lange_Coupler.adsn. Simulates the Lange coupler having the layout of Figure 7.34 and the response in Figure 7.35. 07_Lumped_Branch_Line.adsn. Comparison of the lumped branch coupler of Figure 6.37 with a standard transmission line implementation. Produces the curves of Figure 7.38.

8

Microwave filters 8.1 Introduction Microwave filters are fundamental building blocks in many electronic systems, such as cellular phones, wireless communications, radar, etc. The microwave filter discipline is so broad that no single book can cover it. The limited space of one chapter will be sufficient to introduce some basic concepts of microwave filter design and performances. The filter design involves many theoretical approaches: network theory and synthesis, distributed circuit theory, electromagnetic theory. As well as the theory, both practical and technological aspects must be considered. This chapter presents some theoretical fundamentals of microwave filters and some concepts of microwave filter design. Practical examples complete the description. Some of these are intentionally simple in order to point out some common pitfalls in filter design.

8.2 Definitions The need to filter an electrical signal, in order to clean it of many different disturbances, is as old as electromagnetism itself. The oldest filter synthesis techniques were conceived at the beginning of the twentieth century. They were based on the use of cascaded lumped element cells, to reproduce approximately the properties of distributed networks. The filters obtained consisted of cascades of elementary lumped cells, each matched to its two adjacent ones. Moreover, the first (respectively, last) cell matched the source (load). This basic idea is the foundation of the image parameter design method (see [1–3]): the main advantage of the method is its simplicity; the main disadvantage is the limitation in the performances obtainable. Nowadays the image parameter design method is seldom used, its application being limited to special cases such as periodic structures. Nevertheless, it is believed that its basic concepts can still be useful in a number of emerging applications, such as metamaterials and electronic band gap (EBG) structures, without the need for reinventing new methods. The modern filter design approach is based on a different idea, developed between 1920 and 1930. It consists of approximating the response of an ideal filter with mathematical functions which can be realized with real networks. The theoretical foundation of the method is the well-consolidated theory of


238

MICROWAVE AND RF ENGINEERING Loss-free 2-port network Rs [S]

1

Figure 8.1

2

RL

Schematic principle of a filter.

lumped electrical networks. This theory supplies the techniques to identify the physically realizable networks together with the mathematical procedures to synthesize them.1 Here, we consider a filter as a two-port – ideally loss free – placed between a resistive load RL and a source with internal series resistance RS ,2 according to the configuration shown in Figure 8.1. The filter has the function of transferring power from the source to the load with a frequency-dependent attenuation. The frequency ranges of low or high attenuation3 are called passbands or stopbands respectively. Depending on the type of response, filters are classified into four main categories: low pass, high pass, bandpass and bandstop.4 We can characterize the filter by means of the transmission coefficient or scattering parameter s21.5 It can be demonstrated6 that the square amplitude of the transmission coefficient for any lumped network is a rational function of frequency – more precisely, the ratio between two polynomials of the variable o2 : M X 2

js21 j ¼

i¼1 N X

ai o2i ð8:1Þ bi o2i

i¼1

The quantity (8.1) is referred to as the power transfer ratio of the filter or, when expressed in dB, the power gain or, simply, the gain. Since the filter is loss free, conditions (4.99) apply. We rewrite them as js11 j2 ¼ js22 j2 ;

js11 j2 þ js21 j2 ¼ 1

ð8:2Þ

The filter attenuation is measured in dB, and is defined as L ¼ 10 log10 js21 j2 ¼ 20 log10 js21 j

ð8:3Þ

Other parameters can also be used to characterize the filter response: namely, the passband IL (insertion loss) and the RL (return loss). The former quantity is the maximum value of the insertion loss in the passband h i IL ¼ max 10 log10 js21 j2 ¼ LMAX ð8:4Þ

1

The meaning of the word ‘synthesis’ will be made clear later in this chapter. As we saw in Section 4.5, such a source is equivalent to the incident wave on a transmission line having a characteristic impedance equal to RS . 3 The exact definition of the attenuation is given in footnote 5 of this chapter. 4 Also called notch filters in the case of narrow stopbands. 5 Recall that js21 j2 is the ratio between the active power delivered to the load and the incident one (i.e. the available power). Some authors [2, 3] use the power loss ratio PLR , defined as the ratio between the available power and the power delivered to the load. Another widely used quantity is the attenuation, expressed in dB and defined as L ¼ 10 log10 ðPLR Þ. From these definitions, PLR ¼ 1=js21 j2 ; L ¼ 10 log10 ðPLR Þ . 6 See [4]. 2

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239

Conversely, RL is defined as the minimum absolute value (in dB) of the reflection coefficient in the passband h i h i ð8:5Þ RL ¼ min 10 log10 js11 j2 ¼ min 10 log10 ð1js21 j2 Þ From the definitions (8.4) and (8.5), it follows that IL and RL are related by RL ¼ 10 log10 ð110IL=10 Þ

ð8:6Þ

The first step in the filter design process is the approximation. The approximation problem consists of finding the coefficients of the polynomials of Equation (8.1), giving the best match – according to appropriate criteria – between the responses given by (8.1) and the desired one. Once the polynomials have been determined, network theory allows one to synthesize the filter, i.e. to determine the lumped network (including the values of all its elements) having the response given by the function in (8.1). Suitable frequency transforms allow such a procedure to be extended to distributed constant filters, which is the case of interest for microwave applications. Having said that, the most common synthesis procedure for a generic filter starts from a reference lowdpass filter, called the prototype, from which, after suitable transformations, the filter with the desired response can be obtained. In particular instances, different methods are sometimes employed that allow direct synthesis of the microwave filter.

8.3 Lowdpass prototype The prototype is a lumped constant lowdpass filter, having a power transfer ratio js21 j2 approximating the ideal lowdpass response shown in Figure 8.2 as a function of the angular frequency o0. The ideal power transfer ratio equals one or zero depending on whether the angular frequency o0 is smaller or larger than unity, respectively. The corresponding attenuation is zero (or infinite), for o0 < 1 (or o0 > 1), while the power gain is 0 dB (or 1 dB), respectively. The prototype load resistance is assumed to be unitary, without loss of generality: this is equivalent to normalizing all the impedances by RL. The prototype is therefore nothing but a normalized filter, with unitary load resistance and angular frequency. The real filter has cut-off angular frequency equal to o0 and load resistance RL . It is obtained from the prototype by applying the denormalization procedure. The impedance denormalization simply consists of multiplying all impedances by RL. The frequency denormalization, as we will see later in this chapter, consists of scaling all inductances and capacitances of the prototype in such a way that the corresponding reactances at the cut-off frequency remain unchanged. Briefly, all the prototype inductances and capacitances must be divided by o0.

|s21|2 1

O

Figure 8.2

1

ω′

Response of the ideal lowdpass filter.

240


Five functions like (8.1) are available to approximate the ideal response of Figure 8.2: namely, the Butterworth, Chebyshev,7 Cauer, Bessel and inverse Chebyshev approximations. Here, we will consider the first three, as they are more frequently used with microwave filters.

8.3.1

Butterworth filters

For Butterworth filters, the function (8.1) has the simple form s21; Butterworth ðo0 Þ2 ¼

1 1 þ o0 2N

ð8:7Þ

where N is the filter order and o0 ¼ o=o0 is the normalized angular frequency, so that the cut-off occurs at o0 ¼ 1. We will see later in this chapter that the order of the filter coincides with the number of its elements. As far as the power transfer is concerned, we may note that: .

It is unitary at DC ðo0 ¼ 0Þ.

.

It approaches zero as 1=o0 N when the frequency becomes infinitely large.

.

It equals 1=2 (3.01 dB) at the cut-off angular frequency (o0 ¼ 1) for any order N.

.

All frequency derivatives up to Nth order vanish for o0 ¼ 0. For this reason, Butterworth filters are also referred to as maximally flat.

From Equation (8.6) it follows that Butterworth filters do not present a good matching at the cut-off frequency, where the return loss is only 3.01 dB. For this reason, Butterworth filters are not often used in high-frequency applications.8 Figure 8.3 shows the response of Butterworth filters of orders N ¼ 2 to 7.

8.3.2

Chebyshev filters

The Chebyshev filter offers higher stopband attenuation than its Butterworth counterpart of the same order. Moreover, Chebyshev filters allow the passband attenuation to be controlled as a design parameter. The response is based on the use of Chebyshev polynomials. Such polynomials exploit the property that the cosine of Ny can be expressed as an N-degree polynomial of cosðyÞ, i.e. cosðNyÞ ¼ TN ½cosðyÞ

ð8:8Þ

where TN is the Nth-order Chebyshev polynomial of the variable x ¼ cosðyÞ. We can easily recognize that T0 ðxÞ ¼ 1;

T1 ðxÞ ¼ x;

T2 ðxÞ ¼ 2x2 1

ð8:9Þ

7 We adopt here the most frequently used English transliteration. Other possibilities are Chebysheff or Tchebycheff (German). The latter choice justifies the use of letter T for the homonymous polynomials. 8 A good impedance matching at the ports of the elements of a chain is very important for its performance. To understand this, let us cascade one flat-response amplifier with one filter. The resulting response is reasonably similar to the product of the two responses only if the impedance seen by the amplifier is close to the matched one. A filter return loss of only 3 dB implies that the matching assumption does not hold true, so that the global response of the chain will be quite rippled. The minimum acceptable value for the return loss of the various components depends on the application: as a rule of thumb, from 10 to 12 dB for wide-band components (one octave and more) to 20–25 dB for narrow-band components ðDo=o0 10%Þ.

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241

20 log10[s21(ω)], Butterworth

0 –10 –20 N=2 –30 3

–40 7

–50

6

5

4

–60 0

1

2

3

4

5

ω/ω0

Figure 8.3 Responses of the Butterworth filter for different orders.

Moreover, it is possible to demonstrate the recursive rule TN ðxÞ ¼ 2xTN1 ðxÞ TN2 ðxÞ

ð8:10Þ

The computation of Chebyshev polynomials of any order can thus be performed starting from (8.9) and then applying the recursive relation (8.10). From the point of view of filter design, the interesting properties of Chebyshev polynomials are: (a) jTN ðxÞj 1 for

jxj 1

(b) jTN ðjxj ¼ 1Þj ¼ 1 1 ðodd NÞ (c) jTN ðx ¼ 0Þj ¼ 0 ðeven NÞ (d) limx ! 1 TN ðxÞ ¼ limx ! 1 2N1 xN . Chebyshev polynomials of degrees 1 to 5 are shown in Figure 8.4. Note that, because of property (a), when jxj is less than 1, the Chebyshev polynomial of any degree oscillates between the values 1, touching N1 times the values 1 and 1. Furthermore, because of property (b), the Chebyshev polynomials assume the value 1 or 1 at the extremities of the interval x ¼ 1. The power transfer ratio of Chebyshev filters is expressed as s21;Chebyshev ðo0 Þ2 ¼

1 1 þ e2 TN 2 ðo0 Þ

ð8:11Þ

where N is the filter order, o0 ¼ o=o0 is the normalized angular frequency and e is a parameter determining the maximum passband attenuation. Properties (a), (b) and (c) of Chebyshev polynomials imply the corresponding properties of (8.11): (a0 ) The power transfer ratio oscillates within the interval ½ð1 þ e2 Þ1 ; 1 and the corresponding gain (in dB) ranges from 10 log10 ð1 þ e2 Þ to 0. The quantity RP ¼ 10 log10 ð1 þ e2 Þ is the passband ripple of the Chebyshev response. (b0 ) At the passband edge o0 ¼ 1 the power transfer ratio equals ð1 þ e2 Þ1 for any N.

242


Figure 8.4

Chebyshev polynomials of degree N ¼ 1, 2, 3, 4, 5.

(c0 ) For o0 ¼ 0, the power transfer ratio equals ð1 þ e2 Þ1 or 1 depending on whether N is even or odd, respectively. (d0 ) For o0 ! 1 the power transfer ratio approaches zero as ðo0 Þ2N e2 22ðN1Þ . The factor ðo0 Þ2N is the same as for Butterworth filters, while the factor9e2 22ðN1Þ is responsible for the higher selectivity of Chebyshev filters. From the property (a0 ), it follows that the parameter e determines the passband gain that oscillates between 1 and ð1 þ e2 Þ1 : for this reason, Chebyshev filters are said to be equiripple (or equal ripple). Figure 8.5 shows some examples of Chebyshev filters having passband ripple RP ¼ 0:5 dB ðe 0:349Þ and order N ¼ 2; 3; . . . ; 7. From consideration (a0 ) and from definition (8.11) it follows that, keeping the order constant, the higher the value of e, the higher the passband ripple and the stopband attenuation as well. Therefore, if we want to increase the stopband attenuation without compromising the passband ripple, we must increase the filter order. This is illustrated in Figure 8.6, which shows the gains of two sets of Chebyshev filters with RP ¼ 0.5 dB (solid lines) and 0.05 dB (dashed lines), and orders N ¼ 5, 7, 9, 11. As anticipated, the filters with lower passband ripple also have lower selectivity. Graphs such as those in Figure 8.6 can be used in the synthesis procedure to determine the order of the filter from its specifications.

9

The factor e2 22ðN1Þ is generally less than 1, unless very small values of e are used.

20 log10[s21(ω)], Chebyshev, RP = 0.5 dB

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243

0 –10 –20 N=2 –30 –40 3 –50

7

6

5

4

–60 0

1

2

3

4

5

ω/ω0

Figure 8.5

Responses of Chebyshev filters with 0.5 dB ripple and orders from 2 to 7. 0 RP = 0.5 dB RP = 0.05 dB

–10

20 log10(|s21(ω)|)

–20 –30 5 –40 5 –50 A –60 –70

11

9

11

9

7 7

–80 0.6

1.0

1.4

1.8

2.2

ω/ω0

Figure 8.6 Responses of odd-order Chebyshev prototypes (N ¼ 5 to 11). The ripple is 0.5 dB (solid lines) and 0.05 dB (dashed lines). EXAMPLE 8.1 Let us determine the order of a Chebyshev lowdpass filter having f0 ¼ 500 MHz cut-off frequency and a minimum stopband attenuation of 50 dB at the frequency of 900 MHz. The first step is the frequency normalization, such that the cut-off angular frequency (o0 ¼ 2pf0 ¼ 2p 5 108 ) corresponds to 1. In this way, the stopband frequency of 900 MHz corresponds to the normalized angular frequency o0 ¼ 900=500 ¼ 1:8. Now, in the diagram of Figure 8.6, let us draw a horizontal line corresponding to the minimum stopband attenuation 20 log10 ðjs21 jÞ ¼ 50 dB and a vertical line crossing the normalized angular frequency o0 ¼ 900=500 ¼ 1:8. The two lines intersect at the point A (1.8; 50). All filters whose responses pass above (below) the crossing point A exhibit a stopband attenuation lower (higher) than the required one. If the required passband ripple is 0.5 dB (0.05 dB) the minimum filter order is 7 (9).

244


8.3.3

Cauer filters

The responses of Butterworth and Chebyshev filters are of the all-pole type, since all transmission zeros are located at infinite frequency. Consequently, these filters present a finite attenuation at any finite frequency. The filter selectivity10 can be improved by introducing some transmission zeros at finite frequencies. This idea is applied in Cauer filters, which can be considered as a kind of generalization of Chebyshev filters, obtained by replacing the circular functions with elliptic ones. For this reason, Cauer filters are also known as elliptic filters. The power transfer ratio is s21;Cauer ðo0 Þ2 ¼

1 1 þ e2 FN2 ðo0 Þ

ð8:12Þ

where FN(o0 ) is a rational function having the following properties: (a) Like the Chebyshev polynomials, it oscillates within 1 for jo0 j 1. (b) FN2 ðo0 Þ 1=m0 if o0 2 o1 ¼ 1=m. In the passband (jo0 j 1), Cauer filters show zero attenuation at dN=2e points, while, in the same way as Chebyshev filters, js21 j2 uniformly oscillates between 1 and ð1 þ e2 Þ1 . For even (odd) N we have N/2 ððN þ 1Þ=2Þ attenuation zeros within the interval 0 o0 1. 1=2 In the stopband ðjo0 j 1=m1=2 Þ, js21 j2 oscillates between 0 (infinite attenuation) and 1=ð1 þ em0 Þ. 1=2 At infinite frequency, the power transfer ratio equals 0 ð1=ð1 þ em0 ÞÞ for odd (even) N. Cauer filters allow us to control both the passband (jo0 j 1) and the stopband (jo0 j o1 ) ripples by means of the parameters e and m0 , respectively. Such filters have the drawback that the transmission zeros are located at predetermined frequencies. Another type of filter exists – known as the generalized Chebyshev – exhibiting a uniform passband ripple and having transmission zeros arbitrarily located in the stopband. Unfortunately, since no closed-form expressions exist for the generalized Chebyshev filters, the design can only be performed by numerical methods. Figure 8.7 shows a comparison of fifth-order Butterworth, Chebyshev and Cauer responses. The Butterworth and Chebyshev curves are the same as in Figures 8.3 and 8.5, respectively. Both Chebyshev and Cauer filters have 1 dB passband ripple. It can be seen that the filter selectivity increases from the Butterworth to the Chebyshev to the Cauer. This last has two transmission zeros (attenuation poles) at finite frequencies (oz1 1:13 and oz2 1:51), and a third one at infinity. An expanded view of the passband of the three filters in Figure 8.7 is shown in Figure 8.8. Note that the Butterworth attenuation increases monotonically with the frequency, and reaches the value of 3.01 dB at the cut-off frequency. Both Chebyshev and Cauer attenuations oscillate between 0 and 1 dB, with similar behaviour. In equiripple filters the maximum passband attenuation coincides with the ripple. Now, by virtue of Equation (8.6) which relates the insertion loss and the return loss, passband ripples of 1, 0.5, 0.2, 0.1 and 0.01 dB correspond to return losses of 6.9, 9.6, 13.5, 16.4 and 26.4 dB, respectively. Figure 8.9 shows the return losses of the three filters in Figure 8.8. Note that the uniform passband ripple implies equal local minima of the return loss. The Chebyshev filter is the most frequently used one; compared with the Butterworth, it offers better selectivity and better control of the passband attenuation. The elliptic filter performs better in terms of

10 The selectivity is a parameter determining how fast the filter attenuation increases in the transition from the passband to the stopband.

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245

20 log10[s21(ω)]

0

ω1 = ω0 /0.9

–20

A1 = 20 log10(α1) ≈ –31.6 dB Cauer

–40 Butterworth Chebyshev –60 0

1

2

3

4

5

ω/ω0

Figure 8.7

Responses of fifth-order Butterworth, Chebyshev and Cauer lowdpass filters.

selectivity, but is more difficult to design and realize, therefore its use is limited to the cases where a very sharp selectivity is required to separate closely spaced frequency bands.

8.3.4

Synthesis of the lowdpass prototype

Once the filter response has been selected, be it a Butterworth (Equation (8.7)), Chebyshev (Equation (8.11)) or Cauer (Equation (8.12)) filter, we need to proceed with the filter synthesis: that is, to identify the two-port network of Figure 8.1 possessing such a response.

0 Chebyshev

20 log10[s21(ω)]

Cauer RP = 1 dB

–1

–2

Butterworth

–3 0

1 ω/ω0

Figure 8.8

Expanded view of the passband of Figure 8.7.

246


0

Butterworth –10

Chebyshev Cauer

–40

20 log10[s11(ω)]

20 log10[s21(ω)]

0 –20

–20

–30

–60 0

1

2

ω/ω0

Figure 8.9

Insertion loss and return loss of the filters plotted in Figure 8.8.

The filter synthesis can be reduced to that of its input impedance11 using a method developed by Darlington. In order to do this, we first need to know the reflection coefficient, which is related to the input impedance by the relation Zin ðqÞ ¼ R0

1 þ s11 ðqÞ 1 s11 ðqÞ

ð8:13Þ

where q ¼ s þ jo is the complex frequency12 and R0 is the normalizing impedance. Note that the prototype has R0 ¼ 1, by definition. The first synthesis step consists of determining s11 from js21 j2 . Let us illustrate the procedure. It can be proved that the scattering parameters of a real network satisfy the relation shk ðjoÞ ¼ s*hk ðjoÞ;

h; k ¼ 1; 2

ð8:14Þ

It follows that js11 ðjoÞj2 ¼ s11 ðjoÞs11 ðjoÞ hence, by Equation (8.2), we have s11 ðjoÞs11 ðjoÞ ¼ 1s21 ðjoÞs21 ðjoÞ ¼ 1js21 j2

ð8:15Þ

By applying a mathematical operation known as analytical prolongation we can extend the validity of Equation (8.15) from the imaginary axis (q ¼ jo) to the whole complex plane (q ¼ s þ jo), so obtaining the Feldtkeller equation s11 ðqÞs11 ðqÞ ¼ 1js21 ðqÞj2

ð8:16Þ

11 It can be proved that for a lumped two-port network the input impedance Zin ðqÞ is a real positive function of the complex frequency q ¼ s þ jo, i.e. it is real for real q and positive for s > 0. 12 The complex frequency coincides with the Laplace transform variable, and is typically indicated by lower case s. Here, lower case q is chosen to avoid confusion with the scattering parameters that are also functions of the complex frequency.

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From Equation (8.16) it follows that: 1. The poles of the reflection coefficient coincide with those of the transmission coefficient. 2. The reflection zero frequencies coincide with the frequencies of the unitary transmission coefficient. In the Butterworth case we have N zeros coincident with the origin. In the Chebyshev and Cauer cases, we have N zeros symmetrically placed along the imaginary axis.13 3. The square amplitude of the reflection coefficient includes the poles of both s11 ðqÞ and s11 ðqÞ. We can separate the former from the latter, considering that the poles of s11 ðqÞ present a negative real part. By exploiting the above properties, we can find the poles and zeros of the reflection coefficient, which can then be expressed in the form N Y

s11 ðqÞ ¼ r0 k¼1 N Y

q zk ð8:17Þ q pk

k¼1

where zk and pk are the zeros and poles of s11 , respectively. To complete the description of the reflection coefficient we still need to determine the factor r0. This can be obtained by making Equation (8.16) hold for one convenient value of the q, for instance q ¼ j. Note that (8.16) allows us to find r0 apart from its sign. Equation (8.13) shows that changing the sign of the reflection coefficient corresponds to transforming the input impedance into the input admittance. This is equivalent to swapping the network with its dual network. From a physical point of view, the dual network is obtained from the original one by replacing inductors with capacitors and vice versa, and by replacing series with shunt connections and vice versa. It follows that the dual network exhibits an identical response to the original one, thus a prototype may be equally realized in both forms. The second step of the synthesis procedure consists of determining the network from its input impedance. Without going into details, for our purposes it suffices to say that the procedure consists of N elementary synthesis steps, each reducing the complexity of the network. At each step an elementary term such as sL; sC; sL þ 1=sC; 1=sL þ sC is synthesized as an (inductor, capacitor, series LC cell, parallel LC cell), respectively, and subtracted from the immittance14 being synthesized. The resultant immittance consists of the series (or parallel) of such a term and a residual immittance which is a positive real function of lower order. By successive applications of this procedure, the entire network is synthesized in exactly N steps, giving a ladder network. In the case of all-pole filters (Butterworth and Chebyshev), one of the two ladder networks shown in Figures 8.10a,b is obtained, one being the dual of the other. Such networks are said to be canonical, since they realize the assigned response with the minimum number of elements. In the case of elliptic function filters, the resulting ladder networks have the more complicated structure shown in Figures 8.11a,b. The network in Figure 8.11a (8.11b) derives from Figure 8.10a (8.10b) by replacing the even elements with series (parallel) LC cells. The resonant frequency of each cell realizes one of the finite-frequency transmission zeros of the Cauer filter.

13 For instance, in the Chebyshev response of Figure 8.9, we have that js21 j ¼ 1 (0 dB) at o ¼ 0, 0.588o0, 0.951o0. The figure does not show the negative frequencies; the amplitude response, however, is symmetrical around the y axis. 14 The generic term immittance denotes either the impedance or the admittance.

248

MICROWAVE AND RF ENGINEERING L1 1

L3

LN-2

C2

C4

LN CN-1

2 (a)

L2 1

C1

L4

LN-1

C3

CN-2

CN

2 (b)

Figure 8.10 Canonical all-pole lowdpass filters. The source and the load are not shown; the former is to be connected to port 1, the latter to port 2. The cases illustrated here are for odd N. For even N the last element of the network in (a) (resp. (b)) is a capacitor (inductor).

L1

L3

L N-2 L N

C2

C4

C N-1

L2

L4

L N-1

1

2 (a)

L2

L4

L N-1

C2

C4

C N-1

1

2 C1

C3

C N-2

CN

(b)

Figure 8.11 Canonical elliptic lowdpass filters. The source and the load are not shown, as in Figure 8.10. Illustrated here are the cases for odd N. For even N the last element is a resonant circuit.

As given in Table 8.1, closed-form expressions are available for calculating the filters of Figure 8.10 [2]. Note that the filters are symmetrical, i.e. gk ¼ gNk . Tables listing the values obtained from the formulae15 of Table 8.1 are available in some books (see e.g. [2]). As for elliptic filters, unfortunately no closed-form expressions are available. The synthesis thus requires the use of numerical methods. Element values can be found in [4]. At this point, we must observe that even-order Chebyshev and Cauer filters exhibit non-zero DC attenuation, i.e. js21 ð0Þj < 1. At zero frequency (DC) the filters of Figure 8.10 simply correspond to the direct connection of the output with the input port; therefore a non-zero DC attenuation is achieved if and only if the load resistance differs from the source ðRS 6¼ RL Þ. Such a condition is not desirable in practical cases; therefore most Chebyshev and Cauer lowdpass filters are chosen to be of odd order.

15 The interested reader can use the Mathcad file 02_Low_Pass_Prototype_Coefficients.MCD, which automatically computes all the Butterworth and Chebyshev prototype coefficients of arbitrary order and ripple.

MICROWAVE FILTERS

249

Table 8.1 Formulae for the computation of the prototypes in Figure 8.10. g0 ¼ 1

Butterworth

ð2k1Þp ðk ¼ 1; 3; . . . ; NÞ 2N gn þ 1 ¼ 1 8 e > > b ¼ ln coth > > 40 logðeÞ > > > > > > > > g ¼ sinh b > > < 2N > > 2k1 > 2 kp 2 > ¼ sin ¼ g þ sin a p ; b ðk ¼ 1; 2; . . . ; NÞ > k k > > 2N N > > > > > > 2a 4a a > : g1 ¼ 1 ; gk ¼ k1 k ðk ¼ 2; 3; . . . ; NÞ g bk1 gk1 gk ¼ 2sin

Chebyshev

Lk ¼ Ck ¼ gk

ðk ¼ 1; 2; . . . ; NÞ

Table 8.2 Values of the lowdpass filter elements of Examples 8.1 and 8.2. k¼ gk ¼ gNk

1 1.737

R0 gk o0 gk Ck ¼ CNk ¼ o0 R0 R0 gk Lk ¼ LNk ¼ o0 gk Ck ¼ CNk ¼ o0 R0 Lk ¼ LNk ¼

2 1.258

4 1.344

Figure

8.011012

8.561012

8.10a

20.03109

21.40109

27.65109

3 2.638 41.99109

8.011012

16.801012

8.10b

EXAMPLE 8.2 (continued from Example 8.1) We wish to realize the filter of Example 8.1, choosing the first option: RP ¼ 0.5 dB, N ¼ 7. Applying the synthesis formulae for the Chebyshev prototype (second line of Table 8.1), we get the normalized values16 quoted in the second row of Table 8.2. Now, we choose the load and the source impedance of the filter as RS ¼ RL ¼ 50 O. After denormalizing with respect to both frequency and impedance, we get the following values of the prototype elements: Lk ¼ gk

R0 50 ¼ gk ; o0 2p 50 106

Ck ¼ gk

1 1 ¼ gk o0 R0 2p 50 106 50

The values of the ladder network in Figure 8.10a are quoted in the third and fourth rows of Table 8.2. The fifth and sixth rows list the values of the dual filter in Figure 8.10b. Both networks are seen to be symmetrical.

16 Such values can also be computed by means of the Mathcad file 05_Low_Pass_Example2.MCD, annexed with Chapter 9.

250


8.4 Semi-lumped lowdpass filters The filters considered so far are realized by using lumped elements. In RF and microwave practice, a circuit element can be considered as lumped if its physical size – defined as the maximum among length, width and depth – is sufficiently smaller than the wavelength. In this definition, the wavelength to be considered is the smallest one, corresponding to the highest frequency of interest. Needless to say, the concept of ‘sufficiently smaller’ is quite vague, the definition depending on the required degree of approximation. As a rule of thumb, an element is considered to be lumped when its size is smaller than one-tenth of the wavelength. For instance, a capacitor of size equal to 1 mm behaves as a lumped element up to the frequency fMAX ¼ v=l ¼ 3 108 =0:01 ¼ 30 GHz. In this example we have assumed propagation in a vacuum (v ¼ 3 108 m=s); should a different dielectric be used, the propagation velocity will be pffiffiffiffi reduced by a factor 1= er and thus the maximum frequency. In the subsequent sections of this chapter, we will discuss the techniques for the synthesis of distributed filters, which are more suited and used practically at microwave frequencies. Within certain frequency limits, however, a simpler approach based on so-called semi-lumped elements can be adopted, where inductors and capacitors are realized with short segments of transmission lines having high and low characteristic impedances, respectively. Consider the two-port network shown in Figure 8.12a. It consists of a transmission line segment having characteristic impedance Z0 , propagation velocity v and physical length l; at the angular frequency o, the phase constant is then b ¼ o/v. Using Equation (4.120), the impedance matrix of the network is 2 3 l l cot o 1=sin o 7 6 z11 z12 v v 7 6 7 ð8:18Þ ¼ j Z0 6 5 4 l l z21 z22 1=sin o cot o v v From Equations (8.18) and (4.60), considering the network identity of Figure 4.15, we obtain the equivalent network of Figure 8.12b. The impedances of the two series branches and of the shunt branch are l lo 1 cos o 1 12 sin2 lo v v2 ¼ j Z0 ¼ j Z0 tan Zseries branches ¼ z11 z21 ¼ j Z0 l lo lo v2 sin o 2 sin cos v v2 v2 l Zshunt branch ¼ z21 ¼ j Z0 cot o v

Z11 – Z21

L

Z 0, v

1

2

(a)

1

2

(b)

1

L 2

C

(c)

Figure 8.12 Transmission line segment: (a) symbol and characteristic parameters; (b) equivalent network; (c) simplified low-frequency equivalent network.

MICROWAVE FILTERS

251

Under the assumption of short transmission line (l l ) ol=v p=2), we can approximate the tangent with its argument. Hence the above expressions simplify to Zseries branches ffi j Z0

lo ; v2

Zshunt branch ffi j Z0

v1 lo

The characteristic impedance and the propagation velocity are related to the distributed inductance L0 and capacitance C0 of the line by Z0 ¼

rffiffiffiffiffiffi L0 ; C0

1 v ¼ pffiffiffiffiffiffiffiffiffiffi L0 C0

We then finally obtain the following expressions for impedances of the network in Figure 8.12b: Zseries branches ¼ z11 z12 ¼ j Z0 tan

Zshunt branch ¼ z12 ¼ z21 ¼

l Z0 l o ffij o ¼ joL 2v 2v

j Z0 1 1 ffi ¼ l l joC j o sin o Z0 v v

ð8:19Þ

where we have put L¼

Z0 l L0 ¼ l; 2 2v

C¼

l ¼ C0 l Z0 v

We can easily recognize that the two impedances given by (8.19) represent one inductor having inductance L and one capacitor having capacitance C, respectively. Therefore, if the line length is sufficiently smaller than the wavelength, the simplified equivalent T network of Figure 8.12c approximates the one of Figure 8.12b. Equation (8.19) states that the total inductance (capacitance) of the approximated network equals the global distributed inductance (capacitance) of the transmission line segment. Note also that L (C) increases (decreases) with Z0 . At this point, let us assume that the two ports of the line segment are terminated with impedance ZL. Two interesting cases can be considered, depending on whether the characteristic impedance of the line is much higher or much smaller than the terminating impedance. In the former case (Z0 ZL), the shunt capacitance of the network in Figure 8.12c becomes so small as to be negligible. The equivalent circuit reduces to a series inductance 2L ¼ L0 l. The dual situation occurs when Z ZL : the two series inductors can be neglected and the equivalent circuit reduces to a shunt capacitance C ¼ C0 l. In summary, a short transmission line segment with high (low) characteristic impedance17 is approximately equivalent to a series inductor (shunt capacitor) whose inductance (capacitance) is equal to the line inductance (capacitance) per unit length multiplied by the line length. The design of a semi-lumped lowdpass filter consists of replacing inductors and capacitors in the networks of Figure 8.10 or 8.11(suitably denormalized) with corresponding short segments of high- and low-impedance transmission lines, respectively. Such impedances have to be selected before the physical lengths of the line segments are determined. Although in principle other choices might be possible, in order to simplify the design and to obtain the best stopband performances, the highest (Z0max ) and lowest (Z0min ) impedances that can be realized with the given technology (stripline, microstrip, coplanar line, etc.) are selected. 17

High and low are defined with respect to the terminating impedance.

252

MICROWAVE AND RF ENGINEERING ðLÞ

ðCÞ

The lengths of the transmission line lk and lk corresponding to the inductance Lk and capacitance Ck, respectively, are then given by vM ðLÞ ðCÞ lk ¼ max Lk ; lk ¼ Z0min vm Ck ð8:20Þ Z0 Note that unless the line is TEM, the phase velocity depends on the line geometry. Therefore different impedances generally imply different velocities; that is why in expressions (8.20) two velocities (vM ; vm ) have been used, corresponding to the impedances (Z0max ; Z0min ). The dependence of the velocity on the impedance typically occurs in the microstrip.18

EXAMPLE 8.3 Let us design a semi-lumped lowdpass filter in coaxial cable technology, having a Chebyshev response of order N ¼ 15, passband ripple RP ¼ 0.1 dB, cut-off frequency equal to 7.6 GHz (corresponding to the angular cut-off frequency o0 ¼ 2p 7:6 109 rad/s), and load and source impedances R0 ¼ 50 O. We choose to realize the filter starting from the ladder network of Figure 8.10a (inductive input). It follows that the odd index elements are inductors and the even index ones are capacitors. The synthesis formulae of Table 8.1 yield the prototype coefficients listed in the third row of Table 8.2. After frequency (divide all the values by o0 ) and impedance (multiply the inductances and divide the capacitances by R0 ) denormalization, we obtain the values of the lumped filter quoted in the fourth and fifth rows of Table 8.3. As discussed above, the first synthesis step consists of determining the maximum and minimum characteristic impedance allowed by the given technology. To understand this point better, we need to look at the final aspect of the filter, shown in Figure 8.13. Each low-impedance segment (capacitor) consists of a dielectric-filled coaxial cable with outer diameter douter ¼ 7 mm, inner diameter

Figure 8.13

18

Exploded view of the filter discussed in Example 8.3.

See the graphs in Figure 3.25.

MICROWAVE FILTERS

253

Table 8.3 Values of the filter discussed in Example 8.3. k N þ 1k gk ¼ gNk Lk ¼ LNk Ck ¼ CNk lL;k ¼ lL;Nk lC;k ¼ lC;Nk

1 15

2 14

3 13

4 12

5 11

6 10

7 9

8 8

1.21 1.267 — 2.172 —

1.461 — 0.612 — 0.618

2.166 2.268 — 3.888 —

1.646 — 0.689 — 0.696

2.26 2.366 — 4.056 —

1.678 — 0.703 — 0.709

2.28 2.388 — 4.093 —

1.684 — 0.705 — 0.712

nH pF mm mm

dinner;low ¼ 5 mm and dielectric constant er ¼ 6. The outer diameter has been chosen to be equal to that of the connector so as to minimize the resultant discontinuity. The inner diameter is determined by the technological limitations associated with the realization of the pipe-shaped dielectric cladding, which has to provide the necessary mechanical support and remove the heat produced by the dissipation. The minimum allowed thickness of the dielectric pipe is 1 mm. The minimum impedance and the associated propagation velocity are then found to be 60 7 3 108 ffi 8:42 O; vm ¼ pffiffiffi ffi 1:225 108 m=s: Z0min ¼ pffiffiffi ln 5 6 6 The presence of the dielectric pipe along the whole filter is such that the high-impedance segments are non-homogeneous. The outer diameter is still 7 mm, but the inner diameter has to be minimized to get the maximum possible impedance. The technological constraints require dinner;high ¼ 1 mm. The computation of the impedance and the propagation velocity of the high-impedance segments is not straightforward because of the inhomogeneous filling. Without going into detail, we simply report the final results: 60 7 7 3 108 pffiffiffi ln ffi 47:7 < Z0max ffi 71:45 O < 60ln ffi 116:7; pffiffiffi < vM ffi 1:837 108 m=s < 3 108 1 1 6 6 Applying the design formulae in (8.20), we obtain the lengths of the inductive and capacitive segments, given in the last two rows of Table 8.3. Figure 8.13 shows an exploded view of the filter. It includes: .

Two coaxial input and output connectors (C1, C2).

.

The outer conductor of the coaxial cables, realized with two symmetrical parts B1 and B2.

.

The coaxial cable inner conductor (A), consisting of 15 cylindrical elements, alternatively narrow (inductors) and wide (capacitors).

.

The dielectric pipe (D), which mechanically supports the inner conductor, keeping it coaxial with the outer one.

The filter is assembled by inserting A within D, tightening B1 and B2 around D, and finally fixing the two connectors C1 and C2 at the filter ends. The real dimensions of the realized filter differ slightly from those listed in Table 8.3, the symmetry still being preserved. The reason for this discrepancy is that the semi-lumped design is based on approximate design equations. Moreover, parasitic capacitances are present at the junctions between high- and low-impedance sections. All such second-order effects can be compensated for by changing the element lengths, using computer-based simulation and optimization. The total length of the realized filter, including the connectors, is approximately 50 mm.

254


20 log10(|s21|)

–20

–40

noise floor

measured simulated

–60 lumped-constant –80 0

2

4

6

8

10

12

14

16

18

20

Frequency, GHz

Figure 8.14

Responses of the filters discussed in Example 8.3.

Figure 8.14 shows the responses of the simulated and realized filter. For comparison, the response of the ideal lumped filter is also plotted, showing an increasing stopband attenuation, which becomes infinitely large as the frequency approaches infinity. The response of the semi-lumped filter, on the contrary, is non-monotonic, since the cascade of transmission line segments has no attenuation poles. Finally, we observe that the measured stopband attenuation flattens to about 43 dB at frequencies higher than 10 GHz. This behaviour is not a characteristic of the filter, rather it is due to the noise floor of the test instrument employed.19 The return loss of the lumped filter is 16.4 dB (corresponding to 0.1 dB passband ripple). The simulated response – after optimization – presents a similar value, while the measured filter is slightly degraded: its return loss is about 13 dB (not shown in the figure).

8.5 Frequency transformations The synthesis of highpass, bandpass and bandstop filters can be obtained by applying to the lowdpass prototype suitable frequency transformations of the type o0 ¼ f ðoÞ

ð8:21Þ

0

20

where o is the prototype frequency and o is the frequency of the desired filter. Equation (8.21), the prototype response is modified to

As a consequence of

js21 ðo0 Þj2 ¼ js21 ½ f ðoÞj2 Correspondingly, the reactances of the various prototype o0 L0 ¼ f ðoÞL; 19

21

ð8:22Þ

elements L0 and C0 become

1 1 ¼ o0 C0 f ðoÞC

ð8:23Þ

See Section 17.6. Here we used the abbreviated form ‘frequency’ instead of ‘angular frequency’ or ‘radian frequency’, for the sake of simplicity. The Greek letter o, however, always denotes the angular frequency. 21 In order to avoid confusion, from this point to the end of the section we will mark the quantities relative to the prototype with a prime 0 ; those without a prime refer to the transformed filter. 20

MICROWAVE FILTERS

255

Therefore, the mapping function f ðoÞ not only has to provide the required frequency mapping, but also must be such that the reactances on the right of (8.23) are physically realizable. An almost trivial frequency transform is the frequency denormalization, already discussed in the previous section. The change in the cut-off frequency from one to an arbitrary value can be seen in fact as a simple frequency transformation. If o0 is the filter cut-off, we have o0 ¼ f ðoÞ ¼

o o0

The reactances (8.23) then become o0 L0 ¼ o

L0 ¼ oL; o0

1 1 1 ¼ ¼ o0 C0 oC0 =o0 oC

As previously found, L ¼ L0 =o0 ; C ¼ C0 =o0 are the inductances and capacitances of the frequency denormalized filter. The frequency transformation affects neither the impedance of the filter, nor the passband and stopband attenuations; in particular, it does not change the passband ripple. In addition to the frequency transformation, the synthesis of any of the filters considered here requires also the impedance denormalization. The order of the two operations does not matter. In the remaining sections, we will clarify the application of the above concepts by means of two examples, both based on the fifth-order Chebyshev prototype with 0.1 dB passband ripple RP. We will denote it as the reference prototype. By applying the synthesis formulae of Table 8.1, we get the reference prototype coefficients: g1 ¼ g5 ¼ 1:147, g2 ¼ g4 ¼ 1:371, g3 ¼ 1:975.

8.5.1

Lowdpass to highpass transformation

Apart from the frequency scaling, the simplest frequency transform is the lowdpass to highpass transformation jo0 ¼

1 jo

ð8:24Þ

Equation (8.24) maps the origin into infinity and vice versa, while the cut-off frequency (unitary) is invariant to the transformation.22 Applying the lowdpass to highpass transformation to the reference prototype, we get the response shown in Figure 8.15. Note that the passband extends from the unit frequency up to infinity, while the stopband23 coincides with the interval [0; 1). Moreover, Figure 8.15 shows an expanded view of the passband gain, plotted on the right y axis. The highpass filter exhibits a uniform passband ripple of 0.1 dB, like the reference prototype it is derived from. From the network point of view, the application of Equation (8.24) to the prototype transforms inductors into capacitors and vice versa. Indeed, the respective impedances become ZL ¼ jo0 L0 ¼

L0 1 ; ¼ jo joC ðHPÞ

ZC ¼

1 jo ¼ ¼ joLðHPÞ jo0 C 0 C0

22 Keep in mind that the frequency response of the filter is defined for both positive and negative values of the frequency. Furthermore, the ideal realization of the filter implies that s21 ðoÞ ¼ s21 ðoÞ* . 23 The cited interval includes both the stopband and the transition band. The exact definition of the respective limits depends on the specified stopband attenuation.

0

0.1

–10

0.0

–20

–0.1

–30

–0.2

–40

–0.3

Highpass 20 log10(|s21|)


Highpass 20 log10(|s21|)

256

–0.4

–50 0

1

2

3

4

5

ω

Figure 8.15

Response of the highpass filter derived from the reference prototype.

(HP)

C1

(HP)

1

Figure 8.16

(HP)

(HP)

C3

L2

(HP)

CN-2 CN (HP)

(HP)

L4

L N-1

2

Schematic of the lumped highpass filter derived from the network of Figure 8.10a.

where CðHPÞ ¼ 1=L0 and LðHPÞ ¼ 1=C0 are the capacitor and the inductor corresponding to the prototype inductor L0 and capacitor C0 , respectively.24 From the canonical lowdpass filter of Figure 8.10a we obtain the filter in Figure 8.16, where the component values are the reciprocals of those of the prototype:25 ðHPÞ

Lk

¼

1 Ck0

ðeven kÞ;

ðHPÞ

Ck

¼

1 L0k

ðodd kÞ

ð8:25Þ

The filter obtained has unitary impedance and cut-off frequency; if we want the generic impedance R0 and cut-off frequency o0, we need to denormalize the impedance and frequency. From the expression (8.25), we obtain ðHPÞ

Lk

¼

1 R0 ; Ck0 o0

ðHPÞ

Ck

¼

1 1 1 L0k o0 R0

ð8:26Þ

24 It must be pointed out that the prototype synthesis formulae of Table 8.1 assume o0 ¼ 1 and L0 ¼ gk ; C 0 ¼ gk (with the proper index even or odd for inductors and capacitors). 25 Starting from the dual network of Figure 8.10b, we obtain the dual network of Figure 8.16. In this case, the odd elements are shunt inductors and the even elements are series capacitors. Concerning the component values, Equation (8.25) holds true, but with interchanged even and odd indexes. In any case, the frequency transformation conserves the symmetry of the prototype.

MICROWAVE FILTERS

257

EXAMPLE 8.4 The synthesis of a fifth-order highpass filter having 500 MHz cut-off frequency (o0 ¼ 2p 500 106 rad/s) and R0 ¼ 50 O reference impedance. Starting from the prototype and applying Equations (8.26), we get ðHPÞ

C1

8.5.2

ðHPÞ

¼ C5

¼ 5:551 pF;

ðHPÞ

C3

¼ 3:223 pF;

ðHPÞ

L2

ðHPÞ

¼ L4

¼ 11:607 nH

Lowdpass to bandpass transformation

In high-frequency applications the most interesting type of filter is the bandpass filter. This is because bandpass filters are widely employed to separate radio signals into their different frequency components. The lumped element bandpass filter derives from the lowdpass prototype by the application of the following lowdpass to bandpass transform: o0 o o0 ð8:27Þ jo0 ¼ j o2 o1 o0 o pffiffiffiffiffiffiffiffiffiffiffi where o1 ; o2 are the lower limit and upper limit of the passband, and o0 ¼ o1 o2 (not to be confused with the cut-off frequency of the lowdpass filter). Note that o0 is slightly smaller than the centre of the passband ðo2 þ o1 Þ=2. The two values tend to coincide for very narrow-band responses, i.e. for o2 o1 ðo2 þ o1 Þ=2. By inverting Equation (8.27) we obtain a second-order equation that can be solved in order to write o as a function of o0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 o ðo o Þ o0 ðo2 o1 Þ 2 2 1 2 0 2 o o ðo2 o1 Þo o0 ¼ 0 ) o ¼ þ o1 o2 2 2 This equation shows that each frequency o0 of the prototype maps into two different frequencies of the bandpass response. Table 8.4 shows the corresponding values of o0 and o. The prototype passband o0 2 ð1; 1Þ maps into the two intervals ðo2 ; o1 Þ and ðo1 ; o2 Þ of the bandpass, while the negative portion (o0 < 1) of the prototype stopband transforms into the intervals (1; o2 ) and (o1 ; 0). Similarly the positive stopband frequencies of the prototype (o0 > 1) map into the two intervals (o2 ; 1) and (o1 ; 0). Note that, in contrast with the lowdpass to highpass transformation, which does not modify the cutoff frequency, Equation (8.27) leads to a non-normalized cut-off frequency. Therefore, the filter values obtained from such a transformation need no frequency denormalization. Figure 8.17 shows the response of the bandpass filter resulting from the application of the transform (8.27) to the reference prototype. The black curve, plotted on the left y axis, is the transmission coefficient, expressed in dB, that is the negative of the attenuation. The grey curve, plotted on the right y axis, is the reflection coefficient in dB, coincident with the negative of the return loss. The attenuation of the bandpass filter uniformly oscillates within the passband o 2 ½o1 ; o1 , with the same RP ¼ 0.1 dB ripple as the reference prototype. The frequency behaviour of the reflection coefficient is somewhat complementary to the attenuation in accordance with relation (8.6). When js21 j ¼ 1 ) IL ¼ 0 dB we have js11 j ¼ 0 ) RL ¼ 1 dB, and when IL ¼ RP dB (which is Table 8.4 Mapped points of the lowdpass to bandpass transformation. o0 o

0 o0

1 o1 ; o2

1 o2 ; o1

1 0, 1

þ1 0, þ 1


Bandpass, 20 log10(|s21|)

0

10

s21

–10

0

–20

–10

–30

–20

–40

–30

s11

–50

ω1

ω0

Bandpass, 20 log10(|s11|)

258

–40

ω2

ω

Figure 8.17

Bandpass response after the transformation from the reference prototype.

the maximum passband attenuation) we have the minimum passband return loss RL ¼ 10 log10 ð110RP=10 Þ ffi 16:4 dB. Note also that the return loss exhibits a number of spikes (ideally having infinite amplitude) equal to the filter order, which is five in our case. This observation is the basis for a widely used evaluation criterion: a well-designed and realized bandpass filter shows as many return loss spikes as the order of its prototype. We will see shortly that the number of resonators included in a bandpass filter coincides with the prototype order.26 From the network point of view, Equation (8.27) transforms the impedance of one inductor (capacitor) into that of a series (parallel) resonant LC cell, according to jo0 L0 ¼ jo jo0 C0 ¼ jo

L0 1 o20 L0 1 þ ¼ joLLb þ joCLb o2 o1 jo o2 o1

1 o20 C0 1 C0 þ ¼ joCCb þ o2 o1 joLCb o2 o1 jo

These expressions show that the reactances of the inductors (or, respectively, susceptances of the capacitors) are transformed into series (parallel) LC cells. The component values of the series and parallel LC cells are series LC: parallel LC:

LLb ¼ CCb

L0 ; o2 o1 C0 ¼ ; o2 o1

CLb ¼ LCb

o2 o1 1 o20 L0 o2 o1 1 ¼ o20 C0

ð8:28Þ

It can be seen that both cell types have the same resonant frequency 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ o0 LLb CLb LCb CCb 26 Sometimes the prototype order, or the number of resonators, which is the same thing, is simply referred to as the ‘bandpass’ order. This denomination, however, is somewhat improper.

MICROWAVE FILTERS (BP) (BP)

(BP) (BP)

C1 L1

(BP) (BP)

C3 L3

L(BP) 2

1

259

CN LN

C(BP) L(BP) 2 N-1

C(BP) N-1

Bandpass filter derived from the reference prototype.

Figure 8.18

Remember also that o0 is the frequency corresponding to o0 ¼ 0 in the lowdpass prototype. At the resonant frequency o ¼ o0, the series (parallel) LC cells become short circuits (open circuits). The same things happens with the originating elements of the prototype: at zero frequency (o0 ¼ 0) the prototype inductors (capacitors) become short circuits (open circuits). The above considerations imply that a bandpass filter includes as many resonators as the order of its prototype, the term resonator being used to designate both series and parallel LC cells. This result is even more general and can also be applied to distributed filters. In such a case the term resonator will denote distributed elements or a combination of lumped and distributed elements, such as capacitive coupled lines, open-circuit and short-circuit stubs, capacitively loaded stubs, etc. While the lowdpass to bandpass transform does not need a frequency denormalization, the impedance denormalization is still needed. This simple operation consists of multiplying all inductances and dividing all capacitances – of both the series and parallel LC cells – by the given source and load resistance R0. At the end of the procedure, starting from the reference filter of Figure 8.10a, we obtain the filter shown in Figure 8.18, with the component values ðBPÞ

gk R0 ; o2 o1

ðBPÞ

Lk

¼

Ck

ðBPÞ Lk

o2 o1 R0 ¼ ; o20 gk

¼

ðBPÞ Ck

o2 o1 1 o20 gk R0

ðodd kÞ

1 gk ¼ o2 o1 R0

ðeven kÞ

ð8:29Þ

EXAMPLE 8.5 The synthesis of a bandpass filter with 500–700 MHz passband (o1 ¼ 2p 500 106 , o2 ¼ 2p 700 106 ) and R0 ¼ 50 O impedance, starting from the reference prototype (Chebyshev, N ¼ 5; RP ¼ 0:1 dB). The filter schematic is shown in Figure 8.18. The component values must be computed by applying Equations (8.29) to the prototype coefficients. Table 8.5 lists the resulting values.

Table 8.5 Values of the bandpass filter discussed in Example 8.5. k gk ¼ gNk ðBPÞ Lk ðBPÞ

Ck

¼

ðBPÞ LNk ðBPÞ

1

2

3

4

5

1.147

1.371

1.975

1.371

1.147

3.316

78.583

3.316

45.63

45.63

¼ CNk

1.586

21.824

0.921

21.824

1.586

Series or parallel LC

S

P

S

P

S

nH pF

260


8.5.3

Lowdpass to bandstop transformation

The bandstop filter – also known as a notch in the case of a very narrow passband – shows high attenuation in a small portion of the frequency band and low attenuation elsewhere. As its name suggests, it is used to eliminate unwanted signals falling within a given bandwidth. Thus, the operation of the bandstop filter is the opposite of the bandpass one. The lowdpass to bandstop transformation is jo0 ¼ j

o2 o1 o o 0 2 o2

ð8:30Þ

pffiffiffiffiffiffiffiffiffiffiffi where o1 ; o2 are the lower and upper edges of the stopband, and o0 ¼ o1 o2 approximates the centre of the stopband in the narrow-band case, as pointed out in Section 8.5.2 for the bandpass filter. Equation (8.30) can also be derived by consecutively applying the highpass and bandpass transformations described earlier. In other words, Equation (8.30) follows from (8.27) by substituting jo0 with 1=ðjoÞ. Applying the same procedure in Section 8.5.2, we find that Equation (8.30) maps: .

the stopband edges (o ¼ o1 ; o2 ) into the prototype cut-off frequencies (o0 ¼ 1; 1);

.

the ‘centre’ frequency (o ¼ o0 ) into the prototype infinite frequency (o0 ¼ 1);

.

the minimum and maximum frequencies (o ¼ 0; 1) into the prototype zero frequency (o0 ¼ 0).

Figure 8.19 shows the transmission and reflection coefficients (both in dB) of a bandstop filter derived from the reference prototype. Again, the passband ripple (return loss) is RP ¼ 0.1 dB (16.4 dB), as in the prototype. From the network point of view, applying the same procedures used in Sections 8.5.1 and 8.5.2, we find that Equation (8.30) transforms the series inductors (shunt capacitors) of the prototype into parallel (series) LC cells. Both transformed cells resonate at the frequency o0. Consequently, starting from the

Bandstop, 20 log10(|s21|)

–10

s21

0

–20

–10

–30

–20

–40

–30

Bandstop, 20 log10(|s11|)

10

0

s11 –50

Figure 8.19

ω1 ω0 ω

ω2

–40

Response of a bandstop filter derived from the reference prototype.

MICROWAVE FILTERS (SB)

(SB)

L1

(SB)

1

C1

Figure 8.20

261

(SB)

L3

LN

(SB) C3 (SB) L2 (SB) C2

(SB) CN (SB) LN-1 (SB) CN-1

2

Lumped constant bandstop filter.

network of Figure 8.10a, we obtain the filter of Figure 8.20, having the following component values (denormalized): ðSBÞ

o2 o1 gk R0 ; o20

ðSBÞ

1 1 o2 o1 gk R0

Lk

¼

ðSBÞ Lk

1 R0 o2 o1 gk ðSBÞ ¼ ; Ck ¼ o2 o1 gk o0 2 R0

Ck

¼

ðodd kÞ ð8:31Þ

ðeven kÞ

EXAMPLE 8.6 The synthesis of a bandstop filter having 1.9–2.1 GHz stopband limits (o1 ¼ 2p 1:9 109 , o2 ¼ 2p 2:1 109 ), source and load impedance R0 ¼ 50 O, starting from the reference prototype. The filter has the schematic shown in Figure 8.20. Its components are computed by applying Equations (8.31) to the prototype coefficients. The resulting values are quoted in Table 8.6.

8.5.4

Richards transformation

All the filters considered so far, with the exception of the semi-lumped filters discussed in Section 8.4, are built from lumped constant elements. Therefore, those filters require – by definition – the use of elements having a much smaller size than the minimum wavelength. The high-frequency applications, which this book is focused on, involve quite small wavelengths, so that in most instances the lumped hypothesis cannot be applied. In Section 8.4 we found that a maximum frequency of 30 GHz requires components smaller than 1 mm if the dielectric is air or a vacuum ðer ¼ 1Þ. This is quite an optimistic consideration, since common dielectric materials have much higher permittivities (usually in the range 2–10) than a vacuum; the maximum allowed size is therefore pffiffiffiffi reduced by the factor 1= er.

Table 8.6 Component values of the filter described in Example 8.6. k

1, 5

2, 4

3

gk ¼ gNk

1.147

1.371

1.975

0.457

29.017

0.788

nH

13.878

0.219

8.058

pF

P

S

P

ðSBÞ

¼ LNk

ðSBÞ

¼ CNk

Lk

Ck

ðSBÞ

ðSBÞ

Series or parallel LC

262


The above considerations suggest the use of distributed constant elements for the realization of microwave filters. The main difference between a distributed and a lumped filter is that the former is characterized by trigonometric, thus periodic, functions of o, while the latter depends on rational functions of o. Hence the responses of distributed filters exhibit periodic or pseudo-periodic behaviour. Consequently, it is impossible to realize a distributed constant filter with a genuine lowdpass response, because spurious passbands arise as the frequency increases over some critical values. The Richards transformation, also known as the periodic transformation, allows the lowdpass or highpass lumped filters to be transformed into distributed filters; in other words, into filters built with transmission line segments. The Richards transformation can be conveniently written as po jo0 ¼ ja tan ð8:32Þ 2 o0 where o0 is called the Nyquist frequency and a is a design parameter. Although a could be taken equal to 1, it can be used to facilitate the synthesis of the transformed filter as it affects both the passband width and the impedance level of the transformed filter. When its argument varies in the interval ðp=2; p=2Þ, the tangent assumes all the real values from 1 and þ1, and is periodic with period p. It follows that Equation (8.32) maps the whole o0 axis (1; þ1) into the infinite intervals ðo0 ; þ o0 Þ;

ðo0 ; 3o0 Þ;

...;

½ð2k 1Þo0 ; ð2k þ 1Þo0

where k is any positive or negative integer, including zero. In particular, the key frequencies in the prototype response are mapped as follows: o0 ¼ 0 ! o ¼ 0; 0

2o0 ;

4o0 ; . . . ; 2ko0

o ¼ 1 ! o ¼ o0 ; 3o0 ; 5o0 ; . . . ; ð2k þ 1Þo0 1 1 o0 ¼ 1 ! o ¼ 2o0 tan1 þk p a

ð8:33Þ

Therefore, the frequency response obtained from the Richards transformation is periodic with period 2o0 . The third equation of (8.33) suggests the usefulness of the parameter a to determine, along with o0 , the passband edges of the transformed filter. Indeed a can be used to modify the width of the first passband (k ¼ 0) keeping the frequency periodicity (2o0 ) unchanged. Application of the transformation (8.32) to the lowdpass and highpass prototype27 yields the responses shown in Figure 8.21. The solid (dashed) curve represents the response obtained from the highpass (lowdpass) prototype. Both are periodic with period 2o0 , as anticipated. The dashed curve in Figure 8.21 can represent either a periodic lowdpass or a periodic bandstop response, since it allows DC to be transmitted. Similarly, the solid curve, which stops DC, can represent either a periodic highpass or a bandpass response. In any case, all the passbands and stopbands are centred around an integer multiple of o0. When wide stopband widths are required, filters with different combinations of the parameters a and o0 need to be cascaded in order to match the specifications. When lowdpass or bandpass responses are required with wide stopband widths, many filters, with different combinations of the parameters a and o0 , may need to be cascaded. In such instances, the filter realizing the passband is called the main filter, while the additional cascaded filters are the cleaning filters.

27 In both cases we have used a reference prototype as defined in Section 8.5: Chebyshev, with N ¼ 5 and passband ripple RP ¼ 0.1 dB.

MICROWAVE FILTERS

263

highpass:

0

ω'→ -cot(0.5π ω/ω0)

20 log10(|s21|)

–20

–40

–60

lowpass: ω'→tan(0.5π ω/ω0)

–80 0

Figure 8.21 prototypes.

1

2

3 ω/ω0

4

5

6

Responses of the Richards-transformed filters obtained from the lowdpass and highpass

From the network point of view, Equation (8.32) transforms one inductor (capacitor) into one short-circuit (open-circuit) stub.28 The electrical length of the stub equals p=2 (90 ) at the angular frequency o0. Indeed, by applying the Richards transformation to the inductor impedance and to the capacitor admittance, we get po po jo0 L0 ¼ jaL0 tan ¼ j Z0;L tan ) Z0;L ¼ aL0 2 o0 2 o0 po 1 po 1 jo0 C 0 ¼ jaC 0 tan tan ð8:34Þ ¼j ) Z0;C ¼ 0 2 o0 Z0;C 2 o0 aC As anticipated, the parameter a affects the impedances of the stubs. This impedance modification is the network counterpart of the modification of the passband limits, stated by Equation (8.33). Note that as a consequence of the same Richards transform being applied to all circuit elements, all stubs have the same l/4 electrical length at the frequency o0. For this reason the filters obtained from the Richards transformation are said to be commensurate, and the corresponding responses are periodic. The lowdpass of Figure 8.10a and the highpass of Figure 8.16 (derived from the former) generate the two networks of Figures 8.22a,b, respectively. Finally, after impedance denormalization, Equation (8.34) gives the impedance values of the Richards filters ðLPÞ

Z0;k ¼ agk R0 ðHPÞ

Z0;k ¼

R0 gk a

ðodd kÞ; ðodd kÞ;

ðLPÞ

Z0;k ¼ ðHPÞ

Z0;k ¼

R0 agk

ðeven kÞ

aR0 gk

ðeven kÞ

ð8:35Þ

28 Recall that a stub is a one-port distributed circuit consisting of a length of transmission line terminated at the far end with an open or a short circuit. Correspondingly, the stub is said to be open- or short-circuited.

264

MICROWAVE AND RF ENGINEERING (LP)

Z0,1

(LP)

(LP)

Z0,3

(LP)

Z0,N-2 Z0,N-2

1

2

(LP)

Z 0,2

(HP)

Z 0,1

(LP)

(HP)

Z 0,N-1

(HP)

Z 0,3

(a)

(LP)

Z 0,4

(HP)

Z 0,N-2 Z 0,N-2

1

2

(HP)

Z 0,2

(HP)

Z 0,4

(HP)

(b)

Z 0,N-1

Figure 8.22 Commensurate distributed constant filters: (a) pseudo-lowdpass network; (b) pseudohighpass network.

8.6 Kuroda identities The networks shown in Figure 8.22 theoretically offer the advantage of not requiring the use of lumped elements. Their realization is, however, problematic for two reasons. The first is that they make use of series stubs, which are in practice difficult if not impossible to realize in circuit configurations such as coaxial cable, stripline and microstrip, where a common ground plane exists. On the contrary, series stubs can be realized in rectangular waveguide technology or in two-conductor transmission lines (e.g. coplanar strips). The second difficulty in the realization of the networks in Figure 8.22 is due to the presence of electrical nodes connecting three transmission lines. In practice such nodes cannot be realized physically because of the finite physical size of the actual transmission lines,29 which thus need to be separated by suitable circuit elements. The filter we are considering, however, operates at high frequencies, so that any separating element of finite length would produce phase shifts of the signal, potentially compromising the filter performance. In order to realize the filters of Figure 8.22 physically, we need to introduce some suitable modifications in order to separate the stubs with transmission line segments without altering the filter response, at least the amplitude response. To this end, two approaches are available. The first one, discussed in this section, is based on the Kuroda identities; the second one relies on impedance and admittance inverters and will be presented subsequently in Section 8.7. The Kuroda identities state the equivalence between two networks. There are four Kuroda identities [2], but two of them are sufficient for our needs. Let us consider the two networks shown in Figure 8.23. Network (a) consists of a shunt open stub cascaded with a transmission line segment; network (b) consists of a transmission line segment followed by a series open-circuit stub.30 We can 29 Keep in mind that, by definition, a transmission line, thus a stub, has zero (or very small) transverse dimensions (1-D element). Consequently the length – from port to port – of the two filters in Figure 8.22 is ideally zero. 30 In the literature, the Kuroda identities are sometimes expressed in terms of lumped networks. In this case, inductors (capacitors) replace short-circuit (open-circuit) stubs, and the segment of transmission line with characteristic impedance Zl is replaced by a unit element having the same impedance. The unit element is the inverse Richards transformation of the transmission line length; it is a mathematical entity with no physical lumped element counterpart.

MICROWAVE FILTERS

265

Z s,b Z l,a 1

Z l,b 2

1

2

Z s,a (a)

(b)

Figure 8.23 The first Kuroda identity.

easily demonstrate that the two two-port networks (a) and (b) are equivalent if all their elements have the same electrical length and if the following equalities hold:31 Zl;b ¼

1 1 þ Zl;a Zs;a

1

;

Zs;b ¼ Zl;a Zl;b

ð8:36Þ

Conditions (8.36) constitute the first Kuroda identity. By inverting (8.36), we obtain the second Kuroda identity: Zl;a ¼ Zl;b þ Zs;b ;

Zs;a ¼

1 1 Zl;b Zl;a

1

ð8:37Þ

which simply corresponds to interchanging ports 1 and 2 in both circuits of Figure 8.23. In practice, Kuroda identities allow us to interchange the respective positions of a line segment with a stub, simultaneously replacing a series short-circuit stub with a shunt open-circuit stub and vice versa. In this manner, as we will see shortly, it is possible to transform the network in Figure 8.22a into another one, containing only shunt open-circuit (or series short-circuit) stubs, separated by line segments. Note preliminarily that the amplitude response32 of the filter does not change if we insert any section of matched transmission line33 between the filter input and the source or between the output and the load. The insertion of a matched line causes a phase shift in s21, but does not affect its amplitude. By exploiting this property, we can insert34 bN=2c transmission line segments – all with the same characteristic impedance Z0;1 ¼ Z0;2 ¼ Z0;4 ¼ Z0;5 ¼ R0 and with the same electrical length as the stubs – at both ports of the network in Figure 8.22a. By successive applications of the Kuroda identities, the transmission line sections can be moved within the filter separating consecutive stubs. At the same time all stubs become of the same type, either series or shunt. The procedure is better explained using a specific example.

31 For this purpose it is sufficient to write the ABCD matrices of the three fundamental elements: transmission line segment, series stub and shunt stub. The ABCD matrices of networks (a) and (b) are then obtained as the products of two 2 2 matrices, each correspoding to the proper circuit element. The Kuroda identities are obtained by simply equating the two resulting ABCD matrices. 32 The amplitude response is the only one specified in most cases. In some special applications the phase response (particularly the group delay) is also specified. 33 The term matched denotes here a transmission line having characteristic impedance equal to the source and load impedance R0. We also assume that the load and source have the same impedance. 34 The function bxc returns the lower integer part of x. It is sometimes written as floor(x).

266


EXAMPLE 8.7 The synthesis of a periodic lowdpass filter of order N ¼ 5, using only shunt stubs. We will start from the lumped prototype of Figure 8.10b, which is the dual of Figure 8.10a.35 After applying the Richards transformation, we get the network in Figure 8.24a. The filter amplitude response will not change if we insert at each filter end bN=2c ¼ 2 line segments with impedance equal to R0: Figure 8.24b shows the resulting network. The first application of the Kuroda identities affects the outer stubs of the filter (positions 1 and 5) and generates the network in Figure 8.24c. The second – and final – application of the Kuroda identities involves all the series stubs of the network in Figure 8.24c, and produces the final network shown in Figure 8.24d. ZS2

ZS4

1

2

(a) ZP1 ZS2 Z01

ZP3

ZP5

ZS4

Z02

Z04

Z05

1

(b)

2

ZP1 Z′P1 Z01

ZS2

ZP3

ZP5

ZS4

Z′02

Z′P5 Z′04

Z 05

1

2

(c)

ZP3 Z′01

Z′′02 Z′′04 Z′05

1

2

Z′′P1 Z′P2

ZP3

Z′P4

(d)

Z′′P5

Figure 8.24 Use of the Kuroda identities relevant to Example 8.7: (a) original fifth-order filter; (b) add ½N=2 ¼ 2 transmission line segments at both filter ends; (c) apply the Kuroda identity (Figure 8.23) to the two networks consisting of the line Z0;2 with the parallel stub ZP1 , and of the line Z04 with the parallel stub ZP5 ; (d) apply the Kuroda identity at the four line segments with series stub 0 0 0 0 Þ; ðZ02 ; ZS2 Þ; ðZS4 ; Z04 ½ ðZ01 ; ZP1 Þ; ðZP5 ; Z05 Þ obtaining all parallel stubs. 35 This change in the starting point is required because we need the central element to be a shunt stub. Therefore, if N ¼ 3, 7, 11. . . (N ¼ 5, 9, 13. . .) the starting point is the lumped prototype in Figure 8.10a (8.10b). If we want series stubs only, we must swap the prototype choice.

MICROWAVE FILTERS

267

8.7 Immittance inverters An ideal impedance inverter K, or K inverter, is a two-port network behaving at all frequencies like a l=4 line with impedance Z0 ¼ K: we say that K is the constant of the impedance inverter K. With reference to Figure 8.25a, from the above definition it follows that, when port 2 of the inverter is terminated with the impedance Z2 , the impedance seen at port 1 is Z1 ¼

K2 Z2

ð8:38Þ

In a dual way, we can define the admittance inverter J as a two-port network such that, when port 2 is terminated with the admittance Y2 , the admittance seen from port 1 is

Y1 ¼

J2 Y2

ð8:39Þ

For the sake of simplicity, in the rest of the chapter we will use the generic term immittance inverter to denote either the impedance inverter K or the admittance inverter J. From the above property, it can be seen that the chain or ABCD matrix of an impedance inverter is

A C

0 B ¼ j=K D

jK 0

ð8:40Þ

Similarly, it is found that the impedance and admittance matrices are given by

0 ½Z ¼ jK

jK ; 0

0 ½Y ¼ jJ

jJ 0

ð8:41Þ

If an admittance inverter is considered, Equation (8.40) holds true by just replacing K with 1=J. It is easily recognized that Equations (8.38) and (8.39) directly follow as a consequence of (8.40). The immittance inverter is a loss-free network, since the determinant of the matrix (8.40) is unitary. Properties (8.38) and (8.39) can be summarized by saying that the impedance (admittance) inverter has the basic property of presenting an input impedance equal to the reciprocal of the output one, multiplied by K2 (J2). For instance, if port 2a in Figure 8.25 is terminated by a capacitor C, the input impedance – given by (8.38) – is Z1 ¼

K2 ¼ joK 2 C 1=ðjoCÞ

Therefore, the impedance inverter transforms the output capacitor into an input inductor, having inductance L ¼ K 2 C. In the same way, it can be shown that the inductor is transformed into a capacitor, and an open-circuit stub into a short-circuit stub with the same electrical length (and vice versa).

268


Figure 8.25 Immittance inverters: (a) defining networks; (b) equivalent networks for the impedance inverter; (c) equivalent networks for the admittance inverter; (d) l=4 transmission line impedance inverter. If we terminate the impedance inverter with the series of two impedances Z2,a and Z2,b, the resulting input impedance is Z1 ¼

K2 1 Z2;a Z2;b ) Y1 ¼ ¼ 2 þ 2 Z1 Z2;a þ Z2;b K K

This expression corresponds to the parallel connection of two admittances Z2;a =K 2 and Z2;b =K 2 . Therefore, we can say that the impedance inverter transforms the series connection into a parallel one and vice versa. An important consequence of Equation (8.38) is the equivalence between the two networks of Figure 8.25b, where the immittances Yp, Zs are related by Equation (8.38). The equivalence can be proved by computing the chain matrix of the circuit on the left of Figure 8.25b as the product of three factors: two impedance inverters (first and third factors) and the shunt bipole Yp:

0 j=K

jK 0

1 Yp

0 1

0 j=K

jK 0

1 K 2 Yp ¼ 0 1

ð8:42Þ

MICROWAVE FILTERS

269

The chain matrix of the network on the right of Figure 8.25b is

1 Zs 0 1

ð8:43Þ

Apart from the sign, the matrix (8.43) coincides with (8.42) when Zs ¼ K 2 Yp . The sign inversion implies a p (180 ) phase shift and can be represented by interchanging the output terminals. Such a phase shift is due to the presence of two K inverters, each providing a p/2 phase shift. As in most cases, the phase shift can be neglected if we are not interested in the phase response. By a similar procedure the identity can be proved between the networks in Figure 8.25c. A series impedance Zs can therefore be replaced by a shunt admittance Yp ¼ Zs =K 2 placed between two impedance inverters and vice versa. It is worth observing that the K inverter can also be used to scale Yp by any arbitrary factor A2, simply by changing K into K/A. The equivalences of Figures 8.25b,c can be used to obtain filters with shunt-only (or series-only) elements separated by immittance inverters. This is another way, in addition to the use of Kuroda identities, to overcome the difficulties associated with realization of the networks of Figure 8.22. As an example, Figure 8.26 shows an Nth-order passband prototype realized with N series inductors Li ði ¼ 1; 2; . . . ; NÞ and N þ 1 impedance inverters Ki;i þ 1 ði ¼ 1; 2; . . . ; N þ 1Þ. Since an inverter changes the impedance level by a factor K 2 (or the admittance by a factor J 2 ), the additional degrees of freedom introduced by the addition of the inverters can be exploited to determine arbitrarily the inductance values as well as the impedance level at the source or load terminals. To this end, the impedance inverters have the values indicated in the figure (see [2]). As an alternative, one may arbitrarily fix the values of the inverters, assuming for instance all unitary values: Ki;i þ 1 ði ¼ 1; 2; . . . ; N þ 1Þ. In conclusion, the use of immittance inverters allows us to: 1. Separate contiguous filter elements, thus removing one of the obstacles to realization. 2. Obtain series-only or shunt-only filters, removing the realization problems associated with the networks of Figure 8.22. 3. Alter the impedance levels of the whole filter or of some sections, in such a way as to obtain elements easier to realize. Point 3 specifically distinguishes the immittance inverter technique from that based on the Kuroda identities, as the latter, though mathematically more elegant, often yield impedance values that are difficult or impossible to realize. The inverters, on the contrary, allow for a much higher flexibility. There are several ways to realize an inverter practically. The most obvious one is the quarterwavelength transmission line, shown in Figure 8.25d, with characteristic impedance Zc ¼ K or Zc ¼ 1/J. Note that no physical network can behave as an ideal impedance inverter, but can only represent an approximation valid in a limited frequency band. Other possible realizations of immittance inverters are in the form of T and P networks. By recalling expressions (8.41) and Equations (4.60) and (4.67), it can be seen that any symmetrical lossless T network realizes an impedance inverter if the impedance of the two series branches is the opposite of the shunt

L1 1

K 0,1 =

K0,1

RS L1 , g 0 g1

L2 K1,2

Kk ,k +1 =

LN-1 K2,3

Lk Lk +1 g k g k +1

LN KN-1,N

( k = 1,

, N − 1) ,

KN,N+1

KN −1,N =

2

RL LN g N g N +1

Figure 8.26 Lowdpass filter with series inductors and impedance inverters.

270


2 branch, in such a way that Z11 ¼ Z22 ¼ 0. The resulting impedance inverter is K 2 ¼ Z12 . In a dual fashion, from (8.41) and (4.67) it appears that a P network realizes an admittance inverter if the admittance of the shunt branches is equal to that of the series branch, so that Y11 ¼ Y22 ¼ 0. The resulting admittance 2 inverter is J 2 ¼ Y12 . Such networks cannot be physically realized by themselves, because they involve negative-value elements. However,such elements can be realized virtually, by ‘absorbing’ them within the external circuit. For instance, a negative-value inductor –L1, in series with a positive-value inductor L2, is equivalent to one single inductor, with L2 L1 . The resulting inductor is physically realizable, provided that L2 > L1 . This technique can be applied, for instance, to transform the filter of Figure 8.16 into a filter with series inductors and impedance inverters as shown in Figure 8.26. Apart from the phase, the two filters are fully equivalent in terms of amplitude response. The following example, Example 8.8, illustrates in detail the procedure to obtain a filter made of lines and series stubs only. With a dual procedure, omitted here, a filter made of lines and shunt stubs only can easily be obtained.

EXAMPLE 8.8 The synthesis of a distributed bandpass filter with series open-circuit stubs only, by applying the impedance inverters. The filter uses the reference prototype (Chebyshev, N ¼ 5, RP ¼ 0.1 dB) as starting point. A further specification is that the first passband extends from 2.5 to 7.5 GHz. The reference lowdpass prototype is shown in Figure 8.27a, while its values are listed in the third and fourth rows of Table 8.7. The next step consists of applying the lowdpass to highpass transformation. The resulting network is shown in Figure 8.27b, the element values being listed in the fifth and sixth rows of Table 8.7. Now we apply the Richards transformation (8.32) and impose the passband limits to coincide with the required ones. First of all, since the passband is to be centred on o0 , we have o0 ¼ 2p½ð2:5 þ 7:5Þ=2 109 ¼ 10p 109 rad=s. Once the central frequency has been determined, the parameter a is found by imposing the lower limit of the passband to map into the unitary frequency of the prototype: p p 2p 2:5 109 ) 1 ¼ a tan 1 ¼ a tan )a¼1 9 2 2p 5 10 4 Note that in the upper limit of the passband, we have p 2p 7:5 109 ¼ 1 tan 2 2p 5 109 We have now obtained the network of Figure 8.27c, with the component values listed in the seventh and eighth rows of Table 8.7. As seen in Section 8.5, any bandpass or bandstop filter, obtained from the Richards transformation, is a ladder network with series and parallel resonators in alternate positions.36 By placing an impedance inverter between two adjacent resonators, we obtain a filter with all the resonators of the same type. In our case, if we insert one series open-circuit stub between two impedance inverters with K ¼ 1, we obtain the equivalent of a parallel short-circuit stub with the same characteristic impedance as the original stub. By applying this concept, we obtain the filter of Figure 8.27d, with the component values listed in the ninth and tenth rows of Table 8.7. At this point we must point out that the network of Figure 8.27d is not canonical (like that of Figure 8.27c), since it contains N þ ðN1Þ ¼ 9 elements. The insertion of the impedance inverters, 36

This consideration applies to any type of filter, concentrated or distributed. In the lumped element (distributed element) filters, the resonators are series or parallel LC cells (open-circuit or short-circuit stubs).

MICROWAVE FILTERS

271

in fact, introduces a certain amount of redundancy in the network. Such redundancy can be exploited to satisfy particular design constraints. For instance, we may wish to realize the filter with all equalimpedance stubs. The 11th and 12th rows of Table 8.7 show how to change the impedance inverter constants, so that all inner stub impedances are equal to the outer ones (g1). The distributed network of Figure 8.27d is still ideal, because it includes the impedance inverters that are ideal elements by definition. As anticipated, a transmission line segment is one possible (approximate) realization of the impedance inverter. In contrast with the ideal case, the transmission line segment behaves as an impedance inverter only at the centre frequency where its electrical length equals p=2. Let us now replace the ideal impedance inverters in Figure 8.27d with transmission lines l=4 long at 5 GHz and having Z0 ¼ K. The filter obtained has no ideal elements, but its response is degraded. In particular the passband ripple increases, and consequently the return loss decreases. We can compensate for this effect by optimizing the line and stub impedances. The last two rows of Table 8.7 provide the optimized filter values. Figure 8.28 shows the responses of all the filters discussed in this example. Three pairs of curves are plotted, one for the transmission and one for the reflection coefficient. The thick black curves refer to the filters in Figures 8.27c,d with ideal impedance inverters. Clearly, the responses of these two filters are coincident. The thin black (thick grey) curves are those of the filter with the transmission line segments before (after) optimization. Finally, all the filters discussed in this example are symmetrical, like the prototype they are derived from. Moreover, all the values listed in Table 8.7 are normalized to the reference impedance. If a different impedance level R0 is needed, it is sufficient to multiply all the values by R0.

Table 8.7 Values for Example 8.8. k

1, 5

2, 4

3

gk

1.147

1.371

1.975

ðLPÞ

Lk

g1

ðLPÞ

Ck

ðHPÞ

Ck

g2 1=g1

ðHPÞ

Lk

ðSeriesOpenStubÞ

Zk

1=g3 1=g2

1=g1

ðShuntShortStubÞ

Zk

ðSeriesOpenStubÞ

g3

1=g3 1=g2

Zk

g1

g2

g3

K

1

1

1

g1 rffiffiffiffiffi g1 ffi 0:915 g2

g1

g1

g1 pffiffiffiffiffiffiffiffiffi ffi 0:697 g2 g3

g5 g1 pffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffi g3 g4 g2 g3

0.530

0.530

0.53

1.152

1.122

1.122

ðSeriesOpenStubÞ Zk

K ðSeriesOpenStubÞ

Zk

ðTransmissionlinesÞ

Zk

272

MICROWAVE AND RF ENGINEERING L1

L3

L5

C2

1

(HP)

C4

(HP)

C1

2

(HP)

C3

C5

L(HP) L(HP) 2 4

1

(HP)

Z 0,1

(HP)

(a)

2

(b)

(HP)

Z 0,3

Z 0,5

1

2

(c) (HP)

(HP)

Z 0,2 (HP)

(HP)

Z′0,1

(HP)

Z′0,2

K1

1

Z 0,4

K2

(HP)

(HP)

Z′ 0,3

Z′0,4

K3

Z′ 0,5

K4

2

(d)

Figure 8.27 Application of the impedance inverters to the filter synthesis: (a) lowdpass prototype; (b) highpass prototype; (c) Richards-transformed canonical network; (d) final network with series stubs only. Ideal impedance inverters λ/4 lines λ/4 lines, optimized

0

20

20 log10(|s21|)

–20

0

–30 –20

–40 –50

s11 → –40

–60 0

1

2

3

4

5

6

7

8

9

10

Frequency, GHz

Figure 8.28


20 log10(|s11|)

← s21

–10

MICROWAVE FILTERS

273

Z sa,2 Z sa,4

1

2 (a) Z pc,1 Z pc,3 Z pc,5

Zl 1

Zl 2

Zl 3

Zl 4

1

2

Z s1

Z s2

Z s3

Z s4

(b)

Z s5

Figure 8.29 Schematic synthesis of line-coupled stub filters: (a) canonical ladder network; (b) transformed network by means of l=4 transmission line impedance inverters.

8.7.1

Filters with line-coupled short-circuit stubs

We can design distributed filters having N l=4 short-circuit stubs, separated by (N 1) l=4 transmission lines, by applying a procedure quite similar to the one discussed in Example 8.8. The only necessary change is the starting point, which must be the lumped network of Figure 8.10b, instead of Figure 8.10a. Figure 8.29 shows the network obtained at the last two synthesis steps: Figures 8.29a,b correspond to Figures 8.27c,d, respectively. Alternatively, the same result can be achieved by deriving the dual network of Figure 8.27d: shortcircuit shunt stubs will replace the open-circuit series stubs, and the values listed in Table 8.7 are considered as normalized admittances, instead of impedances. A third option consists of the application of the synthesis formulae of Table 8.8, from [2]. Their derivation, not discussed here, is based on the same principles illustrated in Example 8.8. Table 8.8 Synthesis formulae for the short-circuit stub filter [2]. Passband limits ðf1 ; f2 Þ Impedance inverter constants (coincident with the characteristic impedances of the connecting lines) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 gk gk þ 1 ZN1;N 1 1 g0 gN1 Z1;2 1 1 g2 Zk;k þ 1 ¼ ; ¼ ; ¼ 2 g0 g1 g0 2 g1 gN þ 1 R0 g0 2 g1 R0 R0 k¼2;...;N1 Characteristic impedances of the shunt stubs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi g0 g1 f1 1 2 2 Zs;1;1 ¼ Z1;2 þ tan p Z1;2 R0 f2 þ f1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi g0 g1 pf1 g0 g1 pf1 1 1 2 2 þ Zk;k þ 1 þ Zk1;k Zs;k;k þ 1 ¼ Zk1;k þ tan tan Zk;k þ1 R0 f2 þ f1 R0 f2 þ f1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 gN gN þ 1 g0 g1 f1 g0 g1 f1 2 1 2 KN1;N þ KN1;N Zs:N;N ¼ tan p þ tan p R0 f2 þ f1 R0 f2 þ f1

274


The result obtained using the design formulae of Table 8.8 is considerably more accurate than that obtainable by the simple procedure of Example 8.8. Despite this, an optimization is usually required in order to compensate for the approximations inherent in the realization of the impedance inverters and for the non-idealities of real transmission lines.

EXAMPLE 8.9 Line-coupled short-circuit stub filters typically lend themselves to passbands of the order of one octave. The design procedure discussed above is illustrated here for the example of a microstrip bandpass filter. Let the passband be 12–24 GHz, the filter order N ¼ 5, the passband ripple RP 0.044 dB (corresponding to 20 dB return loss). The reference impedance is R0 ¼ 50 O. The prototype coefficients are quoted in the second row of Table 8.9. The computation of the parameter a gives ! p p 2p 12 109 ¼ a tan ) a ffi 0:577 1 ¼ a tan 2 2p 12 þ2 24 109 3 With the procedure schematized in Figure 8.29, we obtain a symmetrical filter with five shunt shortcircuit stubs, separated by four transmission line segments. Lines and stubs have 90 electrical length (l=4 physical length) at the passband centre frequency of 18 GHz. A procedure similar to the one used in Example 8.8 gives the values (denormalized with respect to 50 O) listed in the third and fourth rows of Table 8.9. Since a l/4 transmission line is a rough and narrow-band approximation of an ideal impedance inverter, the values resulting from the above procedure are inaccurate and need to be corrected by numerical optimization. The optimized values are quoted in the fifth and sixth columns of Table 8.9. As a reference, the values obtained by the synthesis formulae of Table 8.8 are quoted in the last two columns of Table 8.9. Such values are considerably more accurate than those obtained from the simple synthesis method employed in Example 8.8. Figure 8.30 shows the layout of the microstrip filter. As we will explain later, the dimensions of the filter have been modified to compensate for the discontinuity effects. The substrate is h ¼ 508 mm thick with relative permittivity er ¼ 9:9. The ground connections of the stubs are realized by means of rectangular pads, placed on the substrate border and connected to the ground plane by gold ribbons. Looking at the optimized values in Table 8.9, we can see that the Table 8.9 Parameter values of Example 8.9. k Prototype coefficients

gk

Synthesized values

ðstubÞ Zk

1, 5

ðOÞ

2, 4

0.973

1.372

1.803

50 a=ðg1 Þ

50 a=ðg2 Þ

50 a=ðg3 Þ

ffi 29:67

Optimized values

Values from Table 8.8

3

ffi 21:04

ffi 16:01

ðlineÞ Zk

ðOÞ

50

50

50

ðstubÞ Zk

ðOÞ

58.51

29.55

29.85

ðlineÞ Zk

ðOÞ

42.29

40.86

40

ðstubÞ

ðOÞ

66

34

35

ðlineÞ

ðOÞ

42.2

40

40

Zk Zk

MICROWAVE FILTERS

275

outer stubs (k ¼ 1, 5) have roughly twice the impedance of the inner ones (k ¼ 2, 3, 4). This suggests the configuration shown in Figure 8.30, where the inner stubs are realized with two parallel twiceimpedance stubs. The resulting filter presents all the stubs having approximately the same impedance. Also, the lines have approximately the same impedance, as we recognize from Table 8.9. Similar impedances imply similar microstrip widths. The low dispersion in the width values minimizes the sensitivity of the filter with respect to manufacturing tolerances. It is worth observing the different physical lengths of the stubs. This might seem to be in contrast with what we have noted above, since all the stubs should have about the same length since they are all l=4 long at 18 GHz and their impedances and effective permittivities are very close. However, the parasitic reactances associated with the different junctions (T junctions for the outer stubs and crosses for the inner ones) produce different effects and require different compensations. This makes the inner stubs longer than the outer ones. The former have thus been folded in order to place the short-circuit end at the substrate border.37 Figure 8.31 shows the simulated38 response of the filter in Figure 8.30 compared with the ideal line counterpart, using the values listed in the last two rows of Table 8.9. Note that: (a) The transmission coefficient of the microstrip filter is lower than its ideal counterpart. The reasons for this are the finite conductivity of the metal and the non-zero conductivity of the substrate. Furthermore, the attenuation excess is higher in the proximity of the two passband corners39 (12 and 24 GHz). As a secondary effect, the filter loss smoothes the passband return loss peaks (s11 ! 0). (b) The microstrip filter shows a spurious passband around twice the passband centre frequency (f0 ¼ 18 GHz, spurious passband: 34–40 GHz). In the proximity of the spurious passband, the filter should ideally present an infinite attenuation. Indeed, when f ¼ 2 f0 , the stubs are l=2 long and a transmission zero would be expected. Surprisingly, this is not the case with our filter. This is due to the already mentioned effect of the junction discontinuities. The different stubs present, after compensation, similar electrical lengths in the proximity of f0 , but not at 2 f0 . On the other hand, since the lines are also l/2 at 2f0 , they can be seen, apart from a 180 phase shift, as straight connections. Hence, around 2f0 all the stubs, the load and the source are approximately connected together in parallel. As a consequence, as explained next, the resulting susceptance becomes very small or even zero in the proximity of 2 f0 . To clarify this point, consider for simplicity two parallel stubs having slightly different lengths. Both stubs present low reactances around 2 f0 where they are l=2 long, but since their lengths are different, an intermediate frequency fz 2 f0 exists where they have opposite susceptance, so that a parallel resonance (zero susceptance) occurs, thus a high reactance; finally, the filter will exhibit a low attenuation. Figure 8.32 plots the reactances of two equal-length parallel stubs, along with the reactances of two slightly different stubs. It can be seen that around f0 the responses of the two configurations (parallel stubs of unequal or equal lengths) can hardly be distinguished. On the contrary, around 2 f0 , the former (equal-length stubs) present a low impedance, while the latter (unequal lengths) present a high impedance.

37 A different grounding tecnique, such as the plated via hole, would make this arrangement unnecessary, allowing the stub to be grounded at any point of the substrate, not just close to the border. 38 Though only simulated, such results are quite realistic, in the authors’ opinion. 39 It is possible to demonstrate that the filter loss is proportional to the filter group delay (increasing at the passband P edges) by the factor Nk¼0 gk (increasing with N), and by the factor Q1 ðf2 f1 Þ=ðf2 þ f1 Þ, where Q is the quality factor of the filter resonators and f1 ; f2 are the passband limits.

MICROWAVE AND RF ENGINEERING 7 mm

1

2

1 2

RF ports substrate border microstrip lines ground connections

Figure 8.30

Layout of the filter discussed in Example 8.9.

ideal lines microstrip 0

10

s11 →

0

–20

–10 s21

←

–30

s21

–20

–40

–30

–50

–40 0

4

8

12

16

20

24

28

32

36

40

Frequency, GHz

Figure 8.31


20 log10(|s11|)

–10

20 log10(|s21|)

276

Stub reactance

MICROWAVE FILTERS

277

0 a) = λ/4 b) > λ/4 c) < λ/4 d a//a) e b//c) 0

1

2

ω/ω0

Figure 8.32 Reactances of some combinations of short-circuit stubs: curve a, single l=4 stub at the angular frequency o0 ; curve b, single stub slightly longer than l=4 at o0 ; curve c, single stub slightly shorter than l=4 at o0 ; curve d, two parallel stubs both l=4 at; curve e, two parallel stubs, one slightly shorter and the other slightly longer than l=4 at o0 .

8.7.2

Parallel-coupled filters

All the distributed filters considered so far have the same basic structure consisting of stubs and transmission lines in alternate positions. The stubs can be series or parallel, short circuit or open circuit. Stubs and lines all have the same electrical length. The stubs can be viewed as resonators and the lines – or, more generally, the impedance inverters – as coupling elements. This schematization applies to many microwave filters. A microwave filter in fact often consists of N resonators, separated by N 1 coupling elements. Sometimes, two further coupling elements are placed between the source and the first resonator, and between the last resonator and the load. This general structure has many possible realizations; a classical one is the parallel-coupled filter, also known as the edge-coupled filter, which is probably the most popular and cheapest microwave bandpass filter. It can in fact be entirely manufactured using printed circuit technology requiring no ground connections (no via holes, no metal ribbons) unlike the short-circuit stub filter of Example 8.9; moreover, if well designed, it does not require any tuning after fabrication. Figure 8.33 shows a microstrip edge-coupled filter (substrate thickness h ¼ 254 mm and relative permittivity er ¼ 9:9). Two schematizations are possible for the edge-coupled filter: 1. N resonators, each consisting of a l/2 transmission line section open-circuited at both ends and coupled with the adjacent resonator for a l=4 portion of its length. The filter shown in Figure 8.33 has N ¼ 5. 2. N þ 1 cascaded two-port networks of the type shown in Figure 4.25b.40 40

Figure 4.25b shows the schematic of a coupled-line two-port network, together with its line and stub equivalent network. The figure is replicated below. All elements have the same electrical length.

278

MICROWAVE AND RF ENGINEERING 20 mm s 1

2

l

w 1 2

RF ports substrate border microstrip lines ground connections

Figure 8.33

Parallel-coupled microstrip filter.

The edge-coupled filter is a commensurate filter like all the distributed filters considered so far. To avoid any geometrical misalignment of the input and output ports in the filter, which could be a problem in practical applications, the filter layout, as shown in Figure 8.33, includes two bends – one at the input, one at the output – to align the filter ports. Many methods are available for the synthesis of edge-coupled filters. The most frequently used method relies on the design formulae in [10]. Such formulae have been derived by a method based on impedance inverters and are given in Table 8.10. Note that the filter is symmetrical as a consequence of the symmetry of the original prototype. The formulae are claimed to be accurate if the filter fractional bandwidth 2ðf2 f1 Þ=ð f2 þ f1 Þ ¼ ð f2 f1 Þ= f0 is less than 30%. Indeed, wide-band filters require strongly coupled lines (especially the end ones), implying very narrow line spacings. The widest realizable fractional bandwidth therefore depends on the available technology.

EXAMPLE 8.10 41 We illustrate here the design of the filter in Figure 8.33. The passband limits are 9–11 GHz, the source and load impedances are R0 ¼ 50 O, the order is N ¼ 5, and the return loss is 20 dB. The second row of Table 8.11 shows the prototype coefficients.42 The third and fourth rows list the even and odd mode impedances Z0e;k and Z0o;k , computed with the design formulae of Table 8.10. All N þ 1 ¼ 6 coupled-line sections are l=4 at the centre frequency f0 ¼ ð f1 þ f2 Þ=2 ¼ 10 GHz. As in any impedance-inverter-based design, such values are approximate and need an adjustment. The fifth and sixth rows of Table 8.11 list the resulting values. Once the impedances are known, the dimensions of the coupled microstrip sections can be determined by making them have the same even mode impedance, odd mode impedance and electrical length, as required. The last constraint, however, cannot be satisfied in a strict sense,

41 42

See Mathcad file 03_Microstrip_Edge_Coupled_Filters.MCD. The same as those in Example 8.9, given in the second row of Table 8.9.

MICROWAVE FILTERS

279

because, as we know, even and odd modes of coupled microstrips have different permittivities, thus different phase velocities. Fortunately, however, the difference is relatively small and the physical length43 can be computed using either the arithmetic or geometric mean of the two effective permittivities, with similar results. The seventh, eighth and ninth rows of Table 8.11 list the geometrical dimensions wk ; sk ; lk of six l=4 coupled sections of the filter. The electrical parameters of the microstrips do not coincide with the values of the fifth and sixth columns of Table 8.11. Moreover, the segments are slightly shorter than what we could expect considering an electrical length of 90 at 10 GHz. The reasons for such small differences are as follows: 1. The open ends of each coupled-line section are not perfect open circuits. Rather, they present a parasitic capacitive reactance due to the fringing field at the end of the line. This fringing field has the effect – usually denoted as end effect – of producing an increase in the effective length of line, and a decrease in the line impedance44 as well. 2. The junctions between lines of different width (step discontinuities) produce parasitic reactances. 3. The input and output bends at the filter constitute small discontinuities that slightly modify the filter response and must be compensated for. 4. There are different propagation velocities of the even and odd modes. See also footnote 42. Figure 8.34 shows the response of the filter of Figure 8.33 (black curves) together with its ideal counterpart (grey curves). We can observe a similar behaviour as for the filter in Example 8.9: .

The passband attenuation of the microstrip filter is higher than the ideal case, particularly at the passband corners.

.

Unlike the ideal filter, the microstrip filter shows a spurious response around 2f0 . As shown in [8] and [9], this is essentially due to the different phase velocities between the even and odd modes. Nevertheless, even in filters realized with pure TEM lines (coaxial cable, stripline) the end effect is different for both modes. As a consequence, a spurious passband at 2 f0 is present in any edge-coupled filter and the selectivity of the filter upper band edge f > f2 is lower than in the ideal case. Techniques to minimize the spurious passband can be developed, e.g. [11].

For instance, the two inner sections of our filter have w ¼ 0.3 mm and s ¼ 0.443 mm. The associated even and odd mode effective permittivities are ee f f ;e ¼ 6:9; ee f f ;o ¼ 5:73. Calculating the length based on the arithmetic or geometric mean of the two permittivities, we get 43

lartm ¼ vartm =ð4 f0 Þ ¼ c ½0:5ðee f f ;e þ ee f f ;e Þ0:5 ð4f0 Þ1 ffi 2:985 mm or lgeom ¼ vgeom =ð4f0 Þ ¼ c ðee f f ;e ee f f ;e Þ0:25 ð4f0 Þ1 ffi 2:985 mm respectively. 44 The end effect is local. It can be modelled by a lumped capacitor at the open end of the coupled lines. Clearly, the shorter the line, the more pronounced the end effect.

280


Table 8.10 Synthesis formulae for edge-coupled filters. Passband limits ðf1 ; f2 Þ Impedance inverters rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K0;1 2 f1 þ f2 g0 g1 ¼ p R0 f2 f1 Kk;k þ 1 1 f1 þ f2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gk gk þ 1 ðk ¼ 1; . . . ; N1Þ ¼ p f2 f1 R0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KN;N þ 1 2 f1 þ f2 gN gN þ 1 ¼ p R0 f2 f1 Even and odd mode impedances " " 2 # 2 # R0 R0 R0 R0 ; Z0o;k ¼ R0 1 ðk ¼ 0; . . . ; NÞ þ þ Z0e;k ¼ R0 1 þ Kk;k þ 1 Kk;k þ 1 Kk;k þ 1 Kk;k þ 1

Table 8.11 Values of the filter in Figure 8.33. k gk Z0e,k Z0o,k Z0e,k Z0o,k wk sk lk

1, 6

2, 5

3, 4

0.973 95.04 37.76 88 33 0.28 0.075 2.541

1.372 67.47 40.05 73.2 45.4 0.3 0.337 2.515

1.803 62.05 41.98 74.5 51.7 0.3 0.443 2.519

O O O O mm mm mm

0

10

s11

0

–20

–10 ideal microstrip s21

–30

–20

–40

–30

–50

–40 8

10

12

14

16

18

20

22

Frequency, GHz

Figure 8.34

Response of the filter of Figure 8.33.

24

20 log10(|s11|)

20 log10(|s21|)

–10

MICROWAVE FILTERS

C2

C1

l

0

1

2

C3

3

CN–1 CN

N-1

N

1

N+1

2

Figure 8.35

8.7.3

281

Comb-line filter.

Comb-line filters

These are another very popular class of filters. A comb-line filter consists of a sequence of N equal-length coupled resonators made of TEM or quasi-TEM lines, just like the parallel-coupled filter, except that each resonator is short-circuited at one end and grounded at the other end through a capacitor. The filter schematic is shown in Figure 8.35. Input and output are realized through two further non-resonating line sections (0 and N þ 1) coupled to the first and last resonators respectively, realizing at the same time a suitable impedance transformation at each port. In contrast to the parallel-coupled filter, and due to the presence of the capacitors, the electrical length l of the resonators has to be less than l=4 at the centre frequency, a typical value being l=8 or less. Since the second passband occurs when the resonators are more than half a wavelength long, in addition to having a reduced size this type of filter exhibits an extremely wide upper stopband, typically four times the centre frequency or even more. Comb-line filters can be realized using different technologies, such as TEM rectangular lines, stripline or microstrip line. A comb-line filter realized with cylindrical rods in a metal cavity is sketched in Figure 8.36. The gaps between the end of the rods and the cavity wall provide the end capacitances; the latter can be tuned with the screws usually inserted through the cavity wall. Printed circuit technology, mostly microstrip, employs either gap capacitors or discrete surface-mountable capacitors, the latter being especially useful when high capacitances are required. The behaviour of the comb-line filter can easily be analyzed by neglecting any coupling between nonadjacent lines, so that the filter can simply be considered as consisting of N þ 1 pairs of coupled lines (six in the case of Figure 8.36). As described in Section 3.16.2, the coupled-line model is fully characterized in terms of the self- (C 0 i ) and mutual capacitances (C0 i;i þ 1 ) of the lines, as shown in Figure 8.37 in the case of rectangular TEM bars. The filtering properties of the comb-line filter can more easily be understood on the basis of its equivalent circuit. The network identity of Figure 4.25c for the outer pairs of lines, (0 to 1) and (N to N þ 1), and that of Figure 4.25d for the inner pairs can in fact be used to transform Figure 8.34 into the equivalent circuit of Figure 8.38, where all lines and stubs have the same length l. Notice that the filter exhibits transmission zeros for l ¼ l=4 and l ¼ l=2 and that the stubs have an inductive behaviour as long as l < l=4; they resonate with the lumped capacitances at a frequency which is the lower, the higher the capacitance. If the latter were removed (i.e. C ¼ 0) then the resonance would occur at l ¼ l=4 but the circuit would become an all-stop structure. The comb-line filter can be designed using the formulae quoted in Table 8.12, which have been derived from those in [2]. Such formulae are based on the simplified frequency transform quoted in Table 8.12. The characteristic impedances Rak ðk ¼ 1; . . . ; NÞ of the inner lines (considered as uncoupled) affect the unloaded Q factors of the resonators, thus the filter loss. Usually, Rak is chosen to be the same for all inner lines f Rak ¼ Ra 8k 2 ½1; . . . ; N g. A proper choice of Ra must be made depending on the

282

MICROWAVE AND RF ENGINEERING tuning screw locking bolt

l

0

1

2

3

4

5

6

end capacitance RF coaxial connectors

(a)

symmetry planes

b/2

0

1

2

3

4

5

6

b/2

(b) symmetry planes

b/2

0

1

2

3

4

5

6

b/2

(c)

Figure 8.36 Comb-line filter realized with TEM coaxial lines: side view (a) and top view of a filter employing cylindrical rods (b) or rectangular bars (c).

technology employed. As is known, for example, for an air-filled coaxial line, the minimum loss is obtained when45 Ra 76:7 O. Once the lowdpass prototype parameters g0 ; g1 . . . and frequency band have been chosen, Table 8.12 0 returns the self- Ck0 and mutual Ck;k þ 1 capacitances of Figure 8.37 and the end capacitances Ck of 0 Figure 8.38. Depending on the technology employed, Ck0 and Ck;k þ 1 can then be used to compute the cross-sectional dimensions of the coupled lines by appropriate synthesis formulae [2]. Eventually, a fine optimization of the whole structure using full wave models is necessary to compensate for spurious phenomena and the various approximations involved.

45 For any given outer diameter b, the loss is minimum when the ratio of the inner to the outer diameter is lnð3:591Þ ffi 76:71 e0:5 . b=a ffi 3:591; consequently, the minimum loss impedance is Zminloss ¼ 60 e0:5 r r

MICROWAVE FILTERS

283

Table 8.12 Comb-line filter design formulae. ð f1 ; f2 Þ w ¼ 2ð f1 f2 Þ=ðf1 þ f2 Þ 2 f f0 f0 ¼ w f0 y0 < p=2 at f0 ¼ ð f1 þ f2 Þ=2 Ra R0

Passband limits Fractional bandwidth Lowdpass to bandpass transform Line electrical length Isolated line characteristic impedance Reference impedance Normalized parameters b0j ¼

cotðy0 Þ þ y0 sin2 ðy0 Þ R0 ; 2 Raj

j ¼ 1; . . . ; N

Normalized end impedances and inverters rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 gk gk þ 1 K 0 k k þ 1 k¼1;...;N1 ¼ k;k þ 1 ¼ ; w b0k b0k þ 1 R0 pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Line-to-ground capacitances v ¼ 1= m0 e0 er ¼ c= er R0T1 ¼

RT1 g0 g1 ¼ ; R0 w b01

C0 0 ¼

rffiffiffiffiffiffiffiffiffi 1 1 1 ; vR0 R0 T1

C0 1 ¼

R0 TN ¼

RTN gN gN þ 1 ¼ R0 w b0 N

1 R0 1 tanðyÞ 1 þ 0 0 þ C0 0 vR0 Ra1 R T1 K 1;2

1 R0 1 1 C0 k k¼2;...;N1 ¼ 0 þ 0 tanðyÞ vR0 Rak K k1;k K k;k þ 1 C0 N þ 1 ¼

rffiffiffiffiffiffiffiffiffi 1 1 1 ; vR0 R0 TN

C0 N ¼

1 R0 1 tanðyÞ 1 þ 0 0 þ C0 N þ 1 vR0 RaN R TN K N1;N

Mutual capacitances C 0 0;1 ¼

1 tanðyÞ 1 1 C0 0 ; C 0 k;k þ 1 k¼1;...;N1 ¼ ; C 0 N;N þ 1 ¼ C 0 N þ 1 vR0 K 0 k;k þ 1 vR0 vR0

End capacitances Cj ¼

1 cotðy0 Þ ; Raj 2p f0

j ¼ 1; . . . ; N

Characteristic impedances of the equivalent circuit of Figure 8.37: Zi;i ¼

1 ; vC0 i

i ¼ 0; . . . ; N þ 1 Zi;i þ 1 ¼

C'i-2, i-1

1 ; vC0 i;i þ 1

i ¼ 0; . . . ; N

C'i-1,i

C'i, i+1

i-1 C'i-1, i-1

i

C'i+1,i+2 i+1

C'i, i

C'i+1, i+1 ground

Figure 8.37 Line-to-ground and mutual capacitances in coupled coaxial TEM rectangular bars (transverse cross-section).

284

MICROWAVE AND RF ENGINEERING Z 1,2

Z 2,3

Z N-1,N

Z 0,1

1

Figure 8.38

Z N,N+1

C1

C2

C3

Z 0,0

Z 1,1

Z 2,2

C N-1

Z 3,3

CN

2

Z N-1,N-1 Z N,N Z N+1,N+1

Equivalent network of the comb-line filter of Figure 8.37.

EXAMPLE 8.11 We illustrate here the design of a comb-line filter of order N ¼ 5 with Chebyshev response,46 with 26 dB return loss, 9.75–10.25 GHz passband, source and load impedances R0 ¼ 50 O. Assume the filter lines to be l=8 (y0 ¼ p=4 electrical length) at the centre frequency f0 ¼ ð f1 þ f2 Þ=2 ¼ ð9:75 þ 10:25Þ=2 ¼ 10 GHz. The chosen level for the characteristic impedance of all inner lines is Ra ¼ 75 O. The second column of Table 8.13 gives the coefficients gk of the lowdpass prototype. The third and fourth columns give the normalized self- and mutual capacitances of the filter lines, while the lumped end capacitances are quoted in the fifth column. Finally, the characteristic impedances of the equivalent circuit of Figure 8.38 are quoted in the sixth and seventh columns. The coaxial TEM realization is depicted in Figure 8.39, where rectangular bars are used as transmission lines, whereas coaxial input and output ports are applied to the first and last nonresonating lines. The lumped capacitances Ck are realized by the gaps between the bar ends and the outer wall of the filter. To a first approximation they can be calculated as parallel-plate capacitances, i.e. neglecting the edge effect. A comparison between the synthesized equivalent circuit simulation and the EM full wave response of the designed structure is shown in Figure 8.40. Figure 8.41 shows the broad-band response of the filter. Notice that the spurious-free stopband extends up to 40 GHz (four times larger than the filter centre frequency).

Table 8.13 Values of the synthesized filter of Example 8.11. k

gk

0 and 6 1 and 5 2 and 4 3

1 0.7563 1.3049 1.5773

1 Ck0 e1 0 er

0 1 1 Ck;k þ 1 e0 er

5.7452 3.3339 4.4762 4.5762

1.7946 0.3252 0.2252

Ck ðpFÞ

Zk;k ðOÞ

Zk;k þ 1 ðOÞ

0.212 0.212 0.212

65.57 113.00 84.16 82.32

209.91 1158.5 1673.0

46 See the Ansoft file 01_Combline_Filter_Example_11.adsn for analyzing the circuit model of the described filter. Also, the Mathcad file 07_CombLine_Filter_Synthesis.MCD (annexed to Chapter 15) implements the comb-line filter synthesis.

MICROWAVE FILTERS

285

0

10

–20

0

–40

←

CKT

–60

–20

EM s11 →

–80

–100

–10

s21

9

10

11

20 log10(|s11|)

20 log10(|s21|)

Figure 8.39 Coaxial TEM realization of Example 8.11: (a) bottom view; (b) side view; (c) perspective view of the disassembled structure.

–30

12

–40

Frequency, GHz

Figure 8.40

Circuit (CKT) and electromagnetic (EM) simulation of the filter in Example 8.11.

286


20 log10(|s21|)

–20

–40

–60

–80

–100 0

10

20

30

40 *

50

Frequency, GHz

Figure 8.41 Broad-band simulation of the filter in Example 8.11.

Bibliography 1. O. J. Zobel, ‘Theory and design of uniform and composite electric-wave filters’, Bell System Technical Journal, Vol. 2, No. 1, 1923. 2. G. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures, McGraw-Hill, New York, 1964. 3. R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, 1992. 4. A. Zverev, Handbook of Filter Synthesis, John Wiley & Sons, Inc., New York, 1967. 5. P. I. Richards, ‘Resistor transmission line networks’, Proceedings of the IRE, Vol. 30, pp. 217–220, 1948. 6. J. D. Rhodes, Theory of Electrical Filters, John Wiley & Sons, Inc., New York, 1976. 7. I. Hunter, Theory and Design of Microwave Filters, IEE Electromagnetic Wave Series No. 48, London, 2001. 8. G. Bianchi, R. Sorrentino, M. Salerno and F. Alessandri, ‘Image parameter design of parallel coupled microstrip filters’, Proceedings of the 18th European Microwave Conference, Stockholm, Sweden, 1988. 9. G. Bianchi and R. Sorrentino, Electronic Filter Design and Simulation, McGraw-Hill, New York, 2007. 10. S. B. Cohn, ‘Parallel-coupled transmission-line resonator filters’, IRE Transactions on Microwave Theory and Techniques, April, pp. 223–231, 1958. 11. I. J. Bahl, ‘Capacitively compensated high performance parallel coupled microstrip filters’, IEEE MTT-S International Microwave Symposium Digest, Vol. II, pp. 679–682, 1989.

Simulation files Ansoft files 01_Combline_Filter_Example_11.adsn. Analyzes the circuit of the comb-line filter described in Example 8.11.

MICROWAVE FILTERS

287

Mathcad files 02_Low_Pass_Prototype_Coefficients.MCD. Computes the Butterworth and Chebyshev lowdpass prototype coefficients for any order and passband ripple. 03_Microstrip_Edge_Coupled_Filters.MCD. Implements the complete automatic synthesis of microstrip edge-coupled filters, from the specification to the impedances to the microstrip dimensions.

9

Basic concepts for microwave component design 9.1 Introduction This chapter deals with the preliminary concepts needed to explain control and active components, including integrated circuits, nonlinear components, subsystems and test instruments. We will define here the main linear, noise and nonlinear performances of two-port networks. Moreover, many high-frequency components and subsystems consist of many cascaded networks. Therefore, we will give the fundamentals of the method of analysis, based on reducing two cascaded networks into one, in terms of gain, noise and power. It has to be considered that the next applications of the reduction should allow calculation of any cascaded chain with arbitrary numbers of components. Most of the results presented, although widely used in engineering practice, involve approximations; this will be pointed out as much as possible. Finally, Sections 9.6 and 9.7 explain the basic working principles of the most frequently used semiconductor devices, together with their electrical models.

9.2 Cascaded linear two-port networks This section deals with the linear performances of cascaded two-port networks and presents the fundamental equations that determine the linear performances of the resulting chain. A more complex combination can be analyzed in terms of two networks by progressively replacing two cascaded networks with an equivalent one. Within this context, we will consider two networks as equivalent if they present the same network parameters, in particular the scattering matrix.1 Figure 9.1 shows two linear two-port networks, connected in cascade (a), together with their equivalent network (b). The superscripts ‘A’ and ‘B’ denote the parameters of the first and second elements of the cascade, respectively, while ‘A; B’ denotes the parameters of the equivalent network. Furthermore, we will refer to port 1 of network A and port 2 of network B as the external ports, while port 2 of network A and the port 1 of network B will be the interfacing ports. 1

From the identity of the scattering matrix, the identity of all the other matrices of all the possible network representations ([Z], [Y], [ABCD], etc.) follow, if the latter exist. Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-75862-5

290


[S(A)] 1

1

[S(B)] 2

1

[S(A,B)] 2

2

(a)

1

1

2

2

(b)

Figure 9.1 Cascaded linear two-port networks: (a) schematic of the structure; (b) equivalent network. In terms of S parameters, the network in Figure 9.1 gives 8 ðAÞ ðAÞ ðAÞ ðAÞ ðAÞ > ¼ s11 a1 þ s12 a2 > > b1 > > > ðAÞ ðAÞ ðAÞ ðAÞ ðAÞ > > ¼ s21 a1 þ s22 a2 > b2 > > > > ðBÞ ðBÞ ðBÞ ðBÞ ðBÞ > b1 ¼ s11 a1 þ s12 a2 > > > > ðBÞ > ðBÞ ðBÞ ðBÞ ðBÞ > > b2 ¼ s21 a1 þ s22 a2 > > > > > < bðAÞ ¼ aðBÞ 2 1 ðAÞ ðBÞ > > a ¼ b > 2 1 > > > ðA;BÞ > ðAÞ > > b1 ¼ b1 > > > > ðA;BÞ ðBÞ > > b2 ¼ b2 > > > > > ðAÞ > > a1A;B ¼ a1 > > > > : ðA;BÞ ðBÞ a2 ¼ a2

ð9:1Þ

The first two (respectively, the third and fourth) equations of system (9.1) constitute the relation between the incident and reflected waves of network A (B). Moreover, the wave reflected from port 2 of network A coincides with the wave incident on port 1 of network B, and vice versa, as the fifth and sixth equations state. Finally, port 1 (2) of the equivalent network coincides with port 1 of network A (port 2 of network B), and similarly with the corresponding waves, according to the last four equations of system (9.1). In order to determine the scattering parameters of the cascaded network, we have to substitute the fifth and sixth equations into the first four of system (9.1). In this way, we remove the quantities aA2 ; bA2 ; aB1 and bB1 , and can express the remaining waves of networks A and B as " # " # 8 ðAÞ ðAÞ ðBÞ ðAÞ ðBÞ ðAÞ ðBÞ > > bðAÞ ¼ sðAÞ þ s12 s21 s11 aðAÞ þ s12 s11 s22 s12 þ sðAÞ sðBÞ aðBÞ > > 1 11 1 12 12 2 ðAÞ ðBÞ ðAÞ ðBÞ > < 1 s22 s11 1 s22 s11 " # > ðAÞ ðBÞ ðBÞ ðBÞ ðAÞ > > s21 s21 s12 s21 s22 ðBÞ ðAÞ ðBÞ ðBÞ > > b ¼ a þ þ s : 2 22 a2 ðAÞ ðBÞ 1 ðAÞ ðBÞ 1 s22 s11 1 s22 s11 Finally, after simplifying the second coefficient in the first of the above equations, and considering the last four equations of system (9.1), we get the scattering parameters of the equivalent network: 3 2 ðAÞ ðAÞ ðBÞ ðAÞ ðBÞ s12 s21 s11 s12 s12 ðAÞ 2 ðA;BÞ ðA;BÞ 3 6 s11 þ 7 1 sA22 sB11 1 sA22 sB11 7 6 h i s12 s11 7 6 ðA;BÞ 4 5 ð9:2Þ ¼6 S ¼ 7 7 6 ðAÞ ðBÞ ðBÞ ðBÞ ðAÞ ðA;BÞ ðA;BÞ s21 s21 s12 s21 s22 4 s21 s22 ðBÞ 5 þ s22 ðAÞ ðBÞ ðAÞ ðBÞ 1 s22 s11 1 s22 s11

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN

291

Before considering the consequences of identities (9.2) in more detail, we need to consider some important simplified cases: ðAÞ

ðBÞ

1. One of the interfacing ports is perfectly matched, i.e. s22 ¼ 0 or s11 ¼ 0, and the cascaded twoport networks assume the scattering matrix 3 2 ðAÞ ðAÞ ðBÞ s12 s21 s11 ðAÞ ðBÞ h i sðAÞ ¼0 6 sðAÞ þ s s 11 12 12 7 7 1 sA22 sB11 SðA;BÞ 22¼ 6 5 4 ðAÞ ðBÞ ðBÞ s21 s21 s22 or h

2 ðAÞ s i sðBÞ ¼0 6 11 SðA;BÞ 11¼ 6 4 ðAÞ ðBÞ s21 s21

3

ðAÞ ðBÞ

s12 s12 ðBÞ ðBÞ ðAÞ

s12 s21 s22

ðBÞ þ s22 ðAÞ ðBÞ 1 s22 s11

7 7 5

respectively. Thus, if port 2 of network A (port 1 of network B) is perfectly matched, it follows that the input (output) reflection coefficient of the resulting network equals the one of network A (B). Furthermore, in both cases, the forward and reverse transmission coefficients of the resulting network coincide with the products of the corresponding coefficients of the two networks. ðAÞ

ðBÞ

2. One of the two networks is unilateral, i.e. s12 ¼ 0 or s12 ¼ 0, and identities (9.2) simplify to 3 2 ðAÞ s11 0 h i sðAÞ ¼0 6 7 7 ðAÞ ðBÞ ðBÞ ðBÞ ðAÞ SðA;BÞ 12¼ 6 s12 s21 s22 4 s21 s21 ðBÞ 5 þ s 22 ðAÞ ðBÞ ðAÞ ðBÞ 1 s22 s11 1 s22 s11 or

2

ðAÞ ðAÞ ðBÞ

s s s11 ðAÞ 6 s þ 12 21 ðAÞ ðBÞ h i sðBÞ ¼0 6 11 1 s22 s11 6 ðA;BÞ 12 S ¼ 6 ðAÞ ðBÞ 6 s21 s21 4 ðAÞ ðBÞ

1 s22 s11

3 0 7 7 7 7 7 ðBÞ 5 s22

Therefore the other network does not affect the reflection coefficient corresponding to the external port of the unilateral network, and the resulting network is still unilateral. Similar ðAÞ ðBÞ consequences can be derived if we consider the dual case – which is s21 ¼ 0 or s21 ¼ 0 – just by swapping the port index. ðAÞ

ðBÞ

3. One of the interfacing ports is perfectly matched, i.e. s22 ¼ 0 or s11 ¼ 0, and both networks are unilateral, i.e.

ðAÞ s12

¼

ðBÞ s12

¼ 0. The resulting scattering parameters simplify to 2 3 ðAÞ h i s11 0 ðA;BÞ 5 ¼4 S ðAÞ ðBÞ ðBÞ s21 s21 s22

Note that this expression holds true if just one of the interfacing ports is matched; it is not necessary for both ports to match. 4. Both the networks are passive and loss free, thus the network scattering parameters are related as 2 2 2 2 ðAÞ ðAÞ ð AÞ ð AÞ ð AÞ ð AÞ ð AÞ ð AÞ s12 ¼ s21 ; s12 ¼ s21 ¼ 1 s11 ¼ 1 s22 ) s11 ¼ s22

292


and ðBÞ

ðBÞ

s12 ¼ s21 ;

2 2 2 2 ðBÞ ðBÞ ðBÞ ðBÞ ðBÞ ðBÞ s12 ¼ s21 ¼ 1 s11 ¼ 1 s22 ) s11 ¼ s22

From the relation between the transmission and reflection coefficients, we can also write 8 8 h i h i ðAÞ ðAÞ ðBÞ ðBÞ ðAÞ ðBÞ > > > > s ¼ r exp jj ¼ r exp jj s > 11 > 11 11 11 > > > > > > h i h i < < ðAÞ ðAÞ ðBÞ ðBÞ s22 ¼ rðAÞ exp jj22 s22 ¼ rðBÞ exp jj22 and > > > > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i i > > > > 2 2 ðAÞ ðAÞ ðAÞ ðBÞ ðBÞ > > : s12 ¼ s21 ¼ 1 ½rðAÞ exp jj21 : sðBÞ 1 ½rðBÞ exp jj21 12 ¼ s21 ¼ where all the variables rðAÞ ; jð11AÞ ; jð21AÞ ; jð22AÞ ; rðBÞ ; jð11BÞ ; jð21BÞ ; jð22BÞ are real quantities and rðAÞ ; rðBÞ are also non-negative. Substituting the above quantities into expressions (9.2), we obtain the magnitude of the transmission coefficients of the cascaded networks qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 ½rðAÞ 1 ½rðBÞ ðA;BÞ ðA;BÞ h i s12 ¼ s21 ¼ ð AÞ ð BÞ 1 rðAÞ rðBÞ exp j j22 þ j11 Now, for any possible combination of the values jð22AÞ ; jð11BÞ the denominator of the expression above is bounded within the limits nh io ð AÞ ð BÞ 1 rðAÞ rðBÞ 1 rðAÞ rðBÞ exp j j22 þ j11 1 þ rðAÞ rðBÞ Hence, the cascaded two-port networks have a resulting magnitude of transmission coefficient bounded within the limits qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 1 ½rðAÞ 1 ½rðBÞ 1 ½rðAÞ 1 ½rðBÞ ðA;BÞ ðA;BÞ ðA;BÞ ðA;BÞ ¼ s ¼ s s ¼ s21;max 12 21 21;min 1 þ rðAÞ rðBÞ 1 rðAÞ rðBÞ The quantity " 20 log10

ðA;BÞ

s21;max ðA;BÞ

s21;min

#

¼ 20 log10

1 þ rðAÞ rðBÞ 1 rðAÞ rðBÞ

ð9:3Þ

is the ratio between the maximum and the minimum possible amplitude of the transmission coefficient of the cascaded two-port networks when the phases of the reflection coefficients of the interfacing ports vary arbitrarily. It is possible to demonstrate that the resulting network is loss free, like its two components. Thus the minimum amplitude of the transmission coefficient is associated with the maximum reflection coefficient amplitude, hence vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðAÞ 2 on ðBÞ 2 offi u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi u 1 r 1 r t ðA;BÞ ðA;BÞ ðA;BÞ s11;max ¼ s22;max ¼ 1 s21;min ¼ 1 2 ð A Þ ð B Þ ½1 þ r r vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u1 þ 2rðAÞ rðBÞ þ rðAÞ 2 rðBÞ 2 1 þ rðAÞ 2 þ rðBÞ 2 rðAÞ 2 rðBÞ 2 ¼ t 2 2 1 þ 2rðAÞ rðBÞ þ ½rðAÞ ½rðBÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u þ 2rðAÞ rðBÞ þ rðAÞ 2 þ rðBÞ 2 rðAÞ þ rðBÞ rðAÞ rðBÞ 1 ðAÞ t ¼ ¼ ffi r þ rðBÞ ð9:4Þ 2 2 1 þ rðAÞ rðBÞ 1 þ 2rðAÞ rðBÞ þ ½rðAÞ ½rðBÞ


293

Table 9.1 Worst case mismatch combinations of two passive, loss-free, cascaded two-port networks having the same reflection coefficients. rðAÞ ¼ rðBÞ 0.891 0.794 0.708 0.631 0.562 0.501 0.447 0.398 0.355 0.316 0.251 0.200 0.158 0.126 0.100 0.056 0.032 0.018 0.010

20 log10 rðAÞ

ðA;BÞ ðA;BÞ s11;max ¼ s22;max

i h ðA;BÞ 20 log10 s11;max

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 25 30 35 40

0.993 0.974 0.943 0.903 0.854 0.801 0.745 0.687 0.630 0.575 0.473 0.384 0.309 0.248 0.198 0.112 0.063 0.036 0.020

0.057 0.228 0.508 0.890 1.366 1.926 2.560 3.257 4.009 4.807 6.511 8.318 10.195 12.116 14.066 19.007 23.988 28.982 33.980

Table 9.1 shows the results obtained from the worst case mismatch, obtained from the combination of the reflections from the two networks, in the special case of rðAÞ ¼ rðBÞ . The second and fourth columns of the table present the return loss corresponding to the reflection coefficient of the single and cascaded networks, respectively. In engineering practice, it is often assumed that the components to be cascaded are perfectly matched. Under this simplified hypothesis, assuming N components and denoting their scattering parameters with an index from 1 to N, the scattering matrix of the cascaded network is an immediate consequence of Equation (9.2) 3 2 N Y ðk Þ s12 7 0 h i 6 7 6 k¼1 7 ð9:5Þ Sð1...NÞ ¼ 6 N 7 6Y 5 4 ðk Þ s21 0 k¼1

The approximate equation (9.5) neglects the effect of the mismatch between the interfacing ports. Any couple of mismatched interfacing ports contributes to the transmission coefficient by the factor ðkÞ ðk þ 1Þ

½1 s22 s11 1 . Furthermore, the mismatch on the interfacing ports modifies the reflection coefficients of the two-port resulting from two cascaded networks. The approximation involved with the assumption (9.5) consists of replacing (9.2) with 3 2 3 2 ðA;BÞ ðA;BÞ ðAÞ ðBÞ h i s12 s11 0 s12 s12 ðA;BÞ 5 5¼4 ð9:6Þ ¼4 S ðA;BÞ ðA;BÞ ðAÞ ðBÞ s21 s22 s12 s12 0

294


amplitude error |S22aS11b| arg(S22aS11b)

phase error unit circle

Figure 9.2

Graphical vector representation of the denominator of the terms of matrix (9.2).

Comparing (9.6) with (9.2), and focusing on the forward transmission coefficient, we recognize that the approximated approach neglects the denominator of all the terms of the matrix (9.2) ðAÞ ðBÞ

1 s22 s11

ð9:7Þ

Figure 9.2 shows a graphical representation of that neglected factor. ðAÞ ðBÞ ðAÞ ðBÞ It can be seen that, for any given value of the amplitude s22 s11 , the factor 1 s22 s11 can assume2 ðAÞ ðBÞ ðAÞ ðBÞ any amplitude within the limits 1 s22 s11 ; 1 þ s22 s11 , and any phase within the limits ðAÞ ðBÞ ðAÞ ðBÞ tan 1 s22 s11 ; tan 1 s22 s11 . Now, in engineering practice, the amplitude3 – but not the phase – of the network reflection coefficients is usually specified. Moreover, the specification of the transmission coefficient phase is seldom used, as well. The reason for this is because it is difficult to control the phase of the transmission coefficient in the production of the network, and the reflection coefficient phase presents even more difficulties. For instance, if a transmission line is present on the signal path, any variation Dl in its length will cause a variation of bDl on the phase of the transmission coefficient, and double the quantity on the reflection coefficient phase. Examples 9.1 and 9.2 below clarify the effects of the phase shift due to the impedance mismatch.

EXAMPLE 9.1 PHASE MISMATCH CAUSED BY IMPEDANCE MISMATCH Let us consider the simple assembly illustrated in Figure 9.3a. It consists of a Wilkinson power divider (DIV), whose two outputs feed the two external ports (P2 and P3 ) of the assembly through two cables (CB1 and CB2 ). Such an assembly is typically used when a single source (connected to port P1 ) has to deliver its signal to two different loads (connected to P2 and P3 ).

h i h i ðAÞ ðBÞ ðAÞ ðBÞ The minimum or the maximum of the amplitude occurs for arg s22 s11 ¼ p or arg s22 s11 ¼ 0, while the h i ðAÞ ðBÞ phase is minimum or maximum for arg s22 s11 ¼ p=2, respectively. 2

3

In most cases, instead of the reflection coefficient, the return loss is specified: RL ¼ 20 log10 ðjskk jÞ.

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN CB1

Γ1 P1

P2

2

1

3

(a)

P3

Γ2

DIV

CB2

[S(A1)] 2 P1

295

1

1

[S(B1)] 2

P2

CB1 Γ2

(b)

P1

1

[S(B2)]

Γ1 2

[S(A2)]

1

2

P3

CB2

(c)

Figure 9.3 Networks discussed in Example 9.1: (a) block diagram of the assembly; (b) equivalent two-port for the path P1–P2; (c) equivalent two-port for the path P1–P3.

Let us assume that DIV is anpffiffiideal Wilkinson4 power divider with a single section. Two ffi transmission lines with Z0 ¼ 50 2 ffi 70:7 O and the same length connect node 1 with nodes 2 and 3, while an ideal 100 O resistor is placed between nodes 2 and 3. Moreover, the two transmission lines are l=4 at the centre of the operating frequency of the component. That component is electrically and mechanically symmetrical, thus s12 ¼ s21 ¼ s13 ¼ s31 ; s22 ¼ s33 . Figure 9.4 shows the response of the component, assuming a centre frequency of 7.5 GHz. It can be seen that the one-section Wilkinson divider works ideally (s11 ¼ s22 ¼ s33 ¼ s32 ¼ 0; js21 j2 ¼ js31 j2 ¼ 0:5) only at the centre frequency (7.5 GHz in our case). If we accept a return loss5 of about 15.18 dB and an isolation6 of 14.69 dB, the working bandwidth of the components equals one octave (5 to 10 GHz in our case). Now, if CB1 is identical to CB2 , then the symmetry of the assembly follows from the symmetry of DIV. In practical cases, none of these two assumptions is rigorously true. Furthermore, in most cases, the cables are specified in terms of return loss and insertion loss in the frequency band of interest. Sometimes, the maximum difference between the amplitude and/or the phase of the transmission coefficient is also specified. 4 Section 7.7.1 describes the Wilkinson power divider. The shape of the symbol in Figure 9.3 indicates the internal structure of the component. 5 Here, the term return loss denotes the lowest of the quantities 20 log10 ðjs11 jÞ and 20 log10 ðjs22 jÞ ¼ 20 log10 ðjs33 jÞ. Over the considered bandwidth, the worst case input return loss is 20 log10 ðjs11 jÞ ffi 15:18 dB, while the worst case output return loss is 20 log10 ðjs22 jÞ ¼ 20 log10 ðjs33 jÞ ffi 29:16 dB. 6 Here, by the term isolation, we mean the attenuation between the output ports, port 2 and port 3, i.e. the quantity 20 log10 ðjs32 jÞ.

MICROWAVE AND RF ENGINEERING -3.0

10 s21

20 log10(|S21|)

-3.1

0

-3.2

-10 s32 s11

-3.3 -3.4

-20

s22

-3.5

-30 -40

5

Figure 9.4

6

7 8 Frequency, GHz

9

10

20 log10(|s11|), 20 log10(|s22|), 20 log10(|s31|)

296

Response of the ideal single section Wilkinson divider.

In order to focus on the impedance mismatch effects, we will consider DIV as ideal, while CB1 and CB2 are loss free, their transmission coefficients have the same phase, and their return loss is no higher than 20 dB. Furthermore, the two cables have a physical length l ¼ 400 mm and a dielectric constant7er ¼ 2:2. To simplify our analysis, we will assume that the two cables are ideal transmission lines, and that the cause of the finite return loss is the characteristic impedance different from 50 O. The cables present the maximum impedance mismatch (are perfectly matched) when their length equals odd (even) multiples of l=4. ðCB1Þ ðCB2Þ Denoting Z0 and Z0 as the characteristic impedances of CB1 and CB2 , respectively, the maximum amplitude of the reflection coefficients that those lines present to DIV are h h ðCB1Þ i2 ðCB2Þ i2 ½Z ðCB1Þ 2 2 2 Z0 0 Z0 50 50 50 50 rmax;1 ¼ ðCB1 ; rmax;2 h ¼ h i i ðCB1Þ 2 ðCB2Þ 2 ½Z0 Þ 2 2 2 Z0 þ 50 þ 50 þ 50 Z0 50

A return loss of 20 dB corresponds to a reflection coefficient of 20 logðrÞ ¼ 20 ) r ¼ 0:1. Thus the cable impedance can assume any value within the interval h i2 ðCB1Þ Z0 502 0:1 h ; i ðCB1Þ 2 Z0 þ 502

h h

ðCB2Þ

i2

Z0

502

Z0

þ 502

i ðCB2Þ 2

0:1

Solving for the characteristic impedances, the above inequalities give pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ðCB1Þ ðCB2Þ 45:227 ffi 50 9=11 Z0 ; Z0 50 11=9 ffi 55:277 Then, the worst case is when one cable has the minimum impedance and the other has the maximum: ðCB1Þ

Z0

7

pffiffiffiffiffiffiffiffiffiffi ¼ 50 9=11;

Corresponding to Teflon-isolated coaxial cables.

ðCB2Þ

Z0

pffiffiffiffiffiffiffiffiffiffi ¼ 50 11=9

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN 20

-2.5

s21

-3.0

10

s31 -3.5

0

-4.0

-10

s11 s22

s33

-4.5

-5.0 5

Figure 9.5

6

7 8 Frequency, GHz

9

20 log10(|skk|)

20 log10(|s21|), 20 log10(|s31|)

297

-20

-30 10

Simulated response of the network in Figure 9.3a.

The three-port network in Figure 9.3a can be considered as two separate two-port networks, as illustrated in Figures 9.3b,c, respectively. Also, we will neglect the effect of the different loading of the third port of the divider; in other words, we will consider G2 ¼ 0 (G1 ¼ 0) when analyzing the transmission from P1 to P2 (P1 to P3 ). This assumption corresponds to considering DIV as presenting infinite isolation (s23 ¼ 0), while this is true only at the centre frequency. ðAÞ ðBÞ With these different assumptions, we have that the factor 1 s22 s11 has an amplitude bounded within the interval 29:16 20 Amplitude mismatch 2 1 10 20 10 20 ;

1 þ 10

29:16 20 20 10 20

ffi ½0:997; 1:003

corresponding to a maximum amplitude imbalance of 10 log10 ð1:003=0:997Þ ffi 0:061 dB between the paths P1–P2 and P1–P3. The effect of the mismatch on the phase is

29:16 20 29:16 20 Phase mismatch 2 tan 1 10 20 10 20 ; tan 1 10 20 10 20 ¼ ½ 0:2 ; 0:2 The maximum imbalance condition occurs when one port is at its possible minimum and the other is at its maximum; this is why the maximum imbalance variation is double the corresponding mismatch interval. Thus, the maximum phase imbalance is 0:4 . It is possible to describe the network in Figure 9.3a as a circuit simulator.8 Figure 9.5 shows the predicted response of the subsystem. Note that the output return loss (corresponding to s22, s33) is worse

8

See the Ansoft file 01_Power_Divider_with_Cables.adsn.

298


0.5

4

20 log10(|s21/s31|)

20 log10(|s21/s31|)

2 1

-0.5

(180/π) arg(s21/s31)

0 -1

-1.0

(180/π) arg(s21/s31)

3

0.0

-2 -3 10

-1.5 5

6

7 8 Frequency, GHz

9

Figure 9.6 Amplitude and phase matching between the two output ports P2, P3 of the assembly in Figure 9.3.

than that between the power divider and the cables: the peak over the frequency is about 18.25 dB, in good agreement with expression (9.4) ðAÞ r þ rðBÞ ðA;BÞ Output return loss ¼ 20 log10 s11;max ¼ 20 log10 1 þ rðAÞ rðBÞ 0 ¼ 20 log10

@ 10

29:16 20

1 þ 10

þ 10

20 20

29:16 20 20 10 20

1 A ffi 17:47

Furthermore, the transmission coefficient amplitude of the path P1 to P2 differs slightly from that of the path P1 to P3, as expected. More precisely, Figure 9.6 shows the amplitude (left y axis) and the phase (right y axis) matching between the two paths. The resulting peak quantities are 0:2 dB and 1:1 respectively, significantly higher than the values above derived from analytic considerations ð0:06 dB; 0:4 Þ. The reason for such a discrepancy is due to the approximation ignoring the effect of the mismatched cable loading of the output ports of the power divider. This simplification is equivalent to assuming G1 ¼ G2 ¼ 0, as already pointed out. The circuit simulation can also be used to analyze the assembly under the simplifying hypotheses involved with the analytic considerations.9 The resulting amplitude and phase imbalance is 0.05 dB and 0.3 , a little bit less than the worst possible case. Finally, note that both the amplitude and the phase imbalance increase when passing from the centre frequency to the extremes of the frequency band. This is a straightforward consequence of the decreasing output return loss of the power divider.

9

See the Ansoft file 02_Power_Divider_with_Cables_more_Ideal.adsn.


299

EXAMPLE 9.2 MISMATCH RESONANCES The factor (9.7) also plays an important role when the interfacing ports of the two cascaded networks present high values of the reflection coefficient. A typical case is two cascaded filters, which occurs when the second filter is used to clean the spurious passband of the first. In this case, the two filters present a common stopband: in that frequency range, the two reflection coefficients of the ðAÞ ðBÞ interfacing ports have amplitudes close to 1: s22 s11 1. Therefore, if some frequencies h i ðAÞ ðBÞ exist, such that arg s22 s11 ¼ p, then the denominator of the cascaded filters’ transmission coefficient approaches zero, and – simultaneously – the numerator tends to zero as well.10 Consequently, the cascaded filters’ transmission coefficient assumes the form 0=0, and could assume finite non-zero values, despite both filters having s21 0. Under these conditions, the response of the cascaded filters presents some low-attenuation peaks, which we will refer to as mismatch resonances. Here, we will examine an example of such a phenomenon by means of a simple ideal case. The two filters considered in this example are purely ideal; no details about their physical realization will be given. Figure 9.7a shows two cascaded filters with an interposed 50 O transmission line. The presence of the line models any possible physical connection between the two filters, like a printed circuit trace, coaxial cable, male–female connector couple, and so on. As we know, the transmission line can be considered as a part either of the first or the second filter: the result is a modification of the phase but not of the amplitude of the network. The first (second) filter of the network in Figure 9.7a is a bandpass (lowdpass) filter, denoted as BPF (LPF). Figure 9.7b shows the detailed structure of the two filters. BPF is a commensurate Richards filter with three stubs: two open circuit in series and one short circuit in parallel. Its prototype is a third-order Chebyshev one with 0.2 dB ripple: the coefficients are g0 ¼ g4 ¼ 1; g1 ¼ g3 ¼ 1:228; g2 ¼ 1:153. The passband extends from 1.95 to 2.05 GHz, thus f0 ¼ 2 GHz; D f ¼ 100 MHz. Application of the Richards transformation11 gives

po po

tan tan 1 po 2 o0 2 o0 3

¼j ¼j ffi j 39:29 10 tan p o1 p f0 Df =2 jo0 2 o0 tan tan 2 o0 2 f0 From the parameter a and the prototype coefficients12 we obtain the characteristic impedance of the three stubs

p f0 Df =2 g1 ffi 1562:155; Z0;1 ¼ Z0;3 ¼ 50 tan 2 f0

p f0 Df =2 g2 ffi 2:264 Z0;2 ¼ 50 tan 2 f0

Figure 9.8 (solid lines) shows the amplitude response of the ideal Richards filter. It is known that it presents spurious passbands at odd integer multiples of f0. A loss-free two-port network (like a filter is, ideally) has js21 j2 ¼ js12 j2 ¼ 1 js11 j2 ¼ 1 js22 j2 , hence if js11 j ¼ js22 j ! 1 then js21 j ¼ js12 j ! 0. 11 See Section 7.5.4. 12 See Section 7.3.4. 10

300

MICROWAVE AND RF ENGINEERING BPF

Z0=50Ω

P1

LPF P2

(a) BPF Z0,1

LPF Z0,3 Z0=50Ω

L1

L3 C2

P1

L5

L7

C4

C6

7

P2

Z0,2 (b)

Figure 9.7 Cascaded bandpass with lowdpass filter: (a) block diagram; (b) electric circuit with the internal structure of the filters.

The LPF has to clean the spurious response of the BPF, ensuring a minimum stopband rejection of more than 70 dB up to 10 GHz and beyond, although it also slightly increases the overall lowdpass selectivity. The LPF is then a lumped Chebyshev lowdpass one, with a cut-off frequency fT ¼ 2:05 GHz, passband ripple of 0.2 dB, and order N ¼ 7. The prototype coefficients are g0 ¼ g4 ¼ 1;

g1 ¼ g7 ¼ 1:372;

g2 ¼ g6 ¼ 1:378;

g3 ¼ g5 ¼ 2:276;

g4 ¼ 1:5

10

0

20 log10(|s21|)

-40

-10

lowdpass, s21

-20

-60 lowdpass, s11

bandpass, s11

20 log10(|s11|)

0

-20

-30

-80 bandpass, s21

-40

-100 0

2

4 6 Frequency, GHz

8

10

Figure 9.8 Amplitude responses of the filters in Figure 9.7. Solid lines represent the bandpass filter (BPF), dashed lines represent the lowdpass filter (LPF).

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN 20 log10(|s21,assembly |) 20 log10(|s21,bandpass s21,lowdpass|) 20 log10(|s11,assembly |)

0

10 0

mismatch resonance

-40

-10

-60

-20

-80

-30

-100

20 log10(|s11|)

-20 20 log10(|s21|)

301

-40 0

Figure 9.9

2

4 6 Frequency, GHz

8

10

Amplitude response of the filter assembly in Figure 9.7.

From the prototype coefficients, we obtain the ladder network components13 g1 g3 L1 ¼ L7 ¼ 50 ¼ 5:327 nH; L3 ¼ L5 ¼ 50 ¼ 5:327 nH 2pfT 2p fT C2 ¼ C6 ¼

1 g2 ¼ 2:14 pF; 50 2pfT

C4 ¼

1 g4 ¼ 2:329 pF 50 2pfT

Figure 9.8 (dashed curves) shows the transmission and reflection coefficient amplitude of the LPF. Figure 9.9 shows the amplitude response14 of the network in Figure 9.7. In detail: .

The grey solid curve (right y axis) is the reflection coefficient: the resulting passband return loss is about 11.3 dB better than the worst case given by Equation (9.4) ðA;BÞ rðBPFÞ þ rðLPFÞ s11;max ¼ 1 þ rðBPF Þ rðLPF Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ripple BPF Ripple LPF 10 10 þ 1 10 1 10 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0:406 ) return loss 7:82 dB ¼ Ripple BPF Ripple LPF 10 10 1 þ 1 10 1 10

.

The black solid curve (left y axis) is the transmission coefficient of the network in Figure 9.7.

.

The black dashed curve (left y axis) is the product of the two transmission coefficients of the two filters, BPF and LPF.

Comparing the two latter curves, we can see that they are not coincident, because of the factor (9.7). In particular, at the frequency of 2.362 GHz, the combination of the two filters presents a local minimum attenuation of 36.3 dB, rather than of 65.1 dB, as one might expect considering the attenuation of the two filters only. This is exactly a mismatch resonance, as predicted at the beginning of this example. 13 14

For the synthesis of lumped lowdpass Chebyshev filters, see the Mathcad file 05_Low_Pass_Example2.MCD. See the Ansoft file 03_Filter_Assembly.adsn.

302


9.3 Signal flow graphs Signal flow graphs (SFGs) are useful tools for representing and analyzing microwave networks. They are graphical representations of the scattering parameters, and are built from two simple construction rules: 1. The graph resulting from an N-port network has 2N nodes: N for the incident waves and as many for the reflected waves at the corresponding port. 2. The graph resulting from an N-port network has N 2 branches. Each branch is directed from the node representing the incident wave ak ðk ¼ 1; . . . ; NÞ to the one representing the reflected wave bk ðk ¼ 1; . . . ; NÞ. The weight of the branch from node ah to node bk is equal to skh ðh; k ¼ 1; . . . ; NÞ. The weight of the branch is a complex function of the frequency. SFGs can thus be used for networks with an arbitrary number of ports. Figure 9.10a is the signal flow of a two-port network. The SFG resulting from a plurality of networks, somehow reciprocally connected, is obtained by combining all the SFGs of the different networks and reducing the four nodes, relative to the waves at two interfaced ports, to two nodes. If port n of network A is connected to port m of network B, the nodes ð AÞ ðBÞ ðAÞ ðBÞ an ; bm will become the same node, and the same thing happens with bn ; am . Applying this procedure to the network in Figure 9.1, we obtain the SFG in Figure 9.10b. After drawing the couples of coincident nodes as a single node, the resulting final SFG is the one shown in Figure 9.10c. After obtaining the SFG of the arbitrarily complex network, we need to compute the transfer ratio from the generic nodes ah* to bk* ; in other words, we have to determine the scattering matrix of the

s21

a1

b2

s11 (a)

s22

b1

a2

s12

network A a

(A) 1

s11(A) b1(A) (b)

s21

(A)

(B)

(A)

s22

s11 (B)

s12(A) a2(A) = b1 b2(A) = a1(B)

a1

(A)

s11(A) (c)

b1

(A)

network B s21(B) b2(B)

b2(A) = a1(B)

s21 (A) s22

s12(B) b2

(B)

s22

a2(B)

(B)

(B)

(B) s11(B) s22 (A) s12(B) s12 a2(B) a2(A) = b1(B)

Figure 9.10 Signal flow graphs of: (a) a two-port network; (b) two cascaded two-port networks; (c) simplified graph resulting from (b).


303

network. This result can be achieved without any empirical or heuristic method, by applying Mason’s rule. It is possible to demonstrate [1] that the scattering parameter from node ah* to node bk* is h i h i P P P P P1 1 Lð1Þð1Þ þ Lð2Þð1Þ þ P2 1 Lð1Þð2Þ þ Lð2Þð2Þ þ P P P sh*;k* ¼ 1 Lð1Þ þ Lð2Þ Lð3Þ þ ð9:8Þ where: .

.

.

.

The terms P1 ; P2 . . . are the paths from node ah* to node bk* . A path is a set of consecutive branches with the same direction, starting from ah* and arriving at bk* such that any node cannot be touched more than once. The value associated with each path is the product of all its branches. For instance, considering the SFG in Figure 9.10c, and focusing on the nodes aA1 ; bA1 , we have the two paths P1 ¼ sA11 ; P2 ¼ sA21 sB11 sA12 . P The term Lð1Þ is the sum of all first-order loops, defined as the product of the consecutive and codirectional branches, beginning from and ending on the same node. example as PIn the same considered in the previous point, we have only one first-order loop, Lð1Þ ¼ sA22 sB11 . P The terms LðkÞ ðk > 1Þ are the sum of all the kth-order loops, defined as the product of any non-touching first-order loop. P The terms LðkÞðlÞ are the sum of all the kth-order loops not touching the path l.

The SFG in Figure 9.10c has no term of the latter two types; it is therefore easy to apply Mason’s rule to it and reobtain formula (9.1).

9.4 Noise in two-port networks In communication and test equipment, the useful signal is the one that must be transmitted, received, measured, and so on. On the other hand, we define as noise any unwanted signal added to the useful ones. Two main types of signal sources have to be considered: voltage and current noise sources. Noise can be produced either by other electric/electronic equipment which is placed close to the circuit of interest, or by other causes inherent to the devices of the circuit itself. Sometimes noise is simply due to signals considered useful by other users. A typical case occurs when multiple radio emissions in the same frequency band are present in the same space region, and cause reciprocal co-channel interference with each other. Here, we will focus our attention on the noise inherent in the electronic devices themselves. We will see that any real device embodies voltage and/or current noise sources. A voltage (current) noise source is a generator producing a random variable stimulus. Due to its random characteristic, instant-by-instant determination of the noise is impossible; therefore a statistical approach is required.

9.4.1

Noise sources

Given a random varying voltage vn ðtÞ or current in ðtÞ, we can characterize the noise sources in terms of the Fourier transform of the respective time domain expressions: þð1

Vn ð f Þ ¼

þð1

vn ðtÞexpð j2pftÞ dt; 1

In ð f Þ ¼

in ðtÞexpð j2pftÞ dt 1

304


We define the voltage noise density of a noise source in terms of the active power dissipated on a resistor ideally in parallel with the source, within a unit bandwidth. If Vn ð f Þ is the voltage noise density, R is the resistance of the resistor and Pdissipated ð f1 ; f2 Þ is the power dissipated on the resistor in the frequency range ð f1 ; f2 Þ, we have 1 R

ð f2

jVn ðf Þj2 d f ¼ Pdissipated ðf1 ; f2 Þ

ð9:9Þ

f1

If an infinitely narrow band is considered, in other words, for the limit as f1 tends to f2, Equation (9.9) becomes jVn ð f Þj2 ¼ R lim

df !0

Pdissipated ðf d f ; f þ d f Þ 2d f

ð9:10Þ

This equation defines the voltage noise source. Note that the dependence on the load resistance present in Equation (9.10) is only apparent because of the inverse proportionality between power and resistance. Furthermore, if the voltage noise density is constant within a 1 Hz band, it equals the square root of the normalized power in that band. In a similar way, we can define the current noise density as jIn ðf Þj2 ¼

Pdissipated ðf d f ; f þ d f Þ 1 lim d f ! 0 R 2d f

ð9:11Þ

Because of the random characteristic of the noise sources, the functions Vn ðf Þ; In ð f Þ are not directly observable or measurable. Rather, their square amplitude is the only measurable quantity, although the functions themselves are sometimes used in calculations, as intermediate quantities. The simplest case to consider is the resistor. Physical resistors produce random fluctuating voltages, due to the agitation of free charges inside the material of the resistors themselves. As we will see shortly, the power noise associated with these fluctuations is proportional to the absolute temperature of the resistor. The physical root cause of this behaviour is the increasing agitation of the resistor electron with temperature. For this reason the resistor noise is sometimes referred to as thermal noise.15 A detailed explanation of the physics of the phenomenon is beyond the scope of this book, so we will limit ourselves to some fundamental concepts, without giving a demonstration, although it exists. For our applications it is sufficient to know that any resistor has an associated noise source, as Figure 9.11 shows. Any real resistor can be schematized with an ideal noise-free resistor, having the same resistance in series with a noise voltage generator, as Figure 9.11b shows. For the noise source, its square amplitude density is ðresistorÞ 2 V ð f Þ ¼ 4 K T R n

ð9:12Þ

where K ¼ 1:374 10 23 J=K is the Boltzmann constant, T is the absolute temperature of the resistor and R is the resistance of the resistor. Moreover, the amplitude distribution of the voltage noise is Gaussian. More precisely, if we consider a frequency interval ð f1 ; f2 Þ, the observed voltage across the resistor is a random signal, having a Gaussian distribution with zero mean and standard deviation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi svn ¼ 2 K T R ð f2 f1 Þ ð9:13Þ

15 The thermal noise concept is wider than that of resistor noise. For instance, an antenna surrounded by a homogeneous gas at a temperature T picks up noise with a power density K T. It is well known that a directional antenna pointing towards the ground receives higher noise than the same antenna directed towards the sky. Also the noise floor of the Universe corresponds to the absolute temperature of 3 K.


305

R (a) Vn

R (b) R

(c)

in

Figure 9.11 Noise associated with resistors: (a) real resistor; (b) ideal resistor with a series voltage noise source; (c) ideal resistor with shunt current source.

As a consequence of Equations (9.9) and (9.12), we have that any real resistor produces noise, with an associated available power of Resistor noise power ¼ K T B

ð9:14Þ

where B ¼ f2 f1 is the bandwidth considered. Note that the noise power of a resistor depends only on the bandwidth, and not on any particular allocation of it. At the standard temperature of 17 C, corresponding to T ¼ 290:15 K, the resistor noise power density (i.e. the power within 1 Hz bandwidth) is 3:987 10 21 W=Hz, corresponding16 to 113.994 dBm/MHz. Figure 9.11c shows an alternative model for the noisy resistor, which is totally equivalent to the one in Figure 9.11b. Applying the Norton transformation to the latter network, we obtain the first, i.e. in ¼ vn R 1 . The current noise density for the network in Figure 9.11c is then ðresistorÞ 2 1 I ð f Þ ¼ 4 K T ¼ 4 K T G n R

ð9:15Þ

The equivalent network in Figure 9.11b (or c) applies to any resistor included in any circuit model of real capacitors, inductors and stubs, although any ideal purely reactive bipole is noise free. The resistor noise presents a flat voltage density over the frequency. Noise presenting such a characteristic is referred to as white noise. More complicated functions describe the noise associated with more complex devices, such as diodes, transistors, tubes, and so on. Active and nonlinear devices usually include noise contributions that are not white; the most relevant of these noise terms is the flicker noise, which presents a voltage noise density inversely proportional to the square root of the frequency.

9.4.2

Representation of noisy two-port networks

Any two-port network includes all the noise sources associated to all its components: their combination can be schematized with only two noise sources, either at the same port or at both ports of the network. In this regard, it must be considered that two correlated noise excitations are equivalent to a single one equal to their sum. In other words, two correlated noise sources follow the same superimposition rule as for the 16

The thermal noise density is usually rounded to 114 dBm/MHz.

306


usual deterministic excitations. Conversely, the contributions of non-correlated17 noise sources sum quadratically. The equivalent of two voltage (current) noise generators in series (parallel) having a voltage (current) density of Vn1 ð f Þ; Vn2 ð f Þ ðIn1 ðf Þ; In2 ð f ÞÞ is equivalent to a single generator having a voltage or current density of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi or jVn1 ð f Þj2 þ jVn2 ð f Þj2 jIn1 ð f Þj2 þ jIn2 ð f Þj2 The expression for the resulting source is purely real. This is not surprising, in that the only meaningful quantity associated with noise is the power or the normalized power. The effect of all the combined noise sources that could be present inside a two-port network is taken into account by adding a noise term to each equation of the network. Among the many possibilities, we will consider the impedance, admittance and ABCD matrices, which give respectively

V1 ¼ z11 I1 þ z12 I2 þ vn1 V2 ¼ z21 I1 þ z22 I2 þ vn2

ð9:16Þ

I1 ¼ y11 V1 þ y12 V2 þ in1 I2 ¼ y21 V1 þ y22 V2 þ in2

ð9:17Þ

V1 ¼ AV2 þ BI2 þ vn I1 ¼ CV2 þ DI2 þ in

ð9:18Þ

Figure 9.12a shows a noisy two-port network; Figures 9.12b–d show the resulting equivalent networks, obtained from Equations (9.16) to (9.18). All the noise-free two-port networks in Figure 9.12 have the same network parameters as the originating noisy two-port network in Figure 9.12a, apart from the noise, of course. This approach allows us to move the noise sources out of the network, with remarkably simple calculations.

9.4.3

Noise figure and noise factor

From a black-box point of view, the noise factor (F) is the parameter that characterizes the noise performances of the two-port in networks. Figure 9.13a shows a power generator at the input (port 1) of a noisy two-port network. The source model used in Figure 9.13a indicates the noise associated with the series resistance, which is inherently present within the generator. However, for our subsequent considerations a Norton – rather than a Thevenin – schematization is more convenient to use: Figure 9.13b shows the result. As usual, the components of the two schematizations in Figures 9.13a,b are related as Is ¼

Vs ; Rs þ jXs

Gs þ jBs ¼

1 Rs þ jXs

The network in Figure 9.13c derives from the one in Figure 9.13b, by applying the equivalent network in Figure 9.12d – which is based on the ABCD matrix – to the noisy two-port network. The noise-free

17 The hypothesis of non-correlated noise sources applies in most cases. So do all the resistors in a network – including the parasitic ones associated with other components. Sometimes, active device models use both correlated and non-correlated noise sources.

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN I1 V1

307

I2 V2

2

1 noisy network

(a)

I1

Vn1

V1

Vn2

I2

2

1

V2

noise-free network (b)

I2

I1 in1

V1

2

1

in2

V2

noise-free network (c) I1 V1

I2

Vn in

1

2

V2

noise-free network (d)

Figure 9.12 Representations of noisy two-port networks: (a) noisy two-port network; (b) representation of (a) based on the impedance matrix; (c) representation of (a) based on the admittance matrix; (d) representation of (a) based on the ABCD matrix.

two-port network in Figure 9.13c can be modelled as a controlled source with finite input and output impedance, as Figure 9.13d shows. As a first step in our analysis,18 we will determine the parameters of the equivalent noise-free twoport network of Figure 9.13d. The input admittance Yin , presented by that network to the load, is obtained from the ABCD matrix, combined with the condition imposed by the load impedance 8 V1 ¼ AV2 þ BI2 > ( > < V1 ¼ ð A Zload þ BÞI2 I1 D C Zload ) I1 ¼ CV2 þ DI2 ) ¼ Yin ¼ > V B A Zload 1 I1 ¼ ð C Zload þ DÞI2 > : V2 ¼ Zload I2

ð9:19Þ

18 The reader should keep in mind that all the network and noise parameters are functions of the frequency. The dependence is not explicitly indicated, in order to obtain more compact expressions.

308

MICROWAVE AND RF ENGINEERING noisy resistor Xs

Rs

Vnr

+ Vs

1

2

Zoad

2

Zoad

2

Zoad

noisy network (a)

Is

Bs

Gs

inr

1 noisy network

(b) Vn Is

Bs

Gs

inr

in

1 noise-free network

(c)

Vn Is

Bs

Gs

inr

in

Zout 1

Zin Vi

AvVi

2

Zoad

noise-free network (d)

Figure 9.13 Signal power source connected to a noisy two-port network: (a) Thevenin equivalent circuit for the power source; (b) Norton equivalent circuit for the power source; (c) noise representation based on the ABCD parameters; (d) quasi-unilateral equivalent of a noise-free two-port network.

Similarly, combining the ABCD matrix with the input termination due to the source admittance, we obtain the output impedance Zout of the noise-free two-port network 8 V1 ¼ AV2 þ BI2 > > < V2 Zs D B I1 ¼ CV2 þ DI2 ) AV2 þ BI2 ¼ Zs CV2 þ Zs DI2 ) Zout ¼ ¼ > A Zs C I2 > : V1 ¼ Zs I1

ð9:20Þ

Finally, the open-circuit transfer ratio Av is 8 > < V1 ¼ AV2 þ BI2 V2 1 I1 ¼ CV2 þ DI2 ) V1 ¼ AV2 ) ¼ Av ¼ > A V 1 : I2 ¼ 0 Note that the two-port network could be non-unilateral: this means that Yin and Av could depend on Zload .


309

The second step is to compute the contribution of each generator on the input voltage. The voltages produced by the different generators are 1 . Signal current: Vsignal;1 ¼ Is Yin þ Ys 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Resistor noise current: Vnr;1 ¼ 4 K T Gs Yin þ Ys Ys Vn Yin þ Ys 1 . Input noise current: V 0 n;1 ¼ In Yin þ Ys All these individual contributions must be summed using the rules described in Section 9.4.1. In this regard, we will assume that the resistor noise is not correlated with the two-port noise sources. For the two-port network current noise, we will assume that it is partly correlated with the voltage noise. Thus, that noise current is the sum of two terms, one correlated with the noise voltage and one uncorrelated .

Input noise voltage: Vn;1 ¼

In ¼ In;corr þ In;uncorr Furthermore, we can write that the correlated current equals the noise voltage, multiplied by the complex quantity Gg þ jBg, which has the dimension of an admittance; thus

In ¼ Gg þ jBg Vn þ In;uncorr Considering this assumption, and the expressions for the various noise terms, the resulting noise voltage across the input port is h 2 2 i 1 Vn;total 2 ¼ ð9:21Þ 4 K T Gs þ Ys þ Gg þ jBg jVn j2 þ In;uncorr 2 jYin þ Ys j The signal power source includes the thermal noise associated with its conductance; thus any physically existing signal generator is always noisy. The resulting signal to noise ratio (S=N) depends on the bandwidth considered. We can characterize the intrinsic S=N of the signal generator by assuming that it is terminated on its conjugated19 impedance, and considering a narrow bandwidth d f across the signal frequency. The S=N inherent in the power source is I 2 1 s 2 Gs ð9:22Þ S=Ninput ¼ K T df It is easy to see that the ratio (9.22) is independent of the source loading, therefore, in particular, it coincides with the input S=N of the network in Figure 9.13a. The effect of the two-port network on the noise can be characterized by computing S/N at the output, assuming the same narrow bandwidth d f : 2 1 Vsignal;1 2 A2 Zload Vsignal;1 2 v Zload þ Zout ReðZload Þ ð9:23Þ ¼ S=Noutput ¼ 2 Vn;total 2 d f 1 Vn;total 2 A2 Zload d f v Zload þ Zout ReðZload Þ

19 The conjugated matching allows the maximum power transfer ratio. Furthermore, under this condition, the voltage across (the current through) the load equals half of the open- (short-)circuit voltage (current) of the power source.

310


Note that the output S=N depends neither on the noise-free two-port network parameters, nor on the output impedance, in that these parameters affect both the signal and the noise by the same multiplying factor.The noise factor is defined as the ratio of the two ratios (9.22) and (9.23). It is important to clarify that the only input noise to consider is the thermal one associated with the real part of the generator’s immittance, ignoring any other noise contribution: F¼

Ys þ Gg þ jBg 2 jVn j2 þ In;uncorr 2 S=Ninput ¼ 1þ S=Noutput 4 K T Gs

ð9:24Þ

Due to non-correlation between the input and the network noises, the output S/N is always lower (i.e. worse) than the input one. Hence F 1, with the equality applying only for ideal noise-free networks, such as networks with all purely reactive elements. The noise factor (9.24) can be rewritten by using transformed quantities, in order to obtain a more compact and meaningful expression, particularly for high-frequency networks. Exploiting the expressions for the resistor noise (9.12) and (9.15), we can define one equivalent noise resistance for the voltage noise source and one equivalent noise conductance for the uncorrelated noise source Vn;uncorr 2 ¼ 4 K T Rn ;

In;uncorr 2 ¼ 4 K T Gu

ð9:25Þ

Substituting definitions (9.25) into the noise factor (9.24), we obtain F ¼ 1þ

2

2 i G u Rn h þ Gg þ Gs þ Bg þ Bs Gs Gs

ð9:26Þ

The noise factor depends on the noise parameters of the two-port network and on the source impedance. The minimum value of F is the one with the source admittance equal to its optimum value Ys;opt ¼ Gs;opt þ jBs;opt . In order to determine the optimum source admittance, we will begin by considering that the quantity (9.26) is the sum of four non-negative terms, namely 1;

2 Rn Gg þ Gs ; Gs

2 Bg þ Bs ;

Gu Gs

The third term is easily minimized (vanishes) by assuming Bs;opt ¼ jBg . Under this condition, the noise factor (9.26) simplifies to F ¼ 1þ

2 G u Rn Gg þ Gs þ Gs Gs

Now, the noise factor is still positive – more precisely, greater than 1 – and depends solely on the source admittance Gs . Moreover, F tends to infinity for Gs tending either to zero or to infinity. Therefore the optimum source conductance is obtained by minimizing (vanishing) the derivative of F with respect to Gs . The result is

2 Gu Rn rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d 1þ Gg þ Gs þ Gu R G G n s g Gu Gs Gs ¼ ¼ 0 if G ¼ G ¼ þ G2g s s;opt dGs G2s Rn Hence Ys;opt ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gu þ G2g jBg Rn

ð9:27Þ


311

The minimum noise factor of the two-port network is obtained by substituting the admittance (9.27) into expression (9.26)

Fopt ¼ 1 þ 2Rn Gg þ Gs;opt

ð9:28Þ

Finally, we can rewrite the noise factor in terms of the optimum admittance and minimum noise figure. Inverting Equations (9.27) and (9.28), Gs;opt ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gu Gu þ G2g ) ¼ G2s;opt G2g ; Rn Rn

Gg ¼

Fopt 1 Gs;opt 2Rn

Substituting the above expressions into Equation (9.26) we get F ¼ Fopt þ

2

2 i Rn h Gs Gs;opt þ Bs Bs;opt Gs

ð9:29Þ

Note that all three expressions (9.24), (9.26) and (9.29) are functions of six real variables: ðGs ; Bs ; Gg ; Bg ; jVn j2 ; jIn;corr j2 Þ, ðGs ; Bs ; Gg ; Bg ; Rn ; Gu Þ and ðGs ; Bs ; Gs;opt ; Bs;opt ; Rn ; Fopt Þ, respectively. Furthermore, the source conductance and susceptance are always present. In most cases, a logarithmic parameter is used to characterize the noise of two-port networks: that parameter is the noise figure (NF) defined as NF ¼ 10 log10 ðFÞ

ð9:30Þ

Another parameter used to describe the noise in two-port networks is the noise temperature Tnoise , obtained by considering the network as noise free, and attributing the increased output noise to a higher temperature of the input resistance. Figure 9.14a,b show the connections used for the definition of Tnoise . In both these figures the resistor R terminates the input port (1) of a two two-port network. In Figure 9.14a we have the two-port network of interest, with the resistor temperature at its physical value T. The twoport network in Figure 9.14b has the same parameters as the one in Figure 9.14a, but is noise free; its input resistor has the temperature Tnoise . The noise temperature is such that the output noise of both the combinations is the same, thus K T GA F ¼ K Tnoise GA

KT

R

1

(b)

2

K T GA F

2

K TnoiseGA

noisy 2-port network

(a)

K Tnoise

GA, F

R

1

GA, F=1

noise-free 2-port network

Figure 9.14 Setup for the noise temperature definition: (a) noisy two-port network with input resistor at the physical temperature; (b) noise-free two-port network with input resistor at the noise temperature.

312

MICROWAVE AND RF ENGINEERING Sout, Nout

Sin, Nin Zs

+ 1

Vs

GA, F

2

Zoad

(a)

PAV

GA

GA PAV K T B GA F

KTB (b)

Figure 9.15

K T B (GA F -1)

Noisy two-port network (a) and its equivalent block diagram (b).

Therefore Tnoise ¼ F T

ð9:31Þ

Observations: (a) The signal generator considered in the noise factor definition is quasi-ideal, in that it has a finite associated series resistance, and all its noise is due to that resistance. This is not always the case, because most of the real generators have higher associated noise. For instance, if we look at port 2 of any network in Figure 9.13, we can still see a signal generator with a finite series resistance, but its noise is higher than K T B. (b) The definitionof the noise factor– and thus of the noise figure– in terms of a ratio between S/N ratios could cause some misunderstanding. Therefore we must make it clear that the noise produced by any physical two-port network is additive rather than multiplicative. In other words, the two-port network adds noise to the thermal noise floor, and the first – although defined in terms of it – is independent of the latter. Figure 9.15 shows a useful schematization for noisy two-port networks. The configuration in Figure 9.15a comprises one signal power generator and one terminated noisy two-port network. For the available signal power, the two-port network presents at its output port (2) that of thegenerator PAV multiplied byits availablegain GA .On the other hand, independently of the available input noise density, the network adds its own contribution equal to K T ðGA F 1Þ. If the input noise density equals the thermal one, the total output noise is K T GA F. (c) Let us assume the generator in Figure 9.15a has a noise power density equal to K T while the twoport network is a resistive attenuator20 at the same temperature as the generator. Under these assumptions, looking at port 2 of the network, we can still see a power generator with non-zero series resistance at the same temperature as the input generator. Consequently, the attenuator output noise density is K T, as for the input generator. On the other hand, the available signal power at the attenuator output equals that of the input generator multiplied by the attenuator available gain, which is less than 1. If we compute the ratio between input and output S=N, we can 20

For instance, one of the networks described in Section 7.4.


Figure 9.16

313

Schematic for the computation of the noise factor of two cascaded two-port networks.

see that the noise factor of the attenuator equals the reciprocal of its available gain. Therefore, the noise figure of a resistive attenuator is equal to its attenuation.21 The same consideration also applies to many passive matched two-port networks, like lossy transmission lines, filters in their passband, and so on.

9.4.4

Noise factor of cascaded networks

An important application of the schematization in Figure 9.15 is in the analysis of cascaded noisy networks. Figure 9.16 shows two cascaded two-port networks. By definition, the noise factor Fcascaded of the resulting two-port network is the ratio between the input and output S/N, when the input noise is the thermal one Fcascaded ¼

Sin =Nin Sout =Nout

Travelling along the cascade, from the input of the first22 network to the output of the second, we have: .

Sin PAvailable ¼ Nin K T B where PAvailable is the available power of the generator;

.

.

S2 PAvailable GA;1 ¼ N2 K T B GA;1 F1 Sout PAvailable GA;1 GA;2

¼ Nout K T B GA;1 F1 GA;2 þ K T B GA;2 F2 1

Hence Fcascaded ¼ F1 þ

21

F2 1 GA;1

ð9:32Þ

Both these two positive quantities are expressed in dB. When two-port networks are cascaded, common engineering practice defines the network closest to the signal generator (load) as the first (last). 22

314


Equation (9.32) states that the noise factor of the second network contributes to the global noise factor by the term ðF2 1Þ=GA;1 that can be set below any given low value, by providing a sufficient gain of the first network. Under this condition, the noise factor of the first network mainly determines the global one; that is why, when two amplifier stages are cascaded, the stage with the lowest noise must be used as the first one.

9.4.5

Noise bandwidth

The noise can also be used to define a special characteristic of filters or, more generally, of linear two-port networks with frequency-dependent response, known as noise bandwidth (NBW). Figure 9.17 shows the setup for the definition of NBW. The noise generator NG is placed at the input of the filter FILT, which has the output terminated on the termination TERM. The output impedance of NG, the reference impedance of FILT and the impedance of TERM all have the same value Z0 . If Vn ðf Þ is the voltage noise density of NG, the available total noise power at the output of NG is PnAV

1 ¼ 4ReðZ0 Þ

1 ð

jVn ð f Þj2 d f 0 ðFILTÞ

The noise power transferred to the termination is affected by the filter transmission coefficient s21 PðTERMÞ n

1 ¼ 4ReðZ0 Þ

1 ð

ðFILTÞ 2 jVn ð f Þj2 s21 ð f Þ d f

as

ð9:33Þ

0

If Vn ðf Þ ¼ Vn0 is constant over the frequency (white noise), the total noise power on the termination is PðTERMÞ ¼ n

jVn0 j2 4ReðZ0 Þ

1 ð

ðFILTÞ 2 s21 ð f Þ d f

0

Let us consider the special case of an ideal filter, having the rectangular response

js021 ð f Þj ¼ 2

8 > ðFILTÞ > :0

ð9:34Þ

ðFILTÞ 2 where gðFILTÞ is the maximum s21 ð f Þ over the frequency.

If FILT has the response (9.34), the output noise power on the termination is 0

¼ PðTERMÞ n

jVn0 j2 4ReðZ0 Þ

1 ð

js0 21 ð f Þj d f ¼ 2

0

Pin NG

Figure 9.17

jVn0 j2 ðFILTÞ NBW g 4ReðZ0 Þ

Pout FILT

TERM

Setup for the noise bandwidth definition.

ð9:35Þ

0

1.0

-10

|s21|2

1.5

20 log10(|s21|) → 0.5

-20

315

20 log10(|S21|)


2

← |s21|

0.0 0.5

-30 1.5

1.0 f/f0

Figure 9.18

Response and noise bandwidth of a bandpass filter.

The generic noise power (9.33) coincides with the one in the case of the rectangular filter (9.35) if ðTERMÞ

Pn

ðTERMÞ0

¼ Pn

, which implies that NBW ¼

1 ðFILTÞ 2 max s21 ðf Þ

1 ð

ðFILTÞ 2 s21 ð f Þ d f

ð9:36Þ

0

In other words, the noise bandwidth of a given filter FILT is the passband of a rectangular filter, having a passband gain equal to the maximum of FILT and transferring the same noise power to the output. Definition (9.36) applies to any filter response with a finite passband extension, thus lowdpass and bandpass. In these two cases, the parameter f 0 0 is defined as: .

Low pass: f 0 0 ¼ NBW=2. In this way, the function (9.34) defines an ideal lowdpass response extending from zero (DC) to NBW.

.

Bandpass: f 0 0 ¼ f0 . Thus the ideal response (9.34) presents the same centre frequency as FILT.

Figure 9.18 shows the amplitude response in dB (grey curve, right y axis), the power response (black line, left y axis) of a bandpass filter, together with the associated ideal response23(9.34) (dashed curve, left y axis). The specific filter analyzed is a lumped third-order Chebyshev one, with 0.5 dB of passband ripple and a minimum insertion loss of 1 dB, thus gðFILTÞ ¼ 10 1=10 ffi 0:794. The relative equal-ripple bandwidth24 is Df = f0 ¼ Do=o0 ¼ 2=15 ffi 0:133, the relative 3 dB bandwidth is about 0.19 and NBW= f0 ffi 0:196. Thus, the 3 dB bandwidth is wider than the equal-ripple one – as expected – and the noise bandwidth is wider than both.

23

See the Mathcad file 06_Band_Pass_Noise_Bandwidth.MCD. The equal-ripple bandwidth is defined for lossy-free filters, while the 3 dB and the noise bandwidths require no assumption of filter dissipation loss. 24

316


x

Figure 9.19

1

fnon-linear (x)

2

y

Nonlinear instant relation between two variables.

9.5 Nonlinear two-port networks The majority of electrical networks are nonlinear, although their workings can be considered as linear for practical uses. The degree of nonlinearity in high-frequency networks depends on their type. Purely passive components25 exhibit nonlinear behaviour only when the applied power approaches the components’ safety limits. Components including semiconductors or other types of active devices could present significant nonlinear effects even if the input power is quite lower than the maximum. Other types of components, such as detectors, mixers and frequency multipliers, are inherently nonlinear and base their workings on the nonlinearity itself. Whether the nonlinear products are undesired side effects, or constitute the main reason for the component working, some general rules are the basis for both the analysis and design of components. This section deals with instantaneous nonlinear two-port networks, presenting some general concepts and definitions, useful for analysis and design at both the component and system levels. An important property of linear networks is that their response to a sinusoidal excitation is a sinusoid with the same frequency. For instance, if we apply a sinusoidal voltage across a linear bipole, the resulting current is sinusoidal and has the same frequency as the voltage. Conversely, the response of nonlinear networks to sinusoidal excitations is not sinusoidal, as we will see shortly. We can consider the excitation as the input signal of the network, and the response as the output. The case of the two-port network, with the wave incident on one port considered as the input and the wave reflected from the other port as the output, is particularly interesting for high-frequency applications. Given a linear two-port network, if the incident wave on port 1 is a1 ðtÞ ¼ a1;0 cosð2pf0 tÞ then the resulting reflected wave to port 2 is b2 ðtÞ ¼ a1;0 js21 ð f0 Þjcosf2pf0 t þ arg½s21 ð f0 Þg Therefore, there is a frequency-dependent relation between the input and output amplitude and the phase. We define a network as instantaneous if the relation between the input and output is independent of the frequency and the added phase is zero. In our example, the two-port network is instantaneous if js21 ð f0 Þj and arg½s21 ð f0 Þ are independent of f0 and arg½s21 ð f0 Þ ¼ 0. In general, high-frequency two-port networks are simultaneously non-instantaneous and nonlinear. Despite this, a wide approach to analysis and synthesis is based on the separation of the network into simpler subnetworks of two types: linear non-instantaneous or nonlinear instantaneous ones. Here, we will limit consideration to nonlinear instantaneous networks of one single input and one single output. Figure 9.19 shows a block diagram representation of a nonlinear input/output relation. The letter x denotes the input or the independent variable, while the output or the dependent variable is indicated with y. The scheme of Figure 9.19 is quite general, in that x and y could represent voltages, currents, incident or

25

Like those described in Chapter 6.


317

reflected waves. Moreover, those quantities could be at the same port of a bipole or at different ports of a two-port network. As is well known, under some conditions26 which we will assume to be satisfied, a nonlinear function can be approximated by the Taylor series fnonlinear ðxÞ ffi

n X

ck ðx x0 Þk ¼ c0 þ c1 ðx x0 Þ þ c2 ðx x0 Þ2 þ c3 ðx x0 Þ3 þ

ð9:37Þ

k¼0

with ck ¼

1 d k fnonlinear ðxÞ k! dxk x¼x0

The Taylor approximating polynomial (9.37) has a wide range of applicability in high-frequency nonlinear networks. One of the few cases where it does not work is when abruptly quasi-discontinuous nonlinear functions have to be considered. Therefore, very general nonlinear functions can be studied by means of their Taylor approximating polynomials. Subsequent subsections explore the results of sinusoidal excitations of nonlinear networks, exploiting the series expansion in (9.37).

9.5.1

Harmonic and intermodulation products

The present section examines the output of a nonlinear instantaneous network when the input is a single sinusoid or the sum of two sinusoids of different frequency. The trigonometric identities (A.76) to (A.79) show that the expression ½a cosðaÞ þ b cosðbÞk includes terms of the type am bn cosðma nbÞ with m; n ¼ 0; 1; . . . ; k; m þ n k , at least for k 5. Moreover, it is possible to demonstrate that such an assertion holds true for any k, although nonlinearities with order higher than 5 are of limited interest in this context. Equation (9.37) states that the output of the generic nonlinear block in Figure 9.19 is the sum of some powers of the input, multiplied by a constant coefficient. Therefore, if the input is a pure sinusoid, or a combination of two sinusoids, substitution of expressions (9.34) to (9.36) into the function (9.33) results in a combination of many sinusoidal components. Sections 9.5.2 and 9.5.3 will examine these two cases in more detail.

9.5.2

Harmonic distortion

We will consider here the nonlinear response to a purely sinusoidal excitation, showing that the resulting waveform includes components with frequency equal to multiple integers of the one in the excitation. Let us examine the first four terms of Equation (9.37) when the input is a sinusoidal term x ¼ V1 cosðo1 t þ j1 Þ

ð9:38Þ

26 Let f ðxÞ be a function defined over an interval A. For a given x0 2 A let the nth derivative d n f ðxÞ=dxn exist for x ¼ x0 and, furthermore, d n 1f ðxÞ=dxn 1 exist for any x 2 A. Then the following identity applies:

f ðxÞ ¼

n X 1 k d f ðxÞ=dxk x¼x0 ðx x0 Þk þ Rn ðx x0 Þ k! k¼0

where the remainder Rn ðx x0 Þ is an infinitesimal term of order higher than ðx x0 Þn or, equivalently, lim ½Rn ðx x0 Þ=ðx x0 Þn ¼ 0

x ! x0

318


The terms for the second member of expression (9.37) coincide with the first one of the identities (A.82) to (A.84), if we assume a ¼ V1 ; a ¼ o1 t þ j1 ;

b¼0

ð9:39Þ

Substituting (9.39) into (9.37), truncated at the fourth-degree term, and applying the identities (A.82) and (A.85), we get

1 3 3 yðtÞ ffi c0 þ c2 V12 þ c4 V14 þ c1 V1 þ c3 V13 cosðo1 t þ f1 Þ 2 8 4

1 1 1 þ c2 V12 þ c4 V14 cos½2ðo1 t þ f1 Þ þ c3 V13 cos½3ðo1 t þ f1 Þ 2 2 4 þ

1 c4 V14 cos½4ðo1 t þ f1 Þ 8

ð9:40Þ

The first, second, third, fourth and fifth terms of Equation (9.40) are the DC, fundamental, second harmonic, third harmonic and fourth harmonic, respectively. More generally, considering all the terms in the series (9.37), it is possible write yðtÞ ffi

n X

hk cos½ðo1 t þ j1 Þk

ð9:41Þ

k¼0

Thus, the response of a nonlinear network to a sinusoidal excitation is a periodic waveform having the same fundamental frequency as the excitation, but including new sinusoidal components whose frequency is an integer multiple of the fundamental. Those multiple frequency components are the harmonics of the sinusoidal excitation: their frequency is double, triple, quadruple, or any integer multiple of the input one. These distortion products are referred to as the second, third, fourth or the nth harmonic, respectively. Truncation of the series (9.37) to the kth-order term results in a waveform with harmonics up to the same order k. Inspection of expression (9.40) suggests that, for small values of the excitation amplitude V1 , the amplitude of the kth-order harmonic tends to zero as V1k . This assertion holds true for any considered maximum degree in the powers of the series (9.37). For instance, when truncating to fourth order, the amplitudes of the various harmonics, from zero to the fourth, as in Equation (9.40), are: 1 3 . h0 ¼ c0 þ c2 V 2 þ c4 V 4 1 1 2 8 3 3 . h1 ¼ c1 V1 þ c3 V 1 4 1 1 2 . h2 ¼ c2 V þ c4 V 4 1 1 2 2 1 . h3 ¼ c3 V 3 1 4 1 . h4 ¼ c4 V 4 1 8 For all these amplitudes, we have that hk tends to zero as the kth power of the input peak amplitude V1k . Figure 9.20 shows the spectra of the pure sinusoidal input and of the distorted output. Note also that the odd (even) degree term ck xk of the power series (9.33) creates odd (even) harmonics, and affects the amplitude of all the odd (even) harmonics up to order k. In particular, evenorder nonlinearities produce DC components (zero-order harmonic). Furthermore, a generic function f ðxÞ is defined as even or odd if f ðxÞ ¼ f ð xÞ or f ðxÞ ¼ f ð xÞ, respectively. In even (odd) functions having the polynomial form (9.37), only the even-order (odd-order) terms are nonzero. Therefore, even and odd nonlinear characteristics produce harmonics of the same type.


319

x,amplitude

fundamental

O

Frequency

f1 = ω1/2π y,amplitude fundamental 3rd harmonic

DC

2nd harmonic 4th harmonic 5th harmonic f1 = ω1/2π

O

Figure 9.20

9.5.3

2 f1

3 f1

4 f1

5 f1

Frequency

Spectrum of a pure sinusoidal input (x) and the corresponding distorted output (y).

Intermodulation distortion

After considering the nonlinear response to a pure sinusoid, we next examine what happens with a dual tone excitation. Because of the involved complexity of the expressions, we will limit our developments to third order. Nevertheless, some qualitative considerations will be exposed in general cases. A dual tone excitation corresponds to the superimposition of two sinusoidal waveforms, each with arbitrary amplitude, frequency and phase. The relative expression is x2 tone ðtÞ ¼ V1 cosðo1 t þ f1 Þ þ V2 cosðo2 t þ f2 Þ

ð9:42Þ

Again, substituting expression (9.42) into (9.37) – truncated at the term with degree 3 – and applying the identities (A.76) to (A.77), we obtain

1 1 3 3 y2 tone ðtÞ ffi c0 þ c2 a2 þ c2 a2 þ c1 a þ c3 a2 b þ c3 a3 cosðaÞ 2 2 2 4

3 3 þ c1 þ c3 a2 b þ c3 ab2 cosðbÞ þ c2 ab cosða þ bÞ þ c2 ab cosða bÞ 2 4 þ

3 3 3 c3 ab2 cosða þ 2bÞ þ c3 ab2 cosða 2bÞ þ c3 a2 b cosð2a þ bÞ 4 4 4

þ

3 1 1 c3 a2 b cosð2a bÞ þ c2 a2 cosð2aÞ þ c2 b2 cosð2bÞ 4 2 2

þ

1 1 c3 a3 cosð3aÞ þ c3 b3 cosð3bÞ 4 4

where a ¼ V1 ; a ¼ o1 t þ f1 ; b ¼ V2 ; b ¼ o2 t þ f2 .

ð9:43Þ

320


The output waveform (9.43) is the sum of a DC component plus 12 sinusoidal terms, with their frequencies related to the input ones. We can find: 1. Six terms consisting of two sinusoids at the same frequency as the input, together with the respective second and third harmonics ðo1 ; o2 ; 2o1 ; 2o2 ; 3o1 ; 3o2 Þ. 2. Two terms having frequency equal to the sum of and the difference between the input ones ð o1 o2 ; o1 þ o2 Þ. These terms are generated by the quadratic term of the series (9.37) and their amplitude c2 V1 V2 is proportional to both the amplitudes of the input tones. They are denoted as second-order intermodulation products (IM2). 3. Four terms with their frequencies equal to the sum of one input frequency, plus or minus the second harmonic of the other. The relative frequencies are ð o1 þ 2o2 ; o1 2o2 ; 2o1 þ o2 ; 2o1 o2 Þ, with the respective amplitudes 3 4 1 c3 V1 V22 ; 3 4 1 c3 V1 V22 ; 3 4 1 c3 V12 V2 ; 3 4 1 c3 V12 V2 . These components are generated by the cubic nonlinearity, and are denoted as third-order intermodulation products (IM3). The intermodulation (or intermodulation distortion) is denoted by the acronym IMD. Generalizing the considerations of points 2 and 3, we have that a nonlinearity of order k produces intermodulation products of kth order. The respective expressions are constants multiplying terms of the type V1m V2k m cos½m ðo1 t þ j1 Þ ðk mÞ ðo2 t þ j2 Þ where m ¼ 1; 2; . . . ; k 1. Figure 9.21

x, amplitude

O

Frequency

f1 f2 y, amplitude

f1 f2

O f2 f1 2 f1 f2

2 f2 f 1

2 f1 2 f2 f2 f1

3 f1

3 f2

Frequency

2 f1 f2 2 f2 f1

Figure 9.21 Spectrum of a dual tone signal with intermodulation. Black lines with solid arrows are the original components of the signal, grey lines are the second-order products, and black lines with open arrows are the third-order products.


321

shows the intermodulation spectrum obtained from a nonlinear characteristic with second- and thirdorder terms.

9.5.4

Gain compression

So far, we have considered generic nonlinear characteristics and derived the implications of the nonlinearities themselves on combinations of sinusoidal excitations. Now, we will examine the special and important case of the black box of Figure 9.19 representing a two-port network. Furthermore, we will assume that both the input and the output of that network are impedance matched, in order to consider the nonlinear effects separately from the mismatching ones. Under these assumptions, the input and output variables ðx; yÞ can be assumed to represent the input and output voltages of the twoport network. Finally, the two-port network considered is basically linear; the nonlinearities are side effects with increasing importance with high-level signals. Typical examples of such types of networks are amplifiers,27 but also switches, or, more generally, all the control components28 using semiconductor devices like variable attenuators and phase shifters.29 Thus our two-port networks will be excited with sinusoidal signals, and the main output spectral components will be those at the same frequency as the input. For brevity, we will refer to the two-port networks described above as quasilinear. Almost all quasi-linear networks exhibit a compression characteristic, in that for low-amplitude sinusoidal input signals the output is still sinusoidal with the same frequency and amplitude proportional to the input one. Although we did not give a precise definition of low amplitude, we will refer to this working condition as a small signal. In quasi-linear networks under small-signal operation and single tone excitation, the level of the nth harmonic is proportional to the input level raised to the nth power, as in the coefficients hk listed in Section 9.5.2. As the input amplitude rises above a certain limit, the fundamental component of the output increases less than the input, and the harmonic level no longer follows the law described above. This condition is defined as the compression region of the two-port network operation. At high compression level the fundamental output amplitude tends to a constant, almost independently of the input one, defined as the saturation amplitude. Increasing the input amplitude will cause first a decrease in the output fundamental below the saturation amplitude – such a condition represents the overdrive operation – and then permanent damage to the two-port network. As a general rule, it is quite difficult to predict the harmonic amplitudes under high compression and, for stronger reasons, under saturation or overdrive operation. Let us begin by considering a polynomial characteristic of degree 3 fthird order ðxÞ ¼ c0 þ c1 x þ c2 x2 þ c3 x3 As in Section 9.5.2, the coefficient c0 produces a DC output voltage, independently of the input signal. Since real high-frequency quasi-linear networks are normally AC coupled, the eventual DC component is not transmitted. Therefore, we can assume c0 ¼ 0 with no lack of accuracy:

c2 c3 ð9:44Þ fthird order; no DC ðxÞ ¼ c1 x þ c2 x2 þ c3 x3 ¼ c1 x þ x2 þ x3 c1 c1 We can immediately recognize that for sufficiently small values of x, the first-order term dominates the remaining ones, thus lim

x!0 27

fthird order; no DC ðxÞ ¼ c1 x

See Chapter 11. See Chapter 10. 29 Other types of networks (like detectors, mixers and frequency multipliers) are inherently nonlinear, in that their working principles rely on nonlinear effects. 28

322

MICROWAVE AND RF ENGINEERING y=x+c3x3, c3>0

ynon-linear(x)

-0.5

3

0.5

c1 (-c3)

y=x

-0.5

x O

-0.5

3

0.5

c1 (-c3)

-0.5

y=x+c3x3, c3 0; c3 < 0 (thin black dashed line) and c1 > 0; c3 > 0 (thin black solid line). Note that the first case presents a negative compression, in that increasing input amplitude causes an increasing output to input amplitude ratio. 30

Sometimes, SSG is defined in terms of logarithmic units: SSG ¼ 20 log10 ðssgÞ.


323

Note also that for positive x, our curve ( c1 > 0; c3 < 0 ) reaches a maximum, then monotonically decreases up to zero and below. Moreover, the curve is anti-symmetrical (or odd). We can find the position of the maximum and minimum of the function from the vanishing derivative

d fthird order; no DC ðxÞ c3 2 d fthird order; no DC ðxÞ ¼ c1 1 þ 3 x ¼ 0 ) x ¼ Vcritical ð9:48Þ dx dx c1 where Vcritical

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 ¼ 3 c3

Note that the argument of the square root is positive, because c1 and c3 have opposite sign. The relative maximum and minimum of the function are obtained by substituting the derivative zeros (9.48) into the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi function itself. The resulting values are ð 3=4Þc1 c1 =c3 . The cubic function represents the quasi-linear characteristic in a realistic way, provided that the input signal instantaneous amplitude does not exceed the limits (9.48). For higher absolute values, it is more pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi realistic to replace the function with its local maximum and minimum for x > c1 =ð3c3 Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x < c1 =ð3c3 Þ, respectively. The new definition of the nonlinear transfer characteristic is then 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c1 > 3 > > x þ c x j x j c 1 3 > > 3 c3 > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 1 c1 1 c1 fsaturating ðxÞ ¼ ð9:49Þ c1 x > > c 3 3 3 c3 3 > > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 1c 1c > > : c1 1 x < 1 3 3 c3 3 c3 Figure 9.22 also shows the curve (9.49), represented by the thick black line. Both the functions (9.47) and (9.49) are anti-symmetrical (or odd), i.e. f ðxÞ ¼ f ð xÞ. Therefore they produce only even-order distortion; in other words, they distort the positive half wave of the input signal in the same way of the negative one.31 Figure 9.22 also shows the function (9.47) and its saturated counterpart, after the addition of a small second-order coefficient. The result is a slightly non-symmetrical transfer characteristic, with consequent generation of second-order distortion, without significantly affecting the compression performances, as we will see shortly. The function (9.49) is continuous in all of its derivatives (which are non-zero up to third order) within the interval ½ Vcritical ; Vcritical , where Vcritical is defined by Equation (9.48). Within that interval, the function satisfies the requirements for the expansion in a Taylor series, and the series coincides with the originating polynomial c1 x þ c3 x2 . The higher order derivatives of the function (9.49) present a discontinuity at jxj ¼ Vcritical , vanishing for jxj > Vcritical and also being non-zero32 for jxj Vcritical . Outside the above-mentioned interval, the Taylor expansion is therefore no longer applicable. Nevertheless, in many cases, it is still possible to find approximating polynomials for a saturating characteristic of the type (9.49), by applying curve-fitting methods.33 This possibility extends the applicability of the considerations of Sections 9.5.1 to 9.5.4 to the characteristic (9.49). However, if the input signal remains pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi within the limit jxj c1 =ð3c3 Þ, the curve (9.49) is not distinguishable from its originating thirdorder polynomial. 31

See the curves in Figure 9.23, obtained for a purely sinusoidal waveform. For instance, d 2 ðc1 x þ c3 x3 Þ=dx2 ¼ dðc1 þ 3c3 x2 Þ=dx ¼ 6c3 x, which is non-zero 8x „ 0. Conversely, for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jxj > c1 =ð3c3 Þ, the function (9.49) is constant; thus all its derivatives – including the second one – vanish. 33 See the Mathcad file 07_Saturation_Polynomial_Curve_Fit.MCD. 32

324

MICROWAVE AND RF ENGINEERING y(t) ysat 2 V=∞ (square wave)

4 1

V/Vcritical=0.5

ωt=2π

ωt=π

-ysat

Figure 9.23 Output waveforms resulting frompincreasing signals applied to the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amplitude sinusoidal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nonlinear characteristic (9.48)ð Vcritical ¼ 3 0:5 c1 =c3 ; ymax ¼ 2=31:5 c1 c1 =c3 Þ. After clarifying the limitations on the applicability of the function (9.44), we can reconsider the amplitude of the output fundamental signal (9.46). Its explicit expression is h1 ¼ c1 V þ

3 c3 V 3 4

ð9:50Þ

The amplitude (9.50) is zero for two values of the input amplitude, precisely V ¼ 0 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ ð4c1 Þ=ð3c3 Þ. The first of these zeros is quite obvious, corresponding to no input signal. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi However, for positive input amplitudes, 0 < V < ð4c1 Þ=ð3c3 Þ, the fundamental output amplitude h1 initially increases up to a maximum, and then decreases down to zero. The abscissa of that maximum coincides with the positive zero of the derivative of the function (9.50) with respect to V: dðh1Þ 2 ¼0)V¼ dA 3

rffiffiffiffiffiffiffiffiffiffiffi c1 c3

ð9:51Þ

The input amplitude (9.51) slightly exceeds34 the limits (9.48) for the validity of the characteristic (9.44). Thus, we can give three different estimates of the maximum fundamental output amplitude associated with the characteristic (9.48): 1. The most conservative one is based on the maximum validity range of the polynomial (9.44), and gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 1 c1 c1 3 c1 ¼ h1max;1 ¼ h1 V ¼ 3 c3 c3 4

34

The maximum input amplitude is obtained from the vanishing derivative of the output fundamental amplitude, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and is V ¼ ð2=3Þ c1 =c3 ffi 0:667 c1 =c3 , while the range (9.48) gives V c1 =ð3c3 Þ ffi 0:577 c1 =c3 .


325

2. A more realistic value, assuming the input amplitude (9.51), is rffiffiffiffiffiffiffiffiffiffiffi

rffiffiffiffiffiffiffiffiffiffiffi 2 c1 4 c1 ¼ c1 h1max;2 ¼ h1 V ¼ 3 9 c3 c3 3. The third is obtained by using the function (9.49) instead of (9.44). Expression (9.50) is no longer valid. The maximum amplitude of the output fundamental spectral component must be calculated by more complicated considerations. Figure 9.23 shows the output waveform generated by the nonlinear characteristic (9.49) when its input signal is sinusoidal with increasing amplitude. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The input and output normalizing parameters used for that graph are Vcritical ¼ c1 =ð3c3 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ysat ¼ ð2c1 =3Þ c1 =ð3c3 Þ respectively. It can be seen that, for very high input amplitude ðV ! 1Þ, the output signal tends to approximate a square wave having peak amplitude pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2c1 =3Þ c1 =ð3c3 Þ. The amplitude of the fundamental component of that waveform is 4=p times its peak amplitude, therefore the saturated fundamental amplitude becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1max;3 ¼ ½ð8c1 Þ=ð3pÞ c1 =ð3c3 Þ. Determinations 1 and 2 are quite close, while 3 is a little higher

20 log10 h1max;2 =h1max;1 ffi 0:226 dB; 20 log10 h1max;3 =h1max;1 ffi 1:075 dB As discussed, it is possible to find different values for the amplitude of the fundamental spectral component of the output signal in saturated operation. Moreover, the most optimistic value h1max,3 is calculated by assuming that the input amplitude can be arbitrarily increased without damaging the network, and this is not true, of course. A more significant and more univocally determinable parameter is the so-called 1 dB compression point (1dBCP). As seen many times within this section, for increasing input amplitudes V the output fundamental h1 increases less than proportionally to V; rather, the ratio h1=V decreases monotonically with V. Equation (9.47) defines the large signal gain G1 . The third member of that equation is obtained by assuming the characteristic (9.44). Since we assumed c1 > 0; c3 < 0 , G1 decreases with V until the argument of the logarithm is positive. The difference between large- and small-signal gain (expressed in dB) is a meaningful indicator of how much the network is operating under nonlinear conditions:

3 c3 2 SSG G1 ¼ G1 ðV ¼ 0Þ G1 ¼ 20 log 1 þ V 4 c1 The input amplitude which causes the large-signal gain to decrease by 1 dB with respect to the smallsignal one is, by definition, the input 1 dB compression point amplitude. From the definition, it follows immediately that V1dBCP ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 1=20

4 c1 c1 10 1 ffi 0:3808 c3 3 c3

ð9:52Þ

Note that the value (9.52) is within the valid range of the polynomial (9.44). The output fundamental amplitude h11dBCP corresponding to the input V1dBCP is the output 1 dB compression point amplitude. Substituting the value (9.52) into expression (9.50), we obtain h11dBCP ¼ h1ðV ¼ V1dBCP Þ ffi 0:339 c1

rffiffiffiffiffiffiffiffiffiffiffi c1 c3

ð9:53Þ

326


Our simplified cubic and cubic-saturated models do not allow a unequivocal definition of the saturated output level; nonetheless, it is still possible to find relations between the saturated and the 1 dB compression point

h1max;1 ffi 2:13 dB; 20 log h11dBCP

20 log

h1max;2 h11dBCP

ffi 2:35 dB;

20 log

h1max;3 h11dBCP

ffi 3:2 dB

Note that, whatever the definition of the saturated amplitude, the difference between the saturation and the 1 dB compression level is independent of the polynomial coefficients. The assumption 20 logðh1max =h11dBCP Þ 2:5 dB usually gives reasonable accuracy. Most frequently, the compression parameters are used for power – normally expressed35 in dBm rather than voltages. Although we defined the saturated amplitude and the 1 dB compression points in terms of voltage amplitude, a simple relation exists between the two definitions:

1 h12max . Saturated output power, Psat;dBm ¼ 10 log þ 30 10 2 R0 ! 2 1 V1dBcp . Input 1 dB compression power, I1 dBdBm ¼ 10 log þ 30 10 2 R0 2

.

Output 1 dB compression power, O1 dBdBm ¼ 10 log10

1 h11dBcp 2 R0

! þ 30

In these expressions, R0 is the matching resistance of the two-port quasi-linear network. The factor 1=2 is present in the argument of the logarithm because h1; V are peak voltage amplitudes, while the powers are RMS. The additional term þ 30 is to convert the power into dBm.

9.5.5

Intercept points

Compression on the output signal, generation of harmonics, and intermodulation products are phenomena related to the nonlinear nature of a two-port network. In principle, all three phenomena could be used to characterize the network from a nonlinear point of view. However, we have to consider that the model of Figure 9.19 alone is accurate in a few cases of high-frequency two-port networks. The main reason for this lack of accuracy is in the limited bandwidth of the physically existing quasi-linear networks. Looking at the output spectrum in Figure 9.20, we can recognize that the second harmonic falls within the network nominal bandwidth only if its extension is no smaller than an octave: wider bandwidths are required for not filtering out the higher order harmonics. Similar considerations apply to the IM2 products, shown in Figure 9.21. Differently from that, IM3 products fall within the component bandwidth – for any extension of it – providing that f1, f2 fall within the bandwidth and are sufficiently close.36 Therefore, the two-tone excitation can ’ be used to characterize the nonlinear performance of narrow-bandwidth networks. The power in dBm coincides with 10 times the decimal logarithm of the power in mW: PdBm ¼ 10 log10 ðPmW Þ: For instance, let us consider a narrow-band amplifier whose working bandwidth is 0.95–1.05 GHz. Any in-band signal has the second harmonic within the range 1.0–2.1 GHz, well out of the amplifier bandwidth; the higher order harmonics are out of band, for even stronger reasons. The IM2 ðf1 þ f2 Þ product minimum frequency is a little higher than 1.9 GHz ðf1 ¼ 0:95 GHz; f2 ’ 0:95 GHzÞ. The IM2 ðf1 f2 Þ product maximum frequency is 0.1 GHz ðf1 ¼ 0:95 GHz; f2 ¼ 1:05 GHzÞ. Hence all the second-order distortion products are strongly attenuated at the output. Differently, by choosing ð f1 ¼ 0:99 GHz; f2 ¼ 1:01 GHzÞ, some of the IM3 products fall within the network bandwidth, indeed 2f1 f2 ¼ 0:97 GHz and 2 f2 f1 ¼ 1:03 GHz. 35 36


327

A typical dual tone test uses two narrow-spaced sinusoidal signals having the same amplitude. Moreover, in this context we are not interested in the phase of the signals. Hence, the input signal assumes expression (9.42) with the simplifications V1 ¼ V2 ¼ V;

j1 ¼ j2 ¼ 0

The second simplification is always possible, with a proper choice of the time origin, providing that the phases j1 ; j2 are constant. The input signal is x2-tone ðtÞ ¼ V ½cosðo1 tÞ þ cosðo2 tÞ From Equation (9.43), we obtain the IM3 products 3 IM3 ¼ c3 V 3 cos½ðo1 þ 2o2 Þt þ cos½ðo1 2o2 Þt þ cos½ð2o1 þ o2 Þt þ cos½ð2o1 o2 Þt 4 ð9:54Þ All the four IM3 products have the same amplitude, proportional to the cube of the input one; the two underlined terms are the in-band ones. Expression (9.43) has been derived assuming a cubic characteristic of the network. However, all the odd-order terms generate products having frequency equal to o1 2o2 and 2o1 o2 , with amplitude proportional to c5 V 5 ; c7 V 7 . . . . As we know, at low amplitudes the lower order terms dominate. Therefore Equation (9.54) supplies a low-amplitude approximation of the amplitude of the spectral lines at o1 2o2 and 2o1 o2 . Let us focus on one of these two components, with amplitude 3 ð9:55Þ im3 ¼ c3 V 3 4 Figure 9.24 shows the typical compression curves of a quasi-linear two-port network. The power compression curve (thick black) is the fundamental power versus the input one, under a single tone excitation. The IM3 curve (thick grey) is the power of one of the two spectral lines, o1 2o2 or 2o1 o2, versus the power of one of the two equal-amplitude input tones. This curve is close to the

IP2

Output power, 10 dB/div.

1 dB

IP3

1dBCP

fundamental

IM2

IM3 Input power, 10 dB/div.

Figure 9.24 Compression curves of a quasi-linear two-port network.

328


approximation (9.55) (thin black) up to the 1 dB compression point. Figure 9.24 also shows the IM2 curve (thick light grey), although this is less interesting because of the considerations at the beginning of this section.37 Furthermore, Figure 9.24 includes the small-signal approximations (thin black) of the three curves, extended up to high input power operation. An interesting parameter for characterizing the nonlinear performances of a two-port network is the third-order intercept point (IP3), defined as the hypothetical point where the small-amplitude IM3 crosses the fundamental one. The input amplitude VIP3, making the amplitude (9.54) equal to the smallsignal fundamental out amplitude, is sffiffiffiffiffiffiffiffiffiffi ffi 4 c1 ð9:56Þ im3 ¼ c1 V ) VIP3 ¼ 3 c3 From the amplitude (9.55), we can easily define the corresponding powers, expressed in dBm: .

Input third-order intercept point

IIP3dBm ¼ 10 log10 .

Output third-order intercept point OIP3dBm ¼ 10 log10

2 1 VIP3 þ 30 2 R0

2

1 VIP3 þ 20 log10 jc1 j þ 30 2 R0

The factor 1=2 is for the peak-to-RMS conversion, and the additional term þ 30 is peculiar to the dBm unit, as explained in the 1 dB compression point definitions. In a similar way, it is possible to define the second-order intercept points (IIP2, OIP2), although they are less interesting, particularly for narrow-band networks. From the values of (9.56) and (9.52) it is possible to find a relation between IIP3 and I1dB !

V1dBcp 0:3808 20 log10 ð9:57Þ ¼ 20 log10 pffiffiffiffiffiffiffiffi ¼ IIP3 I1dB ffi 9:6 dB VIP3 4=3

9.5.6

Saturation and intercept point of cascaded two-port networks

Sections 9.2 and 9.4.4 dealt with the performances of two cascaded two-port linear networks, in terms of S parameters and noise performances. Here, after defining the principal nonlinear parameters of quasilinear networks, we will examine the nonlinear performances of cascaded networks. Figure 9.25 shows two two-port networks (A and B) in cascade. Each of the two networks consists of three cascaded networks,: two of them linear with a nonlinear one in between. The schematization used in Figure 9.25 somehow resembles the schematic of a single transistor amplifier.38 Looking at network A, the linear network AðA;IÞ and AðA;0Þ could be the input and output matching networks of the amplifier. ðA;IÞ P ðAÞ k and AðA;0Þ could be the nonlinear controlled The nonlinear network k ck x between A generator, contained within the nonlinear model of the transistor.39 The similarity between the schematization in Figure 9.25 and a single transistor amplifier can be increased by considering each reactive linear element of the transistor model as embodied in one of the two of AðA;IÞ and AðA;0Þ .

37

See also Section 9.5.4. See Chapter 11, in particular Section 11.2. 39 See Section 9.6. 38

BASIC CONCEPTS FOR MICROWAVE COMPONENT DESIGN Network A

Network B

[S(A,l)] IN

1

329

2

[S(A,O) ]

(A)

Σk ck x k 2 1

2

1

Figure 9.25

[S

(B,I)

(B)

]

2

1

Σk c k x k 2 1

[S

(B,O)

]

2

1

OUT

Cascaded quasi-linear two-port networks.

The model in Figure 9.25 is more general than a single transistor amplifier, because it can represent any nonlinear non-instantaneous two-port network. On the other hand, networks A and B in Figure 9.25 are purely unilateral, i.e. there is no backward transmission from the output to the input, although it is possible to remove this limitation by making the model more complicated. In any case, the combination of the three subnetworks modelling network A can be used to model: . The input matching, under small-signal operation, by means of AðA;IÞ . . The output matching, under small-signal operation, by means of AðA;0Þ . . The ripple on SSG, by means of AðA;IÞ and AðA;0Þ . . . .

ðAÞ

The distortion performances, through the coefficients ck .

The dependence of the compression characteristic on the frequency, by means of AðA;0Þ . The attenuation of the out-of-band distortion products, by means of AðA;0Þ .

Conversely, the representations in Figure 9.25 cannot model: .

The variation of the input and output matching with the signal amplitude.

.

The variation of the transmission coefficient phase with the signal amplitude, known as amplitude modulation to phase modulation (AM–PM) conversion.

Despite these limitations, the model in Figure 9.25 is sufficiently accurate in the majority of practical cases. In subsequent considerations, we will simplify the analysis of the configuration in Figure 9.25 by ðA;IÞ

assuming perfect input and output matching: s11

ðA;OÞ

¼ s22

ðB;IÞ

¼ s11

ðB;OÞ

¼ s22

¼ 0. For the dependence of ðAÞ

ðBÞ

the nonlinear performances on the frequency, we will consider the coefficients ck ; ck as functions of ðA;OÞ

the frequency, although not explicitly indicated, while s21 ðA;OÞ s21

ðB;OÞ

¼ s21

¼ 1. This last assumption is

ðB;OÞ s21

equivalent to considering ck;A ; ck;B as constant and ¼ as functions of the frequency if the phase of those transmission coefficients is zero. In other words, this model neglects the effects of the phase shift on the different spectral components. The variation of the SSG with frequency can be ðA;IÞ

ðB;IÞ

ðAÞ

ðBÞ

modelled either with variables s21 ; s21 and constants ck ; ck , or vice versa. We will choose the ðA;IÞ ðB;IÞ ðA;IÞ ðB;IÞ latter option, with s21 ¼ s21 ¼ 1. The phase shift associated with s21 or s21 modifies the phase of the input or output spectral components, respectively. A phase shift in the input signal corresponds to a translation of the time origin, if the input is a single or a double tone. For the output, a phase shift is not relevant in this context, in that we are dealing with the amplitudes of the various spectral components.

330

MICROWAVE AND RF ENGINEERING Σk ck(A)xk yA y x

IN

Figure 9.26

s11=s22=0 |s21 |=1 2 1

xB

Σk ck(B)xk y x

OUT

Simplified version of the configuration in Figure 9.25.

In our simplifying hypotheses, the structure in Figure 9.25 simplifies to that of Figure 9.26, consisting of one linear all-pass network placed between two nonlinear networks. The linear network is perfectly matched and loss free; its transmission coefficient could present a frequency-dependent phase. A first approximate idea of the nonlinear performances of the structure in Figure 9.26 can be obtained by using a piecewise approximation of the compression curve of each subnetwork h i 8 ðaÞ ðaÞ ðaÞ > Pin Psat;in < Pin þ SSGðaÞ ðaÞ Pout ¼ ð9:58Þ h i ða ¼ A; BÞ > ðaÞ ðaÞ : PðaÞ P > P out in sat;in where: .

The generic index ða ¼ A; BÞ denotes one of the two networks.

.

SSGðaÞ ¼ 20 log10 ½c1 is the small-signal gain, in dB.

.

2 Psat;out ¼ 10 log10 f½h1ðaÞ max =2R0 g þ 30 is the saturated output power, in dBm.

.

Psat;in ¼ Psat;out SSGðaÞ is the input power level generating the saturated output power in a linear network having the same SSG.

ðaÞ

ðaÞ ðaÞ

ðaÞ

Output power, 10 dB/div.

Figure 9.27 shows the compression curve of a quasi-linear network (thick line), together with its piecewise-linear approximation (9.58).

Psat,out

Psat,in Input power, 10 dB/div.

Figure 9.27

Compression curve (thick line) and its approximating piecewise-linear curve (thin line).


331

The chain in Figure 9.26 saturates if one of its nonlinear subnetworks A or B does. Denoting the overall input power – in dBm – as PIN, network A saturates if PIN > Psat;out;A SSGA . On the other hand, ðBÞ if network A operates linearly, the input power of network B is Pin ¼ PIN þ SSGðAÞ . Therefore, B ðBÞ ðAÞ ðBÞ saturates if PIN > Psat;out SSG SSG . The saturation therefore occurs in the chain of Figure 9.26 ðBÞ ðAÞ ðAÞ if the overall input power exceeds the lower of the two limits PðAÞ SSGðBÞ . sat;out SSG ; Psat;out SSG The saturated input power of the chain is then h i ðA þ BÞ ð AÞ ðBÞ Psat;in ¼ min Psat;out SSGðAÞ ; Psat;out SSGðAÞ SSGðBÞ ð9:59Þ Since we have neglected the mismatch, the SSG of the chain is the sum of the corresponding quantities of networks A and B: h i ðAÞ ðBÞ SSGðA þ BÞ ¼ SSGðBÞ þ SSGðAÞ ¼ 20 log10 c1 c1 The saturated output power is h ðA þ BÞ ðA þ BÞ ð AÞ Psat;out ¼ Psat;out þ SSGðA þ BÞ ¼ min Psat;out þ SSGðBÞ ;

ð BÞ

Psat;out

i

ð9:60Þ

The saturated output power of each nonlinear network can be determined by applying any of the values discussed in Section 9.5.4, h1max,1, h1max,2, or h1max,3; the resulting 1 dB compression point will be 2.13, 2.35 or 3.2 dB lower. A more accurate analysis can be performed by computing the 1 dB compression point of the chain in Figure 9.26. In this regard, it is useful to recall that the input instant amplitude (9.51), relative to the 1 dB compression point, is within the limits (9.47) allowed for the validity of the polynomial (9.44) rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi c1 1 c1 c1 ffi 0:5773 V1dBcp ffi 0:3808 < 3 c3 c3 c3 Hence, we can use the polynomial (9.44) to model the quasi-linear networks A and B. Moreover, in Section 9.5.4, we observed that the 1 dB compression point is barely affected by the second-order ðAÞ

ðBÞ

coefficient, therefore we can assume c2 ¼ c2 ¼ 0. Hence, neglecting the frequency-dependent phase shift introduced by the linear network placed between A and B, we can write 8 ðAÞ ð AÞ y ¼ c1 xin þ c3 x3in > > < A ðBÞ ðBÞ yout ¼ c1 xB þ c3 x3B > > : xB ¼ yA where the superscript ‘A’ or ‘B’ denotes the quantities relative to network A or B, as usual. Removing the intermediate variable xB from the above system of equations, and after some manipulations, we get the output yout as a function of the input xin h i3 h i2 h i2 h i3 ðAÞ ðBÞ ðAÞ ðBÞ ðAÞ ðBÞ ðAÞ ðAÞ ðBÞ ðAÞ ðAÞ ðBÞ ðAÞ ðBÞ yout ¼ c1 c1 xin þ c3 c1 þ c1 c3 x3in þ3 c1 c3 c3 x5in þ3c1 c3 c3 x7in þ c3 c3 x9in The resulting nonlinear characteristic is a ninth-order polynomial. Nevertheless, we can treat the terms with order higher than degree 3 as negligible, because of the relatively small amplitude of the input signal. The result is h i3 ðAÞ ðBÞ ðAÞ ðBÞ ðAÞ ðBÞ ðAþBÞ ðAþBÞ 3 yout ffi c1 c1 xin þ c3 c1 þ c1 c3 x3in ¼ c1 xin þc3 xin ð9:61Þ

332


The approximate characteristic (9.61) of the structure in Figure 9.26 has the same form as Equation (9.44). Therefore, we can compute the input and output 1 dB compression point by applying Equations (9.51) and (9.52) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

u ðAþBÞ u ð AÞ ð BÞ u u c u c1 c1 u ðAþBÞ V1dBcp ffi 0:3808t 1ðAþBÞ ¼ 0:3808u t ðAÞ ðBÞ h ðAÞ i3 ðBÞ ¼ 0:3808u t cð3AÞ c3 c3 c1 þ c1 c3 ðAÞ c1

½cð1AÞ

2 ðBÞ c3

ðBÞ c1

ðAþBÞ

Hence, the input sinusoidal amplitude V1dBcp causing 1 dB of compression in the chain is related to the corresponding quantities of networks A and B as h i2 ðAÞ 2 2 2 V1dBcp;AþB ffi V1dBcp;A þ c1 V1dBcp;B ðAÞ

where c1 is the SSG, in linear units, of network A. ðA þ BÞ Similarly, the output amplitude h11dBcp of the chain at 1 dB of compression is h

i ðA þ BÞ 2

h11dBcp

h i 2h i2 h i2 ðBÞ ð AÞ ð BÞ ffi c1 h11dBcp þ h11dBcp

From the above voltage amplitudes, we obtain the corresponding powers in dBm, which are more frequently used in engineering practice: " ðA þ BÞ I1dBdBm

¼ 10 log10 10

ðAÞ

I1dBdBm 10

O1dBdBm þ SSGðBÞ 10

" ðA þ BÞ O1dBdBm

¼ 10 log10 10

ðBÞ

þ 10

I1dBdBm SSGðAÞ 10

ðAÞ

#

ðBÞ

þ 10

O1dBdBm 10

ð9:62Þ

# ð9:63Þ

At about the third-order intercept point of the chain, we have to compute the intermodulation products resulting from an equal-amplitude dual tone excitation. Let the input signal be xIN ¼ V ½cosðo1 tÞ þ cosðo2 tÞ Then the output of the nonlinear network A coincides with the input of B and is ðAÞ

ð AÞ

yA ¼ xB ¼ c1 V ½cosðo1 tÞ þ cosðo2 tÞ þ c3 V 3 ½cosðo1 tÞ þ cosðo2 tÞ3 Applying the identity (A.77) to the last term, this equation becomes 9 ð AÞ 1 ð AÞ ð AÞ yA ¼ xB ¼ c1 V þ c3 V 3 ½cosðo1 tÞ þ cosðo2 tÞ þ c3 V 3 ½cosð3o1 tÞ þ cosð3o2 tÞ 4 4 3 ð AÞ þ c3 V 3 cos½ð2o1 o2 Þt þ cos½ð2o1 þ o2 Þt þ cos½ðo1 2o2 Þt þ cos½ðo1 þ 2o2 Þt 4


333

The above signal contains the amplified input signal (first term), together with the harmonic (second term) and intermodulation (third term) products. Let us consider the spectral components at the frequency of the input signal and the in-band40 intermodulation components, which have angular frequencies 2o1 o2 ð AÞ and o1 2o2 . The signal output from network A is then the sum of the useful term (I) c1 V ½cosðo1 tÞ þ ðAÞ 3 cosðo2 tÞ, plus the partial intermodulation product (II) c3 V ð3=4Þ fcos½ð2o1 o2 Þt þ ðAÞ ð AÞ cos½ðo1 2o2 Þtg and additional other terms (III) c3 V 3 ð1=4Þ ½cosð3o1 tÞ þ cosð3o2 tÞ þ c3 V 3 ð3=4Þ fcos½ð2o1 o2 Þt þ cos½ðo1 2o2 Þtg. We can approximate the signal output from B by assuming that the network linearly amplifies signals I and II, and generates intermodulation from I. This assumption corresponds to neglecting the intermodulation generated by the intermodulation products. This approach is equivalent to the one used to derive the combined characteristic (9.60), where we neglected polynomial terms higher than degree 3, as we will see shortly. Under our approximation, the overall output signal is ð AÞ ð BÞ

yOUT ¼ yB ffi c1 c1 V ½cosðo1 tÞ þ cosðo2 tÞ 3 ð AÞ ð BÞ h ð AÞ i 3 ð BÞ þ c3 V 3 fcos½ð2o1 o2 Þt þ cos½ðo1 2o2 Þtg þ c3 c1 þ c3 4 Again, we have the amplified input signal, the in-band intermodulation products and other additional – not explicitly written – distortion products. More precisely, the amplitudes of the first two terms are smallsignal approximations of the true quantities. The input third-order intercept point corresponds to the input amplitude which makes the two above quantities equal. That is, h i 3 ðAÞ ðBÞ h ðAÞ i3 ðBÞ h ðA þ BÞ i3 4 ðAÞ ðBÞ ðA þ BÞ ðA þ BÞ 2 c3 ¼ c1 c1 VIP3 ) VIP3 ¼ c3 c1 þ c1 VIP3 4 3

ðAÞ ðBÞ

c1 c1 h i3 ðAÞ ðBÞ ð AÞ ð BÞ c3 c1 þ c1 c3

The input amplitude corresponding to the third-order intercept point is then h

i ðA þ BÞ 2

VIP3

¼

h i h i2 h i 3 c3 3 h ðAÞ i2 c3 ð AÞ 2 ðAÞ ð BÞ 2 þ c1 ¼ VIP3 þ c1 VIP3 ð A Þ ð B Þ 4c 4 c ðAÞ

ðBÞ

1

1

Passing from the voltage input amplitude to the input and output power in dBm, we have ðAÞ

ðA þ BÞ IIP3dBm

ðBÞ

¼ 10 log10 10

IIP3dBm 10

¼ 10 log10 10

OIP3dBm þ SSGðBÞ 10

þ 10

IIP3dBm SSGðAÞ 10

ðAÞ

ðA þ BÞ OIP3dBm

!

ðBÞ

þ 10

OIP3dBm 10

ð9:64Þ

! ð9:65Þ

Note the analogy between formulae (9.64) and (9.65) and the corresponding formulae (9.62) and (9.63). Indeed, it is also possible to derive IIP3 and OIP3 for the cascaded networks, starting from I1dB and O1dB and considering the simple relation (9.57).

40 The considered product is in band, providing o1, o2 are in band and sufficiently close to each other. See also footnote 36.

334

MICROWAVE AND RF ENGINEERING Observations: (a) The compression point and the saturated power of two cascaded two-port networks can be computed by applying any of the Equations (9.60), (9.63) or (9.65), since Psat , O1dB and OIP3 are related as explained in Sections 9.5.4 and 9.5.5. Equation (9.60) gives a more optimistic41 and less realistic result than the other two, which are equivalent; nonetheless, it is useful to dimension the saturated power and the gain of the cascaded components. For instance, let us consider that ðAÞ the two cascaded networks in Figure 9.26 are two amplifier stages. Let Pout;sat ¼ þ 5 dBm be the saturated output power of the first stage, while SSGðBÞ ¼ 8 dB is the linear power gain of the second stage. Equation (9.60) states that stage A or B determines the saturation of the chain if ðAÞ

ðBÞ

ðAÞ

ðBÞ

Pout;sat þ SSGðBÞ ¼ 13 < Pout;sat or Pout;sat þ SSGðBÞ ¼ 13 > Pout;sat. Thus if the saturated outðBÞ

put power of the second stage Pout;sat is lower than 13 dBm, we have early saturation of the chain, ðBÞ

due to the low-power output stage. Conversely, if Pout;sat > 13 dBm, there is no appreciable ðBÞ Pout;sat

improvement in respect of the limit case ¼ 13 dBm, with an associated higher power supply consumption of amplifier B. Therefore, the parameters of our components will be ðBÞ

ðAÞ Pout;sat ¼ þ 5 dBm; SSGðBÞ ¼ 8 dB; Pout;sat ¼ þ 13 dBm. Assuming O1dB ¼ Pout;sat 2:5, as stated at the end of Section 9.5.4, we obtain O1dBðAÞ ¼ þ 2:5 dBm, O1dBðBÞ ¼ þ 11:5 dBm. Applying Equation (9.63), we get

2:5 þ 8 11:5 ðA þ BÞ OIP3dBm ¼ 10 log10 10 10 þ 10 10 ¼ þ 8:49 dBm

which is about 3 dB lower than the one obtained from Equation (9.60), as expected. (b) Although Equations (9.63) and (9.65) are more accurate than (9.60), they still involve many approximations. In particular, the influence of the output loading on the compression characteristic is not considered. In real cases, the input impedance of the second stage is different from a perfectly matched resistor, thus the network A operates under conditions which are different from those used to test its nonlinear performance. Moreover, the linear path between the two nonlinear networks in Figure 9.25 is assumed to have no dispersion. Again, this is not true in most of the cases; rather, all the distortion products at the output of A propagate with different attenuation and velocity. Consequently, they are present at the input of B with different phase and amplitude than we assumed to derive Equations (9.63) and (9.65). Finally, it must be stressed that the model used in Figure 9.26, even after removing the latter approximation, does not take the AM–PM conversion into account.

9.6 Semiconductors devices 9.6.1

Basic semiconductor physics

Figure 9.28 shows the structure and the electrical symbol of the four basic semiconductor devices: (a) junction diode, (b) bipolar junction transistor (BJT), (c) field effect transistor (FET) and (d) metal oxide field effect transistor (MOSFET, or more briefly MOS). Semiconductor devices use material with a high degree of purity, including very small and accurately controlled low impurities. The addition of such substances is usually named doping, the added impurities are the dopant, while the pure semiconductor material is referred to as intrinsic or pure. The atom of the 41 Equation (9.60) is derived by assuming a piecewise-linear approximation of the compression curve: there is no gain compression until the output power reaches the saturated value.


cathode

p

n

anode

c

335

a

(a) e emitter

p

n

n

c

collector b

(b)

base

source

drain

gate

s

P

g

(c)

source

drain

gate

oxide n bulk

d

n

p

n

s

d

g

(d)

Figure 9.28 Semiconductor devices: simplified cross-section (left) and electrical symbol (right): (a) junction diode; (b) BJT; (c) junction FET; (d) MOSFET. intrinsic semiconductor – generally silicon (Si) or germanium42 (Ge) – is tetravalent. Thus, in its pure state, each atom will form four covalent bonds with four neighbouring atoms – all of the same type. The resulting material will assume a crystal structure, with a regular tetrahedron as the elementary cell. The electron couples constituting the covalent bond are strongly tightened, therefore the electrical conductivity of an intrinsic semiconductor is quite low.43 Implantation of the pure semiconductor with low concentrations of pentavalent atoms (phosphorus (P), arsenic (As), antimony (Sb)) perturbs the original regular crystalline structure. Indeed, the fifth electron of the dopant is free from bonds; thus it is available for electrical conduction. The above described operation is n-type doping, the low-concentration dopant (of the order of one atom in 108 of the pure semiconductor) is the n-dopant. The dopant ions, placed within the crystal reticule, are the donor ions. 42 Other more complex materials, like gallium arsenide (GaAs), indium phosphide (InP), aluminium gallium arsenide (AlGaAs), or silicon–germanium (Si–Ge) are in use, particularly for extremely high-frequency devices. However, we will limit our description to the simplest case, since the basic working principle of the different devices does not change with the material used. 43 The resistivity of intrinsic silicon and germanium is about 10 3 O m and 5 10 4 O m. By comparison, the resistivity of copper is 1:7 10 8 O m, four orders of magnitude lower.

336


Similarly, p-doping is possible and consists of adding trivalent dopant (aluminium (Al), gallium (Ga), indium (In)) to the intrinsic semiconductor. In this case, where the trivalent atom is present, one of the four covalent bonds is not saturated. This is equivalent to having one free hole, i.e. one possibility to receive free electrons, available from the covalent bond breaking caused by the thermal agitation. Equivalently, we can consider the p-doped semiconductor as containing positive free charges – known as holes – available for electrical conduction. The absolute value of the hole charge is the same as that of the electron. Finally the ions of the p-dopant are acceptor ions. The free charges in the semiconductor are often called carriers. The electrons (holes) in the n- (p-)region are the majority carriers, while the holes (electrons) are the minority carriers.

9.6.2

Junction diode

The left side of Figure 9.28a schematizes the structure of a junction diode, which is the simplest semiconductor device. Such a device basically consists of a semiconductor crystal: one portion of it is ndoped, and the remaining part is p-doped. The boundary between the two differently doped portions is the junction. Both the n- and p-zones present one metal contact: the cathode and the anode, respectively. Figure 9.29 (top) shows the charge distribution along the diode, in the absence of an external voltage applied to the electrodes. Such a distribution is the result of two contrasting actions. From one side, the electrons of the n-zone tend to migrate towards the p-zone, in order to recombine with the holes, and vice versa. However, this process cannot proceed indefinitely, due to the opposition of the charge concentration around the junction, caused by the recombination itself. Therefore, the charge density along the diode rðxÞ assumes the type of shape shown in Figure 9.29. The consequent electrical static potential can be obtained by solving Poisson’s equation d 2 VðxÞ rðxÞ ¼ dx2 e

ð9:66Þ

where e is the dielectric constant of the material. donors acceptors electrons holes

n zone

p zone

ρ(x)

V(x)

δx

barrier x 0

Figure 9.29 Charge density distribution and electrostatic potential of a p–n junction diode with no applied external bias.


337

Equation (9.66) is a linear second-order equation, thus two conditions must be specified for the solution; here it is assumed that limx ! 1 VðxÞ ¼ 0 and limx ! 1 ½dVðxÞ=dx ¼ 0. The plot in Figure 9.29 shows the potential VðxÞ (bottom curve) obtained by applying Equation (9.66) to the charge distribution rðxÞ (top curve). It can be seen that, sufficiently far from the junction, the electrostatic potential tends to zero in the p-zone and to a finite value – named the barrier – in the n-zone. The positive potential in the n-zone prevents any further migration of electrons towards the p-zone and of holes from the n-zone. Equilibrium is achieved when the forces associated with the chemical bond energy become equal to the electrostatic ones. If no external electric field is applied to the diode, the region across the junction presents no carrier: that region is the depleted zone or the depletion zone of the diode. If we apply a positive voltage between the anode and cathode, some holes will move to the n-zone and some electrons will migrate to the p-zone. This is because the applied voltage opposes the electrostatic voltage. The consequence is a positive current flowing from the anode towards the cathode. This type of diode biasing – positive anode to cathode voltage – is commonly referred to as forward bias. If the applied positive voltage exceeds the barrier, the depleted zone will disappear, and the diode behaves approximately like a resistor, in that any further increase of the forward bias will cause a proportional increase of the current. The ratio between the latter two quantities is the series resistance of the diode, which mainly depends on the contact resistance associated with the electrode metallization. Conversely, if we apply a negative voltage between the anode and cathode – or an inverse bias – then no current flows, because the superimposition of the external voltage on the electrostatic potential increases the effect of the latter, widening the depleted zone. Briefly, a forward- (inverse-)biased diode allows (prevents) the current from flowing through the diode itself. As shown, the reverse-biased diode approximates an open circuit. However, the effect of the depleted zone must be considered too. Looking at curve rðxÞ in Figure 9.29, we can see that it presents two local extremes across the junction: a local maximum in the n-zone and a local minimum in the p-zone.44 Let the distance between those local maxima, along the x axis, be dx as indicated in Figure 9.29. Note also that dx monotonically increases with depletion width. Thus, the charge distribution across the junction resembles that of a parallel-plate capacitor, having its positive (negative) electrode placed in the same position as the positive (negative) local maximum of rðxÞ. Therefore, in the reverse bias, the diode is approximately equivalent to a capacitor, whose capacitance – the junction capacitance – is directly proportional to the junction area and inversely proportional to dx. Consequently, the junction capacitance decreases with the amplitude of the reverse bias, because this widens the depleted zone and thus dx as well. The junction capacitance effect is usually relevant in high-frequency networks. According to the considerations presented so far, a reverse-biased junction is a DC open circuit. However, the charge distribution in Figure 9.29 is a first-approximation model, in that minority carriers are present in the neighbourhood of the junction, although in low concentration. The external applied field moves these carriers, generating a leakage current. At increasing reverse voltages, the velocity of the minority carriers increases and so does their kinetic energy. At high reverse-biased voltages, the kinetic energy of some of the carriers becomes sufficient to create new electron–hole pairs in the collision with some atoms of the reticule, further increasing the current, the carrier energy, and so on. In addition, the newly created electron–hole pairs are also able to create other pairs, after colliding with the atoms of the reticule. This process is known as avalanche, and causes an abrupt increase in the diode current; it occurs if the external applied field rises above a critical limit. The reverse voltage VBREAK , corresponding to that limit, is the breakdown of the junction. The application of a reverse voltage higher than VBREAK could destroy the device, if no current limitation is present on the voltage generator. Breakdown is inherent in

44 The two maxima of opposite sign have the same value if the diode is perfectly anti-symmetrical along the junction.

338


any semiconductor junction device – like all the devices in this section – and assumes relevant importance in high-power applications. With more complex analysis, we find that a junction with a pure metal and a p- or n-doped semiconductor (Schottky junction) exhibits an external behaviour similar to the p–n junction. In addition, the Schottky junction is widely used in high-frequency devices. The rectifying properties of a metaldoped semiconductor junction suggest that the electrode metallizations of a semiconductor device need suitable treatment, in order to avoid forming a rectifying junction with the semiconductor. An in-depth description of the semiconductor device realization process is well beyond the purpose of this book. Therefore we will limit ourselves to mentioning, without any further description, the difference between ohmic and rectifying metal–semiconductor junctions.

9.6.3

Bipolar transistor

Figure 9.28b (left) shows the simplified cross-section of a BJT. It has two p–n junctions; thus n–p–n and p–n–p transistors are possible. For high-frequency applications, n–p–n BJTs are almost exclusively used, therefore we will examine this case only.45 Looking at the structure in Figure 9.28b, we can see three doped zones, two n and one p between them. Each of the three doped zones presents its own metallization, which is the corresponding electrode: from left to light we have the emitter (e), the base (b) and the collector (c). Furthermore, the base region (p) is much thinner than the remaining two (both n). There are four possible combinations for the bias of base–emitter (b–e) and base–collector (b–c) junctions: 1. Active region, b–e is forward biased and b–c is reverse biased. 2. Saturation, both b–e and b–c are forward biased. 3. Interdiction, both b–e and b–c are reverse biased. 4. Inverse active, b–e is reverse biased and b–c is forward biased. Bipolar transistors used in amplifiers and oscillators operate in the active region, saturation and interdiction are used for switches, while inverse active conditions have few or no applications in RF=mWcircuits. Figure 9.30 shows a BJT biased to operate in the active region. For the diode, because of the forward bias, we have a current Ie flowing in the emitter. A small percentage of this current Ib passes through the base, as in a simple diode. Differently from the case of the diode, however, and because the base region is very thin, the majority of the electrons coming from the emitter migrate directly to the collector, attracted by the positive voltage. This happens because the b–c junction is reverse biased, thus a high electrostatic potential is present in the collector; this lets the electrons pass, but hinders the holes from doing the same, as discussed for the diode. Kirchhoff’s law, applied to the network in Figure 9.30, implies that emitter, base and collector currents are related as Ie ¼ Ic þ Ib Furthermore, we have just seen that the majority of the charges moved by the forward bias of the e–b junction flow through the collector; then I e ¼ a Ic

with

a VT

d

oxide ++++++++++++++++ ++++++++++++++++

n

n p

b (a)

Vds

Id

+ +

s Vgs > V T

g

d

oxide ----------------

n

b

++++++++++++++++ ++++++++++++++++

n

p

(b)

Figure 9.32 MOSFET bias schematic: (a) gate voltage below the threshold (no drain current); (b) gate voltage above the threshold (free electrons below the oxide).

342


case, the control voltage Vgs must be negative, in that it has to remove the carriers from the channel and modulate the d–s conduction. In the small-signal regime, the MOSFET works as an inverting VCVS: the common terminal is the source, the input and output ones are the gate and drain, respectively. The low-frequency MOSFET input impedance is higher than that of the JFET. For the high-frequency operation of the MOSFET, the g–s capacitance is particularly important.

9.7 Electrical models of high-frequency semiconductor devices This section deals with the representations of the parameters of semiconductor devices, which are needed for RF and microwave (RF/mW) component analysis and design. For simplicity our discussion is limited to one- and two-port devices, although most of the concepts can be used to model devices with an arbitrary number of ports. Also, most of the proposed representations are not restricted to semiconductor devices, instead most of them can be used for many types of n-port networks. Modern designs are computer based, thus most of the discussed techniques are computer oriented, although some simplified models can be used for initial first-approximation analytic considerations.

9.7.1

Linear models

Most of the characterization of semiconductor devices has an experimental base, although some models can be directly derived from the physics of the devices.50 The simplest case that we can consider is the one for a linear device. Any semiconductor device is inherently nonlinear, but if the amplitudes of the involved RF/mW signals are sufficiently small, it can approximately be considered as linear. The small-signal analysis of a device operating on a given bias working point is equivalent to linearizing the various nonlinear characteristics of the device around that working point. In other words, a device is considered as linear if all its nonlinear characteristics are replaced by linear functions. Figure 9.33 shows a typical setup for the measurement of a linear operating transistor. The transistor is the block indicated by device under test(DUT). Terminal 1 (2) of the DUT could be the base or gate (collector or drain), while the common terminal – emitter or source – is connected to ground. Transistors need a proper DC bias in order to operate as active devices, so the test setup includes two generators with the respective T-bias networks. Each T-bias consists of one series capacitor (C1 ; C2 ) and one shunt inductor (L1 ; L2 ). The capacitors C1 ; C2 have capacitances that are large enough to present a negligible reactance in the RF band of interest. Thus, C1 ; C2 pass the RF signal between the network analyzer(NWA)51 and the DUT, while preventing the direct current from reaching the RF ports of the NWA. Conversely, the inductors pass direct current and present high RF impedances. An ideal T-bias allows the DC generator to bias the DUT without influencing the transmission of the RF signal through the RF path. Practical T-bias networks are not ideal, but their effect on the measurement can be virtually eliminated by means of proper calibration techniques.52 Finally, the two DC generators supply the bias needed by the DUT. In the case of the n–p–n junction, n-channel depletion JFETor n-channel enhancement MOSFET, we have respectively that ‘input terminal bias’ is a positive current generator, a negative voltage generator or a positive voltage generator.

50 The use of physics-based semiconductor models is unusual, due to the complexity of the calculations involved, the lack of accuracy generally inherent with them, and sometimes because some physical and/or chemical parameters of the device are only known approximately. 51 A description of NWA operation can be found in Section 17.6. 52 See Sections 17.6.1 and 17.6.2.


RF port 1

343

RF port 2 Network Analyzer

Tee-bias 1

Tee-bias 2 1 DUT 2

C1

C L2 2

L1

Input terminal bias

Output terminal bias RF path DC path

Figure 9.33

Setup for a two-port linear semiconductor device.

The setup of Figure 9.33 can be reduced to the case of a one-port device – like the diode – just by ignoring one of the two device ports, T-bias, DC generators and NWA RF ports. Similarly the test setup can be extended to a multi-port DUT. Also, if some of the DUT ports need no bias, the relative generator and T-bias can be eliminated, and the unbiased port will be directly connected to the NWA. Once the NWA is calibrated, it is able to measure the biased DUT over a discrete frequency range; the measured data at each frequency can be sent to a computer and stored in a file. There are many possible ways to arrange the measured data in a file, but one of them is particularly important because of its wide use as a file exchange format, particularly for computer-aided engineering(CAE) programs. This format is known as the Touchstone format,53 which consists of a text (ASCII) file containing one header line, some possible comment lines and one data line per test frequency point. The data line has nine (three) columns for a two-port (one-port) network. Moreover, each data line contains vector information on the measured parameters. Any text line beginning with an exclamation mark (!) is a comment, and therefore is ignored by the CAE program. The header specifies the data format and has the following general form: # FREQ_UNIT S FORMAT R IMPEDANCE where: .

FREQ_UNIT is the frequency unit (Hz, kHz, MHz, GHz).

.

FORMAT specifies the scattering parameter format (RI, MA, dB).

.

IMPEDANCE indicates the normalizing impedance of the parameters.

53

The Touchstone file was originally a proprietary file format for the eponymous frequency domain linear circuit simulator from EEsof, launched in 1984. Touchstone is a registered trademark of Agilent Technologies, Inc.

344


Therefore, the Touchstone file can be used to store scattering parameters, and also impedance and admittance parameters, although the first type of data is more frequently used, particularly because S parameters are the ones directly measured by the NWA. To simplify the description of the data line, we will describe the case of a two-port network, and of scattering parameters, giving some generalizations later. Each data line includes nine numbers: the first is the frequency and the following four couples represent one of the network complex parameters s11, s21, s12, s22, respectively, and in this order. Each couple of numbers is the complex value of the respective parameter, in the format specified by FORMAT, i.e. Reðshk Þ jshk j

Imðshk Þ ðh; k ¼ 1; 2Þ

180 argðshk Þ ðh; k ¼ 1; 2Þ p

20 log10 ðjshk jÞ

180 argðshk Þ ðh; k ¼ 1; 2Þ p

ðFORMAT ¼ RIÞ ðFORMAT ¼ MAÞ ðFORMAT ¼ MAÞ

In the latter two cases, the second number is the angle of the complex number, expressed in degrees. If the data represent impedance or admittance parameters, the same structure applies, but with shk replaced by zhk or yhk. In the case of a one-port, the data lines contain three numbers, for instance Reðs11 Þ Imðs11 Þ. The complex representation of the network parameters is needed for those calculations involving the measured device; this implies the use of a vector NWA. If scalar data are available, say js11 j js21 j js12 jjs22 j, they can still be used by replacing the missed phase with dummy numbers, for instance all zeros.54 Such a Touchstone file is still usable by the CAE program, but the resulting computations are quite inaccurate. Sometimes, the S parameter files include the noise parameters55 of the DUT, which have the same complex form as the network parameters in the same file. Table 9.2 shows the Touchstone file of a microwave transistor, also including the noise parameters. A more sophisticated modelling approach than the S parameter table is the derivation of a linear network. The linear model response provides an accurate replica of the originating file. Such linear networks are known as equivalent linear circuits or linear models of the device. The component values of the linear model are computed by exploiting computer optimization programs, and by imposing the S parameters of the network to be as close as possible to the measured data, stored in a Touchstone file. Such a procedure is known as curve fitting. Some techniques are available for facilitating the optimization, usually by supplying approximate initial values of the parameters; they are referred to as parameter extraction techniques.56 The device linear model generally includes many passive linear elements and one controlled generator which is absent in the case of the diode. The type of controlled generator reflects the 54 The addition of the dummy numbers is necessary to keep the number of numbers per line, in order to retain compatibility with the standard. 55 The noise parameters of a two-port network were defined in Section 9.4; Section 17.6.3 describes their measurement techniques. 56 A simple example of the parameter extraction technique is the computation of the resistance in the model in Figure 9.34a from low-frequency scattering parameters. Let us assume that the diode has been measured with the setup in Figure 9.33, and one of the two electrodes (say the cathode) is grounded. The resulting DUT is then a one-port network. From the measured s11 (reflection coefficient) we can compute the corresponding impedance z11 ¼ R0 ð1 þ s11 Þ=ð1 s11 Þ, and from the lowest frequency value of it impose that Rd þ Rs ¼ z11 ðf ¼ fmin Þ, where fmin is the lowest possible measurement frequency. The resistance obtained is the sum of the contact and junction resistances. The first is smaller than or comparable with the latter if the diode is forward or reverse biased. The value Rd þ Rs resulting from the reverse bias is not very meaningful. It could be some tens to millions of the forward bias value, and is totally masked from the other impedances of the model.


345

Table 9.2 Scattering parameters of a microwave FET (GaAs MESFET). ! Transistor measured on 5 May 1997 ! Grounded Source, bias: Vds¼3V, Ids¼10mA # GHz S MA R 50 !FREQ S11-------------- S21------------- S12------------ S22-------2 0.98 –29 3.23 157 0.06 72 0.65 –16 3 0.94 –43 3.13 145 0.08 65 0.64 –24 4 0.92 –57 3.00 135 0.10 56 0.60 –32 5 0.90 –70 2.84 125 0.12 49 0.59 –38 6 0.89 –80 2.67 117 0.14 43 0.58 –44 7 0.87 –91 2.53 108 0.15 36 0.57 –50 8 0.84 –100 2.38 99 0.15 30 0.54 –55 9 0.83 –107 2.23 92 0.16 26 0.54 –58 10 0.82 –116 2.12 86 0.17 23 0.53 –64 !FREQ dB(Fmin) Gamma_opt 2 0.55 0.85 21 4 0.60 0.75 40 6 0.8 0.69 55 8 1.0 0.62 70 10 1.3 0.56 85

Rn/50 (noise parameters) 0.5 0.4 0.4 0.3 0.3

low-frequency behaviour of the device type, for instance we have VCCS for FET, or CCCS for BJT; alternatively a VCCS with a low input resistance can be used for the BJT. Some passive lumped or distributed elements surround the controlled generator and model the high-frequency behaviour of the device. Although most of these elements have a physical explanation, their value is generally obtained by curve fitting. Additional passive elements – usually reactive – can be added to the model, in order to take the package into account. Figure 9.34a,b show one possible linear model for the diode (a) and the FET (b), respectively. Note that the anode–cathode DC resistance in the model of Figure 9.34a equals Rd þ Rs and represents the incremental ratio between the voltage and current variations in the diode. The capacitor Cd mainly represents the junction capacitance, the elements Cp and Ls model the finite length of the device, but can be also used to model the effect of some low-parasitic packages. Figure 9.34b shows the linear model of a chip MESFET, but it could be applied to any FET in a chip. The network consists of a VCCS – whose gain or transconductance is gm – surrounded by some elements, each modelling different parasitic effects: .

Cgs , the gate–source junction plus the interelectrode, the first contribution usually dominating, since the junction is moderately reverse biased for normal FET operation.

.

Cgd , mainly the gate–drain interelectrode capacitance, since the relative junction is strongly reverse biased, in that the drain voltage is positive with respect to the source, which is positive with respect to the gate.

.

Cds , the drain–source interelectrode capacitance.

.

Lg ; Ls ; Ld , the finite device size, in combination with Cgs ; Cds ; Cgd .

.

t, the propagation time of the signal from the gate to the source.

346

MICROWAVE AND RF ENGINEERING Cp Cd Cathode Rs

Rd

Ls

Anode

(a) Lg (130 pH) Rg (0.1Ω)

Cgd (54 fF)

Gate

Ld (160 pH) R d (0.1 Ω) Drain

+ vi -

Cgs (243 fF)

gm •vi (t -τ) (g m =40 mS) Rds (277Ω) (τ =0)

Cds (92 fF)

Ri (3.8 Ω)

Rs (0.1 Ω) L s (36 pH) (b)

Source

Figure 9.34

Linear models for semiconductor devices: (a) diode; (b) FET.

.

Rg ; Rs ; Rd , the contact resistance of the respective electrode.

.

RI , the resistance presented by the channel, which is partially pinched.

Figure 9.34b also shows the numerical value of the model components (numbers in brackets), as obtained by curve fitting with the scattering parameters in Table 9.2. Figure 9.35 shows the scattering parameters of Table 9.2 (solid curves) together with the ones obtained from the network in Figure 9.34b (dashed curves). Figure 9.35 uses a special mixed type of vector representation: the lower half – used for the reflection coefficients – is a standard Smith chart, the upper half – for the forward transmission coefficient – is a polar chart of radius r ¼ 4. The reverse transmission coefficient has a small radius, and is barely distinguishable from the centre of the plot. Such a compact representation is possible when the reflection and transmission coefficients lie on opposite halves of the complex plane. The linear model cannot be more accurate than the S parameters it is computed from. However, some considerations can justify the work needed for computation of the model parameters: 1. The linear model is a compact representation of the scattering parameters. As a comparison, Table 9.2 (excluding the noise parameters) comprises 9 9 ¼ 81 real numbers,57 while the corresponding linear model in Figure 9.34b is defined by its 13 component values. 2. The linear model can easily be used for statistical analysis, just by adding statistical variability to its parameters. 3. Extraction of the linear model is the first step towards computation of the nonlinear model.

57

Nine frequency points, each with nine real numbers, as the FET is a two-port.


347

Figure 9.35 Scattering parameters of a microwave chip GaAs MESFET. Sold lines, measured as in Table 9.2; dashed lines, modelled as in Figure 9.32b. The model in Figure 9.34b can also be used for bipolar transistors, with minor modifications, by connecting one resistor Rbe in parallel with Cgs and short-circuiting RI . The active component of the model becomes a VCCS having finite input resistance Rbe and transconductance gm ; this is equivalent to a CCVS with the same input resistance and current gain b ¼ gm Rbe . If a packaged device must be characterized, rather than a chip one, the models in Figure 9.34 are not sufficiently accurate, due to the parasitics introduced by the package itself. Basically, a package includes a number of terminals – at least one per electrode of the device – to be soldered on the printed circuit board, one or more bonding wires connecting each electrode to the respective terminal(s). Furthermore, the terminals come out from a dielectric package surrounding and protecting the chip with its bond wires.58 Bond wires and terminals introduce additional series inductance to each device electrode, while the mutual couplings between the different wires and terminals generate additional interelectrode coupling capacitance. All these effects can be included in the model, by embedding the models in Figure 9.34 into a more complex network including additional elements to model the package. Figure 9.36 shows such an arrangement for a packaged BJT; other similar combinations can be used for different types of devices.

58

Some more details on packages can be found in Chapter 14.

348

MICROWAVE AND RF ENGINEERING chip network C bc (125 fF)

R b(6.38 Ω) b′

Rc (13.6Ω) c′

+ vi C be (854 fF)

-

gm •v i (t - τ) R be (1618 Ω) (g m =63.3 mS) (τ =1.33 ps)

R ce (43.6 k Ω)

Cce (100 fF)

R e(1.85 Ω)

(a)

e’

C bc1(54 fF)

L b1 (0.96 nH)

L b2 (0.59 nH)

base

chip network b′

C be (495 fF)

e′

Lc1 (0.39 nH)

c′

L e2 (0.26 nH)

Lc2 (0.26 nH) collector

C ce (243 fF)

L e1 (0.65 nH)

(b)

emitter

Figure 9.36

BJT linear models: (a) in-chip; (b) packaged.

The numbers in brackets in Figure 9.36 represent the values obtained by curve fitting with the S parameters of the bipolar transistor MBC13900, manufactured by Freescale Semiconductor. Table 9.3 lists the scattering parameters of the transistor, biased for a collector–emitter voltage of 2 Vand a collector current of 1 mA, as supplied by the manufacturer.59 The curve fitting of the model60 with the data in Table 9.3 gives the response shown in Figure 9.37 (grey lines), while the data are the black lines in the same figure.

9.7.2

Nonlinear semiconductor models

Linear analysis and models do not provide any information about the effect of the excitation amplitude on the network or system response. In other words, the linear network response is, by definition, independent of the power of the applied signal, while networks including semiconductor devices produce nonlinear distortion, as is known. Linear models do not present such distortions, therefore nonlinear models and analysis are required when nonlinear performances – such as harmonic or intermodulation products, compression, output power, etc. – have to be predicted. Diode and/or transistor nonlinear models will be part of the circuit description within a nonlinear simulator. Such a description will include the nonlinear device itself, its matching networks, bias 59 Data taken from the component data sheet, Document Number MBC13900/D Rev. 1.1, 06/2005, Table 7, p.13. Copyright of Freescale Semiconductor, Inc. 2008. Used with permission. 60 See the Ansoft file 04_BJT_Linear_Model.adsn.


349

Table 9.3 Scattering parameters of the bipolar transistor MBC13900. ! MBC13900, Common Emitter, 2V 1mA # GHz S MA R 50 ! S11----------- S21------------ S12--------------S22---------0.1 0.973 –6 3.754 175 0.008 86 0.997 –3 0.5 0.961 –33 3.366 153 0.038 71 0.968 –12 0.9 0.895 –57 3.341 135 0.065 56 0.910 –22 1.0 0.868 –63 3.256 131 0.070 53 0.915 –24 1.5 0.766 –91 2.688 111 0.091 38 0.851 –33 1.9 0.721 –114 2.610 94 0.100 26 0.788 –39 2.0 0.706 –119 2.501 91 0.102 23 0.780 –41 2.4 0.649 –140 2.280 77 0.104 15 0.731 –47 3.0 0.628 –166 1.984 58 0.105 2 0.667 –56 3.5 0.606 173 1.717 45 0.099 –3 0.650 –62 4.0 0.606 155 1.478 33 0.094 –10 0.640 –68 4.5 0.611 138 1.421 21 0.089 –12 0.604 –74 5.0 0.610 122 1.309 9 0.085 –11 0.581 –81

Figure 9.37 Scattering parameters of the BJT MBC13900. Black lines, data measured by the manufacturer (as per Table 9.3); grey lines, response of the model in Figure 9.35. networks together with the respective DC generators, stimulus generators with given frequencies and powers, and load. This setup makes prediction of the power effects possible. The accuracy of the results depends on which one of the models is used. However, the nonlinear analysis is more critical than the linear one: small inaccuracies in the models can lead to gross mistakes in the final result.

9.7.2.1 Diode nonlinear models The simplest device for nonlinear modelling is the diode. Figure 9.38 shows the schematic of a diode nonlinear model. This model is derived from the linear one in Figure 9.34a by replacing the linear resistor Rd with the nonlinear element D, and the linear capacitor Cd with a combination of the linear ðCd;lin Þ and the nonlinear ðCJ Þ capacitor.

350


C d,lin

Cp

CJ D Cathode Idi

Anode Rs

Ls

- V di + - Vd +

Figure 9.38

Diode nonlinear model.

The instantaneous current through the device D is related to the voltage across its terminals as

Vdi Idi ðVdi Þ ¼ IS exp 1 ð9:67Þ n VT where: .

IS is the saturation current of the diode, or the reverse current for high-amplitude negative Vdi .

.

VT ¼ K T q 1 , where q ¼ 1:6022 10 19 C is the electron charge, K ¼ 1:374 10 23 J=K is the Boltzmann constant61 and T is the absolute temperature.

.

n is an ideality factor which usually ranges from 1 to 2.

Equation (9.67) is also known as Shockley’s equation; it can be derived from physical considerations of the semiconductor diode, particularly in the case of n ¼ 1. For the needs of high-frequency circuit analysis, however, this equation is just a mathematical instrument to model the current–voltage relation of the diode. The parameters Is ; n are computed by curve fitting with experimental data. In this respect, it is useful to measure the direct current of the diode at different bias voltages. Under this condition, the external voltage Vd is externally accessible, rather than the internal one Vdi . The two quantities are related as Vd ¼ Vdi þ Rs Idi

ð9:68Þ

Equation (9.68) is valid only for purely DC voltages and currents, in that it neglects the currents flowing through the parasitic capacitances. Conversely, Equation (9.67) refers to the current through D; in this regard, it is more general and applies to instant excitations. Substituting Equation (9.68) into (9.67) we obtain the DC relation of the nonlinear diode model

Idi Vd ðIdi Þ ¼ n VT ln þ 1 þ Rs Idi ð9:69Þ IS The function (9.69) has three arbitrary parameters ð n; IS ; Rs Þ, which could be determined from a minimum number of three different values of the diode current, measured at three different bias voltages. In general, more than three measured couples of ðVd ; Idi Þ are used for the curve fitting, in order to achieve a better accuracy. Moreover, Rs could also be available from the linear model obtained from

61

See also Section 9.4.


351

Id , mA 4 measured Eq.(9.69) Eq.(9.71)

2

0 -5

0

1

Vd , Volt

-2

-4

Figure 9.39

Diode direct current vs. voltage.

high-frequency linear measurements. In general the two values are not coincident. The choice between the value obtained from high-frequency linear and DC measurements needs some trade-off; sometimes an intermediate value is assumed. Figure 9.39 plots the function (9.68) with the parameters obtained by curve fitting62 (grey line) together with some measured data (black dots).63 An alternative way to determine the parameters of Equation (9.69) consists of deriving them from the parameters Rd and Rs of the linear model in Figure 9.34a. This method needs the extraction of some parameter sets for the linear model at different direct currents, and also Rs must hold constant for the different cases. This way, the series resistance Rs of the nonlinear model is coincident with the Rs of the different linear models. Now, if we differentiate the function (9.67) with respect to the voltage, we obtain the differential conductance of the nonlinear device D, which equals the reciprocal of Rs , by definition. Thus

1 qIdi IS Vdi ¼ ¼ exp ð9:70Þ Rd qVdi n VT n VT If two values of the linear resistance Rd;1 ; Rd;2 have been extracted at two different direct currents Idi;1 ; Idi;2 , it is possible to determine the corresponding voltages64 Vdi;1 ; Vdi;2 , by inverting Equation (9.67)

Idi;1 Idi;2 þ 1 ; Vdi;2 ¼ n VT ln þ1 Vdi;1 ¼ n VT ln IS IS Substituting these couples of values into (9.70), we obtain 8

3 2 Idi;1 > >

> n V ln þ 1 T > 6 7 > 1 IS IS > 7 ¼ IS exp ln Idi;1 þ 1 ¼ IS Idi;1 þ IS ¼ Idi;1 þ IS > ¼ exp6 < 4 5 n VT n VT Rd;1 n VT IS n VT IS n VT > > > > > > 1 ¼ Idi;2 þ IS > : Rd;2 n VT Is ¼ 0:1995 10 12 ; n ¼ 1:9271; Rs ¼ 2:864 O. See the SIMetrix file 10_Diode_I_V_Curve.sxsch. 64 Again, voltage Vdi across the nonlinear device D is not accessible from outside the diode. 62 63

352


Dividing the first of the above equations by the second one term by term, we obtain the saturation current

Rd;2 Idi;1 þ IS Rd;2 Rd;2 Rd;2 Rd;2 1 ¼ ) Idi;1 þ IS ¼ Idi;2 þ IS ) IS ¼ Idi;2 Idi;1 1 Rd;1 Idi;2 þ IS Rd;1 Rd;1 Rd;1 Rd;1 Once IS is known, n is easily obtained as 1 Idi;1 þ IS Idi;1 þ IS ¼ )n¼ Rd;1 n VT VT Rd;1 If more than two sets of data are available, a more reliable parameter extraction is possible, using optimization procedures that minimize the error. So far, we have described how to derive the nonlinear model from many linear models obtained at different bias points. The opposite is also possible, by linearizing the nonlinear elements around their working points, and this holds true for any semiconductor device, not just for diodes only. Equation (9.67), combined with (9.68), describes quite accurately the direct current in the forward bias. For negative bias, Equation (9.67) predicts that the negative current increases monotonically with the inverse voltage, and reaches the asymptotic limit IS . This is not the case for a real diode, due to the breakdown, as described in Section 9.6.2. Therefore, for reverse bias, the current through D follows a different equation

Vdi þ VBREAK VBREAK 1þ ð9:71Þ Idi ðVdi < 0Þ ¼ IS exp VT VT where VBREAK is the breakdown voltage of the diode. If the reverse voltage across the internal diode D equals the breakdown value ðVdi ¼ VBREAK Þ, the negative current reaches the value Idi ðVdi ¼ VBREAK Þ ¼ IS VBREAK VT 1 . In general VBREAK is not directly measurable, in that the application of voltages close to VBREAK can easily cause excessive current with permanent damage to the device. Rather, increasing reverse voltages are applied to the diode until the inverse current reaches a value that is measurable, but not destructive to the device, for instance 1 mA. Let Vd;B and Idi;B be the applied voltage and the diode current under that condition. The negative voltage across the internal diode will be Vdi;B ¼ Vd;B Rs Idi;B . From Equation (9.71), we get

Idi;B VBREAK VBREAK ¼ VT ln 1 ð9:72Þ Vdi;B IS VT Once IS has been found, the only remaining unknown in the transcendental equation (9.72) is VBREAK , which can be found by applying numerical methods.65 For instance, let us consider the diode used for the grey curve in Figure 9.39, and assume that we measured Vdi;B ¼ 5 V; Idi;B ¼ 1 mA . Then the numerical solution of Equation (9.71) gives VBREAK ffi 4:422 V. Figure 9.39 (black dashed curve) plots the inverse current of this diode.66 As seen in Section 9.6.2, a reverse-biased diode exhibits a voltage-dependent capacitance, provided that the reverse voltage amplitude is well below the breakdown. In that condition, the current (both DC and AC) through the internal diode D of the model in Figure 9.38 is very low, and the effect of the junction capacitance becomes dominant. The model in Figure 9.38 includes three capacitors: Cp for the package and the coupling between the two leads, Cd;lin for the interelectrode capacitance, and Cj to take the junction capacitance into account. The first two are linear elements, thus their capacitance is independent of the voltage; the last is nonlinear in that it presents a voltage-dependent capacitance. The junction 65

See Mathcad file 08_Diode_Breakdown.MCD. In the SIMetrix file 10_Diode_I_V_Curve.sxsch, the value for the breakdown parameter is slightly different (BV ¼ 4.195), because the SPICE implemented in that program is more complex than we described, in order to take some additional effects into account. 66


353

capacitance decreases as the voltage applied to the diode becomes more negative, as seen in Section 9.6.2. A well-known and accurate formula for the junction capacitance is Cj ðVdi Þ ¼

Cj0 1

Vdi Vf

g

ð9:73Þ

where: .

Cj0 ¼ Cj ðVdi ¼ 0Þ is the zero-bias junction capacitance.

.

Vf is the built-in potential of the junction.

.

g is the capacitance slope parameter.

From Equation (9.72) it follows that the junction capacitance tends to zero at high reverse voltages. Usually, Cj reduces to negligible values at voltage amplitudes much smaller than the breakdown. Nevertheless, the total capacitance in parallel with the internal diode is always non-zero, due to the interelectrode capacitance Cd;lin . Thus the total capacitance in parallel with the internal diode is Cj ðVdi Þ ¼

Cj0 1

Vdi Vf

g þ Cj;lin

ð9:74Þ

The total capacitance given in (9.73) is the nonlinear counterpart of the capacitance Cd present in the model of Figure 9.34a. The four parameters Cj0 ; Vj ; g; Cj;lin can be computed if four values of Cd have been extracted at four different reverse bias points. Let Vtest;k be the DC voltages and Cj ðVtest;k Þ the corresponding capacitance of the linear model, with k ¼ 1; 2;3; 4. By substituting the four couples of measured values into Equation (9.74), we obtain a nonlinear system of four equations with the four unknowns Cj0 ; Vj ; g; Cj;lin :

Cj0 g þ Cj;lin Cj Vtest;k ¼ ðk ¼ 1 . . . 4Þ V 1 Vtest;k j which can be solved with numerical methods. If more than four points are available, a curve-fitting procedure can be implemented, resulting in a more accurate result. An efficient way to compute the junction capacitance parameters can be found in [4], which also describes a method for the direct high-frequency measurement of Cj .

9.7.2.2 FET nonlinear nodels It is possible to obtain the transistor nonlinear models by applying the same basic concepts described for the diode. Basically, the starting point is the linear equivalent circuit. Nonlinear elements replace the linear ones in modelling the effects that are more related to semiconductor devices, while keeping unchanged the secondary parasitic elements, like interelectrode capacitance, bond wire inductances, contact resistances. Figure 9.40a shows the nonlinear model of a high-frequency FET; the typical application of such a model isin chipdepletion devices.The model of a packaged device requires additional linear elements, asin the linear case shown in Figure 9.34b. The network is analogous to its linear counterpart in Figure 9.34b: .

The linear elements Lg ; Rg ; Ls ; Rs ; Ld ; Rd ; RI ; Cds are unchanged,

.

The nonlinear VCCS Ids ðvI ; vds Þ replaces the linear one gm vI and the drain–source resistance Rds .

.

The gate–source and gate–drain capacitances Cgs ; Cds are nonlinear, representing junction capacitances; for this same reason, two diodes Dgs ; Dds are in parallel with the corresponding junction capacitances.

354

MICROWAVE AND RF ENGINEERING Dgd L g Rg

Gate

Ld Rd

Cgd(vds -v i ) + vi Dgs

Drain

+

Cgs (vi ) -

vds Ids(v,v i ds )

- Cds

+

+

Vgs -

Ri

Vds -

Rs Ls Source

(a)

Rd Gate

+ + Vgs -

(b)

+ vds

vi Ids(v,v i ds)

Rs

-

Drain + Vds -

Source

Figure 9.40 Nonlinear equivalent circuit of a FET: (a) complete model; (b) simplified DC model.

The nonlinear capacitances Cgs ; Cds and the associate diodes follow the same equations (9.73) and (9.67), respectively, as for the diode. As can be seen, the capacitance at high levels of reverse bias tends to zero; the residual value due to the interelectrode capacitance can be modelled by two linear capacitors in parallel with the nonlinear ones, although Figure 9.40a does not show them. In the normal operation of a depletion FET, both the gate–source and gate–drain junctions are reverse biased, thus the effect of the interdicted diodes Dgs ; Dds is negligible, unless one of the two junction voltages approaches breakdown. The procedure to compute the parameters of the nonlinear model in Figure 9.40 consists of four main steps: 1. The transistor is measured – usually in the common source condition (source to ground) – at different bias combinations ðVds ; Vgs Þ. The extraction of the linear model for each bias condition is helpful but not essential. 2. The DC curves Ids ðVds ; Vgs Þ of the FET are measured. 3. The nonlinear model parameters are computed by minimizing the deviation between the S parameters measured in step 1 and the ones obtained by linearizing the nonlinear model under the same bias conditions. This step requires the application of numerical methods and sometimes of optimization programs. The linear elements Lg ; Rg ; Ls ; Rs ; Ld ; Rd ; RI ; Cds obtained from the linear models computed at the different bias conditions – if available – constitute a good starting point for the nonlinear model.


355

4. More than the minimum deviation on the S parameters, we also impose that the DC curves obtained from the nonlinear model must be as close as possible to the measured ones. Requirement 3 is sometimes in contrast with 4. In this case, the result is a convenient compromise between the two needs, depending on the application of the model. The extracted nonlinear parameters are such as to minimize the deviation between the S parameters of the nonlinear model and ones measured at the same bias condition. Also, the DC curves of the nonlinear model have to be as close as possible to those of the real device. The nonlinear parameter extractions of capacitors Cgs ; Cds needs a curve-fitting procedure with the linear values obtained at different bias conditions, exactly as described for the diode. Figure 9.41 shows the DC curves of the same GaAs MESFET presenting the S parameters shown in Figure 9.35 and the linear model of Figure 9.34b. The dots represents the measured points, the solid lines are the values obtained from the model, whose parameters have been optimized to fit with the experimental data. The curves in Figure 9.41 have a drain–source voltage continuously varying from 0 to 4 V, while the gate–source voltage equals 0, 0.2 and 0.4 V. Normally the Vgs usable range extends from zero to the pinch-off voltage Vp of the FET: above zero the gate–source is conducting, with a consequent risk of damage; below Vp the drain–source current vanishes, so any further reduction of Vgs produces no appreciable modulation of Ids. Looking at the curves in Figure 9.41, we see that: .

For small Vds (say Vds < 0:5) Ids increases proportionally to Vds , with a slope which decreases with Vgs . That Vds region is sometimes referred to as the triode operation region of the FET, from the homonym thermionic tube device having similar DC curves.

.

For Vds around 1 V, the curves show a knee.

.

For higher Vds, Ids is almost constant, barely depending on the drain–source voltage: again, some authors use another tube device analogy, and name this the pentode region of the FET. Sometimes, the pentode region is also referred to as the current saturation region. Figure 9.41 indicates both the triode and pentode regions, although such delimitation is somewhat qualitative.

measured model (∂Ids/∂Vds) Vds

60 50 triode

pentode region Vgs = 0 V

Ids , mA

40 30

-0.2 V

20 10

-0.4V

0 0

1

2

3

4

5

Vds , V

Figure 9.41

DC curves of the GaAs MESFET presenting the scattering parameters of Figure 9.35.

356


Such behaviour is common to all the transistors considered in this book (BJT, JFET, MESFET, HEMTand MOSFET) and is due to the velocity saturation of the semiconductor carriers. To explain such an effect, we must consider that the carrier mobility67 is not constant, rather it depends on the applied electric field and decreases with it. Thus, if the external field exceeds a critical limit, there is no further increase in the carrier velocity or the velocity is saturated, and the same thing happens with the current. In normal operation, the drain–source voltage falls into the current saturation region; for example, in the FET of Figure 9.41, Vds could range from 2 to 4 V. From a DC point of view, if the FEToperates with its junctions reverse biased – as it usually does – the nonlinear model in Figure 9.40a simplifies to that in Figure 9.40b. In the simplified network we removed the two diodes Dgs and Dgd because they are reverse biased, and therefore do not conduct any appreciable direct current. Moreover, we removed all the capacitors and short-circuited all the inductors and the two gate resistors Rgs and RI , because there is no direct current flowing through the gate. We can now exploit the simplified circuit in Figure 9.40b to explore some details about the nonlinear generator Ids ðvI ; vds Þ. Many authors [5–7] have proposed mathematical expressions for this element, with differing complexity, accuracy and computational difficulty. The Curtice quadratic [7] model gives one of the simplest equations for the drain current: (

2

b vI Vp ð1 þ l vds Þ tanhða vds Þ vI > Vp ð9:75Þ Ids ðvI ; vds Þ ¼

0 vI Vp Note that the variables vI ; vds represent internal voltages of the model, and differ from their external counterparts Vgs ; Vds because of the voltage drop on the resistors Rs ; Rd . Equation (9.75) states that Ids vanishes if the internal control voltage vI becomes lower than the pinchoff voltage Vp . On the other hand, if Vgs is such that vI Vp , this implies that Ids ¼ 0, and consequently the voltage across Rs vanishes. Thus Equation (9.75) also implies that Ids ¼ 0 for Vgs Vp. In normal operation ðvI ; Vgs > Vp Þ, Equation (9.75) is the product of three factors. The third term, tanhða vds Þ, models the knee on the drain current due to carrier velocity saturation.68 The second factor ð1 þ l vds Þ models the slight slope of Ids in the current saturation region. The factor b ðvI Vp zÞ2 describes the dependence of the drain–source current on the gate–source voltage. At relatively high values69 of Vds we have tanhða vds Þ 1, and if the parameter l is such that l vds 1, Equation (9.75) simplifies to the classical equation of the FET in the drain current saturation region

2 Ids ðvI ; vds Þ ffi b vI Vp ¼ Idss

2 vI 1 Vp

ð9:76Þ

where Idss ¼ b Vp2 is the saturation current of the FET, i.e. the current70vI ¼ 0. If we require the model current to present a minimum deviation from the measured values, it is possible to compute the parameters b; Vp ; l; a. The computation has to take into account the voltage drop on the resistors Rs ; Rd . Alternatively, it is possible to impose Rs ¼ Rd ¼ 0: this simplifies the analysis, but also degrades the overall accuracy of the model, particularly at high frequencies. Assuming Rs ¼ Rd ¼ 0, and applying the curve fitting with the DC curves of the FET in Figure 9.41, give the parameter values b ¼ 66 mA V2 , Vp ¼ 0:724 V, l ¼ 0:047 per V, a ¼ 1:494 per V. Similar to what was discussed for the diode, an alternative possibility to determine b; Vp ; l; a consists of deriving the small-signal parameters of the nonlinear model, and requiring that they coincide with the values of the linear model at various bias conditions. This result can be achieved by deriving the 67

See footnotes 46 and 49. The function tanh(x) is zero for x ¼ 0, monotonically increases with x, and asymptotically tends to 1 when x tends to infinity. 69 For example, Vds > 2 V for the FET with the DC curves in Figure 9.41. 70 Manufacturers generally specify Idss as the current for Vgs ¼ 0, which is much easier to measure. 68


357

transconductance and the output resistance of the generator described by Equation (9.74), with the assumptions vI ; Vgs > Vp , tanhða vds Þ 1: gm;nonlinear ¼

qIds ðvI ; vds Þ ¼ 2b vI Vp ð1 þ l vds Þ qvI

2 1 qIds ðvI ; vds Þ ¼ ¼ b vI Vp l Rds;nonlinear qvds

ð9:77Þ ð9:78Þ

The linearized parameters gm;nonlinear ; Rds;nonlinear are functions of the bias condition, and must be as close as possible to the corresponding linear parameters gm ; Rds of the linear model obtained at the same conditions. For each couple of Vgs;k ; Vds;k , the corresponding couple of the internal voltage is71 vI;k ¼ Vg;ks þ Ids;k Rs ; vds;k ¼ Vds;k ðRd þ Rs Þ Ids , where the subscript k ¼ 1 . . . N scans the measured points. Now, substituting the values vI;k ; vds;k into Equations (9.76), (9.77) and equating them to the corresponding linear gm ; Rds, we obtain a system of 2N equations in the three unknowns b; Vp ; l. Hence one single measurement point is not sufficient for solving the system, while two points are too many. Consequently, curve fitting is the most advisable method. The Statz model [6] uses a slightly more sophisticated equation for the drain current 8

2 >

 Vp ð9:79Þ Ids ðvI ; vds Þ ¼ 1 þ y vI Vp >

: 0 vI Vp where

8 vds 3 > > > > > :1

3 vds < a

3 vds a

Comparing Equation (9.79) with (9.75), we can see that the first models a dependence on the saturation current by the gate voltage. Moreover the Statz model replaces the hyperbolic tangent with a piecewise polynomial equation in the velocity saturation model, with a certain reduction in the required computer resources.

9.7.2.3 MOSFET nonlinear models A complete description of the various MOSFET nonlinear models is given in [3], [8] and [9]. For our needs, we will limit ourselves to presenting some basic descriptions of the concepts that will be used to describe the MOSFET operation in some high-frequency circuits. The nonlinear equivalent circuit of Figure 9.40 is basically valid for the MOSFET too, with minor modifications. Figure 9.42 shows a simplified version of the MOSFET nonlinear model under the hypothesis of the substrate connected to the source.72 Comparing the network in Figure 9.42 with the one in 9.40a, we see that: .

The two diodes Dgs ; Dgd from the gate to the source and to the drain disappear, in that the MOSFET has no junction between those two couples of electrodes.

.

A new diode Dds is present between the drain and the source. This diode really models the junction between the drain and the substrate, which is assumed to be connected to ground, like the source.

71 72

Keep in mind that if the linear model is known, Rs and Rd are known as well. As we assumed in Section 9.6.5.

358

MICROWAVE AND RF ENGINEERING Cgd (vds -v i )

L g Rg

Gate

+ + Vgs -

vi Cgs(vi) -

Ld Rd Drain

+ vds Cds Ids(v,v i ds ) -

Rs

+ Dds

Vds -

Ls Source

Figure 9.42

Nonlinear equivalent circuit for a MOSFET.

.

The drain–source capacitance is nonlinear, in that it includes the drain–bulk junction capacitance.

.

The resistor RI is short-circuited.

Note also that the capacitances Cgs ; Cgd remain nonlinear, despite no longer being junction capacitances. A sufficiently accurate drain–source current law for the MOSFET is 8 v2ds 2 > > b v ð V Þ v ðvI > VTH ; vds vI VT Þ > I TH ds > 2 > < ð9:80Þ Ids ðvI ; vds Þ ¼ b > ðvI VTH Þ2 ðvI > VTH ; vds > vI VT Þ > > >2 > : 0 vI VTH where VTH is the threshold voltage of the device, and plays a similar role as the pinch-off voltage for the MESFET. The DC characteristic curves of a MOSFET resemble those of the MESFET. Equation (9.79) implies that the drain–source current exhibits a triode region, a knee and a pentode region. The transition between triode and pentode operation occurs close to the carrier velocity passing from the linear ðvI > VTH ; vds vI VT Þ to the saturated ðvI > VTH ; vds > vI VT Þ regime. All the techniques for the nonlinear parameter extraction described for the MESFET remain valid, thus we will not repeat them.

9.7.2.4 BJT nonlinear models The main nonlinear model for the bipolar transistors is the Gummel–Poon model, as described in depth in [3] and [10]. For our needs, we will just present a basic description of the concepts that will be used to describe the MOSFET operation in some high-frequency circuits. Figure 9.43 shows the nonlinear equivalent network of a bipolar transistor. The two diodes Dbe ; Dbc with the associated junction capacitances Cbe ; Cbc model the base–emitter and base–collector junctions, respectively. The nonlinear controlled generator and the base current follow the laws 2Is exp nFvbeVT vcb rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ic ¼ ð9:81Þ 1 þ VAF 1 þ 1 þ 4 IIKFs exp nFvbeVT

Ic vbe Ib ¼ þ Ise exp bF nE VT

ð9:82Þ


Icc /βR Lb Rb

Base

Dbc Iee /βF

Ld Rd

Cbc

Dbe

359

Collector

Cbe

Ic

Cce

Rs Ls Emitter

Figure 9.43 BJT nonlinear equivalent circuit. 8 7

Ibe = 50 µA

6 40 µA

I ce , mA

5 30 µA

4 3

20 µA

2 10 µA

1 0

0

1

2

3

4

5

Vce, V

Figure 9.44

DC curves of the bipolar transistor MBC13900.

The parameters Is ; Ise ; nF ; nE play a similar role to the saturation current and ideality factor that we already met in describing the diode. The parameter bF is the forward current gain of the transistor, while IKF models the effective current gain Ic =Ib decreasing at high collector currents. The parameter 1=VAF introduces a slope in the collector current, similar to l in the Curtice FET model. Figure 9.44 shows the DC curves73 of the bipolar transistor MBC13900, manufactured by Freescale Semiconductor, as obtained from the nonlinear model supplied by the manufacturer itself.74 Once again, the curves present triode and pentode regions, as with FETs, with the transition region – the knee – located around Vce 0:4. Also, the techniques for the extractions of the BJT nonlinear parameters are essentially the same as for the FET, with just minor modifications.

73

See the SIMetrix file 11_MBC13900_DC_Curves.sxsch. Model parameters taken from the component data sheet, Document Number MBC13900/D Rev. 1.1, 06/2005, Table 9, pp. 18–19. Copyright of Freescale Semiconductor, Inc. 2008. Used with permission. 74

360


Bibliography 1. G. Gonzelez, Microwave Transistor Amplifiers, Prentice Hall, Englewood Cliffs, NJ, 1984, pp. 80–87. 2. H. A. Haus, W. R. Atkinson, G. M. Branch, W. B. DavenportJr, W. H. Fonger, W. A. Harris, S. W. Harrison, W. W. McLeod, E. K. Stodola and T. E. Talpey, ‘Representation of noise in linear twoports’, Proceedings of the IRE, Vol. 48, pp. 69–74, 1960. 3. P. Antognetti and G. Massobrio, Semiconductor Device Modeling with SPICE, McGraw-Hill, New York, 1988. 4. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, Chapter 7, pp. 420–428. 5. A. Materka and T. Kacprzak, ‘Computer calculation of large-signal GaAs FET amplifier characteristics’, IEEE Transactions on Microwave Theory and Techniques, Vol. 33, No. 2, pp. 129–135, 1985. 6. H. Statz, P. Newman, I. W. Smith, R. A. Pucel and H. A. Haus, ‘GaAs FET device and circuit simulation in SPICE’, IEEE Transactions on Electron Devices, Vol. ED-34, No. 2, pp. 160–169, 1987. 7. W. R. Curtice, ‘GaAs MESFET modeling and nonlinear CAD’, IEEE Transactions on Microwave Theory and Techniques, Vol. 36, No. 2, pp. 220–230, 1988. 8. N. Camilleri, J. Costa, D. Lovelace and D. Ngo, ‘Silicon MOSFET, the microwave device technology for the 90’s’, IEEE MTT-S International Microwave Symposium Digest, June 1993, p. 545. 9. D. P. Foty, MOSFET Modeling with SPICE: Principles and Practice, Prentice Hall, Upper Saddle River, NJ, 1997. 10. H. K. Gummel and H. C. Poon, ‘An integral charge control model of bipolar transistors’, Bell System Technical Journal, Vol. 49, May/June, p. 827, 1970.

Related files Ansoft files 01_Power_Divider_with_Cables.adsn. Simulates the assembly in Figure 9.3a, to compute amplitude and phase imbalance between the two output ports. 02_Power_Divider_with_Cables_more_Ideal.adsn. As previous file, but under the simplifying hypothesis that the power divider is perfectly matched. 03_Filter_Assembly.adsn. Analyzes the filter assembly in Figure 9.7. 04_BJT_Linear_Model.adsn. Analyzes the BJT linear model in Figure 9.36, together with the scattering parameters of Table 9.3.

Mathcad files 05_Low_Pass_Example2.MCD. Implements the synthesis formulae for the lowdpass filter in Figure 9.7. 06_Band_Pass_Noise_Bandwidth.MCD. Computes the noise bandwidth of bandpass filters obtained from various approximations. Dissipation loss effects are also included. 07_Saturation_Polynomial_Curve_Fit.MCD. Computes the polynomial approximation of a saturated response.


361

08_IP3_Cubic.MCD. Implements some basic calculations on the single tone and dual tone response of a nonlinear function. 09_Diode_Breakdown.MCD. Computes the diode breakdown parameters, starting from the experimental data.

Simetrix files 10_Diode_I_V_Curve.sxsch. Analyzes the diode to obtain the DC curve in Figure 9.39. 11_MBC13900_DC_Curves.sxsch. Analyzes the bipolar transistor MBC13900, to obtain the DC curves in Figure 9.44.

10

Microwave control components 10.1 Introduction This chapter describes high-frequency components that use diodes and transistors in a passive manner. In other words, the semiconductor devices in the components described do not operate to increase the signal energy, rather they work to modify some signal parameters. Furthermore, the components discussed in this chapter have the common characteristic of not producing nonlinear distortions at high frequency, at least in principle. The importance of the control components comes from their wide use in communication equipment and test instruments. Section 10.2 deals with the switch, which is probably the simplest control component, presenting diode, FET and MEMS components. Section 10.3 illustrates some variable attenuators. Section 10.4 describes the most frequently used types of phase shifters. Within this chapter we will use the words ‘reference impedance’ and ‘50 O’ as synonymous, unless specified differently, since 50 O is the most commonly used impedance.

10.2 Switches Microwave/RF switches are components having n þ 1 RF ports and m control ports. The DC voltage or currents at the different control ports determine the configuration of the connections between the different RF ports. Figure 10.1 shows the electrical symbols for the most common types of switches: in all these components one specific RF port can be connected to one of the remaining ones. In the general case, the connected port is selectable among n, by means of a suitable combination of control voltages: we have the single pole, n throw (SPnT), whose symbol is the one in Figure 10.1c. The simplest SPnT is the single pole, single throw (SPST), having n ¼ 1; Figure 10.1a shows the relative symbol. The RF port P1 is connected or not to the RF port P2 , depending on the combinations of the voltages/currents applied to the control ports V1 to Vm . In the remaining part of this section we will refer to the used combinations of the voltages/currents applied to the control ports as the control stimulus. At this point, we will clarify the meaning of the words connected and non-connected, using the SPST as a reference. In this regard, we will define the SPSTas a linear, passive and reciprocal two-port network.


364


P2

P1

(a)

V1

Vm

P2 P1

P3

(b)

V1

Vm

P2

P3 P1

P4

Pn + 1

(c)

V1

Vm

Figure 10.1 Symbols for some switches: (a) SPST; (b) SPDT; (c) SPnT.

MICROWAVE CONTROL COMPONENTS

365

Furthermore, we will assume that such a network has two working states, low and high attenuation, which are determined by the control stimulus. In the low-attenuation state, the transmission coefficient amplitude is as close to 1 as allowed by the component performances;1 consequently, reflection coefficients present small amplitudes.2 Conversely, the insertion loss in the high-attenuation state is quite high, of the order of 40 dB or more. The reflection coefficient could present either high or low amplitude. In the first case, the reflection coefficient amplitude approaches unity, in the latter case there is a good matching, similar to the lowattenuation state. Those two cases define the reflective or absorptive switch, respectively. The low- and high-attenuation states are sometimes referred to with ON and OFF, respectively. Other possible eponyms for low and high attenuation are transmitting and isolating state, respectively. We will also use this shorter denomination in the case where there is no possibility of confusion with the eponymous states of the switching elements. Also, we will say that in the low- (respectively, high-) attenuation state the RF port P2 is connected to (not connected to, or isolated from) port P1 . The extension of the above concepts to the general case proceeds as follows: (a) SPnT is a linear, passive and reciprocal ðn þ 1Þ-port network, having n meaningful working states, denoted by superscripts 1 to n, respectively. (b) One and only one RF port can be connected to all the remaining ones. Let the first be P1 and the others P2 to Pn þ 1 . We will refer to P1 as the common port, while P2 to Pn þ 1 will be the selectable ports. Sometimes the common port and the selectable port are named ‘input port’ and ‘output port’, respectively, but this is improper, because generators and loads can be connected at any port of the switch. (c) Depending on the control stimulus, SPnT assumes the state k among the possible nðk ¼ 1 . . . nÞ. We will refer to the port Pk þ 1 as the activated or the selected one. In this particular state, the insertion loss between P1 and Pk þ 1 is small, and the return loss at ports P1 and Pk þ 1 , together with the insertion loss from P1 to all other ports but Pk þ 1 , is high: ðk Þ ðk Þ ðk Þ ðkÞ 0 sk þ 1;1 1; s1;1 sk þ 1;k þ 1 0; sm þ 1;1 m6¼k

(d) As can be seen, in any state k the common port P1 and the connected (or the active, or the transmitting) port Pk þ 1 are always impedance matched: all the remaining ports – which are isolated – could be matched or not. In the first (latter) case we have an absorptive (reflective) switch. Some mixed combinations are also possible, with some of the ports P2 to Pn þ 1 presenting low reflection in any state, while the remainder do not. Furthermore, absorptive switches usually present higher ON insertion loss and/or lower OFF insertion loss than their reflective counterparts.3 Further clarifying, we have four main specifications for an SPnT: 1. Insertion loss in the ON state, defined as the insertion loss between the common port P1 and the activated port Pk ðk ¼ 2; . . . ; n þ 1Þ . It is possible that different ports have different specifications of the insertion loss. Sometimes, the insertion loss in the low attenuation is simply referred to as the insertion loss.

1 The insertion loss in the low attenuation results from many factors, such as the maximum working frequency, the relative bandwidth, the required isolation in the high-attenuation state and the technology used. The ON insertion loss of semiconductor-device-based SPST typically falls within the range 1–3 dB. 2 The return loss is usually better than 10 dB in the low-attenuation state. 3 Realized with the same technology, frequency range, switching speed, and so on.

366

MICROWAVE AND RF ENGINEERING 2. Isolation, defined as the insertion loss in the OFF state between port P1 and any of the nonactivated ports Pm þ 1 ðm 6¼ kÞ. The isolation could differ from port to port, as with the insertion loss, and also could change with the selected port. 3. Maximum input power, defined either as the maximum applicable rate without permanently damaging the component, or as the value that guarantees a specified distortion or compression level. The 1 dB compression point is often specified.4 Note that any of the SPnT ports can be used as input or output, at least in principle. For instance, SPnT can be used to select one of the n signals present at the ports P2 to Pn þ 1 , and to deliver it to port P1 , or to distribute the signal at port P1 to one of P2 to Pn þ 1 . The maximum power could depend on the RF port used as input, and could also assume different values in the low- to high-attenuation states. Also, during the transition from low to high attenuation and vice versa, the internal elements of the switch could assume critical states that consistently reduce the power handling. For this reason, the maximum power is normally specified assuming that the switch changes its state with zero power applied (cold switching). The maximum RF power tolerated by SPnT while commuting (hot switching) is quite low, unless the component is specifically designed for that operation. 4. Switching time, defined as the difference tON tDRIVE , where tON denotes the instant when the switch RF output power reaches its steady state value, and tDRIVE is the instant when all the control voltages/currents definitively reach their final value, after the commutation. Sometimes, it is also defined as tDRIVE tOFF switching time, but normally the commutation from low to high attenuation is faster than the opposite case.5

The simplest switch type after the SPST is the single pole, double throw (SP2T) or (SPDT), whose symbol is shown in Figure 10.1b. The importance of the SP2T comes from its wide use as an antenna duplexing device in transmitter–receivers6 and as a building block of more complex control components, such as step attenuators and binary phase shifters.7 From a realization point of view, SPnT generally consists8 of n identical SPSTs, with all their ports P1 connected together, while the ports P2 of the SPST becomes the ports P2 ; P3 ; . . . ; Pn þ 1 of the SPnT. Figure 10.2a shows the connection of n SPSTs to form one SPnT. The above-mentioned n SPSTs are known as the branches of the SPnT. Because of its internal structure, one SPnT can be described in terms of its SPST. Compared with a standalone SPST, the SPnT branch has the additional requirement of presenting a high impedance9 to port P1 when not selected;10 this minimizes the disturbance caused by the non-activated branches on the activated one. Figure 10.2b shows the typical internal structure of an SPST branch. It consists of a series of a switching element (SW1 ) and m shunt switching elements (SW2 to SWm ). A further series element could be present at ports P2 ; P3 ; . . . ; Pn þ 1 : this is not essential for our description, therefore Figure 10.2 does not show it. SW1 can only assume high or low impedance;11 correspondingly SW2 to SWm behave like low or high impedance. The control quantities V1 to Vm must be such as to set the corresponding elements in the described conditions. 4

See Section 9.5.5 for the definition of the 1 dB compression point. The switch is fully transmitting when all of its components reach the corresponding condition. In the ON–OFF transition, the switch is (at least) partially isolating when just a few of its elements begin to assume the condition. Hence the transition OFF–ON (ON–OFF) is determined by the slowest (fastest) device and/or control signal. 6 See Section 15.5.1. 7 See Sections 10.3 and 10.4 below. 8 Other configurations are possible, like multiple cascaded SPDTs. For example, one SP4T can be obtained by cascading one SPDT with two other SPDTs. 9 Ideally, open circuit. 10 Keep in mind that only one of the ports P2 to Pn þ 1 is activated, all the remaining ones being in the OFF state. 11 Excluding the switching transients. 5


P1

1

SPST1

2

P2

1

SPST2

2

P3

1

SPSTn

2

Pn+1

(a)

SPnT

1

P1

367

SW 1

TL1

TL2 SW2

V1

TL3 SW3

V2

TLm

2

P2 , P3 ,.. Pn+1

SWm

V3

Vm

(b)

Figure 10.2

SPnT structure: (a) schematic of the principle; (b) electrical diagram of one SPDT cell.

The network in Figure 10.2b comprises transmission lines and switching elements together with their associated control components, although these are not shown. Many physical devices can be used to realize the switching elements: electronic devices like diodes and transistors, or electromechanical devices. In the first (latter) case some bias networks (electrostatic, electromagnetic or piezoelectric actuators) complete the SPST structure. The most frequently used electronic switching elements are PIN diodes, gallium arsenide MESFETs and MOSFETs.12 Figure 10.3 shows the schematics of some SPST realizations, employing the switching elements that subsequent sections will describe. The SPSTs in Figure 10.3 have one series (SW1 , D1 , Q1 ) and two shunt elements (SW2 , SW3 , D2 , D3 , Q2 , Q3 ); other realizations could present more shunt elements and/or an additional series element close to the RF port P2 . Finally, the schematics in Figure 10.3 include the bias networks needed for the different switching devices. Sections 10.2.1 and 10.2.2 below describe the PIN diode and FET switches, while Section 10.2.3 deals with special electromechanical switches, known as MEMS.

12 Recently, other materials have replaced or been used in combination with arsenide (phosphorus) and gallium (aluminium, indium). Also, more sophisticated and better performing devices (HFET, HEMT, PHEMT) have replaced the MESFET. However, the new devices operate in the same way as the MESFET used as the base for our description.

368

MICROWAVE AND RF ENGINEERING TL 1

SW 1

P1

TL 3

TL 2 SW2

P2

SW3

(a) C1 V1 L2

D1

TL1

P1

TL2A

TL2B

D2

L1

TL3

C2

P2

D3

(b)

TL1

Q1

P1

TL2

Q2

TL3

R1

R2

V1

P2

Q3 R3

V2

(c)

C1 V2 L2 C3

P1

SW1 L1

V1

TL1

TL2A SW2

TL2B

TL3

C2

P2

SW3

C4 (d)

Figure 10.3 (d) MEMS.

10.2.1

SPST with three switching elements: (a) ideal elements; (b) PIN diodes; (c) FETs:

PIN diode switches

The acronym PIN derives from the device structure. It derives from that of the standard p–n (PN) diode, illustrated in Figure 9.28a, with the addition of a layer of intrinsic13 (I) material, placed between the p- and n-zones. PIN diodes present static DC characteristic similar to those of normal junction diodes, which follow relation (9.68). Let us initially consider a normal PN diode, and assume it is biased with a direct current Id0 with a superimposed small sinusoidal current of amplitude is and frequency fs (with os ¼ 2pfs ). The total 13

Not doped.


369

current flowing through the diode is then Id ðtÞ ¼ Id0 þ is cos ðos tÞ

ð10:1Þ

Substituting expression (10.1) into the diode equation (9.68), we obtain the voltage across the diode Id0 is þ cos ðos tÞ þ 1 þ Rs Id0 þ Rs is cos ðos tÞ ð10:2Þ Vd ðtÞ ¼ n VT ln IS IS The voltage (10.2) is non-sinusoidal, but it is still periodic with the same fundamental frequency fs as the current (10.1). Hence, expanding the function (10.2) in Fourier series, we obtain all the harmonics of the diode voltage.14 In our simplified case, is Id0 , we can approximate Equation (10.2) by a small-signal expression. Rearranging expression (10.2), and applying the approximation x 1 ) ln ð1 þ xÞ x, we obtain the small-signal approximation of the diode voltage Id0 n VT Vd ðtÞ ffi n VT ln þ 1 þ Rs Id0 þ þ Rs is cos ðos tÞ ð10:3Þ IS Id0 þ IS The sum of the first two terms of Equation (10.3) equals the DC voltage corresponding to the direct current Id0 . The third term of Equation (10.3) is proportional to the applied sinusoidal current, by the ðDiodeÞ constant RIncremental ¼ n VT =ðId0 þ IS Þ þ Rs . That constant is the incremental resistance of the diode, biased with the direct current Id0 . It coincides with the derivative of the diode voltage (9.69) with respect to the current ðDiodeÞ

RIncremental ¼

@Vd n VT ¼ þ Rs @Id Id þ IS

ð10:4Þ

Equation (10.4) states that the diode incremental resistance decreases with the forward bias current. This conclusion is valid for PIN diodes as well. The two bias conditions that are of interest for the operation of a diode as a switching element are (i) strong forward bias and (ii) reverse bias. In case (i) a forward current much higher than the saturation current flows through the diode, and its incremental resistance becomes h i Id0 IS n VT ðDiodeÞ ðDiodeÞ RIncremental;ON þ Rs ; lim RIncremental ¼ Rs ð10:5Þ Id0 ! 1 Id0 In case (ii), due to the reverse bias, the diode current is close to zero, and the incremental resistance approximates as ðDiodeÞ

I ¼0

RIncremental d0¼

n VT n VT þ Rs Is Is

ð10:6Þ

Assuming the parameters of the diode described in Section 9.7.2 Is ¼ 0:1995 10 12 ; n ¼ 1:9271; Rs ¼ 2:864 OÞ and a forward bias current of 20 mA and zero reverse current, 14 Equivalently, expression (10.2) can be considered as a function of the variable is cos ðos tÞ and expanded in a Maclaurin series

Vd ðtÞ ¼

1 X

ak is k cosk ðos tÞ

k¼0

where ak ¼ ðk!Þ 1 d k ½n VT lnðId0 =IS þ x þ 1Þ þ Rs Id0 þ Rs x=dxk Thus, a1 ¼ n VT =ðId0 þ IS Þ þ Rs , a2 ¼ 0:5 n VT ðId0 þ IS Þ 2 , and so on. Then it is possible to derive the Fourier coefficients from the Maclaurin series by expressing the terms cosk ðos tÞ as combinations of harmonics cosðm os tÞ ðm ¼ 0; . . . ; kÞ .

370


expression (10.4) gives ðDiodeÞ

RIncremental;ON 5:3 O;

ðDiodeÞ

RIncremental;OFF 2:5 1011 O

If the signal current varies sufficiently fast, i.e. for high values of fs , expression (10.4) is no longer valid for PIN diodes, and so it is with its approximations (10.5) and (10.6). The intrinsic region, placed between the P- and N-regions of the PIN diode, causes a lack of validity in Equation (10.4), but still PIN diodes exhibit variable incremental resistance that decreases with the forward direct current. We will not describe the PIN diode physics in depth.15 For our needs it is sufficient to say that, at high frequencies, PIN diodes approximate linear variable resistors much better than PN or Schottky diodes. More precisely, PIN diodes are characterized by the parameter t, called carrier lifetime, which increases with the thickness of the I-region. PIN incremental resistance follows Equation (10.4) only if the current variations are slower than t. In particular, if the current is a direct current with a superimposed sinusoid, like Equation (10.1), then (10.4) is valid only if fs 1=t. Nevertheless, even if fs > 1=t, the two equations (10.5) and (10.6) remain qualitatively valid, in that: (a) Forward-biased PIN diodes present a low incremental resistance, which decreases as the forward current increases, although not proportionally to 1=Id0 and differently from what Equation (10.5) states. Furthermore, Equation (10.5) approximates the ON resistance of the junction diode for small signals, i.e. for Is tending to zero. As Is increases, the nonlinear behaviour of the diode becomes more relevant and consequently the linear approximation loses accuracy. A remarkable property of PIN diodes is the high linearity in the high conduction (or briefly ON) state. If ðDiodeÞ fs 1=t then RIncremental stays almost constant with Is , much more than happens for a junction diode. The DC control current of the PIN diode can be smaller – sometimes by orders of magnitude – than the high-frequency current. (b) Inverse-biased PIN diodes present high incremental resistance16 that is almost independent of the applied reverse voltage, providing that this is higher than a specified limit, which depends on the device. At low frequency the diode is interdicted until the anode–cathode voltage is negative all the time. The second important property of the PIN diode is that it stays in the low conduction (or shortly OFF) state if the anode–cathode voltage becomes positive for short time intervals, in some cases just slightly shorter than half a sine cycle. The concepts (a) and (b) are summarized by saying that PIN diodes present a kind of inertia in the I–V characteristic. The device incremental resistance is not affected by the amplitude of rapidly varying excitations superimposed on the DC bias, rather it tends to remain constant and only depends on the DC bias itself. Such incremental resistance exhibits high linearity with respect to rapid variable excitations, provided that their spectral energy is non-negligible only for f 1=t. The main drawback of the lifetime carrier mechanism is the relatively slow response of the PIN diode to the switching command: PIN SPnT switching time ranges from fractions of a microsecond to a few microseconds. We can now again consider the linear equivalent circuit of the diode, shown in Figure 9.32a. When the diode is strongly forward biased, its series resistance practically coincides17 with Rs , thus the network resulting from the combination of Cd , Rd , Rs almost coincides with Rs only. The maximum working

15

The interested reader can find some more details in [1–3]. ðDiodeÞ Equation (10.4) gives RIncremental;OFF 2:5 1011 O for the diode described in Section 9.7.2, as we have seen. However, the high-frequency OFF incremental resistance is difficult to evaluate, because some other parasitic effects mask it, like the residual capacitance. ðDiodeÞ 17 From Equation (10.4) we have that Rd ¼ RIncremental Rs ¼ n VT =ðId þ IS Þ, which can be arbitrarily reduced by increasing the forward current Id . 16


371

frequency is such that the reactance associated with Ls at that frequency equals Rs ðPIN Þ

fMAX;ON ¼

Rs 2p Ls

ð10:7Þ

Since, usually, the value of (10.7) falls well below the parallel resonance of Ls with Cp , it is pffiffiffiffiffiffiffiffiffiffi ðPIN Þ fMAX;ON ¼ Rs =ð2p Ls Þ 1= 2p Ls Cp In the OFF state, the diode assumes a high series resistance, and the junction capacitance tends to zero due to the wide depleted region across the junction: the linear model in Figure 9.34a simplifies to the capacitor Cp . Usually a resistor is considered in parallel with the capacitor, because no device in any state is purely reactive, and PIN diodes are no exception. Thus, the maximum usable frequency in the OFF state for a given device depends on the circuit impedance R0 and is such that the capacitance reactance equals the latter ðPIN Þ

fMAX;OFF ¼

1 2p Cp R0

ð10:8Þ

Hence, in the ON (OFF) state the PIN diode model is an RL series (RC parallel) cell. SPnT uses the diodes in both the states, thus the maximum working frequency is the lowest value between the ones given by Equations (10.7) and (10.8). The true limit is even lower, due to additional parasitic effects,18 and depends on the accepted maximum insertion loss, minimum return loss and minimum isolation. The minimum usable frequency for a PIN diode is the reciprocal of the carrier lifetime, as seen, with a margin depending on the required linearity. From the considerations so far, it follows that the PIN diode conduction can be changed by applying either a DC voltage or a current to the device. However, although full voltage control is theoretically possible, it is too critical for practical reasons. Indeed, the DC curve in Figure 9.39 presents the wellknown knee, positioned around Vd ¼ 1 V. The voltage to apply to make the diode conducting is slightly higher than the knee value, but a small change in the voltage in that region causes large current variations that can even melt the device. Furthermore, the knee position is also temperature dependent, which makes PIN voltage driving almost impossible. On the other hand, the incremental resistance is easily expressed in terms of the direct current, and if the forward current is above the knee, the voltage across the device is almost constant, even under large variations of such a current. The opposite happens for the OFF state: a small reverse voltage is sufficient, but the corresponding current – negative but very small – is difficult to impress because of the high associated voltage, which can be close to or higher than the breakdown voltage. Thus PIN driving is a mix of current and voltage, to set the device in the ON and OFF condition, respectively. Returning to the schematic in Figure 10.3b, we are now in a position to explain the role of all the components. The inductors L1 , L2 (capacitors C1 , C2 ) provide a low- (high-)impedance path for the direct current and a high (low) impedance for the RF signal. In particular, the capacitor C1 minimizes the sensitivity of driving port V1 to any RF impedance associated with the driving generator, by grounding it at RF. The network configuration makes the DC anode–cathode voltage across D1 always the opposite to that across D2 , D3 . Applying a positive current19 to the port V1 , the diodes D2 , D3 conduct, and their anode–cathode voltage becomes slightly higher than the knee value.20 This small voltage becomes negative across D1 and is sufficient to put and keep it in the OFF state, for property (a) of the PIN diodes. Therefore a positive current injected into port V1 puts the SPST in the isolated state. From the same kind of consideration, we can see that a negative current through V1 sets D1 ON and D2 , D3 OFF, and thus the

18

For instance, bond wires, package parasitics and bias network residual reactance. Of the order of some tens of milliamps. 20 About 1 V, assuming a device with the curve in Figure 9.39. 19

372


SPST in the low-attenuation21 state. Thus a purely current driving realizes a mixed current–voltage driving, due to the voltage stabilization operated by the ON diodes. The bias elements (L1 , L2 , C1 and C2 ) have a highpass behaviour, from the point of view of the RF transmission from P1 to P2 , determining a lower limit on the SPST frequency range, together with t. Figure 10.4 shows a PIN diode SPDT, suitable for a frequency range from 2 to 20 GHz. More precisely, Figure 10.4a is the electrical diagram, which includes two SPSTs of the type shown in Figure 10.3b; here the two inductors L1 of each SPST simplify to one. The driving ports are doubled and renamed as I1 and I2 , considering that the driving excitation is a current. A positive (negative) I1 combined with a negative (positive) I2 enables the RF transmission from P1 to P3 (P2 ). The schematic in Figure 10.4a corresponds to the layout in Figure 10.4b, where: .

The shunt diodes D2A , D3A , D2B and D3B are chip devices, having a size of about 0.4 0.4 mm, with a thickness of about 0.1 mm.

.

The series diodes D1A and D1B are beam-lead devices.

.

The capacitors C1A , C2A , C1B and C2B are chip devices, having a size of about 0.5 0.5 mm, with a thickness of about 0.1 mm.

.

L1 , L2A and L2B are spiral inductors, realized with a wire of diameter of 50–100 mm, wound around a centre diameter of about 0.2 mm; the number of turns is some units.

.

All the transmission lines are realized with microstrips on a substrate having approximately the same thickness as the diodes. The substrate permittivity is such that the lines are slightly wider than the capacitors, in order to allow their assembly.22

Figure 10.4c shows the equivalent network of a shunt diode in the OFF state and of the wires connecting the device to the lines. The component references are those of the diode D3B , but we have the same network for all the shunt diodes. The bond wires that connect the shunt diodes to the lines have a typical diameter of 25 mm, a height above the ground that is slightly greater than the chip thickness, and a length slightly greater than half the chip length. The specific series inductance and the shunt capacitance of each wire are23 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m0 4 h h e0 m0 Lw ¼ ln 2 þ 15; Cw ¼ ð10:9Þ 2 2p Lw d d where d and h are the diameter and height above ground of the wire. With our parameters, we have 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 6 50 10 6 7 4 150 10 Lw ¼ 2 10 ln 2 þ 2 15 ffi 0:6 nH=mm; 50 10 6 150 10 6

Cw ffi 0:017 pF=mm

Two wires – each about 0.3 mm long – connect the diode to its two adjacent lines, so we have a wire inductance Lwire ¼ Lw l ffi 0:09 nH, while the associated capacitance is Cwire ¼ Cw l ffi 0:01 pF. 21 Here we have used the words ‘isolated’ and ‘transmission’ – rather than ‘OFF’ and ‘ON’ – to denote the SPST state, in order to prevent any confusion with the diode states, as sometimes happens. 22 For instance, if the substrate thickness is h ¼ 127 mm and the relative permittivity is er ¼ 2:35, Equation (3.181) gives w ffi 380 mm. A thicker substrate could be used to avoid the capacitor being wider than the line, but involves geometrical discontinuities between the shunt diode anodes and the lines. Such discontinuities can compromise the RF performance of the switch, and can be removed only with complicated and expensive mechanical machining of the ground plane. 23 See [4], equation (7.10).

MICROWAVE CONTROL COMPONENTS IA

IB C1A

C1B

P1 L2A TL3A’

C2A

P2

TL1A

TL1B D1A

D3A

L2B

TL0

TL2A’’

TL2A’

3A

373

TL2B’’

TL2B’

TL3B’’

C2B TL3B’

P3

D1B

D2A

L1

D2B

D3B

(a) I1

I2

P1 C1A

microstrip substrate limit

C1B

TL 0 L2A

P2 TL3A’

TL 2A’

3A

C2A

D3A

L2B

TL 2A’’

TL 1A D2A

TL 2B’’

TL 1B D1A

D1B L1

TL 2B’

TL 3B’ P3

TL 3B’’

D2B

D3B

C2B

GND

(b)

Plane of simmetry

TL2B’

L(3B) L(3B) wire wire

TL3B’’

(D3B)

COFF,3B

(c)

GA

VOFFSET

VCNTRL VBUFF

+ vin

IA gmVb,1

-

GB

VOFFSET VNOR

(d)

+ vin

-

IB gm Vb,2

Figure 10.4 PIN diode SPDT: (a) electrical diagram; (b) layout; (c) parasitic associated with OFF shunt diodes: (d) driver. Passing from a distributed to a lumped24 network, we have the network in Figure 10.4c, where the ðD3BÞ capacitance COFF is the sum of the OFF capacitance of the diode D3B plus those of the two bond wire couples. PIN diodes of the type we are considering – working up to 20 GHz – have a typical ðD3BÞ OFF capacitance of 0.05 pF, thus COFF ffi 0:06 pF. Then the T network consisting of the inductors ð3BÞ ðD3BÞ Lwire and the capacitor COFF can be seen either as a third-order lumped lowdpass filter or approximately as a transmission line. The cut-off frequency25 of the filter is h i ð3BÞ ðD3BÞ 0:5 fcð3BÞ ¼ ð2pÞ 1 Lwire COFF ffi 68:5 GHz

24 This approximation is valid in this context if we assume a wavelength much smaller than the size of the device and its wires, i.e. 0:6 mm l=4 ) f 125 GHz. This condition is well satisfied in our case. 25 The formula can be obtained by applying the image parameter filter design concepts. The interested reader can find the details in [5].

374

MICROWAVE AND RF ENGINEERING Table 10.1 Performances of the SPDT in Figure 10.4. Specification

Value

unit

Frequency range Insertion loss at 2 GHz Insertion loss at 12 GHz Insertion loss at 20 GHz Return loss Isolation Maximum input continuous power Input power at 1 dB compression (pulsed) Switching time (90% of the final output power)

2–20 1.5 1.5 2 12 60 0.5 4 150

GHz dB dB dB dB dB W W ns

having the characteristic impedance ð3BÞ

Z0

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD3BÞ ðD3BÞ 2Lwire =COFF ffi 2 0:09 10 9 =0:06 10 12 ffi 55 O

ð3BÞ

Since Z0 is close to 50 O, the combination of the parasitic elements realizes a good impedance ð3BÞ matching, provided that the maximum frequency of the SPDT is much lower than fc , as it is in our case. Table 10.1 summarizes the fundamental performances26 of the SPDT in Figure 10.4. Note that the destructive input power is smaller than the one causing 1 dB of compression. This means that heating of the devices due to the dissipation loss causes permanent damage to them, at a given input value, which generates a compression of less than 1 dB. For this reason, the 1 dBcp is measured in a pulsed condition, in order to minimize the device heating; the typical pulse width is 1 ms, with a duty cycle of 1%. Figure 10.4d shows the principle of the driver for the control ports of the switch. Such a schematic derives from two considerations: .

The currents I1 , I2 have two possible values each, and the sign of I1 is always opposite that of I2 .

.

An external user is normally interested in changing the SPDT state by using a digital signal, not in taking care of the internal driving signals of the switch. Moreover, one single digital signal (bit) should be sufficient to address the two possible states of the SPDT.

The external digital control signal of the SPDT is VCNTRL and it can be 1 or 0; correspondingly P1 is connected to P3 or P2. The logic gates GA and GB are a buffer and a NOR port, respectively.27 They remove the variations that the external digital control voltage could have. Reasonable values for the output voltages28 VBUFF and VNOR of GA and GB could be 0.5 V and 4.5 V (4.5 V and 0.5 V) when VCNTRL ¼ 1 ð0Þ . The SPDT driving means that IK ðK ¼ A; BÞ has to bias two diodes (D2K , D3K ) when positive and only one diode (D1K ) when negative. Now, assuming that all the diodes present the same incremental resistance at the same DC bias value, with a required forward current for the ON state of

26

The Ansoft file 01_PIN_SPDT.adsn provides a first-level simulation of the SPDT in Figure 10.4 The output of a buffer coincides with the input, but, for each logic state, the output voltage is independent of the input, which can vary within an allowed low or high range. The NOR port behaves with the same modality as the buffer, with the difference that the output voltage corresponds to the logic negation of the input, i.e. it is low (high) when the input is high (low). 28 In order to give a simple description of the circuit we will assume some practical values for the quantities involved with it. However, the numbers used are realistic, but they refer to no real case. 27


375

20 mA per diode, then this must be I1 ¼ 40 mA;

I2 ¼ 20 mA when VCNTRL ¼ 1 ) VBUFF ¼ 4:5 V;

I1 ¼ 20 mA;

I2 ¼ 40 mA when VCNTRL ¼ 0 ) VBUFF ¼ 0:5 V;

VNOR ¼ 0:5 V VNOR ¼ 4:5 V

Thus the output current excursion for each of the two VCVSs is 60 mA, while the voltage excursion at the output of the logic gates is 4 V: then the transconductance has to be gm ¼ 60=4 ¼ 15 mS. Finally the two voltage sources VOFFSET have to shift the characteristic from the gate outputs to the diodes, so as to obtain the above currents. From I1 ¼ 15 10 3 VNOR ¼ 15 10 3 ðVBUFF VOFFSET Þ it follows that VOFFSET ¼ 4:5 40=15 ffi 1:83 V: The driver of Figure 10.4c has many simplifications, useful for describing the working principles and giving an idea of how it is possible to realize a digital controlled switch.29

10.2.2

FET switches

Figure 10.3c shows the schematic of an FET SPST with one series and two shunt transistors. These devices exhibit a variable resistance – between the drain and source – controlled by the DC voltage applied between the gate and source. Thus, FET switches uses the channel resistance to realize controlled variable impedances. At microwave frequencies, the GaAs MESFET30 is the most used device type, while at RF the Si MOSFETs are also used. MESFETs are depletion devices, while MOSFETs could be either depletion or enhancement devices, though the latter case is more frequent. For the switch operation, the main difference between enhancement and depletion devices is the control voltage range. Both MESFET and MOSFET switching elements operate in the triode region, thus with relatively small drain–source voltages.31 Any violation of such a condition makes the device resistance depend on both Vgs and Vds , with consequent nonlinear distortions on the RF signal. Let us assume that MESFET and MOSFET DC curves follow Equations (9.75) or (9.79) and (9.80) respectively. In the linear current region we have that ðMESFET Þ

a vds

1;

ðMOSFET Þ

vds

vI VT

ð10:10Þ

Note that the internal gate–source voltage vI results from the superimposition of the externally applied DC gate–source voltage on the RF voltage resulting from the FET parasitic capacitances. In normal cases the external DC voltage dominates. From the first condition of (10.10) it follows that we can approximate the hyperbolic tangent with its argument h i ðMESFET Þ ðMESFET Þ tanh a vds ffi a vds

29 The switching time has been intentionally omitted from the discussion. We will limit ourselves to mentioning that ON diodes – particularly PIN ones – store carriers that must be removed by fast switching to OFF. The circuit in Figure 10.4c is not able to remove the stored charges, due to the high output impedance inherent in the current generators. Some real drivers approximate current generators with voltage generators having high output resistance, and bypass the latter with suitable speed-up capacitors. 30 Herein ‘GaAs’ and ‘MESFET’ also denote the more evolved materials (AlGaAs, InAsS, InP, etc.) and devices (HEMT, PHEMT, HFET, etc.), which present the same principle of operation. 31 For instance, the FET with the DC curves in Figure 9.41 works reasonably well as a linear switching element if jvds j < 0:5 V.

376

MICROWAVE AND RF ENGINEERING ðMESFET Þ

ðMESFET Þ

Moreover, a vds 1 usually implies that also l vds 1, since usually it is l < a. Thus, in ðMESFET Þ 0 we can approximately rewrite the MESFET equation (9.75) as the gate–source range Vp vI h i h i2 ðMESFET Þ ðMESFET Þ ðMESFET Þ ðMESFET Þ ffi b a vI vI ; a vds Vp a vds

ð10:11Þ

Similarly, for the MOSFET, in the triode region Equation (9.80) simplifies to h i h i ðMOSFET Þ ðMOSFET Þ ðMOSFET Þ ðMOSFET Þ ðMOSFET Þ ¼ b vI vI ; vds VTH vds Ids

ð10:12Þ

ðMESFET Þ

Ids

From Equations (10.11) and (10.12) we can compute the corresponding drain–source incremental resistance h i ðMESFET Þ ðMOSFET Þ ðMOSFET Þ h i2 @I v ; v I ds ds 1 ðMESFET Þ ¼ ffi b vI Vp a ð10:13Þ ðMESFET Þ ðMESFET Þ @vds rds ðMOSFET Þ

1 ðMOSFET Þ

rds

¼

@Ids

h

ðMOSFET Þ

vI

ðMOSFET Þ

; vds

ðMOSFET Þ

@vds

i

h i ðMOSFET Þ ¼ b vI VTH

ð10:14Þ

Note that both the incremental resistances (10.13) and (10.14) depend only on the internal gate–source voltage vI, as needed for a linear operation of the switch. Equation (10.13) states that the MESFET drain–source incremental resistance reaches its minimum ðMESFET Þ

value rds;min

ðMESFET Þ 2

¼ b ½VP

ðMESFET Þ

a when vI

¼ 0, increasing with the negative gate–source

ðMESFET Þ vI

voltage, up to infinity for

Vp. ðMOSFET Þ ¼ VTH Þ down to zero The MOSFET small-signal channel resistance ranges from infinity ðvI ðMOSFET Þ

ðvI ! 1Þ. The ideally zero minimum channel resistance is a peculiarity of the MOSFET with respect to the MESFET. However, real devices have neither zero short-circuit nor infinite short-circuit resistance, due to second-order effects, just electrode resistances and leakage currents, at least. Differently from PIN diodes, FETs have three electrodes: the variable resistance is present between the drain and source, while the control voltage is applied between the gate and source. Similar to what we did for the PIN diode, we can still obtain simple equivalent networks for the drain–source bipole, although the FET model is more complicated than the PIN one. Looking between the drain and source, one can see a parallel RC or series RL in the OFF or ON state, respectively. Compared with the PIN diode, the FET presents three main advantages: (a) The control electrode (gate) is separated from the variable resistance, used as a switching element. This simplifies the bias network design. However, it must be considered that the parasitic capacitances Cgs , Cgd introduce an AC coupling – increasing with the frequency – of the gate with the source and drain. (b) The gate–source control voltage is applied across a reverse-biased junction (MESFET) or on a DC insulated electrode (MOSFET). Thus, an FET switch driver must supply no direct current.32 By comparison, the PIN requires current in the ON state. (c) FET operation does not depend on carrier lifetime Thus the minimum working frequency is theoretically zero, and the switching speed is higher than with the PIN.

32

Some small current is required to charge/discharge the gate parasitic capacitors in the state change transients.


377

And three disadvantages: (d) The FET presents higher (lower) ON (OFF) resistance than the PIN, and consequently FET switches present a higher insertion loss (lower isolation) in the transmitting (isolating) state. (e) Parasitic reactance associated with the FET is higher than with the PIN, thus FET SPnTs have lower maximum working frequency. (f) FET switches generally have lower maximum power handling than PIN ones, in respect of both nonlinear distortions and risk of damage. Figure 10.3c shows the basic schematic of one SPST branch realized with an FET. The bias network is simpler than the one in Figure 10.3b, and consists of one resistor per switching element. Applying zero voltage on the control port V1 and, simultaneously, a value lower than the pinch-off33 on V2 , we have that Q1 conducts and Q2 , Q3 do not.34 In that condition a low attenuation occurs between the RF ports P1 , P2 . The opposite happens when swapping the control voltages, with a consequent high attenuation from P1 to P2 . The determination of the bias resistances R1 to R3 is the result of a compromise between contrasting requirements. The gate is not fully isolated from the drain, rather parasitic capacitances Cgs , Cgd provide a frequency-increasing coupling. Thus a high-frequency signal is present – and dissipated – on the control ports: this crosstalk degrades both the insertion loss and the isolation. Therefore the achievement of the best high-frequency performances implies the use of high-value bias resistors, to minimize the crosstalk. On the other hand, bias resistances must be low enough to charge and discharge the gate parasitic capacitances rapidly, in order to achieve the high switching speed allowed by FET technology. The exact prediction of those performances requires circuit simulations. Nevertheless, some initial ideas can be obtained from relatively simple considerations. Note that the gate capacitance increases with the gate–source voltage, as in Equation (9.73). Therefore, the gate presents the maximum capacitance to ground when the FET is in the conduction state. Typical numbers for a switch operating up to 20 GHz could be Cgs Cds 0:2 pF. Then, the bias resistance has to charge a total capacitance of about 0.4 pF: if R1 ¼ R2 ¼ R3 5 kO we have a time constant of t ¼ Cgs þ Cds Rk 1 ns, which guarantees a settling time of less than 5 ns.35 To put some numbers on the RF performances, an ore resistor with R ¼ 5 kO provides an attenuation of 20 log10 ð5000=100 þ 1Þ ffi 34 dB if placed between a 50 O source and load.36 In the switch case, each couple of gate bias resistors connected to the same driving pad generates a parasitic RF path. That path comprises the FET parasitics from the drain to the gate, from the gate to ground, the two resistors themselves and the RF impedance to ground provided by the driver that applies the control voltage. The last is difficult to determine and is also not repeatable; nonetheless it is certainly different from an open circuit and thus provides an additional – although difficult to predict – attenuation in the parasitic path. Therefore, considering the two resistors and the additional attenuations provided by the above-mentioned elements, each parasitic path generates a crosstalk of the order of twice the case of the 5 kO resistor between the 50 O source and load, with some margin, i.e. in excess of 2 34 ¼ 68 dB. Such attenuation is low enough not to degrade insertion loss and isolation, which is around 40 dB, as is typical for FET switches. 33 The pinch-off voltage used for the Curtice (Equation (9.75)) or Statz model (Equation (9.79)) is around 1 V for the GaAs MESFET; for instance, the example considered in Section 9.7.2.2 has Vp ¼ 0:724 V. However, the drain–source current (and thus the drain–source conductance) does not completely vanish for Vgs ¼ Vp , differently from what the Curtice and Statz models – which are approximate – predict. Usually, the DC gate–source voltage to apply to completely interdict the MESFET falls within the range 2 to 5 V. 34 Here we are considering depletion devices. The enhancement case is similar, just replacing zero voltage with a voltage above threshold and the pinch-off voltage with zero. 35 See [6]. 36 This isolation is the approximated one between the gate and the control port, not the one for the switch.

378


Figure 10.5

GaAs MMIC SPDT: (a) schematic; (b) layout.

The worst case isolation occurs when the parasitic contribution is in phase with the switch main path. If we assume that, without considering the bias network, the isolation is 40 dB, the worst case isolation will be 10 log10 ð10 4 þ 10 6:8 Þ 39:993 dB. The resulting degradation is only 0.007 dB with respect to the ideal bias network case. The parasitic transmission due to the bias network becomes dominant when the main isolation tends to infinity. The parasitic path due to the bias elements also affects the insertion loss in the transmitting state. Assuming that the switch is ideal (0 dB of insertion loss in the transmitting state), the additional loss caused by the signal crosstalk in the bias network is 10 log10 ð1 10 6:8 Þ 6:9 10 7 dB. The transmission lines TL1 to TL3 could have a characteristic impedance of 50 O or higher. The first case occurs when the wire inductance compensates the capacitance to ground of the switching element, as we see for PIN diodes. That condition is generally not verified with FETs, because they present higher parasitic capacitance than the PIN in the OFF state, as already mentioned. Moreover, FET switching elements are frequently used in MMIC or IC technology, where almost no connection inductance is present. Therefore, the lines TL1 to TL3 work like semi-lumped inductors,37 to compensate the parasitic capacitance of Q2 , Q3 in their OFF state; thus the characteristic impedance of those lines is the highest possible. Figure 10.5 shows a schematic (a) and the layout (b) of a GaAs MMIC SPDT. It consists of two identical SPSTs, each with one series and three shunt switching transistors. The subscript ‘A’ (‘B’) denotes the components of the SPST from the RF port P1 to P2 (P3 ). The control voltage V2 (V1 ) applied to the gate of the series FET Q1A (Q1B ) is also applied to the gates of the shunt transistors Q2B to Q4B (Q2A to Q4A ). Moreover, V1 and V2 are complementary in that, when the first is zero, the second assumes a value well below the pinch-off, and vice versa. This arrangement guarantees that one and only one port between the ports P2 , P3 is connected to P1 , as needed.

37 The inductor approximation technique by means of high-impedance lines is based on the semi-lumped concepts discussed in Section 8.4

MICROWAVE CONTROL COMPONENTS 0

379

0 low attenuation→ -2 simulation measurement

-20

-4

-30

-6

-40

-8

← high attenuation

-50

-10 0

Figure 10.6 Figure 10.5.

20 log10(|S21|), ON

20 log10(|S21|), OFF

-10

2

4

6

8 10 12 Frequency, GHz

14

16

18

Predicted (solid lines) and measured (dashed lines) insertion loss of the SPDT in

The transmission lines TL1 to TL3 are narrow microstrips ðw ¼ 10 mmÞ on the GaAs substrate ðh ¼ 100 mm; er ¼ 12:5Þ: Equation (3.183) gives Z0 ffi 92:8 O. The layout in Figure 10.5b shows how such lines have been folded, in order to minimize the chip size, which is about 1 2:8 mm. For the same reason, also the bias resistors R1 to R4 have a snake-like shape. The shunt transistors Q2 to Q4 connect to ground by means of the four via holes G1 to G4: G2 (G3) is shared between Q2 and Q3 (Q3 and Q4 ). Finally, the RF pads of the ports P1 to P3 present two side via holes with two ground probes for the on-wafer RF test.38 Figure 10.6 shows the simulated (solid lines) and measured insertion loss and isolation of the switch in Figure 10.5. The return loss at the two connected ports is better than 10 dB, although not shown. The measured input power for 1 dB compression is around 23 dBm, while the switching time is lower than 5 ns. Comparing these data with those in Table 10.1, we see that the PIN diode SPDT performs better in all the RF situations except the low-frequency limit, while the FET SPDT presents a remarkably shorter switching transient, as expected. However, some FET switches may exhibit long tail effects on the switching transient. Increasing the gate voltage in a series FET from the pinch-off up to zero allows the respective channel to conduct. Unfortunately, at the beginning of the state change, some of the electrons passing through the channel could be trapped in the substrate. In this case, some small portions of the channel remain pinched off – and the series FET does not fully conduct – until all the trapped electrons are liberated. The time needed for complete transmission of the switch could be slightly smaller than 1 s for the insertion loss being within 0.01 dB of its steady state value.39

10.2.3

MEMS switches

MEMS stands for microelectromechanical systems. They are miniaturized 3-D devices based on the mechanical movement of a thin elastic membrane or beam which enables or prevents the propagation of the RF signal along a planar transmission line.40 The latter is most often a coplanar waveguide (CPW) or a 38

See Section 14.5.2. See [7, 8]. 40 MEMS have been developed since the 1970s as mechanical sensors. Their use in high-frequency applications is much more recent and is distinguished by the RF prefix. 39

380


microstrip line and the membrane is in the form of a bridge or a cantilever. The term MEMS switch normally refers to switches where the movable membrane has two fixed positions, namely the up state (membrane suspended) and the down state (membrane collapsed onto the transmission line). In a MEMS varactor, on the contrary, the distance between the movable membrane and the transmission line can be varied in a continuous fashion, so as to realize a variable capacitor. Actuation of MEMS switches can be achieved by various mechanisms, i.e. by electrostatic, electromagnetic, piezoelectric or thermal effects. Although the electrostatic actuation requires high voltages (of the order of 20–80 Vor more), it is actually the most common technique in use today. This is because of the very simple actuation mechanism, the high performance and reliability, the virtually zero power consumption, the small electrode size and the relatively short switching time. We will therefore consider here only electrostatic-actuated devices. There is an ample variety of configurations and geometries that have been proposed and developed for RF MEMS switches. We consider here only the basic and more common configurations that lead to the following categorization: 1. Series and shunt MEMS switches. In the former case, the movable beam is located above an interrupted signal line. In the up state, an open circuit is realized which provides excellent isolation up tovery high frequencies (typical values are 60 dB at 1 GHz and 20 dB at 30 GHz). When the beam is in the down position, the two ends of the interrupted line are electrically connected, resulting in a low insertion loss, typically as low as 0.2 dB in the 0–40 GHz frequency band. In MEMS shunt switches the movable beam is used to short-circuit the transmission line by connecting the centre conductor to ground. In the up state they are extremely low loss (0.05–0.1 dB in the 0–60 GHz frequency band), since the beam hardly affects RF propagation along the line. When the switch is activated (down state), the beam short-circuits the RF signal to ground. This results in a very high insertion loss in a frequency range around the resonant frequency of the bridge (see below). 2. Capacitive and ohmic switches. The contact between the beam and the centre conductor can be either capacitive or resistive (ohmic) depending on whether a dielectric layer (usually a thin passivation layer) is employed to cover the RF signal line. Table 10.2 lists the main performances of the two solutions. 3. Cantilever and bridge (or clamped–clamped) MEMS. The suspended beam may be anchored to the substrate at one end (cantilever beam) or at both ends (clamped–clamped beam). The bridge armatures can be oriented in-line or broadside with respect to the transmission line axis. As an example, Figure 10.7a shows an in-line cantilever beam in coplanar waveguide technology, whereas a broadside clamped–clamped beam is shown in 10.7b. In both examples, the reader can observe underneath the bridges the metal pads used for electrostatic actuation.

Table 10.2 Comparison between ohmic and capacitive switches (after [10], reproduced by permission of Cambridge University Press). Ohmic Contact

On state with metal-to-metal ohmic contact Dependence Insertion loss depends on the contact resistance Operational Broad frequency coverage frequency range (DC–40 GHz) Contact lifetime Short

Capacitive OFF state with capacitive coupling Isolation depends on capacitive ratio between ON and OFF states Not suitable for frequencies lower than 4 GHz Long


381

Figure 10.7 In-line MEMS cantilever beam (a) and broadside MEMS clamped–clamped beam (or bridge) (b) in coplanar waveguide technology, with actuation pads.

This electrostatic actuation is realized by applying a voltage between the beam or membrane and the electrode underneath. The resulting attractive electrostatic force causes the beam to deflect towards the electrode. Using very simple mechanical models, it is found that as soon as the deflection exceeds onethird of the original beam height g0 , the bridge becomes unstable and collapses onto the down-state position. The corresponding voltage is indicated as the pull-in voltage Vpull-in . In some instances, the voltage is applied directly to the centre CPW conductor. This considerably lowers the device complexity since no separate bias circuitry is required to drive the switches, but clearly implies reduced potentialities and creates the need to separate the DC from the RF signals. The pull-in voltage of a simple clamped–clamped beam with central actuation can be expressed as: sffiffiffiffiffiffiffiffiffiffiffiffi 8kg30 Vpull-in ¼ ð10:15Þ 27e0 A where e0 is the vacuum permittivity, A is the overlapping area between the two electrodes, g0 is the initial air gap and k is the spring constant of the movable electrode. With respect to conventional solid state devices, such as PIN diodes or MESFETs, MEMS switches offer numerous advantages, especially for RF and millimetre-wave applications: they show indeed a very low power consumption and high linearity. Typically, they have cut-off frequencies of the order of tens of terahertz and show no intermodulation effects. In addition, low insertion loss and high isolation up to 30 GHz can be achieved by a single MEMS switch. The drawbacks of this technology are essentially due to the limited switching speed, high actuation voltages and, presently, its unachieved full commercial maturity, thus limited reliability. Because of the mechanical inertia, the switching time can hardly be reduced to less than 1 ms. As far as reliability is concerned, the most common failure mechanisms are due to dielectric charging, high current densities and increased temperature in the contact area. RF MEMS can easily be integrated with passive devices, but monolithic integration with active devices is still a problem. Packaging is another critical aspect of this technology, mainly because of the high sensitivity of the suspended membrane to high temperature variations and substrate deformations that may occur when using standard packaging processes. A comparison between MEMS and alternative solid state switches is presented in Table 10.3. The CPW is the printed transmission line technology most suited to the RF MEMS switches since both conductors are located on the same plane. Therefore, we now illustrate in some detail the most popular CPW switches, namely the shunt capacitive and the series ohmic switches.

382


Table 10.3 Comparison of the performance of RF MEMS, PIN and FET switches (after [10]). RF MEMS Insertion loss (dB) Isolation (dB) Linearity (IP3) (dBm) Actuation voltage (V) Switching speed Power consumption Power handling (dBm)

Figure 10.8 attenuation.

0.2–0.5 45 > 65 20–80 < 35 ms 10 nW 38

GaAs FET 0.7–1 25–41 24–70 3–5 < 5 ns < 5 mW 33

GaAs PHEMT 0.6–0.9 28–45 55–72 2–6 < 5 ns < 5 mW 33

GaAs PIN diode 0.3–0.6 11–22 58 5 < 50 ns > 10 mW 36

Si PIN diode 0.4–0.8 12–25 50 5 < 100 ns > 10 mW 38

Capacitive MEMS shunt switch on CPW transmission line: (a) low attenuation; (b) high

10.2.3.1 Clamped–clamped beam shunt capacitive MEMS switch Figure 10.8 shows a clamped–clamped beam shunt capacitive MEMS switch based on CPW technology in the up and down positions. The centre conductor is covered by a thin passivation layer, whose dielectric constant and thickness are er and td . Notice that no actuation pads are present in this example: the switch is actuated by applying a voltage DV directly between the bridge and the centre conductor. Typical parameters are: dielectric thickness around td ¼ 100–200 nm, dielectric constant er ¼ 4:0 7:8, bridge height g ¼ 1–5 mm, bridge length around L ¼ 250–800 mm and bridge width w ¼ 25–180 mm. When no voltage is applied, the bridge stands above the line (up state) at a height g0 and the switch is in the ON state. By applying a voltage greater than the pull-in (DV Vpull-in ), the bridge collapses onto the CPW central conductor, so increasing the bridge capacitance Cb by a factor of 10–100. Such a capacitance has the effect of short-circuiting the microwave signal to ground (OFF state). The switch behaviour can be characterized in terms of two simple equivalent circuits depending on its state, as shown in Figure 10.9. Such equivalent circuits can be derived41 on the basis of simple physical considerations and most often show very good agreement with the measurements. When the membrane is in the up state, the equivalent circuit simply consists of a shunt capacitor Cup. Its value can be calculated by a simple parallel-plate capacitance model plus a parasitic fringing capacitance: Cup ¼ e0

A g0 þ

td þ C fringing er

ð10:16Þ

where A ¼ w W is the overlapping area between the bridge and the RF line and the fringing capacitance C fringing is of the order of 20–30% of the total capacitance. This parasitic capacitance can be calculated more accurately by using electromagnetic simulators. 41

See [11].


383

RF signal Z0

Cup Z0

RF signal

(a)

Z0

Lb Cdown Z0 Rb

(b)

Figure 10.9 Equivalent circuits of the clamped–clamped beam capacitive shunt: (a) CPW switch in the up state (low attenuation); (b) down state (high attenuation). An ideal switch would exhibit zero up-state capacitance: typical values of Cup are of the order of a few tens of femtofarads. Such values are so small that the conduction current flowing through the bridge armature is small enough to be negligible and the equivalent circuit only contains the shunt capacitance Cup . When the membrane collapses into the down state, the bridge capacitance strongly increases from Cup to Cdown . An RF electrical current flows along the bridge with associated resistance Rb and inductance Lb . The bridge capacitance can be expressed simply as Cdown ¼ e0 er

A td

ð10:17Þ

In the down state the fringing capacitance can be neglected because of the small dielectric thickness. On the contrary, the actual capacitance is in practice much lower than predicted by (10.17) because the surface roughness and the imperfect bridge flatness produce small air gaps between the bridge and the passivation layer. Lb and Rb should be calculated by using a parameter fitting from full wave simulations or measured data. Typical Lb values are of the order of a few picohenries, while Rb is of the order of 0.1–0.3 O. Note that in the down state the bridge behaves as a series resonator with impedance given by Zb ¼ Rb þ joLb þ

1 joCdown

ð10:18Þ

At the resonant frequency f0 of the shunt switch f0 ¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p Lb Cdown

ð10:19Þ

the bridge impedance reaches its minimum value Rb and the switch exhibits its maximum isolation. By properly designing the bridge geometry (thus Lb and Cdown ) it is possible to place the resonant frequency in the required operational band so as to obtain very good isolations even at millimetre-wave frequencies. Typical responses of a shunt switch in the up and down positions are shown in Figure 10.10, which also shows a photograph of the structure.

10.2.3.2 Clamped–clamped series MEMS switch In the clamped–clamped beam series switch, the metal membrane is isolated from the coplanar ground planes and suspended above a gap created in the signal line. A series switch in the up and down state is shown in Figure 10.11. Notice, in this case, the presence of the metal pads used for applying the actuating voltage between the bridge and the electrodes underneath. When the beam is in the up state (Figure 10.11a), the gap provides a very high and wide-band isolation from DC up to 40 GHz. In the down state (Figure 10.11b) the beam creates a short circuit across the gap and the RF signal is allowed to propagate along the CPW. The series switch behaviour can be modelled in terms of the two equivalent circuits shown in Figures 10.12a and 10.12b for the down state and up state respectively.42 42

See [11].

384


Figure 10.10 Shunt capacitive MEMS switch: (a) photograph; (b) measured performance of (a) in up state; (c) measured performance in the down state [12]. (Reproduced by permission of EuMA.)

Figure 10.11

Coplanar ohmic contact series switch: (a) down state; (b) up state.

RF signal Coff RF signal (a)

Z0

Z1 Cp

Ron

Coff Z0

Z0

Z1 Cp

R on Z0

(b)

Figure 10.12 Equivalent lumped element circuits of the clamped–clamped beam series ohmic contact switch: (a) down state; (b) up state.


385

Figure 10.13 Capacitive MEMS switch: (a) photograph; (b) measured performance in the up state; (c) measured performance in the down state [3]. (Reproduced by permission of EuMA.) Cp is the series capacitance across the gap and Z1 is the short line section constituted by the beam connecting the two centre conductor ends. The element Ron is the down-state contact resistance between the metal beam and the centre conductor. The most critical aspect of the series switch is the minimization of this series resistance, which can significantly deteriorate the insertion loss. Different solutions for improving the metal-to-metal contacts have been proposed, such as the use of small dimples in the beam bottom surface or the use of noble metals and alloys covering the contact areas. In the up state, the series resistances Ron are replaced by the series capacitances Coff between the open ends and the beam. Such capacitance is the main limiting factor of the isolation of the series switch. Series ohmic contact MEMS switches are often realized by using cantilever beams placed in line with respect to the transmission line. This has the advantage of halving the contact resistance Ron , but, on the other hand, cantilever beams are more difficult to manufacture. As an example, the photograph and the response of a typical series ohmic switch are shown in Figure 10.13.

10.2.4

Alternative multi-port switch structures

This section gives a brief description of some possible configurations of switching networks, at both the circuit and subsystem level. Figure 10.14a shows the schematic of an absorptive SPST. As anticipated, the adjective ‘absorptive’ denotes a switch whose ports are always matched, independently of the selected one. That network has a structure similar to the one in Figures 10.2b and 10.3a. Apart from the number of shunt switching elements – which has no conceptual meaning in this context – the main difference

386

MICROWAVE AND RF ENGINEERING 50 Ω P1

SW1

TL1

TL3

TL2 SW2

SW4

P2

SW3

(a)

P1

P2 (b)

TL2 SW1

TL1 TL3

P3

SW2

Figure 10.14 Alternative switch configurations: (a) absorptive SPDT sections; (b) SPDT realized with shunt elements only. consists of the addition of the series element SW4 , shunted by a 50 O resistor. In the transmitting state SW1 , SW4 (SW2 , SW3 ) are ON (OFF), thus the 50 O resistor is bypassed and has no influence. In the isolated state SW1 , SW4 (SW2 , SW3 ) are OFF (ON), thus, looking at port P2 , we ideally see a 50 O resistor in series with a short-circuit stub TL3. Now, if TL3 is much shorter than a quarter of wavelength, and the ON impedance of SW4 is negligible, or if these elements are such as to compensate each other, then P2 presents an impedance close to 50 O. In other words, P2 is always matched in both the transmitting and isolating state: the SPST in Figure 10.14a is absorptive. If we replace the network in Figure 10.2b with the one in Figure 10.14a the network resulting from Figure 10.14a is an absorptive SPnT. The main drawback of the network in Figure 10.14a is related to the 50 O bypassed switching element SW4 : it presents finite ON resistance and its OFF resistance is closer to 50 O than to infinity. This worsens the insertion loss, without significantly increasing the isolation. All the SPnTs presented so far include series switching elements, which are needed to make the disabled ways present an open circuit at the junction point. The main drawback of the series element is that it ideally has to be totally floating from ground. No electrical conduction path should be present between any of its two terminals and ground: neither AC nor DC, and in no state. Generally, low electrical conduction involves low thermal conduction as well. Thus it is difficult to remove the heat produced by RF and DC power dissipated inside the series switching element. This condition impacts on the maximum non-destructive power43 for the SPnT. The SPDT configuration in Figure 10.14b circumvents this problem by eliminating the series elements. Such a configuration is based on the short- to open-circuit transformation operated by a l=4 transmission line; therefore it is suitable for narrow bandwidths. In order to explain how the circuit works, we will initially assume that: .

The transmission line TL1 is infinitely short.

.

TL2 , TL3 are 50 O l=4 transmission lines at the centre of the working frequency band f0 .

.

The switching elements SW1 , SW2 are ideal, presenting a short circuit (open circuit) in the ON (OFF) state.

43 For instance, the destructive power of the SPDT in Figure 10.4 is not enough to produce 1 dB of compression. In other words, the switch is damaged by a power which is in the quasi-linear range of the component.

MICROWAVE CONTROL COMPONENTS R s= 0.1 Ω 1Ω 2Ω 10

← s21

0

387

-20 0

-10

In-band return-loss -6

-20

s11 →

-60 -80

20 log10(|s31|)

s31→

-4

-40 20 log10(|s11|)

20 log10(|s21|)

-2

-100

-8

-30

-10

-40

-120 2

4

6

Figure 10.15

8

10 12 14 16 Frequency, GHz

18

20

22

24

Quasi-ideal performances of the SPDT in Figure 10.8b.

The SPDT has two possible states: (I) SW1 ON with SW2 OFF; or (II) SW1 OFF with SW2 ON. In condition (I), P2 is isolated from P1 , and, at f0 , TL1 transforms the short circuit produced by SW1 into an open circuit on the common port. This way, P3 is connected to P1 by a simple matched line, with no effect from SW1 and SW2 . If we consider condition (II), we have the same result, just by swapping P2 and P3 . Therefore SPDT in Figure 10.8b has zero insertion loss and infinite isolation in the ideal case and at the centre frequency. As the frequency moves from f0 the unselected way presents a non-zero impedance to the common port, degrading the insertion loss and return loss. Moreover, the switching elements are also non-ideal; rather, they present finite impedances in both of their states, further degrading the performance of the component. In real implementations of the circuit, the transmission line TL1 has the same electrical length and characteristic impedance – lower than 50 O – as TL2 and TL3 . This arrangement improves the matching of the network in a finite frequency bandwidth. The optimum characteristic impedance ZOPT of the lines, for any required bandwidth, can be found by optimization. Note that the result depends only on the relative bandwidth and not on the centre frequency. Figure 10.15 shows the simulated response44 of the SPDT in Figure 10.14b optimized to work in the frequency range 6–18 GHz, with the resulting ZOPT ¼ 33:7 O. The presented curves are obtained by assuming the lines are loss free, the OFF switching element is an ideal open circuit, and the ON impedance is a pure resistance of 0.1, 1 and 2 O. Note that the in-band return loss is slightly better than 17 dB. The achievable return loss of the structure and the optimum line impedance are decreasing functions of the working fractional bandwidth Df = f0 : Figure 10.16 shows those two parameters as obtained by optimization. The performances of a real component are worse than Figure 10.15 shows, due to the finite impedance of the switching elements, the transmission line loss and the bias networks. Nonetheless,

44

See the Ansoft file 02_Shunt_Only_SPDT.adsn.

388

MICROWAVE AND RF ENGINEERING 70 60 In-band Return-Loss 50 40 30 Z OPT 20 10 0 20

40

60

80

100

120

Percent relative bandwidth (100 ∆f/f0 )

Figure 10.16

Optimum parameters for the SPDT in Figure 10.8.

by using PIN diodes as switching elements, reasonable performances across the whole frequency range 6–18 GHz could be: .

maximum insertion loss: 1.5 dB

.

minimum return loss: 14 dB

.

minimum isolation: 25 dB

.

maximum input power: 5 W (non-destructive).

A further advantage deriving from the elimination of the series element is the simplification of the bias network. Figure 10.17 shows some switch architectures using SPDT as building blocks. The configuration in Figure 10.17a uses three SPDTs to realize one SP4T. It can be seen that P1 can be connected to any of P2 to P5 by properly setting SPDT1 to SPDT3. SP4T has four different states. The architecture in Figure 10.11a has three SPDTs, each having two states: the total number of states is then 23 ¼ 8. This discrepancy is only apparent because the network in Figure 10.11a has some redundant states: when SPDT1 is in position 1 (2) the selected port can be only P2 or P3 (P4 or P5), regardless of the position of SPDT3 (SPDT2). However, such state redundancy can be used to increase the isolation at some ports. The configuration in Figure 10.17b uses three reflective SPDTs and two 50 O terminations to realize one absorptive SPDT. All of SPDT1 to SPDT3 are in the same position 1 or 2. In the first (latter) case P2 (P3 ) is selected and P3 (P2 ) is terminated on 50 O: therefore all the ports are matched, independently of the state, as needed. The additional SPDT present in both the paths from P1 to P2 and P3 increases the insertion loss and also the isolation of the architecture. The configuration in Figure 10.17c is known as a transfer switch; it uses four SPDTs, all with the same control. All of SPDT1 to SPDT4 are in the same position 1 or 2. In the first (latter) case P1 is connected to P2 and P3 to P4 (P1 is connected to P3 and P2 to P4). Summarizing, we can say that any couple of non-adjacent ports of the transfer switch can be connected to its left or right neighbour. Figure 10.18 shows one important application of the transfer switch. The proposed arrangement works as a bidirectional amplifier, in that the input can be one between P1 and P2 while the other is the


SPDT1

P1

(a)

SPDT3

1

P2

1

2

P3

2

1

P4

SPDT2

2

P5

50 Ω

2

389

SPDT3

P2 SPDT1

P1

1 1 2 2

SPDT2

P3

50 Ω 1

(b)

P2 1

SPDT1 P1

2

1

SPDT2

2

SPDT3

P3 1

2 (c)

1

SPDT4

P4

Figure 10.17 Special multi-port switch connections: (a) SP4T formed with a branched chain of three SPDTs; (b) absorptive SPDT comprising three reflective SPDTs and two matched terminations; (c) transfer switch. output. Thus, we can amplify from P1 to P2 , or vice versa, by selecting the SPDT position. The architecture in Figure 10.18 requires that the transfer switch isolation be higher than the amplifier gain by a safe margin, in order to avoid instabilities.45

10.3 Variable attenuators Section 6.4 describes fixed attenuators, also providing an analysis and synthesis formulae. Here we will deal with variable attenuators that present a variable attenuation, which depends on some control variables. 45

See Section 8.4.

390


1

P1

2

SPDT2

SPDT3

1

2 P3

SPDT1 2

1

1 SPDT4

AMP1

Figure 10.18

Transfer switch application for a bidirectional amplifier.

High-frequency equipment widely uses variable attenuators to change the output power in RF sources, or to adjust the dynamic range in receivers. Also, variable attenuators have more similarities with switches than with fixed attenuators, which is why we placed such components in this chapter. The two main types of variable attenuators are: .

Continuously variable attenuators, which realize any attenuation within a specified range, with a monotonic relation with one control voltage or current.

.

Step-controlled attenuators (SCAs), which have finite attenuation values, usually set by a digital control word.

Continuously variable attenuators employ PIN diodes (FETs) as current (voltage) variable resistors, similar to the switches described in Section 10.2. The main difference between an SPST and a continuously variable attenuator is that the latter uses all the conduction states of its semiconductor devices, not just the fully conducting and fully interdicted ones. The equivalent networks to consider for the attenuator devices are the linear ones, discussed in Section 9.7.1. The model in Figure 9.34a is suitable for the PIN diode as it is. On the other hand, the FETworks with no DC drain–source voltage in switch and variable attenuators, which then have zero transconductance; this type of operation defines the so-called cold FET. The cold FET linear model derives from the one of Figure 9.34b, after eliminating the VCCS. Attenuator device models cannot be simplified to the simple series or parallel networks considered for the switch, at least not in intermediate conduction states. The effect of all the parasitic elements can be precisely evaluated by computer simulations. However, we can say that variable attenuators employ variable resistors, therefore they work until the parasitic reactance is negligible with respect to the incremental resistance of each device. Section 10.2.1 discussed the relation between incremental resistance and DC (gate–source voltage) current in PIN diodes (FETs). For PIN diodes, we need to add that semiconductor manufacturers provide devices specifically tailored for applications in continuous variable attenuators. The peculiar characteristic of such diodes is their incremental resistance vs. the DC curve, which is as smooth as possible, with no high-slope regions. This minimizes the criticality in the DC setting. The main drawback of attenuator PIN diodes is their relatively slow response to variations in the attenuation-setting DC curves: PIN attenuators are typically one order of magnitude slower than their switch counterparts operating at the same frequency and power.

MICROWAVE CONTROL COMPONENTS (T1)

(T3)

RV

P1

391

RV

P2

(T2) RV

(a) (Π2)

RV

P1

(Π 1)

3) R(Π V

RV

P2

(b) C5

P1

TL1

50 Ω

C50Ω,1 L4 C2

D1

L1

I2 C50Ω,2 50 Ω

TL2

C3

L2

D2

L3

L5

L6

D3

P2

L7

I1 C1

(c)

P1

TL1

50 Ω

C4

TL2

Q1 Q2 R1

(d)

V1

TL3

50 Ω Q4

TL4

P2

Q3 R2

R3

R4

V2

Figure 10.19 Variable attenuators: (a) ideal T network; (b) ideal P network; (c) PIN diode realization; (d) FET realization.

In principle, a continuously variable attenuator is nothing more than a fixed attenuator – in T or P configuration – where controlled variable resistors replace fixed ones. Under this concept, the schematics in Figures 7.5a,b transform into the ones in Figures 10.19a,b. From a realization point of view, each variable resistor in such ideal schematics is one PIN diode or one FET with the associated bias networks. Figures 10.19c,d show two realizations of the T attenuators, based on the PIN and FET, respectively. ðT1Þ ðT2Þ ðT3Þ The diodes D1 , D2 and D3 in Figure 10.19c correspond to RV , RV and RV , respectively, in Figure 10.19a. The various inductors deliver the current to the three diodes as: .

L1 , L2 and L3 for D1

.

L4 for D2

.

L5 , L6 and L7 for D3.

392


The inductances and parasitic impedances of these inductors have a minimum impact on the transmission of the RF signal, similar to the switches discussed in Section 10.2.1. Again, the capacitors C2 and C3 prevent the current to one diode from flowing in other ones or in the RF ports, while C1 , C4 and C5 dampen the residual RF signal on the driving ports to ground. As noted in Section 7.4, T or P attenuators are symmetrical, and so ideally should the network be in Figure 10.13c. Thus D1 should be identical to D3 and their currents should be identical as well. For this reason, the schematic in Figure 10.19c assumes that one single generator supplies the current for the two shunt diodes. The inductors L3 , L5 supply additional RF isolation in the DC path from D1 to D3 . As discussed in Section 10.2.2 for the FET SPST, bias networks introduce crosstalk, which compromises the maximum isolation. Applying the same considerations to the attenuator, we find that bias network crosstalk – which is frequency dependent by definition – compromises the attenuation flatness, particularly at high attenuation. In real cases, D1 and D3 are not perfectly identical; then the lower impedance device would absorb the majority of the current, degrading the circuit symmetry. A simple way to ameliorate this malfunctioning is to add two resistors46 in series with L3 and L5 . The added resistor also provides additional attenuation for the bias network crosstalk. The network in Figure 10.19c presents two RC series cells in shunt with D1 and D3 ; their presence is a consequence of the required resistances at the different attenuations. The two capacitors in series with these resistors are large enough to ensure that the latter are parallel with the series diodes at high frequencies,47 while insulating the resistors from the DC path. Equations48 (7.9) give RT1 ðAÞ ¼ RT3 ðAÞ ¼ 50

1 10 0:05A ; 1 þ 10 0:05A

RT2 ðAÞ ¼ 100

10 0:05A 1 10 0:1A

ð10:20Þ

where A is the attenuation in dB. Now, from Equation (10.20) it follows that RT1 ðA ¼ 0Þ ¼ RT3 ðA ¼ 0Þ ¼ 0; lim RT1 ðAÞ ¼ lim RT3 ðAÞ ¼ 50;

A!1

A!1

lim RT2 ðAÞ ¼ 1

A!0

lim RT2 ðAÞ ¼ 0

ð10:21Þ

A!1

Therefore, the incremental resistance of D2 ranges over all its possible range, from zero to infinity, and the minimum resistance of the devices used determines the upper limit on A. On the other hand, the incremental resistance of D1 and D3 has a much narrower variation range, from zero to 50 O, and its minimum possible value affects the minimum achievable attenuation. Moreover, the useful range of the direct current on D1 and D3 is quite limited, making the setting of a proper value a quite difficult task. The two 50 O resistors alleviate this difficulty, because, now, the parallel resistor of 50 O with the incremental resistance of D1 and D3 , not the latter alone, must be equal to 50 O at the maximum attenuation. Consequently, the two series diodes can use the whole current range, consistently reducing the criticality of the current driving. The main drawback of the added components is the parasitic reactance they involve, which reduces the maximum working frequency of the attenuator. Note that the absorptive SPST in Figure 10.14a also uses a similar configuration in the last series switching element. The variable FET attenuator in Figure 10.19d has a structure similar to the SPST in Figure 10.3c. The main variations consist of the addition of the series FET Q4 close to the RF port P2 , and the two 50 O resistors, placed between the source and drain of the two series elements. The bias configuration in Figure 10.19d is almost the same as the one in Figure 10.3c. The same voltage V1 drives the gates of both Q1 and Q4 ; the considerations on the bias resistors presented in Section 10.2.2 hold true, and will not be

46 The best solution should consist of separating the currents supplied to D1 and D3. This solution, however, complicates the circuit driving, in that it requires three nonlinear dependent currents, instead of two. 47 If flow is the minimum working frequency, then 2p flow C50 O 1=50 must apply. 48 Derived assuming a generic normalization resistance, not simply 50 O as we have now.


393

repeated. The two 50 O resistors in parallel with the series elements work to extend the usable range of the source–drain incremental resistance, in the same way as discussed for the circuit in Figure 10.19c. By comparing PIN and FET continuously variable attenuators, we obtain more or less the same pros and cons as for switches. PINs have better minimum insertion loss, higher maximum attenuation and higher power handling. FETs have faster response and absorb no power from the driver. By comparing SPST and continuously variable attenuators, realized with the same technology and working at the same frequencies, we find that the attenuator has higher minimum attenuation than the absorptive SPST, which is higher than in the reflective SPDT. Moreover, continuously variable attenuators have lower power handling than their switch counterparts, particularly at intermediate attenuation. Indeed, device junctions can rectify RF signals and self-bias the device itself, modifying its incremental resistance. A description of continuously variable attenuators is not complete without considering the problem of DC driving. A detailed examination of the problem and of possible circuit solutions is well beyond the purposes of this book. Nevertheless, some brief considerations will give a picture of the difficulties, and justify the exploration of different attenuator architectures, where possible. Series and shunt elements have to realize precise resistance values at each attenuation, as in Equations (10.15). Given the required incremental resistance, the device characteristic determines the corresponding control quantity. For PIN diodes, inverting Equation49 (10.4) and substituting the result into (10.20), 1 0 B I1 ¼ 2B @

C n VT IS C A; 1 10 0:05A 50 Rs 1 þ 10 0:05A

I2 ¼

n VT IS 10 0:05A 100 Rs 1 10 0:1A

ð10:22Þ

The factor 2 in the first equation of (10.22) takes into account that the current I1 drives both the series diodes D1 , D3 in Figure 10.13c. From a similar procedure, inverting Equation (10.13) or (10.14), and substituting the result into (10.20), we obtain the voltages for the attenuator in Figure 10.19d, realized with a MESFET or MOSFET ðMESFET Þ

V1

1 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Vp ; 1 10 0:05A a b 50 1 þ 10 0:05A

ðMOSFET Þ

V1

¼

1 1 þ 10 0:05A þ VTH ; b 50 1 10 0:05A

ðMESFET Þ

1 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Vp 10 0:05A a b 50 1 10 0:1A

ðMOSFET Þ

¼

V2

V2

1 1 10 0:1A þ VTH b 50 10 0:05A

ð10:23Þ

ð10:24Þ

where we have considered that V2 drives two transistors (Q2 and Q3 ) whose parallel realizes the shunt resistance RT2. Although Equations (10.4), (10.13) and (10.14) are quite approximate, none of the relations (10.17) to (10.19) give linear or simple driving laws. Furthermore, all the latter three equations contains nonlinear parameters of the devices, thus the driving curves can vary from piece to piece. Figure 10.20a shows a block diagram of a possible analogue driver, where two nonlinear transfer characteristics y1 and y2 generate the driving quantities from the external control voltage. The design, setup and alignment of the driver in Figure 10.20a are quite difficult. Furthermore, the differences from device to device may require manual adjustments to each individual attenuator, on a case-by-case basis. Additional difficulties for the attenuator driving arise from the temperature dependence of the relations (10.22) to (10.24). Equation (10.22) explicitly includes the temperature-dependent factor 49 Equation (10.4) is not rigorously valid for PIN diodes. Nevertheless, it has a simple analytic form and gives a qualitatively valid relation between the current and incremental resistance of the device.

394


VCONTROL= vin

y1 (vin)

V1, or I1

y2 (vin)

V2, or I2

(a)

DWCONTROL

ROM 1

D/A 1

V1, or I1

ROM 2

D/A 1

V2 , or I2

ROM1

D/A1

V1, or I1

ROM2

D/A1

V2, or I2

(b)

VCONTROL

A/D

(c) analog signals

digital signals

Figure 10.20 Variable attenuator drivers: (a) full analogue; (b) for digital control; (c) digital with analogue input control. VT , and the parameters a; b; Vp ; VTH , of Equations (10.23) and (10.25), which are temperature dependent as well. Figure 10.20b shows a less critical solution, assuming that the attenuator control is a digital word rather than a continuously varying voltage. The combination of such a driver with a continuously variable attenuator realizes an SCA. Each digital control word corresponds to a nominal attenuation. Typically 00 . . . 0 ð11 . . . 1Þ corresponds to the minimum (maximum) attenuation with a linear staircase relation. The two digital-to-analogue converters (DACs) or (D/As) D=A1 , D=A2 generate the corresponding series and shunt driving quantity required. Two read-only memories ROM1 , ROM2 implement the nonlinear relations (10.22) to (10.24): their input (address) is the external control word, while the output (data) is the binary representation of the corresponding driving quantity. The main advantage of the digital solution is that any adjustment requires changed data to be stored in memory, rather than substitution of the components as in the analogue case. Normally, a computer-aided alignment procedure finds the optimum data to store in the ROM, by setting a value, measuring the attenuator and adjusting – if needed – before storing. One additional advantage of the digital solution with respect to the analogue one is the possibility to implement temperature-dependent look-up tables, by including in the ROM address some bits connected to a temperature sensor.50

50

See [9] for more details.

MICROWAVE CONTROL COMPONENTS 1

395

s11(RV) s 12 (RV) s21(RV) s 22 (RV)

RV, 2

(a) hybrid1 1

P1

hybrid2 2

0°

1

RV,1

90° 90°

50Ω 3

0°

4

2

0°

50Ω

90° 90°

3

0°

4

P2

RV,2 (b)

P1 s21(R1)

0.5

a1

0.5

s22 (R1) j

0.5

s11(R1)

j

0

0.5 0.5

b1 0.5

s12(R 1) s21 (R2)

P2 0.5

b2

0.5

0

j

0.5

Figure 10.21 graph of (b).

j

0.5

s11(R 2) 0.5

(c)

s22(R2)

s12 (R2)

a2 0.5

Shunt resistor only attenuator: (a) basic cell; (b) matched attenuator; (c) signal flow

Figure 10.20c shows a digital configuration emulating an analogue one. One analogue-to-digital converter (ADC) or (A/D) transforms the external continuously varying control voltage into a corresponding digital word. From the output of the A/D up to RF devices the diagram in Figure 10.20c coincides with the one in Figure 10.20b. The configuration in Figure 10.14c circumvents the difficulties involved with the analogue nonlinear transfer characteristic, but realizes a staircase rather than a true continuous relation between the control voltage and attenuation. When digital or pseudo-linear control is not acceptable, and the high-frequency bandwidth is relatively narrow, a solution with shunt elements only – and thus with a single driving quantity – can be used. If we simply remove the series resistors from the T attenuator we obtain the simple two-port network in Figure 10.21a. Its scattering matrix derives from Equation (7.4) with the simplification RT1 ¼ RT3 ¼ 0 1 s11 ðRV Þ s12 ðRV Þ 50 2RV ¼ ð10:25Þ s21 ðRV Þ s22 ðRV Þ 2RV þ 50 2RV 50 From this equation it follows that the network in Figure 10.21a achieves any transmission coefficient between zero and one when the shunt resistance varies from zero to infinity. Unfortunately, the reflection

396


coefficient is also a function of the shunt resistance, and becomes unitary if RV ¼ 0. Inverting the relation between s21 and RV in Equation (10.25) and substituting the result into s11 in the same equation, we obtain s11 ¼ 1 s21

ð10:26Þ

which implies that the return loss decreases with the attenuation and becomes better than 10 dB, i.e. js11 j 10 0:5 ffi 0:316 when s21 1 10 0:5 ffi 0:684, corresponding to a maximum attenuation of A ¼ 20log10 ðjs21 jÞ ffi 3:3 dB. Such an attenuator should have quite a few applications because of its poor impedance matching and/ or limited attenuation range. Figure 10.20b shows an arrangement with two attenuators of the type in Figure 10.20a, placed between two 90 hybrid couplers. That configuration inherently provides low (ideally zero) reflection coefficients at the RF ports P1 and P2 as can easily be demonstrated by analyzing the SFG of the network, shown in Figure 10.20c. The reflection coefficient at P1 is, by definition, the ratio of the reflected (b1 ) and incident (a1 ) waves to that port, when the incident wave on P2 is zero. From a1 to b1 we have two paths, each having three branches. The resulting reflection coefficient is then pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð10:27Þ s11 ¼ 0:5 s11 RV;1 0:5 þ j 0:5 s11 RV;2 j 0:5 ¼ s11 RV;1 s11 RV;2 The anti-symmetry of the network implies that the reflection coefficient at P2 is s22 ¼ s22 RV;2 s22 RV;1

ð10:28Þ

Equations (10.27) and (10.28) imply that if RV;1 ¼ RV;2 then s11 ¼ s22 ¼ 0. Thus, if the two shunt resistors are identical, ports P1 , P2 of the network in Figure 10.20b are perfectly matched. For the transmission coefficient, we still have two paths, each having three branches, which give pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi s21 RV;1 þ s21 RV;2 ð10:29Þ s21 ¼ 0:5 s21 RV;1 j 0:5 þ j 0:5 s21 RV;2 0:5 ¼ j 2 Thus, the resulting transmission coefficient is the arithmetic mean of those of the two unmatched attenuators.51 If RV;1 ¼ RV;2 from Equation (10.29) it follows that the attenuation of the network in Figure 10.20b coincides with that of the circuit in Figure 10.20a, apart from the dissipation loss of the two hybrids, which have been not considered. The derivation of Equations (10.28) and (10.29) implies no assumptions on the S parameters of the two two-port networks placed between the two hybrids. Then, in the arrangement of Figure 10.20b, with two identical two-port networks placed between the two 90 hybrid couplers: .

The reflection coefficient at the RF ports P1 , P2 is zero.

.

The magnitude of the transmission coefficient from P1 , P2 equals that of the two identical two-port networks.

These properties have important applications in amplifiers, as Section 11.5.4 will show. The configuration with 90 hybrids has the important property of requiring one single control voltage or current, while guaranteeing good impedance matching. Its main drawback is the need for two hybrid couplers, which are significantly larger than the conventional series elements, introduce additional loss and – above all – are inherently limited in bandwidth. Section 6.7 illustrates two types of 90 hybrids which are widely used in engineering practice: the branch line and the Lange coupler,52 which have a fractional bandwidth53 of about 20% and 100%, respectively. 51 The Ansoft file 03_Shunt_Resistor_Attenuator.adsn provides a simple description of the attenuator in Figure 10.15b, assuming a coupled line realization of the hybrids. 52 See Section 7.7.3 53 The fractional bandwidth is defined as 100 Df =f0 , where D f ¼ f2 f1 , f0 ¼ ðf2 þ f1 Þ=2 and f1 ; f2 are the limits of the bandwidth.


P1

SW1

SW3

RT,1

RT,3

RT,2

397

P2

SW2 (a) SW2

P2

P1 R Π,1

R Π,2

SW1

R Π,3 SW3

(b)

AdB 1

1

P2

P1 2

(c)

Figure 10.22

2

vCONTROL

Binary attenuators: (a) T cell; (b) P cell; (c) switched cell.

The combination of a continuously variable attenuator with a digital driver, like that in Figure 10.20b, is an SCA, as anticipated. Here, however, we will consider other attenuator circuits which inherently have discrete attenuation values; all of them are combinations of fixed attenuators and switches. Figure 10.22 shows three configurations of attenuators with only two possible attenuation values, or binary attenuators. All of them exhibit an attenuation variable from a minimum (ideally zero) and a specified maximum (ideally flat over the frequency) value. In more detail, Figure 10.22 shows: (a) Binary T attenuator. The two switching elements SW1 and SW3 are always in the same state, which is opposite to that of SW2 . If SW1 ¼ SW3 ¼ ON and SW2 ¼ OFF then the two series resistors RT,1, RT,3 are short-circuited and the shunt resistor RT,2 is disconnected from ground, working like an open circuit. The insertion loss in this condition is ideally zero, but in real cases is higher, due to the parasitic elements in the resistors and switching elements with the associated bias networks. Under the opposite condition, SW1 ¼ SW3 ¼ OFF and SW2 ¼ ON , the series resistors are not bypassed and the shunt resistor is connected to ground. Thus, the attenuator works like a passive fixed one. The series resistance of SW2 sets a lower limit on the effective shunt resistance, thus an upper limit on the attenuation. The parasitic-caused loss sets a lower limit on the minimum attenuation. (b) Binary P attenuator. This is the dual of (a), with a consequent opposite switch actuation logic. The same considerations for the parasitic effect apply, and will not be repeated.

398

MICROWAVE AND RF ENGINEERING (c) Switched binary attenuator. Two SPDTs sharing the same control signal select two possible paths between the two RF ports: one path has one attenuator with attenuation AdB , the other path is a direct connection between the two switches. If the two SPDTs are in position 1, the attenuation between the two RF ports is AdB plus the one resulting from the two switches. In the non-attenuating position (both SPDTs in position 2), the insertion loss is the sum of those of the two SPDTs. In the attenuated position (1), there is, however, a parasitic transmission via the non-selected ports of the SPDT, due to the finite isolation of the latter. That parasitic transmission has to present an attenuation much greater than AdB , otherwise the relative attenuation between the two states is not controllable.54 Therefore the isolation of each SPDT must be much higher than AdB =2. This configuration offers the maximum design flexibility: the fixed attenuator could be of any type, the switching elements do not affect the attenuation flatness, and there is no theoretical limit on the maximum attenuation, provided there is enough isolation in the switches. The main drawback is the presence of two SPDTs on the signal path, even at no nominal attenuation. The associated loss is between 2 and 6 dB, depending on the switch technology and on the required maximum attenuation.55

The binary switches can be used as standalone, when only one selectable attenuation is needed, or in combination with other similar components, to build a multiple step attenuator. Figure 10.23a shows an arrangement with N cascaded binary attenuators, each of them having a separate control signal, or bit. All the cells in this schematic are of the type in Figure 10.22c, although any of the three solutions of Figure 10.22 could be used. Thus the schematic in Figure 10.23a is a purely conceptual representation: each cell has two states, one with minimum attenuation (ideally zero) and one with a specified attenuation; furthermore, there is one single command signal per cell. Let us assign the logical values 0 and 1 to each control quantity VCNTRL;k ðk ¼ 1; 2; . . . ; N Þ . If VCNTRL;k ¼ 1 or 0 the SPDT of the kth cell is in position 1 or 2, and thus in the first (latter) case we insert an attenuation of 2k 1 AdB (0) inside the path from P1 to P2 . The two extreme conditions are VCNTRL;1 ; VCNTRL;2 ; . . . ; VCNTRL;N ¼ 0; 0; . . . ; 0 and VCNTRL;1 ; VCNTRL;2 ; . . . ; VCNTRL;N ¼ 1; 1; . . . ; 1. In the first case, the only loss in the RF path is the one coming from the switches;56 in the latter case, there is an additional loss of ! N X N 1 k1 AdB ¼ 2N 1 AdB AMAX ¼ AdB þ 2AdB þ þ 2 AdB ¼ 2 ð10:30Þ k¼1

Also, it is easy to recognize that, by changing the control bits, it is possible to obtain 2N discrete attenuation values bound between zero and AMAX , and spaced by the step AdB . Furthermore, the resulting attenuation is proportional to the digital word VCNTRL;1 ; VCNTRL;2 ; . . . ; VCNTRL;N , which therefore assumes the function of a digital control word. The architecture in Figure 10.23a is elegant, requires no complicated drivers, is inherently temperature stable and produces lower nonlinear distortion than the solutions based on continuously variable attenuators. Its main drawback is its minimum insertion loss, caused by 4 N cascaded SPDTs, or by 3 N resistor-removing switching elements. That loss depends on many factors, like the technology used, the working frequency and the maximum attenuation; as a rule of thumb, we can assume from 1 to 7 dB per binary cell. 54 Crosstalk varies with frequency, therefore it causes frequency-dependent variations on the attenuation, degrading the flatness over frequency. 55 Maximum attenuation affects the switch isolation, as shown. The higher the attenuation, the higher the required isolation, then the higher the number of switching elements and, finally, the higher the insertion loss. Therefore binary cells with small values of AdB require low isolation from the switches, and potentially allow low attenuation in the minimum loss state. 56 Or, more generally, the residual minimum attenuation of all the binary cells, if some of them have the configuration in Figure 10.16a or 10.16b.

MICROWAVE CONTROL COMPONENTS A dB

399

2 N-1AdB

2•A dB

1

1

1

1

1

1

2

2

2

2

2

2

P1

P2

VCONTROL,1

(a)

VCONTROL,2 2

3 •AdB

3•2• AdB

2•AdB

2•2•A dB

2

1

P1 4

1 2 3

VCONTROL,N

A dB

2 3

4

1

4

1 2 3

2

1•2•AdB

2 3

P2 4

VCONTROL,1 VCONTROL,2 VCONTROL,3 VCONTROL,4

(b)

Figure 10.23

Multi-bit digital attenuators: (a) with binary cells; (b) with quaternary cells.

The configuration in Figure 10.23b works in the same way as the one in Figure 10.22a, at least from an external point of view, but alleviates the minimum loss problem by halving the number of switches along the chain. The building block of the architecture in Figure 10.23b is a four-state (or quaternary) attenuator, rather than a binary one, and has two control bits. One quaternary cell is equivalent to two cascaded binary cells, both realizing all the integer multiples from zero to three of an assigned attenuation. Therefore, the configuration in Figure 1023b derives from the one in Figure 10.23a by regrouping two adjacent binary cells into a quaternary one. The minimum loss in the solution in Figure 10.17b is slightly higher than the one in Figure 10.17a, because the SP4T has higher loss than the SPST realized with the same technology. Figure 10.24 shows the attenuation of a 4-bit SCA, implementing the architecture in Figure 10.23b, with two quaternary cells, each with PIN diode SP4T. The working frequency range is 2–20 GHz and the displayed attenuation is normalized to the reference 0 dB state. In the minimum attenuation, the insertion loss of the attenuator is about 6, 5 and 7 dB at 2, 12 and 20 GHz, while the return loss is better than 12 dB in all the states. The 1 dB compression input power is about 1 W ( þ 30 dBm) and the switching time is less than 150 ns, from 50% of the control bit voltage to 90% of the RF output power. The whole discussion on variable attenuators pointed out some key aspects of their working principles and characteristics: 1. A variable attenuator realizes some attenuation values from zero to an assigned maximum. The zero attenuation state is characterized by a non-zero insertion loss, thus a minimum insertion loss has to be specified for the component. 2. The attenuation at each state differs from the nominal value and is frequency dependent as well. Therefore attenuation error and flatness over the frequency must be specified.

400

MICROWAVE AND RF ENGINEERING 0 -4

20 log10(|s21/s21,REF|)

-8 -12 -16 -20 -24 -28 -32 2

Figure 10.24

4

6

8


16

18

20

Measured response of a 4-bit SCA, having the configuration of Figure 10.23b.

3. Variable attenuators based on continuously variable high-frequency devices are inherently temperature dependent, and this behaviour needs to be specified. 4. The SCA implementing the architectures in Figure 10.23 could be non-monotonic. For instance, let us consider the first case, and assume we pass from the attenuation of ð1 þ 2 þ 22 þ þ 2N 2 ÞAdB to the next step, which is 2N 1 AdB . The two values are obtained by totally different chains. If the cells from 1 to N 1 present higher attenuation than their nominal value, the last cell presents lower attenuation than the nominal one. In this case, it could happen that the attenuation decreases instead of increasing, passing from the first to the second state. The monotonic behaviour is a requirement to be specified, if needed. 5. As for any component employing semiconductor devices, variable attenuators produce nonlinear distortion. A maximum input power exists that guarantees a specified distortion level, and a nondestructive limit exists as well. 6. Variable attenuators require a finite settling time, as any variable-state component. 7. Variable attenuators vary not only the amplitude but also the phase of the transmission coefficient with the control signal. If not specified otherwise, the phase variation is virtually out of control. Attenuators with constant phase for different attenuations are known as phase invariant. The design of a phase-invariant attenuator is particularly difficult, especially in the case of the continuously variable type. A relatively easier possibility consists of using a solution based on binary cells similar to the one in Figure 10.23c. They can be made phase invariant by inserting an additional two-port network in one of the two paths, in order to make the argðs21 Þ of the two paths as similar as possible. Furthermore, the two SPDTs have to be symmetrical with respect to the two selectable ports. This way, the argðs21 Þ of the binary cell does not change with the switch position, i.e. with the attenuation, and so it is for the complete attenuator, as required for a phase-invariant one.

10.4 Phase shifters Section 7.5 deals with fixed phase shifters, which are two-port networks presenting a controlled phase of the transmission coefficient over the frequency. Here, we will deal with variable phase shifters that are


401

required when signals travelling along different paths must be aligned in time. An important application of variable phase shifters is the phased array antenna;57 in that application, the radiation diagram of the antenna rotates if the RF signals of the different radiating elements are properly delayed. In the rest of the section, the denomination ‘phase shifter’ will be used instead of ‘variable phase shifter’ for brevity and since there will be no possibility of confusion with the circuits considered in Section 7.5. By definition, a phase shifter is an N-state two-port network which is ideally matched and loss free in all its N states. Denoting the phase shifter states by superscripts ‘(1)’ to ‘(N)’, we have ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðN Þ

ðN Þ

s ¼ s ¼ s ¼ s ¼ ¼ s11 ¼ s22 ¼ 0 11 22 11 22 ð1Þ ð1Þ ð2Þ ð2Þ ðN Þ ðN Þ s12 ¼ s21 ¼ s12 ¼ s21 ¼ ¼ s12 ¼ s21 ¼ 1

ð10:31Þ

Assuming state (1) as the reference, we have that the transmission coefficient phase in the various states, normalized to the reference, assumes a known frequency dependence. That is, we have h i ð2Þ ð1Þ Arg s21 =s21 ¼ f2 ð f Þ h i ð3Þ ð1Þ Arg s21 =s21 ¼ f3 ð f Þ ð10:32Þ .. . h i ðN Þ ð1Þ Arg s21 =s21 ¼ fN ð f Þ We can classify the phase shifters according to their phase characteristics fk ð f Þ, obtaining: I. True phase, if fk ð f Þ is a constant over the frequency. II. True delay, or true time delay (TTD), if fk ð f Þ is proportional to the frequency. As their names suggest, true-phase (true-delay) phase shifters realize a phase shift (delay) that is constant with frequency and can be set with suitable driving quantities. Another possible classification of phase shifters is based on their circuit configuration: III. Transmission phase shifter, which is a linear loss-free matched two-port network having one or more variable elements. IV. Reflection phase shifter, which is a special case of a transmission phase shifter with a structure including one 90 hybrid. It exploits one 90 hybrid coupler to transform the reflection coefficient s11 of two identical bipoles into the transmission coefficient s21 ¼ js11 . The transmission coefficient obtained then has the same magnitude as the bipole reflection coefficient (which is unitary if the bipole is reactive) and the same phase (besides a constant difference of p=2). V. Analogue or continuously variable phase shifters, which are characterized by the fact that the phase shift can assume any intermediate value between a minimum and maximum value. Typical cases of analogue phase shifters employ varactors: when the reverse diode voltage – which is the control voltage of the phase shifter – varies within a given interval, the varactor capacitance, and thus the relative phase characteristic, uniformly and continuously vary from zero to the maximum. Figure 10.30b shows the schematic of a phase shifter using two varactors.

57

See Section 15.5.1.1.

402

MICROWAVE AND RF ENGINEERING L

L

P1

P2 C

(a)

L

2L

2L

L

P1

P2 C

C

C

(b)

Figure 10.25 True-delay transmission phase shifter: (a) elementary cell; (b) multiple cascaded cells. VI. Digital or stepped phase shifters, if the phase shift can assume only n discrete values. VII. Binary phase shifters, which are special cases of digital phase shifters, having an integer power of two different states, n ¼ 2N BIT , where the integer number N_BIT is the number of bits of the digital control word. All the configurations III to VII can realize any of the responses I and II. However, this section shows only some of the possibilities.

10.4.1

True-delay and slow-wave phase shifters

Figure 10.25a shows a simple transmission-type phase shifter. Its two identical capacitors are variable with an external control. We can consider the network in Figure 10.25a as a lumped approximation of a transmission line segment. The lumped nature of the network, however, introduces a cut-off frequency. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The characteristic impedance, propagation delay and cut-off frequency are equal to Z0 ¼ 2 L=C , pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0:5 58 t ¼ 2 L C and 2pfT ¼ ðL CÞ respectively. Now, with a proper choice of L, and if C varies within a sufficiently narrow interval, Z0 is such as to hold the network matched, the delay varies pffiffiffiffi proportionally to C and the cut-off frequency remains above the maximum working frequency. For instance, if L ¼ 0.46 nH and C 2 ½0:1; 0:5 pF, we have Z0 2 ½95:92; 42:9 O;

t 2 ½9:59; 21:45 ps;

fT 2 ½23:47; 10:49 GHz

59

A circuit analysis of the network with the above values shows that the return loss is always better than 10 dB, from zero up to 8 GHz. Choosing the minimum capacitance as the reference state, the expected relative delay of the network is Dt ¼ tðC ¼ 0:5 pFÞ tðC ¼ 0:1 pFÞ ffi 21:45 9:59 ffi 11:86 ps. Figure 10.26 shows the result obtained from the circuit analysis. The transmission delay of the network in Figure 10.25a can be increased by cascading multiple cells, then obtaining the network in Figure 10.25b. A variation of the same concept is to use one of the lowdpass filters presented in Chapter 8. A Chebyshev filter60 could be one possibility. It presents a good matching, but also large variations of the group delay, within its passband. Therefore, only the first lower portion of the passband can be used in this application. Conversely, we have the Bessel filter, which exhibits a flat group delay, but a poor passband matching. For opposite reasons, we arrive at the same conclusion: that is, to use filters having a cut-off frequency much higher than the maximum working frequency of the phase shifter. 58

The cut-off frequency considered here is the image parameter value. See [5]. See the Ansoft file 04_True_Delay_Phase_Shifter.adsn. 60 See the Mathcad file 08_Chebysheff_Lowpass_Filter_Delay.MCD. 59


403

20 C=0.5 pF

Relative delay, ps

16

C=0.4 pF 12 8

C=0.3 pF

4

C=0.2 pF

0

C=0.1 pF

-4 0

2

Figure 10.26

P1

4 6 Frequency, GHz

8

10

Relative delay of the phase shifter in Figure 10.25a. Ll Cl

Ll CS Cl

Figure 10.27

Ll CS Cl

Ll CS Cl

Ll

P2

CS

Slow-wave phase shifter.

The networks in Figure 10.25 usually have a smaller length61 than transmission lines presenting the same delay. The apparent increase in the component length can be thought of as due to a reduction in the wave propagation inside the component itself, with respect to the transmission line. For this reason these types of networks are also referred to as slow-wave phase shifters. Alternatively, slow-wave phase shifters can be seen as variants of the loaded-line phase shifter, described in Section 6.5.1. It consists of a transmission line periodically loaded with electronically tunable reactive loads, which affect the phase velocity of the travelling signal. The loads are usually realized as parallel capacitors connected through diodes or MEMS switches. Figure 10.27 shows the lumped element circuit model of a capacitively loaded slow-wave phase shifter, the transmission line being represented by the distributed inductance Ll and capacitance Cl . When put at a distance d which is small with respect to the wavelength, the capacitors realize a distributed capacitance CS =d that adds up to the distributed line capacitance. The phase constant thus becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ o Ll ðCl þ CS =d Þ ð10:33Þ The periodic load therefore has the effect of reducing the phase velocity of the signal, producing, with respect to the unloaded case, a phase shift rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CS 1 1þ 1 ð10:34Þ Df ¼ lDb ¼ bl l d Cl pffiffiffiffiffiffiffiffiffi where bl ¼ o Ll Cl . The capacitance CS can vary in a continuous or discrete fashion, by employing, for example, varactor diodes, semiconductor switches or MEMS switches. 61

But also higher loss, due to the higher current densities involved with the reduced size.

404


As observed for the circuit in Figure 10.25a, the added capacitance simultaneously modifies the phase velocity and the characteristic impedance of the line, so producing a mismatch of the device. A possible countermeasure consists of simultaneously loading the line with series inductors in such a way as to keep the same L/C ratio as the unloaded line. This is, of course, at the price of a more complicated device.62 Depending on whether the loading capacitance is allowed to vary in a discrete or continuous fashion, the reflection architecture is amenable for realizing digital or analogue phase shifters.

10.4.2

Reflection phase shifters

The schematic in Figure 10.28a shows the principle of a reflection phase shifter. In practical realizations, the 90 hybrid coupler can assume the aspect of a Lange or branch-line coupler, as explained in Sections 6.7 and 6.8. The choice between the two cases depends on the available technology and on the required bandwidth. We can analyze the network in Figure 10.28a by means of its corresponding SFG in Figure 10.28b. Using the same method as for the variable attenuator in Figure 10.21b, we obtain pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi s11 ¼ 0:5 G1 0:5 þ j 0:5 G2 j 0:5 ¼ G1 G2 ð10:35Þ s21 ¼

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi G1 þ G2 0:5 G1 j 0:5 þ j 0:5 G2 0:5 ¼ j 2 s22 ¼ s11 ¼ G2 G1

ð10:36Þ ð10:37Þ

Thus, if the two bipoles are identical ðG1 ¼ G2 ¼ GÞ, it follows that the network in Figure 10.28a is perfectly matched ðs11 ¼ s22 ¼ 0Þ, and the transmission coefficient coincides with the bipole reflection coefficient multiplied by the imaginary unit ðs21 ¼ j GÞ. The multiplying coefficient j comes from the 90 phase shift of the hybrid, and has no effect on the amplitude of the transmission coefficient, while making argðs21 Þ ¼ argðGÞ þ p=2. Moreover, if the bipoles are purely reactive ðjGj ¼ 1Þ, then the phase shifter in Figure 10.27a is loss free ðjs21 j ¼ 1Þ, as in an ideal phase shifter. Therefore, varying one or more components inside both the bipoles, we change the phase of the transmission coefficient in the network. Figure 10.27c shows a simple case where the two reactive bipoles consist of variable capacitances.63 The reflection coefficient of a capacitor, having capacitance C, connected to ground is GC ¼

ZC R0 ðjoCÞ 1 R0 ¼ ¼ expðjjÞ ZC þ R0 ðjoCÞ 1 þ R0

with

j ¼ 2tan 1 ðR0 C oÞ

ð10:38Þ

The reflection coefficient has unitary amplitude, as expected. The reflected wave has the same amplitude as the incident one, while its phase presents an additional shift, tending to zero or to p for zero or infinite capacitance, respectively. Therefore, by varying C within an interval, we have a corresponding variation in the phase shift of the network in Figure 10.28c. Such a phase variation is frequency dependent, because it is the reflection coefficient (10.38). Nevertheless, a careful choice of the capacitance variation range can optimize the phase flatness over the frequency range of interest. Figure 10.29 shows the relative phase shift of the network when C ranges from 0.1 to 0.589 pF, using C ¼ 0:1 pF as reference. It can be seen that the simple ideal network approximately realizes a true-phase response within the frequency range 10–15 GHz.

62 63

See [13]. See the Mathcad file 09_Reflection_Phase_Shifter.MCD and the Ansoft file 05_Reflection_Phase_Shifter.adsn.

MICROWAVE CONTROL COMPONENTS P1

1

P2

3

0°

405

0° 90° 90°

Γ1

2

Γ2

4

(a)

P1 0.5

a1 j

b1

0.5

a1

0.5 j

0.5

b1 (b)

P1

Γ1

0.5

Γ2

0.5

P1

1

P2

3

0°

0° 90° 90°

2

4

C

C

(c)

Figure 10.28 shifter.

Reflection phase shifter: (a) basic configuration; (b) SFG; (c) variable-capacitor phase

The phase shifters in Figures 10.25a, 10.27 and 10.28c use variable capacitors to implement variable reactive elements. One possible way to realize variable-voltage-controlled capacitors is to exploit the junction of a reverse-biased diode. To this end, manufacturers provide specially tailored devices, known as varactors, or sometimes varicaps (from the contraction of variable reactors, or variable capacitors). Varactors are diodes designed to present a controlled C–V characteristic following Equation (9.74), with precise and stable parameters ð Cj0 ; Vj ; g; Cj;lin Þ and with a series resistance as small as possible. Figure 10.30b shows a phase shifter deriving from, and more realistic than, the one in Figure 10.28c, which is also shown in Figure 10.30a as a reference. The network working frequency range is 6–18 GHz.

406


Relative phase shift, deg

112.5 90.0

C = 0.589 pF

67.5

C = 0.369 pF

45.0

C = 0.247 pF

22.5

C = 0.164 pF

0.0

C = 0.1 pF

-22.5 10

11

1

P2

3

0°

1

P1

2

0° hybrid1

4

C

15

4

hybrid1

CL1

90° 90° 2

14

Relative phase shift of the network in Figure 10.28c.

Figure 10.29

P1

12 13 Frequency, GHz

TLA,2

C

P2

3

TLA,1 TLB,1

TLB,2

TLA,3 TLB,3

(a)

DV,A

DV,B LBias,A LBias,B

CBias,A

C Bias,B

(b)

V1 Figure 10.30

Variable-capacitance reflection phase shifter: (a) principle; (b) circuit implementation.

A complex network, including one varactor DV;K with its bias elements CBias;K ; LBias;K and three transmission lines TLK;1 ; TLK;2 ; TLK;3 (with K ¼ A or B), replaces each capacitor to ground. The bias network operates in the same way as already seen for PIN diode switches and attenuators. The capacitors CBias;K provide a low impedance to ground at high frequencies, while being a DC open circuit, the inductors LBias;K feed the DC voltage to the varactor anodes while minimizing the RF leakage to the driving port. The two short-circuit stubs TLK;2 work as RF elements, and simultaneously provide a DC return for the varactors. The three transmission lines not only flatten out, but also reduce, the phase shift produced by the varactor capacitance.


407

Table 10.4 Optimized values of the phase shifter in Figure 10.24b. TL1;K TL2;K TL3;K CV ¼ ½0:04; 0:517 pF

Z0 ¼ 46:4 O Z0 ¼ 113 O Z0 ¼ 33:8 O

yð f ¼ 12 GHzÞ ¼ 54:9 yð f ¼ 12 GHzÞ ¼ 102 yð f ¼ 12 GHzÞ ¼ 21:2

112.5 0.517 pF

Phase shift, deg

90.0 67.5

0.344 pF

45.0

0.221 pF

22.5

0.124 pF Cv = 0.04 pF

0.0 -22.5 6

8

Figure 10.31

10


16

18

Phase shift of the circuit in Figure 10.30b.

Instead of an ideal hybrid coupler, the network in Figure 10.24b uses a matched coupled-line section, having Z0e ¼ 137 O; Z0o ¼ 2500=Z0e ffi 18:25 O and being l/4 at the centre of the working frequency of 12 GHz. This latter circuit element is a first-order model for a Lange coupler, ignoring the mode dispersion, the dissipation loss and the bond wire inductance. The three transmission line parameters and the capacitance range are computed by a circuit optimization, producing the results in Table 10.4. Figure 10.31 shows the phase response64 of the optimized network, normalized to the minimum capacitance (0.04 pF) state.

10.4.3

Stepped phase shifters

One possibility of realizing stepped phase shifters is to use a continuously variable phase shifter – for example, the circuit of Figure 10.30b – in combination with a D/A-based driver. The N BIT-bit digital control word at the A/D input transforms into one of the 2N BIT possible control voltages, and then the corresponding phase shift, which can thus assume 2NB discrete values. This configuration is the phase shifter counterpart of the variable attenuator using the driver of Figure 10.20b. Alternatively, it is possible to implement the variable reactance required for the circuits of Sections 10.4.1 and 10.4.2 with multiple switched reactive elements. The solution inherently realizes – frequency by frequency – only discrete reactances and consequently phase shifts. A discrete variable reactance can be realized with multiple switchable series or parallel reactive elements, which could be capacitors, inductors or combinations of the two. Figures 10.32a,b show a series and parallel switchable capacitor, respectively. By replacing the capacitors in Figure 10.32 with inductors, resonating cells or 64

See the Ansoft file 06_Analog_Phase_Shifter.adsn.

408


(a)

(b)

Figure 10.32 Switched capacitor realization of variable capacitances: (a) series configuration; (b) parallel configuration. transmission lines, we obtain variable inductors, reactive LC bipoles or stubs, respectively. This possibility increases the design flexibility. The capacitors in Figure 10.32 can be inserted or removed by means of electrically controlled switching elements: namely, PIN, FET or MEMS devices. They are no different from those considered in Sections 10.2, and no further description is required. Stepped phase shifters can also be realized with groups of n two-port networks, selectable through couples of SPnT. If n ¼ 2, or n ¼ 4, these configuration are the counterparts of the attenuators in Figure 10.23a, or 10.23b, respectively.

10.4.4

Binary phase shifters

All the configurations described in Section 10.4.3 can also be used to realize binary phase shifters, as a consequence of the fact that binary phase shifters are particular cases of the stepped ones. However, the peculiarity of the binary phase shifter is that the number of its states is an integer power of 2: that is, n ¼ 2N BIT , where the integer number N BIT is the number of bits. Furthermore, the difference between the phase shift at the next two states is constant. This property can be exploited to realize the phase shifter by cascading N BIT cells, each with two phase shifts (or with 1 bit). Figure 10.33a shows the structure of a two-state (1-bit) binary phase shifter. The two two-port networks NTA , NTB placed between the two SP2Ts are ideally impedance matched. The two SP2Ts are impedance matched too, and their transmission coefficient phase from the common to the selected port does not change from position 1 to 2. This latter assumption is always satisfied, in that we can consider any phase difference between the SPDT paths as belonging65 to NTA or NTB. Under these hypotheses, the S parameters of the phase shifter, when the two SPDTs are in position 1 or 2, are respectively " ð1Þ ð1Þ # s11 s12 0 1 ðSPDT1;position 1Þ ðAÞ ðSPDT2;position 1Þ s21 s21 ¼ s ð1Þ ð1Þ 1 0 21 s21 s22 65

In addition, SPDTs are usually symmetrical, e.g. Figures 10.4b, 10.5b, and 10.14b.


409

φA- φB= ∆φ NTA φA,1

1

1

P1

P2 φB,1

2

SPDT1

(a) φA,1 - φB,1 = ∆φ 1

P1 2

(b)

φA,1 φB,1

φA,2- φB,2 = 2•∆φ 1

1

2

2

V1

φA,2 φB,2

2

NTB

SPDT2

V1 φA,N_BIT - φB,N_BIT = 2

•∆φ

N_BIT-1

1

1

2

2

V2

φA,N_BIT φB,N_BIT

1

P2 2

VN_BIT

TL2 TL1

TL3

∆l/2

1

1

P1

P2 2

TL4

2

SPDT1

(c)

Figure 10.33

SPDT2 V1

Binary phase shifters: (a) single bit cell; (b) multi-bit architecture; (c) TTD binary cell. "

ð2Þ

s11

ð2Þ

s21

ð2Þ

s12

ð2Þ

s22

#

¼

0 1 ðSPDT1;position2Þ ðBÞ ðSPDT2;position2Þ s s21 s21 1 0 21

The relative phase shift is then " # " # " # ð1Þ ðSPDT1;position 1Þ ðAÞ ðSPDT2;position 1Þ ðAÞ s21 s21 s21 s21 s ¼ jA jB ¼ Dj arg ð2Þ ¼ arg ðSPDT1;position 2Þ ðBÞ ðSPDT2;position 2Þ ¼ arg 21 ðBÞ s21 s21 s21 s21 s21

ð10:39Þ

where fA ; fB are the transmission coefficient phases of the two two-port networks placed between the two SPDTs. Equation (10.39) states that a proper design of the two fixed two-port networks allows us to obtain any required phase characteristic from the phase shifter in Figure 10.33a: true phase or true delay. The configuration in Figure 10.27 may also be employed to realize a binary phase shifter by cascading slow-wave sections, each section realizing the appropriate bit. Similarly, 1-bit phase shifters can also be obtained from any of the configurations in Figures 10.25 and 10.30b. In this latter case the control voltage has to assume two different values. Compared with these cases, the design of the network in Figure 10.33a is simpler, and gives better phase accuracy, because the two networks NTA , NTB can be synthesized independently of each other. On the other hand, the two SP2Ts in the architecture in Figure 10.27a make the insertion loss more critical than in the other solutions.

410


Figure 10.34

Photograph of a binary TTD MEMS phase shifter.

Figure 10.33b shows N BIT cascaded binary cells, realizing a multiple bit phase shifter. The relative phase shift of each cell is double (half) that of its adjacent one towards the RF port P1 (P2 ). Note the similarity of the structure in Figure 10.33b with the one for the attenuator in Figure 10.23a. in From similar considerations66 as those used for the attenuator, we conclude that the architecture Figure 10.33b has n ¼ 2N BIT states with a relative phase shift from zero up to 2N BIT 1 Df, with a step of Df. Again, Df could be either constant or proportional to the frequency, for a true-phase or a true-delay response. Figure 10.33c shows one particular 1-bit TTD cell, known as a switched-line phase shifter. Conceptually, it consists of two pieces of matched transmission lines, of different length, placed between two SP2Ts. The phase shift produced is proportional to the difference Dl between the physical length of the delay (TL1 ,TL2 and TL3 ) line and that of the reference line (TL4 ). Clearly, the longest line needs to be folded in order to reach the same SP2Ts as the shortest one, which is why Figure 10.33c represents the longest line with three cascaded segments. Figure 10.34 shows the layout of a 3-bit phase shifter employing three cascaded cells of the type in Figure 10.33c. The SP2Ts use MEMS devices. The labels on the first cell (on the left) indicate the four transmission lines (TL1 to TL4 ) and the control voltages (V1a and V1b ). The RF ports of the chip are indicated by P1 and P2 . The component in Figure 10.34 is designed to operate in the Ka band (centre frequency is 20 GHz). The simulated response is shown in Figure 10.35. As the linear phase variation shows, the device is a TTD. Due to reactive parasitic phenomena, the device does not operate properly above 24 GHz. Figure 10.36 shows an alternative example of a binary phase shifter cell, obtained by switching between a lowdpass (lag) and a highdpass (lead) filter. There are many possible design techniques for this network, from network synthesis to circuit optimization. We will present a simple method, based on a simple combination of filter synthesis and computer optimization.67 ðLPÞ ðHPÞ Let f1 ; f2 be the limits of the working frequency bandwidth and fT ; fT the cut-off frequencies of the lowdpass and highpass filter, respectively. Since the network must be impedance matched over all the ðLPÞ ðHPÞ frequency range in both its states, then fT f2 and fT f2 must apply. We will use a Chebyshev ðLPÞ ðHPÞ ¼a prototype with the same order and ripple for both the filters, and set fT = f2 ¼ f1 = fT with a 1. This way, the lowdpass maps f2 to the same prototype frequency as the highpass does with f1. Then, we will choose a passband ripple that is satisfactory for our impedance matching

66 67

Just swapping the relative attenuation and the relative phase. See the Mathcad file 10_Lowpass_Highpass_Pase_Shifter.MCD and the Ansoft file 07_PH22d5.adsn.


411

450

Phase shift, °

360

270

180

90

0 15

Figure 10.35

17


23

25

Simulated response of the MEMS phase shifter in Figure 10.34. L1

L3 C2

1

LN CN-1

1

P2

P1 2

2

C1

C3 L2

CN LN-1

vCNTRL

Figure 10.36

Highpass, lowdpass binary phase shifter.

requirements, for instance RP ¼ 0:01 dB, corresponding to a return loss68 of about 26.4 dB. Finally, we set an odd number for the filter order; for simplicity our choice will be N ¼ 3. Now we have only the design parameter a, and therefore we can scan it to find the required phase shift in the frequency range of interest. Let us assume: .

f1 ¼ 1 GHz;

.

N ¼ 3;

.

Df ¼ 22:5 .

f2 ¼ 2 GHz

RP ¼ 0:01 dB

68 The final return loss of the cell derives from the combination of the filters with the two SPDTs. Precisely two cascaded two-ports each having a return loss of 10, 15, 20 and 25 present additional phase ripples of 5.7 , 1.8 , 0.6 and 0.2 ; see Equation (9.7) and relative comments in Section 9.2.

412


By scanning a from unity to increasing values, we obtain the optimum value aDf ¼ 4:101. From this, and applying the synthesis procedures69 for the lowdpass ðL1 ¼ L3 ¼ 0:61 nH; C2 ¼ 0:377 pFÞ and highpass ðC1 ¼ C3 ¼ 20:747 pF; L2 ¼ 33:633 nHÞ filters, the resulting phase shift presents a maximum error of 0.7 over the whole frequency range.

10.4.5

Final considerations on phase shifters

Now we can focus on the deviations of the real phase shifter from the ideal case: 1. No phase shifter presents a perfect impedance match, even within its working bandwidth. Therefore we need to specify a minimum in-band return loss. 2. Real phase shifters present a non-zero insertion loss, which also varies with the frequency and from state to state. Therefore, we will have to have an insertion loss mask and a specified insertion loss variation with the state. The counterpart of this latter parameter in variable attenuators is that they change their phase at different attenuation. This behaviour is usually non-specific, and often also not well known. 3. The relative phase characteristic, in both the true-phase and true-delay cases, is not uniform with the frequency. Consequently, the mean, maximum and minimum, phase shift or delay value must be specified over the frequency for each state. Sometimes RMS error is also specified. Furthermore, the interactions between the unmatched phase shifter and its unmatched generator and load cause additional phase ripple, as addressed in Section 9.2. 4. Phase shifters present nonlinear distortions like any component with semiconductor devices. At high input power, we have both amplitude compression and phase variations. Therefore, a further specification is the maximum input power (operational and not damaging). 5. Similar to switches and attenuators, phase shifters do not react to their driving signal instantaneously. We then have a specified settling time for a giver phase error. 6. Finally, all the specifications 1 to 5 are also temperature dependent, and there are, of course, working and surviving temperature ranges for the phase shifter.

Bibliography 1. G. Hiller, ‘Designing with PIN diodes’, MA-COM Application Note AG-312. 2. R. H. Caverly and G. Hiller, ‘Distortion properties of MESFET and PIN diode microwave switches’, IEEE MTT-S Symposium Digest, pp. 533–535, 1992. 3. R. H. Caverly, ‘Distortion modeling of PIN diode switches and attenuators’, IEEE MTT-S Symposium Digest, pp. 957–960, 2004. 4. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, Chapter 7, pp. 420–428. 5. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, Chapter 2, pp. 57–63. 6. G. Bianchi and G. Pinto, ‘DC-18 GHz Ga-As MMIC SPST and SPDT’, Military Microwave Conference, Brighton, UK, 1992. 69

See Chapter 7.


413

7. M. Golio, The RF and Microwave Handbook, CRC Press, Boca Raton, FL, 2001, Chapter 8.7. 8. ‘Agilent FET switch speed and settling time’, Application Note #57 – Rev. A, July 2002. 9. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, Chapter 7, pp. 432–434. 10. S. Lucyszyn, Advanced RF MEMS, Cambridge University Press, 2010. 11. M. Rebeiz, RF MEMS Theory, Design and Technology, John Wiley & Sons, Inc., Hoboken, NJ, 2003. 12. S. Di Nardo, P. Farinelli, F. Giacomozzi, G. Mannocchi, R. Marcelli, B. Margesin, P. Mezzanotte, V. Mulloni, P. Russer, R. Sorrentino, F. Vitulli and L. Vietzorreck, ‘Broadband RF-MEMS based SPDT’, Proceedings of the 36th European Microwave Conference, Manchester, UK, September 2006. 13. A. Ocera, P. Mezzanotte and R. Sorrentino, ‘Design of tunable phase shifters by the image parameter method’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-54, No. 6, pp. 2383–2390, 2006.

Related files Ansoft files 01_PIN_SPDT.adsn. Provides a first-level simulation of the SPDT in Figure 10.4, to compute insertion loss, isolation and return loss. 02_Shunt_Only_SPDT.adsn. Provides a first-level simulation of the SPDT in Figure 10.8, to compute insertion loss, isolation and return loss for different values of the diode ON resistance. 03_Shunt_Resistor_Attenuator.adsn. Provides a simple description of the attenuator in Figure 10.15b, assuming a coupled-line realization of the hybrids. It allows computation of the attenuation and return loss for any incremental resistance of the diode. 04_True_Delay_Phase_Shifter.adsn. Analyzes the phase shifter in Figure 10.19a and produces the curves in Figure 10.20. 05_Reflection_Phase_Shifter.adsn. Analyzes the phase shifter in Figure 10.21c and produces the curves in Figure 10.22. 06_Analog_Phase_Shifter.adsn. Analyzes the phase shifter in Figure 10.24b and produces the curves in Figure 10.25. 07_PH22d5.adsn. Analyzes the lowdpass, highpass phase shifter in Figure 10.27.

Mathcad files 08_Chebysheff_Lowpass_Filter_Delay.MCD. Performs some analyses and optimizations on the application of a Chebyshev filter as a delay line. 09_Reflection_Phase_Shifter.MCD. The equivalent of Ansoft file 05 above, implemented with a mathematical rather than circuit analysis method. 10_Lowdpass_Highpass_Pase_Shifter.MCD. Analyzes, optimizes and synthesizes the lowdpass, highpass phase shifter in Figure 10.27.

11

Amplifiers 11.1 Introduction This chapter deals with transistor-based RF and microwave amplifiers. Such devices are sometimes referred to as solid state amplifiers (SSAs), in order to distinguish them from vacuum tube device circuits. High-frequency amplifiers are too wide a topic for one single chapter, and probably for one single book as well. Here, we will limit our descriptions to the main concepts and to the most used configurations. The largest part of this chapter (Sections 11.2 to part of 11.4) describes the classical one-transistor amplifier with one input and one output matching network. In this architecture there are three important cases: generic or small-signal amplifiers, low-noise amplifiers and power amplifiers. Small-signal and low-noise amplifiers assume a linear operation of the transistor. In the first case, what matters is the gain, thus the best possible impedance matching of the transistor input and outputs. In lownoise amplifiers, the input is matched for the minimum noise figure, all the gain optimization relying on the output. Power amplifiers use the transistor in the nonlinear region, the output matching tries to exploit best the transistor power and any improvement in the gain is based on the input only. Sections 11.2 and 11.3 deal with small-signal and low-noise amplifiers respectively. In order to clarify the design concepts expressed in these sections, Section 11.4 presents the design of a trial amplifier, step by step. Section 11.5 discusses some techniques for power amplifier design. Section 11.6 describes some non-classical amplifier configurations, which are particularly suitable for integrated circuit technology. Section 11.7 completes the chapter describing two amplifiers taken from real engineering practice

11.2 Small-signal amplifiers Figure 11.1 shows the basic structure of a typical high-frequency amplifier. The active device Q1 assumes the symbol of a FET, but it could also be a bipolar or MOS device, with minor changes, limited to the DC circuitry. For the active device connection, the most used one is the common source/common emitter, although common base/gate or common drain/collector are sometimes also used. Furthermore, most of the real amplifiers use one single active device, but some applications employ multiple devices in Darlington or cascode connection.


416

MICROWAVE AND RF ENGINEERING input matching

L gg Input

1

2

C dd L dd

R gg

R dd

+ Vgg input bias

Figure 11.1

output matching

Q1

C gg

1

2

Output

+ V dd output bias

Principle of a single stage high-frequency amplifier.

We will begin our descriptions by referring to the configuration in Figure 11.1 as it is, because it is the simplest one. Subsequent sections will discuss some variants. Beginning with Q1 , we can see two bias networks, one at the input and one at the output, each consisting of one capacitor, one inductor and a voltage generator1 with a series resistor. The capacitors present infinite impedance at DC and negligible impedance at RF, and the opposite applies for the inductors. This way, the two DC generators can bias the respective transistor electrodes neither disturbing the high-frequency elements nor being disturbed by them. The two DC generators Vgg ; Vdd , combined with their respective series resistors Rgg ; Rdd , are the Thevenin models of any real voltage or current source. The two matching networks modify the impedances of the transistor as seen by the input and output into values as close as possible to 50 O. In many real cases, matching and bias networks are not distinct, rather they share one or more component that simultaneously performs matching and bias functions. Amplifier design normally has two phases: choice of the most appropriate device; and matching/bias network design. The decision about the best transistor for a given application is based not only on electrical parameters, but also on cost considerations. Usually the best performing devices present the highest cost. On the other hand, better performing transistors could require simpler – thus cheaper – matching networks, lower criticality in production, and so on. The first evaluation of a potential device relies on the manufacturer’s data, which are generally sufficient for the complete design. Few particularly critical applications require a more accurate device characterization: in these cases the designer may need to measure the transistor and extract a specifically tailored model. The matching network design is strictly related to the required performances of the amplifier. Design techniques are available for maximum gain, best gain flatness over the frequency, best impedance matching, minimum noise figure, maximum power, maximum linearity, and combinations of these. Besides those purely electrical performances, the yield of the amplifier in production is assuming a growing importance. From the design point of view, this implies that the network must present minimum sensitivity of its performance to the component tolerances. This additional requirement stimulated the CAE techniques known as yield analysis and yield optimization.

11.2.1

Gain definitions

There are three main definitions of the amplifier power gain that are in use in engineering practice: 1. Transducer power gain: (GT ) is the ratio between the power delivered to the load (PL) and the one available from the generator (PAVS ).

1 In this chapter we will always use the word ‘generator’, instead of ‘source’, in order to avoid confusion with the eponymous FET electrode, as in Section 10.2.

AMPLIFIERS

417

2. Operating power gain: (GP ), also known simply as power gain, is the ratio between the power delivered to the load (PL ) and the one at the amplifier input (Pin ). 3. Available power gain: (GA ) is the ratio between the available powers of the amplifier (PAVN ) and of the generator (PAVS ). Thus, in formulae, GT ¼

PL ; PAVS

GP ¼

PL ; PIN

GA ¼

PAVN PAVS

ð11:1Þ

Figure 11.2a shows the generic diagram of a transistor connected to an input generator and an output load. The generator and load reflection coefficients are Gs ; GL respectively, while the superscript ‘(Q)’ denotes the transistor parameters. The generator and load in Figure 11.2a could embed their respective matching networks, because of the generic value assumed for Gs and GL . Figures 11.2b,c show the signal flow graphs (SFGs) corresponding to the schematic of Figure 11.2a, assuming that the generator is terminated with a generic load Gl . We will use those graphs to derive the explicit expressions of the power gains (11.1). Beginning with Figure 11.2c, the power delivered from the generator to the load equals the square amplitude of the incident wave 1 2 2 jbs j Pl ¼ jal j2 ¼ ð11:2Þ 1Gs Gl Γin

Γout

(Q)

[S ] generator

(a)

1

Γs

load

2

ΓL

transistor

(Q) a1 s21

bs

b2

s11(Q) Γs

s22(Q) ΓL b1

(b)

a2

al

bs

Γs

(c)

(Q)

s12

Γl

bl

Figure 11.2 Structures for the computation of the power transfer ratio of a two-port network: (a) electrical schematic; (b) amplifier signal flow graph; (c) generator signal flow graph.

418


The third term of Equation (11.2) is easily obtained by applying Mason’s rule to the SFG of Figure 11.2c. The available power, by definition, is the power delivered to a load conjugated/matched to the generator. Thus Equation (11.2) returns the generator available power after the substitution Gl ¼ G*s , where the asterisk denotes the complex conjugate:2 PAVS ¼

1 1jGs j2

jbs j2

ð11:3Þ

The power delivered to the load is the difference between the amplifier output reflected and incident wave square amplitude ð11:4Þ PL ¼ jb2 j2 ja2 j2 ¼ jb2 j2 1jGL j2 The terms (11.3) and (11.4) involved with the transducer power gain definitions include the variables bs to b2 . The ratio between these two variables can be computed by analyzing the SFG in Figure 11.2b. The definitions given in Section 9.3 involve the following contributions to the transfer ratio (9.8): ðQÞ

.

The only path from bs to b2 is P1 ¼ s21 .

.

.

There are no first-order loops that do not touch P1 . X ðQÞ ðQÞ ðQÞ ðQÞ There are three first-order loops; their sum is Lð1Þ ¼ s11 Gs þ s22 GL þ s21 GL s12 Gs . X ðQÞ ðQÞ The only second-order loop is Lð2Þ ¼ s11 Gs s22 GL .

.

There are no higher order loops.

.

Hence, the output reflected wave is ðQÞ

b2 ¼

s21 bs ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ 1 s11 Gs þ s22 GL þ s21 GL s12 G þ s11 Gs s22 GL

ð11:5Þ

Substituting Equations (11.3) to (11.5) into the first equation of definitions (11.1), we have 2 ðQÞ 2 jb j2 1jGs j2 s21 1jGs j PL 2 GT ¼ ¼ 1jGs j2 ¼ h i PAVS jbs j2 1 sðQÞ G þ sðQÞ G þ sðQÞ G sðQÞ G þ sðQÞ G sðQÞ G 2 s L L 12 s s 22 L 11 22 21 11 Rearranging the denominator, we obtain the more compact formula 2 ðQÞ 2 1jGL j2 s21 1jGs j G T ¼ h 2 ih i ðQÞ ðQÞ ðQÞ ðQÞ 1s11 Gs 1s22 GL s12 Gs s21 GL ðQÞ

ð11:6Þ ðQÞ

Further forms of the transducer power gain can be obtained by replacing s11 or s22 with the reflection coefficients indicated in Figure 11.2a as Gin and Gout respectively. From (9.2) we have ðQÞ

Gin ¼ s11 þ

ðQÞ ðQÞ

s21 s12 GL ðQÞ

1s22 GL

;

ðQÞ

Gout ¼ s22 þ

ðQÞ ðQÞ

s21 s12 Gs ðQÞ

1s11 Gs

ð11:7Þ

The conjugate of a complex number is defined as ða þ jbÞ* ¼ conjða þ jbÞ ¼ a jb, see also the definition associated with the identity (B.5). It is also ða þ jbÞ* ða þ jbÞ ¼ a2 þ b2 . 2

AMPLIFIERS

419

2 ðQÞ Dividing and multiplying the denominator of Equation (11.6) by the factor 1s22 GL , and rearranging the expression, we obtain 2 1jG j2 1jGs j2 ðQÞ L GT ¼ 2 s21 2 ðQÞ sðQÞ G sðQÞ G 1s22 GL s 21 L ðQÞ 12 1s11 Gs ðQÞ 1s22 GL By virtue of the first equation in (11.7), we can write the second term of the second denominator as ðQÞ

ðQÞ

s12 Gs s21 GL ðQÞ

1s22 GL

ðQÞ ðQÞ ¼ Gin s11 Gs ¼ Gin Gs s11 Gs

Then, the transducer gain becomes GT ¼

1jGs j2 ðQÞ 2 1jGL j2 s21 2 ðQÞ j1Gin Gs j2 1s G 22

ð11:8Þ

L

2 2 ðQÞ ðQÞ With a similar procedure, but inserting 1s11 GS instead of 1s22 GL , we obtain 1jGs j2 ðQÞ 2 1jGL j2 GT ¼ 2 s21 ðQÞ j1Gin GL j2 1s11 Gs

ð11:9Þ

The three expressions (11.6), (11.8) and (11.9) give three points of view about the effect of the various reflection coefficients on the transducer power gain. As for the other two power gain definitions, we have that the amplifier input power is the difference between the generator available power and the reflected power. Considering this, the first two equations of (11.1) give GP ¼

PL PAVS PL PAVS 1 ¼ ¼ GT ¼ GT PIN PIN PAVS PIN 1jGS j2

Substituting these expression into Equations (11.8) and (11.9), we obtain GP ¼

2 1jG j2 2 1jG j2 1 ðQÞ ðQÞ L L s ¼ s 21 2 2 21 j1Gout GL j2 ðQÞ ðQÞ j1Gin Gs j2 1s22 GL 1s11 GS 1

ð11:10Þ

The amplifier available power is the one delivered to a matched load; hence, the available power gain coincides with the value given by Equation (11.9) when GL ¼ G*out . Thus 1jGs j2 ðQÞ 2 1 GA ¼ 2 s21 ðQÞ 1jGout j2 1s11 Gs

ð11:11Þ

All the expressions derived for the transistor power gain apply to any linear two-port network as well, simply by replacing the transistor S parameters with those of the network. In particular, if we replace the scattering parameters of the transistor with those of the complete amplifier, and put Gs ¼ GL ¼ 0, expressions (11.9) to (11.11) give the amplifier gain on the 50 O generator and source: ðAmpÞ 2 2 s 21 ðAmpÞ GT ¼ GP ¼ s21 ; GA ¼ 1jGout j2

420


where the superscript ‘(Amp)’ denotes the amplifier scattering parameters, which are combinations of the input matching network, transistor and output matching network by the next applications of Equation (9.2).

11.2.2

Stability

An important property of non-unilateral transistors is their potential instability. Particular combinations of input and output terminating impedances may cause the transistor to oscillate. Oscillators are the topic of Chapter 12 which shows how to exploit transistor instability for design purposes. Here, we will limit ourselves to showing how to recognize if a two-port network is stable or not, and what the fundamental stabilization techniques are. A generic two-port network is unconditionally stable if, on terminating its port 1 (respectively, 2) with any passive bipole, the impedance seen at port 2 (1) is still that of a passive bipole.3 A unstable twoport network does not necessarily oscillate, but it could oscillate in some specific cases, as discussed in Chapter 12. We can define the stability condition of the transistor, in reference to Figure 11.2a, by applying expressions (11.7). Assuming that the terminating bipoles are passive, then jGs j 1;

jGL j 1

The transistor is unconditionally stable if, for any phase of the terminating reflection coefficients, the values resulting from Equations (11.7) have magnitude no greater than 1: ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ s21 s12 GL ðQÞ s21 s12 Gs ð11:12Þ 1; s22 þ 1 s11 þ ðQÞ ðQÞ 1s GL 1s Gs 22

11

Instead, if the inequalities (11.12) are not satisfied for at least one couple of values ðGs ; GL Þ, then the transistor is potentially unstable. In general, physical devices satisfy the condition (11.12) only in some frequency bands. The terminating reflection coefficients that are stable or unstable for the device define the input and output stability region. We can determine the stability regions by finding their limit values, in mathematical terms, by solving Equations (11.12) with the equals sign. The two inequalities (11.12) have the same structure: we can obtain each of them from the other by swapping the index 1 and 2 of the scattering parameters and Gs for GL. For this reason, we will develop our calculations on the first inequality, the other case being analogous. Starting from the first inequality (11.12) with the equals sign, and squaring both terms, we have i ðQÞ h s 1sðQÞ G þ sðQÞ sðQÞ G 2 L L 11 22 21 12 ¼1 ðQÞ 1s22 GL which is equivalent to 2 h i 2 ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ s11 þ s21 s12 s11 s22 GL ¼ 1s22 GL which can be rewritten in a more compact way as 2 2 ðQÞ ðQÞ s11 D GL ¼ 1s22 GL

3 We can define a passive bipole in terms of its immittance ½ReðZÞ; ReðYÞ 0 or reflection coefficient ðjGj 1Þ, where clearly Z, Y and G are the impedance, admittance and reflection coefficients of the bipole.

AMPLIFIERS

421

where ðQÞ ðQÞ

ðQÞ ðQÞ

D ¼ s11 s22 s21 s12

ð11:13Þ

is the determinant of the transistor scattering matrix. For any complex number Z this is jZ j2 ¼ Z Z * , thus ðQÞ

ðQÞ*

ðQÞ

ðQÞ*

½s11 D GL ½s11 D* G*L ¼ ½1s22 GL ½1s22 G*L Expanding the above expression, and regrouping terms in GL, we can write 2 h 2 h i i ðQÞ ðQÞ ðQÞ* ðQÞ* ðQÞ 2 ðQÞ s11 þ s22 s11 D GL þ s22 s11 D* G*L þ jDj s22 GL G*L ¼ 1 After dividing both sides by the coefficient of GL G*L , this equation becomes h

2 i h i* ðQÞ ðQÞ ðQÞ* ðQÞ ðQÞ* 1s11 s22 s11 D GL þ s22 s11 D G*L * þ GL GL ¼ 2 2 ðQÞ ðQÞ jDj2 s22 jDj2 s22

Adding the term 2 2 h i h i* ðQÞ ðQÞ ðQÞ* ðQÞ ðQÞ* s22 s11 D s22 s11 D jDj2 s22 to both sides, we get h i h i* h i h i* ðQÞ ðQÞ* ðQÞ ðQÞ* ðQÞ ðQÞ* ðQÞ ðQÞ* s22 s11 D s22 s11 D s22 s11 D GL þ s22 s11 D G*L þ þ GL G*L 2 2 2 ðQÞ 2 ðQÞ jDj s22 jDj2 s22 2 h i h i* ðQÞ ðQÞ ðQÞ* ðQÞ ðQÞ* 1s11 s22 s11 D s22 s11 D ¼ 2 þ 2 2 ðQÞ ðQÞ jDj2 s22 jDj2 s 22

which can be rearranged as 9 8h 9 8h 2 i* i h i h i* ðQÞ ðQÞ* ðQÞ* ðQÞ ðQÞ* ðQÞ ðQÞ* > > > => = < sðQÞ < sðQÞ 1s11 s22 s11 D s22 s11 D 22 s11 D 22 s11 D 2 þ G L 2 þ G*L ¼ 2 þ 2 2 > > > ;> ; jDj2 sðQÞ : jDj2 sðQÞ : jDj2 sðQÞ ðQÞ 22 22 22 jDj2 s 22

Note that the first term of the above equation is the product of two complex conjugate numbers, which can be replaced with the square amplitude of those numbers themselves, giving 2 h 2 ðQÞ ðQÞ* i* ðQÞ ðQÞ* 2 ðQÞ s22 s11 D 1 s s s D 11 22 11 2 þ GL ¼ 2 þ 2 2 ðQÞ ðQÞ ðQÞ jDj2 s22 jDj2 s22 jDj2 s22 It is possible to write the above equation in a more compact way, by reducing the second term to a single fraction

422


2 h 2 2 2 2 h i i h ðQÞ ðQÞ* i* ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ* ðQÞ* ðQÞ 2 2 ðQÞ s22 s11 D D D j j j j s11 s22 þ s11 s22 þ s22 s11 D s22 s11 D* 2 þ GL ¼ 2 2 ðQÞ ðQÞ jDj2 s22 jDj2 s 22

Exploiting the identity jZ j2 ¼ Z Z * again, we can rewrite the numerator of the second term as 2 h h i h i 2 2 ðQÞ ðQÞ* i* ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ* ðQÞ* ðQÞ* ðQÞ* s22 s11 D s þ s11 s22 s s s s s s s 11 22 12 21 11 22 12 21 2 þ GL ¼ 2 2 ðQÞ ðQÞ jDj2 s22 jDj2 s 22

h

i h i ðQÞ ðQÞ ðQÞ* ðQÞ* ðQÞ* ðQÞ* ðQÞ* ðQÞ* ðQÞ ðQÞ ðQÞ ðQÞ s11 s22 s11 s22 s12 s21 s11 s22 s11 s22 s12 s21 þ 2 2 ðQÞ jDj2 s22 Expanding the products of the numerator of the second term and simplifying opposite terms, we get finally 2 2 h ðQÞ ðQÞ* i* s22 s11 D sðQÞ sðQÞ 12 21 ð11:14Þ 2 þ GL ¼ 2 ðQÞ 2 ðQÞ jDj2 s22 jDj s22 Equation (11.14) describes a circle in polar coordinates, in the plane of the complex variable GL, with centre and radius respectively ðQÞ ðQÞ ðQÞ* ðQÞ * s s 12 21 s s D CL ¼ 22 11 2 ; rL ¼ ð11:15Þ 2 ðQÞ ðQÞ jDj2 s22 jDj2 s22 This circle delimits the stability region of the load reflection coefficient: it is the load stability circle or the output stability circle. With the same calculations,4 and starting from the second inequality (11.12), we derive the centre and radius of the input stability circle ðQÞ ðQÞ ðQÞ* ðQÞ s12 s21 s11 s22 D* Cs ¼ ð11:16Þ 2 ; rs ¼ 2 ðQÞ ðQÞ jDj2 s11 jDj2 s11 The scattering parameters of the transistor are complex functions of the frequency, and thus the stability circles are frequency dependent as well. All the points on the external perimeter of the output (input) stability circle represent values of the output (input) reflection coefficients, making the reflection coefficient seen at the transistor input (output) have unitary amplitude. The identification of the stable regions depends on determining if the stable points are the ones inside or outside the stability circle. In this regard, the centre of the Smith chart is a useful marker. Let us consider the output stability, and assume that the transistor input reflection coefficient has ðQÞ amplitude not greater than one s 1 . This means that a 50 O termination ðGL ¼ 0Þ on the 11

transistor output represents a stable load, by definition of s11; then the Smith chart origin is a stable point for the output. Consequently, if the output stability circle surrounds the centre of the Smith chart, the 4

Or swapping the index 1 and 2 in the scattering parameters of Equation (11.15).

AMPLIFIERS

423

ðQÞ unstable region is outside that circle, and vice versa. Conversely, if s11 > 1, the output stable points are external or internal to the stability circle, depending on whether the latter encircles the centre of the ðQÞSmith chart or not. Applying the same considerations to the input stability, we obtain that if s 1 sðQÞ > 1 the input stable points are internal (external) or external (internal) to the input 22 22 stability circle if that includes the point GS ¼ 0 or not. Under this hypothesis, the reference reflection coefficients are stable points by definition: in other words, the transistor is stable on 50 O. ðQÞ ðQÞ Most transistors present s11 1; s22 1, and thus they are stable on 50 O for both input and output. ðQÞ ðQÞ If s11 j 1; s22 j 1 and the stability circles are totally outside the unit ratio Smith chart, it follows that any passive termination is stable for the device: in this case the transistor is unconditionally stable. We can plot the stability circles5 at all the frequencies of interest and check the stability of the device: the further their distance from the Smith chart external circumference ðjGj ¼ 1Þ, the more stable the device. The Kurokawa criterion provides an alternative method for the stability analysis. It is possible to demonstrate that the transistor is unconditionally stable if its scattering parameters satisfy the inequalities ðQÞ ðQÞ 1s11 s22 þ jDj2 K¼ > 1; jDj < 1 ð11:17Þ ðQÞ ðQÞ 2s12 s21 where D is the determinant (11.13) of the transistor scattering matrix. Again, we can plot6 the quantities K and D: the transistor is stable according to whether the first (latter) quantity is greater (smaller) than 1. In principle, to speak about different degrees of stability on an unconditionally stable device could sound like nonsense. However, this is not the case, due to differences from device to device, temperature variations and the uncertainty of the transistor parameters due to measurement errors. Slightly different assemblies could also modify the transistor scattering parameters. In normal engineering practice, designers allow some margin on the stability, in order to override the above-mentioned tolerance effects. So far we have described the stability analysis of a transistor, but the relative considerations apply to any linear two-port network: all the expressions (11.15) to (11.17) remain valid, just by replacing the transistor scattering parameters with those of the network. In particular, the stability analysis applies to the amplifier in Figure 11.1. Any two-port, obtained by cascading an arbitrary number of purely reactive networks at the input and output of the transistor, conserves the stability properties of the transistor itself: the cascade is unconditionally stable or not; so is the transistor. This assertion can easily be justified by considering that: 1. A two-port network consisting of an arbitrary number of purely reactive two-port networks is still purely reactive, or loss free. 2. If the transistor is unstable, values of GS or GL exist inside the unit ratio Smith chart that violate one of the conditions (11.12). Any purely reactive two-port network moves the unstable points from inside the unit ratio Smith chart to different zones on the same chart. 3. The consequence of 1 and 2 is that an amplifier having the structure of Figure 11.1, with arbitrarily complex purely reactive matching networks, has the same transistor stability.7

5 All the microwave circuit simulators usually include a stability circle plotting capability. Usually, they also mark the stable region, inside or outside the circle. 6 All the microwave circuit simulators are capable of plotting K and D. 7 In considerations 1 to 3, we assumed that the bias network has no influence at high frequencies.

424


Most real transistors are not unconditionally stable, therefore they must be stabilized, and purely reactive matching networks are not allowed. In a few cases it could be possible that potentially unstable amplifiers, employing unstable transistors and loss-free matching networks, are usable. This happens when the amplifier is intended to operate with specified input and output impedances, falling well inside the stability regions. Such a condition, however, does not occur frequently. Unstable amplifiers are critical for normal use, in that the effective input and output reflection coefficients are only approximately known. Stabilization of the amplifier requires the insertion of resistances inside the matching networks. The amplifier must be unconditionally stable at any frequency, not just in the working band. Outof-band instabilities8 can lead to out-of-band oscillations that intermodulate with the wanted signal producing spurs, or even saturate the amplifier and totally degrade its gain. In addition, terminating impedances at out-of-band frequencies are generally more uncertain than in-band ones, which increases the out-of-band stability requirements.

11.2.3

Matching networks

Having defined the power gain, we can now describe the matching networks which maximize the power transfer from the generator to the transistor, and from this to the load. Chapter 6 presents the most important matching networks and techniques. Matching networks for amplifier applications, however, need careful attention and approaches. Usually the input generator and the output load are resistive, while the transistor presents reactive and frequency-dependent impedances at its electrodes. If we look at the matching network from the active device (a) or from the opposite side (b), we have two different but equivalent descriptions of their operation: (a) The input (output) network transforms the generator (load) impedance in the conjugate of the transistor input (output) one. (b) The input (output) network transforms the transistor input (output) in the conjugate of the generator (load) impedance one. This second definition could omit the adjective ‘conjugate’, since load and generator are usually resistive, as written above. The matching network design can use many different methods and circuit topologies: the result could be a lumped,9 distributed or mixed, purely reactive or lossy network, with low-pass or bandpass response. One of the key parameters for the matching network design is the working bandwidth of the amplifier. As a general rule, wider bandwidths require more complicated networks. Usually, amplifier designers prefer reactive to resistive matching, due to the signal loss involved with the latter. However, Fano’s equations10 state an insurmountable limit for the return loss achievable for a given transistor impedance with a loss-free network. Therefore reactive matching networks are advisable, in principle, until the required combination of return loss and bandwidth is incompatible with the Fano limits. In any case, we must consider that complex matching networks potentially give better matching, as close as possible to the Fano limit, but also are usually more critical and tolerance sensitive. One additional case, which requires resistive matching, occurs when the transistor is potentially unstable, if the amplifier must be absolutely stable. Sections 11.2.4, 11.3 and 11.5 subsequently present some considerations of the matching network design oriented towards maximum gain, minimum noise figure, maximum output power and amplifier stability. 8

Typically, they occur at lower frequencies, where the transistor gain is higher. Sometimes lumped elements (capacitors and inductors) are semi-lumped elements, obtained by applying the method described in Section 6.5. 10 See Section 6.2. 9

AMPLIFIERS

11.2.4

425

Maximum gain impedance matching

In order to examine the effect of the matching networks, we initially assume their removal. The network in Figure 11.1 simplifies to the transistor with its bias networks, which do not affect the RF signal, by hypothesis. Denoting the transistor parameters with superscript ‘(Q)’, Equation (11.6) with Gs ¼ GL ¼ 0 then gives the power gain of the simplified amplifier 2 ðQÞ GT;mismatched ¼ s21 ð11:18Þ In this simplified configuration, the transistor input does not absorb all the power available from the generator, because of its non-zero reflection coefficient. The ratio between the available power from the generator and the one absorbed by the transistor input is PAV;generator ¼ Pin;transistor

1 2 ðQÞ 1s11

ð11:19Þ

Similarly, the output mismatch prevents the transistor output from transferring all its available power to the load. The ratio between the power available from the transistor output and the one transferred to the load is PAV;transistor ¼ Pload

1 2 ðQÞ 1s22

ð11:20Þ

After the transistor is input and output matched, the two ratios (11.19) and (11.20) become unitary. ðQÞ Assuming that the two matching networks are loss free and that the transistor is unilateral ðs12 ¼ 0Þ, the matched power gain increases by the two above-mentioned ratios, becoming 2 1 1 ðQÞ GT;matched ¼ ð11:21Þ 2 s21 2 ðQÞ ðQÞ 1s11 1s22 It is also possible to derive Equation (11.21) from (11.9). From assuming the transistor to be unilateral, it ðQÞ ðQÞ follows that that GIN ¼ s11 ; GOUT ¼ s22 , and therefore Equation (11.9) becomes 1jGS j2 ðQÞ 2 1jGL j2 GT ¼ 2 s21 2 ðQÞ ðQÞ 1s11 GS 1s22 GL

ð11:22Þ

From Equation (11.22), it follows that the maximum transducer gain is achieved when the factors 2 ðQÞ 2 ðQÞ 1jGs j2 Þ 1s11 Gs ; 1jGL j2 1s22 GL are at their maximum. Now, the two above-mentioned quantities are functions of the complex variables Gs ; GL and have the form 1jGj2 j1s Gj

2

¼

1jGj2 j1jsj jGjexpf j ½argðGÞ þ argðsÞgj2

with

0 jGj 1

ð11:23Þ

For any given jGj, the denominator of the factor (11.23) is minimum – then the whole function is 2 maximum –2when argðGÞ ¼ argðsÞ. Under this condition, the factor (11.23) becomes 1jGj 1jsj jGj , which is a function of the real variable jGj, rather than of the complex variable G. Moreover, lim 1jGj2 ð1jsj jGjÞ2 ¼ 1 and lim 1jGj2 ð1jsj jGjÞ2 ¼ 0 jG j ! 0

jG j ! 1

426


Thus the maximum of the factor (11.22) falls somewhere within the range 0 jGj 1, and can be computed by the vanishing derivative i d h 1jGj2 ð1jsj jGjÞ2 ¼ 2ðjGjjsjÞð1jsj jGjÞ3 ¼ 0 if d jGj

jGj ¼ jsj ðQÞ*

ðQÞ*

Therefore, the first and third factor of the gain (11.22) reach their maxima when Gs ¼ s11 ; GL ¼ s22 ðQÞ 2 1 ðQÞ 2 1 and those maxima are equal to 1s11 ; 1s22 , respectively. The more the transistor is mismatched, the higher is the increase of gain produced by the matching networks. Equation (11.21) gives the absolutely maximum power gain achievable with perfectly matched devices using loss-free matching networks. Such a condition is theoretically possible at discrete frequencies by means of one of the networks described in Chapter 6, dimensioned to transform ðQÞ ðQÞ s11 ; s22 into 50 O at the frequency of interest. However, real transistors are never unilateral, rather their reverse transmission coefficient is nonzero, although its amplitude is smaller than that of the forward transmission coefficient.11 One consequence of this situation is that the input (output) reflection coefficient of the transistor depends on the impedance closing the output (input). The transistor [S] parameters are, by definition, ðQÞ ðQÞ measured with the ports terminated on 50 O. Then s11 ðs22 Þ is the input (output) reflection coefficient of the transistor, when the output (input) is closed on 50 O. Therefore, if we begin by matching the input, ðQÞ we have to design a network that transforms s11 into 50 O. Now, if we consider the effect of the input ðQÞ matching network, we can see that the output reflection of the device is no longer s22 . Nevertheless, we still could match the new output reflection coefficient into 50 O. Unfortunately, the output matching network modifies the transistor input reflection coefficient, degrading the input matching. To correct ðQÞ this, it should be possible to modify the input matching network in order to take the modified s11 ðQÞ into account, but this will change s22 , and so on. We get the same conclusion if we start from the output matching. Considering that one possibility to match a bilateral transistor is to proceed back and forth by modifying the input and output matching networks, such a procedure normally converges after some iterations and could be quite tedious, particularly for broad-band amplifiers. A more efficient and scientific approach is simultaneous conjugate matching. Equations (9.2) or (11.7), specialized to our case, give the input and output reflection coefficients of the transistor, which are respectively ðQÞ

ðQÞ

GIN ¼ s11 þ

ðQÞ ðQÞ

s21 s12 GML ðQÞ 1s22 GML

;

ðQÞ

ðQÞ

GOUT ¼ s22 þ

ðQÞ ðQÞ

s21 s12 GMS ðQÞ

1s11 GMS

ð11:24Þ

where GMS ; GML are the reflection coefficients presented to the transistor input and output by the respective matching networks. Input and output are simultaneously matched if GMS ¼ G*IN ;

11

GML ¼ G*OUT

For example, Table 9.2 gives . ðQÞ ðQÞ s12 s21

f ¼2 GHz

. f ¼10 GHz ðQÞ ðQÞ 53:8s12 s21 ffi 12:5

and the transistor in Table 9.3 has . ðQÞ ðQÞ s12 s21

f ¼1 GHz

. f ¼10 GHz ðQÞ ðQÞ 46:5s12 s21 ffi 15:4

ffi

ffi

ð11:25Þ

AMPLIFIERS

427

Combining the coefficients (11.24) with conditions (11.25), and after some manipulation, we obtain the reflection coefficients present at the transistor terminals for simultaneous matching qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 B21 4jC1 j2 B2 B22 4jC2 j2 ; GML ¼ ð11:26Þ GMS ¼ 2C1 2C2 where 2 2 ðQÞ ðQÞ B1 ¼ 1 þ s11 s22 jDj2 2 2 ðQÞ ðQÞ B2 ¼ 1 þ s22 s11 jDj2 C1 ¼

ð11:27Þ

ðQÞ ðQÞ* s11 s22 D ðQÞ

ðQÞ*

C2 ¼ s22 s11 D and D is the quantity (11.13). The two equations in (11.26) return two different values each; however, only one of them has amplitude less than 1, and therefore is the good one. Moreover, if the transistor is potentially unstable, then both couples of values in (11.26) have unitary amplitude.12 This prevents design of the conjugate matching network, as we can easily demonstrate. Let us consider for instance the output matching network in the schematic of Figure 11.1. Let Gload be the generator reflection coefficient, clearly with jGload j < 1, and let us denote the scattering parameters of the output matching network by the superscript ‘(out)’. Neglecting the bias network, or considering it as part of the matching network, Equation (9.4) gives the maximum amplitude of the reflection coefficient seen by the transistor output ðoutÞ s22 þ jGload j Gout;P1 ðoutÞ 1 þ s22 jGload j where the equals sign applies for purely reactive matching networks ðoutÞ only. The above relation implies that Gout;P1 ¼ 1 if and only if s22 ¼ 1. Again, this is possible only if ðoutÞ ðoutÞ the matching network is loss free, and implies that s21 ¼ s12 ¼ 0. In other words, there is no transmission between the two ports of the network. Thus, a matching network can present a purely reactive load to the transistor only by isolating it from the load, and then transferring no power at all. Equivalently, we could assume that the load is purely resistive – any series reactance is considered as part of the matching network – and normalize the S parameters on the load resistance. In this case, the reflection coefficient seen by the transistor output coincides with the input one of the matching network, and has unitary amplitude only if the two ports of the matching network are isolated. Note that we assumed a generic value for the load reflection coefficient: the load impedance has no effect on our considerations, unless it is purely reactive by itself; in that case no active power transfer is possible to the load, independently of any other circumstances. The same considerations apply to the input matching network. The conclusion is that simultaneous conjugate matching is impossible if the transistor is potentially unstable. Once the reflection coefficients GMS ; GML are known, by means of Equations (11.26) and (11.27) we can dimension the matching network in the same way as for a unilateral transistor having G*MS ; G*ML as input and output reflection coefficients, respectively. 12

See the Mathcad file 07_Simultaneously_Conjugated_Matching.MCD.

428


Substituting the coefficients (11.26) into the gain (11.6), we obtain the maximum transducer power gain of a non-unilateral perfectly matching transistor

GT;MAX

2 ðQÞ 2 1jGML j2 s21 1jGMS j ¼ 2 ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ ðQÞ 1 s11 GMS þ s22 GML þ s12 s21 GMS GML þ s11 GMS s22 GML

ð11:28Þ

As can be seen, the coefficients GMS ¼ G*IN and GML ¼ G*OUT given by Equations (11.26) are not physically realizable if the transistor is unstable; then the maximum gain (11.28) is not achievable. The maximum gain for a potentially unstable transistor can be found by combining Equations (11.17) and (11.26) with (11.28), and after some manipulation the resulting expression is GT;MAX ¼

js21 j2 js12 j

2

pffiffiffiffiffiffiffiffiffiffiffiffi ðK K 2 1Þ

ð11:29Þ

The transducer power gain (11.29) is a monotonically decreasing function of the stability factor, and reaches its maximum at the minimum stability condition ðK ¼ 1Þ. The resulting value is the maximum stable gain GMSG ¼

js21 j2 js12 j2

ð11:30Þ

The value in (11.30) is the maximum theoretically achievable by the amplifier in Figure 11.1, if the matching networks present the minimum possible dissipation loss needed for transistor stabilization. Amplifiers employing potentially unstable transistors need to be stabilized, particularly when the termination impedances are not exactly known, or are variable.13 In the latter case, one of the possible terminating impedances could be such that the reflection coefficient seen by the transistor comes from the stability region, with the consequent risk of oscillations. Stabilization prevents the use of purely reactive matching networks, in that they can move the unstable regions from one portion of the Smith chart to another, always remainining within the unit circle ðjGj ¼ 1Þ. Instead, we need to push the unstable regions out of the Smith chart. Thus, lossy matching networks are required. Therefore, input and/or output matching networks must include resistors: they could be either lumped or loss elements in non-ideal reactive elements. Fano limits do not apply to resistive matching networks. Therefore any return loss is theoretically achievable over arbitrarily wide bandwidths. On the other hand, good matching is obtained at the expense of amplifier gain, since resistive matching networks absorb part – sometimes relevant – of the power. The gain with simultaneous matching is the maximum possible: as the generator (load) reflection coefficient differs from GMS ðGML Þ, the gain decreases accordingly. The terminating reflection coefficients producing the same gain constitute loci in the G plane, having the form of circles, known as constant gain circles. The complete treatment of the gain circles involves many cases: input and output, transducer, available and operating power gain with unilateral, bilateral stable and unstable transistors. The interested reader can find more details in [1]; also most of the high-frequency circuit simulators can plot all the gain circles. Here, we will limit our presentation – without demonstrations – to the case of the available gain of a perfectly output matched amplifier, with a bilateral stable transistor. Under these assumptions, the available and transducer power gain coincide. The constant available loci are circles in the input reflection

13 Several factors determine the impedance variability, like realization tolerances or variable working conditions. Amplifiers connected to multiple sources and/or loads, selectable by means of switches, is a typical case of variable termination.

AMPLIFIERS coefficient plane ðGS Þ having centre and radius vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 GA ðQÞ* ðQÞ * u12K sðQÞ sðQÞ GA þ sðQÞ sðQÞ 2 GA 4 2 s11 s22 D t 12 21 12 21 2 ðQÞ ðQÞ ðQÞ s21 s21 s21

; RA ¼ CA ¼ 2 GA ðQÞ 2 G 1 þ 2 s11 jDj2 ðQÞ A 2 ðQÞ 1 þ s11 2 jDj s21 ðQÞ s21

429

ð11:31Þ

where GA is the available gain obtained when the input reflection coefficient moves around the circle. Constant gain circles give an intuitive representation of the gain sensitivity to the input output reflection coefficients: a Smith chart region populated with many (few) gain circles indicates a high (low) sensitivity. Normally, the transistor forward transmission coefficient and maximum gain decrease with frequency, with a slope of about 20 dB/decade. This behaviour has two main effects on the matching network design: 1. If caution is not used, the amplifier gain tends to increase at low frequency. Sometimes, the lowfrequency gain could be higher than the in-band value. Under this condition, out-of-band signals – including noise – can reduce the amplifier’s dynamic range, by causing intermodulation with the useful signals. The extreme case is that relatively strong low-frequency input signals saturate the amplifier, totally compromising its working. Therefore, high-pass matching networks are preferable in respect of low-pass ones, because the lack of mismatch associated gain decreases the gain at frequencies lower than the minimum operating one, contrasting the natural increase of the transistor gain. 2. The natural decrease of the gain at low frequencies, combined with the limits on wide-band impedance matching stated by the Fano inequalities, lead to an interesting design strategy for broad-band amplifiers. The transistor is matched at the maximum frequency only; at lower frequencies, a controlled mismatching compensates for the device gain increase and flattens it out.

11.3 Low-noise amplifiers Amplifiers add noise to the amplified signal. It is important that the added noise is as low as possible, particularly if the signals to be amplified are weak. We can express this concept in quantitative terms, by saying that the noise figure14 of the amplifier must be kept to a minimum. The noise factor of a generic linear two-port network is given by Equation (9.29). We can rewrite it as F ¼ Fmin þ

2 Rn YS Yopt ReðYS Þ

ð11:32Þ

where YS is the admittance terminating the network input. The second term in the final member of Equation (11.32) is always non-negative, because so the noise resistance Rn and the generator admittance ReðYs Þ are. Thus, the noise factor reaches its minimum Fmin if the above-mentioned term vanishes, which means YS ¼ Yopt . The parameter Rn determines how fast the noise factor increases for a given deviation of the generator admittance from Yopt . Note that the noise factor depends on the input admittance only, not on the output. All the above considerations apply to any linear two-port network, in particular to transistors and amplifiers. However, in the case of a complete amplifier, the only relevant parameter is the noise figure on 50 O, since that is the working impedance. The case of a single transistor is different, because the 14

See Section 9.4.2 for the definition of noise figure and noise factor.

430


Noise figure meter

SPDT 1

C gg

Γs Stub tuner

Cdd

DUT

Ldd

Lgg

2 Rdd

Rgg

+ +

Vdd

Vgg output bias input bias

Network analyzer

Figure 11.3

Setup for the measurement of transistor noise parameters.

complete noise parameter set is needed to get the best noise performance in the complete amplifier. Design of a low-noise amplifier (LNA) having the configuration of Figure 11.1 consists of designing an input matching network that presents an admittance to the transistor input as close as possible to Yopt . Equation (11.32) suggests a method for the measurement of the noise parameters of a two-port network, and of a transistor in particular, based on multiple noise factor measurements with different input impedances. Figure 11.3 shows the test set needed to implement the noise parameter measurement of the transistor.15 The noise figure meter measures the transistor noise figure with different input loads, provided by the stub tuner. The network analyzer measures the impedance presented to the transistor input. The two T-bias networks16 do not affect the measurement because they are calibrated out. Possible differences in the two paths of the SPDT are also taken into account by specific calibration techniques. The measurement procedure consists of the following steps: 1. With the SPDT in position 1, modify the generator admittance by means of the stub tuner, to minimize the noise figure measured by the noise figure meter. The corresponding noise factor is Fmin . 2. Keep the stub tuner in the condition found in 1 and the SPDT in position 2. The admittance measured by the network analyzer is Yopt . 15 Section 17.6 describes network analyzer noise and the noise figure meter, while Section 6.5 describes the matching network with line and stub. The stub tuner has the structure of Figure 6.11, stub and line impedances are usually 50 O, but the stub length y2 and its position along the line y1 vary with micrometric adjustments. The most sophisticated devices include computer-controlled motors to change y1 and y2 . Sometimes, a double stub tuner is used, which has two variable stubs in fixed positions, like the one in Figure 6.13. Both types of devices are capable of presenting any real part immittance, at least in principle. 16 Bias networks are sometimes referred to as T-bias, in this context.

AMPLIFIERS

431

3. Normally 1=Yopt is different from 50 O. In order to exploit this difference, measure the noise figure on 50 O, F50 O . Set SPDT in position 2, and adjust the stub tuner to minimize the reflection coefficient measured by the network analyzer. In that condition, move the SPDT to position 1: the measured noise factor coincides with F50 O . 4. Equation (11.32) states that the noise resistance is a function of the parameters taken at the points 1 to 3, i.e. R0 Rn ¼ ðF50 O Fmin Þ 1R0 Yopt 2 Note that when the SPDT is in position 2, the voltage Vdd is applied to the network analyzer test port. This could be dangerous for the instrument, depending on the voltage and the instrument type itself. Therefore, it is advisable to switch Vdd off, before setting the SPDT on position 2. We can develop more considerations of the low-noise amplifier by expressing the noise factor in terms of reflection coefficients rather than admittances. The generator and the optimum admittances are related to their respective coefficients as Ys ¼

1 1Gs ; R0 1 þ Gs

Yopt ¼

1 1Gopt R0 1 þ Gopt

Substituting the above expressions into Equation (11.32), we obtain Gopt Gs 2 Rn F ¼ Fmin þ 4 R0 1jGs j2 1 þ Gopt 2

ð11:33Þ

For any noise factor F > Fmin, Equation (11.33) returns the values of the reflection coefficient Gs that produce F. All the values of Gs corresponding to a constant F > Fmin lie on circles, known as constant noise circles. With a procedure similar to the one used for the stability circles we can determine centre CF and radius RF of the constant noise circles, finding Gopt 2 FFmin R0 1þ 1 þ Gopt 4 Rn 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FFmin R0 2 1 þ Gopt u u 1Gopt 4 Rn u RF ¼ 2 t1 þ FFmin R0 FFmin R0 1 þ Gopt 2 1þ 1 þ Gopt 4 Rn 4 Rn CF ¼

ð11:34Þ

As seen for the constant gain circle, a region of the plane Gs densely populated by noise circles denotes a high sensitivity of the noise factor to the input reflection coefficient. Notice that GMS does not coincide with Gopt , therefore the input matching network designed for the minimum noise factor does not give the maximum gain, and vice versa. Thus, input matching network design can pursue the minimum noise, the maximum gain or a compromise between them. However, the lack of gain associated with the minimum noise figure involves an input mismatch. For instance, if the gain associated with Gopt is 1 dB lower than the maximum, then the input return loss for the minimum noise is 10 log10 ð1101=10 Þ 6:9 dB. This relatively bad input matching could compromise the performance of a chain including the amplifier, due to the amplitude and phase ripple associated with the input mismatch, as discussed in Section 9.2. A common technique for moving the maximum gain point closer to the minimum noise one consists of adding a series inductor to the ground connection of the source or emitter. The stabilization – if needed – of a transistor for LNA application has to be done preferably with resistors in the output matching network, rather than in the input one. Equation (9.32) together with some

432


simplified considerations gives a reasonable justification for that criterion. Let us initially cascade a resistive attenuator to the transistor input: the resulting noise factor from Equation (9.32) is Fattenuator þ transistor ¼ Fattenuator þ

Ftransistor 1 GA;attenuator

Passing from the noise factor to the noise figure, and remembering that resistive attenuators have the noise figure coincident with the attenuation (both in dB), we get NFattenuator þ transistor ¼ ILattenuator þ NFtransistor

ð11:35Þ

where ILattenuator is the attenuation in dB of the attenuator. Equation (11.35) states that an attenuator cascaded with the input of a linear two-port network increases the resulting noise figure by its attenuation. Now, in order to evaluate the effect of the loss in the output matching network, let us decide to connect the same attenuator as above in cascade with the transistor output. The resulting noise figure will be h i NFtransistor þ attenuator ¼ NFtransistor þ 10 log10 1 þ 10ILattenuator =10 1 10ðGA;transistor NFtransistor Þ=10 ð11:36Þ Both Equations (11.35) and (11.36) present an additional positive term to the transistor noise figure. Thus, the noise figure (11.36) is smaller than the noise figure (11.35) if h i 10 log10 1 þ 10ILattenuator =10 1 10ðGA;transistor NFtransistor Þ=10 < ILattenuator which occurs if GA;transistor > NFtransistor . Therefore, attenuation in the output degrades the noise performances of the amplifier less than the same attenuation placed in the input, if the gain of the transistor exceeds its noise figure. Note that the transistor gain considered in Equation (11.36) includes the contribution of the matching networks, thus it is typically higher than the noise figure. However, in all these considerations we used the same input and output attenuation: transistor stabilization on the input matching could require quite different attenuation than in the output. The last aspect to consider is the bias point of the transistor. The noise parameters of the transistor are bias dependent, like the scattering parameters. Usually, the bias point giving the minimum possible noise figure is different from the one giving the maximum gain, and from the one giving the maximum power as well. Consequently, the choice of bias point is the result of a compromise among the various performances, similar to the input reflection coefficient.

11.4 Design of trial amplifier This section presents the application of the concepts discussed in Sections 11.2 and 11.3 to the design of a trial amplifier. The transistor of our amplifier is a MESFET having the scattering parameters listed in Table 9.2. First of all we will examine the stability of the transistor over all its specified frequency range, from 2 to 10 GHz. Equation (11.17) applied to the parameters in Table 9.2 gives17 K < 1 over all the frequency range, with a minimum K ¼ 0:11. Equivalently, we can examine the stability circles18 plotted in Figure 11.4. They invade the upper half of the Smith chart, particularly the output ones that occupy almost all the inductive portion of the Smith chart. 17 See the Mathcad file 07_Simultaneously_Conjugated_Matching.MCD and the Ansoft file 01_Unilateral_ and_Bilateral_Matching.adsn, graph ‘A_Stabilities’. 18 See the Ansoft file 01_Unilateral_and_Bilateral_Matching.adsn, graph ‘H_Minimum_Noise_Figures’.

AMPLIFIERS

433

Figure 11.4 Input (black) and output (grey) stability circles of the transistor having the scattering parameters as in Table 9.2. The next step in our design consists of stabilizing the transistor. As a first attempt, we can see the effect of a resistive attenuator placed at the input and at the output of the transistor. From the analysis of the two cascaded components, it follows that the minimum input (output) attenuation for critical stability is 2 (10). This result is in good agreement with the disposition of the transistor stability regions, and confirms that the output is more critical for the stability. Equation (11.35) implies that 2 dB of attenuation at the transistor input degrades its noise figure exactly by the same amount: NFattenuator þ transistor ¼ 2 þ NFtransistor We can approximately predict the noise figure degradation due to a 10 dB output attenuator, by means of Equation (11.36). Assuming some reasonable numbers, like GA;transistor 12 dB and NFtransistor 1 dB, the result is h i NFtransistor þ attenuator ¼ NFtransistor þ 10 log10 1 þ 1010=10 1 10ð122Þ=10 ffi NFtransistor þ 2:79 Thus the configuration with resistors in the input matching network is more promising, in terms of gain (8 dB better) and in terms of noise figure (0.79 dB better) than the one with resistors in the output matching network. Moreover, the output power will also be higher, due to the absence of losses from the transistor output to the load. Let us momentarily consider the transistor and the input stabilizing elements, by ignoring bias and matching networks, and short-circuiting the source inductor Ls1. The subnetwork that we initially consider consists of the transistor together with the resistors Rg1 ; Rg2 and the capacitors Cg1 ; Cg2 . The resulting two-port network has the node n1 as input and the collector of Q1 as output. For the capacitors, Cg2 ¼ 10 pF is large enough to consider the node n3 as a high-frequency ground, Cg1 ¼ 1 pF gives a relatively low impedance at 8 GHz, while reducing the low-frequency gain.19 The low RF impedance to 19 The reactance associated with 10 pF at 2 and 10 GHz is 1=ð2p 2 109 10 1012 Þ ffi 8 O and 1:6 O respectively; the series reactance associated with 1 pF is 10 times higher.

434


Figure 11.5 Input stability circles of the transistor of Table 9.3, with (black) and without (grey) stabilization resistors at the input.

ground of the node n3 allows us to use it to inject the gate bias voltage, with a minimum perturbation of the high-frequency working of the matching network, while Cg1 isolates the gate from the amplifier input at DC. The values Rg1 ¼ 10 O; Rg2 ¼ 150 O have been found by tuning. them up to reach stability while minimizing the loss. By comparison, formulae (7.9) give20 for a 50 O T-type attenuator Rg1 ffi 5:7 O; Rg2 ffi 215 O: our network has slightly lower series and shunt resistance, then comparable attenuation as well. Figure 11.5 shows the effect of Rg1 ; Rg2 ; Cg2 on the input stability circles that fall completely outside of the Smith chart, so the output ones do too, although they are not shown. Figure 11.6 shows the electrical schematic of the complete amplifier. 20

See Mathcad file 08_Passive_T_Attenuator.MCD.

AMPLIFIERS

435

NT2 : output bias and matching NT1 : input stabilization, bias, and matching TL g1

Input TL g2

n1Cg1 n2 Rg1 Rg2 n3

TLd1

Q1

n4

TLd2

n6 Cd1

Output

TLd3 Z g1

Ls1

Zg1

Cg2

n5 Cd2

+ Vdd

+ Vgg

Figure 11.6

Electrical diagram of the amplifier.

After stabilizing the transistor, we can consider the conjugate matching coefficients, in particular the values at 8 GHz. Figure 11.7 shows the scattering parameters of the stabilized transistor, their complex conjugates and the conjugate matching coefficients. Note that the latter two couples are slightly different, since the two-port network is non-unilateral. Figure 11.8a shows the constant gain and constant noise figure circles as black and grey curves, respectively. As anticipated in Section 11.3, the input reflection for the maximum gain is different from the one for the minimum noise. More precisely, the minimum noise figure of 4.3 dB corresponds to a gain slightly higher than 7.4 dB, while the maximum gain of 9.3 dB corresponds to a noise figure slightly higher than 5.5 dB. The input matching design can pursue the minimum noise, the maximum gain or any compromise between them. In particular, the input return loss for the minimum noise is about 10 log10 ð110ð7:49:3Þ=10 Þ ffi 4:5 dB, which is critical in most applications. The addition of one inductor in series with the source moves the optimum gain closer to the optimum noise, and a better compromise is achievable. The schematic in Figure 11.6 shows that component, labelled as Ls1 . The source series inductor is a form of feedback that reduces not only the transistor gain, but also the instability, then, after less attenuation is required in the matching networks, partially compensates the lack of gain. The choice of Ls1 is the result of some attempts with some values, each of them considering the positions of GMS ; Gopt ; GT;max ; and minðKÞ, where the last parameter indicates how much loss is required to stabilize the transistor. A good solution was found to be Ls1 ¼ 200 pH. This choice allows a reduction in the series and increases the shunt resistance in the input matching network: their values become Rg1 ¼ 5 O; Rg2 ¼ 300 O. Figure 11.8b shows the gain and noise circles of the two-port network consisting of the transistor with the added series source inductor and the stabilizing elements. Comparing Figure 11.8b with Figure 11.8a, we can see that GMS ; Gopt are now closer, the minimum noise figure being 2.8 dB instead of 4.3 dB, due to the reduced input attenuation, and the maximum gain being 8.8 dB instead of 9.3 dB. Summarizing, the source inductor improved the minimum noise figure by 1.5 dB while degrading the maximum gain by 0.5 dB only. The new gain–noise compromises are 8.8 or 7.7 dB of gain with 3.6 or 2.8 dB of noise figure. The input return loss for the minimum noise is then 10 log10 ð110ð7:78:8Þ=10 Þ ffi 6:5 dB, or 2 dB better than without the inductor, but still not good enough for many applications. Therefore we decided to pursue the maximum gain (perfect input matching) and accept the noise figure of 3.6 dB, 0.8 worse than in the optimum case. At this point, the design of the matching networks is quite straightforward. Beginning with the input, looking at the node n1, we have GMS ¼ 0:025 þ j 0:738, corresponding to the admittance YMS ¼ 5:69 103 j 0:019. The imaginary part of YMS can be realized with either a shunt inductor or a short-circuit stub shorter than l=4. We chose the second option, with characteristic impedance Z0 ¼ 70 O, its electrical length at 8 GHz being found by imposing the stub susceptance to be equal to the imaginary part of YMS . Therefore

436


Figure 11.7 Reflection coefficients of the transistor in Figure 11.6 together with its stabilization elements. Input at the node n1, output at the transistor drain, and Rg1 ¼ 10 O; Rg2 ¼ 150 O; Cg1 ¼ 1 pF; Cg2 ¼ 10 pF; Ls1 ¼ 0. ½ j Z0 tanðyÞ1 ¼ j ImðYMS Þ ) y ¼ tan1 ð1=0:019 1=70Þ, corresponding to an electrical length of 37.66. After connecting the parallel short-circuit stub, the imaginary part of YMS vanishes, and the resulting 0 ¼ 1=ReðYMS Þ ¼ 175:62 O. That resistive value can be easily matched with a l=4 impedance is ZMS 0 transmission line transformer, whose characteristic impedance is the geometric mean of ZMS and 50 O: the resulting value is 93.7 O. With reference to Figure 11.6, the characteristic impedance and electrical length of TLg1, TLg2 are 93:7 O; 90 and 70 O; 37:66 , respectively. The amplifier now includes the complete input matching and bias network. Considering the two-port network having the amplifier input as input and the transistor collector as output, we have GMS ¼ 0:406 þ j 0:537, corresponding to an impedance of ZMS ¼ 42:58 þ j 83:82. As in the input matching network, the impedance to be realized is inductive. In this case, we will use a series inductor, instead of a shunt one, realized with a short high-impedance line, applying the semi-lumped concepts.21 We will use 21

See Section 7.5.

AMPLIFIERS

437

Figure 11.8 Constant gain (black) and constant noise (grey) circles of the transistor in Figure 11.6 together with its stabilizing elements. The input of the two-port network is the node n1, the output is the transistor drain Cg1 ¼ 10 pF; Cg2 ¼ 10 pF: (a) Rg1 ¼ 10 O; Rg2 ¼ 150 O; Ls1 ¼ 0; (b) Rg1 ¼ 5 O; Rg2 ¼ 300 O; Ls1 ¼ 200 pH.

438


10

2.0

10

6

0 -5

s22 4

0 NF

NF, dB

20 log10(|s21|)

s11

-10 -15

K 2

-5

-20

1.5

K

s21

5

20 log10(|s11|), 20 log10 (|s11|)

5

-25 0

-10 6

7

8 Frequency, GHz

Figure 11.9

9

-30

1.0

10

Performances of the amplifier in Figure 11.6.

a 90 O transmission line (TLd1 ), tuning its length until GMS falls on the zero-reactance line, obtaining an 0 0 ffi 42:58. Note that ZMS does not electrical length of about 40 . The resulting transistor impedance is ZMS exactly coincide with ReðZMS Þ, due to the unwanted shunt reactance associated with the semi-lumped inductor. Before completing the matching network design by adding a l=4 transformer (TLd2 ), we need to add the bias network. For this reason, we place a 1 pF capacitor ðCd1 Þ between the transformer and the amplifier output. This element involves a slight modification of both TLd1 and TLd2, with the resulting characteristic impedances and electrical lengths 90 O; 41 and 35:5 O; 90 , respectively. The addition of a drain bias element completes the amplifier. That structure consists of the l=4 90 O line TLd3 and the 10 pF capacitor to ground Cd2 . For the same reasons as for the input matching network, the node n5 is a high-frequency ground. Thus, TLd3 transforms the low impedance on n5 into high impedance in shunt with the node n6 . The consequent effect on the output matching network is negligible at 8 GHz, while the direct current can flow from the bias port Vdd to the transistor drain. Figure 11.9 shows the performances of the complete amplifier. Note that the stability factor is always greater than 1, at 8 GHz the gain22 is 8.8 dB and the noise figure is 3.6, as expected. Moreover, the minimum noise figure is reached at around 9 GHz, due to the better noise input matching at that frequency. Finally, the amplifier presents the maximum gain at a frequency slightly lower than 8 GHz, despite the use of high-pass elements in the matching networks, due to the natural high-frequency roll-off of the transistor gain. Figure 11.10a shows one of the possible physical layouts23 of the amplifier, realized with hybrid technology on an alumina substrate, having er ¼ 9:8; h ¼ 254 mm. The transistor and the capacitors are chip devices, connected to the microstrip lines by means of bond wires.24 The resistors are realized directly on the substrate, using thin-film technology. Via holes realize all the required ground connections. All the transmission lines used for the amplifier have five different characteristic impedances: 93.7, 90, 70, 50 and 35.5 O. A numerical inversion of the analysis formula (3.224) gives the corresponding From 7.5 to 8.5 GHz, 7:9 < 20 log10 ðs21 Þ < 8:9 and 3:5 dB < NF < 4:2 dB. See also the Ansoft file 02_Bilateral_Matching_Layout.adsn. 24 The bond wires add a series inductance, which can be characterized as shown in the reference [2]. 22 23

AMPLIFIERS

439

Figure 11.10 Physical realizations of the schematic in Figure 11.6: (a) straightforward solution; (b) more compact configuration.

widths 0.044, 0.05, 0.11, 0.254 and 0.477 mm. Entering these widths into Equation (3.182), we get the effective relative dielectric constants of 6.1, 6.1, 6.4, 6.8 and 7.2. From the electrical length at a specified frequency, and the effective permittivity, we obtain the physical length of all the transmission lines lphysical ¼

lelectrical ð f0 Þ 75 1 pffiffiffiffiffiffiffi 90 f0 ee f f

Once all the physical lengths for all the transmission lines are determined, the amplifier design is still not complete, because of the parasitic effects associated with T junctions, step discontinuities, finite impedance of the via holes, bond wire inductance,25 and so on. The final design step consists of 25

The scattering parameters of the transistor include the bond wires, but this is not the case with the capacitors.

440


analyzing the circuit26 and modifying the microstrip parameters until the response of the circuit is reasonably close to the ideal case. The layout in Figure 11.10a has a total size of 11 7 mm, including two 50 O lines at input and output. Figure 11.10b shows a more sophisticated realization with a more compact layout, having a size of 7 5 mm: it is derived from the one in Figure 11.10a by folding TLg1 and replacing the output transformer TLd2 with a semi-lumped realization of the network in Figure 6.15a. For the latter element, TL0d2 is the combination of TLd1 and the transmission line corresponding to Lb , and the 50 O open stub TL0d2 realizes a capacitive impedance to ground, corresponding to Cb .

11.5 Power amplifiers The definition of a power amplifier (PA) is somewhat arbitrary. A widely used – although not rigorous – criterion is that a power amplifier is an amplifier whose output power is close to its 1 dB compression point. Under that condition, the nonlinear aspects of the active device are not negligible. By contrast, Sections 11.2 to 11.4 assumed that the transistor is a purely linear device. These sections limit consideration to the influence of DC generators and bias networks as elements of the matching networks, with possible side effects on the gain and noise figure. Moreover, the only effect of the bias point is to change the scattering and noise parameters of the transistor. Finally, the transistor DC power is solely dependent on the DC bias point, not on the RF signal amplitude. The main limitation of this approach is that it gives no idea of how to determine the maximum RF output that the amplifier is able to deliver. In the present section we will examine some of these missed aspects, analyzing how transistor characteristics, working point and load impedance determine the amplifier output power. Subsequently, some basic concepts on the different working classes of power amplifiers will be given. Complex amplifier configurations complete the section.

11.5.1

Output power optimization with negligible transistor parasitics

This section describes some basic techniques to get the maximum RF output power from a given transistor. The concepts presented follow from analyzing how transistors convert DC into RF power, and the associated limitation factors. Moreover, this section assumes that the transistor output current is nonzero for the entire cycle of the RF signal. Such operation is usually referred to as class A. Momentarily we will use this denomination for brevity. Section 11.5.5 below deals with the different output current waveform types, or amplifier classes. In order to investigate the effects of transistor characteristics, DC working point and output load, we will initially consider the structure in Figure 11.11 derived from the one in Figure 11.1. As in the original network, the symbol for the transistor in Figure 11.11 is one FET, but it represents a generic device. In the BJT case, base current Ib , collector current Ice and collector–emitter voltage Vce replace gate–source voltage Vgs , drain current Ids and drain–source voltage Vds , respectively. In the following considerations, the variables Vds;ce ; Ids;ce will denote the generic drain–source or collector–emitter voltage and current, respectively. Similarly, the variable xin represents either the gate–source voltage or the base current, depending on the actual case.

26

See the Ansoft file 02_Bilateral_Matching_Layout.adsn.

AMPLIFIERS

441

+ Ldd

Vq

Isupply

Cdd

ILOAD

I ds , Ice Cgg ZS

Vgs, I b

Vds, Vce Q1

VLOAD RLOAD

L gg +

VRF

Figure 11.11

Vgg

Conceptual configuration of a single transistor amplifier.

Compared with the network in Figure 11.1, the one in Figure 11.11 has some simplifications and additional specifications: 1. The two resistors in series with the two DC generators are not indicated. 2. The bias networks are purely ideal, in that the two inductances Lgg ; Ldd and the two capacitances Cgg ; Cdd assume infinite value. This way, the inductors (capacitors) offer zero (infinite) DC impedance and infinite (zero) impedance at high frequencies. 3. Input and output matching networks are not indicated. 4. The input sinusoidal RF generator with its series impedance Zs delivers enough power to the transistor input to modulate the output current sufficiently. 5. The resistor RLOAD models the impedance – assumed to be resistive – that the output matching network presents to the transistor output. 6. The transistor is purely unilateral, and no reactive parasitic elements connect with its output. 7. The transistor DC curves are perfectly horizontal in their pentode region. In a MESFET or BJT, this means that l ¼ 0 or VAF ¼ 1, respectively, as from the model equations (9.75), (9.79) or (9.81). 8. The triode region of the device degenerates into the single point Vds;ce ¼ 0. In a MESFET modelled by Equation (9.75) or (9.79), this is equivalent to assuming the parameter a tends to infinity. 9. The Ids;ce corresponding to equal increments of xin are equispaced. In other words, we will assume the transistor output current is linearly dependent on the input excitation, Ids;ce ¼ B1 xin þ B0 , where B1, B0 are two constants. 10. The transistor input behaves like a linear network. Assumptions 1, 3 and 4 are just to simplify our description, and have no important implication. The simplification 2 implies no lack of generality, in that any non-ideal element in the bias networks can be considered as being part of the corresponding matching network. The remaining hypotheses are needed to allow a relatively easy analytical treatment of the problem.

442

MICROWAVE AND RF ENGINEERING Ice , Ids

Ice , Ids

Increasing Ib , Vgs

Ix = Vq /R LOAD + Iq 2I q Iq α

Vq

χ

β

β

t

α, χ

2 Vq Vce, Vds Vx = Vq + Iq R LOAD V ce, Vds

α β, χ

t

Figure 11.12 DC curves of the transistor used in Figure 11.11 (upper left), output current (upper right) and voltage (lower left) waveforms of the same transistor. Figure 11.12 shows the DC curves of our hypothetical quasi-ideal device, based on assumptions 7 to 9. The transistor derived from those hypotheses is an almost ideal VCCS or CCCS. The non-ideality is that it can only supply positive currents and only if a positive voltage is present on its output terminal. Therefore, the network in Figure 11.11 behaves as a linear one until the transistor output voltage and current are non- negative. Under this condition, we can analyze the circuit by superimposing the effects due to DC and RF generators. It is possible to replace the second part of hypothesis 6 with the assumption that the output matching network – combined with the transistor parasitics – presents a resistive impedance to the transistor-controlled generator. According to assumptions 4 and 8 to 10, the transistor output current is the sum of a DC term due to the input bias, plus one sinusoidal term due to the RF generator, i.e. Ids;ce ðtÞ ¼ Iq þ Ipeak sinðo0 tÞ

ð11:37Þ

The term Iq in the above expression is the quiescent current of the amplifier, defined as the one that Vq supplies in the absence of an RF input signal. Our approximated model is valid until Ids;ce 0, therefore the quiescent current must not be less than the RF peak current, Iq Ipeak . Under this hypothesis, the direct current of a class A amplifier is independent of the input signal, and thus from the output power. The DC block capacitor Cdd allows only high-frequency current to flow through. Then the current through RLOAD , which coincides with the one through Cdd , is ILOAD ðtÞ ¼ ILOAD;peak sinðo0 tÞ

ð11:38Þ

The current supplied by the DC generator Vq, which coincides with the one through the inductor Ldd , is the sum of the quantities (11.37) and (11.38). Moreover, no RF current can flow through the inductor Ldd , because of specification 2. Thus Isupply ðtÞ ¼ Ids;ce ðtÞ þ ILOAD ðtÞ ¼ Iq þ ðIpeak þ ILOAD;peak Þ sinðo0 tÞ ¼ Iq

ð11:39Þ

Equation (11.39) implies that Ipeak ¼ ILOAD;peak , consequently the load current in Equation (11.38) becomes ILOAD ðtÞ ¼ Ipeak sinðo0 tÞ ð11:40Þ The load voltage equals the current (11.40) multiplied by the resistance RLOAD itself. Then, applying the superimposition of the effects due to Vq and VLOAD to the transistor output voltage, we have Vds;ce ðtÞ ¼ Vq RLOAD Ipeak sinðo0 tÞ

ð11:41Þ

AMPLIFIERS

443

Similar to what was considered for Equation (11.37), our model is valid until Vds;ce 0. Thus, RLOAD Ipeak Vq must apply and consequently the load peak voltage must not be greater than Vq as well. Extracting the factor Ipeak sinðo0 tÞ from Equation (11.41) and substituting the result into (11.38), we obtain the relation between the transistor output current and voltage Ids;ce ðtÞ ¼ Iq þ

Vq Vds;ce ðtÞ RLOAD RLOAD

ð11:42Þ

Expression (11.42) is the well-known load line equation, which describes a straight line passing through the point ðVq ; Iq Þ. The intersections of the load line with the coordinate axes Vds;ce ðIds;ce Þ are found by imposing Ids;ce ¼ 0 ðVds;ce ¼ 0Þ; then the resulting values are Vx ¼ RLOAD Iq þ Vq ðIx ¼ Iq þ Vq R1 LOAD Þ. Figure 11.11 shows three load lines, together with the corresponding transistor voltage and currents, for three cases: (a) RLOAD < Vq =Iq . In this case the quiescent current is the limiting factor for the RF output peak voltage, which has a maximum value of Iq RLOAD . The corresponding RF output power is PLOAD;a ¼ ðIq RLOAD Þ2 =ð2 RLOAD Þ ¼ 0:5 Iq2 RLOAD . Therefore, the amplifier output power increases with the load resistance. (b) RLOAD > Vq =Iq . In this case, the load peak voltage limitation comes from the output supply voltage. The maximum output peak voltage equals Vq , with a resulting power of PLOAD;b ¼ 0:5 Vq2 R1 LOAD . The output power decreases with the load resistance, opposite to case (a). (c) RLOAD ¼ RLOAD;w ¼ Vq =Iq . In this case the two limiting factors (a) and (b) give the same result. The maximum power is PLOAD;w ¼ 0:5 Iq2 RLOAD ¼ 0:5 Vq2 R1 LOAD ¼ 0:5 Vq Iq . Combining the three output power expressions, obtained for (a) to (c), we get

PLOAD;max

8 2 Iq RLOAD > > > > > 2 > > > > < I2R Vq2 Vq Iq q LOAD ¼ ¼ ¼ > 2 2 R 2 LOAD > > > > 2 > V > q > > : 2 RLOAD

Vq RLOAD < Iq

Vq RLOAD ¼ Iq

Vq RLOAD > Iq

ð11:43Þ

Equation (11.43) states that the optimum load for the maximum output power is the one as in case (c). Thus, in our simplified hypotheses, the output matching network designed to maximize the output power has to transform the 50 O into RLOAD;w ¼ Vq =Iq . Under this condition, Vds;ce ranges from zero to 2 Vq , and this maximum must be lower than the device breakdown, with some margin. Therefore the DC voltage applied to the transistor output electrode has to be smaller than half of the breakdown voltage of the drain–gate or collector–base junction. Similarly, the output current voltage is a sinusoid with zero as the minimum and 2 Iq as the maximum. Thus, the quiescent current needs to be lower than the maximum allowed current for the transistor, as specified by the manufacturer. Note that, if the load resistance becomes higher than RLOAD;w , then the positive peak of Vds;ce becomes higher than 2 Vq , with a consequent risk of permanent damage to the device. From Equation (11.39) and subsequent considerations, it follows that the DC output current from the generator Vq coincides with Iq, independently of RLOAD . The DC power absorbed from the amplifier – also known as DC input power – is then PDC ¼ Vq Iq

ð11:44Þ

444


The quantity in (11.44) neglects the RF and DC power absorbed from the input, which is usually a small fraction of PDC itself. One important performance of PAs is their capability to deliver high RF power with a DC power consumption that is as small as possible. The quantitative parameter for this characteristic is the drain or collector efficiency or, more briefly, the efficiency, defined as Z¼

PLOAD PDC

ð11:45Þ

Another important definition is the power added efficiency (PAE), which takes the RF input power into account

PLOAD PIN PLOAD G1 1 P PLOAD ¼ ¼ 1 Z ð11:46Þ PAE ¼ PDC PDC GP The reason for the adjective ‘added’ is because the definition of PAE removes the input contribution to the RF power; PAE tends to Z when the operating power gain tends to infinity. Usually both the quantities in (11.45) and (11.46) are expressed in per cent. In our case, Equations (11.43) and (11.44) give maximum RF power and DC power of the amplifier. In the best case, for RLOAD ¼ RLOAD;w ¼ Vq =Iq and with the output current at its maximum swing, we also have the maximum efficiency of 50%: Zmax ¼

Vq Iq =2 ¼ 50% Vq Iq

The total DC input power is equal to the sum of the RF power delivered to the load plus the one dissipated by the transistor. This latter is

PLOAD Z ð11:47Þ PDC ¼ 1 Vq I q Ptransistor ¼ PDC PLOAD ¼ 1 100 PDC From Equations (11.43) to (11.47) it follows that the amplifier efficiency vanishes if there is no input signal. Under that condition, the transistor power dissipation is at its maximum, Ptransistor;max ¼ Vq Iq , and decreases up to half this value when the RF output power is at its absolute maximum. Note that a PA with RLOAD ¼ RLOAD;w ¼ Vq =Iq presents a low-output return loss. In our simplified model, the transistor DC curves are perfectly horizontal; then the associated device output resistance Rds;ce ¼ ð@Ids;ce =@Vds;ce Þ1 becomes infinite. Consequently, the corresponding reflection coefficient is always unitary (return loss equal to 0 dB), independently of any finite load resistance. In real cases the situation is less critical, but the small-signal return loss obtained from a maximum power design is always poor, as Section 11.5.2 points out.

11.5.2

Output power optimization in presence of transistor parasitics

The considerations based on hypotheses 1 to 10 give some general ideas about PA design. Unfortunately, no practical case satisfies those assumptions. In particular, transistors present parasitics associated with their output and an input junction. The transistor output parasitic reactances introduce a phase shift between the output voltage and current, while the input junction makes the transistor working point change with input power. We will describe both these effects by means of a trial amplifier operating at 1 GHz, and based on the transistor27 MBC13900. Figure 11.13 shows the schematic of a simple amplifier operating at 1 GHz and designed with the method described in Section 11.5.1. Thus, a 50 O RF generator directly connects to the base through the 27

See Section 9.7.

AMPLIFIERS +

Vq = 5 V

L dd = 1µH Iq = 5 mA

Cgg = 1 nF

vLOAD

Cdd = 1 nF

Rs = 50 Ω VRF

445

RLOAD= Vq / I q = 1 kΩ

Ib0 = 33 µA

(a)

Vq = 10 V

Rdd= 1 kΩ

+

Ldd= 1 µH vLOAD

Cgg = 1 nF

Cdd = 1 nF

Rs = 50 Ω VRF

RLOAD= Vq / Iq = 1 kΩ

Ib0 = 33 µA

(b)

Rdd= 1 kΩ Rgg2 = 4.92 kΩ

L dd= 1 µH

Vq = 10 V +

Iq = 5 mA

Rgg1 = 1 kΩ L gg = 1 µH Rs = 50 Ω

vLOAD

Cdd = 1 nF

RLOAD = Vq / Iq = 1 kΩ

Cgg = 1 nF

VRF Collector current servo controller LPF1

(c)

i

Iq = 5 mA +

Cdd = 1 nF Cgg = 1 nF

G•vin

+ Vref = H•Iq vLOAD

RLOAD= Vq / Iq = 1 kΩ

Rs = 50 Ω VRF

H•i

Ldd = 1 µH

Vq = 5 V

+ vin -

+

Ib0

(d)

Figure 11.13 PAs designed with the method of Section 11.5.1: (a) with open-loop bias; (b) with collector DC power limitation; (c) as (b) but with self-stabilizing collector current; (d) with automatic collector current control.

446


DC blocking capacitor Cgg ¼ 1 nF, which presents a negligible reactance of less than 0.16 O at 1 GHz. The absence of an input matching network has negligible effect on the output power. Its only effect is that a relevant portion of the input power is reflected back to the generator, with a consequent gain reduction. On the other hand, we are assuming that the input power is sufficient to drive the transistor at the desired level. The only input bias element is the current generator Ib0 ¼ 33 mA, which determines the quiescent current Iq ¼ 5 mA while behaving like an open circuit at 1 GHz. The DC collector voltage is Vq ¼ 5 V. From the considerations developed in Section 11.5.1, it follows that the maximum output load is RLOAD ¼ Vq =Iq ¼ 1 kO. The output matching28 is needed to transform 50 O into 1 kO The simplest possibility is the l/4 transmission line transformer, discussed in Section 6.3. Unfortunately, the required pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi characteristic impedance is Z0 ¼ 1000 50 ffi 223:6 O, which is too high to be realized in most technologies.29 More practical solutions30 could be: 1. The lumped network in Figure 6.15a ðCa ¼ 0:73 pF; La ¼ 36:513 nHÞ. 2. The lumped network in Figure 6.15b ðCb ¼ 0:694 pF; Lb ¼ 34:683 nHÞ. 3. The distributed network in Figure 6.11ðZ 01 ¼ Z 02 ¼ 50 O; y1 ¼ 77:4 ; y2 ¼ 13:24 Þ. Port 1 of networks 1 and 2 connects with the 50 O load, while port 2 is connected to the transistor collector through the capacitor Cdd ¼ 1 nF; the opposite connection applies to network 3. Furthermore, networks 1 and 3 present high attenuation at low frequency, compensating the natural roll-off of the transistor gain, as discussed in Section 11.2. On the other hand, we are considering PAs, which tend to operate in the nonlinear regime: then, the low-pass nature of network 2 could be useful for filtering out the harmonics. In any case, the resistor RLOAD in Figure 11.13 schematically represents the 50 O load with its matching network. We will consider neither the response of the latter nor its filtering effect on the output signal. For the output bias network, the inductor Ldd ¼ 10 mH (capacitor Cdd ¼ 1 nF) ensures high (low) reactance at 1 GHz, enough to approximate RF open (short) circuit. Figure 11.14 (dashed curves) shows the computed output waveforms31 of the amplifier in Figure 11.13, with three different available input powers: 36 dBm (linear operation), 10 dBm (moderate compression) and 4 dBm (saturation). Note that the output waveform increasingly differs from a sinusoid at high amplitudes. The output voltage vLOAD across the resistor RLOAD differs from the collector–emitter voltage by the voltage across Cdd , which is almost constant and equal to Vq . At the maximum input power vLOAD presents a more pronounced saturation in the lower than the upper waveform. The positive peak exceeds Vq , in qualitative agreement with the assumptions of Section 11.5.1. They imply that the collector–emitter voltage is non-negative and can reach any positive amplitude due to the extra voltage produced by Ldd . In real cases the minimum collector voltage is typically less than 1 V, and the maximum is limited by the collector–emitter breakdown. The waveforms in Figure 11.14 correspond to the load curves of Figure 11.15 (solid black lines). Figure 11.15 also includes the transistor DC curves, the ideal load line32 (11.42) and two hyperbolas, 28

See the Ansoft file 03_Matching_Networks_1000_to_50_Ohm.adsn. Figures 3.32 and 3.36 show a maximum realizable characteristic impedance about 200 O for stripline and microstrip, respectively. For the coaxial cable, a diameter ratio follows from Equation (3.208) as pffiffiffiffi pffiffiffiffi b=a ¼ expð2pZ0 m0:5 e0:5 Þ ¼ expð223:6=60Þexpð er Þ ffi 41:5 expð er Þ 29

In the most favourable case of er ¼ 1, and assuming b ¼ 10 mm, we have a ¼ 0.24 mm, which is very critical to realize. 30 See the Ansoft file 03_Matching_Networks_1000_to_50_Ohm.adsn. 31 See the SIMetrix file 15_MBC13900_LOAD_ Line_1GHz.sxsch. 32 See the SIMetrix file 14_MBC13900_LOAD_Line.sxsch, which performs a low-frequency (1 MHz) analysis of the amplifier, in order to eliminate the effects of the transistor parasitic reactance.

AMPLIFIERS

447

0.4 0.2

PAV,in = -36 dBm

0.0 -0.2 -0.4 0.0

0.5

1.0

1.5

2.0

Load voltage, V

8 4

PAV,in= -10 dBm

0 -4 -8 0.0

0.5

1.0

1.5

2.0

8 4

PAV,in= -4 dBm

0 -4 -8 0.0

0.5

1.0

1.5

2.0

Time, ns

Figure 11.14 Output voltage of the amplifiers in Figures 11.13a (dashed) and Figure 11.13c (solid), for three different available input powers.

Ptransistor = 50 mW

15

100 mW Ibe = 75 µA

-4 dBm 60 µA 10 45 µA Ice , mA 35.3 µA

small signal load line 5

30 µA

PAV,in = -36 dBm 15 µA -10 dBm 0 0

5

10 V ce , V

Figure 11.15

ideal load line 15

Load lines of the amplifiers in Figures 11.13a (black) and Figure 11.13c (grey).

448


Vce Ice ¼ Ptransistor , where Ptransistor is the power dissipated by the transistor, equal to 50 and 100 mW. Each point on a given hyperbola represents the corresponding transistor power dissipation. Beginning from the lower input power, we can see that the load curves are similar to ellipses rather than straight lines, due to memory effects in the transistor output reactance that introduce phase shifts between the collector voltage and current. As the driving level increases, the load lines become more distorted and move away from the ideal line, moving towards higher transistor dissipation. Consequently, the direct current of the amplifier increases at high driving levels. In general cases, on amplifiers operating in the compression region, the direct current changes, decreasing or increasing. It depends on the input driving power because of the base–emitter (gate–source) junction33 in the BJT (MESFET), which rectifies the input RF signal. The rectification produces direct currents (voltages) that are superimposed on the ones impressed by the input bias network, and modifying the amplifier direct current. The transistor working area moving towards higher power could be dangerous for the transistor, especially for the BJT. Indeed, BJT collector current for a given base current increases with transistor temperature. Therefore, a high input RF level could spark off a destructive process: the current increases for the high driving, the transistor temperature increases consequently, by causing a further current rise, and so on. Figure 11.13b shows a simple solution for this problem: the DC generator Vq0 ¼ 10 V in series with the resistor Rdd ¼ 1 kO replaces the DC generator Vq ¼ 5 V. At low input RF power, when the direct current is 5 mA, the voltage drop on Rdd is 5 V, thus the voltage applied to the collector is 5 V, as in the amplifier in Figure 11.13a. Therefore, at low RF power the two amplifiers in Figures 11.13a,b are totally equivalent. Now, the combination of Vq0 in series with Rdd is a generator having an available power of PAV;DC ¼ 0:25 V q0 2 =Rdd ¼ 25 mW. That maximum power is transferred to the load (the amplifier) when its equivalent load resistance is equal to Rdd . This condition verifies when the collector voltage and current are 5 V and 5 mA, respectively. In other words, the transistor absorbs a maximum power of 25 mW with no input signal; in all other conditions, the power dissipated on the transistor is equal or lower. This solution guarantees transistor safety, but at the expense of the overall efficiency of the amplifier, in that Rdd dissipates about34 the same power as the amplifier. The efficiency of the amplifier in Figure 11.13b is half the one in Figure 11.13a. The amplifier in Figure 11.13c presents an improved base bias scheme with respect to the arrangement in Figure 11.13b. The two resistors Rbb1 ; Rbb2 form a voltage divider that increases the base current monotonically with the collector voltage, which decreases with the collector current itself. This way, the bias scheme realizes a kind of automatic stabilization of the collector current. Clearly, the inductor Lbb isolates the voltage divider from the base at RF, preventing the latter from loading the transistor input. Figure 11.13d proposes a more sophisticated approach to transistor biasing. The controlled generators CCVS1 and VCCS1 , together with the low-pass filter LPF1, form a servo-control loop with Q1 . The output voltage of CCVS1 is proportional to the amplifier current, LPF1 eliminates the alternate components of that voltage, and the error amplifier VCCS1 drives the base of Q1 with a current proportional to the difference between its two input voltages. If the loop is stable, these two voltages are ðAmpÞ close, then H IDC ffi Vre f , and consequently ðAmpÞ

IDC

ffi

Vre f H

ð11:48Þ

were H is the gain of CCVS1 and Vre f is a suitable DC voltage. Equation (11.48) states that the direct current of the amplifier in Figure 11.13c depends only on two parameters of the automatic control circuit, not on the RF power. In other words, the servo control compensates the self-biasing of the transistor input junction. 33

This is in direct opposition with assumption 10 in Section 11.3.1. The power dissipated on Rdd is exactly the same as on the amplifier in the case of no input signal, when Iq ¼ 5 mA. As can be seen, high RF input levels affect Iq and thus the percentage power on Rdd . 34

15

20

10

19

5

18

0

17

-5 -10 -15 -20 -40

16

(a) (b) (c) (d)

Output power

449

Gain, dB

Output power, dBm

AMPLIFIERS

15

Gain

14 13

-35

-30

-25

-20

-15

-10

-5

0

Input power, dBm

Figure 11.16 Compression curves of the amplifiers in Figure 11.13a (thick solid), Figure 11.13b (dashed) and Figure 11.13c (thin solid).

The configuration in Figure 11.13d applies also to FET amplifiers; in this case a VCVS replaces VCCS1 . The grey lines in Figure 11.15 are the load lines of the amplifier in Figure 11.13d, computed35 for the same frequency (1 GHz) and input power (36, 10 and 4 dBm) as for the amplifier in Figure 11.14a. Note that all the load lines of the amplifier in Figure 11.13d remain more or less concentric and closer to the ideal one, different from the case of the amplifier in Figure 11.13a. Figure 11.14 shows the output waveforms of the amplifiers in Figures 11.13a (dashed lines) and Figure 11.13d (solid lines) at the same three input power levels as for the load lines. Figure 11.14 does not show the output waveforms of the amplifier in Figures 11.13b,c, which are close36 to the ones of the amplifier in Figure 11.13d. Figure 11.16 shows the compression curves37 of the amplifiers in Figure 11.13a (solid thick), Figure 11.13b (dashed), Figure 11.13c (solid thin) and Figure 11.13d (solid thick with square). The three black curves are the fundamental output power versus the available power from the input generator, while the grey lines are the differences between these quantities, namely the transducer power gains of the amplifiers. The small-signal performances of the three amplifiers are coincident, say up to an input power of 15 dBm. At higher power the curves diverge. Furthermore, the amplifier in Figure 11.13b presents an oversaturation characteristic: the fundamental output power decreases – instead of increasing – as the input power exceeds 6 dBm. This behaviour makes the input driving of the input amplifier quite critical, as it almost prevents the saturated output power from being achieved. However, by modifying the values of the amplifier in Figure 11.13b from Vq0 ¼ 10 V; Rdd ¼ 1 kO to Vq0 ¼ 5 V; Rdd ¼ 0 the amplifier gradually transforms into the one in Figure 11.13a; intermediate combinations of Vq0 ; Rdd give the corresponding intermediate compression curves. The solution in Figure 11.13c eliminates the oversaturation from the output power compression curve, but does not alleviate the efficiency reduction related with the collector series resistance. 35

See the SIMetrix file 17_MBC13900_LOAD_Line_1GHz_DC_Servo.sxsch. See the SIMetrix file 16_MBC13900_LOAD_Line_1GHz_RL.sxsch. 37 See the SIMetrix files 18_MBC13900_1GHz_PS.sxsch, 19_MBC13900_1GHz_RL_PS.sxsch, 20_MBC13900_ 1GHz_RL2_PS.sxsch and 21_MBC13900_1GHz_DC_Servo_PS.sxsch, which analyze the compression curves of the amplifiers from Figures 11.13a–d. 36

450


The amplifier in Figure 11.13a has the highest output power ðPout;saturated;a ffi 12:3 dBmÞ among the ones in Figure 11.13. On the other hand, the amplifier in Figure 11.13d is the one working closer to the conditions of Section 11.5.1, its saturated output power being Pout;saturated;c ffi 9:7 dBm, while the value predicted from the simplified model of Section 11.5.1 is Pout;max;c ¼ 10 log10 ðVq Iq =2Þ þ 30 ¼ 10 log10 ð5 53 =2Þ þ 30 ffi 11 dBm. The 1.3 dB missed output power is due to the transistor parasitic reactance and to the minimum collector–emitter voltage, which is about 0.42 V, as in the abscissa of the highest power load (grey) line in Figure 11.15. The transistor parasitic reactance makes the 1 kO resistor not the optimum one, while the minimum collector–emitter voltage reduces the output voltage swing by twice its value. Considering this, the expected output power is P0out;max;c ¼ 10 log10 ½0:5 ðVq 0:42ÞIq þ 3 ffi 10:6 dBm. The high power of the amplifier in Figure 11.13a – higher than the limit (11.43) – is achieved at the expense of higher collector current, and the risk of transistor damage is also present. The configuration in Figure 11.13d is likely to be the best compromise among output power, current consumption and transistor safe operation. However, besides circuit complexity, the main drawback of that solution is its settling time. A detailed description of the automatic current control is beyond the scope of this text, but we will present some qualitative considerations of the dynamic performance of the system. The schematic in Figure 11.13d is purely conceptual, since the real controlled generators typically use operational amplifier circuits, present offset, drift over time and temperature, their frequency responses are low-pass type and therefore interact with the response of LPF1, and so on. The gain G of VCCS1, together with the cut-off frequency and response of LPF1, are the design parameters of the automatic control, and determine value, precision, stability and settling time of the amplifier current. The current control loop analysis is a finite bandwidth feedback system, and thus it presents a finite settling time, typically ranging from hundreds of nanoseconds to milliseconds. Therefore, the amplifier is not immediately operating after switching on the power supply. Moreover – and probably more important – if the RF input signal amplitude experiences a sudden change, a transient irregularity will affect the output until the loop sets, particularly when passing from linear to nonlinear operation or vice versa. Figure 11.17 shows the envelopes of input and output waveforms of the amplifier in Figure 11.13d. Here, the term envelope denotes the curve passing through the local maxima and minima of an oscillating waveform. In more quantitative terms, let us consider the most general RF waveform, having both amplitude and phase modulation vRF ðtÞ ¼ aðtÞcos½o0 t þ yðtÞ where aðtÞ; yðtÞ present slower fluctuations with respect to the RF oscillation period T0 ¼ 1= f0 ¼ 2p=o0 , or, equivalently, their spectral components have a much narrower bandwidth than f0 . It is always possible to write vRF ðtÞ in such a way that the amplitude coefficient aðtÞ is non-negative jaðtÞjaðtÞ vRF ðtÞ ¼ jaðtÞjcos o0 t þ yðtÞ þ p 2jaðtÞj This expression points out that the modulated RF signal consists of oscillations having a time-varying amplitude jaðtÞj, which is the envelope of the signal. Figure 11.17 shows the envelopes of the RF signals – instead of the signals themselves – because the RF signals are not distinguishable, having 500 oscillations during the time span of 500 ns. The parameters38 of the control loop are H ¼ 103 ; G ¼ 100 mS, and LPF1 is a first-order low-pass filter with a cut-off frequency of about 16 kHz. The output signal envelope in Figure 11.17 needs about 400 ns to reach its steady state shape; before that time, undesired amplitude modulation is present. Therefore, the amplifier in Figure 11.13d is not suitable for RF signals having fast amplitude modulating signals. 38

See the SIMetrix file 22_MBC13900_1GHz_DC_Servo_Burst.sxsch.

AMPLIFIERS

451

Figure 11.17 Response of the amplifier in Figure 11.13c to a stepped increase in RF input amplitude: input and output envelope.

11.5.3 Load pull Sections 11.5.1 and 11.5.2 considered the theoretical maximum output power in a simplified amplifier and some deviations from ideality, respectively. The present section completes the discussion, describing a general method to find the optimum load that gets the maximum power from a given transistor, at specified frequency and bias conditions. This method is known as load pulling, and can be applied not only to maximize the output power, but also to other nonlinear performances, like linearity, efficiency, harmonic content, and so on. Furthermore, the load pull can be based on either measurements or simulations. Figure 11.18 shows the experimental load-pull setup. Two DC generators and T-bias determine the required transistor DC working point, with negligible RF effects. Again, the bias networks are generic and the transistor can be a bipolar or field effect device, although a MESFET symbol is used. One RF generator drives the transistor input through the RF path of the input T-bias. In some cases, an input matching network is placed between the RF generator and transistor input. Unilateral design is only one possible for such a network, in that the output load of the transistor is variable by definition, as we will see shortly. In any case, the input RF generator – matched or not – has enough power to drive the transistor. The transistor output is connected to a variable load through the RF path of the output T-bias. The variable load consists of one stub (or double stub) tuner, one SPDTand one directional coupler. The SPDT and directional coupler present a negligible attenuation. A matched power meter terminates the stub tuner, making the resulting impedance not purely reactive, while measuring the transistor output power. A spectrum analyzer,39 connected to the coupled port of the directional coupler, picks up a small fraction (of the order of 10 to 30 dB) of the transistor output signal. It also checks for possible transistor oscillations. A vector network analyzer40 is also connected to the SPDT, and measures the impedance seen by the transistor. One ammeter, measuring the drain current, completes the setup. 39 40

See Section 17.4. See Section 11.6.2.

452


Spectrum analyzer

Cgg

RF generator

L gg

SPDT 1

C dd A 1

DUT

L dd 2

Rgg

Rdd

A3 Directional coupler

ΓL Stub tuner

Power meter

+

Vgg input bias

Ammeter

+

Vdd

A2

output bias

Figure 11.18

Network analyzer

Test setup for the transistor load pull.

The load-pull measurements includes two steps: 1. With the SPDT in position 1, the RF input power and the stub tuner are adjusted to maximize the power measured by the power meter. Once a maximum has been found, the output spectrum must contain the input frequency and its harmonics. Any different spectral component reveals the presence of oscillations. If this is case, the load found must be rejected, because the oscillations corrupt the useful signal and give an optimistic measured power. Once we find an oscillation-free optimum load, we lock the tuner and record output power, harmonic levels and transistor current. After that, the drain DC generator is switched off ðVdd ¼ 0Þ, before passing to the next step. 2. With the SPDT in position 2 and the output transistor unbiased, the network analyzer measures the complex reflection coefficient presented to the transistor output as in the previous step. This requires the two paths A1–A3 and A2–A3 to be either equal or accurately calibrated out. As in step 1, we record the measured reflection coefficient at the test frequency. The generation of theoretically arbitrary reflection coefficients requires the SPST to present a low insertion loss, otherwise the impedance that the transistor sees is independent of the stub tuner regulations. Therefore electromechanical devices are the preferred choice. The result of this test procedure is the optimum load impedance for the maximum output power at a given frequency. Alternatively, the same procedure can be applied to optimize the efficiency, the linearity or any other nonlinear performance, including combinations. Once the optimum load is known, the matching network has to transform the 50 O into that load. Sometimes, the load-pull procedure includes additional steps, in order to determine the transistor nonlinear performances for a load impedance that is slightly different from the optimum. The result is a nonlinear performance as a tabular function of the load impedance or reflection coefficient, which are complex variables. Contour plots are the most used representation for this two-variable function. Depending on the nonlinear performance of interest, we have plots of output power, efficiency, 1dBCP, and so on. In modern and sophisticated load-pull test sets, the stub tuner includes motorized computerdriven controls. A computer program oversees the whole measurement procedure, generating output load and contour plots. Figure 11.19 shows the load-pull contour plot of the amplifier in Figure 11.13a; the normalization impedance for that Smith chart is Rnorm ¼ RLOAD;opt ¼ Vcc =Iq ¼ 1 kO, the approximated optimum load given by the method of Section 11.5.1. The marked points on each line represent constant load reflection

AMPLIFIERS

Figure 11.19

453

Load-pull contours of the amplifier in Figure 11.13a.

coefficients giving the same output power. Clearly, the load-pull measurements are finite in number, and consequently the contour lines are piecewise linear. A Smith chart region densely populated with contour lines denotes a high sensitivity of the nonlinear performance to the output load. The effective optimum load impedance in Figure 11.19 is ZLOAD;opt ¼ ð0:472 þ j 0:198ÞRnorm ¼ ð472 þ j 198Þ O, while ZLOAD ¼ Rnorm ¼ 1 kO corresponds to an output power approximately 1.5 dB lower than the maximum. The reactive part of ZLOAD;opt somehow compensates for the transistor parasitic reactance, and the real part is lower than Vcc =Iq ¼ 1 kO, due to the collector mean current increasing at high output power, as pointed out in Section 11.5.2. Note also that the optimum load in Figure 11.19 lies on the inductive half plane of the Smith chart, in that it compensates for the transistor output capacitance. The load-pull method is quite general: for instance, it could also be used to find the load for the maximum linear gain. In this case, the contour lines are the constant output gain circles. A more recent design method is based on nonlinear simulations with suitable transistor nonlinear models. Some methods reproduce the test bench in Figure 11.18 on the simulator, finding the optimum load and sometimes the load contours. After that the output matching network design continues exactly as with the experimental load-pull design method. Such a method is sometimes referred to as simulated load pulling. In other cases, the designer describes the whole amplifier in the nonlinear simulator, including the input matching network, transistor nonlinear model and output matching network. The parameters of the last element are then tuned and/or optimized until the optimum is achieved. Both experimental and computer simulation design methods present advantages and disadvantages. The experimental method is usually more accurate and less flexible, in that the obtained optimum load refers to a well-defined performance, which is not well related to the other ones: for instance, the optimum

454


load for power could be quite different from the one for the efficiency. Moreover, the amplifier nonlinear performances do not exclusively depend on the fundamental frequency load. Rather, the harmonic load is also influential, and this is difficult to control independently from the one at the fundamental frequency. However, some electronic computer-controlled variable loads have the capability to generate arbitrary loads at different harmonics, though they are of course quite expensive. On the contrary, simulation is more flexible, in that the same nonlinear model can be used to optimize any nonlinear performance. Moreover, the complete amplifier analysis automatically takes the harmonic load into account. The main disadvantage of the simulation is its lower precision inherent in the nonlinear simulation itself; sometimes it does not even converge, particularly in high-compression conditions. As a final observation on the nonlinear design of the output matching network, we have to note that the optimum output power load does not coincide with the conjugate load considered in Section 11.2.3. In this regard, let us again consider the transistor DC curves in Figure 11.15. As we found, the load resistance from case (c) is RLOAD ¼ 1 kO. On the linear side, linearizing the DC curve in the working point Vq ¼ 5 V; Iq ¼ 5 mA, we get the small-signal output resistance of the transistor Rce ffi DVce =DIc ffi 6:57 kO, as Figure 11.15 shows. The corresponding return loss is 20 log10 ½jðRLOAD Rce Þ= ðRLOAD þ Rce Þj ffi 2:66 dB, better than the one obtained from the simple model of Section 11.2.3, but not good enough. In more realistic analyses and in practical cases, the small-signal output return loss of a power amplifier typically ranges from 4 to 8 dB.

11.5.4

Balanced amplifiers

Figure 11.20a shows an amplifier configuration known as a balanced amplifier. It consists of two ideally identical amplifiers placed between two hybrid couplers. Compared with single amplifiers, the balanced amplifier ideally exhibits the same gain and double the power: the real resulting gain and output power are lower than the ideal case, due to the losses in the hybrid couplers. Typical hybrid couplers present a dissipation loss in the range of 0.1–0.5 dB. The resulting balanced amplifier presents a gain of 0.2–1 dB lower and an output power of 2.9–2.5 dB higher than its two single amplifiers. Together with increasing output power, balanced amplifiers could have additional advantages depending on the hybrid coupler type, like better impedance matching, higher stability and reduction of some nonlinear distortion products. The networks in Figure 11.20a labelled HYB1 and HYB2 are hybrid couplers, and their fourth port is terminated with the matched impedance. Their scattering matrix is 2 3 0 1 e jj h i h i ð11:49Þ SðHYB1Þ ¼ SðHYB2Þ ¼ a 4 1 0 0 5 e jj 0 0

The phase f may assume the values pffiffiffi 0; p=2 or p for a 0 , 90 or 180 hybrid, respectively. In the ideal case of a loss-free coupler, a ¼ 1= 2. The terminated hybrids can be more synthetically represented with the structures in Figure 11.20b, which shows the same S parameters as in Equation (11.49). The two amplifiers A1 and A2 are identical in the ideal case, but not in practice. The superscript in ðA1Þ brackets denotes the S parameters of the component itself: for instance, s21 will denote the forward transmission coefficient of amplifier A1. Figure 11.20c shows the SFG corresponding to the network in Figure 11.20b. The scattering parameters of the balanced amplifier can be found by analyzing that graph. The input reflection coefficient is, by definition, the ratio between the input reflected and incident waves, with no incident wave on the output. Under these conditions, there are two paths, of three branches each, from a1 to b1 : ðA1Þ ðA2Þ they are, respectively, a; s11 ; a and a e jj ; s11 ; a e jj . The input reflection coefficient of the balanced amplifier is then h i b1 ðA1Þ ðA2Þ ðA1Þ ðA2Þ ¼ a s11 a þ a e jj s11 a e jj ¼ 2a2 s11 þ e j2j s11 s11 ¼ a 1 a2 ¼0

AMPLIFIERS 1

2

0

1

0

π−φ

455

2 π−φ

φ

φ

0

0

4

3

4

1

√0.5 2

3

3

(a)

√0.5⋅exp(jφ)

√0.5⋅ exp(jφ) √0.5 2

3

1

(b)

(A1)

s21

0.50.5

a1

s11(A1) s22(A1) 0.5

0.5

exp(jφ)

b1

0.5 0.5

0.5

s

Figure 11.20

0.5

0.5

0.5

exp(jφ)

s11(A2) s22(A2)

(c)

exp(jφ)

(A1) 12

s21(A2) 0.5

0.5

s12

(A2)

0.50.5 exp(jφ)

0.50.5

Balanced amplifier: (a) block diagram; (b) equivalent diagram; (c) SFG.

The remaining three scattering parameters can be found with the same procedure, resulting in h i 3 2 ðA1Þ ðA2Þ ðA1Þ ðA2Þ s12 þ s12 e jj s11 þ e j2j s11 h i 7 6 ð11:50Þ SðBALAMPÞ ¼ a2 4 h 5 i ðA1Þ ðA2Þ ðA1Þ jj j2j ðA2Þ s21 þ s21 e s22 þ e s22 ðA1Þ

ðA2Þ

ðA1;A2Þ

If the two single amplifiers are identical, i.e. if shk ¼ shk ¼ shk for h; k ¼ 1; 2, then Equation (11.50) simplifies to 2 3 ðA1;A2Þ jj j2j ðA1;A2Þ h i 1 þ e ½ s 2s e 11 12 5 ð11:51Þ SðBALAMPÞ ¼ a2 4 ðA1;A2Þ jj ðA1;A2Þ 2s21 e ½1 þ e j2j s22 The forward gain amplitude of the balanced amplifier, as coming from Equation (11.51), is then ðBALAMPÞ s ¼ 2a2 sðA1;A2Þ : each decibel of dissipation loss in the hybrid couplers affects the balanced 21 21 amplifier gain by 2 dB.

456


Note that we assume the two couplers are identical and the two amplifiers are potentially different. This is equivalent to assuming that the two amplifiers are identical and using two different scattering matrices for the two couplers, in order to model any imbalance between the two paths of the balanced amplifier. pffiffiffi If the two hybrid couplers are loss free, a ¼ 1= 2, then Equation (11.51) further simplifies to 3 2 1 þ e j2j ðA1;A2Þ ðA1;A2Þ e jj s12 s11 h i 6 7 2 7 ð11:52Þ SðBALAMPÞ ¼ 6 5 4 j2j 1 þ e ðA1;A2Þ jj ðA1;A2Þ e s21 s22 2 Equation (11.52) states that the full–ideal balanced amplifier has the same amplitude gain as that of its two single amplifiers. Let us apply the models in Section 9.5 to analyze the nonlinear performance of the ideal balanced amplifier. Following the method described there, we will assume that A1 and A2 are perfectly matched and unilateral. Under these assumptions, the ratio of output to input voltage of the amplifiers coincides with their transmission coefficient. Let the balanced amplifier input voltage be vin ðtÞ ¼ Vi0 cosðot jÞ ðA1;A2Þ The constant j ¼ arg s21 has been added in order to simplify the expressions for output signals, and the origin of the time axis changed accordingly. HYB pffiffiffi 1 distributes the input voltage to the amplifier inputs. The input voltage amplitude equals Vi0 = 2 for both amplifiers, due to the power division factor of the hybrid coupler. Furthermore, the input signal of A2 is phase shifted by f. Then the input voltages of the amplifiers A1 ; A2 are respectively 1 ðA1Þ vin ðtÞ ¼ pffiffiffi Vi0 cosðot fÞ; 2

1 ðA2Þ vin ðtÞ ¼ pffiffiffi Vi0 cosðotf þ jÞ 2 ðA1;A2Þ

Let us assume that A1 ; A2 have the voltage transfer characteristic (9.44), with s21 ¼ c1 e jj . If the input voltage is low enough to keep the amplifiers within their monotonic region,41 then the two amplifier output voltages are 2 3 Vi0 Vi0 Vi0 ðA1Þ vout ðtÞ ¼ c1 pffiffiffi cosðotÞ þ c2 pffiffiffi cos2 ðotÞ þ c3 pffiffiffi cos3 ðotÞ 2 2 2 2 3 Vi0 Vi0 Vi0 ðA2Þ vout ðtÞ ¼ c1 pffiffiffi cosðot þ fÞ þ c2 pffiffiffi cos2 ðot þ fÞ þ c3 pffiffiffi cos3 ðot þ fÞ 2 2 2 Expanding the cosine powers, and eliminating the constant terms from the quadratic coefficient, we get " 3 #

3 Vi0 3 Vi0 c2 Vi0 2 1 V ðA1Þ pffiffiffi cosð2otÞ þ c3 pi0ffiffiffi cosð3otÞ cosðotÞ þ vout ðtÞ ¼ c1 pffiffiffi þ c3 pffiffiffi 4 2 2 4 2 2 2 " 3 # 2 3 Vi0 3 Vi0 c2 Vi0 1 Vi0 ðA2Þ p ffiffi ffi p ffiffi ffi p ffiffi ffi þ c3 cosðot þ fÞ þ vout ðtÞ ¼ c1 cosð2ot þ 2fÞ þ c3 pffiffiffi cosð3ot þ 3fÞ 2 4 2 4 2 2 2 The pffiffiffioutput hybrid combines the two signals at the output of A1 ; A2 by multiplying their amplitudes by 1= 2 and phase shifting the components of the first by f. The resulting output voltage of the balanced amplifier is then 41

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From Equation (9.48) it follows that this condition occurs if Vi0 = 2 c3 =3c1 , see also Figure 9.22.

AMPLIFIERS ðBALAMPÞ

vout

3 c2 ðtÞ ¼ c1 þ c3 Vi02 Vi0 cosðot þ jÞ þ pffiffiffi Vi02 ½cosð2ot þ jÞ þ cosð2ot þ 2jÞ 8 4 2 1 3 þ c3 Vi0 ½cosð3ot þ jÞ þ cosð3ot þ 3jÞ 16

457

ð11:53Þ

The voltage large-signal gain of the balanced amplifier is the coefficient of the fundamental frequency component of the output signal c1 þ ð3=8Þc3 Vi02 ; this has the same form as Equation (9.50), but with the coefficient c3 =2 replacing c3 . Consequently, the resulting 1 dB compression point of the balanced amplifier is " rffiffiffiffiffiffiffiffiffi 2 # 1 c1 1 ðBALAMPÞ þ 30 ð11:54Þ I1dBdBm ¼ 10 log10 0:538 c3 2 R0 while the 1 dB compression power of the single amplifiers is " rffiffiffiffiffiffiffiffiffi 2 # 1 c1 1 ðA1;A2Þ I1dBdBm ¼ 10 log10 þ 30 0:3808 c3 2 R0

ð11:55Þ

The I1dB point of the balanced amplifier is 3.01 dB higher than those of its two amplifiers, and the same applies to the output 1 dB compression point, saturated power, intercept points, and so on. After deriving the general properties of the balanced amplifier, we can consider the three relevant cases used, classified according to the particular phase f of the hybrid coupler: 1. Hybrid coupler in phase or 0 balanced amplifier, f ¼ 0. Equations (11.51) and (11.52) give the scattering matrix of the in-phase balanced amplifier 2 3 3 pffiffiffiffiffi 2 ðA1;A2Þ ðA1;A2Þ a¼1= 0:5 ðA1;A2Þ ðA1;A2Þ h if¼0 s s s s 11 12 11 12 5 ¼ 4 5 SðBALAMPÞ ¼ 2a2 4 ð11:56Þ ðA1;A2Þ ðA1;A2Þ ðA1;A2Þ ðA1;A2Þ s21 s22 s21 s22 In the ideal case, the balanced amplifier exhibits the same scattering parameters as its two single amplifiers. For the nonlinear performances, Equation (11.53) implies that the output signal has single tone excitation

3 c2 V 2 c3 Vi03 ðBALAMPÞ ðtÞ ¼ c1 þ c3 Vi02 Vi0 cosðotÞ þ pi0ffiffiffi cosð2otÞ þ vout cosð3otÞ ð11:57Þ 8 2 2 4 2 Compared with the single amplifiers, the 0 balanced amplifier has the second and third harmonics attenuated by 3.01 and 6.02 dB, respectively, for the same input signal. More generally, considering a higher order polynomial for the voltage transfer, we have that the kth harmonic is attenuated by the factor 0:5k1. The main advantage of the 0 balanced amplifier is that its hybrid couplers can easily be realized in printed circuit technology and with multi-octave working bandwidths, for instance with Wilkinson dividers.42 2. Hybrid coupler in quadrature, j ¼ p=2, or 90 balanced amplifier. The scattering matrix is 2 3 3 pffiffiffiffiffi 2 ðA1;A2Þ a¼1= 0:5 ðA1;A2Þ 0 js12 0 s12 f¼0 2 5 ¼ 4 5 ð11:58Þ SBalancedAmpli fier ¼ 2a 4 ðA1;A2Þ ðA1;A2Þ s21 0 js21 0 42

See Section 7.7.1.

458

MICROWAVE AND RF ENGINEERING Equation (11.58) shows the main advantage of the 90 balanced amplifier: its input and output reflection coefficients are zero, independently of the ones of A1 ; A2 , provided that the latter are equal. This remarkable property of the 90 balanced amplifier has many applications, in LNAs, broad-band amplifiers and PAs, with the output matched for the maximum power. All these three types of amplifiers present mediocre input or output impedance matching: LNAs because Gopt 6¼ GMS ; broad-band amplifiers because their input is matched at the highest frequency only, to compensate the transistor gain roll-off with frequency; and PAs because the load for maximum power is different from the one for impedance matching. In all three cases the 90 balanced configuration provides a perfect impedance matching. The main drawback of this solution with LNAs is that the input coupler degrades the noise figure by its loss. That degradation is of the order of 0.1–0.5 dB, as anticipated in the initial part of this section. The 90 balanced connection can be considered as a kind of lossy matching technique. The power reflected by the two single amplifiers dissipates on the hybrid couplers’ terminations. As a side effect, the balanced amplifier shows a better stability than the single amplifiers. The Kurokawa conditions (11.17) applied to the scattering parameters (11.58) give, in the ideal case, ðA1;A2Þ ðA1;A2Þ 1 ðA1;A2Þ ðA1;A2Þ s21 s21 s12 þ s12 ðA1;A2Þ ðA1;A2Þ K¼ > 1; jDj ¼ s12 s21 xmin Þ where xin ¼ vgs or xin ¼ ibe for the FET or BJT, respectively. The factor GL represents the linear gain of the transistor in the given representation and network embodiment. The input transistor excitation is a sinusoid with a superimposed DC bias xin ðtÞ ¼ xDC xpeax cosðo0 tÞ

ð11:63Þ

The corresponding output current has one of the shapes in the upper right corner of Figure 11.21. Each of those three curves – labelled A, B or C – represents a particular amplifier operation, known as class A, class B or class C. Here, we will analyze these operation classes, assuming for the transistor the same hypotheses as in Section 11.5.1, particularly hypothesis 6. 44

The Mathcad file 09_Class_A_B_C.MCD contains some analyses of the PA considered in this section.

460

MICROWAVE AND RF ENGINEERING (xin - x min) G L ids,ce

ids,ce

Isat

(iii)

(i) A Isat /2 C B

B

A

C xmin

t C

B

xin = vgs, vbe

C

B t

B

A

θc (ii)

xin

Figure 11.21 Class A, B and C amplifier diagrams: (i) typical output current transfer characteristic of a microwave transistor; (ii) input excitations; (iii) output current. Substituting Equation (11.63) into (11.62), we obtain ( 0 ðxin xmin Þ ids;ce ðtÞ ¼ ðxDC xmin ÞGL ðxpeak xmin ÞGL cosðo0 tÞ ðxin > xmin Þ

ð11:64Þ

In one period ð0 o0 t < 2pÞ of the RF signal, we can replace the conditions xin > xmin ; or xin xmin, = ½pyc =2; p þ yc =2, respectively, where the parameter yc is the with o0 t 2 ½pyc =2; p þ yc =2 or o0 t 2 conduction angle, given by

xmin xDC yc ¼ 2p 2cos1 ð11:65Þ xpeax The transistor output current minimum and maximum limits are respectively Imax ¼ ðxDC þ xpeax xmin ÞGL ;

Imin ¼ ðxDC xpeax xmin ÞGL

ð11:66Þ

Combining Equations (11.64) to (11.66), in order to eliminate the quantities xmin ; GL ; xDC ; xpeax ; Imin , we obtain

8 yc > >

cos þ cosðo0 tÞ > > yc yc 2 > >

; p þ I t 2 p o > 0 < max yc 2 2 cos 1 ids;ce ðtÞ ¼ ð11:67Þ 2 > > >

> > yc yc > > :0 = p ; p þ o0 t 2 2 2

AMPLIFIERS

461

I ds,ce(t) Imax Isat θsat

θc

0

π

2π

ω0 t Imin

Figure 11.22

Transistor output current in a class A, B or C amplifier.

The transistor output current follows the law (11.67) until the corresponding voltage reaches negative values along a portion of the RF cycle. This occurs when the load resistance exceeds a given value, and causes the positive peaks of ids;ce to clip at the value Isat , as we will see later in this section. The black curve in Figure 11.22 is a graphical representation of the waveform (11.67) when yc < 2p, the dashed curve representing the case of yc ¼ 2p. We have a waveform like the one shown by the grey solid line in Figure 11.22 when Imax > Isat , although we will not discuss this case in depth. Combining selected parts of the first and third curves, we obtain the remaining case ðyc ¼ 2p; Imax > Isat Þ. The conduction angle defines the operation class of the amplifier. Assuming no saturation on the positive current peak, i.e. Imax Isat , we have: .

Class A, yc ¼ 2p. This is the case for all the amplifiers considered in Sections 11.2 to 11.5.1. In other words, class A amplifiers work with the input signal falling entirely within the linear region of the transistor transfer characteristic. Equivalently, we can say that the conduction angle of class A amplifiers equals 2p. The amplifier discussed in Section 11.5.1 works with the input signal placed right in the middle of the transistor linear region, xDC ¼ ðxmax þ xmin Þ=2; xpeak ðxmax xmin Þ=2. When the latter inequality is satisfied with the equals sign, the class A amplifier exhibits its maximum efficiency and output power, as pointed out in Section 11.5.1.

.

Class B, yc ¼ p. The transistor output current is non-zero for exactly half a cycle of the RF signal. Class B is the limit case for an amplifier with no drain/collector current without an input RF signal.

.

Class AB, p < yc ¼ 2p. Pure class B operation is quite theoretical, due to the lack of transistor gain in the low-current region. Figure 11.21 shows a representation of that phenomenon: the grey curve in Figure 11.21(iii) is the response of the non-ideal transfer characteristic in Figure 11.21(i), grey curve, to the input excitation (B) of Figure 11.21(iii). For this reason, the input working point of class AB amplifiers is not exactly placed on the conduction knee ðxmin Þ, but shifts towards a slightly higher value. This improves the gain flatness in the low input amplitude range, but also generates a small quiescent current flow in the absence of the input signal. Thus, class AB amplifiers present a conduction angle p < yc < 2p, intermediate between classes B and A.

462


Vq

+

Output matching network

Isupply L dd

Cdd

IAC

I FUNDAMENTAL n:1

VAC

Ids , Ice Cgg

Vds , Vce Q1 ZLOAD

Vgs, Ib Lgg

ZS

C3

CN

L2

L3

LN

LN+1

T1

50Ω

T1

50Ω

2

RLOAD = n 50 Ω 2 f0

+ V RF

C2

3 f0

N f0

Vgg

(i)

Vq +

Isupply Ldd

L 2 I LOAD

C2

Ids , Ice Cgg

Vds , Vce Q1

Vgs , Ib

ZS

Lgg

VRF

Vgg

RLOAD

C1

+ (ii)

Vq

+

Output matching network

Isupply Ldd

Cdd

L3

Vds , Vce Q1

C3

IFUNDAMENTAL n:1

VAC

Ids , Ice Cgg

Vgs, Ib

ZS

Lgg

VRF

Vgg

CN

L2

LN

LN+1

RLOAD = n2 50 Ω +

(iii)

C2

2 f0

N f0

Figure 11.23 Schematics of PA principle: (i) class A, B or C; (ii) class E; (iii) class F.

.

Class C, yc < p: Small conduction angles minimize the amplifier’s efficiency at the expense of its linearity, as we will see shortly.

Class B and C amplifiers produce periodic non-sinusoidal output transistor current, differently from class A ones. For this reason, the output matching network of class B or C amplifiers not only presents a specific impedance to the transistor at the fundamental frequency, as usual, but also has additional requirements at harmonic frequencies. Figure 11.23(i) shows the principle of a class B or C amplifier. The output

AMPLIFIERS

463

matching network includes: .

One ideal transformer, to translate the 50 O load into any different required resistance.

.

A plurality (ideally an infinite number) of series LC cells each resonating at one harmonic frequency, in order to shunt all the current harmonics to ground.

.

The inductor LN þ 1, which parallel resonates with all the LC cells at the fundamental frequency, in order to cancel their reactance.

By removing all the series LC cells and the shunt inductor LN þ 1, the schematic in Figure 11.23 simplifies to that in Figure 11.11. Thus, the schematic in Figure 11.23 is a generalization of that in Figure 11.11, which is why the caption in Figure 11.23 describes a class A, B or C amplifier. From a more quantitative point of view, the transformer ratio n transforms the 50 O load into any required resistance RLOAD n¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RLOAD =50

ð11:68Þ

The inductances of the series LC cells relate to the respective capacitances as Lh Ch ¼ ðko0 Þ2 ;

h ¼ 2; 3; . . . ; N

ð11:69Þ

At the fundamental frequency all those cells are below their resonant frequency, thus they present a capacitive reactance, and the same thing applies to the resulting bipole. Consequently, the inductor LN1 can resonate out the reactance of the LC series cells. The admittance that the matching network presents to the transistor output is YLOAD ð joÞ ¼

N N X X 1 jCh o 1 1 ¼ þ þ ¼ ZLOAD ð joÞ h¼1 1Lh Ch o2 jLN þ 1 o RLOAD h¼1

jCh o 1 1 þ

2 þ jL R o N þ1 LOAD o 1 ho0

At the fundamental frequency, it must be equal to the reciprocal of the optimum resistance. From YLOAD ð jo0 Þ ¼ 1=RLOAD it follows that LN þ 1 ¼

N X Ch o20 1h2 h¼1

!1 ð11:70Þ

The transistor output waveforms of the amplifier in Figure 11.23 can be computed by the same considerations as used in Section 11.5.1. The amplifier in Figure 11.23 differs from that in Figure 11.11 because, in the first, the transistor output current is non-sinusoidal and the load impedance is frequency dependent. However, the transistor output current is periodic, with the same period as the input signal; thus, we can expand it in a Fourier series, obtaining ids;ce ðtÞ ¼

1 X

ik cosðko0 tÞ

k¼0

with

i0 ¼

o0 2p

2p o ð0

Ids;ce ðtÞ dt 0

and

ih ¼

o0 p

2p o ð0

Ids;ce ðtÞcosðko0 tÞ dt ðh ¼ 1; 2 . . .Þ 0

ð11:71Þ

464


Now, since the matching network shunts all the harmonics to ground, the first two terms of series (11.71) are of particular interest. The respective amplitude coefficients are

yc 2p p yc þ 2o o o 0 0 0 ð ð cos þ cosðo0 tÞ o0 o0 2

dt i0 ¼ Ids;ce ðtÞ dt ¼ Imax yc 2p 2p 1 cos 0 yc p 2 o0 2o0 p y yc p c yc o0 þ o0 þ ð 2o0 ð 2o0 cos o0 Imax 1 2

¼ dt þ cosðo0 tÞ dðo0 tÞ Imax yc yc 2p 2p cos 1 p yc 1 p yc cos 2 2 o0 2o0 o0 2o0 ð11:72Þ

yc

cos Imax I 1 yc yc 2 yc þ max

sin p þ ¼ sin p yc yc 2p 2p 2 2 cos 1 1 cos 2 2

yc yc yc 2sin cos Imax 2 2

¼ yc 2p cos 1 2

yc 2p p yc o o0 þ ð0 ð 2o0 cos þ cosðo0 tÞ o0 o0 2

Ids;ce ðtÞcosðo0 tÞ dt ¼ Imax cosðo0 tÞ dt i1 ¼ yc p p cos 1 0 yc p 2 o0 2o0 yc p p þ yc o0 þ o0 ð 2o0 ð 2o0 Imax yc Imax cosðo0 tÞ dðo0 tÞ þ cos2 ðo0 tÞ dðo0 tÞ cos p 2 p yc yc p p o0 2o0 o0 2o0

¼ yc 1 cos 2 yc yc p p p þ yc o0 þ o0 þ o0 ð 2o0 ð 2o0 ð 2o0 yc 1 1 cos cosðo0 tÞ dðo0 tÞ þ cosð2o0 tÞ dðo0 tÞ þ dðo0 tÞ 2 2 2 yc yc yc p p p Imax o0 2o0 o0 2o0 o0 2o0

¼ yc p 1 cos 2

yc yc yc 1 yc sin p þ sin p þ ½sinð2p þ yc Þsinð2pyc Þ þ cos Imax 4 2 2 2 2

¼ yc p cos 1 2

yc yc 1 yc 2 cos sin þ sinðyc Þ þ Imax 2 2 2 2

¼ yc p cos 1 2 yc 1 Imax 2 2 sinðyc Þ

ð11:73Þ ¼ yc p 1 cos 2

AMPLIFIERS

465

Note that the quantity Imax is the positive peak of the transistor output current, as Figure 11.22 shows. The inductor Ldd transmits the continuous current while presenting infinite impedance at the fundamental frequency, and at its harmonics a fortiori. On the contrary, the capacitor Cdd stops the direct current while passing the RF current. Therefore, the only component of the transistor output current to pass through Ldd is the DC one, and thus the DC generator supplies the current isupply ¼ i0. Then, the current balance at the transistor output implies that iAC ðtÞ ¼ ids;ce ðtÞisupply ¼

1 X

ik cosðko0 tÞi0 ¼

k¼0

1 X

ik cosðko0 tÞ

k¼1

By hypothesis, the output matching network shunts to ground all the current harmonics except the fundamental. The voltage across the transformer input is then vAC ðtÞ ¼ RLOAD i1 cosðo0 tÞ ¼ n2 50 i1 cosðo0 tÞ The voltage across Cdd is constant and equal to Vq , as a consequence of the infinite value of Cdd itself, hence the transistor output voltage is vds;ce ðtÞ ¼ Vq þ vAC ðtÞ ¼ Vq þ RLOAD i1 cosðo0 tÞ The idealized transistor amplifies only if vds;ce 0, thus the maximum peak-to-peak excursion of the output voltage is 0 vds;ce ðtÞ 2Vq . Consequently, it must be jRLOAD i1 j Vq . The resistance for the maximum output power is then Vq ð11:74Þ RLOAD;PMAX ¼ i1 If the load resistance exceeds the limit (11.12), the transistor will work differently from our hypothesis. In particular, when vds;ce becomes negative, the transistor can supply no current, and saturation occurs on the latter, as Figure 11.12 shows (upper left graph, grey line). The RF output power is maximum when the transistor output voltage swing is maximum, which occurs if the load resistance assumes the value in (11.74). Note that the optimum load resistance depends on the conduction angle. The RF power delivered to the load is 32 yc 1 sinðy Þ c 7 RLOAD 6 7 6Imax 2 2

¼ 5 yc 2 2 4 p 1 cos 2 2

PLOAD ¼ RLOAD

i12

ð11:75Þ

Substituting Equation (11.74) into (11.75), we obtain the maximum RF power deliverable to the load

PLOAD;MAX

yc 1 Imax 2 2 sinðyc Þ

¼ Vq yc 2p 1 cos 2

ð11:76Þ

Note that, if yc ¼ 2p, Equation (11.76) coincides with the second value of Equation (11.43), as expected.

466


100

1.0

90

70 0.6

η ( θ c)

60

η(θc)

Pout(θc)/Pout(2π)

80 0.8

0.4 50 Pout (θc)/Pout(2π)

0.2

40

0.0 0

90

180

270

30 360

-1

180 π θc , deg.

Figure 11.24 Maximum output power (black) and maximum power added efficiency (grey) for the amplifier in Figure 11.23. The power delivered from the output DC voltage generator is

PDC ¼ Vq i0 ¼ Vq

Imax 2p

yc cos

yc yc 2 sin 2 2

yc cos 1 2

ð11:77Þ

The resulting efficiency is Z¼

1 RLOAD Imax yc sinðyc Þ 1 sinðy Þ yc c

yc yc yc 2 Vq 2p 2 yc cos 1 cos 2sin 2 2 2

ð11:78Þ

Figure 11.24 plots the quantities (11.76) and (11.78) as the black and grey lines, respectively. The output power is normalized to the power of a class A amplifier having the same transistor maximum current. Figure 11.24 shows that Z increases for small conduction angles, up to a theoretical 100% at yc ¼ 0. On the other hand, the output power decreases with the conduction angle as well and vanishes if yc ¼ 0. Observations: (a) From the considerations developed so far, it follows that the conduction angle is a function of the input peak amplitude xpeak and mean value xDC , where the latter depends on the input bias. Furthermore, the combination of Vq ; RLOAD and yc determines the maximum peak output current without saturation, maximum output power and efficiency. (b) The maximum theoretical efficiency of class A, B and C amplifiers is 50%; 50p=2 ffi 78:54% and 100%, providing that their load resistances assume the value in (11.74). Classes A and B present the same maximum output power, while a 100% efficiency in class C implies zero output power. Assuming a conduction angle of yc ¼ 2p=3 (120 ), a class C amplifier presents an

AMPLIFIERS

467

efficiency of 89.68%, while its output power equals 61.2% of a class A amplifier having the same maximum current. (c) For a given Imax , Equation (11.76) reaches its maximum if yc ffi 4:278 ð245:1 Þ, which is PLOAD;MAX ð4:278Þ PLOAD;MAX ð4:278Þ ¼ ffi 1:073 PLOAD;MAX ðpÞ PLOAD;MAX ð2pÞ with the associated efficiency Zðyc ¼ 4:278Þ ffi 65%. (d) The input amplitude required to drive the amplifier is proportional to the xpeak value associated with a given output power. Substituting Equation (11.65) into (11.66), we obtain the input peak level as a function of output current peak and conduction angle xpeax ¼

1 I max yc GL 1 cos 2

Assuming yc ¼ 2p; yc ¼ p and yc ¼ 2p=3, we have the values for class A, B and one case of C operation xpeax;A ¼

1 Imax ; 2 GL

xpeax;B ¼

Imax ; GL

xpeax;C ¼ 2

Imax GL

Class A, B and C amplifiers require input excitation amplitudes that are in the ratio of 1, 2 and 4, in order to produce the respective maximum output power. Despite this, class C power is 61.2% of class A or B power. Consequently, small conduction angles are possible only if the transistor has enough gain, otherwise the power consumed to produce the driving signal compromises the overall efficiency. (e) Equation (11.74) with yc ¼ p gives the output power of a class B amplifier as a function of the input excitation amplitude PLOAD;B ¼

RLOAD;B G2L x2peak 8

Thus, a purely ideal class B amplifier is linear, in that its output power at the fundamental frequency is proportional to the square amplitude of the input sinusoid. This characteristic is peculiar to class A and B amplifiers: any conduction angle different from 180 or 360 leads to a nonlinear relation between input and output power. In particular, class C amplifiers produce no output signal until the input positive peak exceeds xmin. The Fourier coefficients (11.71) for class B amplifiers are

ih;B

o0 ¼ p

2p o ð0

0

Imax ¼ 2p

o0 Ids;ce ðtÞcosðo0 tÞ dt ¼ p

1 2ðp

Imax cos½ðh þ 1Þx dx þ 2p

3 2p

and vanish for odd h.

3p 2ð o0

Imax cosðo0 tÞcosðho0 tÞ dt 1p 2o0

1 2ðp

cos½ðh1Þx dx ¼ 3 2p

Imax 1 þ ð1Þh p 1 h2

468


(Q1)

vds,ce1 ids,ce 2 1

Vgs1, Ib1

nin VRF

Q1

Tin

(Q1)

Vgs2 , Ib2

(Q2)

2

iout= [i ds,ce - i ds,ce ] n out

1 1

j Zout

50 Ω

nout

(Q2) vds,ce2 ids,ce

3 Tout

Q2

3 +

+

Vgg

(a)

nout

nin

j Zin

50Ω

2

Vq

Iout(t) AB B O

T/2

t T

(b)

Figure 11.25

Push–pull amplifier: (a) schematic diagram; (b) output current waveform.

Thus, class B amplifiers present only even-order nonlinear distortion, which can be cancelled with a 180 balanced configuration. Moreover, even-order intermodulation products are unlikely to fall in the working bandwidth,45 particularly in narrow-band circuits: in this case the output filter cancels both the even harmonics and the intermodulation products. However, it must be considered that real class B amplifiers produce output current waveforms like the grey curve in Figure 11.21(iii), rather than half sinusoids, due to the drop in transistor gain at low output currents. Therefore, real class B amplifiers are not linear and produce odd harmonics as well, although relatively small amounts of quiescent current (class AB) can alleviate the problem, particularly with the 180 balanced configuration. (f) So far, we have compared different amplifiers using the same transistor by assuming they present the same maximum instantaneous current. A different comparison, based on device dissipation, is possible. Clearly, this criterion results in more efficient amplifiers presenting lower transistor dissipation to produce the same output RF power, or, equivalently, they produce more RF power for a given device power. No conduction angle is preferable in respect of the transistor voltage swing, in that the maximum power is always achieved when vds;ce swings from zero to 2Vq : Equations (11.74), (11.76) and (11.78) follow exactly from this assumption. From observation (e) it follows that class B amplifiers are particularly suitable for application in 180

balanced amplifiers; Figure 11.25 shows one implementation of such a configuration. In that network the two ideal transformers Tin and Tout – having transformer ratios nin and nout respectively – replace the two 180 hybrids of Figure 11.20a. 45

See Section 9.5.

AMPLIFIERS

469

If nin ¼ nout ¼ 1 the scattering matrix of Tin and Tout is almost coincident with Equation (11.48); the only difference is that s23 ¼ s32 6¼ 0. However, the transmission between ports 2 and 3 of the transformers is not relevant for the subsequent considerations of this section. If nin ; nout 6¼ 1, Tin and Tout match the input and output 50 O into the required impedance in combination with the two reactances Zin ; Zout . The circuit in Figure 11.25a consists of two identical amplifiers – built around the transistors Q1 and Q2 respectively – biased for class B operation. The input transformer excites the amplifier inputs with two signals having the same amplitude but opposite phase. Then, the output currents of the two transistors are half sinusoids with a reciprocal phase shift of half a period. Expanding the transistor output currents as a Fourier series and using the coefficients as in observation (c), and considering that the odd coefficients are zero, we have ðQ1Þ

ids;ce ðtÞ ¼

1 X GL xin GL xin GL xin 2 cosðo0 tÞ þ cosð2ko0 tÞ p 2 p 14k2 k¼1

The output current from the transistor Q2 has a similar expression, but with the time shifted by half a period

T p ðQ2Þ ¼ Ids;ce;1 t ids;ce ðtÞ ¼ Ids;ce;1 t 2 o0

X

1 GL xin GL xin p GL xin 2 p cos 2ko t ¼ cos o0 t þ 0 p 2 p 14k2 o0 o0 h¼1 ¼

1 X GL xin GL xin GL xin 2 cosð2ko0 t 2kpÞ cosðo0 t pÞ þ p 2 p 1 4k2 h¼1

¼

1 X GL xin GL xin GL xin 2 cosð2ko0 tÞ þ cosðo0 tÞ þ p 2 p 1 4k2 h¼1

The push–pull amplifier output current is then h i ðQ1Þ ðQ1Þ iout ðtÞ ¼ ids;ce ðtÞ ids;ce ðtÞ n2 2 ¼ n22 GL xin cosðo0 tÞ

ð11:79Þ

Expression (11.79) states that the current flowing in the load is purely sinusoidal, and its amplitude is proportional to that of the input signal. Of course, this applies similarly to the output voltage, and to the ðQ1Þ ðQ2Þ transistor output voltages ids;ce ðtÞ; ids;ce ðtÞ, due to the output transformer. Therefore the push–pull class B amplifier is linear and distortion free, and both of its transistors work as in a conventional single-ended configuration: the output current waveform is a half sinusoid, while the voltage is sinusoidal. Note that this result is achieved without any filtering structure in the output matching network, differently from the network in Figure 11.23(i). As can be seen, for each half cycle one of the two transistors supplies the current to the load. This kind of operation resembles that of two woodcutters using a two-handled saw to fell a tree: for this reason the configuration in Figure 11.25a is also known as push–pull. Compared with each of its two single-ended amplifiers, the circuit in Figure 11.25 absorbs double the supply current and – neglecting the output transformer loss – delivers double the RF power. Therefore the push–pull class B amplifier has the same efficiency as the single-ended one. The 180 balanced (or push–pull) class B amplifier is the most efficient solution among all the possible linear amplifiers. So far, we have considered the effects of the drop in gain in the low-current region of the transistor, already mentioned within the description of class AB operation. If the input bias of the transistors Q1 ; Q2 falls somewhere around the point marked B in Figure 11.21(i), the transistor output current waveform differs from a half sinusoid. Rather, it is distorted, mainly in the proximity of zero current, as the grey curve in Figure 11.21(iii) shows. If the transistor’s working point is too close to point C in

470


Figure 11.21(iii) – or, equivalently, if the quiescent current is too low – then the above-mentioned distortion becomes relevant, and the load current assumes the shape shown in Figure 11.25(ii) by the grey curve, marked as B. That waveform is a sinusoid, distorted in the zero- crossing region: for this reason it is referred to as crossover distortion. Crossover distortion compromises the linearity and causes odd-order nonlinear distortion in the amplifier. Class AB operation, with a judicious choice of quiescent current, can alleviate the problem, while minimizing degradation of the efficiency. The technique consists of ðQ1Þ ðQ2Þ producing ids;ce ; ids;ce that are both non-zero in a proper interval across t ¼ T=2. This way, both the transistors contribute to the output current when working in the respective low-output-current region, with the gain increasing by a factor of 2. Class A to C amplifiers have one common working principle: part of the output bias generator power produces RF power on the load at the fundamental frequency, while the transistor dissipates the remaining part. Therefore, amplifier efficiency is maximum when the transistor dissipation is minimum, and the latter is the average over one period of the product of transistor output voltage and current: ðQ1;Q2Þ PDISSIPATED

ðT 1 ðQ1;Q2Þ ðQ1;Q2Þ ¼ v ðtÞids;ce ðtÞ dt T ds;ce 0

The transistor dissipation can be increased by ensuring that vds;ce is low when ids;ce is high and vice versa. In the ideal case, the transistor dissipation is zero. This occurs if vds;ce > 0 ðids;ce > 0Þ implies that ids;ce ¼ 0 ðvds;ce ¼ 0Þ. The basic idea behind class D operation is that abrupt variations in vds;ce and/or ids;ce from zero to their maximum reduce the time when they are both non-zero, thus minimizing the transistor dissipation. The schematic showing the principle of a class D amplifier is very similar to the one in Figure 11.25, with the addition of a low-pass or bandpass network between the reactance Zout and the 50 O load. The two transistors are biased in order to have no quiescent current without input signal, similar to class B or C. The input signal is strong enough to overdrive the two transistors that essentially work as switches: in the OFF (ON) state this is vds;ce > 0; ids;ce ¼ 0 ðvds;ce ¼ 0; ids;ce > 0Þ. The transistors of an ideal class D amplifier change their ON/OFF state in a very short time; in other words, their transition time is zero. Under that hypothesis, in the first half of the cycle, for 0 < t < T=2, Q1 is OFF and Q2 is ON. The opposite happens in the second half of the cycle, for T=2 < t < T. Hence, at any instant ðQ1Þ ðQ1Þ ðQ2Þ ðQ2Þ vds;ce ids;ce ¼ vds;ce ids;ce ¼ 0, the two transistors dissipate no power. During the transition time, of course, the transistor output voltage and current are both non-zero, with a consequent power dissipation in the device. Without the need for further analyses, we can state that the current to the load is periodic with the same period as the exciting signal. The above-mentioned output low-pass or bandpass filter eliminates the harmonics, so that the output signal is sinusoidal. Therefore, an ideal class D amplifier presents unitary efficiency (PAE ¼ 100%), until the transition time becomes negligible. Thus the transistor transition time must be zero, or at least much shorter than the fundamental period. This requirement limits the maximum frequency for class D operation to a few tens of megahertz. Figure 11.23(ii) shows a more interesting solution for RF/microwave applications, known as a class E amplifier. The schematic does not explicitly show the output transformer matching the 50 O load into the required impedance, but we will assume it is present if needed. The basic idea is to use the capacitor C1 across the output of the transistor to minimize the effects of the finite transition time of the transistor. The complete analysis of the circuit is too long to fit in this chapter,46 so we will mention just the assumptions made and the final design equations: 1. The inductor Ldd is large enough that only the direct current isupply flows through it. 2. The loaded quality factor of the resonant circuit consisting of L2 and C2 is high enough to ensure that the current iLOAD is sinusoidal with the same period as the exciting signal. This also implies 46

The interested reader can find more details in [3, 4] and references therein.

AMPLIFIERS

471

Q1 ON

vout Q1 OFF

vds,ce

ids,ce 0

1 Time, ns

Figure 11.26

2

Class E amplifier waveforms.

that the resonant circuit removes all the harmonics. However, the resonant frequency of the network is slightly different from o0 , as we will see shortly. 3. The transistor Q1 works like a perfect switch: in the OFF (ON) state its output current (voltage) is zero. 4. The transistor output capacitance, if any, is independent of the voltage, and is lower than C1 , which represents the combined added and transistor output capacitance. Under these hypotheses, the operation of a class E amplifier is relatively straightforward. Figure 11.26 shows the waveforms relative to one simulated circuit.47 When the switch is ON, vds;ce ¼ 0, and all the direct current from the power supply flows through the transistor. As soon as the switch becomes OFF, the direct current begins to charge the capacitor C1, but the resonant circuit makes the voltage go back to zero. With a proper design of the circuit, vds;ce becomes zero immediately before the transistor goes ON again, thus minimizing the transistor power dissipation during the transient. It is possible to demonstrate that the amplifier efficiency is maximum if in the state change instant the transistor output voltage and its time derivative are zero vds;ce ðt ¼ T=2Þ ¼

dvds;ce ðt ¼ T=2Þ ¼0 dt

This result can be achieved with the value combination L2 ¼ QL

RLOAD ; o0

C1 ¼

p2 p 1 þ1 ; 2 RLOAD o0 4

C2 ¼

1 1:42 1 þ QL 2:08 o20 L2

ð11:80Þ

where the design parameter QL is the loaded quality factor of the resonant circuit consisting of L2 and C2 .

47

See the SIMetrix file 24_Class_E_GHz.sxsch.

472

MICROWAVE AND RF ENGINEERING With the values from (11.78), we get: .

Z ¼ 100%.

.

Output power: PLOAD ¼

p2

Vq2 Vq2 8 ffi 0:577 þ 4 RLOAD RLOAD

Class B amplifiers have PLOAD ¼ 0:5Vq2 =RLOAD . .

Transistor peak voltage: 1

Vds;ce;peak ¼ 2p sin

rffiffiffiffiffiffiffiffiffiffiffiffiffi! 4 Vq ffi 3:56Vq p2 þ 4

Class A to C amplifiers have Vds;ce;peak ¼ 2Vq . .

Transistor peak current: " Ids;ce;peak ¼ isupply

.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi

# p2 1 1þ þ 1 1 2QL 4

If QL tends to infinity, and since isupply ¼ PLOAD =Vq , we have Ids;ce;peak ffi 2:86PLOAD =Vq , thus class B amplifiers present a peak current of Imax ¼ 4PLOAD =Vq .

Normally, the filtering action of the resonator L2 ; C2 is not sufficient to attenuate the harmonics as required. Therefore one additional low-pass or bandpass filter is often placed between the resonator and the load. This modifies the circuit waveforms, and usually requires an adjustment of the values of C1 ; C2 ; L2 , normally performed with CAE tools. Figure 11.23(iii) shows the principle of a class F amplifier. Its output matching network shunts to ground all the harmonics but the third one; at that frequency it presents a high impedance to the transistor, unlike class A to C amplifiers. From a conceptual point of view, the parallel resonator L3 ; C3 in Figure 11.23(iii) replaces the series resonator L3 ; C3 in Figure 11.23(i). The transistor output voltage of class F amplifiers is then the sum of two sinusoidal components, one at the fundamental frequency and one at the third harmonic. The resulting waveform resembles a square wave. Consequently, vds;ce is higher (lower) when ids;ce ¼ 0 ðids;ce > 0Þ, with a resulting increased efficiency. Figure 11.27 shows the typical waveforms of a class F amplifier.48 Class F operation requires a conduction angle slightly smaller than p, in order to ensure that a third harmonic is present49 on the output current and that ids;ce ¼ 0 when vds;ce > 0, for a higher efficiency. The transistor behaves as a variable-current generator, producing a waveform similar to class B operation. The output matching network makes the load current sinusoidal, and the parallel resonator L3 ; C3 provides a third harmonic on the transistor output voltage. If this component has 1/9 the fundamental amplitude, and the proper phase, it flattens the output voltage and reduces its maximum amplitude by a factor of 8/9, as Figure 11.27 shows (black solid line in the upper plot). This allows an increase in the load resistance by the reciprocal of the same factor, and hence an increase in the maximum swing of vds;ce (grey solid line in the upper plot of Figure 11.27). Consequently, 48

See the Mathcad file 10_Class_C_F.MCD. Keep in mind that yc ¼ p implies that all the even-order harmonics of the output current are zero, as in observation (e) of this section. 49

AMPLIFIERS

473

Vq + R LOAD i1 cos(ω 0 t) Vq + RLOAD [ i1 cos(ω0 t) - (i3 /9) cos(3ω0t)] Vq + (8/9) i1 RLOAD

vds,ce (t)

RLOAD(i3 /9) cos(3ω 0t)

Vq - (8/9) RLOAD i1

ids,ce (t)

0

1

2

ω0 t/(2 π)

Figure 11.27

Class F amplifier waveforms.

the output power and efficiency increase by the same factor, becoming respectively 9 Vq 9 Vq i2 9 9p ; PLOAD;F ¼ RLOAD;F 1 ¼ Vq Imax ; PAEF ¼ ffi 88:3% RLOAD;F ¼ ¼ 8 i1 4 Imax 32 2 32 Class F is a kind of enhancement of class B that theoretically increases output power and efficiency by about 10% through careful design of the output matching network and the transistor working point. Moreover, class F operation relies on a precise third-harmonic value, therefore it works in a narrow range of the input power. Note that the load of a class F amplifier differs from the class B one mainly in its impedance on the third harmonic. That difference increases the maximum output power and efficiency by about 10%; it is reasonable to assume that the wrong third-harmonic impedance could decrease the above parameters by the same quantity. This gives a still approximate, but more precise, idea than the one in Section 11.5.3 of the importance of the harmonic impedance on amplifier performance.

11.5.6

Amplifier linearization

The description of class A to F amplifiers, given in Section 11.5.5, shows that linearity and efficiency are difficult to conciliate. Furthermore, when a low distortion level is required, the amplifier must work in class A or B and well below its maximum power, with a consequent degradation of its efficiency. This section presents some configurations to increase PA linearity by keeping the efficiency constant and/or vice versa. The overall result is lower distortion with a given DC power, although with a more complex circuit than the one in Figure 11.1. The various proposed configurations achieve this result by applying different approaches. Solutions in Sections 11.5.6.1 and 11.5.6.2 correct the PA nonlinearity by adding suitable amounts of distortion, while the one in Section extends the efficient working of the PA over a relatively wide output power range. Finally Sections 11.5.6.4 and 11.5.6.5 present two ways to use saturated amplifiers to produce signals with controlled time-variable amplitude.

11.5.6.1 Predistortion The discussion in Section 9.5 pointed out that the distortion level at the output of a nonlinear two-port is an increasing function of the ratio between the actual and the 1 dB compression power of the network.

474

MICROWAVE AND RF ENGINEERING Predistorter D1

Input

1

2

D2 R1

1

Figure 11.28

4

3 R2

2

5

Output

3

Amplifier linearization based on predistortion.

Amplifiers are no exception to this rule. More quantitatively, Equation (9.56) states that the input power for the third-order intercept is 9.6 dB lower than the one corresponding to 1 dB of compression. Therefore, if the input power equals I1dBdBm the third-order distortion products of the amplifier are 9:6 2 ¼ 19:2 dB below the carriers. If, for instance, a distortion level of 40 dB is required, the input power must be ð4019:2Þ=2 ¼ 10:4 dB below P1dBdBm. Considering that the saturated power is about 2.5 dB higher than the 1 dB compression point, the corresponding output power is 10:4 þ 2:5 ffi 13 dB below the maximum considered in Sections 11.4 and 11.5: the efficiency becomes 5% of the maximum value (11.78). Figure 11.28 shows a conceptually simple solution to increase the linear range of the amplifier closer to the 1 dB compression point. The basic idea is to place a nonlinear two-port between the RF generator and the amplifier input, to compensate the transfer characteristic of the amplifier itself. The network within the dashed rectangle in Figure 11.28 is one of many possible implementations of the predistorter. Its working principle can easily be explained by considering the diode I–V curve in Figure 9.37. Neglecting the breakdown, the diode current presents a knee for forward voltages VTH that is about 1 V in the graph of Figure 9.39. In order to develop simple expressions, we will approximate the diode characteristic with a two-segment piecewise-linear curve: the diode current is zero when its forward voltage is less than VTH ; above that limit, the diode is constant and equal to VTH . With this assumption, the voltage transfer characteristic from nodes 2 to 3 becomes 8

R2 R1 þ R2 8 > > > V2 < VTH < R2 V ðjV V j < V Þ < R þ R V2 R1 2 2 3 TH 1 2

V3 ¼ R1 þ R2 ¼ : > R 1 þ R2 > V2 VTH ðjV2 V3 j VTH Þ > V2 VTH : V2 VTH R1 Thus, the predistorter exhibits a low gain A1 A2 R2 =ðR1 þ R2 Þ when the input amplitude is less than VTH ðR1 þ R2 Þ=A1 and a high gain A1 A2 elsewhere. The two amplifiers A1 ; A2 are supposed to be linear for any input power within the operative range; their gains are additional parameters to adjust the predistorter characteristic. In real networks, the transition between low gain and high gain is not abrupt as in our simplified analysis, rather the transition zone presents a smoothed knee.50 By experimenting with the design parameters A1 ; A2 ; R1 =R2 it is possible to shape the predistorter characteristic to be the complement of the PA one. Figure 11.29 shows the compression characteristics of a PA (thick grey), predistorter (thin black) and their combination (thick black). As reference, Figure 11.29 also shows the linear characteristic (black dashed). The predistorter increases its gain in correspondence to the PA compression: the negative compression of the first compensates the latter. Of course, no predistortion can increase the PA saturated power, but it can improve the overall linearity up to an input level close to the saturation point, as in Figure 11.29. 50

See the SIMetrix file 23_Diode_Predistorter_Curve.sxsch.

AMPLIFIERS predistorter

475

linear

Output power, dBm

linearized

power amplifier

Input power, dBm

Figure 11.29

PA, predistorter and global compression curves.

TL 1

5

3 1 6

3

φ PH1

1

IN

2 DIV1

7

φ PH2

ATT2

AMP2

ATT1

TL 2

2

COUP1

4

OUT COUP2

COUP3

AMP1

Figure 11.30

Feedforward linearization of a PA.

The application of a properly designed predistorter51 can increase the linear range of the PA up to few decibels below the 1 dB compression point, with third-order distortion products in the range of 40 dB. As a drawback, the predistortion technique requires that the low- to high-gain transition occurs at a precise level which depends on the I1dB and SSG of the PA. Both these two parameters vary with frequency, temperature and from piece to piece: the network adjustment could be critical.

11.5.6.2 Feedforward Figure 11.30 shows a distortion-reducing configuration based on adding to the output signal the opposite of the distortion products of the PA. This solution is known as feedforward. The following considerations are a simplified analysis of the block diagram in Figure 11.30, under the hypothesis that all the blocks are perfectly impedance matched. The block AMP1 is the PA to be linearized. The diagram also includes one additional amplifier (AMP2 ), three directional couplers (COUP1 , COUP2 and COUP3 ), two phase shifters (PH1 ; PH2 ), two attenuators (ATT1 ; ATT2 ), two transmission lines (TL1 ; TL2 ) and one power divider (DIV1 ) used as a power adder. 51

See the Mathcad file 11_Predistortion.MCD.

476


The coupled output ports of COUP1 and COUP2 provide an attenuated sample of the input and output signals of AMP1 , respectively; the difference between a proper combination of these two signals is the amplifier distortion. Let the input voltage be VIN ; then the resulting amplifier output is qffiffiffiffiffiffiffiffiffiffiffi ðAMP1Þ V2 ¼ j 1c21 s21 VIN þ DIST ðAMP1Þ

where s21 is the transmission coefficient of AMP1 , c1 is the coupling coefficient of COUP1 and DIST is the nonlinear distortion of the PA. In the block diagram in Figure 11.30 there are two signal paths from the input to node 6. One of them includes AMP1 , and therefore transmits the associated distortion. The other path consists of linear networks only: COUP1 , COUP2 , PH1 and ATT1 . The characteristics of the various block are such that the two above-mentioned paths present the same amplitude gain and a relative phase shift of 180 , and this occurs if qffiffiffiffiffiffiffiffiffiffiffi ðTL1Þ ðDIV1Þ ðAMP2Þ ðPH1Þ ðATT1Þ ðDIV1Þ c1 s21 s13 ¼ j 1c21 s21 c2 s21 s21 s12 ð11:81Þ where .

c2 is the coupling coefficient of COUP2 .

.

s12 and s12 respectively.

.

s21 ; s21 ; s21 respectively.

ðDIV1Þ

ðTL1Þ

ðDIV1Þ

ðAMP2Þ

are the transmission coefficients of DIV1 from its ports 2 to 1 and 3 to 1,

ðPH1Þ

ðATT1Þ

and s21

are the transmission coefficients of TL1 , AMP2 , PH1 and ATT1 ,

This way, the signal at node 6 is proportional to the PA nonlinear distortion: ðPH1Þ ðATT1Þ ðDIV1Þ s21 s12 DIST

V6 ¼ c2 s21

AMP2 , ATT2 and PH2 amplify the error signal V6 and send it to the output in phase opposition to the PA nonlinear distortion, in order to cancel it. The nonlinear distortion of AMP1 is then present on the PA output (node 2) itself and on the power divider common port (node 6). If the two paths from nodes 2 and 6 to the output have the same gain amplitude and are 180 out of phase, the result is cancellation of the distortion at the output. The distortion cancellation implies that qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi ðPH1Þ ðATT1Þ ðDIV1Þ ðPH2Þ ðATT2Þ ðAMP2Þ ðTL2Þ s21 s21 s12 s21 s21 s21 c2 ¼ 1c22 s21 1c23 ð11:82Þ ðPH2Þ

ðATT2Þ

ðAMP2Þ

ðTL2Þ

; s21 and s21 are the transmission coefficients of PH2 , ATT2 , AMP2 and TL2 , where s21 ; s21 respectively, and c3 is the coupling factor of COUP3 . Equations (11.81) and (11.82) relate the transmission coefficients of the various blocks in ðAMP1Þ and Figure 11.30. These relations must be verified within the PA working bandwidth, with s21 ðAMP2Þ varying over the frequency. Consequently, TL1 and TL2 could not be simple transmission lines: in s21 the general case, they are linear two-port networks with a specified transmission coefficient phase (or ðATT1Þ ðATT2Þ and s21 could be frequency dependent. group delay) over the frequency. Similarly, s21 Furthermore, it is required that the amplifier AMP2 operates linearly. In respect of this consideration, the choice of the coupling coefficient c2 results from a compromise between two opposite requirements. High c2 implies high attenuation on the direct path, and thus high loss on the PA output power. Conversely, low c2 implies high attenuation from the output to AMP2 to the overall output, and therefore higher output power required from AMP2 . For example, if c2 ¼ 100:5 ð10 dBÞ, and neglecting the loss due to TL2 and the direct path of COUP2 , the loss from AMP1 to the overall output is 10 log10 1c23 ffi 0:46 dB. Correspondingly, the

AMPLIFIERS

477

linear output power of AMP2 must be 10 dB higher than the distortion power produced by AMP1. If the feedforward correction has to work up to the 1 dB compression output power of AMP1 O1dBðAMP1Þ , then the corresponding third-order distortion is about 18.2 dB lower than O1dBðAMP1Þ . Thus, the linear range of AMP2 should be no lower than O1dBAMP1 18:2 þ 10 dB ¼ O1dBAMP1 8:2 dB. At this level, if, for instance, O1dBðAMP2Þ ¼ O1dBðAMP1Þ , AMP2 itself produces third-order intermodulation products at a level of 2 8:2 18:2 dB ¼ 34:6 dB below O1dBðAMP1Þ. COUP3 attenuates them by an additional 10 dB. The resulting distortion on the feedforward system is then 44.6 dBc below the carrier. The overall DC output power of this specific feedforward-corrected amplifier is twice that of a single amplifier; in addition the output power is 0.45 dB lower than AMP1 itself. Nevertheless, the solution is still convenient if linearity is a requirement. By comparison, a simple amplifier with no correction requires an output 1 dB compression power of O1dBðAMP1Þ þ ð44:6 18:2Þ=20:45 2 dB ¼ O1dBðAMP1Þ þ 12:3 dB: the corresponding DC power is 101:23 =2 ffi 8:49 times higher than the solution in Figure 11.30. The simplified description of the feedforward working principle pointed out, however, that the distortion cancellation relies on equal-amplitude gain and phase of different RF paths. Each of them can vary over frequency, temperature, time and from piece to piece. Therefore, some parameters of the correction branches are adjustable, usually PH1, ATT1 , PH2 and ATT2 . Sophisticated implementations of the feedforward configuration include automatic computer control of those components, which periodically measures the distortion and applies the required adjustments.

11.5.6.3 Doherty amplifier Figure 11.31 shows an arrangement known as the Doherty amplifier,52 from the name of its inventor. The network in Figure 11.31 resembles the 90 balanced amplifier, but it works differently, as we will see shortly. The diagram in Figure 11.31a includes one main amplifier, AMP1 , operating at low input power, and one peak amplifier, AMP2 , which adds its contribution only at high input power. AMP1 operates in ðAMP1Þ class B53 and then behaves as linear from a small input power up to a maximum PIN;max . AMP2 operates in ðAMP1Þ ðAMP1Þ class C, and its output current is zero if PIN < PIN;max . Therefore, in the range of 0 < PIN < PIN;max , the Doherty amplifier works like the schematic in Figure 10.32b: the output of the inoperative AMP2 is an open circuit, and the l/4 transmission line presents a resistive load of 4RLOAD to the output of AMP1 . The ðAMP1Þ latter is designed such that the resistance 4RLOAD equals the best value (11.74) when PIN ¼ PIN;max ; in ðAMP1Þ this condition AMP1 presents its maximum efficiency of 78.54%. Let Imax be the output current peak of AMP1 in this condition. By virtue of Equations (11.72) and (11.73)

y ¼p yc yc c 1 1 cos cos V 2 2 q ðAMP1Þ p ¼ 1 Vq ¼ i1 Imax p¼ yc 1 4RLOAD yc 1 2 RLOAD sinðyc Þ sinðyc Þ 2 2 2 2 ðAMP1Þ

the corresponding output power is POUT ¼ Vq2 =ð8RLOAD Þ. ðAMP1Þ As PIN increases above PIN;max , AMP1 saturates if AMP2 is not present. Instead, at that point, AMP2 ðAMP2Þ begins to deliver RF output power, increasing with PIN . At the input power PIN ¼ PIN;max the peak amplifier gets close to saturation. If the two amplifiers present the same maximum power, at that point they deliver the same signal to the load, because the two paths from the input to the load (node 5) have the same phase.54 Therefore, we can consider the two amplifiers as delivering half of the current to the load. Thus each of them sees a resistance equal to 2RLOAD as Figure 11.31c shows. In this condition AMP1 52

More details about the Doherty amplifier can be found in the references [5] and [6]. The main amplifier can also work in class A, with subsequent minor modifications which remain qualitatively valid. 54 Each path includes one amplifier and a 90 phase shift. 53

478


√0.5

4

2

-j √0.5

5 TL1 Z 0 = 2 R LOAD θ= π/2

1

R LOAD

3 (a)

2

in

Z L= 4 RLOAD 1

√0.5

4

2

-j √0.5

5 TL 1 Z 0 = 2 R LOAD θ=π/2

1

RLOAD

3 (b)

2

in

ZL= 2 RLOAD 1

√0.5

4

2

-j √0.5

1

5 TL1 Z0= 2 RLOAD θ=π/2

3 (c)

in

2 R LOAD

2 R LOAD 2

Z L = 2 RLOAD

Figure 11.31 Doherty amplifier: (a) schematic; (b) equivalent circuit for low-output-power operation; (c) equivalent circuit for high-power operation. ðAMP1Þ

works linearly up to a current peak equal to 2Imax , due to reduced load resistance from 4RLOAD to ðAMP2Þ ðAMP1Þ 2RLOAD . Its maximum output power is then POUT ¼ Vq2 =ð4RLOAD Þ ¼ 2POUT . In the same time, AMP2 delivers the same power to the load, by hypothesis. The total maximum power is than POUT;max ¼ ðAMP1Þ ðAMP1Þ 4POUT;1 POUT;max ¼ 4POUT , corresponding to an input power of 4PIN;max . Thus, the Doherty amplifier ðAMP1Þ efficiency monotonically increases with input power from zero ðPIN ¼ 0Þ to 78.54% ðPIN ¼ PIN;max Þ, then it stays almost constant for another 6 dB of increasing input level. More precisely, the efficiency in the higher range is ideally higher than 78.54%, in that it results from a combination of two amplifiers, one class B and one class C. Therefore, the Doherty configuration offers nearly the same efficiency as class B, but for a wider output power range, still ensuring the linearity.

AMPLIFIERS 4

479

5 DET1

LPF1 Vcc

1

IN

2

3

OUT

COUP1 (a)

AMP1

AMP2

envelope

carrier RF signal

(b)

Figure 11.32 PA architecture with envelope elimination and restoration: (a) block diagram; (b) waveforms.

11.5.6.4 Envelope elimination and restoration The idea behind envelope elimination and restoration is to use a saturated PA to amplify an RF signal having a time-varying amplitude. The advantage of using a saturated amplifier is that it can be a highly efficient class C, D, E or F one. Figure 11.32a shows the conceptual block diagram of such a solution, and Figure 11.32b shows the main associated waveforms. The power amplifier AMP1 has enough gain so that its output amplitude is constant for any amplitude of the input signal within a specified range. The detector DET1 in combination with the low-pass filter LPF1 produces an output DC voltage proportional to the envelope of the RF input signal or to the square55 of it. That voltage (node 5) is then used as the drain/collector voltage of AMP1 , usually after amplifying it with a high-efficiency, low-frequency amplifier. This way, the PA works always at its maximum power, with a consequent high efficiency. Besides complexity, the main critical point of the architecture in the figure is the unwanted phase modulation associated with the change in PA drain/collector voltage, which adds undesired phase modulation to the output signal. Furthermore, the high gain required to saturate the PA for a wide range of input amplitudes can introduce further phase modulation and overdrive56 phenomena, with a reduction in the output power over the highest range of the input amplitude. 55 56

For a detailed explanation of the detector working, see Section 13.2. See the dashed curve in Figure 11.16.

480


Vpeak cos(ω 0 t+φ1)

φ1

√2 Vpeak A(t) cos[ ω 0t + θ(t)] 1 2

IN

φ2

Vpeak cos( ω 0t+φ2 ) 2

A(t) θ(t) ROM1

D/A1

LPF1

ROM2

D/A 2

LPF2

Figure 11.33 LINC block diagram.

11.5.6.5 LINC The acronym LINC stands for linear amplification with nonlinear components; it allows the application of two saturated amplifiers to generate an arbitrarily modulated RF signal. Figure 11.33 shows the principle of such a configuration. The power divider DIVIN distributes the input signal to the phase shifters PH1 and PH2 that feed the two power amplifiers AMP1 and AMP2 . Both AMP1 and AMP2 work in saturation, therefore their output power is constant and independent of the input one. On the contrary, the phase of the two output signals changes according to the respective input phase shift. The Wilkinson divider DIVOUT combines the two amplifiers’ output signals to the output. Assuming that the two amplifiers are equal, and that DIVOUT is ideal, the global output voltage is Vpeak Vpeak vRF ðtÞ ¼ pffiffiffi cos½o0 t þ f1 ðtÞ þ pffiffiffi cos½o0 t þ f2 ðtÞ 2 2

ð11:83Þ

where Vpeak is the output peak voltage of the two amplifiers, o0 is the angular frequency of the RF signal, and j1 ðtÞ; j2 ðtÞ are the phase shifts of PH1 and PH2 , respectively. We can see intuitively that if f1 ðtÞ ¼ f2 ðtÞ, the schematic in Figure 11.33 coincides with a 0

balanced amplifier with a phase shifter at the input: that configuration can only produce phase modulation. Similarly, if f1 ðtÞ ¼ f2 ðtÞ, the resulting output phase is always zero, for symmetry reasons. In this second case, the output amplitude can vary from zero to 2Vpeak , when f1 ðtÞf2 ðtÞ ¼ 2f1 ðtÞ varies from zero to p. More quantitatively, manipulations based on Equations (A.86) to (A.93) allow us to rewrite expression (11.83) as vRF ðtÞ ¼ aðtÞ cos½o0 t þ yðtÞ

ð11:84Þ

with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðtÞ ¼ fcos½f2 ðtÞf1 ðtÞ þ 1g2 þ sin2 ½f2 ðtÞf1 ðtÞ;

yðtÞ ¼ tan

1

sin½f2 ðtÞf1 ðtÞ cos½f2 ðtÞf1 ðtÞ þ 1

AMPLIFIERS

481

Expression (11.84) reveals that the output signal of the arrangement in Figure 11.33 presents both amplitude and phase modulation. Conversely, it is also possible to determine the phase shifts to apply, in order to get the required modulation. Inverting the coefficients of expression (11.84), we have 2 1 A ðtÞ 1 þ yðtÞ; j2 ðtÞ ¼ j1 ðtÞ2 tan1 ftan½yðtÞ þ j1 ðtÞg j1 ðtÞ ¼ cos1 ð11:85Þ 2 2 Thus, from the required amplitude aðtÞ and phase yðtÞ, Equation (11.85) returns the corresponding phase shifts to apply. In principle, Equation (11.85) could be implemented with analogue nonlinear networks that use aðtÞ; yðtÞ as input and j1 ðtÞ; j2 ðtÞ as output quantities to control two continuously variable phase shifters. However, analogue implementation is critical for many reasons: namely, component tolerances, stability over time and temperature, circuit complexity, and so on. Moreover, the phase shifter control law itself is nonlinear.57 For these reasons, Figure 11.33 proposes a digital approach: aðtÞ; yðtÞ are two digital words, and the read-only memories ROM1 and ROM2 numerically implement the relations (11.85). The two digital-to-analogue converters D=A1 and D=A2 transform the respective input digital words into proportional voltages. Finally, the two low-pass filters LPF1 and LPF2 smooth the staircase D/A output voltages, by eliminating its high-frequency components.58 One of the additional advantages of the digital solution, in comparison with the analogue one, is that it can easily implement any complicated function, including the nonlinear control law of the phase shifter. The LINC architecture allows the use of saturated – thus potentially highly efficient – amplifiers to generate arbitrarily modulated RF signals. The main drawback of this solution is that it relies on the phase and amplitude matching between the two paths from the input to the output. This requires that the two amplifiers have the same saturated power and the same transmission coefficient phase, that the two branches of DIVOUT are amplitude and phase matched ðs12 ¼ s13 Þ, and that the two phase shifters present the same phase with the same control voltage. The two branches of DIVIN must be phase matched only: small variations in their output power are cancelled by the saturated amplifiers. Note that all the above requirements apply for all the phase shift combinations, over frequency, temperature and time, and from piece to piece. Phase accuracy is particularly difficult to maintain under the various conditions.

11.5.7

Additional PA issues

This section completes the discussion on PAs, mentioning two additional important points, namely stability and spectral regrowth. Section 11.2.2 has already considered amplifier stability, supplying some important tools like stability circles and the Kurokawa criterion. Both these methods could apply to PAs, at least in principle. In practice, however, PAs are inherently nonlinear; definitions of large-signal scattering parameters are then possible, but neither unique or so meaningful. Consequently, the calculation of stability circles and/or the Kurokawa parameters is difficult and the result is not very meaningful. Moreover, in the highcompression regime the transistor behaves quite differently from its linear model. Some imperfectly designed PAs present instabilities, typically in the form of a tendency to oscillate at frequencies lower than the working range. The elimination of the instability in PAs is a cumbersome task for the designer. Theoretical analyses do not give reliable results, unless very accurate – and maybe specially tailored as well – nonlinear models are available for the transistor. The alternatives to the theoretical approach are long and patient experimental adjustments of the circuit on the test bench. As a general rule, the stabilizing elements are lumped and/or distributed resistors, preferably placed in the input matching network so as not to affect the output power. This criterion is diametrically opposite to the one given in Section 11.3 on LNAs, in that the PA (LNA) optimizes the power transfer from the transistor (generator) to the load (transistor). 57 58

See Section 10.4. See [8] for further details.

482


Normalized PA spectra, dBm/Hz

20 f1 f0

0

B C

f2

-20 ∆f

2 f1 - f2

-40

2 f2 - f1

-60

-80 0.5

Figure 11.34

1.0 f/f0

1.5

Spectral re-growth in PA output spectrum.

Spectral regrowth is inherent in any nonlinear component when its input spectrum has energy spread over a relatively wide frequency range. Spectral regrowth in PAs is particularly of interest, because its reduction is expensive in terms of efficiency and/or circuit complexity, as we will see shortly. Typical wide-band signals are RF signals modulated with complex schemes, particularly with high-speed digital transmissions, such as WCDMA.59 One other typical case is when the nonlinear component deals with adjacent multiple modulated carriers. Figure 11.34 shows a typical wide-band input (grey) and output (black) spectrum of a PA.60 The useful spectrum occupies the bandwidth Df located across the centre frequency f0. We can consider that spectrum as the superimposition of many tones having frequencies within the range ð f0 D f =2; f0 þ D f =2Þ; Figure 11.34 reveals the components at frequencies f1 and f2. Now each couple of such discrete approximated lines produces intermodulation products at the nonlinear component output. In particular, if the nonlinearity is of third order, the intermodulation products ð2 f1 f2 ; 2 f2 f1 Þ have a frequency close to f0 and fall on the sides of the useful frequency range, disturbing the adjacent channels. The only way to reduce the side emission is to increase the linearity. As we know, linear operation requires that the component works at an output power much lower than its compression point. This implies overdimensioning of the component with associated high DC consumption. The PA is usually the last link in the chain, thus the one handling the highest RF and DC power: any increase in its maximum output power implies large amounts of additional DC power. This consideration justifies a specific discussion of spectral regrowth dealing in PAs, and emphasizes the importance of the techniques discussed in Section 11.5.6. Section 9.5.6 defined the AM–PM conversion referring to generic two-port nonlinear networks. PAs pertain to that class of components and present AM–PM conversion, which could be relevant in the power range around 1dBCP.

11.6 Other amplifier configurations This section deals with amplifier configurations that are different from the classical one of Figure 11.1. Most of the solutions presented are widely used in MMIC and or RFIC technology. 59 60

Wide-band Code Division Multiple Access; it is the modulation scheme used for UMTS. See the Mathcad file 12_Spectral_Regrowth.MCD.

AMPLIFIERS

1

IN

Vgg

Q1

RF

LF

4

L1

Ldd

L gg

5

483

OUT

Vdd

3 2

(a) RF

IN

G

D + v Cgs - in

OUT

gmvin

Q1

(b)

Figure 11.35

11.6.1

Feedback amplifier: (a) principle; (b) simplified small-signal equivalent network.

Feedback amplifiers

Feedback is widely used in low-frequency amplifiers for its associated advantages, like flattening of the frequency response, reduction of distortion, low sensitivity of the amplifier performance to the transistor parameters, and so on. Microwave applications of feedback amplifiers are less common, but still the technique offers some benefits that we will briefly discuss here. Detailed analyses of the various feedback configurations can be found in electronics books;61 for our needs it is sufficient to recall some basic principles. Feedback consists of injecting a portion of the output signal of the amplifier to its input. We get positive or negative feedback if the injected signal produces an output in phase or in phase opposition to the original output signal. Due to the phase relation between the output and injected signal, negative feedback reduces the overall gain and its dependence on the amplifier. On the contrary, positive feedback increases the overall gain, and causes instability if the gain from the input to the injected signal (the loop gain) is unitary. Figure 11.35a shows the principle of a microwave feedback amplifier. It includes the input ðC1 ; C2 ; Lgg Þ and output ðC3 ; C4 ; C5 Ldd Þ bias networks. The usual considerations on the bias networks apply to these elements: all the capacitors (inductors) are open (short) circuits for the direct current and short (open) circuits at RF. Moreover, since the gate absorbs no current from Vgg , in some case it is possible to replace the inductor Lgg with a cheaper high-value resistor. A complete analysis of the circuit in Figure 11.35a is possible, but its results are complicated and probably not very meaningful. However, a simplified approximate equivalent network allows more practical analyses that are valid qualitatively in the real case. The results obtained from the simplified model are the starting point for a computer circuit analysis and optimization of the amplifier. The simplified network in Figure 11.35b is derived from the one in Figure 11.35a by eliminating the bias networks, replacing the transistor with its simplified linear model, and short-circuiting the feedback inductors ðLF ; L1 Þ. 61

The interested reader can find more details about feedback and its effects on amplifiers in [9].

484


The amplifier in Figure 11.35b clearly presents feedback, operated by the resistor RF . Moreover, the drain signal voltage is 180 out of phase with respect to that of the gate, hence the feedback is negative. From relatively simple calculations, we obtain the forward transmission and the two reflection coefficients of the network in Figure 11.35b s21 ð joÞ ¼

2R0 ðRF gm 1Þ 2R0 þ RF þ R20 gm

1 R0 ðR0 þ RF Þ 1 þ jo Cgs 2R0 þ RF þ R20 gm

s11 ð joÞ ¼

RF R20 gm joR0 ðR0 þ RF ÞCgs 2R0 þ RF þ R20 gm þ joR0 ðR0 þ RF ÞCgs

s22 ð joÞ ¼

RF R20 gm joR0 ðR0 RF ÞCgs 2R0 þ RF þ R20 gm þ joR0 ðR0 þ RF ÞCgs

ð11:86Þ

where R0 is the reference impedance. The transmission coefficient has the form of a first-order low-pass function, with the low-frequency coefficient s21 ð0Þ ¼

2R0 ðRF gm 1Þ 2R0 þ RF þ R20 gm

js21 ð0Þj monotonically increases with RF and tends to 2R0 gm for RF tending to infinity, which corresponds to applying no feedback. This confirms that the feedback is negative in the amplifier of Figure 11.35b. At very low frequencies, the reflection coefficients (11.86) simplify to s11 ð jo ¼ 0Þ ¼

RF R20 gm ; 2R0 þ R20 gm þ RF

s22 ð jo ¼ 0Þ ¼

RF R20 gm 2R0 þ R20 gm þ RF

ð11:87Þ

From both the equations in (11.87) it follows that the amplifier in Figure 11.35b is perfectly matched at low frequency if RF ¼ R20 gm . Under this condition, the amplifier gain is s21 ð joÞ ¼

1 R0 gm s21 ð0Þ ¼ R0 Cgs 1 þ j o 1 þ jo oT 2

ð11:88Þ

Equation (11.88) describes a low-pass behaviour, with the DC gain s21 ð0Þ ¼ 1R0 gm and the first1 order angular cut-off frequency oT ¼ 2R1 0 Cgs . Therefore, the transistor transconductance has to be sufficient to generate enough gain, and the gate–source capacitance needs to be such that oT is much higher than the required bandwidth. If we consider a 50 O working impedance, and the transistor in Figure 9.32b, we have RF ¼ 100 O; s21 ð0Þ ¼ 1. In other words, the resulting amplifier has unitary gain. In cases like this, it is possible to increase RF in order to increase the gain at the expenses of degradation in the port matching. Note that this assertion is not in contrast with the considerations in Section 11.2, because of the reduction in gain involved with feedback: increasing RF corresponds to reducing the feedback amount. Now we can describe the role of the feedback inductors. LF reduces the feedback at high frequencies by increasing the feedback impedance from the drain to the gate, in order to compensate for the natural roll-off of the transistor. L1, in combination with the transistor output capacitance, provides additional input/output phase shift. This transforms the feedback at high frequencies, from negative to positive, with a further increase in the gain. A properly designed feedback amplifier can exhibit higher gain than a transistor at high frequencies, despite the absence of matching networks. An additional advantage of the configuration in Figure 11.35a is that negative feedback improves the stability. Positive feedback occurs only in the upper frequency range, where the transistor presents the

AMPLIFIERS

485

lowest gain, and therefore is less liable to instability. Consequently, the two-port network in Figure 11.35a usually has better stability – higher K – than its transistor. The configuration in Figure 11.35a is also suitable for realizing wide-band amplifiers,62 with working bandwidths wider than 100% and with a maximum frequency of 18 GHz. Both monolithic and discrete realizations of feedback amplifiers are possible. However, the feedback inductances are typically fractions of nanohenries, thus critical for discrete realizations, particularly at the highest frequencies.63 MMIC technology allows the realization of semi-lumped inductors with almost arbitrarily low values and minimum interconnection-related parasitics. In this sense MMIC is the technology of choice in high-frequency feedback amplifiers.

11.6.2

Distributed amplifiers

The travelling wave amplifier (TWA), also known as the distributed amplifier, was originally developed around 1948 as a wide-band amplifier for the vertical channel of oscilloscopes.64 The original circuit used valve devices (pentodes) and had a maximum frequency of 140 MHz. In the early period of MMIC technology, around 1980, the TWA awakened interest among designers, because of the possibility it offered to realize broad-band amplifiers, with bandwidths from a few hundred megahertz up to 20 GHz or more. The minimum frequency of the TWA is ideally zero, but the bias networks determine its real value. Figure 11.36a shows the small-signal principle of a TWA, without the bias networks. It includes a number N of FETs, which are represented by a simplified linear equivalent circuit consisting of a VCCS with input and output capacitance. Each FET input and output terminal connects to the corresponding terminal of the adjacent FET terminal via two series inductors. Starting from the input, the inductors Lg , together with the capacitors Cgs , form a ladder LC network, comprising N equal cascaded cells, each having the schematic of Figure 11.36b, with L ¼ Lg ; C ¼ Cgs . Such a ladder network is the lumped approximation of a transmission line having the characteristic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi impedance Zg ¼ 2 Lg =Cgs . If both ends of that line are matched, i.e. Zg ¼ Rg ¼ R0 ¼ 50 O, a forward wave propagates from the amplifier input to the termination Rg . Consequently, the voltages across the gate–source junctions of the N transistors have thepsame amplitude and a progressive delay from the first ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi to the last one: the first transistor has a delay tg ¼ 2Lg Cgs, the second has double that quantity, the third three times, and so on. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi On the output, we have another lumped line, having characteristic impedance Zd ¼ 2Ld =Cds . The line is matched if Zd ¼ Rd ¼ R0 . In the following considerations, we will denote the two lumped transmission lines as the gate line and drain line. Q1 injects a current proportional to the input incident wave into the drain line. That current propagates through the drain line in both directions, towards the output and the drain termination Rd. The path from the input to Q1 to the drain terminations includes one section of the gate line and one of the drain line; its delay is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ð11:89Þ t1;RD ¼ 2Lg Cgs þ 2Ld Cd The path from input to output passing Q1 includes one section of the gate line and N 1 sections of the drain line, with the resulting delay pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11:90Þ t1;OUT ¼ 2LgCgs þ ðN1Þ 2Ld Cds

62 The Ansoft file 04_Feedback_SSA.adsn provides a simple design of such an amplifier, with a gain of about 6 dB from 2 to 10 GHz. See also [10, 11] for other designs. 63 In the circuit analyzed in the Ansoft file 04_Feedback_SSA.adsn, LF ¼ 1 nH and L1 ¼ 0:47 nH, with a maximum frequency of 10 GHz. Higher maximum frequency involves lower inductance values. 64 See [12, 13].

486

MICROWAVE AND RF ENGINEERING Ld

Ld

Ld

Ld

Ld

Ld

OUT

Rd G1

D1

Cgs Lg

+ vin -

G2

Cds

Lg

Cgs

Lg

GN

D2 + vin -

g mvin

g mvin

Cds

Cgs

Lg

Lg

DN + vin -

gm vin

Cds

Lg

IN

Rg

(a)

Lg , Ld Lg , Ld 1

2 Cgs, Cds

(b) TLd01

Ldd

Vdd

TLd12

TLd23

TLd34

TLd45

TLd56 OUT

dd2

Rdd

TLd1

TLd2

TLd3

TLd4

TLd5

dd1

Q1

Q2

Q3

Q4

Q5

TLg12

TLg23

TLg34

TLg45

TLg56

IN

Cgg1

TLg01

Rg Rgg

(c)

Vdd

Cgg2

Figure 11.36 TWA: (a) principle; (b) lumped transmission line cell (low pass); (c) complete schematic of a five-transistor distributed amplifier. Considering the contributions of the other transistors, both the delays from the input to the gate and from the drain to the drain terminations increase with the transistor index. For the kth transistor we then have pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11:91Þ tk;RD ¼ k 2Lg Cgs þ 2Ld Cds while the delay from the input to output is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tk;OUT ¼ k 2Lg Cgs þ ðN kÞ 2Ld Cds

ð11:92Þ

The gate and the drain line have the same propagation constant if pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Lg Cgs ¼ 2Ld Cds

ð11:93Þ

The input/output delay is independent of the transistor pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tk;OUT ¼ N 2Ld Cds ¼ N 2Lg Cgs

AMPLIFIERS

487

In other words, all the transistor current contributions are in phase at the amplifier output. Conversely, the delays to the terminal vary from transistor to transistor, therefore the current sum on the drain termination tends to cancel. Under the condition (11.93), a forward wave from the drain termination to the output corresponds to a forward wave from the input to the gate termination. With the approximations involved in the lumped approximation transmission lines, the TWA gain is js21 j ¼ N

gm R0 2

ð11:94Þ

The gate and drain lines are lumped ladder low-pass networks,65 which present a cut-off frequency fT ¼

1 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ 2p 2Lg Cgs 2p R0 Cgs 2p R0 Cds

ð11:95Þ

The ladder networks approximate a transmission line up to fT that also coincides with the maximum working frequency of the amplifier. The schematic in Figure 11.36a implies a direct connection of gates and sources to the inductors of the respective lines. Discrete realizations always present additional series elements between the transistor terminal and the junction between the two inductors: at least a bond wire in hybrid technology, or a bond wire and package lead with package devices. The series-added element degrades the amplifier performance – particularly the maximum frequency – making monolithic realization more convenient. In this regard, Equation (11.95) implies that the TWA’s maximum frequency decreases linearly with the transistor capacitance and thus with the transistor periphery.66 On the other hand, Equation (11.94) states that the gain is directly proportional to the transconductance and then to the transistor periphery. The FET periphery determination is based on the maximum amplifier frequency, by Equation (11.95), while the number of transistors follows from the gain, by Equation (11.94). Thus, many transistors – each with a low individual gain – contribute to the overall gain; this is how the denomination ‘distributed amplifier’ arose. Figure 11.36c shows the complete schematic of a real TWAwith five transistors. The inductors of the gate and drain lines are realized with high-impedance lines, applying the semi-lumped technique. Since Cgs Cds ,67 the distributed capacitance on the drain inductors is higher than on the gate ones, so as to compensate for the difference. Thus, the drain inductors are realized with lower impedance lines than the gate ones. If we consider the complete transistor model – which includes some resistors – and the dissipation loss of the transmission lines, we find that the gain has a negative slope with the frequency. The additional lines TLd1 to TLd5 introduce resonance peaks that compensate for the losses, making the gain flat or even slightly increase with frequency. The drain bias network consists of three capacitors Cdd1 to Cdd3 and one inductor Ldd ; it applies the voltage Vdd to all the drains, without sending DC to the output and to the drain termination. The gate bias network is simpler, consisting of two capacitors Cgg1 , Cgg2 and one resistor Rgg. Such a simplification is possible because there is no current flowing through the gate–source junctions. Again, the same voltage Vgg applies to the gates of all the transistors and is DC isolated from the RF input. The bias networks present a high-pass behaviour and determine the minimum frequency of the TWA, which is zero in principle. Figure 11.37 shows the layout of a monolithic realization of a TWA having the schematic in Figure 11.36c; the die has a size of 2.2 1.7 mm. Note that the components described in Section 14.3 realize all the circuit elements: printed resistors, overlap capacitors, spiral inductors, via hole, etc. The five transistors are four-finger 4 50 mm devices. Many of the transmission lines are folded, to minimize the circuit size. Note also that TLg12 , TLg23 , TLg34 , TLg45 have neither the same length nor double TLg01 and TLg56 , differently from the principle in Figure 11.36a. The reason for the difference comes from 65

More precisely, they are constant-k low-pass filters with T sections, see [14] for more details. The FET periphery is a parameter that quantifies the size of the transistor. See Section 14.3.6 for further details. 67 For instance, in the model of Figure 9.34b, Cgs ¼ 243 fF; Cds ¼ 92 fF. 66

488


Layout of a 2–18 GHz GaAs MMIC TWA, having the schematic of Figure 11.36c.

Figure 11.37 10

20 log10(|s21 |), dB

8

6

best case nominal

4 worst case measured 2 0

Figure 11.38

2

4

6


14

16

18

20

Simulated (black) and measured (grey) gain of the amplifier in Figure 11.37.

a circuit optimization to flatten the gain over the frequency, taking all the parasitics into account: bends on the lines, couplings between adjacent lines, parasitics associated with the passive components, bias networks and the complete model of the transistor. Figure 11.38 shows the simulated and measured gain of the amplifier in Figure 11.37. The two dashed curves are the minimum and maximum simulated gain when the circuit parameters vary within their tolerance range. Note also that the gain has a slightly positive slope with the frequency, to compensate the loss of the passive68 components of the chain where it is placed. 68 Transmission lines and passive components usually exhibit higher loss at high frequency. See also Chapters 3 and 10.

AMPLIFIERS

11.6.3

489

Differential pairs

In modern high-density assemblies, it frequently happens that amplifiers are placed near potential sources of disturbance, like crystal oscillators. The output frequency of those devices is normally some tens of megahertz. Nonetheless, their harmonic content could still be relevant at microwave frequencies. The injection of such disturbances in the amplifier input can degrade the overall system performance, particularly with high-sensitivity receivers.69 The ports of the amplifier considered so far have one common terminal to ground. Consequently the rejection of disturbances can only rely on the shielding of the amplifier input from the various critical zones of the PCB. Differential amplifiers have a greater immunity to disturbing signals, because their ports are couples of terminals with opposite signal with respect to ground, and, clearly, the output difference is proportional to the input one. Consequently, the output tends to cancel any externally induced EM field, which tends to excite the same disturbing signal on the two input terminals. A typical implementation of the differential amplifier is the differential pair (DP), also known as the emitter-coupled pair. The DP is an important building block in RFICs and this section will examine its amplifier application, although it can also work as an oscillator or mixer.70 We will mainly consider BJT-based circuits, like the one in Figure 11.39. Nonetheless, the general conclusions hold true also for FET implementations, albeit with different formulae. First we will examine the static nonlinear transfer characteristic of the DP, then the high-frequency linear operation. A complete analysis involving nonlinear and high-frequency effects requires computer simulation and suitable models.71 Application of the model in Section 9.7.2.4 (DC) to the analysis of the circuit in Figure 11.39 leads to complicated and not very meaningful expressions. Thus, in order to simplify the calculations reasonably and to obtain understandable results, the model needs the following simplifications: .

Both IKF and VA are infinite.

.

bF 1 so that Ib is negligible in respect of Ic and then Ie ¼ Ic þ Ib ffi Ic .

.

Transistor Q1 is identical to Q2 .

.

1 1. The base–emitter junctions of Q1 and Q2 are forward biased, such that exp Vbe;k n1 F VT

With these assumptions for the transistor Q3 and its associated circuitry, we can write ( " ðQ3Þ # ) " ðQ3Þ # Vbe Vbe ðQ3Þ ðQ3Þ ðQ3Þ ðQ3Þ 1 ffi Is exp ðQ3Þ I0 ¼ Ie ffi Ic ¼ Is exp ðQ3Þ nF VT nF VT From this, it follows that the base–emitter voltage of the transistor Q3 is " # ðQ3Þ Ic ðQ3Þ ðQ3Þ Vbe ffi nF VT ln ðQ3Þ Is ðQ3Þ

The voltage balance in the mesh including Vbe , Vee , Re and the base–emitter junction of Q3 imply that ðQ3Þ

Vbe

ðQ3Þ

¼ Vb

ðQ3Þ

Vee Re3 IeðQ3Þ ffi Vbe

Vee Re IcðQ3Þ

69 Consider, for instance, that the sensitivity of a GSM receiver is about 103 dBm. This means that the minimum signal to the input of the receiver LNA is even lower, due to the loss of the components between the antenna and LNA. 70 See Sections 12.5.6, 12.8.3 and 13.3.9. 71 For static, linear RF and large-signal RF analysis of a DP based on the BJT used in Section 9.7, see the files 25_MBC13900_Differential_Pair.sxsch (SIMetrix), 05_BJT_Differential_Pair.adsn (Ansoft) and 26_MBC13900_ Differential_Pair_Spectrum.sxsch (SIMetrix), respectively.

490

MICROWAVE AND RF ENGINEERING Vcc plane of symmetry

(Q2)

(Q1)

Rc =R c

Rc =R c

(2)

(2)

Vin(1)

Vout

Q1 Q2

(Q2)

(Q1)

Vout

Rc (1) Vin

(1) Vout

Rc (2)

Vout

Q1

(2)

Vin

Q2

(2)

Vin

I0 (Q3)

Ic Q3

= I0 (Q3)

Vb

(b)

Re (a)

Vee

(Q2)

(Q1)

Rc

(1)

(1) Vin

Vout

Rc (2)

Vout

Q1

(2)

Vin

Q2 I0 /2

I0 /2

(c)

(Q2)

(Q1)

Rc

(1)

(1) Vin

Vout

Rc (2)

Vout

Q1

(2)

Q2

Vin

(d)

Figure 11.39 DP: (a) BJT implementation; (b) linear equivalent circuit of (a); (c) even mode decomposition of (b); (d) odd mode decomposition of (b). Then, combining the two above equations, we have " # ðQ3Þ Ic ðQ3Þ ðQ3Þ ðQ3Þ Re Ic þ nF VT ln ðQ3Þ ffi Vb Vee Is

ð11:96Þ ðQ3Þ

ðQ3Þ

ðQ3Þ

Now, the first term of Equation (11.96) is the voltage drop across Re , while the term nF VT ln½Ic =Is represents the base–emitter voltage of Q3 . The second term has a logarithmic dependence on the collector ðQ3Þ ðQ3Þ ðQ3Þ ðQ3Þ current of the same transistor, and a convenient design can make nF VT ln½Ic =Is Re Ic . Thus, ðQ3Þ the collector current of Q3 is almost constant and slightly smaller than the value ½Vc Vee =Re , which depends on one resistance and two applied voltages. Therefore, the transistor Q3 and the associated circuitry behave like a current generator, with a current liable to be changed by changing the voltage applied to its base.

AMPLIFIERS

491

This current generator is one of the many possible implementations; the interested reader can find more circuits in [11]. ðQ3Þ One single generator having current Ic can then replace Q3 , Re and the two voltage generators Vee , ðQ3Þ Vb in the circuit in Figure 11.39a. The emitter and collector voltages of Q1 and Q2 are ( " ðQkÞ # ) " ðQkÞ # Vbe Vbe ðQkÞ ðQkÞ ffi Ic ¼ Is exp ðQkÞ Ie 1 ffi Is exp ðQkÞ ðk ¼ 1; 2Þ ð11:97Þ nF VT nF VT The sum of the emitter currents of Q1 and Q2 equals the collector current of Q3 , thus ( " ðQ1Þ # " ðQ2Þ #) Vbe Vbe þ exp ðQ2Þ I0 ¼ Ie;1 þ Ie;2 ¼ Is exp ðQ1Þ nF VT nF VT

ð11:98Þ

The base–emitter voltages of Q1 and Q2 relate to the input voltages as ðQ1Þ

Vbe

ðQ2Þ

Vbe

ð1Þ

ð2Þ

¼ Vin Vin ¼ DVin

ð1Þ ð2Þ The quantity DVin ¼ Vin Vin is the differential input voltage. ðQ1Þ ðQ2Þ ðQ2Þ to (11.98), assuming nF ¼ nF ¼ nF and eliminating Vbe , we

ð11:99Þ

Substituting Equation (11.99) in-

obtain " ðQ1Þ #

V DVin DVin I0 ¼ Is exp be exp þ 1 ¼ IcðQ1Þ exp þ1 nF VT nF VT nF VT ðQ1Þ

A similar relation can be obtained by eliminating Vbe . Rearranging the expressions, we have IcðQ1Þ ¼

I0

; DVin þ1 exp nF VT

ðkÞ

ðkÞ ðkÞ

The output voltages are Vout ¼ Vcc Rc Ic ð1Þ

Vout ¼ Vcc

IcðQ2Þ ¼

I0

DVin þ1 exp nF VT

ðk ¼ 1; 2Þ, thus

Rc I0

; DVin exp þ1 nF VT

ð1Þ

ð2Þ

Vout ¼ Vcc

Rc I0

DVin exp þ1 nF VT

ð11:100Þ

ð2Þ

Then, the differential output voltage DVout ¼ Vout Vout as a function of the differential input is Rc I0 Rc I0

DVin DVin 1 þ exp 1 þ exp nF VT nF VT

1 DVin 1 DVin Rc I0 exp Rc I0 exp 2 nF VT 2 nF VT

¼ 1 DVin 1 DVin 1 DVin 1 DVin exp exp þ exp þ exp 2 nF VT 2 nF VT 2 nF VT 2 nF VT

1 DVin 1 DVin exp exp 2 nF VT 2n V

F T

¼ Rc I0 1 DVin 1 DVin exp þ exp 2 nF VT 2 nF VT

1 DVin ¼ Rc I0 tanh 2 nF VT

DVout ¼

ð11:101Þ

492


Vcc , Vdd

Vout,2 (BJT)

Rc I0

Vcc - Rc I0

Vout,1 (BJT) ∆Vin

-Rc I0

∆Vout BJT (FET)

A v V in 0.5

-A v = (n F VT )-1 I0 Rc, (2 β I0)

Figure 11.40

Rd

Transfer characteristics of the amplifier in Figure 11.39a.

Figure 11.40 plots the transfer characteristics72 from the differential input to the single-ended outputs (11.100) and to the differential output (11.101). Note that all three curves resemble the function (9.48) plotted in Figure 9.20. The differential input to differential output is specifically interesting for the disturbance rejection associated with it, as mentioned at the beginning of this section. Expanding expression (11.101) as a Maclaurin series, we get DVout ¼

1 Rc I0 1 Rc I0 1 Rc I0 DVin þ DVin3 DVin5 þ . . . 2 nF VT 24 ðnF VT Þ3 240 ðnF VT Þ5

ð11:102Þ

Expression (11.102) reveals that the DP has no even-order distortion,73 like the 180 balanced amplifier. The small-signal voltage gain at low frequency is the first-order coefficient of the above expression, which ðQ3Þ ðQ3Þ ðQ3Þ is proportional to the current I0 ¼ Ie ffi Ic that monotonically increases with the voltage Vbe : SSVGDP ¼ lim

DVin ! 0

DVout 1 Rc I0 ¼ 2 nF VT DVin

ðQ3Þ

Thus, Vbe is a control voltage for the amplifier gain; this is an additional interesting feature of the circuit in Figure 11.39a. Truncating the polynomial (11.102) at the third-degree term, and applying Equation (9.51) to it, we approximate the DP 1 dB compression input peak voltage qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDPÞ V1dB ¼ 4 ð1101=20 ÞnF VT An exact numerical computation74 gives a slightly higher value 1.08 times the value above, corresponding to 0.671 dB.

72 The SIMetrix file 25_MBC13900_Differential_Pair.sxsch provides a DC analysis of the transfer characteristic of a DP having the schematic in Figure 11.39a and employing the bipolar transistor considered in Section 9.7.4.2. 73 See the SIMetrix file 26_MBC13900_Differential_Pair_Spectrum.sxsch. 74 See the Mathcad file 13_Tanh_and_Cubic_1dBCP.MCD.

AMPLIFIERS

493

DPs can also work with FETs, and exhibit similar behaviours than BJT-based ones. The calculation of the FET DP transfer characteristic requires a procedure similar to the one used for deriving Equation (11.101): .

Q1 , Q2 are field effect devices. Source, gate and drain replace emitter, base and collector, ðQ1Þ respectively, and also the corresponding voltages and currents. Two drain resistors Rd ; ðQ2Þ ðQ1Þ ðQ2Þ Rd ¼ Rd replace the collector resistors Rc ; Rc ¼ Rc . Furthermore, we will denote the positive supply voltage by Vee instead of Vcc .

.

Equation (9.74) with a tending to infinity (full pentode operation) and l ¼ 0 (perfectly horizontal DC curves) applies to Q1 and Q2 .

With these assumptions, an FET DP has the differential input to output characteristic rffiffiffiffi

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 I0 I0 > 2 > b 2 DV DV < R DV j j > d in in in jDVin j I0 > > : Rd I0 jDVin j DVin b

ð11:103Þ

A comparison between Equations (9.75) and ((9.80) reveals that a MESFET and MOSFET have the same DC behavior in the pentode region. Thus, Equation (11.103) also applies to MOSFET DPs, after replacing b with b/2. The maximum output swing is Rd I0 and can be as large as Vdd , assuming that Rd I0 ¼ Vdd . Figure 11.40 also plots Equation (11.103) in the case of maximum output swing (thick grey curve). Analytic expansion of the function (11.103) – dueptoffiffiffiffiffiffiffiffiffi its piecewise definition – as a Maclaurin series is pffiffiffiffiffiffiffiffiffi only possible within the range I0 =b DVin I0 =b: sffiffiffiffiffiffi sffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffi b b b b ð11:104Þ DV 3 þ Rd DV 5 þ . . . DVout ¼ Rd 2bI0 DVin þ Rd 2 2I0 in 16I0 2I0 in From the series (11.104) follows that FET DPs have only odd-order distortion, like a BJT, and this holds pitffiffiffiffiffiffiffiffiffi true also if jDVin j > I0 =b, because of the odd symmetry of the function (11.103): 8DVin 2 R; pffiffiffiffiffiffiffiffiffi DVout ðDVin Þ ¼ DVout ðDVin Þ. The first-order coefficient Rd 2bI0 is by definition the small-signal voltage difference gain of an FET DP. It decreases with I0 , as with a BJT, but with a square root – rather than linear – law. Figure 11.39b shows the small-signal network equivalent to the DP in Figure 11.39a. This representation assumes the usual linear simplifications (i), (ii) and one additional simplifying hypothesis (iii): (i) Short (open) circuits replace the DC voltage (current) generators. (ii) The linear model or S parameter sets replace the respective transistor. (iii) The current generator based on transistors Q3 is assumed to be ideal. The application of Bartlett’s theorem to the symmetric network in Figure 11.39b allows a straightforward analysis. With simple manipulations, it is possible to show that any input excitation of the DP is the superimposition of even and odd mode ones 8 ð1Þ ð2Þ ð1Þ > V V V Vin;2 SVin DVin > ð1Þ > ¼ þ < Vin ¼ in þ in þ in 2 2 2 2 2 2 ð2Þ ð1Þ ð2Þ ð2Þ > > ð2Þ Vin V V V SVin DVin > : Vin ¼ þ in þ in in ¼ 2 2 2 2 2 2

494

MICROWAVE AND RF ENGINEERING ð1Þ

ð2Þ

ð1Þ

ð2Þ

The quantities SVin ¼ Vin þ Vin ; DVin ¼ Vin Vin are the common and the differential input voltages, respectively. We define the condition with the same voltage simultaneously applied to the ð1Þ ð2Þ two inputs ½Vin ¼ Vin ; DVin ¼ 0 as even, or symmetrical, or common-mode excitation. The opposite ð1Þ ð2Þ case ½Vin ¼ Vin ; SVin ¼ 0 defines the odd, or anti-symmetrical, or differential-mode excitation. The applicability of the effect of superimposition follows from the linearity of the network. Thus the output resulting from any input situation is the sum of the response to even and odd mode excitation. The network in Figure 11.39b presents a plane of symmetry: it then consists of two mirror symmetric halves, each with some nodes intersecting the symmetry plane. For Bartlett’s theorem, in the even (odd) excitation, analyzing the full network is equivalent to separately analyzing the half networks, where the nodes intersecting the symmetry plane are left open (shorted to ground). Thus, in the symmetric and anti-symmetric excitation, the network in Figure 11.39b simplifies to the ones in Figures 11.39c,d, respectively. Figure 11.39c shows that the DP has no common-mode gain, in that the emitter has infinite impedance to ground. On the other hand, in the differential mode, the DP is equivalent to two separated amplifiers, each including one transistor, as Figure 11.39d shows. All the considerations developed in Sections 11.2 and 11.3 therefore remain applicable.

11.6.4

Active loads

The technology of integrated circuits (ICs) usually allows the realization of inductors, but they present low Q factors and occupy large areas of the chip. For instance, the inductor Ldd in Figure 11.37 is considerably larger than a transistor. For this reason, IC designers try to use as few inductors as possible. The need for inductors comes from matching and bias networks. The design techniques to eliminate inductors are combinations of: .

Accepting some interstage mismatch, and avoiding accumulation of the resulting gain ripple, with maxima and minima conveniently located. Normally, the interconnections inside the IC are very short, so the frequency distance between two mismatch gain peaks is relatively high.

.

Using the transistor periphery as a design parameter, in order to minimize the impedance mismatch between the interfacing ports of two adjacent stages.

.

Putting the matching network outside the chip, for instance in the module that includes and encloses the chip.

.

Using resistors instead of inductors in bias networks, wherever this is possible – in practice in gate/base bias.

.

Replacing the drain/collector bias inductor with a current generator, similar to the one in Figure 11.39a. Transistor-based current generators in this application are known as active loads.75

Current generators behave similarly to inductors, presenting high RF impedance and allowing direct current to flow; their application as DC-feed elements in bias networks is attractive in terms of chip area occupation. However, the price to pay for the reduction in size is a reduction in amplifier efficiency by around 50%. The reason for this degradation is because active loads have no reactive energy to accumulate and return to the network, differently from inductors. To see this, it is sufficient to reconsider the network in Figure 11.11 and replace the drain inductor Ldd with a current generator Iq. The latter works if the voltage across its terminals Vq Vds;ce is higher than a specified value, which is zero in the ideal case. This means that the drain/collector voltage can swing from zero to Vq. Thus the load peak voltage is Vq =2, with a peak current of Iq. The resulting maximum RF output power is PDC ¼ Vq Iq =4 (with a load

75

One simple current generator is the current mirror, described in the reference [15].

AMPLIFIERS

495

ZF (CF) I1

I2

I1

V1

V2 = V1AV

I2

V1

AV

V2 = V 1 A V AV

(a) C4

Vdd L4

Q2

C5

P2

L3 C1

P1

Q1

L1

Vg2 C3

Vg1 (b)

C2

Z d1 Cgd

P1

G1 + vin -

(c)

Cgd D1

gm vin

S1

Rds

G2

D1 + vin -

gm vin

P2

Rds

S2

Figure 11.41 Cascode configuration: (a) Miller’s theorem; (b) electrical diagram of two FETs in cascode connection; (c) approximated linear equivalent network of (b). resistance then of 0:5 Vq =Iq ) and the DC power is PDC ¼ Vq Iq . Then the maximum efficiency of a class A PA with active load is 25%, one-half of the solution with the inductor as drain/collector DC feed.

11.6.5

Cascode configuration

Figure 11.41b shows a cascode configuration consisting of one common source/emitter transistor (Q1 ) followed by a common gate/base one (Q2 ). Figure 11.41b also includes the transistor bias networks. Compared with a single transistor, the cascode employing two transistors with the same type as the single one has lower input capacitance (i), higher output resistance (ii) and slightly lower transconductance (iii). Here, we will give some qualitative justifications of these assertions, using a simplified linear FET model. The same considerations apply to the BJT case, with minor modifications. The FET linear model in Figure 9.34b includes three capacitors Cgs ; Cds and Cgd . Cgs and Cds approximately coincide with the input and the output capacitance, respectively, while Cgd contributes to increase both. The validity of this assertion is based on the network equivalence in Figure 11.41a, known as Miller’s theorem.76 The block with the symbol of an amplifier in Figure 11.41a is a linear two-port network presenting a voltage gain AV with given input and output terminations. On the left of Figure 11.41a, the feedback 76

See [16].

496


bipole ZF is connected between the input and output of AV . The input generator and the amplifier output deliver the currents V1 V1 V2 V1 ð1 AV ÞV1 þ ¼ þ Zin ZF Zin ZF

1 1 V2 V2 V2 V1 V2 AV I2 ¼ þ ¼ þ ZL ZF ZL ZF I1 ¼

where Zin ; ZL are the input and the termination impedances of the amplifier, respectively (not shown in Figure 11.41a). Therefore, the feedback impedance increases the input and output current with the terms 1 ð1AV ÞZF1 V1 and ð1A1 V ÞZF V2 . Thus, from the input and output current point of view, the presence of the feedback bipole is equivalent to shunting the amplifier input and output with two bipoles, having 1 impedance ð1AV Þ1 ZF and ð1A1 V Þ ZF . In other words, the two networks on the left and right sides of Figure 11.41a are equivalent. Note that the feedback bipole also affects the voltage gain AV if the input generator or the amplifier output impedances are non-zero: the AV value to consider is the one with the actual terminations, including ZF . Applying Miller’s theorem to Cgd in the model of Figure 9.34b, and neglecting the gain modification due to the feedback bipole, we have ZF ¼ ð jo Cgd Þ1 , with the associated increase in input and output capacitance DCin ¼ ð1AV ÞCgd and DCout ¼ ð1A1 V ÞCgd . The exact determination of the voltage gain AV involves the analysis of a complex network; however, some simplifying hypotheses give a qualitative expression. The first transistor in Figure 11.41c uses a simplified version of the model in Figure 9.34b, with only the transconductance, the output resistance, and Cgd . Zd1 representing the output termination of Q1 : in Figure 11.41c it is the input impedance of the common gate Q2 , but we can use a generic value to analyze the single common source stage. In this simplified hypothesis,77 AV ¼ gm Rds jjZd1 , then the approximate FET input and out capacitance are respectively

1 Cin Cds þ ð1 þ gm Rds jjZd1 ÞCgd ; Cout Cds þ 1 þ ð11:105Þ Cgd gm Rds jjZd1 The values in (11.105) depend on the impedance Zd1 of the output termination of the transistor Zd1. As a reference, the low-frequency conjugate matching ðZd1 ¼ Rds Þ and the parameters of the model in Figure 9.34b give Cin 243 þ 299 fF; Cout 92 þ 64 fF. Thus, the Miller capacitance Cgd contributes to the overall input capacitance by about 55%. Equation (11.41) highlights that minimization of Zd1 minimizes the feedback contribution to the input capacitance: this is exactly what the common gate Q2 provides. Further simplifying the second transistor model, and considering its transconductance and output resistance only, its input impedance is Zd1 ¼

Rds þ RL 1 þ gm Rds

where RL is the resistance terminating the drain of Q2 . Again assuming RL ¼ Rds , and from the parameters in Figure 9.32b, Zd1 ffi 45:9. With this drain impedance, the input capacitance of Q1 becomes Cin 243 þ 139 fF, which is about 70% of the standard common source. A relative simple analysis of the network in Figure 11.41c, with all capacitances removed, shows that the DC output impedance of the cascode is Rds ð2 þ gm Rds Þ, which is higher than Rds, as anticipated. Similarly, the equivalent cascode transconductance is ½1ð2 þ gm Rds Þ1 gm . Assuming the model parameters in Figure 9.32b, we have ð2 þ gm Rds Þ1 ffi 0:076; then the transconductance 1 1 The quantity Rds jjZd1 ¼ ðR1 denotes the impedance resulting from the parallel connection of the ds þ Zd1 Þ two bipoles. 77

AMPLIFIERS

497

of the cascode is slightly smaller than that of the single transistor in the common source configuration. The considerations developed so far show the qualitative trend of the cascode configuration. Nevertheless, they give a good design starting point, despite the many approximations with them. Clearly, accurate and reliable predictions require circuit simulation78 and suitable models, as usual. The application of a cascode in amplifiers having the structure in Figure 11.1 could be problematic, due to the difficulty of matching the high cascode output resistance. On the other hand, different amplifier types take advantage of that characteristic. One typical case is the TWA: the output resistances of its transistor cause dissipation loss in the output line, and consequently a drop in the gain at high frequency. Increasing the transistor output resistance means reducing the loss along the drain line, and facilitates the achievement of a flat gain over the frequency. It is implicit from our – albeit approximate – analysis that the connection between the drain of Q1 and the gate of Q2 in the circuit of Figure 11.41b has to be as short as possible. Also, the gate of Q2 needs to see a low RF impedance to ground. Consequently, IC technology is particularly suitable for the cascode.

11.7 Some examples of microwave amplifiers This section presents two practical amplifier realizations with relatively high performances. The first one is a millimetre-wave amplifier, the second is a low-noise amplifier. Both the circuits use as many printed circuit elements as possible, in order to reduce the number of components and consequently the assembly time. Apart from the cost reduction, printed circuit elements are attractive also because they generally offer more constant and predictable performances than their discrete counterparts.

11.7.1

Two-stage millimetre-wave amplifier

The amplifier presented in this section is realized in hybrid microwave integrated circuits (MICs) technology; it uses two identical transistors to realize two cascaded stages. The working frequency bandwidth ranges from 34 to 36 GHz, with a nominal gain of 13 dB. Its design is a compromise among port impedance matching, noise figure and output power. Figures 11.42a,b show the electrical diagram and the layout of the amplifier, respectively. The input matching network of Q1 has the same principle79 as in Figure 6.11, with the coupled line section CL1 working as both a l=4 transformer and DC block. CL1 is a coupled-line section having even and odd mode impedance Z0e ; Z0o respectively, and electrical length80 of about p/2 at the centre frequency of 35 GHz. For the network identity in Figure 4.25b, at 35 GHz, CL1 is equivalent to a single transmission line having a characteristic impedance of Z0 ¼ ðZ0e Z0o Þ=2. The combination of the open stub ST1, resistors R1 to R3, transmission line TL1 and capacitor C1 works as the gate bias network of Q1 and as part of its input matching network. It cancels the input susceptance of Q1 ; then CL1 matches a resistive impedance into 50 O. The capacitor C1 is a short circuit to ground at RF, but allows the DC voltage Vg1 to reach the gate of Q1 . The combined action of C1 and R3 ensures a sufficient decoupling between the DC gate bias port and the RF input port of the amplifier. The 78 See the Ansoft file 06_Cascode.adsn for the analysis of the scattering parameters of a cascode using two transistors having the model in Figure 9.32b. 79 In this case, the gate of Q1 and the amplifier input connect to ports 2 and 1 of the matching network, respectively. The stub cancels the transistor input reactance and the line works as a standard l=4 transformer, placed between the resistive load and the source. 80 For microstrip coupled lines there is no precise definition of the electrical length, because the two modes propagate with different velocity. However, it is possible to use an equivalent propagation velocity equal to the arithmetic or geometric mean between the even and odd mode propagation velocity, if the two values are not too different.

498

MICROWAVE AND RF ENGINEERING Vdd,1

C2

TL1

R4 Q1

CL2

R1

CL1

C4

TL4

TL2

ST1 IN

Vdd,2

TL3 R3

Vgg,1

TL 8

ST2 TL 5

TL9

TL 6 Q2 R5

OUT

CL3

TL7

R2

R6

R7

Vgg,2

C1

C3

(a)

Carrier

C2 ST1 CL1

Q1

IN

R1 TL3

Vgg,1 R3

Vdd,1 TL4

ST2

C1

Q2

R4 TL5

TL2 CL2 R2

C4

substrate

Vgg,2

R5 TL7 R7

Vdd,2 TL9 TL 8 CL 3

OUT

R6

C3

(b)

Figure 11.42

Millimetre-wave (34–36 GHz) amplifier: (a) electrical diagram; (b) layout.

input bias and matching network include three resistors (R1 to R3 ), which increases the input return loss and improves the stability, but degrades the noise figure. The capacitor C4 shorts one terminal of the transmission line TL9 to ground. Thus, TL9 behaves like an RF short-circuit stub, and also provides a path to the drain of Q2 for its DC voltage Vd2 . The output matching network works similarly to the input one: TL9 and C4 cancel the output reactance of Q2 , and the coupled lines CL3 provide DC isolation and complete the matching, passing from a resistive value to 50 O. The output matching and bias network is the only one without resistors, in order to minimize the loss on the output power, although Q2 is matched for the maximum gain and not for the maximum power. The interstage matching network essentially consists of two cascaded networks of the abovedescribed type. The additional resistor R4 completes the amplifier stabilization. The microstrip elements of the amplifier’s matching networks are realized on three alumina rectangles having h ¼ 254 mm and er ¼ 0.9: Figure 11.42b indicates them with three dashed rectangles. All the resistors are thin-film ones, and the transistors and capacitors are chip devices connected with

AMPLIFIERS

499

16

20 log10(|s21|)

14

12 simulation 10 measurement 8

6 32

Figure 11.43

33


37

38

Simulated (solid) and measured (dashed) gain of the amplifier in Figure 11.39.

gold bond wires. The three alumina rectangles and the chip devices are brazed on one metal carrier that works as a common ground plane and mechanical support as well. The carrier size is about 6 4 mm. Figure 11.43 shows the simulated (thick curve) and the measured (thin curve) SSG of the amplifier. The difference between the two curves is less than 2 dB over all the frequency range 32–38 GHz and becomes less than 1 dB in the specified bandwidth 34–36 GHz. Within this latter range, the measured amplifier exhibits an input and output return loss of better than 10 dB, a noise figure of about 5 dB and saturated output power of about 10 mW (þ 10 dBm).

11.7.2

Low-noise amplifier

Figure 11.44 shows the schematic of an LNA81 working in the frequency range from 10 to 12 GHz. It is realized with surface mount device (SMD) technology on a soft microstrip substrate ðh ¼ 508 mm; er ¼ 3:5Þ with a via hole to ground. The matching/bias networks of this amplifier use basically the same philosophy and the same circuit elements as the ones for the amplifier in Section 11.7.1. Differently from that case, the present LNA includes radial stubs (RS1, RS2) as RF short circuits to ground. Radial stubs behave approximately like open stubs with low characteristic impedance. For our needs it is sufficient to remember that radial stubs present a low impedance to ground over a wider range than conventional l/4 open stubs. Furthermore, the capacitors C1 , C2 (C3 , C4 ) complete the RF decoupling of the input (output) DC bias port from the corresponding RF port. The resistors R1 , R2 stabilize the amplifier by providing a suitable dissipation loss at the transistor input and output. Figure 11.45 shows a photograph of the realized amplifier. Note that Q1 presents a non-negligible series impedance to ground, due to three contributions: the bond wires connecting the chip terminal to the leads inside the transistor package, the non-zero length of the leads and the via hole inductance. Therefore, our circuit exploits the connection parasitics to implement the series source inductance technique discussed in Sections 11.3 and 11.4. Figure 11.46 shows the measured performances of the amplifier. In the nominal range of 10–12 GHz, we have a gain of about 12 dB, an input and output return loss of better than 10 dB, and a noise figure of 81 The amplifier described in this section is the property of RF Microtech. Photograph, descriptions and data are published by permission.

MICROWAVE AND RF ENGINEERING R2

TL9

RS2

TL 6 TL1

TL 7

OUT

Q1

TL5 C1

TL3 R1 RS1

C2

Figure 11.44

Figure 11.45

CL2

LNA schematic.

Photograph of the LNA having the schematic in Figure 11.44.

15

s21

10

2.0

10

1.5

0

NF

5

1.0

s11

0

-5 9

Figure 11.46

10


NF, dB

s22 -10

0.5

-20

0.0

-30

20 log10(|s11|), 20 log10 (|s11|)

Vgg

TL4

Vdd C4

TL2

IN CL1

TL10 C3

TL 8

20 log10(|s 21|)

500

14

Measured performances of the LNA in Figure 11.45.

AMPLIFIERS

501

better than 0.85 dB with a minimum of 0.58 dB at 10.6 GHz. Note the effectiveness of the source inductance in optimizing the compromise between noise figure and input return loss.

Bibliography 1. G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, Prentice Hall, New York, 1984, sections 3.4 to 3.8, pp. 102–125. 2. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 7.2.1, pp. 423–426. 3. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, Chapter 10, pp. 349–400. 4. N. O. Sokal and A. D. Sokal, ‘Class E: a new class of high efficiency tuned single-ended power amplifiers’, IEEE Journal of Solid-State Circuits, Vol. SC-10, No. 3, pp. 168–176, 1975. 5. W. H. Doherty, ‘A new high efficiency power amplifier for modulated waves’, Proceedings of the IRE, Vol. 24, No. 9, pp. 1163–182, 1936. 6. R. J. McMorrow, D. M. Upton and P. R. Maloney, ‘The microwave Doherty amplifier’, IEEE MTT Symposium Digest, pp. 1653–1656, 1994. 7. D. R. Cox, ‘Linear amplification with nonlinear components’, IEEE Transactions on Communications, December, pp. 1942–1945, 1974. 8. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, Chapter 8, pp. 462–484. 9. K. B. Niclas, W. T. Wilser, R. B. Gold and W. R. Hitchens, ‘The matched feedback amplifier: ultrawide-band microwave amplification with GaAs MESFET’, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-28, pp. 285–2945, 1980. 10. A. M. Pavio, S. D. Carter and P. Saunier, ‘A monolithic multi-stage 6-18 GHz feedback amplifier’, IEEE Microwave and Millimeter-Wave Symposium, 1984, pp. 45–48. 11. P. R. Gray, P. J. Hurst, S. H. Lewis and R. G. Meyer, Analysis and Design of Integrated Circuits, John Wiley & Sons Inc, New York, 2001, Chapter 8, pp. 553–622. 12. E. L. Ginzton, W. R. Hewlett, J. H. Jasberg and J. D. Noe, ‘Distributed amplification’, Proceedings of the IRE, August, pp. 956–969, 1948. 13. N. B. Schrock, ‘A new amplifier for milli-microsecond pulses’, The Hewlett-Packard Journal, Vol. 1, No. 1, pp. 1–4, 1943. 14. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 2.3, pp. 57–66. 15. P. R. Gray, P. J. Hurst, S. H. Lewis and R. G. Meyer, Analysis and Design of Integrated Circuits, John Wiley & Sons, Inc., New York, 2001, Chapter 4, pp. 253–298. 16. J. M. Miller, ‘Dependence of the input impedance of a three-electrode vacuum tube upon the load in the plate circuit’, Scientific Papers of the Bureau of Standards, Volume 15, Government Printing Office, Washington, DC, 1920, pp. 367–385.

Related files Ansoft files 01_Unilateral_and_Bilateral_Matching.adsn. Analyzes the unilateral and bilateral matching of the transistor having the scattering parameters as in Table 9.3. The unilateral device is an ideal twoport network having the same scattering parameters as the bilateral one, except for the backward transmission coefficient, which is zero.

502


02_Bilateral_Matching_Layout.adsn. Generates a microstrip layout from the optimized design obtained in file 01. 03_Matching_Networks_1000_to_50_Ohm.adsn. Provides various impedance matching networks for the amplifier in Section 11.5.1. 04_Feedback_SSA.adsn. Analyzes a simplified feedback amplifier, based on the transistor model in Figure 9.32b. 05_BJT_Differential_Pair.adsn. Provides the high-frequency scattering parameters of the amplifier in Figure 11.39a, from the differential input to the differential output. 06_Cascode.adsn. Compares the single FET with the cascode scattering parameters, assuming that the single and the cascode ones are of the same type as the linear model in Figure 9.35a.

Mathcad files 07_Simultaneously_Conjugated_Matching.MCD. Provides some support calculations for amplifier 1. 08_Passive_T_Attenuator.MCD. Synthesizes the resistors of a T attenuator. 09_Class_A_B_C.MCD. Performs numerical computations of most of the parameters discussed in Section 11.5.5. 10_Class_C_F.MCD. Performs numerical computations of most of the parameters discussed in Section 11.5.5. 11_Predistortion.MCD. Analyzes a simple predistortion amplifier linearization, as in Section 11.5.6.1. 12_Spectral_Regrowth.MCD. Computes the spectral regrowth associated with a cubic compressing characteristic. 13_Tanh_and_Cubic_1dBCP.MCD. Computes the 1 dB compression point of cubic and hyperbolic tangent compressing curves.

SIMetrix files 14_MBC13900_LOAD_Line.sxsch. Simulates a quasi-static load line of the amplifier in Figure 11.13a by means of a low-frequency (1 MHz) transient analysis. 15_MBC13900_LOAD_ Line_1GHz.sxsch. Simulates the load curves at 1 GHz of the PA in Figure 11.13a at different output powers. 16_MBC13900_LOAD_Line_1GHz_RL.sxsch. Simulates the load curve at 1 GHz of the PA in Figure 11.13b at different output powers. 17_MBC13900_LOAD_Line_1GHz_DC_Servo.sxsch. Simulates the load curve at 1 GHz of the PA in Figure 11.3d at different output powers. 18_MBC13900_1GHz_PS.sxsch. Simulates the compression curve of the PA in Figure 11.13a. 19_MBC13900_1GHz_RL_PS.sxsch. Simulates the compression curve of the PA in Figure 11.13b. 20_MBC13900_1GHz_RL2_PS.sxsch. Simulates the compression curve of the PA in Figure 11.13c. 21_MBC13900_1GHz_DC_Servo_PS.sxsch. Simulates the compression curve of the PA in Figure 11.13d. 22_MBC13900_1GHz_DC_Servo_Burst.sxsch. Analyzes the RF burst response of the amplifier in Figure 11.13d. 23_Diode_Predistorter_Curve.sxsch. Computes the transfer characteristic of a diode predistorter of the type considered in Section 11.5.6.1. 24_Class_E_GHz.sxsch. Simulates voltage and current waveforms of a quasi-ideal class E amplifier. 25_MBC13900_Differential_Pair.sxsch. Generates the static transfer characteristics of a BJT DP. 26_MBC13900_Differential_Pair_Spectrum.sxsch. Computes the output spectrum of the same DP as in file 24, with a sinusoidal input.

12

Oscillators 12.1 Introduction Oscillators produce the high-frequency signals that the other microwave components process. Therefore, oscillators are probably the most important high-frequency components: without them there is no RF and no microwaves. From the conceptual point of view, oscillators consist of a series or parallel resonant circuit, which is the resonator, and some active circuitry that compensates for the resonator loss. Section 12.2 deals with the general working principles of the oscillators. Sections 12.3 and 12.4 analyze the negative resistance and the positive feedback approaches, respectively. Section 12.4 also shows the equivalence between these two approaches. Section 12.5 considers the most common oscillator configuration, presenting some analytic techniques which are suitable for the simplest cases. Section 12.6 illustrates a trial design in order to clarify the concepts and the implications involved with the previous sections. Section 12.7 illustrates the principal oscillator specifications and their relations with the design parameters. Section 12.8 presents some circuits that are widely used in engineering practice. The description of a real oscillator design, presented in Section 12.9, completes the chapter.

12.2 General principles As anticipated in the introduction, the resonator is one of the two main oscillator elements. Chapter 5 described resonators in depth, while here the word ‘resonator’ denotes a reactive bipole, presenting a zero for the imaginary coefficient of its immittance at a certain frequency f0, known as the resonant frequency of the resonator. We have a series or parallel resonator in the case of a reactance or susceptance zero, respectively. In ideal resonators, not only the imaginary coefficient of the immittance but also its real part are zero at the resonant frequency. The resonator is also known as the resonating circuit, while the parallel resonator is also referred to as the anti-resonant circuit. Also, some authors use the words ‘resonator’ and ‘tank’ interchangeably. Figure 12.1 shows two of the simplest series (a) and parallel (b) resonating circuits: one or the other of these two RLC bipoles can approximate any arbitrarily complex reactive bipole realizing a resonator, in the neighbourhood of its resonant frequency.1 Note also that the network in Figure 12.1a is the dual of the 1

See also Chapter 5 and [1].


504


Cs

Ls

Rs Lp

Cp

Rp

(a)

(b)

Figure 12.1

Resonating circuit: (a) series; (b) parallel.

one in Figure 12.1b. Thus the equation for the latter also applies to the first, by swapping voltages across with currents through, and, consequently, Cs ; Ls ; Rs with Lp ; Cp ; R1 p , respectively. For this reason, we will derive all the expressions for the anti-resonant circuit; the other case is obtained by duality. The circuit in Figure 12.1b includes no current generator, thus the sum of the currents through the three components Lp ; Cp and Rp is zero. From this, the voltage vðtÞ across the parallel RLC circuit follows the differential equation 1 Lp

ðt vðtÞ dt þ 1

1 dvðtÞ vðtÞ þ Cp ¼0 Rp dt

Differentiating both sides, we have 1 1 dvðtÞ d 2 vðtÞ þ Cp vðtÞ þ ¼0 Lp Rp dt dt2

ð12:1Þ

Equation (12.1) is a homogeneous linear second-order differential equation, its general solution being a linear combination of particular ones of the type eat . The substitution vðtÞ ¼ eat into Equation (12.1) gives 1 at a at e þ e þ a2 Cp eat ¼ 0 Lp Rp

ð12:2Þ

The function eat is always non-zero, therefore Equation (12.2) implies that 1 a þ þ a2 Cp ¼ 0 Lp Rp Thus, we have an ordinary second-degree equation in the variable a, which has the solutions s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 1 ð12:3Þ a¼ 2Rp Cp 2Rp Cp Lp Cp The general solution of Equation (12.1) is a linear combination of two functions of the type eat , where a is given by Equation (12.3), and assumes the form vðtÞ ¼ v1 expða1 tÞ þ v2 expða2 tÞ

ð12:4Þ

As anticipated, a1 ; a2 are the two values obtained from expression (12.3) considering the signs ‘ þ ’ and ‘’, respectively. Note that the function (12.4) is the solution of Equation (12.2) in all the cases, with the exception of a1 ¼ a2 , which occurs when Lp ¼ 4R3p Cp . However, we will ignore this single exception, in that it is not interesting for our considerations, as we will see shortly. The determination of the coefficients v1 ; v2 follows from the initial conditions on the function and its derivative with respect to time. If they are both zero, i.e. vðt ¼ 0Þ ¼ dvðtÞ=dtjt¼0 ¼ 0, the function (12.4) degenerates into vðtÞ ¼ 0. In all the other cases, we have a non-trivial evolution over time. The simplest

OSCILLATORS initial condition to consider consists of assuming a non-zero initial voltage dvðtÞ ¼0 vðtÞjt¼0 ¼ v0 ; dt t¼0

505

ð12:5Þ

Imposing the condition (12.5) on Equation (12.4) gives 8 vðt ¼ 0Þ ¼ v1 þ v2 ¼ v0 > < > dvðtÞ ¼ a1 v1 expða1 tÞ þ a2 v2 expða2 tÞj ¼ a1 v1 þ a2 v2 ¼ 0 : t¼0 dt t¼0 The above two equations form a non-homogeneous linear system with the two unknowns v1 ; v2 , and their solution is v1 ¼

a2 v0 ; a2 a1

v2 ¼

a1 v0 a1 a2

Finally, substituting the above coefficients into Equation (12.4), and rearranging the result, we get the solution t ð12:6Þ vðtÞ ¼ A exp sinðO t þ fÞ t where V0 A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; L 1 4CppR2 p

t ¼ 2Cp Rp ;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 O¼ ; Lp Cp ð2Cp Rp Þ2

1

j ¼ tan

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4Cp R2p 1 Lp

Let us define the (angular) resonant frequency o0 ¼ 2pf0 and the quality factor Q, sffiffiffiffiffiffi Cp 1 o0 ¼ pffiffiffiffiffiffiffiffiffiffi ; Q ¼ Rp Lp Lp Cp

ð12:7Þ

Note that the resonant frequency (12.7) coincides with the susceptance zero of the circuit in Figure 12.1a, as expected. Moreover, the quality factor Q is the ratio between the susceptance of one of the reactive elements o0 Cp ¼ ðo0 Lp Þ1 and the conductance at the resonant frequency.2 The parameters of Equation (12.6) can be rewritten in terms of the resonant frequency and quality factor as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Q2 1 2Q 2Q ð12:8Þ 4Q2 1 A ¼ v0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t ¼ ; O¼ o0 ; f ¼ tan1 o0 2Q 4Q2 1 Assuming non-zero initial conditions different from (12.5), the resonator voltage over time always assumes the form (12.6), but with different values of A and f. The same applies to the response of the antiresonant circuit to a current step applied to its terminals. Similar results also arise from any finite and time-limited excitation. If jQj > 0:5 – lower values are of no interest in this context – then radian frequency O is a real number, and the function (12.6) describes oscillations having an exponential envelope with the time constant t. The first three plots of Figure 12.2 – a, b and c – show the three important cases of vðtÞ, when Q > 0:5; Q < 0:5; Q ¼ 1. In these plots, the function (12.7) is a thick solid curve, while the thin dashed lines show the envelope A expðt=tÞ. Strictly speaking, the waveforms in Figure 12.2 are not periodic, in that their peak amplitude varies over time, with the exception of the case in Figure 12.2a (Q ¼ 1). 2

See also Section 5.3.

506

MICROWAVE AND RF ENGINEERING Q=∞ 0

t

(a) Q 0.5

0

t

(c) amplitude-depending Q 0

t

(d)

Figure 12.2 Switch-on transients of a resonant circuit: (a) infinite Q (no resistance); (b) negative Q; (c) positive Q; (d) amplitude-depending Q.

All physical resonators present finite and positive Q. The case of infinite quality factor denotes the ideal resonator, consisting of purely reactive elements ðRp ¼ 1Þ. Clearly, ideal resonators and purely reactive elements as well are not realizable. If, absurdly, they were, they would still be of no practical use, because any attempt to couple them externally causes energy transfer from inside to outside the resonator. In other words, any load somehow applied to the resonator introduces a parallel resistance, Q becomes finite and the oscillation amplitude decays over time. Oscillation negative-Q resonators increase their amplitude over time. This implies that the resonator energy also increases over time, and thus that energy feeds to the resonator externally. In other words, a resonator with negative quality factor has to include active elements. Thus, negative-Q resonators produce increasing amplitude oscillations starting from any non-zero initial condition or from finite excitations. This mechanism realizes oscillator startup, and the initial resonator stimulus could be a switch-on transient of the oscillator, or even the circuit noise. Oscillators, however, have to produce waveforms with amplitude and frequency ideally constant over time. None of the conditions considered so far give satisfactorily working oscillators. Finite-Q resonators do not behave as oscillators: positive and negative ones exhibit exponentially decaying and increasing oscillation amplitude over time, respectively. In the first case the oscillation is simply extinguished; the second case involves arbitrarily high voltages, and is then not realizable. On the other hand, a loss-free resonator is still not adequate, despite its theoretically constant amplitude, because any perturbation applied to the resonator (variations in the load, noise, disturbances) can cause amplitude variations, and the circuit is incapable of restoring the initial condition.

OSCILLATORS

507

Id (1) (2)

∆ Id

(3)

Lb

(4)

Cb

+ Vb

Vd

t

Id

+ Vd

(2)

R NEG

Lp

Cp

Rp

-

(1)

(1) (4)

Vd

(a)

(b)

Figure 12.3

∆Vd

Oscillator with negative resistance element: (a) schematic; (b) voltages and currents.

Therefore oscillators require a combination of Q < 0:5; Q > 0:5 and Q ¼ 1 for small, large and steady state amplitudes. Moreover, Q1 has to decrease monotonically with oscillation amplitude, in order to stabilize the oscillation amplitude. This way, the negative Q increases the oscillation amplitude from an initial non-zero arbitrarily small perturbation. As the amplitude increases, any further increase becomes progressively more difficult. At a given amplitude vosc , Q becomes infinite, causing the amplitude neither to increase nor decrease; vosc is then the equilibrium point of the system. As the oscillation amplitude – for any external or internal perturbation – increases above (respectively, decreases below) vosc , then Q becomes positive (negative), compensating the effect of the perturbation. Figure 12.2 – plot d – shows a typical oscillator startup waveform. Initially the amplitude increases exponentially, as with negative Q. Consequently, Q becomes less negative, making the amplitude increment less evident. Finally, the oscillation amplitude stabilizes and Q becomes infinite. At this point, for any increment (decrement) of the output amplitude, Q assumes finite positive (negative) values, opposing the variation and stabilizing the amplitude. We define the initial part of the oscillator switch-on transient, when it works as a negative-Q resonator, as the startup. The time when the oscillation reaches and keeps constant amplitude is the steady state. Figure 12.3a shows the principle of a circuit implementing the mechanism described above. That oscillator comprises the anti-resonant circuit Lp ; Cp ; Rp and the active device RNEG with its associated bias elements Vb ; Lb ; Cb , which produces the amplitude-dependent negative resistance. In our simple schematization, the resistor Rp represents the loss in the resonator and the effect of any load coupled to it; Lb ðCb Þ presents very high (low) impedance at the oscillation frequency.3 In addition, we will assume that RNEG presents no parallel susceptance. This involves no loss of generality, in that any non-zero parallel susceptance can be considered as included in the resonator, by modifying some of the values Lp ; Cp ; Rp . Figure 12.3b shows the Id –Vd characteristic of the device RNEG , one portion of the curve presenting a negative slope that corresponds to an incremental negative resistance. For the same reasons as considered in Section 11.5.1, the voltage Vd can swing from Vb down to zero and up to a positive value Vmax , which depends on Rp . If Vmax 2Vp then the voltage across RNEG is a sinusoid with both mean and peak values equal to Id –Vd. We will not describe the internal structure of the

3

Similar to the components Ldd and Cdd in Figures 11.11 and 11.13.

508


active device; however, the characteristic in Figure 12.3b resembles that of a tunnel, Gunn or IMPATT diode.4 Figure 12.3b also shows four different possible sinusoidal voltages across RNEG , having increasing amplitude, and labelled (1) to (4). The negative conductance DId =DVd monotonically increases with the oscillation peak-to-peak amplitude DVd : it is negative for (1) and (2), zero for (3) and positive for (4). Consequently, the resonator sees the active devices as conductance that is negative for small DVd , decreases with DVd and becomes positive for DVd > DVdð3Þ. The current in the active device has a nonlinear dependence on the voltage. Nonetheless, it is still possible to use the parameter DId =DVd as a large-signal conductance, in order to develop some qualitative considerations.5 Thus, the resulting resonator parallel resistance is approximately 1 R1 eq ¼ ðRp þ DId DVd Þ. If Req is negative for small values of DVd , then the circuit in Figure 12.3a satisfies one of the conditions required to operate as an oscillator: the small-amplitude Q is negative. Then oscillations with angular frequency O will arise from any non-zero initial condition and exponentially increase their amplitude. As DVd approaches DVd;g ; R1 eq vanishes, and the resonator Q becomes infinite, stabilizing the oscillation amplitude6 at a value close to DVd;g : any variation of DVd above or below that value is contrasted by the Q variations associated with the function Id ðVd Þ. In the steady state condition Q tends to infinity, thus the oscillation frequency O coincides with o0 ¼ ðLp Cp Þ1 . If the active device characteristic is such that DId =DVd < R1 p , even for small DVd , then stable oscillations cannot take place. Nevertheless, it is still possible to circumvent the obstacle, by inserting a transformer between Cb and the resonator, in order to increase the negative conductance seen by the resonant circuit.7 Therefore an active device having the characteristic of the form in Figure 12.3b is suitable for working as a negative resistance in an oscillator employing a parallel resonating circuit. For duality reasons, series resonating circuits require a negative resistance – rather than a negative conductance – decreasing with oscillation amplitude.

12.3 Negative resistance oscillators This section shows that a negative conductance device in parallel with an anti-resonant circuit restores the energy lost at each cycle due to the loss internal to the resonator and to the energy transfer to the load. Moreover, we have seen that a negative conductance that decreases with the oscillation amplitude stabilizes the amplitude itself. By duality, the same considerations apply to a negative resistance in series with a series resonant circuit. The present section analyzes the more general case of both resonator and active device presenting complex impedance, deriving the conditions for the oscillations in a more formal way.8 Figure 12.4a shows the principle of a negative resistance oscillator. The resistor RL is the load, which is somehow coupled to the resonator. The two-port includes the resonator and its coupling/matching networks to load and the active circuit, and is placed between those elements. Any non-zero series reactance of the load is considered as part of its coupling network to the resonator. The active circuit has complex admittance with negative real part. The two-port network also includes the bias elements for the active device, which are not explicitly represented in Figure 12.4a. Figure 12.4b shows the equivalent circuit of the schematic in Figure 12.4a. The bipole YðoÞ is the complex admittance presented from the resonator to the active circuit. The current generator cðtÞ provides 4 The SIMetrix file 07_Negative_R_I_V_Curve.sxsch provides a macroscopic model of an Id ðVd Þ characteristic of the type plotted in Figure 12.3b. 5 Section 12.4 provides more rigorous analyses. 6 See the SIMetrix file 09_Negative_R_Oscillator.sxsch, which provides a switch-on transient analysis of an oscillator having the configuration in Figure 12.3a. 7 Equivalently, we can also say that the transformer decreases the conductance seen by RNEG. 8 See also [2].

OSCILLATORS

509

Y(ω)

resonant circuit with matching and coupling networks

RL load

active circuit -G + jB

(a) + i(t) c(t) (b)

Figure 12.4

Y(ω)

v(t)

I(t)

Y′ = -G′ + jB′

-

Negative resistance oscillator: (a) principle; (b) equivalent network.

the initial stimulus that triggers the oscillations: we will use it to describe the network voltages and currents, assuming no steady state impressed current. If the network in Figure 12.4b produces periodic oscillations with constant amplitude, as required for an oscillator, then the voltage across the active device and the linear bipole is vðtÞ ¼ A cosðot þ jÞ ¼ RefA exp½ jðot þ jÞg ¼ RefVg

ð12:9Þ

where j represents slow fluctuations of the phase over time. The active device current is IðtÞ ¼ G0 A cosðot þ fÞB0 A sinðot þ fÞ ¼ RefY 0 Vg

ð12:10Þ

Computation of the resonator current requires the calculation of the derivative of the function (12.9) with respect to time dvðtÞ dA d½cosðot þ fÞ dA dðot þ fÞ ¼ cosðot þ fÞ þ A ¼ cosðot þ fÞA sinðot þ fÞ dt dt dt dt dt dA df ¼ cosðot þ fÞA sinðot þ fÞ o þ dt dt dA df Re exp½ jðot þ fÞg þ A o þ ¼ Re j exp½ jðot þ fÞg dt dt The quantities A; o; f; t are real, and given two arbitrary complex numbers z1 ; z2 , then Reðz1 Þ þ Reðz2 Þ ¼ Reðz1 þ z2 Þ, hence dvðtÞ dA df ¼ Re exp½ jðot þ fÞ þ Re jA o þ exp½ jðot þ fÞ dt dt dt dA df exp½ jðot þ fÞ þ jA o þ ¼ Re exp½ jðot þ fÞ dt dt df 1 dA ¼ Re j o þ þ A exp½ jðot þ fÞ dt A dt Now, differentiation over time in the time domain corresponds to multiplication by the factor jo in the frequency domain. Thus, we can consider the factor

510

MICROWAVE AND RF ENGINEERING dj 1 dA oþ þ dt jA dt

as a generalization of the angular frequency that takes into account the amplitude and phase fluctuations. The generalized angular frequency coincides with o in the case of zero fluctuations. Application of the generalized angular frequency to YðoÞ consists of operating the substitution dj 1 dA j oþ þ ! jo dt A dt The first-order Taylor series of the resonator admittance is " # df j dA dYðoÞ df j dA df j dA 2 ¼ YðoÞ þ þo Y oþ dt A dt do dt A dt dt A dt where oðxÞ denotes an infinitesimal term9 of higher order than x. Assuming slow amplitude and phase fluctuations, i.e. dj 1 dA ; o dt A dt then the Taylor series gives a good approximation of the resonator admittance:

df j dA dGðoÞ df 1 dA dGðoÞ 1 dA dBðoÞ df ffi GðoÞ þ þ BðoÞ þ j BðoÞ þ Y oþ dt A dt do dt A dt do A dt do dt Thus, the resonator current is iðtÞ ¼ RefY Vg

dGðoÞ df dBðoÞ 1 dA ¼ GðoÞ þ þ A cosðot þ fÞ do dt do A dt

dGðoÞ 1 dA dBðoÞ df þ A sinðot þ fÞ BðoÞ do A dt do dt

ð12:11Þ

The impressed current cðtÞ is the sum of the resonator and active circuit current cðtÞ ¼ IðtÞ þ iðtÞ ¼ RefðG0 þ jB0 ÞVg þ RefY Vg Expanding the expression for cðtÞ, we have

dGðoÞ df dBðoÞ 1 dA þ A cosðot þ fÞ cðtÞ ¼ GðoÞG0 þ do dt do A dt

dGðoÞ 1 dA dBðoÞ df þ A sinðot þ fÞ BðoÞ þ B0 do A dt do dt

ð12:12Þ

Multiplying both terms of Equation (12.12) once by cosðo t þ jÞ, once by sinðo t þ jÞ and integrating the results over one oscillation period T0 ¼ 2p=o0 , we obtain 8

ðt > > dGðoÞ df dBðoÞ 1 dA T0 > 0 > > cðtÞcosðot þ fÞ dt ¼ GðoÞG þ þ A > > do dt do A dt 2 < tT0 ð12:13Þ t

ð > > dGðoÞ 1 dA dBðoÞ df T0 > 0 > > cðtÞsinðot þ fÞ dt ¼ BðoÞB þ A > > do A dt do dt 2 : tT0

9

Equivalently, limx ! 0 ½oðxÞ=x ¼ 0.

OSCILLATORS

511

where we have assumed that both j and A need a much longer time than T0 to produce relevant fluctuations: j and A behave approximately like constants, during the integration time. The two equations in (12.13) constitute one linear non-homogeneous system with two unknowns, ð1=AÞdq=dt; dj=dt, that can be found by solving the system. For our needs, it is sufficient to solve the system (12.13) in a situation close to steady state, rather than in the general case. In that condition the two integrals in the first terms of system (12.13) vanish, because the oscillation period is close to T0 and c(t) is zero as well. Then, the solution of the system is 8 dYðoÞ2 1 dA dBðoÞ > dGðoÞ > > ½GðoÞ þ G0 þ ½BðoÞ þ B0 ¼ < do A dt do do ð12:14Þ 2 > > > dYðoÞ df ¼ dGðoÞ ½GðoÞ þ G0 dGðoÞ ½BðoÞ þ B0 : do dt do do Equations (12.14) determine the conditions for the presence of stable oscillations together with their amplitude and frequency. In the true steady state, there are neither amplitude nor phase fluctuations, and the impressed current is zero. Then df 1 dA ¼ ¼ dt A dt

ðt

ðt cðtÞcosðot þ fÞ dt ¼ tT0

cðtÞsinðot þ fÞ dt ¼ 0 tT0

Under these conditions, Equations (12.14) simplify to GðoÞG0 ¼ 0 BðoÞ þ B0 ¼ 0

ð12:15Þ

Equations (12.15) express the conditions for free steady state oscillations, and determine the respective amplitude A0 and angular frequency o0. If the amplitude A has a small deviation dA from its steady state value A0 , it affects the active circuit admittance, but not that of the resonator, which is linear by definition. Then the quantities of Equations (12.15) become 8 @G0 @G0 > > dA ¼ dA < Gðo0 ÞG0 ðA0 þ dAÞ ¼ Gðo0 ÞG0 ðA0 Þ @A @A ð12:16Þ 0 0 > > : Bðo0 Þ þ B0 ðA0 þ dAÞ ¼ Bðo0 Þ þ B0 ðA0 Þ þ @B dA ¼ @B dA @A @A where the partial derivatives in the terms of Equations (12.16) are evaluated at the point A ¼ A0 . Substituting the quantities (12.15) into the first equation of (12.14), and considering that A ¼ A0 þ dA and A0 is constant, thus dA=dt ¼ dðdAÞ=dt, we obtain a differential equation for dA

dYðoÞ2 1 dðdAÞ dBðoÞ @G0 dGðoÞ @G0 dA þ ¼0 do @A do @A do A dt

ð12:17Þ

The stability of the oscillation amplitude implies that dA decays over time, and this is possible if and only if

@G0 dBðoÞ @B0 dGðoÞ > 0 @A do @A do

ð12:18Þ

Equations (12.15) and (12.18) give the conditions to be satisfied in the design of a negative resistance oscillator. They generalize and formalize the concepts discussed in Section 12.2. The condition (12.15) has a clear interpretation: in the steady state the resonator admittance has to be equal and opposite to that of the active element.

512


If the active element has no susceptance, we can see that conditions (12.15) and (12.18) coincide with the corresponding ones of Section 12.2: .

The second equation in (12.15) simplifies to BðoÞ ¼ 0, which means that the resonator has zero susceptance at the oscillation frequency. Thus the resonator behaves like an anti-resonant circuit, and its resonant frequency coincides with the oscillation one.

.

Anti-resonant circuits present negative (positive) susceptance below (above) their resonance, thus in the neighbourhood of o0 this is dBðoÞ=do > 0. From this condition and from Equation (12.18) it follows that @G0 =@A < 0. Thus, the negative conductance absolute value must decrease with the oscillation amplitude.

.

The first equation in (12.15) gives BðoÞ þ B0 ¼ 0. This means that in the regime condition the active circuit exactly compensates the resonator loss, or that the Q resulting from the combination of the two bipoles is infinite.

.

From the last two points, it follows that the negative conductance amplitude is higher than, equal to or lower than the resonator conductance at oscillation amplitudes lower than, equal to or higher than the stable value A0 , respectively. This way, the active circuit tends to correct any positive or negative deviation of A from A0 .

All the considerations and equations of this section also apply to a series resonator, just by swapping the electrical quantities and the respective dual ones: current with voltage, admittance with impedance, susceptance with reactance, conductance with resistance, series with parallel, and vice versa.

12.4 Positive feedback oscillators Positive feedback is a method – alternative to the negative resistance – for oscillator analysis and design. The main difference between the two design methods consists of their applicability: the most convenient one depends on the specific circuit, models and CAE program. However, despite their different points of view, the two methods give identical results when both are applicable. This section presents the basic principle of positive feedback, using the intuitive approach as in Section 12.2, showing the equivalence of the two methods. Figure 12.5 shows the block diagram of a feedback system, without specifying the quantities at the input and output of its various blocks: they could be voltages, currents or convenient linear combinations of them. The diagram in Figure 12.5 includes three blocks: one adder, one amplifier and one selective network. The amplifier gain av ðoÞ is the forward gain of the loop, while bðoÞ is the feedback gain. The adder

a v(ω) IN

OUT

u

β(ω)

Figure 12.5

Block diagram of a feedback system.

OSCILLATORS

513

output is the sum of its two inputs, the amplifier gain is frequency dependent in the general case, and the selective network presents a gain peak in a narrow band, as its symbol suggests. Under proper conditions the arrangement in Figure 12.5 works as an oscillator, and the quantity applied to the node IN plays the same role as the excitation current in the diagram of Figure 12.4b. From an intuitive point of view, we can consider that the feedback introduces a portion of the amplified input signal back to the system input. The path from IN to the injection point u presents a gain equal to the product of the forward gain and the feedback gain GL ðoÞ ¼ av ðoÞbðoÞ

ð12:19Þ

The quantity (12.19) is the loop gain of the block, and is the gain from the node IN to node u when the mesh is cut at the point marked with a cross. Each time the signal loops, the adder output increases by a quantity that increases with the open-loop gain: this process is known as regeneration. If one frequency o0 exists such that arg½GL ðo0 Þ ¼ 0, three cases are possible that correspond to the three conditions of Section 12.2 on the quality factor: 1. jGL ðo0 Þj < 1: the output signal consists of damped oscillations having exponentially decaying amplitude over time. This case corresponds to 0:5 < Q < 1. 2. jGL ðo0 Þj ¼ 1: in this case the output signal consists of constant-amplitude oscillations, and corresponds to Q ¼ 1. 3. jGL ðo0 Þj > 1: all the regenerative contributions are in phase and present increasing amplitude over time. Thus, the oscillator produces oscillations at a frequency close to o0 and with exponentially diverging amplitude, no matter how small the initial input signal, as for 1 < Q < 0:5. In more quantitative terms, the application of Mason’s rule to the diagram in Figure 12.5 gives the closed-loop gain OUT av ðoÞ av ðoÞ ¼ ¼ GCL ðoÞ ¼ IN 1av ðoÞbðjoÞ 1GL ðoÞ

ð12:20Þ

If GL ðo0 Þ ¼ 1, then the function (12.20) becomes infinite. This is equivalent to saying that a non-zero signal at the angular frequency o0 is present at the output even with zero input signal. Therefore the loop gain of the diagram in Figure 12.5 must be unitary (higher than 1), in order to fulfil the condition for stable oscillations (startup). From this two relations follow: jGL ðo0 Þj 1;

arg½GL ðo0 Þ ¼ 0

ð12:21Þ

Conditions (12.21) are equivalent to 20 log10 ½jGL ðo0 Þj 0;

arg½GL ðo0 Þ ¼ 0

ð12:22Þ

Conditions (12.21) with the equals sign, obtained for the positive feedback oscillator, correspond to the ones in (12.15), derived for the negative resistance oscillator. Usually the feedback network is passive and linear, and so it is for its transfer function bðoÞ. Consequently, the first condition in (12.21) requires the amplifier gain to decrease with the output amplitude. Usually amplifiers naturally satisfy such a condition, as discussed in Chapter 11. The intuitive concepts above correspond to the Nyquist stability criterion, which we can shortly describe as follows. First, a system is stable in the strict sense if its response to any time- and amplitudelimited excitation tends to zero, within a finite time. An oscillator is an unstable system by definition.

514


Then, it is required to extend the function (12.20) from the purely imaginary variable jo to the complex variable10 s ¼ s þ jo. Circuit theory states that the network function of a lumped element network is a rational function of the variable s, with real coefficients. Furthermore, if the network includes distributed elements, they can be approximated by finite numbers of lumped elements, at least in the frequency band of interest. Now, the closed- and open-loop gains are network functions, hence they are real rational functions of the variable s, either approximately or exactly, depending on whether the network includes distributed elements or not. The poles of GCL ðsÞ determine the stability of the system, and coincide with the zeros of 1GL ðsÞ, unless some of them cancel with the zeros of aðsÞ, which is unlikely. System theory tells us that the system is unstable if the poles of its transfer functions fall in the right half plane of the variable s, or, equivalently, if their real part is positive. Assuming that GCL ðsÞ has no positive real part poles, as it does normally, the Nyquist criterion states that the system is unstable if its loop gain, evaluated on the whole imaginary axis 1 < o < 1, describes a curve encircling the critical point 1 þ j0. The Nyquist plot is a Cartesian diagram that plots the imaginary coefficient of the loop gain versus its real part, equivalently. It can also be seen as a polar representation of the loop gain, as we can see in Figure 12.13, which is the Nyquist diagram of the oscillator presented in Section 12.6. The real nature of the transfer function coefficients implies that GL ðoÞ ¼ conj½GL ðoÞ, and thus the Nyquist diagram is symmetrical in respect of the real axis, as Figure 12.13 also shows. Now, if the loop gain encircles the critical point, this means that the function GL ðoÞ crosses the real axis to the right of the critical point. This also means that a frequency o0 exists such that the loop gain fulfils conditions (12.21). As a final remark on the Nyquist criterion, note that it is normally used to check the stability of feedback systems; on the contrary, in the oscillator context, it is used to make sure that the system is unstable. Moreover, feedback is usually negative in feedback systems. Thus the block diagram considered in the system theory has a 180 phase inversion in the direct path: this corresponds to considering bðoÞ instead of bðoÞ, while the critical point becomes 1 þ j0. Sometimes, Equations (12.21) or (12.22) are also referred to as the Barkhausen oscillation criterion. Figure 12.6a shows a simple feedback oscillator. The input voltage generator VIN represents the initial condition of the circuit, and is used for intermediate calculations, but does not physically exist. The transfer function from the input to the output voltage is VOUT ¼ VIN

a0 RP

1 Ra0F

1 joLp

þ

1 joLp

1 Rp

þ

þ 1 Rp

1 RF

þ

þ joCp

1 RF

1

þ joCp

1

ð12:23Þ

The function (12.23) has the same structure as the function (12.18), the forward gain and the open-loop gain being av ðoÞ ¼

a0 RP

1 1 1 þ þ þ joCp joLp Rp Rp

1

;

GL ðoÞ ¼

a0 RF

1 1 1 þ þ þ joCp joLp Rp Rp

1

Figures 12.6b,c clarify the physical meaning of the above parameters: .

av ðoÞ is the voltage gain from VIN to VOUT when the loop is open by cutting the mesh m1 and connecting the right terminal of the feedback resistor RF to ground. Note that this connection makes the resistor left terminal equal to zero, as required to disable the feedback.

10 Such a complex variable is sometimes referred to as the Laplace variable, from the well-known eponymous transformation.

OSCILLATORS

515

RF

a0

Rp +

Lp

VOUT

Cp

VIN (a)

RF

a0

Rp +

Lp

VOUT

Cp

VIN (b)

RF V1 Lp

Rp

a0

Cp

VOUT

(c)

RF

Lp

Rp

a0

Cp

VOUT

(d)

(e)

Figure 12.6 Basic oscillator: (a) principle; (b) loop forward gain; (c) open-loop gain; (d) negative resistance analysis; (e) SFG corresponding to (d).

.

GL(o) is the loop gain as obtained by cutting the mesh m1 and calculating the voltage gain from the right side (V1 ) to the left side (VOUT ) of the cut.

.

1 bðoÞ ¼ GL ðoÞa1 v ðoÞ ¼ RP RF is the feedback gain.

The loop gain presents zero phase for o ¼ o0 ¼ ðLp Cp Þ0:5, the corresponding amplitude is GL ðo0 Þ ¼ av Rp ðRp þ RF Þ1 , and the first condition in (12.21) requires that av

Rp þ RF Rp

ð12:24Þ

The factor Rp ðRp þ RF Þ1 is the voltage gain from V1 to the amplifier input in Figure 12.6c, computed at the frequency o0. At that frequency the resonating circuit admittance is purely real, and the passive

516


network behaves like a voltage divider having the series resistor RF and the shunt resistor Rp. Therefore, the maximum open-loop gain is higher or lower than 1 if the amplifier voltage gain exceeds or does not the reciprocal of the voltage divider ratio. In other words, the circuit in Figure 12.6a works as an oscillator if the small-signal voltage gain of the amplifier is higher than the resistive loss.11 We can reconsider the network in Figure 12.6a from the negative resistance point of view. There are many possible ways to carry out such an analysis; here we will begin by writing the time domain differential equation of the network, then we will apply Miller’s theorem. Equation (12.23) can be rearranged as

1 1av av 1 þ joLp þ ð12:25Þ þ ð joÞ2 Lp Cp VOUT ðoÞ ¼ joLp VIN ðoÞ Rp RF Rp Equation (B.18) implies that the factor j2p f ¼ jo in the Fourier domain corresponds to the derivative in the time domain. Thus, from Equation (12.25), it follows that the differential equation with the corresponding time domain voltage functions vIN and vOUT is 1 1av dvOUT ðtÞ d 2 vOUT ðtÞ av Lp dvIN ðtÞ þ ¼ Lp vOUT ðtÞ þ þ Lp Cp RF Rp Rp dt dt2 dt If the input voltage is zero, as assumed, the above differential equation simplifies to 1 1av dvOUT ðtÞ d 2 vOUT ðtÞ þ ¼0 Lp þ Lp Cp vOUT ðtÞ þ Rp dt dt2 RF

ð12:26Þ

Equation (12.26) has the same form as (12.1), but with a different first-order coefficient. Therefore, the output voltage of the oscillator in Figure 12.6a coincides with the function (12.6), just by replacing the resistance Rp with the equivalent parallel resistance RP in the expressions for its parameters sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Cp R2P V0 1 1 1 A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t ¼ 2Cp RP ; O ¼ ; j ¼ tan1 2 L Lp Lp Cp ð2Cp RP Þ 1 p 2 4Cp RP

1 o0 ¼ pffiffiffiffiffiffiffiffiffiffi ; Lp Cp

sffiffiffiffiffiffi Cp Q ¼ RP Lp

where

RP ¼

1 1a0 þ RF Rp

¼

RF jjRp 1a0

ð12:27Þ

The third term of Equation (12.27) highlights that the circuit in Figure 12.6a behaves like an anti-resonant circuit Lp ; Cp ; Rp with the additional shunt resistance ð1a0 Þ1 RF . This resistance is exactly the one resulting from the combination of the feedback resistor and the amplifier, as a consequence of Miller’s theorem.12 Note that ð1a0 Þ1 RF is negative if and only if a0 is greater than 1. Also, at high output amplitudes, the amplifier compression reduces the gain, and consequently the negative conductance, stabilizing the oscillation amplitude. We can get the same result by analyzing the network in Figure 12.6a as Figure 12.6d suggests. By cutting the mesh m3, it is possible to consider separately the admittances of the resonator and that of the active device. On the right of the cut we have one RLC parallel resonator, having admittance YLC . Miller’s 11 The SIMetrix file 08_Positive_Feedback_Oscillator.sxsch analyzes the schematic in Figure 12.6a in the time domain showing the oscillation arising. The saturating amplifier characteristic is modelled with the hyperbolic tangent transfer characteristic. 12 See also Section 11.6.5.

OSCILLATORS

517

theorem applied to the subnetwork on the right of the cut implies that, looking at the right of the cut, we will see the negative resistance RIN. From Equation (12.15), it follows that stable steady state oscillations occur if the resonator admittance is the opposite of the active device one YLC ðo0 Þ þ YIN ðo0 Þ ¼ 0

ð12:28Þ

Note that, for the circuit in Figure 12.6d, Re½YLC ðo0 Þ þ YIN ðo0 Þ ¼ R1 P and Im½YIN ðo0 Þ ¼ 0. Now, the criterion developed in Section 12.2 implies that the circuit in Figure 12.6a behaves like an oscillator at the frequency o0 if R1 P is negative for small output amplitudes and increases towards positive values. Therefore the complex equation (12.28) corresponds to the two scalar conditions Re½YLC ðo0 Þ þ YIN ðo0 Þ 0;

Im½YLC ðo0 Þ þ YIN ðo0 Þ ¼ 0

ð12:29Þ

The first condition in (12.29) applies with the equals or less than sign in the steady state or startup condition, respectively, while the second condition in (12.29) has to be satisfied for any output amplitude. From the first condition in (12.29) and Equation (12.26) it follows that for oscillation startup and stabilization, a0 ðRp þ RF ÞR1 p : this coincides with the inequality in (12.24), hence positive feedback and negative resistance points of view give the same result when applied to the same circuit. Conditions (12.29) can be rewritten as jYLC ðo0 ÞYIN ðo0 Þj 0;

arg½YLC ðo0 ÞYIN ðo0 Þ ¼ 0

ð12:30Þ

The inequality in (12.30) has the same form as relations (12.21): both include one condition on the amplitude and another on the phase of the quantity of interest. The separation between resonator and active device in Figure 12.6d also offers an alternative analysis possibility. Considering the reflection coefficients, rather than admittances, the network in Figure 12.6d corresponds to the SFG in Figure 12.6e, where GLC (GIN ) is the reflection coefficient of the resonator (active device). The simple diagram in Figure 12.6e is a loop, and its gain is the product of its two branches GL ðoÞ ¼ GLC ðoÞGIN ðoÞ

ð12:31Þ

The loop gain (12.31) is the quantity (12.19) expressed in terms of incident and reflected waves, and the SFG in Figure 12.6e corresponds to the generic diagram of Figure 12.5. The conditions (12.21) and (12.22), applied to the reflection coefficients, become respectively jGLC ðo0 ÞGIN ðo0 Þj 1;

arg½GLC ðo0 ÞGIN ðo0 Þ ¼ 0

20 log10 ½jGLC ðo0 ÞGIN ðo0 Þj 0;

arg½GLC ðo0 ÞGIN ðo0 Þ ¼ 0

ð12:32Þ ð12:33Þ

The same conditions, with the equals sign, can be derived by considering that the reflection coefficients relate to the respective admittances as GLC ¼

1R0 YLC 1R0 YIN ; GIN ¼ 1 þ R0 YLC 1 þ R0 YIN

Now, from condition (12.15) it follows that YLC ðoÞ þ YIN ¼ 0, then GLC GIN ¼

1R0 YLC 1R0 YIN 1R0 YLC 1 þ R0 YLC ¼ ¼1 1 þ R0 YLC 1 þ R0 YIN 1 þ R0 YLC 1R0 YLC

as anticipated. Observations: (a) All the conditions (12.22), (12.30) and (12.33) have the form jHðo0 Þj 0; arg½Hðo0 Þ ¼ 0, where the generic function HðoÞ represents one of GL ðoÞ, ½YLC ðoÞ þ YIN ðoÞ, or GLC ðoÞGIN ðoÞ. In the remaining part of this chapter, we will address the first and second condition of the couple as the amplitude and phase conditions, respectively. Note that the phase

518

MICROWAVE AND RF ENGINEERING condition determines the oscillation frequency, while the amplitude condition is a requirement for the energy that the active device must supply in order to sustain the oscillation. (b) Conditions (12.21), (12.29) and (12.33) are rigorously derived for the steady state operation of the oscillator, where the first of them apply with the equals sign. In that case the three conditions are totally interchangeable and give the same result. The addition of the inequality sign in the amplitude condition is based only on qualitative considerations. (c) The assumption of different negative resistance, gain or reflection coefficient in the active device, for different output amplitudes, involves nonlinear operation of the device. In the steady state, the oscillator active device works under compression, with the consequent generation of harmonics. Conversely, resistance, gain and reflection coefficient are inherently parameters of linear devices. Amplitude-depending linear parameters are an artifice which can be useful for practical design purposes. It consists of ignoring all the harmonics and considering thePratio between the fundamental components. For instance, if the amplifier P input signal is k vin;k cosðko0 t þ fin;k Þ, the output is k vout;k cosðko0 t þ fout;k Þ, with nonnegative amplitude coefficients, and the corresponding amplitude-dependent linear gain is ðvout;1 =vin;1 Þexp½ jðfout;k fin;k Þ. (d) The active device of the oscillator in Figure 12.6a has no reactance, thus b; YIN ; and GIN are independent of the frequency. Consequently, the network fulfils more restricting conditions than (12.22), (12.30) and (12.33): at the angular frequency o0, not only does arg½Hðo0 Þ ¼ 0, but also jHðo0 Þj is a maximum. In such cases, during startup, as the output amplitude increases, jHðo0 Þj decreases, tending to Hmax , while arg½Hðo0 Þ holds at zero. (e) In general, if the active device includes reactive elements, arg½Hðo0 Þ ¼ 0 does not imply that jHðo0 Þj is a maximum. Moreover o0 depends on the transistor gain, i.e. on its compression degree. Hence, during startup the oscillation pseudo-period is not constant.

Linear design of oscillators must fulfil one of the condition couples (12.22), (12.30) or (12.33), depending on the particular design approach. At the desired oscillation frequency, the linear analysis of the circuit must satisfy the phase condition. Furthermore, the circuit must respect the amplitude condition within a certain margin, in order to take into account temperature variations, differences from device to device and assembly tolerances. Passing from the startup to the steady state, a simple choice consists of assuming that jHðo0 Þj decreases while arg½Hðo0 Þ stays constant: this implies that the final output frequency coincides with the one obtained from the linear analysis. Unfortunately, this assumption is true only if the active device has no reactive elements, which happens in no practical high-frequency case. In general, assuming the inequality for the amplitude condition, Equations (12.22), (12.30) or (12.33) satisfy the phase relation at different frequencies. The simplified analysis of the standard oscillator configurations, the application of the three design methods to a trial oscillator design and their comparison with a nonlinear approach will clarify the point in the next two sections.

12.5 Standard oscillator configuration The configuration in Figure 12.6a is quite ideal, requiring an ideal voltage amplifier. An additional drawback of this solution is the loading effect on the resonator due to the feedback resistor that degrades Q, as we will see in Section 12.7. In real oscillators one transistor replaces the ideal amplifier and the feedback elements are usually reactive ones: the transistor parasitic capacitances work as feedback elements in many cases.

OSCILLATORS

519

Figure 12.7 shows some of the most widely used configurations for transistor oscillators. For the sake of simplicity, all the schematics in Figure 12.7 omit the output network and use FETs, although the standard configurations may also use bipolar devices. This section presents some linear analyses of the standard configurations, based on the application of simplified linear models for the transistor. Real circuits involve more complex models for both the

Ld M

Q1 Lp

Rp

Cp

+ Vdd

(a)

Q1 + Lg

Lp

Rp

Lp2

Rp

Cp

Cp Vdd

(b)

Q1 M

+ Lp1

Vdd

(c)

Q1 Lp

Rp

Q1 Ls Cs Rs

Cp1 + Cp2

Vdd

(d)

Cp1

Rp + Vdd

Cp2

(e)

Lp

Rp

Cp

Cd

Cp

Rp Lp

Lp

Q2

Q1

Cd

Q1

Rp Cp/2 Rp

Cd

Lp

Cd Q2

+ I0 (f)

+

Vdd

I0

Vdd

(g)

Figure 12.7 Standard oscillator configuration: (a) inductive coupled; (b) inductive gate feedback; (c) Hartley; (d) Colpitts; (e) Clapp; (f) cross-coupled differential oscillator; (g) alternative configuration for (f).

520


transistor and the resonator. Nevertheless, the approximate analysis results are useful starting points for design. Sections 12.5.1 to 12.5.5 present the calculation of the loop gain for all the circuits in Figure 12.7. The technique used to open the loop consists of cutting the connection between the gate and the control port of the linear VCCS of the transistor model, placing an AC generator across the VCCS control port and computing the resulting voltage between the disconnected gate and the source. The voltage loop gain is the ratio between the latter and the first of the quantities mentioned above. Since all the standard oscillators shown in Figure 12.7 use lumped elements only – including the elements of the transistor model – the resulting open-loop gain is always a rational function of the variable jo. Consequently, the loop voltage gain has the form GL ðoÞ ¼

Nu ðjoÞ De ðjoÞ

ð12:34Þ

P P where Nu ðjoÞ ¼ k bk ðjoÞk ; De ðjoÞ ¼ k ak ðjoÞk are polynomials of the variable jo, with real coefficients ak ; bk 2 R. In order to determine if the circuit works as an oscillator, we need to check if the function (12.34) satisfies conditions (12.21). Here, we will introduce a formal method for such verification. After separating the real and imaginary parts of both numerator and denominator, the function (12.34) becomes Nu ðjoÞ Re½Nu ðjoÞ þ jIm½Nu ðjoÞ GL ðoÞ ¼ ¼ De ðjoÞ Re½De ðjoÞ þ jIm½De ðjoÞ The real (imaginary) parts of the numerator and denominator comprise the even- (odd-)order monomials of the respective polynomials X X X b ðjoÞk ¼ b ðjÞk ok ¼ b ð1Þk=2 ok Re½Nu ðjoÞ ¼ Nu;r ðjoÞ ¼ k;even k k;even k k;even k X X X Re½De ðjoÞ ¼ De;r ðjoÞ ¼ a ðjoÞk ¼ b ðjÞk ok ¼ b ð1Þk=2 ok k;even k k;even k k;even k X X X Im½Nu ðjoÞ ¼ Nu;i ðjoÞ ¼ j k;odd bk ðjoÞk ¼ b ðjÞk1 ok ¼ b ð1Þk ok k;odd k k;odd k X X X Im½De ðjoÞ ¼ De;i ðjoÞ ¼ j k;odd ak ðjoÞk ¼ a ðjÞk1 ok ¼ a ð1Þk ok k;odd k k;odd k Using these quantities, multiplying the numerator and denominator of the function (12.34) for the complex conjugate of the denominator and simplifying, we obtain Nu;r ðjoÞDe;r ðjoÞ þ Nu;i ðjoÞDe;i ðjoÞ þ j½Nu;i ðjoÞDe;r ðjoÞNu;r ðjoÞDe;i ðjoÞ GL ðoÞ ¼ ½De;r ðjoÞ2 þ ½De;i ðjoÞ2 Starting with the phase condition, the argument of the function (12.34) vanishes if ½Nu;i ðjoÞDe;r ðjoÞNu;r ðjoÞDe;i ðjoÞ ¼ 0

ð12:35Þ

Equation (12.35) is generally a high-order polynomial and thus has many solutions o0 , which can be analytically computed in a few special cases. In more general cases, a solution is possible with numerical methods, or by introducing some simplifications in the network. Substituting one solution into the function (12.34), we obtain GL ðo0 Þ ¼

Nu;r ðjo0 ÞDe;r ðjo0 Þ þ Nu;i ðjo0 ÞDe;i ðjo0 Þ ½De;r ðjo0 Þ2 þ ½De;i ðjo0 Þ2

¼ Re½GL ðo0 Þ

ð12:36Þ

This expression is real by definition of o0 . However, the oscillation conditions (12.21) state a stricter requirement: the loop gain must be not only real, but also positive and no smaller than 1, at the oscillation frequency. This eliminates the solutions of Equation (12.35) with GL ðo0 Þ ¼ Re½GL ðo0 Þ < 1 from the list of the potential oscillation frequency.

OSCILLATORS

521

If one unique o0 exists such that Re½GL ðo0 Þ 1, then it is the oscillation frequency of the circuit. If more than one such solution exists, the circuit presents more true possible oscillation frequencies. In this case, the output frequency can jump from one value to another, after an abrupt change of the load impedance, switching the oscillator off and on again, for temperature changes, and so on. Such a malfunctioning is sometimes referred to as mode jumping, and must absolutely be avoided.

12.5.1

Inductively coupled oscillator

The inductively coupled oscillator is the simplest to analyze among the ones in Figure 12.7a. Figure 12.8 shows the various simplification steps needed to calculate the loop gain, in order to determine the oscillation conditions. Figure 12.8a shows the complete electrical diagram of the oscillator, including the output bias network ðVdd ; Ldd ; Cdd Þ and load ðRdd Þ. We will assume the bias inductor (capacitor) presents a much higher (lower) impedance than the load resistor, at the oscillation frequency. Therefore, the linear equivalent circuit of Figure 12.8b replaces Ldd and Cdd with the open circuit and short circuit, respectively. Moreover, a simplified linear model – consisting of a VCCS with transconductance gm and input capacitance Cgs – models the transistor. Figure 12.8c shows the next step, which consists of opening the loop by cutting the mesh m1 inside the transistor, breaking the connection between the gate and the VCCS control pin. One AC voltage generator connects to the VCCS control port. The desired loop gain is the ratio between the voltage across the transistor input capacitance and the impressed AC voltage. The load resistor in Figure 12.8b is in series with a current generator, which does not affect the network equations, so Figure 12.8c short-circuits that resistor. Also, the inductor Ld is the only component directly connected to the current generator, hence Figure 12.8c inverts the terminals of both those components with respect to Figure 12.8c. This double inversion does not alter the network equations, but allows a further simplification in the circuit, as Figure 12.8d shows: the mutually coupled inductors Lp ; Ld are replaced by their equivalent network consisting of three single inductors in a T connection. Similar to RLOAD in Figure 12.8b, Ld is now in series with a current generator, and can be short-circuited, as Figure 12.8e shows. The final transformation consists of passing from the Norton configuration of the current generator in parallel with the inductor M to the Thevenin equivalent one. The current generator gm vi with impedance jo M transforms into a voltage generator jo M gm vi , with the same impedance in series, as shown in Figure 12.8f. Note that the new network in the figure has two inductors M; Lp M in series: they are equivalent to a single one, having inductance L ¼ M þ Lp M ¼ Lp . Hence, the transformation reduces the number of nodes in the network and consequently further simplifies our calculations. The calculation of the loop gain of the network in Figure 12.8f is straightforward. Basically, from the input to the output we have multiplication by a constant and a voltage division factor: R

p Rp jj joðCp1þ Cgs Þ vgs gm vi joðCp þ Cgs ÞRp þ 1 ¼ jo M ¼ jo M g GL ðoÞ ¼ m R vi vi joLp þ Rp jj joðCp1þ Cgs Þ joLp þ joðC þ Cp ÞR p

gs

p

þ1

Rationalizing the denominator and simplifying, we obtain GL ðoÞ ¼

1þ

Lp Rp

gm M jo jo þ ðCp þ Cgs ÞLp ðjoÞ2

ð12:37Þ

The function (12.37) has a purely imaginary numerator; it fulfils the phase condition coming from Equation (12.21) if its denominator is purely imaginary as well. Therefore the zero open-loop phase angular frequency o0 is the one vanishing real part of the denominator. Equation (12.35) becomes 1ðCp þ Cgs ÞLp o2 ¼ 0

522

MICROWAVE AND RF ENGINEERING Cdd

Ld

VOUT RLOAD

L dd M

Q1 Rp

Lp (a)

+ Vdd

Cp

Ld G

M

Lp

Rp

Cgs

Cp

+ vgs -

m1 + vi -

Q1

D

VOUT

m i

RLOAD S

(b)

vgs

M Ld

+ Lp

Cp

Rp

Cgs

(c)

Ld-M Lp-M M

Rp

Cp+Cgs

Lp-M

+

+ Rp

Cp+Cgs

Lp-M Lp

+ vi -

m i

+ vi -

m i

vgs

(e)

M

m i

vgs

(d)

M

+ vi -

vgs + Rp

Cp+Cgs

(f)

+ vi -

+ m i

Figure 12.8 Inductively coupled oscillator: (a) electrical diagram; (b) simplified linear equivalent circuit; (c) simplification of (b) with the loop opened; (d) as (c) but with the coupled inductors replaced by their equivalent network; (e) elimination of the inductor in series with the current generator; (f) transformation of gm vi ; M from the Norton to the Thevenin canonical form. and its solution is ðInductively coupledÞ

o0

¼ ½ðCp þ Cgs ÞLp 0:5

ð12:38Þ

The two opposite solutions (12.35) denote the same oscillation mode, in that sinðo0 tÞ ¼ sinðo0 t þ pÞ. In other words, changing the sign of the frequency corresponds to a constant phase shift, which is of no interest within this context. Therefore, we will consider the positive solution only.

OSCILLATORS

523

The loop gain at o0 is real and positive, as required GL ðo0 Þ ¼ gm Rp L1 p M ¼ jGL ðo0 Þj GL ðo0 Þ 1 implies that coupledÞ 1 R1 gðInductively m p Lp M

ð12:39Þ

The transistor input capacitance directly affects the oscillation frequency. If we use a bipolar device instead of an FET, the transistor input resistance is also not negligible. Nevertheless, the above formulae hold true, just by considering the transistor input resistance Rbe in parallel with the one of the resonant circuit Rp : in other words, replacing the latter with Rp jjRbe ¼ Rp Rbe ðRp þ Rbe Þ1 .

12.5.2

Inductive gate feedback oscillator

Figure 12.9 shows some more details of the inductive gate feedback oscillator, already presented in Figure 12.7a, which is widely used at high frequencies. Moreover, Section 12.6 below uses the circuit in Figure 12.7b as an application example of different design techniques. + Ldd

Vdd

Cdd

Q1

VOUT RLOAD

Lp

Cp

Lg

Rp

(a) m1

G

D

+ Lg

Vgs

-

+ vi -

Cgs

Lp

VOUT RLOAD

gm Vi

S

Cp

Rp

(b)

G

vgs

Lg Lp

Cp

Rp

D

+

vi gm Vi

Cgs

S (c)

Figure 12.9 Gate inductive feedback oscillator: (a) electrical diagram; (b) simplified linear equivalent circuit; (c) rearrangement of (b) for open-loop gain computation.

524


Figure 12.9a shows the electrical diagram of the circuit, including output bias and load, as in Figure 12.8a. From that schematic, we obtain the simplified linear equivalent circuit in Figure 12.9b, under the same assumption as the ones used in passing from Figure 12.8a to Figure 12.8b. The analysis technique used in Section 12.5.1 also applies to this case. However, the network in Figure 12.9b is slightly more complicated, and so it is with the relative computations. These complications come from the transistor having no terminal to ground. In cases like this, it is helpful to use the technique known as ground redefinition, which consists of changing the voltage reference node, from ground to a different one, in order to simplify the network equations. In principle, loop gain analysis consists of cutting the mesh at the point m1, applying an AC voltage generator at the transistor input (right side of the cut) and computing the resulting voltage across Cgs : the quantity vgs =vi is the desired open-loop gain. One difficulty in this calculation comes from vgs ; vi having no terminal at the ground potential. A simpler calculation is possible by changing the reference node of the network from the physical ground to a more convenient one.13 For instance, using the transistor source as reference, both the voltages vgs ; vi simplify to voltages with respect to the new reference. Figure 12.9c shows the network in Figure 12.9b after the mesh cut and the source is used as the common node: an open isosceles triangle indicates this new reference. Note that, in Figure 12.8b, the load resistor is in series with a current generator, therefore a short circuit replaces RLOAD in Figure 12.9c. The calculation of the voltage transfer ratio vgs =vi of the network in Figure 12.9c is relatively straightforward and gives GL ðoÞ ¼

gm Lp jo vgs ¼ 4 X vi ak ðjoÞk

ð12:40Þ

k¼0

with a0 ¼ 1;

a1 ¼ Lp =Rp ;

a2 ¼ ðCp þ Cgs ÞLp þ Cgs Lg ;

a3 ¼ Cgs Lg Lp =Rp ;

a4 ¼ Cgs Lg Cp Lp

In order to determine if the circuit in Figure 12.9a works as an oscillator, we have to check if the function (12.40) satisfies conditions (12.21). The condition (12.35) applied to the function (12.40) gives a0 a2 o2 þ a4 o4 ¼ 0

ð12:41Þ

Equation (12.41) is of second order in the variable o2 , and its solutions are h

ðInductive gate feedbackÞ o0

i2

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a2 1 a2 2 a0 1 1 1 1 ¼ ¼ þ þ a4 2 Cgs Lg Lg Cp Cp Lp 2 a4 4 a4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 2 1 þ þ 4 Cgs Lg Lg Cp Cp Lp Cgs Lg Cp Lp

ð12:42Þ

Substituting one of the values (12.42) into the function (12.40) and imposing that it must not be smaller than 1 we have a1 þ a3 ð joÞ2 gm Lp The loop gain amplitude is then not smaller than 1 if ðInductive gate feedbackÞ gm

13

See also [3, 4].

Cgs Lg 2 1 o Rp Rp

ð12:43Þ

OSCILLATORS

525

Now, gm and Lp are both positive quantities. Thus, the above condition can be satisfied only if 1 o pffiffiffiffiffiffiffiffiffiffiffiffi Cgs Lg

ð12:44Þ

Therefore, in our simplified model, the minimum potential oscillation frequency coincides with the resonant frequency of the gate inductor with gate–source capacitance. The tank contributes to the determination of the actual oscillation frequency by affecting solutions (12.42). Equation (12.42) gives two possible oscillation frequencies; however, usually the one with the minus sign falls below the limit (12.44) and therefore is not a true oscillation frequency. In particularly inaccurate designs, it is possible that the circuit presents two true possible oscillation frequencies, with the consequent risk of mode jumping. In the inductive gate feedback oscillator, the frequency (12.42) satisfying the phase condition does not give the maximum loop gain amplitude,14 as observation (e) in Section 12.4 anticipated, differently from the circuits in Figures 12.6 and 12.7a.

12.5.3

Hartley oscillator

Figure 12.7c shows the Hartley oscillator, where the feedback signal coming from the source is applied to an intermediate tap point of the tuning inductor. The two resulting inductors Lp1 and Lp2 can be mutually coupled or not. Applying the techniques used in Sections 12.5.1 and 12.5.2, we obtain a loop gain having the form of Equation (12.34) with a third-order numerator and a fourth-order denominator, with all nonzero coefficients ak and bk . Starting from such an expression, the determination of oscillation frequency and minimum transconductance is quite cumbersome. However, a further simplification in the linear equivalent circuit – consisting of neglecting the input capacitance of the transistor – gives GL ðoÞ ¼ gm

M 2 Lp1 Lp2 ð joÞ2 þ Cp ðM 2 Lp1 Lp2 Þð joÞ3 Rp Lp1 þ Lp2 þ 2M jo þ Cp ðLp1 þ Lp2 þ 2MÞð joÞ2 Rp

M jo þ 1þ

ð12:45Þ

The numerator and the denominator of the function (12.45) are respectively

M 2 Lp1 Lp2 ð joÞ2 þ Cp ðM 2 Lp1 Lp2 Þð joÞ3 Nu ðoÞ ¼ gm M jo þ Rp De ðoÞ ¼ 1 þ

Lp1 þ Lp2 þ 2M jo þ Cp ðLp1 þ Lp2 þ 2MÞð joÞ2 Rp

With these quantities, Equation (12.35) becomes a second-order algebraic equation in the variable o2 , and its solutions are the possible oscillating frequencies. However, the expressions obtained from that procedure are neither simple nor meaningful. In the simplified case of Rp tending to infinity we have ðHartleyÞ

o0

1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cp ðLp1 þ Lp2 þ 2MÞ

Substituting the value (12.46) into the function (12.45), we obtain GL ðo0 Þ ¼ gm

jRp ðM þ Lp1 ÞðM þ Lp2 ÞðM 2 Lp1 Lp2 Þ

ð12:46Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lp1 þ Lp2 þ 2M Cp

jðLp1 þ Lp2 þ 2MÞ2

14 See for instance the Mathcad file 03_Inductive_Gate_Feedback.MCD. Assuming Lp ¼ 0:5 nH, Cp ¼ 2 pF, Rp ¼ 60 O, Lg ¼ 1:2 nH and Cgs ¼ 0:243 pF, we have o0 ¼ 2p 9:320 791 109 , while the maximum loop gain amplitude occurs for o ¼ 2p 9:320 833 109 , which is slightly different from o0 . Bigger differences come from more accurate transistor models.

526


The corresponding amplitude is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L þ L þ 2M ½Rp ðM þ Lp1 ÞðM þ Lp2 Þ2 ðM 2 Lp1 Lp2 Þ2 p1 Cp2p jGL ðo0 Þj ¼ gm ðLp1 þ Lp2 þ 2MÞ2 which is greater than 1 if ðHartleyÞ gm

ðLp1 þ Lp2 þ 2MÞ2 ½Rp ðM þ Lp1 ÞðM þ Lp2 Þ2 þ ðM 2 Lp1 Lp2 Þ2

Lp1 þ Lp2 þ 2M Cp

ð12:47Þ

The inequality (12.47) gives the approximate minimum transconductance satisfying the first condition in (12.21) for the Hartley oscillator.

12.5.4

Colpitts oscillator

Figure 12.7d shows the principle of a Colpitts oscillator. The Colpitts configuration is the dual of the Hartley one with M ¼ 0:Cp1 ; Cp2 and Lp in Figure 12.7d replacing the corresponding elements Lp1 ; Lp2 and Cp in Figure 12.7c, respectively. Compared with the Hartley circuit, the Colpitts solution has the advantage of exploiting the transistor input capacitance as a feedback element. In fact the capacitance Cp1 in Figure 12.7d is the sum of the transistor input capacitance Cgs (always present) and an additional external one (present if required). The network in Figure 12.7 omits drain bias circuitry and output load – as for all the networks in Figure 12.7 – and a DC path to ground for the source current, which usually is a series RL element between the source and ground. Both negative resistance and positive feedback analysis methods apply to the circuit in Figure 12.7d with relatively easy calculations. Here we will follow the second approach by applying the techniques introduced in Sections 12.5.1 to 12.5.3 and assuming that: .

The transistor has the same simplified linear model as in Section 12.5.2.

.

The bias networks are the same as in Section 12.5 above, and the associated hypotheses also apply.

.

The source DC path to ground presents infinite impedance at the oscillation frequency.

The voltage loop gain from the transistor control port to the voltage across Cp1 has the form of Equation (12.34) GL ðoÞ ¼ gm

1þ 3 X

Lp Rp

jo

ð12:48Þ

ak ðjoÞk

k¼1

with a1 ¼ Cp1 þ Cp2 ;

a2 ¼ ðCp1 þ Cp2 ÞLp =Rp ;

a3 ¼ Cp1 Cp2 Lp

The condition (12.35) applied to the function (12.48) becomes # #

"X

"X 3 3 Lp Lp k k Im gm 1 þ jo Re ak ðjoÞ ¼ Re gm 1 þ jo Im ak ðjoÞ Rp Rp k¼1 k¼1 which simplifies to Lp a2 o2 ¼ a3 o2 a1 Rp

OSCILLATORS

527

From this, it follows that (" ðColpittsÞ

o0

¼

1

# )12 Lp 1 1 1 1 1 þ þ Cp1 Cp2 Lp R2p Cp1 Cp2

ð12:49Þ

Substituting the value (12.49) into the function (12.48), we obtain the zero-phase loop gain GL ðo0 Þ ¼

gm Cp1 Cp2 R2p ðCp1 þ Cp2 ÞLp ¼ jGL ðo0 Þj Rp ðCp1 þ Cp2 Þ2

The above real quantity exceeds 1 if gðColpittsÞ m

12.5.5

ðCp1 þ Cp2 Þ2 Rp Cp1 Cp2 R2p ðCp1 þ Cp2 ÞLp

ð12:50Þ

Clapp oscillator

The Clapp oscillator, shown in Figure 12.7e, derives from the Colpitts circuit with the addition of a capacitor in series with the inductor. We can also consider the Clapp oscillator as a combination of the series resonator consisting of Ls ; Cs and Rs with the active element Q1 and the feedback capacitors Cp1 ; Cp2 . For this reason, the resonator loss is represented by a series resistor, rather than a parallel one, like all the other circuits in Figure 12.7. Proceeding as in Section 12.5.4, we have the Clapp oscillator voltage loop gain GL ðoÞ ¼

3 X

gm Cp

ð12:51Þ k

ak ð joÞ

k¼1

with a1 ¼ Cs Cp2 þ Cs Cp1 þ Cp1 Cp2 ;

a2 ¼ Rs Cs Cp1 Cp2 ;

a3 ¼ Cs Cp1 Cp2 Ls

The condition (12.35) is particularly simple to apply to the function (12.51): the argument of the function is zero if the imaginary part of its denominator vanishes. The corresponding equation is a1 ð joÞ þ a3 ð joÞ3 ¼ 0 Excluding the trivial solution ðo ¼ 0Þ, the potential oscillation frequency is ffi rffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 1 1 1 1 ðClappÞ ¼ þ þ o0 ¼ a3 Cs Cp1 Cp2 Ls

ð12:52Þ

Passing to the limit as Cs tends to infinity, Equation (12.52) coincides with the potential oscillation frequency of the Colpitts oscillator in the loss-free case. Substitution of the value (12.52) into the function (12.51) gives GL ðo0 Þ ¼

gm Ls Cs ¼ Re½GL ðo0 Þ Rs Cp1 Cp2 þ Cs Cp2 þ Cs Cp1

The second condition in (12.21) implies that jGL ðo0 Þj 1, consequently gðClappÞ m

Rs Cs Cp1 þ Cs Cp2 þ Cp1 Cp2 Ls Cs

ð12:53Þ

528


12.5.6

Differential oscillator

Figure 12.7f shows the differential oscillator, which uses a differential pair15 as amplifier and two identical parallel resonating circuits. This circuit is particularly important for its diffused application in RFIC, and offers two interesting performances: .

It delivers two 180 out-of-phase oscillating voltages, one for each transistor drain (or collector).

.

If the circuit is perfectly symmetrical, the difference between the two voltages above has no even harmonics.

Here, we will limit our analysis to a perfectly symmetrical case: the dashed line in Figure 12.7f shows the symmetry plane of the circuit. The discussion in Section 11.6.3 showed that the DP has no gain in the common mode, therefore the circuit can only oscillate in the differential mode. In the latter case, the symmetry plane behaves like a ground and the circuit in Figure 12.7f is completely equivalent to the one in Figure 12.7g. Figure 12.10a shows the complete circuit – including the loads RLOAD and the respective coupling capacitors Cdd – of the oscillator in Figure 12.7f. Proceeding in the same way as in Sections 12.5.1 to 12.5.4, we pass from the network in Figure 12.10a to the one in Figure 12.10b: the DC voltage (current) generator is short-circuited (open-circuited) and the coupling capacitors have negligible impedance with respect to RLOAD at the oscillation frequency. The loads directly affect the resonant circuits’ Q: the effective parallel resistance is now R0 p ¼ Rp jjRLOAD ¼ Rp RLOAD =ðRp þ RLOAD Þ, which is smaller than Rp . The loop opening consists of cutting the meshes m1; m2 and placing two AC voltage generators of equal amplitude and 180 out of phase across the transistors’ control ports, and then computing the voltages on the two gates. The configuration of the input generators and the assumed perfect symmetry of the network ensure that the circuit operates under a purely anti-symmetric excitation.16 Figure 12.10c shows the simplified linear network for the loop gain analysis. From each AC voltage generator to the respective transistor drain, we have !1 jo Lp 1 1 Cd Cgs VOUT;k ¼ þ þ jo Cp þ jo Id;k ¼ gm vi;k L joLp R0p Cd þ Cgs 1 þ 0p jo þ Cp0 Lp ð joÞ2 Rp

with k ¼ 1; 2

and

Cp0 ¼ Cp þ

Cd Cgs Cd þ Cgs

From the drain of Q1 ðQ2 Þ to the gate of Q2 ðQ1 Þ a voltage divider is present, consisting of the feedback capacitance Cd and the transistor input capacitance Cgs . Thus, the voltage gain from each transistor gate to the internal control port of the other one is vgs;1ð2Þ ¼

1þ

Lp Rp

jo Lp

Cd

jo þ Cp0 Lp ð joÞ2 Cd þ Cgs

gm vi;2ð1Þ

The desired loop gain is the ratio between the differential gate and the input excitation AC voltages: GL ðoÞ ¼

15

vgs;2 vgs;1 ¼ vi;1 vi;2 1þ

Lp Rp

jo Lp

Cd

jo þ Cp0 Lp ð joÞ Cd þ Cgs

gm

ð12:54Þ

Section 11.6.3 deals with the differential pair as amplifier. The ideal DP has zero common-mode gain, as Section 11.6.3 showed. Therefore symmetrical excitations produce no effect on the ideal DP. 16

OSCILLATORS

Lp

Rp

Cp

Cp

529

Rp Lp

VOUT,1

VOUT,2

Cdd

Cd

Cd

Q1

Cdd Q2

RLOAD

+

RLOAD Vdd

I0 (a)

Lp

R′p

Q1

VOUT,1 Id,1

Cp

Cp

Cd

Cd m1

D

+ v1,1 -

gmVi,1

G

G

+ vgs,1 Cgs -

Cgs

m2 + vgs,2 -

R′p L p VOUT,2 Id,2 Q2 D

+ vi,2 -

S

I d,2

VOUT,2 Cd

gmVi,2

Cp

S

gmVi,2

(b) vgs,2

Cd VOUT,1 Id,1 D

Cgs

Lp

R′p

Cp gmVi,1

vgs,1

D

+ v1,1 -

+ vi,2 -

R′p Lp

Cgs

(c)

Figure 12.10 Differential oscillator: (a) electrical diagram; (b) simplified linear equivalent circuit; (c) network for open-loop analysis. After some simple manipulations, we can rearrange the loop gain as GL ðoÞ ¼ where ðDPÞ o0

1 ffi; ¼ pffiffiffiffiffiffiffiffiffiffiffi C0 p Lp

1 o Q j o0

1þ

1 o Q j o0

2 GL0 þ j oo0

sffiffiffiffiffiffiffi C0 p ; Q¼Rp Lp 0

GL0 ¼ R0 p

ð12:55Þ

Cd gm Cd þ Cgs

ð12:56Þ

It is easy to recognize that at the angular frequency o0 the function (12.55) becomes real and positive with GL ðo0 Þ ¼ GL0 Therefore, the circuit in Figure 12.10a produces stable amplitude oscillations at the angular frequency o0 if the parameter GL0 is not smaller than 1. This implies a corresponding condition on the transistor

530


transconductance gðDPÞ m

Cgs 1 1þ Cd R0 p

ð12:57Þ

If we consider a bipolar DP instead of an FET one, the analysis is slightly more difficult due to the nonnegligible input resistance of the amplifier. The loop gain of a differential oscillator BJT DP can be derived from expression (12.54) by replacing the gate–source admittance with the base–emitter one jo Cgs ! jo Cbe þ 1=Rbe The above transformation is equivalent to replacing the quantity Cgs by the quantity Cbe þ ðjo Rbe Þ1 in Equation (12.55). After applying the condition (12.35) and developing the relative calculations, we find the oscillation frequency and the minimum transconductance, which are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R L L C ðCd þ Cbe Þ2 u 1 þ 1 Lp 1 þ Rbep Cd þ R1be Rpp 1 þ CCbed þ R2p 1 þ Cdp Cd Rbe Rp Cd þ Cbe u be ; gm o0 ¼ t L ðCd þ Cbe ÞRp þ Rbep Cp þ CCd dþCCbebe Lp It is easy to verify that the above quantities coincide with (12.56) and (12.57) respectively, when Cbe ¼ Cgs and for Rbe tending to infinity, as expected.

12.6 Design of a trial oscillator The analytical methods presented in Sections 12.2 to 12.5 are more suited to understanding how an oscillator works than for real design purposes. Their main common limitation is related to the simple models for the resonator and for the active device. Simplified analytical methods are also useful for the synthesis in the design, beginning by helping to find initial reasonable parameter values and/or to choose the most suitable transistor. On the other hand, numerical computer-based analyses have virtually no limitation on model complexity, and allow very accurate predictions – provided that the models are accurate as well – but they are of little or no help in the initial orientation of the design. This section shows how to exploit the above illustrated design principles, in order to set up accurate computer simulations, starting from some initial analytical computations. The design presented is not a state-of-the-art circuit, and also includes some ideal elements in the resonator and in the bias network. However, the addition of more detailed models for all the circuit elements does not significantly improve the accuracy of the proposed analyses. Let us design a 7 GHz ðoosc ¼ 2p 7 109 Þ oscillator with the inductive gate feedback topology and using the FET of Section 9.7ðgm ¼ 40 mS; Cgs ¼ 243 fFÞ. Equation (12.44) determines the minimum potential oscillation frequency of the circuit for a given gate inductance, or the latter quantity from the first one. We need the minimum potential oscillation 1 2 frequency to be lower than oosc , thus Lg > Cgs oosc ffi 2:13 nH. The simplified linear model used in Section 13.5.2 neglects many additional parasitics, therefore it is reasonable to assume that the effective capacitance to consider is much higher than Cgs . Consequently, the required gate inductance will be lower than 2.13 nH. Also, the neglected parasitics shift the effective oscillation frequency down towards lower frequencies. Let us temporarily assume that the anti-resonant circuit parameters are Cp ¼ 2 pF; Rp ¼ 500 O; Lp ¼ 1 nH. Equations (12.7) give the resonant frequency and the quality factor of the resonator17

17

See also the Mathcad file 03_Inductive_Gate_Feedback.MCD.

OSCILLATORS 1 o0 ¼ pffiffiffiffiffiffiffiffiffiffi ffi 2p 3:559 GHz; Lp Cp

531

sffiffiffiffiffiffi Cp ffi 22:4 Q ¼ Rp Lp

Note that: .

The resonant frequency of the tank is distant from our target of 7 GHz, but, again, the frequency shift due to the active device must be considered.

.

Normally the known parameter is the resonator’s Q, while the parallel resistor is computed consequently; however, the above value is quite realistic for a lumped element circuit.

þ Assuming Lg ¼ 1:8 nH, Equation (12.42) gives the possible oscillation frequencies of oosc ¼ 9 9 2p 8:161 10 ; oosc ¼ 2p 3:319 10 . The second value is well below the minimum potential þ oscillation of the active element, and should not cause any mode jumping. Instead, oosc is not only within the oscillation range of the active circuit, but also much greater than o0 , as predicted. At this point, analytical calculations can be only slightly more accurate, at the expense of complexity and intelligibility in the formulae involved. Therefore, it is more convenient to pass to computer-aided circuit analysis, first linear and then nonlinear. Figure 12.11b shows the complete linear equivalent circuit of the oscillator in Figure 12.11a: the transistor is replaced by its linear model of Figure 9.34b, and the drain voltage DC generator is

+ Vdd Rp(500 Ω)

Q1

L dd (15 nH) Cdd (47 pF) VOUT RLOAD (50 Ω)

L p(1 nH)

Cp (2 pF)

L g(1.8 nH)

(a) β) Cut for open-loop gain Q1

Ldd (15 nH)

Lg (130 pH) Rg (0.1 Ω) Lg (1.8 nH)

Ld (160 pH) Rd (0.1 Ω)

Cgd (54 fF)

+ vgs Cgs (243 fF) -

+ vi -

gmvi (t-τ) (gm=40mS)

Rds (277 Ω)

Cds (92 fF)

Cdd (47 pF) VOUT RLOAD (50 Ω)

Ri (3.8 Ω) Rs (0.1 Ω) Ls (36 pH) YIN, ΓIN α, χ) Cut for admittance and reflection coefficient

YLC, ΓLC L p (1 nH)

(b)

Cp(2 pF)

Rp(500 Ω)

Figure 12.11

Trial oscillator: (a) electrical diagram; (b) equivalent linear circuit.

532


short-circuited. Figure 12.11b also shows two possible cutting lines, which can be used for different types of linear analysis: (a) Breaking the connection between the resonator and the active circuit, and computing the admittances YLC and YIN seen on opposite sides of the cut. These are the admittances of the resonator and of the active devices, as defined in Sections 12.2 and 12.3. According to the condition (12.29), the circuit oscillates at the frequency oosc;a such that Im½YLC ðoosc;a Þ þ YIN ðoosc;a Þ ¼ 0, if simultaneously Re½YLC ðoosc;a Þ þ YIN ðoosc;a Þ 0. The circuit analysis returns oosc;a ¼ 2p 6473:1 109 with Re½YLC ðoosc;a Þ þ YIN ðoosc;a Þ ffi 79 103 O1 . (b) Breaking the connection between the VCCS and the gate–source capacitance, and computing the complex voltage transfer ratio GL ðoÞ ¼ vgs =vi , which is the oscillator voltage loop gain. The fulfilment of the

condition (12.22) requires that a value oosc;b exists, such that 20 log10 GL ðoosc;b Þ 0 and arg½G L ðoosc;b Þ ¼ 0. The result in our case is oosc;b ¼ 2p 6823:556 109 and 20 log10 GL ðoosc;b Þ ¼ 13:5 dB. (w) An alternative technique consists of computing the reflection coefficients corresponding to the admittances (12.33) it follows that, at the oscillation frequency, this is (a). From the relations 20 log10 GLC ðoosc;w ÞGIN ðoosc;w Þ 0 and arg½GLC ðoosc;w ÞGIN ðoosc;w Þ ¼ 0. The computer-aided analysis gives oosc;w ¼ 2p 7351:802 109 and 20 log10 GLC ðoosc;w ÞGIN ðoosc;w Þ ¼ 8:15 dB. Figure 12.12 shows the results of all the three analyses above.18 Note that: .

None of the quantities ReðYLC þ YIN Þ; jGL j; jGLC GIN j reach their local maxima at the respective oscillation frequencies, as observation (e) in Section 12.3 anticipated.

.

The three linear methods give different oscillation frequencies.

However, the three values (a), (b) and (w) coincide in the critical case of the amplitude condition satisfied with the equals sign. In order to show the validity of this assertion, we can start from (b), which is the simplest critical amplitude condition to obtain. In that network, the transistor transconductance affects the amplitude but not the phase of the voltage loop gain. Thus, downscaling the transconductance19 as 1 g0m ¼ gm GL ðoosc;b Þ ffi 40 1013:5=20 ffi 8:455 mS analysis (b) gives GL ðoosc;b Þ ¼ 1 while arg½GL ðoosc;b Þ ¼ 0 holds. Using the reduced gain transistor linear model, the oscillation frequencies from (a) and (w) also coincide with the value (b) oosc ¼ oosc;a ¼ oosc;b ¼ oosc;w ¼ 2p 6823:556 109 with Re½YLC ðoosc Þ þ YIN ðoosc Þ ¼ 20 log10 ½jGL ðoosc Þj ¼ 20 log10 ½jGLC ðoosc ÞGIN ðoosc Þj ¼ 0 and Im½YLC ðoosc Þ þ YIN ðoosc Þ ¼ arg½GL ðoosc Þ ¼ arg½GLC ðoosc ÞGIN ðoosc Þ ¼ 0 More generally, if we change the transistor parameters, in order to fulfil the oscillation conditions at the critical value, in any of the analyses (a) to (w), then the three methods satisfy the phase condition at the same frequency and the amplitude condition with the equals sign. Unfortunately, oscillators cannot work

18 See the ANSOFT file 01_Open_Loop_Rneg_Oscillator.adsn and the SIMetrix file 10_Oscillator_AC_ OpenLoop.sxsch. 19 See the ANSOFT file 02_Open_Loop_Rneg_Oscillator_Critical_Gain.adsn.

OSCILLATORS

YLC+YIN

200

Re(YLC+YIN)

533

Im(YLC+YIN)

(α)

0

-200

5

6

7

8

9

10 180

20

(β)

dB(GL)

5

ang(GL) -180

0

5

6

7

8

9

10 180

20

dB(ΓLCΓIN)

ang(GL)

0

10

0

0

ang(ΓLCΓIN)

dB(ΓLCΓIN)

-180

-20

5

6

7

8

9

(χ) ang(ΓLCΓIN)

dB(GL)

15

10

Frequency, GHz

Figure 12.12 Linear analysis of the oscillator in Figure 12.11: (a) sum of resonator and active network admittance; (b) loop voltage gain; (c) product of resonator and active network reflection coefficients. in the critical condition, because any perturbation (increase of the amplitude from the small initial value, temperature, tolerance, etc.) will cause the loss to exceed the power supplied by the active device, and stop the oscillations. Moreover, the steady state oscillation amplitude is stabilized by the transistor compression mechanism, as explained in Sections 12.2 to 12.4. This condition is inherently nonlinear, and consequently the application of the transistor linear model is quite inaccurate. Nevertheless, we can formulate three simple hypotheses on the large-signal behaviour of the transistor, which could allow us to use one of the results (a) to (w) to predict the steady state oscillation frequency. We can conjecture that at the oscillation frequency, and passing from small- to large-signal operation, the transistor behaviour is such that: (a0 ) ImðYLC þ YIN Þ holds at zero, while ReðYLC þ YIN Þ becomes less and less negative, up to zero. (b0 ) argðGL Þ is constantly zero, while jGL j progressively decreases down to one. Note that this corresponds to saying that the compression affects only the transconductance. (w0 ) GLC GIN behaves like GL in (w0 ). None of the assumptions (a0 ) to (w0 ) is rigorous, although (b0 ) looks more reasonable, particularly for low compression. This is not our case, which presents 13.5 dB of loop gain excess in the small-signal condition. At this stage of our design, we could decide to trust one of the techniques (a) to (w) with the corresponding additional assumption (a0 ) to (w0 ), and adjust the circuit to obtain the desired output frequency of 7 GHz. This can be achieved by tuning some of the parameters Lg ; Lp ; Cp .

534

MICROWAVE AND RF ENGINEERING Re(ΓLCΓIN) ρ=|ΓLCΓIN|

ω0

Figure 12.13

Nyquist diagram corresponding to the plot of Figure 12.11c.

For completeness, Figure 12.13 shows the Nyquist diagram of the reflection coefficient loop gain, which coincides with the polar plot of the functions GLC GIN and conjðGLC GIN Þ. Note that it encircles the critical point, as expected. A more accurate approach consists of passing to the nonlinear simulation, by restoring the DC generators and using a nonlinear transistor model, in our case the one of Figure 9.38. A transient analysis20 of the circuit shows a steady state oscillation frequency of 5.267 GHz, which is about 23% lower than the mean of the values (a) to (w). The various linear methods give oscillation frequencies that change by less than 13% from the minimum to the maximum. This clearly indicates that it is not worth refining the linear design. It is possible to increase the output frequency by lowering one or more of the feedback inductance, resonator inductance and resonator capacitance. In the present design, we tuned the feedback inductance to Lg ¼ 866 pH, and obtained the required simulated21 output frequency of 7 GHz. Figure 12.14 plots the simulated output voltage of the final oscillator, showing the initial switch-on transient, which takes about 10 ns, and seven periods of the steady state waveform, corresponding to a time span of 1 ns. The initial part of the waveform is a result of the oscillation rising up and of the bias network transient. The output waveform is quite different from a sinusoid, confirming that the design circuit works under a heavy compression regime, as expected from the high margin on the amplitude condition achieved in (a) to (w).

12.7 Oscillator specifications Sections 12.2 to 12.6 described the basic theory of oscillator operation, analysis and design. An ideal oscillator produces a sinusoidal output voltage with constant amplitude and frequency, independently of the supply voltages – at least if they stay within specified limits – and from the load impedance. Now it is time to consider the possible deviations of the oscillator performance from the ideal case. The output voltage of an ideal oscillator is vðidealÞ ðtÞ ¼ vpeak cosðoosc tÞ osc 20 21

See the SIMetrix file 11_Oscillator_Transient.sxsch. See the SIMetrix file 12_Oscillator_Transient_II.sxsch.

ð12:58Þ

OSCILLATORS

535

10

initial transient

steady-state

VOUT , V

5

0

-5 0

5

29

30

Time, ns

Figure 12.14

Switch-on transient of the oscillator in Figure 12.7a.

while real oscillators produce voltage waveforms of the type vðrealÞ osc ðtÞ ¼

NH X k¼1

vp;k ðtÞcos½k oosc t þ fn;k ðtÞ

ð12:59Þ

The signal (12.59) consists of a number NH of harmonically related spectral components. If the amplitude vp;k and the phase term fn;k of the kth harmonic are constant over the time, the function (12.59) is the Fourier series expansion of the oscillator output voltage. The subsequent discussion focuses on the fundamental component vp;1 ðtÞ cos½oosc t þ fn;1 ðtÞ, except from the harmonic level specification. The main specifications which characterize the oscillator performance are as follows: 1. Fundamental output power.: The output voltage is periodic, but in general not sinusoidal. The useful power is the one relative to the spectral line at the fundamental frequency. Assuming that vp;1 ðtÞ is independent of time, the fundamental output power is

where YLOAD

1 P1 ¼ v2p;1 Re½YLOAD 2 is the load admittance at the fundamental frequency.

2. Harmonic level.: This parameter defines the harmonic content of the output signal. The harmonic level is usually expressed in logarithmic units, relative to the fundamental, or dBc. In many cases a particular harmonic is specified, e.g. the second or the third harmonic; in other cases it is the global harmonic power that matters. Assuming that vp;k are independent of time, the kth harmonic level and the total harmonic level are respectively vp;k PH;k ¼ 20 log10 dBc ðk ¼ 2; 3 . . .Þ vp;1 1 0 NH X 2 v C B B k¼2 p;k C C PH ¼ 10 log10 B B v2 CdBc @ p;1 A

536

MICROWAVE AND RF ENGINEERING RF m1 Lp

Rp

αv

Cp

Zu VOUT

Z LOAD (a) RF

Lp

Rp

V2

Cp

V1

αv

Zu VOUT

Z LOAD

ZLCP (b) V1

αv

Zu

RF V2 ZLOAD

Z LCP

(c)

Figure 12.15 Oscillator load-pull analysis: (a) basic oscillator schematic; (b) open-loop network of (a); (c) rearrangement of (b).

3. Pulling.: The oscillator output frequency is not independent of the load. For instance, let us reconsider the simple circuit in Figure 12.6a and assume that the amplifier has finite output impedance Zu . Figures 12.15a,b show the resulting new schematic and the relative open-loop network, respectively. Rearranging the schematic in Figure 12.15b, and indicating the impedance of the resonant circuit as 1 1 1 ZLCP ðoÞ ¼ joCp þ þ Rp joLp we obtain the network in Figure 12.15c, which is an ideal voltage amplifier, followed by a fourelement ladder network. Its voltage transfer ratio is V2 ¼ av V1

ðRF þ ZLCP ÞZLOAD RF þ ZLCP þ ZLOAD ðRF þ ZLCP ÞZLOAD RF þ ZLCP þ ZLOAD þ Zu

ZLCP ZLOAD ZLCP ¼ av RF þ ZLCP ðRF þ ZLCP ÞZLOAD þ ðZLOAD þ RF þ ZLCP ÞZu

This equation points out that the load impedance ZLOAD influences the loop gain and, consequently, the oscillation frequency. Particularly harmful impedances can cause mode jumping, or even stop the oscillation. A simple case of the latter type is the short circuit ZLOAD ¼ 0 ) V2 =V1 ¼ 0. In engineering practice, the pulling is specified through the load reflection coefficient. For a given reflection coefficient amplitude, and for any arbitrary phase, the output frequency must keep within specified limits. Figure 12.16 shows the test setup, which supports that type of specification. The variable load is generated by means of a variable-length transmission line, while a directional coupler samplesthe outputfrequency. If the transmission line is perfectly matched ðs11 ¼ s22 ¼ 0Þ and loss free js21 j2 ¼ 1 , then it modifies the phase but not the amplitude of the load reflection coefficient. The directional coupler, even in the ideal case, adds a phase term yc, and reduces the amplitude of the reflection coefficient by the square amplitude of its forward transmission coefficient 1c2 . Normally, the directional couplers used

OSCILLATORS

537

Frequency meter

θ

Oscillator Under Test

directional coupler

variable length transmission line

mismatched load

Γ3 =(1-c2) ρLexp(jθ0+2θ+2θc Γ2 = ρLexp(jθ0+2θ)

Figure 12.16

Γ1 = ρLexp(jθ0)

Oscillator pulling test set.

for this test have a coupling coefficient lower than 10 dB, which means that 1c2 1100:1 ¼ 0:9. Now, it is easy to see that, when the electrical length of the transmission line varies from its initial value y to y þ p, the load presented to the oscillator describes a circle on the Smith chart, with centre at the origin and radius rMAX ¼ ð1c2 ÞrL . The corresponding measured maximum and minimum frequency must be within the specified limits.22 4. Pushing.: The supply voltage also affects the output frequency. The pushing specification determines by how much the output frequency is sensitive to a given voltage variation. High pushing values are harmful because any ripple present on the supply voltage frequency modulates the oscillator, producing unwanted sideband spur lines.23 5. Stability.: This defines how constant the output frequency is over time. It is possible to define short-, medium- and long-term stability, with time intervals between two measurements that range from seconds to years. 6. Phase noise.: We can consider the phase noise24 as a characterization of the very short-term stability of the oscillator, but for its particular importance in radar, radio communication and test instruments it needs to be discussed separately. Considering vp,1 as constant, and passing from the angular frequency o to the standard frequency f, the fundamental component of the oscillator signal is v1 ðtÞ ¼ vp;1 cos½2pfosc t þ fn;1 ðtÞ

ð12:60Þ

22 The SIMetrix file 13_Oscillator_Transient_Pulling.sxsch provides the pulling analysis of the oscillator discussed in Section 12.6. With a load of 40 O corresponding to rMAX ¼ 1=9 and to 19.1 dB of return loss, the output frequency varies from 6.753 to 7.367 GHz. 23 The SIMetrix file 14_Oscillator_Transient_Pushing.sxsch provides the pushing analysis of the oscillator discussed in Section 12.6. When the supply voltage varies from 3.5 to 4.5 V, the output frequency changes from 7.158 to 6.905 GHz. 24 See the reference [5]–[7] for further details about phase noise.

538

MICROWAVE AND RF ENGINEERING and the instant phase associated with that signal is yðrealÞ osc ðtÞ ¼ 2pfosc t þ fn;1 ðtÞ, with the ðrealÞ

corresponding instant frequency fosc ðtÞ ¼ fosc þ ð2pÞ1 djn;1 ðtÞ=dt. By comparison, the ideal

ðtÞ ¼ 2pf osc tosc and fosc , respectively, as in oscillator instant phase and frequency are yðidealÞ osc Equation (12.58). The instant fluctuations of the phase – and of the frequency as well – produce sideband noise spectra in the output signal. From Equation (12.60) it follows that v1 ðtÞ ¼ vp;1 cosð2pfosc tÞcos½fn;1 ðtÞvp;1 sinð2pfosc tÞsin½fn;1 ðtÞ

Now, for small phase fluctuations, as they are normally, jn;1 ðtÞ p=2; cos½jn;1 ðtÞ 1; sin½jn;1 ðtÞ jn;1 ðtÞ, and the above equation simplifies to v1 ðtÞ vp;1 cosð2pfosc tÞvp;1 sinð2pfosc tÞfn;1 ðtÞ

then

ð12:61Þ

Moving from the time domain to the Fourier domain, we have V1 ð f Þ vp;1

dð f þ fosc Þ þ dðf fosc Þ Fn;1 ð f þ fosc ÞFn;1 ðf fosc Þ þ vp;1 2 2j

ð12:62Þ

where the upper case quantities are the Fourier transforms of the corresponding lower case ones, and dðoÞ is the Dirac pulse. From fn;1 ðtÞ 2 R, it follows that Fn;1 ð f Þ ¼ conj½Fn;1 ðf Þ, thus Fn;1 ðf Þ ¼ Fn;1 ðf Þ and consequently also jV1 ð f Þj ¼ jV1 ð f Þj. Therefore, since we are only interested in the noise power, we will limit our consideration to the positive part of the oscillator output spectrum. The black curve in Figure 12.17 represents the amplitude of the spectrum (12.62). It consists of two symmetrical components, located at o ¼ oosc , each one being the sum of a carrier and two noise proportional to the phase fluctuation spectrum. The quantity sidebands, 2 ðvp;1 =2Þ2 Fn;1 ð fm Þ ð2 RLOAD Þ1 is the noise power coming from a rectangular passband filter, with 1 Hz bandwidth and centre frequency fm distant from fosc : it is the single sideband phase noise power density at the offset frequency fm. On the other hand, ðvp;1 =2Þ2 ð2 RLOAD Þ1 is the power associated with one of the two symmetrical spectral lines of the carrier. The phase noise is the ratio, usually expressed in logarithmic units referred to the carrier power, between the carrier and the noise Lm ð fm Þ ¼ 20 log10 Fn;1 ð2p fm Þ dBc=Hz ð12:63Þ The black curve in Figure 12.18 is the measured phase noise diagram of a 2.4 GHz oscillator.

|V(ω )|

0.5 v1δ(ω+ω0)

0.5 v1δ(ω−ω0)

v1|Φn1(Ω)|

0.5 v1|Φn1(ω +ω0)|

0.5 v1|Φn1(ω −ω0)|

O

Figure 12.17 Oscillator output amplitude spectrum.

ω

OSCILLATORS

539

-60

-80 L m(fm ), dBc

A2 fm-2 -100

A3 fm-3

-120

A0 -140 1k

Figure 12.18

10k

100k Fm’ Hz

1M

10M

Phase noise spectrum of a microwave oscillator.

We are now in a position to investigate the causes that generate phase noise inside the oscillator. Let us reconsider the block diagram in Figure 12.5, and assume that it represents an oscillator of the type in Figure 12.6a.25 From Equation (12.23) and with a few simple manipulations, we obtain the loop gain as GL ðoÞ ¼ GL0 2 jo o0

1 jo Q o0

þ

1 jo Q o0

ð12:64Þ

þ1

where GL0

Rp ¼ GL ðo0 Þ ¼ av ; Rp þ RF

1 o0 ¼ pffiffiffiffiffiffiffiffiffiffi ; Cp Lp

RF Rp Q¼ Rp þ RF

sffiffiffiffiffiffi sffiffiffiffiffiffi Cp Cp ¼ RF jjRp Lp Lp

The gain loop reaches its maximum at o ¼ o0 , where its phase vanishes. We know that o0 is also the regime oscillation, and in that condition the loop gain is unitary. The Taylor series, truncated to the first term, of the loop gain (12.64) is d½GL ðoÞ oo0 ðoo Þ ¼ 1j2Q GL ðoÞ GL ðo0 Þ þ GL0 0 do o¼o0 o0 Considering that in the steady state the loop gain is unitary, the corresponding closed-loop gain is 2 2 av o0 ¼ a2 1 ð12:65Þ jGCL ðoÞj2 ¼ v 4Q2 oo 1G ðoÞ L

0

Passing from the angular frequency to the regular frequency, and since o0 ¼ 2pf0 ; oo0 ¼ 2pð f f0 Þ ¼ 2pfm , Equation (12.65) becomes 2 1 f0 ð12:66Þ jGCL ð fm Þj2 ¼ a2v 2 4Q fm 25 We need this assumption because of the simple expression of the loop gain of those circuits; however, it is possible to extend our conclusion to a more general case. Alternatively, it is also possible to consider the circuit in Figure 12.7a, which presents a similar loop gain expression.

540

MICROWAVE AND RF ENGINEERING Now, from Equations (12.61) to (12.63) and (12.66), and with some more considerations of the circuits in Figures 12.5 and 12.6c, we can compute the phase noise of those oscillators. Equation (12.66) is the square magnitude of the output to input voltage ratio of the circuit in Figure 12.6c. The input voltage is the oscillation voltage across the resonator plus the thermal noise. If the amplifier is noise free, and vL is the peak amplitude of the oscillating voltage across the resonator, the phase noise is 2 R Rp RF 2 1 f0 1 LOAD Lm ð fm Þ ¼ 10 log10 2 2 KT a dBc=Hz av vL Rp þ RF v 4Q2 fm RLOAD where K is the Boltzmann constant and T is the absolute temperature, as defined in Section 9.4.1. Now, PS ¼ 0:5 v2L ðRp þ RF Þ=ðRp RF Þ ¼ 0:5 v2L =ðRp jjRF Þ is the power of the oscillating signal dissipated on the resonator, thus, neglecting the amplifier noise, the oscillator phase noise is " # 1 KT f0 2 dBc=Hz ð12:67Þ Lm ð fm Þ ¼ 10 log10 2PS 4Q2 fm A complete consideration of all the noise contributions in oscillators is too large to fit within this chapter; however, it is self-evident that the amplifier increases the phase above the value (12.67). The minimum increase to consider is the noise factor of the amplifier. Moreover, the amplifier works under compression. Therefore it operates quite nonlinearly, and all its low-frequency noise components intermodulate with the carrier, producing the so-called upconverted noise. Equation (12.67) then becomes " # 1 F 0 KT f0 2 fc p floor Lm ðfm Þ ¼ 10 log10 dBc=Hz ð12:68Þ 1þ þ 2PS 4Q2 fm fm Posc where: F 0 is the noise factor of the amplifier, conveniently increased, in order to keep the upconverted noise into account; fc is the flicker noise corner, defined as the frequency at which the flicker noise equals the thermal noise; p floor is the thermal noise floor of the amplifier; and POSC is the oscillator power delivered to the load. Thus, the best (low) phase noise involves: high Q factors, high RF power on the resonator26 and minimum noise on the active device. Note also that the intrinsic Q (or the unloaded Q) of the resonator does not matter. Rather, what it is relevant is how the resonator shapes the loop gain or the loaded Q factor. At this point we can see that the feedback resistor RF in the circuit of Figure 12.6a degrades the loaded Q of the resonator and, consequently, the phase noise. For this reason, all the standard configurations presented in Section 12.5 use reactive feedback elements rather than resistive ones. Also, high resonator power can cause highly nonlinear operation of the active device, and then high levels of upconverted noise. In other words, F0 and Ps in Equation (12.68) are not independent, and increasing Ps can involve increasing F0 as well. Therefore, sophisticated designs require accurate evaluations of the nonlinear noise performances of the active device. We can rewrite Equation (12.68) as A3 A2 ð12:69Þ Lm ð fm Þ ¼ 10 log10 3 þ 2 þ A0 fm fm

26 This means that the peak oscillating voltage across (current through) the parallel (series) resonator of an oscillator must be maximized to achieve the minimum phase noise.

OSCILLATORS

541

Equation (12.69) indicates that the oscillator phase noise consists of three terms, one constant and the other two inversely proportional to the square and to the cube of the offset frequency. Precisely, A0 ¼

p floor ; Posc

A2 ¼

1 F 0 KT 2 f ; 2PS 4Q2 0

A3 ¼

1 F 0 KT 2 f fc 2PS 4Q2 0

As confirmation, the phase noise plot in Figure 12.18 plots the best fit curve (thick grey line) of the form (12.69) and its three contributions (thin grey lines) separately: the distance between the measured and the model curve is always less than 2 dB. It is possible to extend the validity of expressions (12.68) and (12.69) to general types of oscillators, with arbitrary loop gain functions. The function (12.64) has two important properties that are useful for this: one applies to the amplitude, the other to the phase. Precisely, the 3 dB bandwidth limits of the loop gain are delimited by two frequencies o1 , o2 such that GL ðo1 Þ2 GL ðo2 Þ2 1 ¼ G ðo Þ G ðo Þ ¼ 2 L 0 L 0 thus o1 ¼ o 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4 Q2 1 ; 2Q

o2 ¼ o0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4 Q2 þ 1 2Q

Then, the 3 dB bandwidth of the loop gain (12.64) is Do3dB o2 o1 1 ¼ ¼ o0 o0 Q The argument of the function (12.64) is

o0 o arg½GL ðoÞ ¼ tan1 Q o o0 Differentiation of the angular frequency gives d farg½GL ðoÞg ¼ do o 2 o0

o 2

þ1 Q o 2 i2 o0 2 þ 1 o0 Q o0

h

Then, at the oscillation frequency, dfarg½GL ðoÞg Q ¼ 2 do o 0 o¼o0 Thus Q¼

o0 1 dfarg½GL ðoÞg ¼ o0 Do3dB 2 do o¼o0

ð12:70Þ

Figure 12.19 shows the amplitude (black line) and the phase (grey line) of the function (12.64). Figure 12.19 also shows the 3 dB bandwidth and the phase slope at o0 .


20 log10 [|GL(ω)/GL(ω0)|]

3

d{phase[GL(ω)]}/dω

90

0

∆ω3

-3

0 -6 ←

→

-9

(180/π) arg[GL(ω)/GL(ω0 )]

542

-90

-12 1 ω /ω0

Figure 12.19

Graphical representation of the loop gain (12.64).

High Q involves large variations in both amplitude and phase of the loop gain, as the oscillation frequency tends to change from its nominal value: these variations tend to oppose the frequency change that generated the variations themselves. Thus high Q implies low phase noise, in agreement with Equation (12.68) as well. Equation (12.70) supplies the generalization criterion we are looking for: the 3 dB bandwidth and the phase slope of the loop gain at the frequency satisfying the conditions in (5.16). The exact prediction of the oscillator phase noise is a difficult problem, and requires accurate nonlinear models for the active device and its noise behaviour. However, combinations of linear and transient nonlinear circuit analyses give reliable, albeit approximate, solutions by exploiting Equation (12.69) and some empirical assumptions. In this regard, let us reconsider the oscillator described in Section 12.6. The graph in Figure 12.12b gives27 o0 1 d farg½GL ðoÞg o0 ffi 2p 6:824 109 ; ffi 14:6; o0 ffi 9:4 2 Do do 3dB

o¼o0

The last two of the above quantities are not coincident, and their substitution into Equations (12.70) gives two different quality factors. Then we will use the geometric mean of the two and assume Q ¼ 11:7. Now, from the transient simulation we have that the oscillator output power is Posc ffi 79 mW, and the power dissipated on the resonator is Ps ffi 1:6 mW. Assuming a thermal noise floor of 130 dBm/Hz, a flicker noise corner frequency fc ¼ 20 kHz, and an equivalent noise factor F 0 ¼ 10, we have A0 ¼ 1:26 1015 ; A2 ¼ 1:13; A3 ¼ 2:26 104 . Then the oscillator phase noise at 10 kHz, 100 kHz and 10 MHz offset from the carrier is 74.7, 98.7 and 139 dBc/Hz, respectively. Further than thermal and upconverted transistor noise, additional causes can degrade the oscillator phase noise. Any noise superimposed on the supply voltage modulates the frequency – and thus the phase – of the oscillator, generating additional phase noise. Noise on the gate bias voltage is particularly harmful, in that it modulates the gate–source capacitance, which has a direct impact on the output frequency. 27

See the Mathcad file 04_Oscillator_Phase_Noise.MCD.

OSCILLATORS

543

7. Amplitude noise.: This is the noise associated with variations of the fundamental amplitude. Amplitude noise generates symmetrical noise sidebands around the carrier, similar to the phase noise, but associated with amplitude – rather than with phase – modulation. However, the amplitude noise is usually lower than the phase noise by orders of magnitude and is also significantly less important in the most common applications. Therefore, we will not discuss amplitude noise any further. Note that the performances 1 to 4 can be improved from outside the oscillator: .

The output power can be increased/decreased with amplifiers/attenuators.

.

Low-pass and/or bandpass filters can reduce the harmonic content of the output signal.

.

Combinations of attenuators, amplifiers and isolators, placed between the oscillator output and the load, improve the isolation of the first from the last, and reduce the pulling.

.

Additional voltage regulators decrease the oscillator’s internal voltage variations, minimizing the pushing.

Conversely, stability, phase noise and amplitude noise can only be improved from inside the oscillator, by using better components and/or a more accurate design. However, variable-frequency oscillators28 can be stabilized by locking their frequency to a more stable one.29

12.8 Special oscillators All the circuits considered in Sections 12.2 to 12.7 are quite general or simplified types of oscillators. Summarizing these topics, we can say that any oscillator basically consists of two parts: one negative resistance device and one resonator. The first device could be either a transistor with its associated feedback elements, or a special two-terminal device, like a tunnel, Gunn or INPATT diode. The resonator is a reactive element – with the highest possible Q – presenting a zero in the impedance (series resonator) or in the admittance (parallel resonator) at one particular frequency. Many solutions are available for the resonator, depending on the required electrical performance, size and cost; the subsequent sections will describe the most important ones.

12.8.1

Lumped element and transmission line oscillators

From the conceptual and design points of view, the simplest possible resonator uses lumped elements. However, such elements can be realized to a reasonable approximation up to relatively low frequencies. For instance, at the frequency of 5 GHz the vacuum wavelength is 60 mm. The maximum allowed size is much less than a quarter wavelength, and is further decreased in that the propagation medium has a relative permittivity greater than a vacuum. Therefore, the true minimum size for a ‘lumped’ element operating at 5 GHz depends on the desired approximation;30 a rough estimate would be 1 mm. An additional limiting factor for lumped element resonators is their low quality factor, of the order of a few tens at frequencies above 1 GHz.

28 Section 12.8 below deals with voltage-controlled oscillators, which are particular types of frequency-variable oscillators. 29 Section 15.5.2 deals with such systems, known as phase-locked loops (PLLs). 30 See also Section 14.2.

544

MICROWAVE AND RF ENGINEERING + Ldd (15 nH)

Vdd Rp(500 Ω)

Cdd (47 pF) VOUT

Q1

RLOAD (50 Ω)

(a)

L p (1 nH)

Cp (2 pF)

Lg (866 pH)

TL'dd Vdd TL dd Q1 VOUT CL dd

(b)

TL LP

TL CP

RLOAD (50 Ω)

TL g

Figure 12.20 Lumped and distributed element oscillators: (a) lumped element network, as obtained from the design discussed in Section 12.6; (b) distributed element transformation of (a).

The first way to increase the frequency is to realize the reactive elements with semi-lumped components, applying the semi-lumped technique discussed in Section 7.5. Figure 12.20 shows the application of that technique to the oscillator discussed in Section 12.6. Figure 12.20a shows the circuit with the component values as obtained after the final adjustment after the nonlinear analysis. Figure 12.20b shows the resulting schematic after replacing all the lumped components with distributed ones. Beginning with the inductors to ground, we can realize shunt inductors with high-impedance shortcircuit stubs. For this specific case, we can also derive a more accurate approximation than for the series inductor consider in Section 8.4. The stub has the same impedance as the inductor having one terminal to ground if ! c jZ0 tan pffiffiffiffiffiffiffiffi lstub oosc ¼ joosc L ee f f where: .

Z0 , ee f f and lstub are the characteristic impedance, the effective relative permittivity and the physical length of the stub, respectively.

.

c ¼ 3 108 m=s is the speed of light in a vacuum.

.

L is the inductance of the inductor.

Solving the above equation for lstub, we obtain the design equation to pass from the inductance to the stub physical length pffiffiffiffiffiffiffi ee f f 1 oosc lstub ðLÞ ¼ tan1 L ð12:71Þ Z0 c oosc

OSCILLATORS

545

Assuming that our transmission line is microstrip with h ¼ 0:508 mm; er ¼ 9:9, and the minimum realizable width is wmin ¼ 0:1 mm, then Equation (3.183) gives31 Z0 ¼ 90:07 O, and from Equation (3.182), ee f f ¼ 6:095. Then, we can realize the inductors Lp and Lg with short-circuit stubs that present the same impedance at the oscillation frequency. With our transmission line parameters and inductance values ðLp ¼ 1 109 ; Lg ¼ 866 1012 Þ, Equation (12.71) returns lstub ðLp Þ ¼ 1:255 mm and lstub ðLg Þ ¼ 1:105 mm. In principle, the same calculation also applies to the drain bias inductor32 Ldd , and gives lstub ðLdd Þ ¼ 3:965 mm. However, we can exploit the distributed nature of the stub to realize a perfect open circuit – rather than an approximate one – at the oscillation frequency. This means that the stub has to be l=4 at oosc , and its 1 physical length is 0:25 c e0:5 ¼ 4:34 mm. This way, the drain bias stub presents an open e f f ð2p oosc Þ circuit at the oscillation frequency. Moreover, at all the even harmonics the stub length is an even multiple of l=2; then the stub shorts all the even harmonics to ground, improving specification 2. Thus, in the transmission line version of the oscillator in Figure 12.20, the short-circuit stubs TLLP and TLG replace the inductors Lp and Lg , respectively, while the transmission line TLdd replaces the inductor Ldd . Passing to Cp , the open stub TLCP replaces it. Imposing that the stub must have the same impedance as the capacitor at the oscillation frequency, we obtain pffiffiffiffiffiffiffi ee f f 1 tan1 ðoosc Z0 CÞ ð12:72Þ lstub ðCÞ ¼ c oosc Many choices are possible for the stub impedance. Here, we are mainly interested in the reactance at one specific frequency, therefore we can use a 50 O impedance, which is realizable in the given substrate and gives a good shape factor for the stub. With our substrate, the width of the 50 O line is 0.489 mm, correspondingly ee f f ¼ 6:621, and Equation (12.72) gives lstub ðC ¼ 2 1012 Þ ¼ 3:571 mm. The circuit in Figure 12.20b also includes the additional open stub TL0dd (grey) connected to the drain bias node. This stub is l=4 at oosc and provides a short circuit at the bias point, independently of the DC generator RF impedance, which is difficult to control. The stub’s physical length is 1 0:25 c e0:5 e f f ð2p oosc Þ ; if its characteristic impedance is 50 O, the resulting length is 4.164 mm. Application of the network identity of Figure 4.25b offers the possibility to replace even the series DC block capacitor Cdd with a coupled-line section that is l=4 at oosc . The two-port network on the left of the figure is equivalent to two open series stubs with an interposed transmission line, and all those elements have the same electrical length as the coupled-line segment. When the coupled lines are l=4, the two stubs present zero series impedance, and then the network simplifies to a l=4 transmission line having Z0 ¼ ðZ0e Z0o Þ=2. Therefore, the network in Figure 4.25b behaves like a DC block and does not affect the propagation of the signal if its electrical length is 90 at the oscillation frequency and its even and odd mode impedances are such that Z0e Z0o ¼ 100

ð12:73Þ

Under that condition, the only free parameter Z0e or Z0o must be chosen in order to guarantee the physical realization of the coupled lines. The constraint (12.73) usually gives narrow width and gaps. Setting w ¼ s, and calculating33 the value fulfilling the condition (12.73), we obtain w ¼ s ¼ 59 mm, which is still realizable if the technology is relatively good. From the effective permittivities34 for even and odd modes ðee f f ;e ¼ 6:426; ee f f ;o ¼ 5:467Þ it follows that the physical length of the coupled lines 0:25 c ðee f f ;e ee f f ;o Þ0:25 ð2p oosc Þ1 ¼ 4:401 mm. 31

See the Mathcad file 05_Microstrip_Analysis_Synthesis.MCD. Keep in mind that the DC voltage generator behaves like a short circuit at RF. Therefore, from the high-frequency point of view, Ldd has one terminal to ground, like Lp and Lg . 33 See the Mathcad file 06_Coupled_Microstrips.MCD. 34 If microstrip coupled lines are not TEM, then they present two different propagation velocities for the even and odd modes, and so it is for the effective permittivities. Consequently, the electrical length is not exactly defined; here we used the geometric mean between the even and odd relative permittivities. 32

546


The oscillator in Figure 12.20b consists of one transistor and some printed circuit elements. Therefore, it requires a minimum number of assembly operations, with consequent low cost and high repeatability of circuit performance. A slightly different realization technique consists of realizing the resonant circuit with a single shortcircuit (open-circuit) stub for a parallel (series) resonant circuit. Using this option, the open stub TLLP in Figure 12.20b is no longer present. Requiring the stub to have the same impedance as the resonant circuit at resonance means that the stub must be l=4 when o ¼ ðL CÞ0:5 . This determines the electrical length, but not the characteristic impedance of the stub, which needs an additional constraint. For instance, we can impose that, for the parallel (series) resonating circuit, the derivative of the admittance (impedance) of the stub with respect to o must coincide with that of the LC element. For the parallel resonant circuit, we have (

) 1 d joL þ joC d jZ10 cot p2 oo0 1 j p 1 po 2 ¼ 1 þ cot ¼ 2 þ jC ¼ o L Z0 2 o0 2 o0 do do pffiffiffiffiffiffiffiffiffi At o ¼ o0 ¼ 1= L C the above equation simplifies to rffiffiffiffi p L Z0 ¼ 4 C

ð12:74Þ

For a series LC network, by duality we can swap L with C, the short circuit with an open stub and Z0 with its reciprocal, obtaining rffiffiffiffi 4 L Z0 ¼ ð12:75Þ p C Application of Equation (12.74) to the resonant circuit of Figure 12.20a gives Z0 ¼ 17:562 O and, consequently, ee f f ¼ 7:874, l ¼ 7:51 mm. If the frequency is not too high, the resonator can be realized with a coaxial line rather than a microstrip one: the associated Q passes from an order of magnitude of some tens to some hundreds. Observations: (a) The network in Figure 12.20b replaces all the lumped elements of the network in Figure 12.20a with distributed ones. The only exception is the resistor Rp, which is not a physically existing component, it just models the resistive losses in Lp and Cp . The two stubs TLLP and TLCP present non-zero dissipation loss, although no specific element models the loss. (b) In general, a distributed element oscillator obtained from a lumped element one has a slightly different oscillation frequency, due to the approximations inherent in the design equations. Therefore the distributed element circuit needs additional tuning to obtain the desired oscillation frequency. Further tuning is required after considering other second-order effects, like connection parasitics and transmission line discontinuities. A different design approach does not start from a lumped design, but considers distributed elements from the small-signal analysis phase. (c) The use of single stubs instead of open- and short-circuit combinations leads to longer structures and usually to higher Q factors. The greater length is a disadvantage at low frequency, where the component sizes are usually large, but can become an advantage at high frequencies, where dimensions that are too small could be difficult to realize. (d) The distributed elements present multiple resonant frequencies: the risk of mode jumping is then more prevalent than in lumped realizations, and requires additional care during the design and fabrication.

OSCILLATORS

12.8.2

547

Cavity oscillators and dielectric resonator oscillators

The highest quality factors – thus the lowest phase noise – are obtained from cavity resonators. Section 5.6 described how to compute the resonant frequency, the Q factor and the equivalent circuit of some cavity resonators. In particular, a loop coupled with a cylindrical cavity behaves like a parallel RLC network, at least in a frequency interval placed across one of its resonant frequencies. Thus, in principle, we can replace the lumped parallel resonator in all the oscillators considered in Sections 12.2 to 12.6 with a loop-coupled cavity. The values of Q that are typically obtained with this solution are of the order of a few thousand. The main drawback of cavity resonators is their size, which is comparable with the wavelength. Dielectric resonators (DRs) are special types of cylindrical resonators, built with high-permittivity, low-loss dielectric materials. Due to the high dielectric constant, the DR size is much smaller than that of the equivalent metal cavity. Also, the absence of metal similarly eliminates the loss: the quality factor of a DR typically ranges from 5 103 to 104 . From a circuit point of view,35 placing a DR close to a microstrip produces an attenuation peak at the resonant frequency of the DR itself. This is equivalent to cutting the microstrip at the point where the DR is placed (defined by the intersection of the line passing from the centre of the DR with the microstrip axis) and inserting a parallel RLC network between the two sides of the cut. Figure 12.21a shows the microstrip with the coupled DR and Figure 12.21b shows the equivalent network. Oscillators using DRs are usually referred to as dielectric resonator oscillators (DROs). Figure 12.22 shows the schematics of two oscillators using a DR coupled with a microstrip. The one in Figure 12.22a is an inductive gate feedback. The oscillator in Figure 12.22b is a Colpitts36 configuration, where the gate–source capacitance of the transistor realizes the capacitance Cp1 , and Cp2 is realized by the open stub TLCP , which is shorter than l=4 at the oscillation frequency. Both the DROs in Figure 12.22 are reflection-type DROs, in that they exploit the reflections along the line, caused by the DR. For both the circuits, the length of the line to the left of the DR (l2 ) is not relevant, in that the line is impedance matched and presents constant impedance, independently of its length. Conversely, the length l1 is a design parameter of the circuit: .

On the left of the dashed line we have the parallel RLC network due to the DR, in series with a 50 O resistor to ground, with corresponding reflection coefficient GLC .

.

At the right end of the microstrip, looking towards the transistor we see a negative resistance element. The corresponding reflection coefficient is GIN with jGIN j > 1.

.

Between the resonator and the negative resistance there is a 50 O line having the physical length l1 that can be adjusted to modify the phase of GIN , in order to fulfil the condition (12.32).

Another possibility for DRO design is to use the DR coupled with two microstrip lines. The network can be analyzed in a relatively simple way in the symmetrical case of the DR centre having the same distance from the two lines, in terms of even and odd modes.37 Figure 12.21c shows the disposition of the resonator and Figure 12.21d illustrates the equivalent network, which is an extension of the one in Figure 12.21b. Terminating the ports P2 and P4 on 50 O while using P1 as input and P3 as output, the resulting two-port network has a passband response with the peak gain at the resonant frequency of the DR. This passband response can be used to realize a transmission-type DRO, like the one in Figure 12.23a. Again, in that circuit, moving the DR from the left to the right of the picture we simultaneously change l2 ; l4 (which are

35

See [8]. See also Section 12.5.4. 37 See [9]. 36

548


DR puck P1

P2

(a) Lp Rp

P1

P2

Cp (b) P1

P2

symmetry plane DR puck P3

P4

(c)

P1

T2

P2

Lp Rp Cp

(d)

P3

P4 T1

Figure 12.21 Dielectric resonator coupled with microstrip lines: (a) dielectric resonator coupled with a single microstrip; (b) equivalent circuit for (a); (c) dielectric resonator symmetrically coupled with two microstrip lines; (d) equivalent circuit for (c).

not significant) and l1 ; l3. If the amplifier is perfectly 50 O matched at input and output, the only significant phase on the loop path is the sum of the one of the amplifier itself plus the contribution of the lines l1 ; l3. Therefore, in moving the DR from the left to the right, one position exists such that the loop phase is unitary. In this condition, if the amplifier gain is sufficient to overcompensate the losses, the circuit in Figure 12.23a oscillates at the resonant frequency of the DR.

OSCILLATORS

549

DR puck + L dd

Vdd

Cdd

Q1 50 Ω termination

VOUT

microstrip line

RLOAD (50 Ω) Lg

(a)

2

1

DR puck + Vdd

L dd Cdd VOUT

Q1 50 Ω termination

RLOAD (50 Ω)

microstrip line L gg

(b)

Figure 12.22

2

1

TLCP

Reflection-type DRO: (a) inductive gate feedback; (b) Colpitts.

The network in Figure 12.23b is derived from the one in Figure 12.23a by eliminating the lines to the right of the DR and neglecting the bend discontinuities of the microstrip connected to the amplifier input. With further rearrangements, the circuit in Figure 12.23b can be transformed into the one in Figure 12.23c, which highlights the similarity of the transmission DRO with the inductively coupled oscillator, discussed in Section 12.5.1. DROs present very low phase noise, below 100 dBc/Hz at 10 kHz offset from the carrier. As a drawback, DRO design could be critical, considering the high number of resonant frequencies of the cavity resonators in general, and of the DR in particular, with the associated risk of mode jumping.

12.8.3

Voltage-controlled oscillators

An important class of oscillator is given by the voltage-controlled oscillators (VCOs) that produce a variable frequency depending on a suitable control voltage. In most common cases VCOs use the junction capacitance of a semiconductor device (varactor) to implement voltage control of the output frequency. Less frequently, other devices are used: electromechanical and/or piezoelectric controlled reactive elements (capacitors, inductors, coaxial resonators), ferrite resonators (YIG), gyrator simulated variable inductors, etc. Here, we will only consider varactor VCOs. Section 9.7.2.1 described the nature and voltage dependence of a reverse-biased junction capacitance, which depends on the reverse bias voltage as in Equations (9.72) or (9.73). Varactors are special types of junction diodes, designed to work reverse biased, and presenting a controlled voltage–capacitance curve, together with a low series resistance. Section 10.4 described the application of varactors to continuously variable phase shifters.

550


50 Ω termination 3

DR puck αv

VOUT microstrip line

(a)

1

TL2

RLOAD (50 Ω)

2

T2 50 Ω

2

Lp Rp Cp

αv

TL1 VOUT T1

RLOAD (50 Ω)

1

(b)

TL1 αv

VOUT 50 Ω

1

T2 50 Ω

T1 Lp

Rp

TL 2

Cp 2

(c)

Figure 12.23 Transmission-type DRO: (a) principle; (b) equivalent network obtained from the model in Figure 12.21d; (c) rearrangement of (b). For VCO application, it is sufficient to replace the capacitor of the resonant circuit with a varactor and its associated bias circuitry, to obtain a variable output frequency, depending on the reverse bias applied to the junction. Figure 12.24 shows three typical VCO circuits. The one in Figure 12.24a uses inductive gate feedback, and is widely used at microwave frequencies in both MIC and MMIC technologies. The circuits in Figures 12.24b,c are common building blocks of RFICs operating below 5 GHz. In all three networks in Figure 12.24, a series element ZT is placed between the tuning port and one of the varactor electrodes. The requirements on ZT are that it has to transmit the DC voltage while presenting high impedance at the oscillation frequency, in order to minimize the resonant circuit loading. It must be considered that the varactor works reverse biased. Therefore only a small current flows through ZT . Consequently, a simple high resistance – of the order of some kilohertz – could be sufficient.

OSCILLATORS Vdd

551

Cdd2 Ldd Cdd

Q1 VTUNE ZT

Out

DV1 Lg Cp2 Lp

(a) Vdd Cdd2 Rcb VTUNE ZT

Ls

Cs2 Q1 CF,be

DV1

Cdd

CF,ee R be

Out

Ree L ee

(b) Vdd Lp

DV1

DV2

Lp

Cdd2

VTUNE ZT

Cd Q1

Vbb

+ Out -

Cd Q2

Q3 R ee

(c)

Figure 12.24

The VCO: (a) gate inductive feedback; (b) Clapp with BJT; (c) differential pair with BJT.

The network in Figure 12.24a is a straightforward transformation of the one in Figure 12.11a or Figure 12.20a. The only modification consists of replacing Cp with the varactor DV1 in series with the high capacitance Cp2 and the addition of the element ZT . Usually Cp2 presents low impedance to ground at the oscillation frequency. The network in Figure 12.24b implements a Clapp configuration38 with a bipolar transistor.39 The capacitor Cs2 isolates the varactor from the transistor base in direct current.

38

See also Section 12.5.6. See the SIMetrix file 15_MBC13900_VCO_Clapp.sxsch. It uses the transistor chip model as in Section 9.7.2.4 and a varactor with Cj0 ¼ 4 pF, Vj ¼ 0:8 V and g ¼ 0:5. 39

552


The circuit in Figure 12.24c is a direct derivation of the differential oscillator in Figure 12.7f. The varactors DV1 ; DV2 replace the two fixed capacitors Cp , and the differential pair uses two bipolar transistors instead of field effect devices.40 Differently from the two remaining VCOs in Figure 12.24, the latter present a balanced or differential output between the collectors of the transistors Q1 and Q2 . The transistor Q3 with Ree works like a constant-current generator, with the voltage Vbb derived from Vcc by the IC’s internal circuitry. Another peculiarity of the circuit in Figure 12.24c is that its varactors have the cathode connected to Vcc and the anode connected to the tuning voltage. This means that the tuning voltage cannot exceed Vcc , in order to keep the varactors reverse biased. Other solutions are of course possible, by complicating the bias scheme of the two varactors. VCOs have five additional specifications in addition to the ones illustrated in Section 12.7: 8. Output frequency range.: This is the interval between the minimum and maximum output frequency, for the tuning voltage varying within its allowed range. 9. Modulation sensitivity.: This is defined as the derivative of the output frequency with respect to the tuning voltage, and is usually expressed in MHz/V. The output frequency vs. tuning voltage characteristic of an ideal VCO is a straight line, with the two quantities related as fout ¼

Kv VTUNE þ f0 2p

In this case, the modulation sensitivity is constant over the tuning range and equals Kv =ð2pÞ. Unfortunately, real VCOs present a different tuning characteristic. Usually the varactor capacitance tends to saturate at high reverse bias, and the same thing happens with the VCO tuning curve at high tuning voltages. Moreover, other unwanted factors – like the output load varying with the frequency – introduce further irregularities on the tuning curve. Consequently, the modulation sensitivity is not constant over the tuning range, but varies between specified minimum and maximum values. As an example, Figure 12.29 below shows the tuning curve (thick black curve, left y axis) and the corresponding modulation sensitivity (thick grey curve, first y axis on the right) of a measured VCO. 10. Output power flatness.: The fundamental output power is not constant over the tuning range, rather it varies within a specified range. As an example, Figure 12.29 below plots the output power versus the tuning voltage (black dashed curve, second y axis on the right) of a measured VCO. 11. Settling time.: This parameter expresses how fast the output frequency changes, after the tuning voltage passes from one value to a different one. Normally, the settling time is specified for a step in the tuning voltage from two defined voltages V1 to V2 and for a given error in the output frequency. If f1 (f2) is the asymptotic frequency corresponding to V1 (V2 ), the VCO response to a step in the tuning voltage is not a step, due to the time needed to charge the varactor and all the capacitances’ direct current connected to it. For instance, in the VCO in Figure 12.24a, the tuning voltage must charge DV1 and Cp2 through the impedance ZT . Additional causes, like internal charge inertia of the varactor, can further increase the settling time. 12. Post-tuning drift.: In the previous description we mentioned an asymptotic frequency, meaning that if the tuning voltage becomes constant, the VCO reaches a stable frequency within a sufficiently long time. This is not true, in general, due to thermal causes internal to the oscillator. Not only the output power, but also the transistor-dissipated DC power are not constant with the output frequency. This causes the output frequency to drift after a change in the tuning voltage. 40 See the SIMetrix file 16_MBC13900_VCO_Differential.sxsch, which uses the same transistor chip model as in Section 9.7.2.4.

OSCILLATORS

553

Going back to the step response as defined in specification 11, when VTUNE remains at V1 for a long time, the transistor’s DC power is PDC1 and its temperature is T1. After the step, the transistor’s DC power tends to PDC2 and its temperature to T2, which means that during the transient the transistor temperature slowly changes from T1 to T2, affecting the output frequency. One of the techniques used to improve the post-tuning drift consists of maximizing the thermal conduction between the transistor and the external temperature, and stabilizing the latter. Normally, all the performances 8 to 12 vary with the temperature, from piece to piece, and 9 to 12 with the tuning voltage, too. The specification numbers are the worst case for all the possible combinations.

12.8.4

Push–push oscillators

Push–push oscillators are special arrangements consisting of two synchronous oscillators producing the same frequency but with a reciprocal time shift of half a period of the fundamental. Their importance comes from their capability to generate a given output frequency, starting from two single oscillators working at half of that frequency. This feature extends by one octave the application of any type of resonator and/or active device. Denoting the two oscillator output currents by superscripts ‘(1)’ and ‘(2)’ respectively, then NH NH NH X X X ð1Þ ð2Þ ak ½Ip cosðoosc tÞk ; id ðtÞ ¼ ak ½Ip cosðoosc tpÞk ¼ ak ½Ip cosðoosc tÞk id ðtÞ ¼ k¼0

k¼0

k¼0

Summing the two currents, the odd-order products cancel ð1Þ

ð2Þ

id ðtÞ þ id ðtÞ ¼

NH X k¼0

ak Ipk ½1 þ ð1Þk ½cosðoosc tÞk ¼ 2

NH=2 X m¼0

a2m Ip2m cos2m ðoosc tÞ

ð12:76Þ

As discussed in Section 9.5.2, the term 2a2m Ip2m cos2m ðoosc tÞ includes a sinusoidal component at the frequency 2moosc. Thus, summing the output currents of two synchronized oscillators operating with a relative 180 phase shift on the fundamental frequency, we cancel the odd harmonics – including the fundamental – from the output spectrum. This is the opposite of what happens with the push–pull configuration, which cancels the even harmonics.41 The same conclusion is reached by considering the voltage instead of the current, or any other linear combination of these two quantities. Therefore, with the push–push configuration, two oscillators at the frequency oosc produce a fundamental output frequency of 2oosc . The simplest push–push oscillator is the differential one,42 if we take the sum of the two drain/ collector output voltages instead of their difference, as considered in Sections 12.5.6 and 12.8.3. Figure 12.25 shows a push–push arrangement employing two VCOs of the type in Figure 12.24a. The dashed line indicates the symmetry plane of the two oscillators placed above and below that plane. Compared with the single oscillator, the push–push one has two main differences: 1. The gate inductor Lg of each oscillator does not connect to ground, but with the terminal of the corresponding inductor of the other half oscillator. Their junction point has a DC path to ground through the bipole Zgg . 2. There is a single common bias network for the two oscillators, consisting of Ldd =2, 2Cdd and 2Cdd2 . This is equivalent to connecting the corresponding bias elements of each oscillator in parallel. The transistor output currents sum by the resistors Rdd . If the two oscillators are synchronized and 180 out of phase, then the symmetry plane behaves like a short circuit to ground, and they can work approximately like a single one, apart from some nonlinear 41 42

See Sections 11.5.4 and 11.6.3. See Section 11.6.3.

554

MICROWAVE AND RF ENGINEERING ZT

DV 1

id(1)

Q1

Cp2 Lp VTUNE

Lg

Rdd

Zgg

2 Cdd iLOAD Out

Lg ZT

Rdd

DV 2 Q2

Cp2 Lp

Figure 12.25

id(2)

Ldd /2 Vdd 2 Cdd2

Push–push VCO.

effects. If the two oscillators are in phase, the symmetry plane becomes an open circuit, and the impedance to ground of each gate becomes joosc Lg þ 2Zgg , which stops the oscillation. Therefore the structure in Figure 12.25 supports anti-phase oscillations and blocks in-phase ones. Finally, the current to the load is the opposite of the sum of the AC components of the two transistor currents, and includes only even harmonics, as predicted. Figure 12.26 shows two fundamental periods of the two transistor output currents of the oscillator in Figure 12.25 at one tuning voltage, and their combination to the load.43 Note that two periods of the single oscillators produce four cycles of the push–push recombined signals. An ideal push–push oscillator, consisting of two identical mirrored parts, produces no odd harmonics. This is not the real case; in practice the fundamental component is typically 10–15 dB below the second harmonic, and must be filtered out. Since the third harmonic is also present, it is impossible to filter out those unwanted spectral lines in a push–push VCO having an output frequency range ratio higher than 1.5 : 1. It is theoretically possible to extract higher order harmonics from push–push oscillators, for instance the fourth harmonic, but this involves increasing difficulties in removing the adjacent unwanted spectral lines. 80

Output currents, mA

60

Id1

Id2

40

20

0

ILOAD -20 0.0

0.5

1.0

1.5

f0 t

Figure 12.26 43

Currents in a push–push oscillator.

See the SIMetrix file “17_Oscillator_Transient_II_Push_Push.sxsch”.

2.0

OSCILLATORS

555

One additional advantage of push–push oscillators derives from the fact that the symmetry line – which passes to the output port – is at zero volts for the fundamental frequency voltage. This means that the output port is a virtual short circuit to ground for the fundamental component. Therefore any output load does not affect the impedance seen by the two oscillators at the fundamental frequency, which is the most influential point for the oscillation frequency. Consequently, push–push oscillators present a low but non-zero pulling factor, because their symmetry is not perfect, and the harmonic loading slightly affects the oscillation frequency.

12.8.5

Amplitude-stabilized oscillators

In all the oscillators considered in Sections 12.2 to 12.8.4 it is implicit that the active device compression is the mechanism that stabilizes the oscillation amplitude. In the startup time there is an excess of negative conductance,44 or equivalently of loop gain; in the steady state, the gain exactly compensates the loss. The startup loop gain excess ensures a convenient margin on the variations of the loss in passive elements and of the active element gain. The variations to consider are the temperature, from piece to piece, and those due to the variability of the assembly process. Thermal compensation or stabilization of the oscillator can minimize the first effect, and individual tuning of the circuit reduces the margin needed for the latter two. However, a margin of a few decibels in the small-signal loop gain is at least required. This means that the active devices operate under a few decibels of compression in the steady state. Such a non-negligible degree of compression implies (i) the generation of harmonics, (ii) variation in the transistor DC working point, and (iii) upconversion of the low-frequency components of the active device noise. We have already met effect (i) in Section 11.5, dealing with power amplifiers, particularly on the circuit in Figure 11.13a. At high output swings, the oscillating voltage modifies the transistor bias, moving the direct current towards values that are different from the static point, in way that is difficult to predict. In extreme cases, the direct current can increase to become destructive for the transistor. The solution to this problem could be stabilization of the DC bias with a servo control as in Figure 11.13d, the current limitation as in Figure 11.13b or the DC feedback as in Figure 11.13c. Equivalent solutions can be found for FET oscillators. The harmonic generation problem (ii) is in principle always solvable by means of suitable low-pass and/or bandpass filters at the output. If the oscillator is a VCO with an output frequency range close to or wider than one octave, filtering becomes possible by means of a filter bank or a tunable filter, rather than with one single filter. The phase noise increase associated with the upconverted low-frequency noise is impossible to correct outside the oscillator, as mentioned in Section 12.7. When low-phase noise is a requirement, transistor saturation must absolutely be avoided, by keeping the transistor linear and with the proper gain. The scheme in Figure 12.27 shows a possible implementation of the concept. The block diagram uses a differential VCO, but it applies to any oscillator. In the figure, Q1 and Q2 form the DP, the varactors DV1 , DV2 biased at the voltage VTUNE Vdd contribute to the determination of the output frequency, together with Lp ; Cd , and the transistor parasitics. The differential amplifier AMP1 provides a single-ended output voltage, which could be another DP with the output on one collector.45 The transistor Q3 with the resistor Ree work as avariable-current source, with a current I0 increasing with the voltage on the base Q3 itself. As shown in Section 11.6.3, the SSG of the DP is proportional to the current,46 thus by varying thevoltage Vbb it is possible to control the gain of the amplifier. This is the specific function of the control loop formed by the directional coupler COUP1 , the detector DET1 , the low-pass filters LPF1 , LPF2 , and the VCVS AMP2 . The detector DET1 produces an output voltage47 proportional to the fraction of the output power extracted from the coupled port of COUP1 44 More precisely, we have excess negative conductance or resistance depending on whether the resonator is a parallel or series one, as explained in Sections 12.2 and 12.3. 45 As in the circuit of Figure 13.24a. 46 See Equations (11.102) for BJT DPs or (11.104) for FET DPs. 47 See Section 13.2.

556

MICROWAVE AND RF ENGINEERING Vdd Lp

DV1

DV2

Lp

Cdd2

VTUNE ZT

Cd

AMP1 P0 + + (1) vin A v vin -

Cd

Q1

Q2

c2 P0

COUP1

VDET

I0

Out

LPF2 VREF AMP2 + + vin (2) A v vin -

DET1

Vbb

Q3

POUT=(1-c2) P0

LPF1

Ree

Figure 12.27

DP VCO with automatic amplitude stabilization. VDET ¼ c2 gP0

where g is the detector sensitivity, c is the coupling coefficient of COUP1 and P0 is the RF output power of AMP1 . LPF1 cleans out all the residues of the RF signal from the detector output. LPF1 also contributes to limit the bandwidth of the control loop48 together with LPF2 . The control loop increases Vbb – thus the DP current and then its gain – when the output power tends to decrease and vice versa. Therefore, the arrangement in Figure 12.27 provides automatic control of the active devices in the oscillator. This control system – instead of the compression mechanism – stabilizes the output amplitude. The output voltage of AMP2 is proportional to the difference between the DC component of the detector voltage and the reference voltage VREF . The DC component of that voltage is exactly Vbb ð2Þ 2 Vbb ¼ að2Þ v ðVREF VDET Þ ¼ av ðVREF c gP0 Þ

and consequently the overall RF output power is POUT ¼ ð1 c2 ÞP0 ¼

" # 1 c2 1 Vbb V REF ð2Þ c2 g av

ð2Þ

If the loop is stable, and av is sufficiently high, we find that the second term in the square brackets becomes negligible and the output power simplifies to POUT ffi

1 c2 1 VREF c2 g

ð12:77Þ

Equation (12.77) states that the output power of the oscillator in Figure 12.27 depends only on the control loop parameters. With a suitable combination of g; c and VREF , it is possible to keep POUT at such a value that the DP and AMP1 operate linearly, while simultaneously ensuring that the loop gain is constantly unitary. Under this condition, the DP automatically compensates the loss in the resonant circuits. It operates linearly in 48

Such a control loop is a special case of automatic level control (ALC), see also Section 15.6.3.

OSCILLATORS

557

both the startup and steady state, regardless of temperature, device-to-device variation and even tuning voltage. Note also that the automatic amplitude stabilization schematized in Figure 12.27 applies also to any transistor oscillator. The only additional requirement for the oscillator is the possibility to change the transistor gain by a control voltage. Equation (9.76) states that the FET gain is monotonically decreasing with the gate–source DC voltage, which can be used as the gain control quantity. Similarly, the BJT decreases its gain with the base current, which is the gain control parameter: in this case one VCCS replaces AMP2 in Figure 12.27; all the other considerations hold true.

12.9 Design of a push– push microwave VCO This section describes one realized circuit,49 taken from the engineering experience of one of the authors (GB). It is a VCO with an output frequency range of 15.5–17.5 GHz. The design presented exploits many of the techniques discussed in Sections 12.3 to 12.8.4: it is an inductive gate feedback push–push VCO. Figure 12.28 shows the layout of the VCO, realized in MIC technology with the same microstrip substrate ðh ¼ 254 mm; er ¼ 9:8Þ as the amplifiers in Sections 11.4 and 11.7.1. For the device technology, we have: .

Four chip capacitors ð2 Cp2 ; Cdd and Cdd2 Þ; having the shape of a parallelepiped of approximate size 0.5 0.5 0.1 mm. The three capacitors having one terminal to ground ð2 Cp2 and Cdd2 Þ are directly brazed on the carrier, which works as the common ground and mechanical support of the circuit, simultaneously.

.

Two chip FETs ðQ1 ; Q2 Þ; having the shape of a parallelepiped of approximate size 0.3 0.4 0.1 mm.

.

Two varactors ðDV1 ; DV2 Þ; having the bottom electrode (the cathode) directly brazed on the top electrode of the respective bypass capacitors Cp2 . Their size is about 0.3 0.3 0.1 mm.

Cp1

VTUNE

Cdd2

Vcc bond wires

DV1 Zgg Lg Lg

Q1 Q2

Rdd

Lp

chip devices Out

Rdd L Cdd p

DV2 Cp2

thin film resistors microstrip lines microstrip substrate carrier

Figure 12.28

49

Layout of a microstrip push–push VCO having the schematic in Figure 12.25.

The circuit was aligned and tested by Raffaele Masina, who is gratefully acknowledged.

558


500

0

400

16

300 →  →

←

15

200 5

6

7

8

9

10

11

12

-4

-8

Output power, dBm

17

Modulation sensitivity, MHz/V

Output frequency, GHz

→

-12

13

Tuning voltage, V

Figure 12.29 .

Measured performances of the VCO in Figure 12.28.

Three printed resistors ð2 Rdd ; and the one associated with Zgg Þ; realized with thin-film resistor (TFR) technology.50

All the connections to components, microstrips and ground are realized with gold bond wire, having a diameter of 18 mm. The overall size of the carrier is 6 8 mm. The circuit schematic corresponds to Figure 12.25, where all the inductors ð2 Lg ; 2 Lp ; and Ldd Þ are realized with the semi-lumped technique, as discussed for this application in Section 12.8.1. The bipole Zgg , having high RF and low DC impedance to ground, consists of four cascaded elements: one small (0.1 0.1 mm) TFR, one bond wire, one printed spiral inductor and another bond wire to ground. The top electrodes of Cp2 (brazed to the respective varactor cathode) are DC connected to the tuning port via two bond wires and one U-shaped transmission line. The low impedance to ground of Cp2 , combined with the relatively high attenuation of the line, are sufficient to ensure enough RF isolation between the two resonators, and from these to the tuning port. The carrier assembly in Figure 12.28 is cascaded with another one containing one 6 dB attenuator and one edge-coupled filter ðN ¼ 3Þ, not shown. The attenuator is necessary in order to provide a low pulling. The filter reduces the unwanted spectral components ðf0 ; 3 f0 ; 4 f0 . . .Þ to a level lower than 60 dBc. From outside, the VCO in Figure 12.28 appears like a standard VCO, although the output frequency is equal to twice the internal oscillation one. Figure 12.29 shows the measured performances of the VCO. From the left to the external right y axis, we have: .

The output frequency vs. tuning voltage curve (thick black line, left y axis), also known as the tuning curve. With a tuning voltage from 5 to 13 V, the output frequency covers the range 15.1–17.9 GHz, which exceeds the specification by a margin of 400 MHz per side.

.

The modulation sensitivity (thick grey curve, internal right y axis), which is the derivative of the above quantity, as explained in Section 12.8.3. This quantity ranges from 235 to 421 MHz/V. It

50

See Section 14.3.3.

OSCILLATORS

559

decreases at high tuning voltages, due to saturation of the capacitance vs. reverse bias voltage, also anticipated in Section 12.8.3. .

The output power vs. tuning voltage (thin black line, external right y axis), which varies from 4 to 0 dBm over the tuning range. The output power variation can be reduced either with an automatic control – like the one described in Section 12.8.5 – or by cascading a saturating amplifier. Note that in the latter case the low-noise upconversion is not as relevant as within the oscillator, because the saturating active element would be external to the oscillator loop.

Bibliography 1. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, sections 4.4.4.3, 4.4.4.4 and 4.4.5, pp. 241–248. 2. K. Kurokawa, ‘Some basic characteristics of broadband negative resistance oscillator circuits’, Bell System Technical Journal, July–August, pp. 1937–1955, 1969. 3. S. Alechno, ‘Analysis method characterizes microwave oscillators’, Microwaves & RF, November, pp. 82–86, 1997. 4. G. Bianchi, Phase-Locked Loop Synthesizer Simulation, McGraw-Hill, New York, 2005, section 2.4, pp. 61–79. 5. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, section 8.17, pp. 283–295. 6. R. L. Howald, ‘Gauge oscillator spectral density and noise functions’, Microwaves & RF, March, pp. 113–120, 1994. 7. B.-G. Goldberg, ‘Oscillator phase noise revisited – a heuristic review’, RF Design, January, pp. 52–64, 2002. 8. Y. Komatsu and Y. Murakami, ‘Coupling coefficient between microstrip line and dielectric resonator’, IEEE Transactions on Microwave Theory and Techniques, Vol. 31, No. 1, pp. 34–40, 1983. 9. G. Bianchi and G. Pinto, ‘A simple and accurate model for a dielectric resonator symmetrically coupled with two microstrips’, Proceedings of the 19th European Microwave Conference, London, pp. 1189–1194, 1989. 10. A. A. Sweet, MIC & MMIC Amplifier and Oscillator Circuit Design, Artech House, Boston, MA, 1990.

Related files Ansoft files 01_Open_Loop_Rneg_Oscillator.adsn. Implements all the three linear analyses (a) to (w), of the circuit discussed in Section 12.6. 02_Open_Loop_Rneg_Oscillator_Critical_Gain.adsn. As file 01, but with the transistor transconductance reduced, in order to satisfy the amplitude condition with the equals sign.

Mathcad files 03_Inductive_Gate_Feedback.MCD. Implements the inductive gate feedback analysis formulae of Section 12.5.2 and numerically computes the frequency for the maximum loop gain.

560


04_Oscillator_Phase_Noise.MCD. Manipulates the AC loop gain obtained from a linear simulation to compute the parameters of Equation (12.70) and compute the oscillator phase noise. 05_Microstrip_Analysis_Synthesis.MCD. Analyzes and synthesizes isolated microstrips, also imple menting some of the calculations for Section 12.8.1. 06_Coupled_Microstrips.MCD. Analyzes coupled microstrips, and implements some of the calculations for Section 12.8.1.

SIMetrix files 07_Negative_R_I_V_Curve.sxsch. Provides a current versus voltage law that gives a negative resistance characteristic of the type in Figure 12.3b. 08_Positive_Feedback_Oscillator.sxsch. Analyzes the transient of the circuit in Figure 12.6a showing the oscillation arising. 09_Negative_R_Oscillator.sxsch. Describes an oscillator having the schematic as in Figure 12.3a, using device 5 as a negative conductance device. The analysis gives the switch-on transient voltage across the resonator. 10_Oscillator_AC_OpenLoop.sxsch. Provides the linear AC voltage loop gain of the oscillator discussed in Section 12.6. 11_Oscillator_Transient.sxsch. Includes the nonlinear transient analysis of the same oscillator as file 10. 12_Oscillator_Transient_II.sxsch. Same as file 11, but after adjusting the feedback inductance, in order to get an output frequency of 7 GHz. 13_Oscillator_Transient_Pulling.sxsch. Pulling analysis of the oscillator in file 12. 14_Oscillator_Transient_Pushing.sxsch. Pushing analysis of the oscillator in file 12. 15_MBC13900_VCO_Clapp.sxsch. Analyzes the circuit in Figure 12.24b, providing a nonlinear transient swept over the tuning voltage. 16_MBC13900_VCO_Differential.sxsch. Analyzes the circuit in Figure 12.24c in the same way as file 16. 17_Oscillator_Transient_II_Push_Push.sxsch. Analyzes the same circuit as file 12, but doubled and rearranged in a push–push configuration.

13

Frequency converters 13.1 Introduction This chapter deals with the three components that base their working on nonlinear effects: detectors, mixers and frequency multipliers. Such components have the common characteristic of generating an output frequency that differs from the input one. Section 13.2 describes the detectors which produce a DC output voltage proportional to the RF input power or to the RF peak voltage. They are mainly used for measurement purposes and sometimes in demodulator applications. Mixers are fundamental building blocks in modern high-frequency receiver, transmitter and test instruments. All these applications exploit mixers to convert a signal into an equivalent one with a translated frequency which is easier to process. Section 13.3 illustrates various types of mixer, employing diodes, FETs and BJTs in single-ended or balanced configuration. Finally, Section 13.4 presents some basic concepts on frequency multipliers that are sometimes used to generate high-frequency signals starting from lower frequency ones.

13.2 Detectors In this context, the word ‘detector’ denotes a nonlinear two-port having a high-frequency input and a video output. The video output produces a DC output voltage that increases with the input amplitude. We have linear or quadratic detectors if the DC output voltage is proportional to the amplitude of the sinusoidal input voltage or to its square, respectively. Quadratic and linear detectors are also known as square-law and envelop detectors, respectively. Both the detector types exploit the asymmetric conduction of a two-terminal device or the asymmetric transfer characteristic of a three-terminal one. The most common detector type employs semiconductor diodes as nonlinear devices, although transistor-based designs are also possible. The explanation of the square-law or envelope operation of a diode detector requires us to reconsider the diode DC characteristic presented in Section 9.7.2.1. As discussed, Equation (9.66) gives the relation between the diode current and the voltage drop across the internal diode D in the model of Figure 9.36, while Equation (9.69) expresses the overall voltage versus current of the diode. Detector analysis requires


562

MICROWAVE AND RF ENGINEERING 2x10-4

equation (13.1)

Id, A

1x10-4

1x10-4

5x10-5

Load line: I = I0-Vd /Rload

0

0.0

0.2

0.4

0.6

0.8

1.0

Vd , V

Figure 13.1

Forward current characteristic of a Schottky diode.

the current versus voltage function. Unfortunately, Equation (9.69) cannot be analytically inverted. For this reason we will develop two different expressions for the diode current as a function of the voltage, valid for small and large currents, respectively. Figure 13.1 plots1 the DC curve of a Schottky diode suitable for detector applications, having the parameters IS ¼ 250 109 A; n ¼ 1:1; Rs ¼ 100 O: this section uses this diode as a reference many times. The curve in Figure 13.1 has the same shape as the one in the upper right quadrant of Figure 9.39, although the conduction knee and the slope are quantitatively different. Looking at the curve in Figure 13.1, it seems that the diode presents no appreciable conduction when the voltage applied to its terminals is lower than 0.1 V. Consequently, for any applied excitation below that limit, the diode behaves like an open circuit, and the only consequence of its presence is due to its parasitic reactances: in other words, the diode behaves like a linear reactive element. In reality, this conclusion is not exact, in that the diode exhibits some non-negligible current also below the conduction knee. Figure 13.2 illustrates this assertion, by plotting the same curve (thick black line) as the one in Figure 13.1, but with logarithmic axes, in order to make the small and large current regions both visible. It is possible to develop exact and approximated equations for the curves in Figures 13.1 and 13.2 from the model in Figure 9.38, by removing its capacitors and short-circuiting the inductor. Moreover, under DC operation, the diode current Id coincides with Idi . Then, from Equation (9.69) it follows that Id Vd ðId Þ ¼ n VT ln þ 1 þ Id Rs ð13:1Þ IS Equation (13.1) has no analytic inversion, exactly like Equation (9.69). For small currents, the voltage drop across Rs is negligible with respect to Vdi , therefore we can assume Vd ffi Vdi . Consequently, Equation (9.67) gives the required current versus voltage expression of the diode Vd Id ðVd Þ ¼ Is exp 1 ð13:2Þ n VT Note that Equation (13.2) corresponds to setting Rs ¼ 0 in Equation (13.1) and inverting the result, which becomes possible with that simplification. Expanding the function (13.2) into a Maclaurin series, we get

1

See the Mathcad file 02_Diode_Curves.MCD.

FREQUENCY CONVERTERS

563

10 -1 equation (13.3)

10 -2 10 -3 Id , A

equation (13.4)

IS Vd n-1 VT-1

10 -4 10 -5

equation (13.1)

10 -6 10 -7 10 -8 10-3

Figure 13.2

10-2

10-1 Vd , V

100

101

Exact and approximated DC characteristics of a Schottky diode. Is Vd IS Vd 2 k Id ðVd Þ ¼ V ffi IS þ n VT 2 n VT k!nk VTk d k¼1 1 X

ð13:3Þ

The expression in the third term of Equation (13.3) is the Maclaurin series truncated to the second degree. This type of approximation is accurate enough for the present case, which assumes small current through, and then small voltage across, the diode. For the opposite case, at high forward currents, the voltage across D becomes negligible with respect to the drop across Rs : the first (respectively, latter) increases logarithmically (linearly) with the current. At high reverse bias, the diode current tends to IS , which is usually close to zero. Hence, at large voltages, the diode current tends asymptotically to the piecewise-linear curve 8 < Vd ðVd > 0Þ Id ðVd Þ ffi Rs ð13:4Þ : 0 ðV 0Þ d

The graph in Figure 13.1 plots the forward bias portion of functions (13.3) and (13.4), together with the linear term in the series of the first. Observations: (a) For small Vd , Equation (13.3) approximates the diode characteristic well. (b) At high Vd , the diode DC curves asymptotically tend to the piecewise-linear function (13.4). (c) In the region where Equation (13.3) approximates the diode curve well, its second-order term assumes smaller values than the first-order one. Sections 13.2.1 and 13.2.2 below exploit the approximated expressions (13.3) and (13.4) to analyze the quadratic and envelop operation of the diode detector, respectively.

13.2.1

Quadratic diode detector

Figure 13.3a shows the principle of a diode detector. It includes: .

The RF input generator with its series impedance ZG , the matching network NT1 .

.

The diode D1 .

564


I0 RF generator

NT1

D1

L bias

ZG 1

VG

2

L in

Cout

VDC R load

(a)

D1 Cp

Cd + Vdi CJ

RF generator

NT1

Id Ls

Rs

ZG 1

VG

I0

2

L bias

D

- Vd +

L in

Cout

VDC R load

extended matching network

(b)

D Z G′ VG′

Id

VDC

+ Vdi -

I0

(c)

D

Id

Rs V RF rdi

Rload

I0

(d)

VG′

Rs IOUT

I0’

Rload

VDC

-Vdi

ZG′

Cout

rdi

VDC I0

R load

(e) rd VOUT

VDC Rload

(f)

Figure 13.3 Diode detector: (a) electrical diagram; (b) extension of the matching network to the diode parasitic elements; (c) equivalent network of (b); (d) DC equivalent input of (c); (e) RF input and DC output equivalents of (c); (f) Thevenin scheme of the detector output circuit.


565

.

The bias current generator I0 with associated components Lin , Lbias , which have negligible DC impedance while behaving approximately like open circuits at RF.

.

The output capacitor Cout, which presents negligible RF impedance and is a DC open circuit.

.

The resistor Rload , which represents the input impedance of the circuit following the detector, and which usually is an amplifier.2

The presence of Cout – which is sometimes referred to as the RF-short – ensures that the RF voltage is applied to the diode, with no drop in the output circuit. Conversely, the inductors Lin , Lbias minimize the RF influence of the bias generator while allowing the DC bias current to flow through the diode. At first glance, Lbias looks like a redundant element, as it is in series with a current generator, but this not the case: the RF admittance of real DC generators is not only non-zero, but also difficult to predict or specify. Therefore, Lbias is needed for isolating the bias generator from the RF network. The inductor Lin has the dual function of Cout , and is also known as the DC-short, of the detector. The function of the current biasing is to move the working point of the diode close to the conduction knee, as we will see shortly in detail. Finally, the matching network maximizes the power transfer from the RF generator to the diode. From observation (c) above, it follows that for small voltages applied to the diode, it behaves almost linearly, with some second-order effect related to its quadratic nonlinearity. Moreover, small voltages also mean small voltage variations, therefore the junction capacitance of the diode behaves approximately as linear. Thus, for relatively small voltages, the superimposition of the effects is applicable, in particular to the RF and DC excitation. For our analysis it is convenient to consider the parasitic elements of the diode and Lin as part of the matching network. Figure 13.3b shows the concept of the ‘extended matched network’, enclosed within the dashed polygon, which includes NT1 itself, the diode parasitics and Lin . In this schematization, Cout is simply a short circuit to ground for the RF. The extended matching network transforms the voltage VG and the series impedance ZG of the generator into V 0 G and Z 0 G , respectively, where Z 0 G is as close as possible to the diode RF resistance. Figure 13.3c shows the detector network as resulting from this transformation. In the simplified case of negligible diode reactance and real ZG , NT1 simplifies to a transformer. Clearly, the matching network presents a DC short circuit to the cathode of the internal diode D, due to the presence of the inductor Lin. The direct current flowing through D1 equals a fraction of I0 , in that part of the latter flows through the load resistance. Let us call I 0 0 the portion of I0 flowing through the diode. The current I 0 0 results from the analysis of the network in Figure 13.3d, which is the DC equivalent of the circuit in Figure 13.3c. The direct current through the diode coincides with the ordinate of the diode DC curve with the load line,3 having the equation id ¼ I0 vd R1 load Assuming the diode parameters, which generate the curve in Figures 13.1 and 13.2, and I0 ¼ 50 mA; Rload ¼ 20 kO, the resulting DC diode4 is I 0 0 ffi 42:51 mA. Figure 13.1 also shows the load line (dashed segment) together with its intersection with the diode DC curve.

2 The amplifier following the detector takes the name of video amplifier, which presents a bandpass from DC to some megahertz. 3 Section 11.5 introduced the load line concept for power amplifiers. In that case the bias network consists of a voltage generator with a series inductor. Here the bias network consists of the current generator I0 with the resistor Rload in parallel. Passing from the Norton to the Thevenin representation, that bias network is equivalent to a voltage generator V0 ¼ Rload I0 with Rload in series. Therefore the load line is a segment in the VdId plane passing through the points ðRload I0 ; 0Þ and ð0; I0 Þ. 4 See the Mathcad file 02_Diode_Curves.MCD.

566

MICROWAVE AND RF ENGINEERING Given the direct current through the diode, Equation (13.1) gives the corresponding DC voltage I0 þ 1 þ I 0 0 Rs vd;0 ¼ n VT ln IS

The diode voltage is DC with a superimposed RF component vd ðtÞ ¼ vd;0 þ vd;RF ðtÞ

ð13:5Þ

where vd;RF ðtÞ ¼ vR cosðoR tÞ. Therefore, the diode voltage varies within the range ½vd;0 vRF to vd;0 þ vRF , and in our hypothesis the approximation (13.3) is valid in that range. This also implies that the voltage drop across Rs is negligible with respect to the one across D. Consequently, 0 I0 þ1 ð13:6Þ vd;0 ffi n VT ln IS For stronger reasons Equation (13.2) – which only neglects the effect of Rs – is valid, and gives the diode current vd;0 þ vd;RF id ffi Is exp 1 ð13:7Þ n VT Since the voltage across the diode has a non-zero DC component, the more general Taylor series is more convenient than the series (13.3) for analyzing the circuit. The Taylor series of the function (13.7) with initial voltage Vd,0 and truncated to the second-order term is ( ) vd;0 vd;0 vd;RF ðtÞ 1 vd;0 vd;RF ðtÞ 2 ð13:8Þ þ exp 1 þ exp id ðtÞ ffi Is exp 2 n VT n VT n VT n VT n VT Substituting the quantity (13.6) into the series (13.8) gives the result vd;RF ðtÞ I 0 0 þ IS vd;RF ðtÞ 2 id ðtÞ ffi I 0 0 þ ðI 0 0 þ IS Þ þ n VT n VT 2

ð13:9Þ

The current (13.9) is the sum of three terms: .

I 0 0 is constant and coincides with the bias current.

.

ðI 0 0 þ IS Þðn VT Þ1 vd;RF ðtÞ is proportional to the RF voltage.

.

0:5ðI 0 0 þ IS Þðn VT Þ2 v2d;RF ðtÞ is proportional to the square of the RF voltage, i.e. to the RF power. This term explains how the circuit in Figure 13.3a works like a quadratic detector.

The linear term of Equation (13.9) corresponds to the linear incremental resistance of the diode biased with the current I0 . Equation (10.4) gives that quantity: rd ¼

qvd n VT ¼ 0 þ Rs ¼ rdi þ Rs qid I 0 þ IS

ð13:10Þ

The linear resistance (13.10) coincides with the ratio of the voltage and current in the linear term of the series (13.9) if we neglect Rs , as done. Note that rd represents both the RF resistance seen by the RF generator in Figure 13.3c and the DC resistance seen by the output load; the latter is also known as video resistance. However, the diode parasitic reactances do not significantly affect the video impedance, because of its low frequencies, but modify the RF impedance. Thus the quantity (13.10) is more meaningful for the video output than for the RF input. With the diode and the bias point of Figure 13.1, we find that rd ¼ rdi þ Rs ffi 662:18 þ 100 ¼ 762:18.


567

Regarding the output, the second term in the series (13.9) produces no direct current and can be neglected. The only relevant terms are the constant one and the quadratic one. Substituting expression (13.5) into the current (13.9), and exploiting the trigonometric identity (A.82), we have id ðtÞ ffi I 0 0 þ

I 0 0 þ IS 1 I 0 0 þ IS 2 1 I 0 0 þ IS 2 þ v þ v cosðo tÞ þ v cosð2oR tÞ R R R n VT 4 ðn VT Þ2 4 ðn VT Þ2 R

ð13:11Þ

The current (13.11) is the sum of four terms. The last two oscillate at RF and at its double, respectively: they are short-circuited to ground by Cout and produce no effect on the output. The first term I 0 0 is constant and independent of the RF signal, as stated. The second term produces a DC output current proportional to the RF input power iOUT ¼

1 I 0 0 þ IS 2 v 4 ðn VT Þ2 R

ð13:12Þ

In order to complete the square-law diode detector treatment, we need to consider the effects of the diode incremental resistance in more depth. The diode resistance (13.10) consists of the sum of two terms. The first one is rdi ¼ n VT ðId þ IS Þ1 and pertains to the device D of the nonlinear diode model. The second term, Rs , is simply the linear resistance associated with the diode. We can move Rs out of the extended matching network – although it is embedded within it – with some approximations. After this transformation, the network in Figure 13.3c becomes equivalent to the two networks of Figure 13.3e. On the left, there is the RF network, comprising the matched RF generator and one voltage divider, consisting of the series resistance Rs and the shunt resistance rdi . The latter only represents the useful part of the nonlinear device resistance, while Rs causes dissipation loss. The right side of Figure 13.3e shows the equivalent output network of the detector, which includes three current generators: the biasing one with the first two terms of Equation (13.11). Again rdi is in parallel with the current generators, while Rs increases the output resistance of the detector without modifying its output voltage. The RF voltage amplitude across rdi in the RF network is then 0 rdi vR ¼ V G 0 Z G þ Rs þ rdi Equation (13.12) gives the corresponding detector DC output current, which is 2 1 I 0 0 þ IS 0 rdi iOUT ¼ V G 0 4 ðn VT Þ2 Z G þ Rs þ rdi The available power from the RF generator is PAV;RF ¼

jV 0 G j2 jVG j2 ¼ 0 8 ReðZ G Þ 8 ReðZG Þ

where the second identity applies if the whole extended matching network (excluding Rs ) is purely reactive, in that it does not affect the generator available power. The detector output current is then iOUT ¼ 2

I 0 0 þ IS

r2di ReðZ 0 G Þ

ðn VT Þ jZ 0 G þ Rs þ rdi j2 2

PAV;RF

ð13:13Þ

Passing from the Norton schematization (right side of Figure 13.3e) to the Thevenin one, we obtain the detector output network of Figure 13.3f. The Thevenin output voltage is vOUT ¼ I0 rd þ I 0 0 rdi þ iOUT rdi

ð13:14Þ

568


The first two terms of Equation (13.14) are constant and independent of the RF power. The second term is the useful one, and equals ðDETÞ

vOUT ¼ 2 where

I 0 0 þ IS

r3di ReðZ 0 G Þ

ðn VT Þ jZ 0 G þ Rs þ rdi j2

g¼2

2

PAV ¼ g PAV

r3di ReðZ 0 G Þ

I 0 0 þ IS

ðn VT Þ jZ 0 G þ Rs þ rdi j2 2

ð13:15Þ

ð13:16Þ

The parameter g is the sensitivity of the detector, which is normally expressed in mV/mW. Its value ranges from a few hundreds to some thousands, depending on the design, the diode, the frequency range, the RF bandwidth, and so on. If Rs is much smaller than rdi , there are three important simplified cases to consider: there is no matching network (1); RG ¼ rdi (2); and Z 0 G ¼ rdi (3). In all the three cases the output resistance (or video resistance) of the detector is ROUT;1 ¼ ROUT;2 ¼ ROUT;3 ¼

n VT I 0 0 þ IS

1. The matching network is not present, i.e. the RF generator has resistive impedance RG , and is directly connected to Lin . The detector sensitivity is 2 n VT I 0 0 þ IS 2RG g1 ¼ 2 n VT RG þ In0 0þVTIS 2. RG is negligible with respect to rdi . For instance, in the case plotted in Figure 13.1, n VT ðI 0 0 þ IS Þ1 ¼ 662:18 O, which is much greater than the standard 50 O of most RF generators. The sensitivity simplifies to g2 ffi

2RG n VT

which is independent of the bias current. 3. The extended matching network matches the generator impedance exactly to the diode resistance ðZ 0 G ¼ rdi Þ. The resulting sensitivity is 1 1 g3 ¼ 0 2 I 0 þ IS In this case, the detector sensitivity decreases with the bias current. However, the high sensitivity achieved with small or zero bias is also associated with a high output resistance, which forms a voltage divider with Rload , reducing the effective output voltage, and partially compromises the sensitivity. A more precise parameter to evaluate that performance is the available output DC power of the detector, which is equal to VOUT;DET 2 1 g PAV;RF 2 I 0 0 þ IS 1 1 1 PAV;DC;3 ¼ ¼ 3 ¼ 0 P2 ROUT;2 8 I 0 þ IS n VT AV;RF 2 2 n VT This equation implies that the available DC output voltage is inversely proportional to the bias current. However, a reason for not recommending low bias currents is the difficulty in matching very high resistances to the standard 50 O and the Fano bandwidth limitation due to the diode reactance. For instance, the diode used for Figure 13.1 with a bias current of 0, 25 and 50 mA exhibits a series resistance of about 1:1 105 , 1:1 103 and 563 O. With these bias values, assuming a total parallel capacitance of the diode CJ þ Cd þ Cp ¼ 1 pF, and neglecting Ls, Fano’s equation (6.1) gives the maximum bandwidth


569

of 3.8, 387.3 and 770.7 MHz for 10 dB of return loss. Furthermore, the bias current, thus low or high output resistance, is not advisable in that it limits the so-called detector video bandwidth(VBW). That parameter defines how fast the output voltage reacts to an abrupt variation in the RF voltage amplitude. More precisely, if vRF is not constant, but varies over time, the detector output voltage produces a variable output voltage over time as well. If the detector has no output bandwidth limitation, its output voltage is vOUT;IDEAL ðtÞ ¼

ROUT I 0 0 þ IS 2 v ðtÞ 4 ðn VT Þ2 RF

The output of the real detector is different from the ideal case; in particular it tends to cancel the fast variation of the RF amplitude: vOUT;REAL ðtÞ 6¼

ROUT I 0 0 þ IS 2 v ðtÞ 4 ðn VT Þ2 RF

From a quantitative point of view, the video transfer function of the detector is defined as HDETECTOR ð f Þ ¼

F½vOUT;REAL ðtÞ F½vOUT;IDEAL ðtÞ

ð13:17Þ

where the notation Yð f Þ ¼ F½yðtÞ indicates the Fourier transform of the function y(t), as in the definition (B.1). Normally, the video transfer function can be approximated with a first-order low pass, due to the RC cell formed by Cout, the detector video resistance and the load resistance HDETECTOR ð f Þ ¼

1 f 1 þ j VBW

ð13:18Þ

where VBW ¼ ½2pðROUT jjRload ÞCout 1 . The minimum RF determines a lower limit on the detector output capacitance, in that it has to provide reasonably low RF impedance to ground. For instance, if oRF ¼ 2p 109 (1 GHz), the capacitor reactance equals 5 O (one-tenth or the standard RF impedance) if Cout > 31:8 pF. Moreover, the video bandwidth decreases with the video resistance, while the output capacitance cannot. If the load resistance equals the video resistance (maximum video power transfer) the video bandwidth becomes 1 1 11 1 1 n VT 1 ¼ ¼ þ R ð13:19Þ VBW ¼ s p ROUT Cout p rd Cout p I 0 0 þ IS Cout Equation (13.19) states that the matched detector presents a video bandwidth increasing with the bias current. With the diode used for Figure 13.1, with the bias currents of 0, 25 and 50 mA considered above, and assuming an output capacitance of 47 pF (which is a standard value), the matched video bandwidth becomes 0.06, 5.5 and 10.2 MHz, respectively. From a design point of view, the bias current follows from a compromise among sensitivity, difficulty of RF matching the detector and video bandwidth. Typically the bias current is of the order of some tens of microamperes, although some special devices – named zero-bias detector diodes – present parameters which fit with zero-bias operation. The detector current bias generates a DC offset voltage at the output,5 which comes from the first two terms of Equation (13.14). In addition, that offset is temperature dependent, as it is for the diode equation. The DC offset is a problem in many applications – the arrangement in Figure 13.4 shows one possible solution. That scheme ideally consists of two identical detectors, biased with the same current. One of them – the hot detector – is connected to the RF generator, the other – the cold detector – is not. The hot (cold) detector consists of the diode D1 (D2 ) and of its surrounding components. The circuit output 5 For instance, the example in Figure 13.1 presents 0.1498 Vat the output, with no RF signal applied. See also the Mathcad file 02_Diode_Curves.mcd.

570


I0 NT1

RF generator

D1

ZG 1

VG

2

Lbias v (1) DC

Rload

(1)

Lin

C out

(2)

vDC= A V (vDC - vDC )

+ AV Rload I0 D2

Figure 13.4

AMP1 (2)

vDC

Schematic for the detector voltage offset cancellation.

voltage is the difference between the hot and cold detector output, generated by the VCVS AMP1 . The cold detector only works in direct current, and then it does not include some components: the matching network is absent, none of the capacitors are in place and all the inductors are short-circuited. If Rload is the resistance of the two inputs of AMP1 , the hot and cold detector output voltages are ð1Þ

vDC ¼ ðI0 rd þ I 0 0 rdi þ gPAV;RF Þ

Rload ; ROUT þ Rload

ð2Þ

vDC ¼ ðI0 rd þ I 0 0 rdi Þ

Rload ROUT þ Rload

The hot and cold detectors have the same DC offset, which is cancelled from their difference, at the output of the circuit h i Rload ð1Þ ð2Þ vDC ¼ vDC vDC AV ¼ g AV PAV;RF ROUT þ Rload Thus, the circuit in Figure 13.4 behaves like an offset-free detector, from the external point of view. Clearly, the difference amplifier AMP1 introduces noise and further limits the video bandwidth of the circuit.

13.2.2

Envelope detectors

The analysis developed in Section 13.2.1 is accurate until the diode voltage – DC with superimposed RF – remains within the validity range of the approximation (13.3). The graph of Figure 13.2 shows that the quadratic approximation applies up to a maximum total voltage of about 50 mV. This section considers the opposite condition, where the voltage swing widely exceeds the conduction knee. In this case, the diode behaves approximately like a resistor having resistance Rs when the voltage across its terminals is positive, and like an open circuit in the opposite case. Equation (13.4) describes such behaviour, and Figure 13.2 shows the asymptotic validity of that approximation. This detector operation requires large RF voltage amplitudes; then the influence of the bias current becomes negligible. For our needs the schematic of a detector working in the linear region simplifies to the one shown in Figure 13.5a. Again, the matching network modifies the voltage amplitude and the series impedance of the RF generator, as considered in Figure 13.3c. The equations arising from the exact analysis of the detector under large RF voltage operation are quite complicated and difficult to derive. However, with some simplifying assumptions, the task becomes considerably easier, and the accuracy does not degrade significantly. The required simplifying hypothesis is to assume simply that the capacitor Cout is large enough to keep the detector output voltage at a constant value vDC , and that this value is smaller than the RF peak voltage.

FREQUENCY CONVERTERS RF generator

D1

vRF

Z G′

vDC

id

V G′

571

Cout

Rload

(a) V vR vDC

T/2 O

t1

T t

t2 T/4

vRF (t) (b)

Figure 13.5

Envelope diode detector: (a) electrical diagram; (b) voltage waveforms.

Let the detector input voltage be6 vRF ðtÞ ¼ vR sinðoR tÞ

ð13:20Þ

with vRF > vDC . From Equation (13.4) it follows that the current flowing through the diode is non-zero only when the RF input voltage exceeds the DC output one, i.e. 8 < vRF ðtÞvDC ðvRF > vDC Þ id ðtÞ ¼ Rs : 0 ðvRF vDC Þ Equation (13.20) implies that VRF > VDC if t1 < t < t2 , with 1 p 1 p 1 vDC 1 vDC sin sin t1 ; t2 ¼ ¼ t1 ¼ oR oR oR oR vR vR Figure 13.5b plots one period of the input RF voltage together with the output DC voltage, also showing the two limits of the conduction time interval. The mean of the diode current over a complete RF cycle is ðt2 ðt2 1 oR vR oR vR id ¼ id ðtÞ dt ¼ sinðoR tÞ dt ¼ ½cosðoR t1 ÞcosðoR t2 Þ T 2p Rs 2p Rs t1 t1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vR 1 vR 1 vR ð13:21Þ 1sin2 ðoR t1 Þ ½cosðoR t1 ÞcosðpoR t1 Þ ¼ cosðoR t1 Þ ¼ ¼ 2p Rs p Rs p Rs s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 vR vDC 1 ¼ p Rs vRF 6 Equation (13.20) uses the sine instead of the cosine, differently from Equation (13.5) used for the analysis of the quadratic detector. The only reason for this difference is to simplify the treatment. It is possible to pass from the cosine to the sine with a simple translation of the time axis.

572


The initial assumption on the output capacitor implies that the current on the load is almost constant and equal to vDC =Rload . The DC balance requires that the quantity (13.21) equals the load current, therefore sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 1 vR vDC vDC 1 ¼ p Rs vR Rs Squaring both sides, and extracting vDC , we obtain the envelope detector output voltage as a function of the RF peak voltage " # 1 2 Rs 2 p þ1 vR ð13:22Þ vDC ¼ Rload Usually pRs ¼ Rload , for instance with our reference diode Rs ¼ 100 O, while a typical video load resistance could be 5 kO, therefore Equation (13.22) simplifies to vDC ffi vR

ð13:23Þ

Both the analyses on the square law and on the envelope detector, presented in Sections 13.2.1 and 13.2.2, are approximate. In particular, they do not consider nonlinearity in the junction capacitance, and give no precise idea about the respective limits of validity. Moreover, the simplified cases 1 and 2 of Section 13.2.1 ignore the diode parasitic reactance completely. Nonlinear computer simulation, however, is possible, in order to take all those missed factors into account for the complete range of the RF input voltage.7Figure 13.6 shows the result of such CAE analysis, compared with the approximations given in Sections 13.2.1 and 13.2.2: the agreement is reasonably good in the respective valid ranges, despite the analytic expressions ignoring the diode parasitic reactance.

quadratic detector

101

DC output voltage, V

100 10-1 envelope detector 10-2 10-3 10-4 10-5 10 -8

10 -7

10 -6

10 -5 10 -4 10 -3 Available input power, W

10 -2

10 -1

Figure 13.6 Nonlinear CAE analysis of a diode detector (thick solid curve) and approximate squarelaw and envelope analyses (thin dashed lines).

7 See the SIMetrix file 06_Schottky_Detector_Power_Sweep.sxsch. RF is 1 GHz. The diode parameters are IS ¼ 250 109 A; n ¼ 1:1; Rs ¼ 100 O, as for Figures 13.1 and 13.2, with Cj0 ¼ 0:3 pF; Vj ¼ 0:5; g ¼ 0:5 for the junction capacitance: care must be taken not to confuse the junction capacitance exponent with the detector sensitivity, despite the fact that they are denoted by the same Greek letter. The simulation considers neither diode parasitics, nor the matching network.


13.2.3

573

FET detectors

The nonlinear characteristic of the devices used in quadratic-law detectors has to be – exactly or approximately – a second-order polynomial, as Section 13.2.1 showed. Diode detectors exploit the exponential characteristic (9.67), which relates the device current and voltage, and can be approximated with the second-order polynomial (13.3) for small voltages. Another important device exhibits a squarelaw characteristic: it is the FET, as in Equations (9.75), (9.76) and (9.79), although in the FET case the quadratic law relates quantities at different electrodes. Figure 13.7a shows the schematic of the FET detector, which is the counterpart of the diode-based arrangement of Figure 13.4. In both cases the single detector is a biased device that generates offset voltage even without an RF input signal. Therefore, the difference amplifier eliminates the offset by subtracting the voltage coming from a cold detector. The hot (cold) detector in the schematic of Figure 13.7a consists of Q1 (Q2 ) and its surrounding components. As usual, the gate inductors Lgg (capacitors Cgg ) present high (low) impedance at RF, while passing (blocking) the direct current. The gate generators Vgg bias the gates at a voltage slightly higher than the pinch-off, such that the drain current is of the order of 1 mA, or less. The matching network NT1 is placed between the RF generator and the hot FET. The matching network is redundant for the cold FET, whose gate is RF terminated on R0 . In principle the RF termination of the cold FET input is arbitrary. Nonetheless, the cold FET gate RF termination is close to 50 O, for stability reasons, in that FETs are usually stable8 on 50 O. The voltage generators Vdd bias the drains, in combination with the resistors Rdd . Since the drain current is some hundreds of milliamps, Rdd ranges around 1 kO, so that its voltage drop is of the order of a few volts. The value of Vdd exceeds the drop on Rdd by some volts, in order to make the FET operate in the pentode region. The series resistors Rout – which are close to 50 O – and the capacitors Cout terminate the transistor output on stable impedance. Moreover, the combination Rout , Cout is a first-order lowdpass filter that removes the highfrequency components – namely, signal and noise – from the detector outputs. Our analysis of the circuit in Figure 13.7a will use the simplified equation (9.76), assuming all its associated assumptions as valid. In particular: .

The gate voltage is always higher than the pinch-off. This is always true for the cold FET, being guaranteed by the design; in the hot device, this implies small RF voltages.

.

The transistors operate in the pentode region, as guaranteed by the bias parameter design.

.

The drain current is independent of the drain voltage; in other words, the Ids curves are horizontal in the pentode region. This condition is approximately satisfied considering the low drain currents, and Equation (9.78) implies that the current slope is proportional to the current itself.

The voltage on the gate of Q1 is the superimposition of the RF voltage with Vgg , while the gate voltage of Q2 coincides with Vgg . The drain currents of the two transistors follow from Equation (9.76) by approximating the control voltage vI with the gate–source voltage, which coincides with the gate voltage vgs since the source is grounded. The result is 2 2 vRF Vgg Vgg ð1Þ ð2Þ ids ffi Idss 1 ; ids ffi Idss þ1 ð13:24Þ Vp Vp Substituting the sinusoidal function (13.20) of the RF signal into the first equation in (13.24), we have Idss Vp þ Vgg Vp þ Vgg 2 ð1Þ ids ðtÞ ffi 2 v2R sin2 ðoR tÞ2Idss v sinðo tÞ þ R R Vp Vp2 Vp Vp þ Vgg 1 Idss 2 1 Idss 2 ð2Þ ¼ v v cosð2oR tÞ2Idss vR sinðoR tÞ þ ids ðtÞ 2 Vp2 R 2 Vp2 R Vp2

8

See the discussion in Section 11.4.

574

MICROWAVE AND RF ENGINEERING +

Rdd

Vdd Rin

(2)

vc

Cgg

Rout v (2) DC

Q2

Cout

Lgg Vgg

+ Rload

+ RF generator ZG

Vdd

NT1 1

v

2

RF

RF

Cgg

VG

Vgg (a)

(2)

Rdd (1)

gg

+

(1)

(1)

VDC= AV [vDC - vDC ]

Rout v DC

Q1

Cout

Rload

Lgg AMP1 +

+ Vdd

(k) vc

Rdd Rout

(k)

(k)

vDC = [ Rload /(Rload +R out)] vc

Qk

+

Rload

+

′

Rdd ′= Rdd //(Rload +Rout ) (k)

vc

Qk

+

(k = 1,2)

(b)

+

Rdd

(FET)

, s11(FET)

Zin

Vdd

Rout

ZG′ VG′

Q1

Figure 13.7

Cout

Rload

Cgg Lgg Vgg

(c)

(1)

vDC

ZG′, ΓG′

+

FET detector: (a) principle: (b) equivalent DC networks; (c) equivalent RF network.


575 ð2Þ

The drain current of the two transistors is periodic with the fundamental period T ¼ 2p o1 R ; ids ð1Þ is constant, while ids is the sum of two constant terms plus two sinusoidal ones. These latter are short-circuited to ground by Cout, and thus do not contribute to the output voltage. Therefore, the voltage ð1Þ ð1Þ ð1Þ vDC depends only on the sum of the constant terms of ids , which coincides with the mean value of ids itself over time. Figure 13.7b shows the DC equivalent network for the computation of the output voltage of the two transistors. Applying Thevenin’s theorem to the network consisting of the resistors Rdd , Rout , Rload and the voltage generator Vdd , we obtain a simpler equivalent drain bias network, still consisting of one voltage generator, but with only one series resistor. Thus the network in the upper portion of Figure 13.7b simplifies to the one in the lower portion of the same figure, where ðRload þ Rout ÞRdd Rload R0 dd ¼ Rdd jjðRload þ Rout Þ ¼ ; V 0 dd ¼ Vdd Rload þ Rout þ Rdd Rload þ Rout þ Rdd The drain voltages of the two transistors are then Rload ðRload þ Rout ÞRdd ðkÞ 0 0 vðkÞ i c ¼ V dd R dd Ids;k ¼ Vdd Rload þ Rout þ Rdd Rload þ Rout þ Rdd ds where ðkÞ

ids ¼ T 1

ðT 0

ðk Þ

ids ðtÞ dt

ðk ¼ 1; 2Þ

ðk ¼ 1; 2Þ ð1Þ

ð2Þ

denotes the mean value of the drain current over one period, and clearly ids ¼ 0:5 Idss Vp2 v2R þ ids . From the network on the upper portion of Figure 13.7b it is easy to see that Rload ðkÞ vDC ¼ vðkÞ ðk ¼ 1; 2Þ Rload þ Rout c Thus the amplifier output voltage is h i AV Rload Rdd Idss 2 ð2Þ ð1Þ v vDC ¼ vDC vDC AV ¼ 2 Rload þ Rout þ Rdd Vp2 R

ð13:25Þ

Now, Rdd is of the order of a few kilohms, while Rout is close to 50 O and Rload is around 10 kO, as specified in the description of Figure 13.4; thus in many cases, Equation (13.25) simplifies to AV Idss vDC ffi Rdd 2 v2R ð13:26Þ 2 Vp Equations (13.25) and (13.26) state that the circuit in Figure 13.7a works like a quadratic detector, providing that the RF voltage peak is such that Vgg vR Vp . This implies that the maximum RF peak voltage is vR;max ¼ Vgg Vp . The determination of the FET detector sensitivity, however, requires the expression of the RF peak voltage in terms of the generator available power. The required expression derives from the analysis of the RF equivalent network of the hot detector, shown in Figure 13.7c. In that network, V 0 G and Z 0 G are the generator parameters as modified by the matching network, which has been removed, similarly as for Figure 13.3c. The RF available power is PAV;RF ¼

1 V 0G2 8 ReðZ 0 G Þ

where V 0 G is the RF peak voltage of the RF generator. Passing from the generator impedance to the corresponding reflection coefficient, Z 0 G ¼ ð1 þ G0 G Þð1G0 G Þ1 R0 , and with some manipulations, the available power expression becomes PAV;RF ¼

1 V 0 2G 12ReðG0 G Þ þ jG0 G j 2 8 R0 1jG0 G j

2

ð13:27Þ

576


The RF peak voltage on the gate of Q1 is v2R ðFETÞ

ðFETÞ

¼

Z ðFETÞ 2 2 in V 0 G ðFETÞ Z 0 þ Z G in

ðFETÞ

where Zin ¼ ½1 þ s11 =½1s11 R0 is the RF impedance presented to the generator from the gate of Q1 , which corresponds to the FET input reflection coefficient9 at the specific bias point.10 Again, the gate RF peak voltage, written in terms of reflection coefficients, becomes i h 1 þ sðFETÞ ð1G0 Þ 0 G V G 11 ð13:28Þ vRF ¼ ðFETÞ 0 2 GG 1s 11

Extracting V 0 G from Equation (13.27) and substituting the result into (13.28) yields h i ðFETÞ 2 0 2 1 þ 2Re sðFETÞ þ s 1 G j j G 11 11 2R0 PAV;RF v2RF ¼ 2 h i ðFETÞ ðFETÞ 2 0 0 12Re s G G þ s11 jG G j 11

ð13:29Þ

Substituting expression (13.29) into Equation (13.25) and extracting the ratio between the output DC voltage and the available RF power from the result, we obtain h i ðFETÞ 2 0 2 1 þ 2Re sðFETÞ þ s 1 G j j G 11 11 VDC Rload Rdd Idss R0 ¼ g ¼ AV ð13:30Þ 2 h i 2 ðFETÞ 0 2 PAV;RF Rload þ Rout þ Rdd Vp ðFETÞ 0 þ s G G 12Re s j j G G 11 11 The sensitivity of the circuit in Figure 13.7 depends on linear and nonlinear FET parameters, the input matching network, bias parameters and the gain of the differential amplifier AMP1. It is possible to remove the contribution of this last factor by assuming AV ¼ 1. Expression (13.30) is difficult to interpret. Two special cases are more meaningful: ðFETÞ

1. The generator is conjugate matched to the FET, G0 G ¼ conj½s11 , so the sensitivity becomes h i ðFETÞ 2 ðFETÞ þ s 1 þ 2Re s 11 11 Rload Rdd Idss gmatched ¼ AV R0 2 2 ðFETÞ Rload þ Rout þ Rdd Vp 1s11 2. The matching network is not present, the 50 O generator connects directly to the gate, G0 G ¼ 0, and the sensitivity becomes

h i Rload Rdd Idss ðFETÞ 2 ðFETÞ 2 ðFETÞ þ s11 R0 ¼ 1s11 gmatched g50 O ¼ AV 1 þ 2Re s11 Rload þ Rout þ Rdd Vp2 ðFETÞ

Clearly g50 O gmatched , and the equals sign applies only if s11

¼ 0.

Observations: (a) The analytical considerations developed in this section contain some approximations, as usual, particularly in the treatment of the FET input impedance, which has been assumed linear. More The components Rout and Cout terminate the FET output on 50 O at RF. This assumption involves some further approximations, in that the RF voltage makes the gate–source voltage swing over the quadratic region of the Ids ðVgs Þ law, in contrast with the small-signal hypothesis implicit with the scattering parameter concept. 9

10


577

precise results require the application of nonlinear computer analyses and the availability of accurate models. (b) Equation (9.76), which is the starting point of all the discussion, applies more or less approximately to the various real existing transistors. Its accuracy – besides the pentode approximation with perfectly horizontal DC curves – changes from transistor to transistor and with the bias point. (c) If the instant gate–source voltage becomes smaller than Vp , then the drain current at that instant becomes zero, and so it remains if Vgs becomes further negative. Thus, at high RF amplitudes the FET detector behaves similarly to an envelope detector in the negative half wave of the signal, while continuing to follow the quadratic law in the positive half wave. The resulting vDC versus PAV;RF characteristic has a slope that is intermediate between the two cases, with the DC voltage proportional to PaAV;RF with 1 < a < 2. (d) The drain voltage of each FET can swing from zero up to a maximum of V 0 dd ¼ ðRload þ Rout þ Rdd Þ1 Rload Vdd , which is (usually slightly) smaller than Vdd . Therefore, ð1Þ the overall output voltage saturates when vc approaches one of its limits, independently of any other condition. (e) The differential structure of the schematics in Figure 13.7a, also used in Figure 13.4, theoretically cancels the DC offset. It keeps this feature also over time and temperature, providing that the components of the two detectors are as similar as possible; that condition is better satisfied in monolithic realizations, where the hot and cold detectors are realized in close regions of the same chip. However, the sensitivity of the hot detector itself is temperature dependent, and the differential structure does not compensate this effect at all. When temperature stability is required, additional techniques have to be used, usually based on a temperature-variable bias. Figure 13.8 shows the CAE simulated11vDC versus PAV;RF characteristic of an FET detector having the schematic of Figure 13.7a, without any matching network. The working frequency is 1 GHz. Figure 13.8 also includes the curve of the diode detector plotted in Figure 13.6. On the right y axis (grey curves), Figure 13.8 displays the ratio VDC =PAV;RF , which coincides with g in the quadratic-law region. Note: .

The higher sensitivity of the FET detector, compared with the diode one.

.

The intermediate slope region of the FET detector within the interval PAV ð0:01; 1Þ mW, as predicted in observation (c) of the present section,

.

The saturation of the FET characteristic above the upper limit of the previous interval, in agreement with observation (d).

13.3 Mixers A frequency mixer or, in short, a mixer is a nonlinear three-port network. Two of the ports are inputs: the radio frequency (RF) and the local oscillator12(LO); the output is usually called the intermediate frequency (IF) port. At the first level of ideal schematization, a mixer works as a product detector, in that 11

See the SIMetrix file 06_FET_Detector_Power_Sweep.sxsch. The term local oscillator comes from the application of mixers in radio receivers. The oscillator that provides the signal for the mixer is inside the receiver, i.e. ‘local’, while the transmitter oscillator is ‘remote’. 12


104 103

γ, FET γ, diode

VDC , mV

102 101

103 102 101

VDC, FET

100

100

10-1

VDC, diode

10 -1

γ, mV/mW

578

10-2

10 -2 10 -3 10-6

10-5

Figure 13.8

10 -4

10-3

10-2 10-1 100 PAV,RF , mW

101

10 2

10-3 103

FET and diode detector characteristics, computed at 1 GHz.

the output signal is the product of the two input ones. Figure 13.9a shows the standard symbol for a mixer; the cross inside the circle is a reminder of the ideal schematization of the component. Within the context of this section the term signal is a generic denomination for a voltage, current or any linear combination of the two, such as incident or reflected waves.

VRF

VIF

R L

(a)

VLO MIXER1

(1) VRF

γRF(1)

(1) IF

γ

R L

(1)

VIF

(1)

γLO

Power divider Local oscillator (2) LO

γ

(2)

VRF

γRF(2)

L R

γIF (2)

VIF

(2)

MIXER 2 (b)

Figure 13.9

Mixer fundamental elements: (a) symbol; (b) double mixer connection.


13.3.1

579

Product detector

Starting from the product detector schematization, and assuming that the input signals are sinusoidal, the signals applied at the RF and LO input respectively are vRF ðtÞ ¼ vR cosðoR t þ jR Þ;

vLO ðtÞ ¼ vL cosðoL t þ jL Þ

ð13:31Þ

The IF output proportional to the product of the two inputs is thus vIF ðtÞ ¼ 2kIF vRF ðtÞvLO ðtÞ ¼ 2kIF vR vL cosðoR t þ jR ÞcosðoL t þ jL Þ ¼ kIF vR vL cos½ðoR þ oL Þt þ jR þ jL þ kIF vR vL cos½ðoR oL Þt þ jR jL

ð13:32Þ

where the constant 2kIF has a suitable dimension to fit the output with the inputs and the fourth term follows straight from the trigonometric identity (A.72). Equation (13.32) states that the IF signal is the sum of two sinusoidal conversion products, expressed by the two cosine terms in the fourth term of Equation (13.32). The frequency and the instant phase of the first (second) IF sinusoidal component equals the sum (difference) of the two input frequencies and phases, respectively: for this reason, they are often referred to as the sum and the difference conversion product. Generally, only one of the two signals is useful, and a filter is provided to eliminate the unwanted one. The sum frequency is always higher than the RF one, thus when the sum product is the useful one, the mixer is sometimes referred to as an upconversion mixer. In the opposite case, the denomination downconversion mixer is not always appropriate, in that the difference frequency is lower than the RF one if and only if oL < 2oR . The cosine is invariant to a sign inversion of its argument, thus the difference frequency does not change if oL ¼ oR Do. Equation (13.32) also shows that the amplitude of both the IF terms is proportional to the product of the two input amplitudes. However, normally the LO amplitude is constant over time. Moreover, in real mixers, the IF amplitude variations are less than proportional to variations in the LO amplitude,13 providing that the latter are within given limits. Hence, we can embed the LO amplitude within the constant KIF ¼ vL kIF , and rewrite the IF output signal as vIF ðtÞ ¼ KIF vR cos½ðoR þ oL Þt þ jR þ jL þ KI vR cos½ðoR oL Þt þ jR jL

ð13:33Þ

One important mixer specification is the conversion gain – usually expressed in dB – which is the ratio between the power of the IF product (sum or difference) and the RF power. If the signals considered in the above equations are of the same type, for instance voltages, the constant KIF is a pure number. Furthermore, if RF and IF have the same normalization impedance, the conversion gain is 2 2 PIF K v ¼ 10 log10 IF2 R ¼ 20 log10 ðKIF Þ dB ð13:34Þ CGdB ¼ 10 log10 PRF vR Frequently, the conversion gain is negative, and its opposite is specified: the conversion loss. In principle, both the RF and LO phases ðfR ; fL Þ vary over time, and both variations impact on the IF signal. In particular, if the LO signal is affected by phase noise, this adds to the conversion products as phase noise with the same level and spectrum. If the LO phase is constant, then a suitable choice of the time origin is possible to make fLO ¼ 0, and the IF signal further simplifies to vIF ðtÞ ¼ KIF vR cos½ðoR þ oL Þt þ fR þ KIF vR cos½ðoR oL Þt þ fR

ð13:35Þ

Equation (13.35) reveals that both the sum and the difference signals are sinusoids having amplitudes proportional to the RF one. Furthermore, any phase variation in the RF signal transfers to the conversion product with no modification. In other words, the IF products are frequency-translated replicas of the RF signal: any amplitude or phase14 modulation on the RF is reproduced in the IF, with ideally no distortion. This is a key property of mixers. It is used in receivers to convert the receiving variable-frequency signal 13 14

Sections 13.3.2, 13.3.4, 13.3.8 and 13.3.9 explore this topic in more detail. Which also includes frequency modulation, as usual.

580


into a fixed-frequency one, by tuning the LO frequency.15 On the other hand, transmitters use the frequency translation to convert a modulated signal in order to generate the radiated modulated signal.16 That relation between the RF and IF of a product detector resembles linearity, with the exception that the input and output frequencies are different. We will say that a mixer operates linearly when the IF and RF amplitudes are proportional, although the mixer is inherently nonlinear. The product detector is a linear mixer in the above-specified sense. Real mixers are only approximately linear, and only for input RF levels lower than a specified limit which depends on the accepted deviation from linearity. Thus, mixers produce harmonic IF components if the RF is a single tone, and intermodulate with multiple tones; moreover, they add noise to the converted signal. Subsequent sections give some more details on these aspects. The block diagram in Figure 13.9 shows a mixer application with one single LO that provides the signal for two mixers by means of a power divider. Two linear two-port networks are placed between the power divider outputs and the LO ports of the two mixers. They model a constant phase/amplitude difference between the two LO signals for the two mixers. Other four linear two-port networks model the physical paths to the RF ports and from the IF ports. The signal transfer factor of the six two-ports is indicated by the complex variables ðkÞ

ðkÞ

ðkÞ

gR ¼ aR þ jbR ;

ðkÞ

ðkÞ

ðkÞ

gL ¼ aL þ jbL ;

ðkÞ

gI

ðkÞ

ðkÞ

¼ aI þ jbI

ðk ¼ 1; 2Þ

For each network, the ratio of the respective output to the input signal is the exponential of the transfer coefficient: h i h i h i i h i h ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðX ¼ R; L; I; k ¼ 1; 2Þ exp gX ¼ exp aX þ jbX ¼ exp aX exp jbX ¼ AX exp jbX Note that the amplitude and phase factors of the different paths have to be considered at different frequencies. Let the RF signals be i i h h ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ vRF ðtÞ ¼ vR cos oR t þ jR ; vRF ðtÞ ¼ vR cos oR t þ jR Ignoring the amplitude difference on the two LO paths, the resulting IF signal is i o nh ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ vIF ðtÞ ¼ KIF AR AI vR cos oR þ oL t þ jR þ jL þ bR þ bL þ bI i o nh ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ þ KIF AR AI vR cos oR oL t þ jR jL þ bR bL þ bI i o ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ oR þ oL t þ jR þ jL þ bR þ bL þ bI i o nh ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ þ KIF AR AI vR cos oR oL t þ jR jL þ bR bL þ bI

ð2Þ

ð2Þ ð2Þ ð2Þ

vIF ðtÞ ¼ KIF AR AI vR cos

nh

ð13:36Þ

A comparison between the two sum and difference products, as expressed by Equation (13.35), reveals that: .

The resulting phase shift on the sum component equals the sum of the three phase shifts ðkÞ

ðkÞ

ðkÞ

bR þ bL þ bI LO or IF path. .

15 16

ðk ¼ 1; 2Þ, exactly as if the same total phase shift is totally placed in the RF, ðkÞ

ðkÞ

ðkÞ

The phase shift of the difference signal is bR bL þ bI ðk ¼ 1; 2Þ. In this case the phase shift in the LO path cancels an equal phase shift in the RF or IF path.

See Section 15.4. See Section 15.3.


581

.

The sum IF signals conserve the amplitude, frequency and phase relations that exist between the original RF signals, apart from a constant difference.

.

The difference components always conserve the amplitude relation between the two inputs. The ð1Þ ð2Þ conservation of frequency and phase requires oR oL to have the same sign as oR oL .

.

Any phase modulation present in the LO signal directly affects both the IF outputs.

13.3.2

Single-ended diode mixers

The product detector is an ideal component but difficult to realize, particularly at high frequency. Nevertheless, it is possible to exploit the nonlinear relation between current and voltage in a diode to implement an equivalent function. The simplest possible real mixer exploits the nonlinear characteristic of one single diode. A mixer using only one nonlinear device is also known as a single-ended mixer (SEM). Such a mixer consists of one diode and some passive linear circuitry that applies the sum VRF þ VLO of the RF and LO voltages across the diode terminals. If the diode current–voltage follows the law17 in (9.67), then the corresponding series (13.3) includes one term proportional to ðvRF þ vLO Þ2 ¼ v2RF þ 2vRF vLO þ v2LO The second term on the r.h.s. of the above expression is exactly the product of the RF and LO voltages, which generates the sum and difference IF signals. Clearly, the SEM also produces many unwanted signals, as a consequence of the infinite terms in the Maclaurin series. Figure 13.10 shows one possible SEM implementation. The directional coupler COUP1 applies the combined RF and LO signals to the anode of the diode D1 . The matching networks NT1 and NT2 match the diode impedance into 50 O at the RF, LO and IF frequencies, respectively. It is possible to consider the parasitic elements of the diode as parts of NT1 or NT2 by applying the same considerations used for the detector.18 Usually, NT2 embodies a filtering network that attenuates the unwanted (sum or difference) conversion product. The reactive impedance Z1 (Z2 ) ideally presents an open (short) circuit at RF and LO frequencies and a short (open) circuit at IF. The role of Z1 and Z2 is very important: they ensure that the RF and LO voltage is fully applied to the diode, with a minimum drop along the path. Similarly, the IF component of the diode current flows directly to the IF port, with no dispersion in the RF and LO. The schematic presented in Figure 13.10 is not unique. The configuration is convenient when the RF and LO frequencies are relatively close – say when their difference is of the order of 10% – and when the wanted IF component is the difference. Under this condition, the IF is a small fraction of the RF and LO frequencies, and Z1 , Z2 are relatively easy to realize. The main drawback of the configuration in Figure 13.10 is the loss of the LO power due to the directional coupler. Its coupling factor is normally in the range of 3–10 dB, for physical realization reasons, and to limit the RF insertion loss19 in the direct path, which correspondingly becomes 3–0.9 dB. For higher distances between RF and LO, different solutions become possible to combine RF and LO, for instance based on filters. In this case, however, the impedances Z1 , Z2 could be more critical to realize. If Equation (13.31) expresses the external applied RF and LO signals, and NT1 perfectly matches the diode into 50 O at the respective frequencies, the voltage on the anode of D1 is vR p v1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi2ffi cos oR t þ fR þ c1 vL cosðoL t þ fL Þ ð13:37Þ 2 1c1 where c1 is the coupling factor of COUP1 . 17

Which is the same as Equation (13.2). See Figures 13.3a–c and the relative comments. 19 Keep in mind that the RF is close to the LO frequency by hypothesis. 18

582

MICROWAVE AND RF ENGINEERING COUP1 v1

RF

1

v2

2 NT1

R0

D1

IIF

1 Z1

Z2

2

IF

NT2

LO

Figure 13.10 Single-ended diode mixer.

Then, NT1 modifies the output amplitude and phase of the RF and LO components of the input voltage, which is then v2 ðtÞ ¼ v0 R cosðoR t þ j0 R Þ þ v0 L cosðoL t þ j0 L Þ

ð13:38Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffi v0 R ¼ aR vR = 1c1 2 ;

j0 R ¼ jR

p þ yR ; 2

v0 L ¼ aL c1 vL ;

j0 L ¼ jL þ yL

and the quantities aR ; aL ; yR ; yL are amplitude and phase modification factors due to the matching network. If port 1 of NT1 is not perfectly matched, the quantities aR ; aL ; yR ; yL assume different – but still constant – values, and Equation (13.38) remains valid. Here and in most diode mixer analyses in this chapter, we will assume that the IF voltage is much smaller than the RF and LO ones. Therefore, since Z2 is a short circuit for the RF and LO signals, the voltage across the diode coincides with v2. Also, at this point we can ignore the diode series resistance Rs and all the parasitic elements, in that their effect could only be to alter further the amplitude and phase of the RF and LO voltages. Equivalently, we can consider Rs and the other parasitic elements of D1 as part of the matching networks, as we did for the detector. The diode current is the sum of ideally infinite sinusoidal components: those of them with frequency within the IF range – determined by Z2 and NT2 – flow towards the load, producing a proportional voltage. The remaining part of the diode current shunts to ground through Z2 . From the above considerations, it follows that the series (13.3), which results from the diode relation (9.66), becomes id ðtÞ ¼

1 X

Is vk ðtÞ kVk 2 k!N T k¼1

ð13:39Þ

where the ideality factor is indicated by the capital letter N instead of n, because in the standard denomination of mixer quantity lower case n has a different use. The term with k ¼ 1 represents a linear contribution to the diode current, and does not generate any mixing product. The terms with k > 2 generate nonlinear distortions and the quadratic term (k ¼ 2) is the useful one for mixer applications. In expression (13.39) the diode voltage to consider is the sum of the RF and LO voltages. Hence, substitution of the RF and LO expressions (13.38) into Equation (13.39) gives id ðtÞ ¼

1 X

Is ½v0 R cosðoR t þ j0 R Þ þ v0 L cosðoLO t þ j0 L Þk k!N k VTk k¼1

ð13:40Þ


583

The term for the current (13.40) with k ¼ 2 expands to Is ½v0 R cosðoR t þ j0 R Þ þ v0 L cosðoL t þ j0 L Þ2 2N 2 VT2 Is Is Is 2 2 ¼ v0 þ v0 þ v0 R v0 L cos½ðoR þ oL Þt þ j0 R þ j0 L 4N 2 VT2 R 4N 2 VT2 L 4N 2 VT2 Is Is 2 v0 R v0 L cos½ðoR oL Þt þ j0 R þ j0 L þ v0 cos½2ðoR t þ j0 R Þ þ 4N 2 VT2 4N 2 VT2 R Is 2 þ v0 cos½2ðoL t þ j0 L Þ 4N 2 VT2 L

ð13:41Þ

The first and the second terms in the third term on the r.h.s. of expression (13.41) are two sinusoidal currents, having frequency oRF oLO and amplitude proportional to vR : the desired one flows through the IF load, producing a voltage proportional to Is vR vL cos½ðoRF þ oLO Þt þ fR þ fL 4N 2 VT2

or

Is vR vL cos½ðoRF oLO Þt þ fR fL 4N 2 VT2

The proportionality constant depends on Z2 , NT2 and the impedance of the IF load. The components Z2 and NT2 also filter – or cancel in extreme cases – some the unwanted signals coming from expression (13.40) out of the IF port. Thus, the current on a diode subjected to two summed sinusoidal voltages contains one term proportional to the product of the two voltages. Consequently, considering the voltages as inputs and the current as output, the diode behaves like a product detector, with some added spurious spectral components. Any nonlinear device having either two or three terminals can potentially present similar results, providing that the Maclaurin series of its nonlinear relation has a non-zero second-degree coefficient. Newton’s binomial formula20 applied to the terms of the sum (13.40) and with a few manipulations gives the current from the cathode of D1 towards the IF port iIF ðtÞ ¼

1 X k X Is 1 m km v0 ðtÞ v0 LO ðtÞ k V k ðkmÞ!m! RF N T k¼1 m¼0

ð13:42Þ

with v0 RF ðtÞ ¼ v0 R cos½oR t þ j0 R ;

v0 LO ðtÞ ¼ v0 L cos½oL t þ j0 L

From the identities (A.82) to (A.85) it follows that v0 m v0 km RF ðtÞ; LO ðtÞ are the sums of some harmonically related sinusoidal components, having a maximum frequency of m oR ; ðkmÞ oL , respectively.21 Furthermore, the identity (A.72) implies that the product between sinusoidal terms produces couples of sinusoids with frequency equal to the sum and the difference of the two factors. Hence, the terms of the nested sums (13.42) produce multiple spectral components proportional to functions of the type v0 R v0 L cos½ðm oR þ n oL Þt þ m j0 R þ n j0 L m

20

n

ð13:43Þ

That is, ða þ bÞk ¼

k X

k X k! am bkm ¼ ðkmÞ!m! m¼0 m¼0

k am bkm m

21 The identities (A.82) to (A.85) have been derived for a cosine power with a maximum exponent of 5, but it is possible to show that the assertion holds true for any integer power.

584


where m; n are positive or negative integer numbers with jmj; jnj ¼ 0; . . . ; k and jmj þ jnj k. The number k is ideally infinite, but the terms (13.43) have proportionality coefficients that decrease with k. Therefore, truncation of the external sum to a maximum finite order of 3 to 5 usually gives enough accurate results. Moreover, high k values imply high-frequency products and then lack of model accuracy. The spectral IF components (13.44) are the mixing products of the mixer, while m and n are the RF and LO harmonic orders or harmonic index, or simply index, respectively. Among the mixing products, the two with jmj ¼ jnj ¼ 1 are the most useful or desired ones; all the others – which are usually unwanted22 – are the mixer spurs. Moreover, the even-k terms of Equations (13.42) also produce DC components, with amplitude proportional to v0 kR and v0 kL , and mixing products having the same ðoR oL Þ frequency, like the desired ones, of the type ðv0 R v0 L þ v0 R v0 L Þcos½ðoR oL Þt þ j0 R j0 L k1

k1

ð13:44Þ

The spurs (13.44) have a special role, by affecting the amplitude of the main conversion products, generating compression at increasing RF power and making their amplitude not proportional to vL , as anticipated in Section 13.3. Section 13.3.9 gives more details on mixer spurs. Diode mixers work reasonably the same as in the above description until the LO voltage across the diode exceeds the RF one. The useful dynamic range of the RF input power depends on the acceptable deviation from ideal behaviour. Typically, the RF power incident on the diode must be at least 3 dB lower than the LO one.

13.3.3

Singly balanced diode mixers

Figure 13.11 shows two examples of singly balanced mixers (SBMs). Both the circuits consist of a combination of two SEMs, their RF and LO having equal amplitude and suitable reciprocal relations between the respective phases. One hybrid coupler distributes the RF and LO signals to the two diodes. More precisely, the circuit in Figure 13.11a is a 180 SBM, while the one in Figure 13.14b is a 90 SBM. Regarding the circuit, the main difference between the two configurations is the type of hybrid junction, which is 180 or 90 , as the definition suggests. That circuit difference implies different overall performances, with advantages and disadvantages in both cases, as we will see shortly. ðAÞ The two SBMs in Figure 13.11 have the same general structure. The matching networks NT1 , ðBÞ ðAÞ ðBÞ NT1 , NT2 and the reactive bipoles Z1 , Z1 , Z2 play the same role as in the SEM of Figure 13.10. In ðAÞ ðBÞ particular, Z1 , Z1 and Z2 provide a short circuit at the IF, RF and LO, respectively, while approximating the open circuit in the swapped case. This section analyzes the two mixers in Figure 13.11 under the same hypotheses as assumed for the SEM in Section 13.3.2. That is: .

RF and LO signals have the same expression (13.31).

.

Equation (13.38) is valid for the diode current versus voltage law.

.

Rs and the other diode parasitic elements are considered as part of the matching networks.

.

The IF voltage v5 is much smaller than v3 and v4, therefore Vd1 ffi v3 and Vd2 ffi v4 .

.

Additionally, we will assume here that the corresponding elements of the two SEMs are identical, ðAÞ ðBÞ i.e. NT1 is identical to NT1 , D1 is identical to D2 and Z1 A ¼ Z1 B .

22

With the exception of the subharmonically pumped mixers, described in Section 13.3.5.

FREQUENCY CONVERTERS HYB1 1

RF

0°

4

v1

1

0°

3

0°

v2

2

1

Z

v5

1

2

Z2

IF

NT2

D2

v4

2

iIF

(A) 1

NT1(A)

180° LO

D1

v3

2

585

Z1(B)

NT1(B) (a)

HYB1 1

RF

0°

4

v1

1

90° LO

3

0°

v2

2

1

Z

(B)

NT1

iIF

v5

Z2

1

2

IF

NT2

D2

v4

2

(b)

Figure 13.11

(A) 1

NT1(A)

90°

D1

v3

2

Z1(B)

Singly balanced diode mixer: (a) with 180 hybrid; (b) with quadrature hybrid.

13.3.3.1 Diode 180 SBM Under these hypotheses, and temporarily assuming that the hybrid is perfectly terminated on its four ports, the voltages at ports 1 of the two RF and LO matching networks are vRF ðtÞ vLO ðtÞ v1 ðtÞ ¼ pffiffiffi þ pffiffiffi ; 2 2

vRF ðtÞ vLO ðtÞ v2 ðtÞ ¼ pffiffiffi þ pffiffiffi 2 2

ð13:45Þ

The matching networks modify the amplitude and the phase, and the respective output voltages are then v3 ðtÞ ¼ v0 RF ðtÞ þ v0 LO ðtÞ;

v4 ðtÞ ¼ v0 RF ðtÞ þ v0 LO ðtÞ

ð13:46Þ

where vR v0 RF ðtÞ ¼ aR pffiffiffi cosðoR t þ jR þ yR Þ; 2

vL v0 LO ðtÞ ¼ aL pffiffiffi cosðoL t þ jL þ yL Þ 2

and the quantities aR ; aL ; yR ; yL are the amplitude and phase modifications due to the matching network. ðAÞ ðBÞ Note that if the ports 1 of NT1 and NT1 are not perfectly matched, similar factors must be considered when passing from Equation (13.31) to (13.44). The parameters v0 R ; v0 L ; j0 R ; j0 L represent quantities of the same type as vR ; vL ; fR ; fL of Equation (13.31). Therefore, the voltages v3 , v4 are linear combinations of the RF and LO voltages with constant modification factors for amplitude and phase. Moreover, the RF and LO components of v3 and v4 maintain the same amplitude and phase relations as they would have at the outputs of a matched hybrid. Assuming that the positive terminal is the anode, the voltage across D1 (D2 ) coincides with v3 (v4), because Z2 is a short circuit at RF, LO. Therefore, substituting the voltages (13.46) into the series (13.3),

586


we obtain the current through each diode ð1Þ

id ðtÞ ¼

1 X

Is ½v0 RF ðtÞ þ v0 LO ðtÞk ; kV k k!N T k¼1

ð2Þ

id ðtÞ ¼

1 X

Is ½v0 RF ðtÞv0 LO ðtÞk kVk k!N T k¼1

The IF voltage is proportional to the current towards the IF port, which equals the difference between the currents through the two diodes ð1Þ

ð2Þ

iIF ðtÞ ¼ id ðtÞid ðtÞ ¼

o Is n 0 k k 0 0 0 ½v ðtÞ þ v ðtÞ ½v ðtÞv ðtÞ RF LO RF LO k k!N k VT k¼1

1 X

Then, applying Newton’s binomial formula and after a few manipulations, the current towards the IF port becomes iIF ðtÞ ¼

1 X k X Is 1ð1Þkm 0 m km v ðtÞ v0 LO ðtÞ N k VTk ðkmÞ!m! RF k¼1 m¼0

ð13:47Þ

The terms in the sums of (13.47) vanish when the LO order k m is even. Therefore the SBM in Figure 13.11a suppresses the mixing products with even LO order; this is the main advantage of the 180 SBM over the SEM. In particular, the first three groups of terms in Equation (13.47) are iIF ðtÞ ¼

i 2Is 0 2Is Is h 2 3 v LO ðtÞ þ 2 2 v0 RF ðtÞ v0 LO ðtÞ þ 3 3 3v0 RF ðtÞ v0 LO ðtÞ þ v0 LO ðtÞ þ n VT n VT 3n VT

ð13:48Þ

Equation (13.48) states that the IF output of the 180 SBM with the configuration in Figure 13.11a presents no components at RF. Swapping the RF with the LO signal in the same circuit, the result suppresses the even-order RF products,23 in particular the LO signal.

13.3.3.2 Diode 90 SBM The 90 phase shift relation is difficult to write in general terms in the time domain. Therefore, the derivation of the IF current will follow a slightly different procedure than the one used for the SEM and 180 SBM. Under the hypotheses defined in Section 13.3.3 and considering the phase relations inherent to the 90 hybrid, the voltages on the anode of D1 and the cathode of D2 are respectively v3 ðtÞ ¼ v0 R cosðoR t þ j0 R Þ þ v0 L sinðoL t þ j0 L Þ v4 ðtÞ ¼ v0 R sinðoR t þ j0 R Þ þ v0 L cosðoL t þ j0 L Þ pffiffiffi pffiffiffi where v0 R ¼ aR vR = 2; j0 R ¼ jR þ yR ; v0 L ¼ aL vL = 2; j0 L ¼ jL þ yL , as in Equation (13.46). The current from the junction of the two diodes towards the IF port is ð1Þ

ð2Þ

iIF ðtÞ ¼ id ðtÞid ðtÞ ¼

1 X

1 X Is Is k v ðtÞ ½v4 ðtÞk 3 kVk kV k k!N k!N T T k¼1 k¼1

ð13:49Þ

ð13:50Þ

Substituting expressions (13.49) into Equation (13.50) gives ¼ iIF ðtÞ

1 X Is k k 0 k 0 0 0 0 0 0 0 ½v cosðo tþj Þþv sinðo tþj Þ ð1Þ ½v sinðo tþj Þþv cosðo tþj Þ R L R L R L R L R L R L k!N k VTk k¼1 23

As is easily found by swapping the indexes R and L in expressions (13.46) to (13.48).


587

Applying Newton’s binomial formula, we have iIF ðtÞ ¼

1 X k X

Is k N VTk k¼1 m¼0

1 m km v0 v0 fm;k ðtÞ m!ðkmÞ! R L

ð13:51Þ

where hm;k ðtÞ ¼ cosm ðoR tþj0 R Þsinkm ðoL tþj0 L Þð1Þk sinm ðoR tþj0 R Þcoskm ðoL tþj0 L Þ

ð13:52Þ

The trigonometric identity sinðaÞ ¼ cosðap=2Þ allows us to rewrite the terms in (13.52) as p p hm;k ðtÞ ¼ cosm ðoR tþj0 R Þcoskm oL tþj0 L ð1Þk cosm oR tþj0 R coskm ðoL tþj0 L Þ 2 2 The highest frequency spectral components of each cosine power have a frequency equal to the argument of the cosine multiplied by the respective exponent. Multiplication of these components generates terms of hm;k having the form h pi Am;m cosðmoR tþmj0 R ÞAmk;mk cos ðkmÞoL tþðkmÞj0 L ðkmÞ 2 and p k ð1Þ Am;m cos moR tþmj0 R m Amk;mk cos½ðkmÞoL tþðkmÞj0 L 2 The factor An;p is the coefficient of the expansion of the cosine power cosn ðaÞ ¼

n X

An;p cosðpxÞ

p¼1

From the trigonometric identity (A.69) and since ð1Þk cosðaÞ ¼ cosðaþkpÞ, it follows that the highest frequency generated components of terms of fm;k can be rewritten as fm;k ðtÞ ¼

þ Am;m Akm;km 3 þ ðtÞ cos Y ðtÞþk pmp cos Y 2 2 n h Am;m Akm;km pio þ cos½Y ðtÞcos Y ðtÞþk þ 2 2

ð13:53Þ

where p 2 p Y ðtÞ ¼ ½moRF ðkmÞoLO tþmfR ðkmÞfL þðkmÞ 2 Y þ ðtÞ ¼ ½moRF þðkmÞoLO tþmfR þðkmÞfL ðkmÞ

Expressions (13.53) represent – apart from a proportionality constant – the mixing products having RF order equal to m and LO order equal to n ¼ ðkmÞ, respectively. The mixing product, having orders m;n, vanishes if the arguments of the two cosines in the first term of the function (13.54) differ by an even multiple of p. This happens if 3k2m ¼ 4hðk ¼ 4hÞ for the mixing products corresponding to Y þ ðY Þ, where h is any positive or negative integer number, including zero. From these considerations, and since the LO order is n ¼ ðkmÞ, it follows that the 90 SBM suppresses the mixing products with RF and LO orders such that ðY þ Þ

ðY Þ

mþ3n ¼ 4h; mþn ¼ 4h where h ¼ 0;1;2... is any positive or negative integer number, including zero.

ð13:54Þ

588


Both the relations in (13.54) are valid after swapping the index m with n. For the second relation, the above assertion is obvious; for the first one, its demonstration requires some effort. Let us assume that a couple of numbers m, n satisfy the first relation in (13.54) with swapped index, namely 3m þ n ¼ 4h, where h is a positive, negative or zero integer. From 3m þ n ¼ 4h it follows that m þ 3n ¼ 4h þ 2ðnmÞ. Now if n is odd (even), 3m also must be odd (even), because their sum 3m þ n is even by hypothesis. Therefore the difference nm is even and can be written as twice a suitable integer h0, which can be positive, negative or zero, as nm ¼ 2h0 . Consequently, 3m þ n ¼ 4h ) m þ 3n ¼ 4ðh þ h0 Þ But 4ðh þ h0 Þ is an arbitrary integer, like h, and therefore our couple of indexes m; n satisfy the first condition in (13.54), thus it pertains to a suppressed product, which completes the proof. The symmetry of the relations in (13.54) is a particular aspect of the interchangeability of RF and LO in the 90 SBM that comes from the symmetric structure of the circuit in Figure 13.11b. One important suppressed mixing product of the 90 SBM is the sum one, which has m ¼ n ¼ 1, as follows from the first relation in (13.54) by putting h ¼ 1. We can also obtain the lowest order mixing products of the 90 SBM by considering the quadratic term of Equation (13.51) i Is h 0 2 0 02 0 iIF ðtÞ ¼ v cosð2o t þ 2j Þv cosð2o t þ 2j Þ R L R L L 2N 2 VT2 R Is 0 0 þ 2 2 v R v L sin½ðoL oR Þt þ j0 L j0 R N VT 1 X k X Is 1 m km fm;k ðtÞ ð13:55Þ v0 v0 þ k m!ðkmÞ! R L k N VT k¼1 m¼0 ðk6¼2Þ

The first term in Equation (13.55) comprises the second harmonic of the RF and LO signals; the second term is the difference IF product. No sum frequency and no DC components are generated. The inversion of the polarity of D2 in the circuit of Figure 13.11b corresponds to changing the sign of ð2Þ ð2Þ both id and vd . Equation (13.51) is valid, just with a different expression for its base functions, which become fm;k ðtÞ ¼ cosm ðoR t þ j0 R Þsinkm ðoL t þ j0 L Þ þ sinm ðoR t þ j0 R Þcoskm ðoL t þ j0 L Þ

ð13:56Þ

Proceeding in the same way as we did for deriving the conditions (13.54), we find that the mixer in Figure 13.11b with inverted polarity on D2 suppresses the mixing products with orders ðY þ Þ

mn ¼ 2ðh þ 1Þ;

ðY Þ

m þ n ¼ 2ðh þ 1Þ

ð13:57Þ

The symmetry between RF and LO in relations (13.57) is even more evident than it is for relations (13.54). The second relation in (13.57) with h ¼ 1 states that m ¼ n ¼ 1 is a suppressed product. In other words, the 90 SBM with one inverted diode suppresses the IF difference product. Compared with its 180 counterpart, the 90 SBM presents two main advantages. The first of them is that wide-band – one octave and more – microstrip realization of the 90 hybrid is possible by means of the relatively simple Lange coupler. Conversely, the rat-race24 is a 180 hybrid which is even simpler to realize, but its relative bandwidth is of the order of 20%, more or less equivalent to the one of the branchine coupler. Second, the RF and LO ports of the 90 hybrid are always matched, providing that its two

24

See Section 7.7.2.


589

PCB

RF

A

Z1

HYB 1

D1 IF

D2

B

Z1

LO printed circuit metallization via holes bond wires semiconductor devices (beam lead) RF ports

Figure 13.12

Microstrip realization of a 90 SBM.

diodes and matching networks present the same impedance at ports 2 and 4 of HYB1 , for the 90 hybrid properties presented in Sections 10.325 and 11.5.4.26 The main disadvantage of the 90 SBM compared with the 180 one is in the less interesting spur suppression characteristics of the first. In other words, the spurs suppressed by the 180 configuration are usually more harmful, in most applications. Figure 13.12 shows the layout of a microstrip realization of a 90 SBM, designed to operate with RF and LO frequency within the range 6–18 GHz and with an IF bandwidth of 0.5–3.5 GHz. The microstrip substrate has a thickness of h ¼ 508 mm and dielectric relative permittivity er ¼ 9:8. The whole circuit ðAÞ ðBÞ occupies an area of about 11 10 mm. HYB1 is a Lange coupler. The short-circuit stubs Z1 and Z1 realize the low (high) impedance to ground at RF/LO (IF); they are a quarter wavelength at the centre RF/LO frequency of 12 GHz. These elements also provide some impedance matching. The IF network consisting of the open stubs ST1 to ST4 and of the transmission line TL1 implements the functions of the 25 26

See Equations (10.22), (10.28) and (10.29) together with the associated comments. See point 2, in particular.

590


Conversion gain, dB

0

-5

measured -10 simulated

-15 0

1

2

3

4

5

IF frequency, GHz

Figure 13.13

Performances of the mixer in Figure 13.12.

bipole Z2 and of the matching network NT2 in Figure 13.11b. Each couple of identical stubs ST1, ST2 and ST3, ST4 behaves as a single stub with half characteristic impedance. These stubs, together with the high characteristic impedance27TL1 , form a semi-lumped third-order lowdpass filter with a cut-off frequency of 3.7 GHz, 200 MHz higher than the maximum IF. Finally, the diodes D1 and D2 are beam lead devices. Figure 13.13 shows the simulated and measured conversion gain of the mixer in Figure 13.12. The LO frequency is 12 GHz with a power of 10 dBm and the RF sweeps from 12.1 to 16.1 GHz. The measured conversion loss is 7.2 dB with a flatness of 0.85 dB over the IF bandwidth. Different combinations of RF and LO frequency such that 0:5 109 joR oL j=ð2pÞ 3:5 109 and 6 109 oR =ð2pÞ; oL =ð2pÞ 18 109 give similar curves.

13.3.4

Doubly balanced diode mixers

The doubly balanced mixer(DBM) consists of a combination of two 180 SBMs with phase relations between the respective RF, LO and IF signals so as to obtain cancellation of the mixing even-order LO and RF products. Figure 13.14a shows the principle of a diode DBM. The four diodes D1 to D4 in that schematic form a ring: the anode of Dk connects to the cathode of Dk þ 1 ðk ¼ 1 to 3Þ, and the anode of D4 connects to the cathode of D1 . Such a configuration is usually named diode ring or diode (ring) quad. Differently from the schematics in Figures 13.10 and 13.11, the DBM in Figure 13.14a explicitly shows neither matching networks nor shunt impedances, in that such elements are supposed to be embedded within the hybrid couplers. With the same hypotheses as in Section 13.3.3, the voltages across the four diodes are ð1Þ

ð2Þ

ð3Þ

ð4Þ

vd ¼ v0 RF v0 LO ; vd ¼ v0 RF þ v0 LO ; vd ¼ v0 RF v0 LO ; vd ¼ v0 RF þ v0 LO

ð13:58Þ

Although expressions (13.58) omit an indication of their time dependence, the quantities v0 RF , v0 LO are functions of time: their expressions are of the type (13.46). The IF current is proportional to the difference between the current entering into the two ports of HYB2 connected with the diode ring ð1Þ

ð2Þ

ð3Þ

ð4Þ

iIF / id þ id id id 27

ð13:59Þ

The physical stub width is w ¼ 100 mm and the consequent characteristic impedance is Z0 ¼ 91:5 O.


591

HYB1 1

RF

0°

4 D2

D1 IIF+

0° 180° 3

0°

D3

HYB2 4

D4

2

1

0°

3

IF

0° IIF

-

180° 2

4

0°

4 180° 0°

0°

0° 1 HYB3

3 R0 (a)

RF

LO

1

TL1 , λ/4

2

C1

BALUN2 D2

L1

2

L2

3

D1

D3

D4

C2

TL2 , 3λ/4

TL 3 , λ/4

1

IF (LO)

L3 L4

3 TL 4 , 3λ/4

LO (IF)

(b)

CL1 , λ/4 RF

CL 2 , λ/4

C1 L1 L2

IF (LO)

D2

D1

D3

D4

L3 L4

C2 (c)

Figure 13.14 schematic.

LO (IF)

DBM: (a) principle; (b) narrow-band simplified schematic; (c) wide-band simplified

592


Again, HYB2 can modify the amplitude and/or phase of each sinusoidal component of the quantity (13.59), therefore the proportionality constant is complex and frequency dependent. Substituting expressions (13.58) into the second member of Equation (13.3), we have ð1Þ

ð2Þ

ð3Þ

ð4Þ

id þ id id id ¼

1 X

1 X Is Is k 0 0 ðv v Þ þ ðv0 RF þ v0 LO Þk RF LO kVk kVk k!N k!n T T k¼1 k¼1

1 X

1 X Is Is ðv0 RF v0 LO Þk ðv0 RF þ v0 LO Þk k k kVk k!n V k!n T T k¼1 k¼1

The above expression is the sum of four sums of infinite binomial powers of the type ðv0 RF v0 LO Þk . The application of Newton’s formula with some rearrangement gives ð1Þ

ð2Þ

ð3Þ

ð4Þ

id þ id id id ¼

1 X k h i X Is 1 m km ð1Þkm þ ð1Þm ð1Þk 1 v0 RF v0 LO k k ðkmÞ!m! N V T k¼1 m¼0

Now we can rewrite the quantity between square brackets as ð1Þkm þ ð1Þm ð1Þk 1 ¼ ½ð1Þkm 1 ½1ð1Þm Therefore the IF current is iIF ðtÞ /

1 X k X Is ½ð1Þkm 1 ½1ð1Þm 0 m km v R ðtÞv0 L ðtÞ k k ðkmÞ!m! N V T k¼1 m¼0

ð13:60Þ

The terms of the nested sums (13.60) vanish for either even n ¼ km or m. Therefore the DBM suppresses all the even-order (both RF and LO) mixing products. A different analysis of the schematic in Figure 13.14a is possible under the hypothesis of large LO amplitude. If the LO voltage across the diodes is much greater than the diode conduction knee and the RF voltage is much smaller than the LO, then the four diodes work like switches activated by the LO signal. Therefore, if the LO voltage is positive, D2 and D4 conduct, while D1 and D3 present high series resistance, and the opposite happens in the negative LO half cycle. Thus, at each LO half cycle there is an inversion of the connected couples of ports between HYB1 and HYB2 . This corresponds to saying that the IF signal equals the LO one with a 180 phase modulation at the LO frequency. In more quantitative terms vIF ðtÞ ¼

vLO ðtÞ < 0 v0 RF ðtÞ ¼ v0 RF ðtÞ v0 RF ðtÞ vLO ðtÞ > 0

1 vLO ðtÞ > 0 ¼ v0 RF ðtÞ swLO ðtÞ 1 vLO ðtÞ > 0

ð13:61Þ

Equations (13.61) assume that the diodes behave as ideal switches, presenting zero and infinite series impedance in the ON and OFF states, respectively. In the real case this is not true, and the IF voltage expression will include a proportionality constant smaller than 1. The function swLO in the third equation of (13.61) is a square wave having unitary amplitude and the same instant phase of the LO voltage. Hence swLO is periodic with the same period TL ¼ 2p=oL , like the LO voltage, and can be expanded into the Fourier series swLO ðtÞ ¼

1 X 4 ð1Þk

p 2k þ 1 k¼0

cos½ð2k þ 1ÞðoL t þ j0 L Þ

Substituting expression (13.62) into Equation (13.61), we obtain

ð13:62Þ

FREQUENCY CONVERTERS vIF ðtÞ ¼ v0 RF ðtÞ ¼ v0 R

1 X 4 ð1Þk

p 2k þ 1 k¼0

1 X 2 ð1Þk1 k¼0

v0 R

p 2k þ 1

cos½ð2k þ 1ÞðoL t þ j0 L Þ

cosf½oR þ ð2k þ 1ÞoL t þ j0 R þ ð2k þ 1Þj0 L g

1 X 2 ð1Þk1 k¼0

593

p 2k þ 1

cosf½oR ð2k þ 1ÞoL t þ j0 R ð2k þ 1Þj0 L g

ð13:63Þ

Equation (13.63) is totally independent of the LO amplitude, and includes only mixing products with unitary RF order and odd LO order n ¼ 2k þ 1. Moreover, the amplitude of the various spurs decreases proportionally to the respective LO order. The conversion gain of the DBM with large LO amplitude and ideal diodes (zero ON impedance and infinite OFF impedance) follows from the first terms ðk ¼ 0Þ of the sums (13.63) 2 CGdB ¼ 20 log10 ffi 3:92 dB p The DBM presents the best performances in terms of spur suppressions. For this reason some diode manufacturers produce diode rings in one single package, specifically for DBM applications. An additional feature of the DBM is that the RF, LO and IF ports are interchangeable, at least in principle. In other words, it is possible to use any of the three ports as input for the strong signal (originally LO), weak signal (originally RF) and output (originally IF). Clearly the passive circuitry is designed to work on a determined bandwidth, thus the signals at each port must have a frequency falling within a specified range. Moreover, different utilization of the ports leads to different suppression properties of the mixer. The main drawback of the schematic in Figure 13.14a is that it needs three hybrid junctions, but simplified solutions are possible in some situations. One relatively common case is when the LO frequency is much smaller than RF, and, consequently, the latter is close to IF. The simplification follows from considering the role that the hybrid junctions play in the circuit of Figure 13.14a. Precisely: .

HYB1 applies the RF voltage between the junctions of D1 , D2 and D3 , D4 as a voltage, with opposite polarity.

.

HYB2 converts the differential voltage between the junctions of D1 , D4 and D2 , D3 into the groundreferenced IF voltage

.

HYB1 , HYB2 and HYB3 together distribute the LO voltage across D1 , D3 and D2 , D4 with opposite polarity for each couple.

Now, if oL ¼ oR , and consequently oI ¼ joR oL j oRF, the RF bandwidth is sufficiently narrow. Therefore four transmission lines, four inductors and two capacitors can replace the three hybrid junctions, as Figure 13.14b shows. The assemblies of two transmission lines TL1 , TL2 and TL3 , TL4 form – as their global indication suggests – a balanced-to-unbalanced (balun) network. All the four pffiffiffi transmission lines present a characteristic impedance of Z0 ¼ 2R0 ffi 70:7 O. TL1 and TL3 (TL2 and TL4 ) are one-quarter (three-quarters) of a wavelength at the nominal operation frequencies, which are oR for TL1, TL2 and oL for TL3, TL4 . This way, the scattering matrix of BALUN1 and BALUN2 is, at the respective working frequencies, pffiffiffi pffiffiffi 3 2 0 j 2 j 2 h i 1 7 6 pffiffiffi ð13:64Þ SðBALUNÞ ¼ 4 j 2 1 1 5 2 pffiffiffi j 2 1 1

594


Equation (7.41), rewritten below for convenience, gives 2 0 h i j 6 1 ðHYBRIDÞ 6 ¼ pffiffiffi 4 S 0 2 1

the scattering matrix of the 180 hybrid 3 1 0 1 0 1 07 7 ð13:65Þ 1 0 15 0 1 0

A comparison between the matrices in (13.64) and (13.65) reveals that ðBALUNÞ

s11

ðBALUNÞ s12

ðHYBRIDÞ

¼ s11 ¼

¼ s41

ðBALUNÞ

¼ s21

ðBALUNÞ

¼ s31

ðBALUNÞ

¼ s13

s13 s12

¼0

ðBALUNÞ s21

ðHYBRIDÞ

¼ s14

ðHYBRIDÞ

ðHYBRIDÞ

¼ s12

ðHYBRIDÞ

pffiffiffi ¼ j 2 pffiffiffi ¼j 2

ð13:66Þ

ðBALUNÞ

Now, the identities (13.66) are sufficient for mixer operations. Moreover, the four inductors L1 to L4 distribute the LO voltage to the various diodes, with the proper amplitude and phase, as do the combinations of the three hybrids in the circuit of Figure 13.14a. The inductors L1 to L4 present low impedance at the LO frequency and high impedance at RF and IF, while the opposite happens with the two capacitors C1 , C2 . Most applications of the circuit in Figure 13.14b exploit the interchangeability of the DBM between LO and IF. In that case, the LO is applied at port 1 of BALUN2 , while the IF port is placed at the junction between L1 and L2 , as in the indications in brackets in Figure 13.14b. However, also in this second type of application the frequency passing through the inductors has to be much smaller than the one regarding the baluns, i.e. oI ¼ oR ; oL . The arrangement in Figure 13.14c works on the same principle as the one in Figure 13.14b, although with a potentially wider RF bandwidth. The improvement in the new solution consists of using wide-band baluns, consisting of a coupled-line section of one-quarter wavelength at the centre working frequency. If the even mode impedance of the pffiffifficoupled lines is much higher than R0 (ideally infinite) and if the odd mode impedance is Z0o ¼ R0 = 2 ffi 35:35 O, then the working bandwidth of the balun becomes very wide – up to several octaves.28

13.3.5

Subharmonically pumped mixers

The subharmonically pumped mixer or subharmonic mixer(SHM) operates with a half-frequency LO signal. It employs a special singly balanced configuration, which exploits nonlinear devices with antisymmetrical characteristics. The simplest of such devices consists of two identical diodes anti-parallel connected, i.e. the two diodes are in parallel with the anode of one device connected to the cathode of the other one and vice versa. SHM is particularly useful at millimetre waves, where the LO signal is difficult to generate. One possible implementation of the SHM consists of replacing the single diode D1 in the SEM of Figure 13.10a with two identical anti-parallel diodes D1 and D2 , as shown in Figure 13.15a. Figure 13.15b presents a more common architecture, which represents the dual network of the one in Figure 13.15a: Z 0 1 (Z 0 2 ) presents low (high) impedance at RF and LO, and high (low) impedance at IF. For the remaining aspects, NT1 (NT2 ) matches the combination of the diodes and Z 0 1 (Z 0 2 ) into 50 O at RF, LO (IF) frequency, and the diode parasitics are parts of the above elements, as usual. The analysis of the SHM in Figure 13.15b is relatively simple, by adopting the same assumptions used for Sections 13.3.3 to 13.15b, and with the simplified notation of Section 13.3.4. The interested 28

See the Ansoft file 01_Coupled_Lines_Balun.adsn.


595

D2 COUP1 v1

RF

1

NT1

R0

(a)

iIF

v2

2

Z1

1

D1

2

Z2

IF NT2

LO

COUP1

NT1

RF

1

Z1′

2

iIF

v2

NT2

Z2′ 1

2

IF

id R0 D1 (b)

D2

LO

CR

LR

CI

LI

RF

IF LL

D1

D2

CL

(c)

LO

Figure 13.15 SHM: (a) arrangement derived from Figure 13.10; (b) dual network of (a); (c) simplified diagram of (b).

reader can easily repeat the procedure to analyze the configuration in Figure 13.10, with few changes. By hypothesis, the diodes D1 and D2 follow the law (13.3), so their current relates to the respective voltage as 1 1 X X Is h ð1Þ ik Is h ð2Þ ik ð1Þ ð2Þ id ¼ ; id ¼ ð13:67Þ vd v k k k!N VT k!N k VTk d k¼1 k¼1 ð1Þ

ð2Þ

The anti-parallel connection of the two diodes implies that v2 ¼ vd ¼ vd , and that the current flowing ð1Þ ð2Þ globally through the anti-parallel pair is id ¼ id id . Therefore, considering the anti-parallel pair as a

596


single device, the resulting voltage–current relation is ð1Þ

ð2Þ

id ¼ id id ¼

1 1 i X X Is Is Is h k k k k v2 v ðv Þ ¼ 1ð1Þ 2 2 k!N k VTk k!N k VTk k!N k VTk k¼1 k¼1 k¼1

1 X

ð13:68Þ

The factor ½ð1Þk 1 that multiplies all the terms of the sum in (13.68) vanishes for even k, hence this sum contains only odd-index terms. By replacing the generic index k with the odd index 2i þ 1, Equation (13.68) becomes 1 X 2Is id ¼ v2i þ 1 ð13:69Þ 2i þ 1 V 2i þ 1 2 T i¼0 ð2i þ 1Þ!N The relation (13.69) contains only odd powers of the voltage, thus it is anti-symmetrical, in that id ðvd Þ ¼ id ðvd Þ. Such a property is fundamental for SHM operation. For the specifications assumed on Z 0 1 and Z 0 2 , the spectral components of id at IF can only flow through Z 0 2 , towards the IF load. Consequently, the IF output voltage contains all and only the spectral components of id falling in the IF bandwidth, each multiplied by a frequency-dependent complex constant. Therefore, by replacing v2 with the linear combination of RF, LO voltages, and adopting the simplified notation of Section 13.3.4, the current through the anti-parallel pair is 1 X 2Is ðv0 RF þ v0 LO Þ2i þ 1 id ¼ 2i þ 1 V 2i þ 1 ð2i þ 1Þ!N T i¼0 Applying Newton’s formula to the binomial power ðv0 RF þ v0 LO Þ2i þ 1, the above equation becomes id ¼

1 2i þ1 X X i¼0

Is 2 m 2i þ 1m v0 v0 2i þ 1 V 2i þ 1 m!ð2i þ 1mÞ! RF LO N T m¼0

ð13:70Þ

The exponents m; n ¼ 2i þ 1m in the monomials of the sum in (13.70) correspond to mixing products having m; n as RF and LO harmonic order, respectively, and the sum m þ n ¼ 2i þ 1 is always odd. Therefore, the mixer in Figure 13.15 suppresses the mixing products such that the sum of their RF and LO orders is even. Consequently, the first nonlinear term of the IF current is the one with degree equal to 3. Isolating the term proportional to v0 RF v0 2LO from the nested sums (13.70) yields " # 1 2i þ1 X X Is Is 2 0 02 0 m 0 2i þ 1m id ¼ 2i þ 1 2i þ 1 v RF v LO þ v v 2i þ 1 V 2i þ 1 m!ð2i þ 1mÞ! RF LO N VT T i¼0 m¼0 N m;2i þ 1m6¼1;2

ð13:71Þ Substituting expressions (13.38) into the first term of the second member of Equation (13.71), we obtain Is 0 0 2 v Rv L 3 N VT3

¼ ¼

Is 0 1 2 v cosðoR t þ j0 R Þ v0 L ½cosð2oL t þ 2j0 L Þ þ 1 3 R 3 2 N VT Is 0 02 v R v L fcos½ðoR þ 2oL Þt þ j0 R þ 2j0 L þ cos½ðoR 2oL Þt þ j0 R 2j0 L g 4N 3 VT3

Is 2 v0 R v0 L cosðoR t þ j0 R Þ 2N 3 VT3

ð13:72Þ

The first term of the third member of Equation (13.72) represents the useful mixing products of the SHM. Their frequency is the sum and difference between the RF and twice the LO one and their amplitude is proportional to the RF one, as anticipated. More generally, the term of order ð1; nÞ produces the spectral components Is An;n 0 0 n v R v L fcos½ðoR þ n oL Þt þ j0 R þ n j0 L þ cos½ðoR n oL Þt þ j0 R n j0 L g ð13:73Þ ðN VT Þn þ 1 n! with An;n =n! ¼ 1=4; 1= 192; 1=23 040 . . . for n ¼ 2; 4; 6 . . ..

FREQUENCY CONVERTERS RF filter

IF filter VRF

RF

597

VIF

R

IF

L

VLO

Local oscillator

Figure 13.16

Basic mixer application in radio receivers and transmitters.

Thus, it is possible to design the SHM to operate with any even harmonic of the LO, although with conversion loss and sensitivity to the LO amplitude increasing with n. Figure 13.15c shows a simplified solution, which is possible if oR ; n oL , and the desired solution between ðoR n oL Þ are sufficiently spaced, and if the anti-parallel pair impedance is acceptably close to 50 O. The three series resonant circuits in series with the mixer ports are tuned to the respective frequencies LR CR ¼ o2 R ;

LL CL ¼ o2 L ;

LI CI ¼ o2 I

This way, the RF and LO voltages apply across the diodes and the IF spectral components of the diode pair current flow through the IF port, with minimum dispersion.

13.3.6 Image reject mixers Figure 13.16 shows a typical application of mixers in receivers and transmitters. In the first case, the RF filter has a centre frequency equal to oR of the signal to be received and the mixer converts the received signal into a different – usually lower – frequency oI ¼ joR oL j. The IF filter passes the frequencies close to oI within a specified bandwidth, and removes all the remaining spectral components. If a variable frequency has to be received, the LO frequency is variable and the mixer converts different selectable RFs into a fixed one.29 The main drawback of this configuration is that, for any given oL , the mixer converts to the same IF two different RFs: oR ¼ oL oI . For example, if oL ¼ 2p 109 and oI ¼ 2p 108 , the receiver in Figure 13.16 potentially receives both 0.9 and 1.1 GHz. The RF filter must then pass the desired one of the two RF signals and stop the other, whose frequency is called the image frequency. The difference between the desired receiving frequency and its image is 2oI . Transmitter applications30 of the scheme in Figure 13.16 use a relatively low-frequency RF-modulated signal and upconvert it to the final transmitting frequency. In this case, the IF is usually higher than the RF, and could be variable; if so it is the LO. Similar to the receiver, for any RF and LO frequencies, the conversion generates two IFs, oI ¼ oL oR , which differ by 2oRF from each other. Sometimes, also the undesired frequency coming from the upconversion is referred to as the image frequency. The RF (IF) filter has to reject the image frequency, and this could be difficult if the IF (RF) centre frequency is low. If the receiving (transmitting) bandwidth is wider than 2oI ð2oR Þ, one single filter cannot eliminate the image frequency, which falls in the passband: therefore many switched filters or a variable31 filter are required. In such situations the image reject mixer (IRM) could be a convenient solution. 29

This type of receiver architecture is known as superheterodyne, see Section 15.4.2 for further details. See Section 15.3 for more details. 31 See [4]. 30

598


MIX1

I

R L

HYB1

RF DIV1

HYB2

LO

0°

0°

IFA

90°

90°

90°

90° 0°

0°

3

IFB

R0 L R

Q

MIX2 I-Q MIXER

Figure 13.17

The IRM.

The IRM is a balanced mixer configuration which employs two mixers (usually DBMs) and two 90 hybrid junctions, in order to achieve image suppression. Figure 13.17 shows the basic IRM configuration. The assembly inside the grey rectangle consists of two mixers having the same RF and LO in quadrature, known as a quadrature mixer or IQ mixer, and is extensively used in modern radio equipment.32 The following analysis considers all ideal components: the power divider and the hybrids are perfectly amplitude and phase balanced, while the mixers are product detectors. The effect of amplitude and phase imbalances will be separately considered in Section 13.3.8. From the nonlinear perspective, MIX1 and MIX2 suffer from the limitations described in Sections 13.3.2 to 13.3.5, although we will not consider their implications in the IRM.

13.3.6.1 Downconversion IRM From expressions (13.31) and considering the transmission coefficients of DIV1 and HYB1 , the RF and LO voltages of the two mixers are vR ð1Þ MIX1 vRF ¼ pffiffiffi cosðoR tÞ; 2 vR ð2Þ MIX2 vRF ¼ pffiffiffi cosðoR tÞ; 2

vL ð1Þ vLO ¼ pffiffiffi cosðoL tÞ 2 vL ð2Þ vLO ¼ pffiffiffi sinðoL tÞ 2

Consequently, the difference IF products for the two mixers are KIF ð1Þ vIF ¼ pffiffiffi vR cos½ðoR oL Þt; 2

KIF ð2Þ vIF ¼ pffiffiffi vR sin½ðoR oL Þt 2

where KIF is the conversion gain of the two mixers, which is assumed to be independent of the LO amplitude.

32

See Sections 15.3.3, 15.4.2, 15.4.3, 15.3.4, 17.7.1 and 17.7.3.


599

Rearranging the IF voltages such that their frequency is always positive, and since cosðaÞ ¼ cosðaÞ; sinðaÞ ¼ sinðaÞ, we have the following two cases: ð1 þ Þ

oR > oL :

vIF

oR < oL :

vIF

ð1Þ

KIF ¼ pffiffiffi vR cosðjoR oL jtÞ; 2 KIF ¼ pffiffiffi vR cosðjoR oL jtÞ; 2

ð2 þ Þ

vIF

ð2Þ

vIF

KIF ¼ pffiffiffi vR sinðjoR oL jtÞ 2 KIF ¼ pffiffiffi vR sinðjoR oL jtÞ 2

The two outputs of the arrangement in Figure 13.17 become ðAÞ

ðBÞ

vIF

vIF

ðA þ Þ

o R > oL

vIF

o R < oL

vIF

ðAÞ

ðB þ Þ

¼ KIF vR cosðjoR oL jtÞ vIF

ðBÞ

¼0

vIF

¼0

ð13:74Þ

¼ KIF vR sinðjoR oL jtÞ

Therefore, the output IFA (IFB ) transmits the IF signal when the RF is higher (lower) than the LO one. Connecting to the output corresponding to the desired conversion product, while terminating the other one with a matched load, we have a mixer that converts to IF only one of the two possible RFs, which is an IRM by definition.

13.3.6.2 Upconversion IRM The case of the IF sum product is simpler to analyze, in that it is always oR þ oL > 0. The output signals of the two mixers are KIF ð1Þ vRF ¼ pffiffiffi vR cos½ðoR þ oL Þt; 2

KIF ð2Þ vRF ¼ pffiffiffi vR sin½ðoR þ oL Þt 2

ð13:75Þ

Consequently, the output voltages of the IRM are ðAÞ

vIF ¼ 0;

ðBÞ

vIF ¼ KIF vR sin½ðoR þ oL Þt

ð13:76Þ

Thus, if oL > oR the output IFB transmits the sum but not the difference product. The same result with oL < oR can be obtained by swapping the RF and LO signals.

13.3.6.3 Image-enhanced mixers The IRM described in Sections 13.3.6.1 and 13.3.6.2 is a relatively complex circuit, and offers limited image suppression, due to the different amplitude and phase that are inevitably present within the circuit.33 For this reason, the IRM is not widely used, rather radio equipment generally uses filters to remove the image frequency, if possible. If not – typically because the image frequency is close to the desired one – then the IRM remains the only option. The treatment in Sections 13.3.6.1 and 13.3.6.2 shows that the image spectral component is transmitted to one of the two outputs IF1 or IF2: the relative energy is therefore dissipated on the output termination. Sophisticated design techniques recover the image energy to increase the conversion gain of the circuits in Sections 13.3.2 to 13.3.5. The resulting circuit, known as the image-enhanced mixer(IEM), has some image rejection combined with reduced conversion loss. In order to illustrate the IEM mechanism better, we need to consider that the frequencies of the main conversion product and of the corresponding image are oR oL and oR oL , respectively. The IEM technique basically consists of terminating the diodes with convenient reactive impedance at the image frequency. This reflects the image energy back to 33

See Section 13.7.7 below for a quantitative description.

600


the diodes. This way, the image spectral component mixes with the LO second harmonics and generates the additional spectral component oR m oL 2oL ¼ oR oL , whose frequency coincides with the desired signal. If the phase of that reconverted mixing product coincides with the main one (generated by the direct mixing between the RF and LO), the amplitude of the main conversion product will increase, as required. Simultaneously, the reactive termination at the image frequency reduces the image level at the IF port, as anticipated. TheavailablespaceinthisbookisinsufficienttodescribetheIEMinmoredepth.However,fromthisshort discussion it follows that the image termination of the IF port influences the mixer conversion loss. Reactive values with the proper phase can reduce the conversion loss by some decibels. Finally, the image remix mechanism also affects the nonlinear performances of the mixer, potentially increasing the spur level as well.

13.3.7

Suppression in presence of amplitude and phase imbalance

Sections 13.3.3 to 13.3.6 showed that the balanced mixer cancels some mixing spurs by producing two identical and 180 out-of-phase spectral components. However, real hybrid couplers are not perfectly balanced: the reflected waves at the two non-isolated ports have different amplitude, and their phase difference is not exactly 0 , 90 or 180 , despite any expectation. Moreover, the two (in the SBM) or four (in the DBM) diodes are also not identical. Therefore, a more accurate analysis should take those amplitudes and phase imbalance into account. This is possible, but unfortunately the resulting expressions are complicated and their meaning is difficult to understand. It is more convenient to develop some more general considerations of the signal resulting from the superimposition of two sinusoids u1 , u2 having the same frequency but different amplitude and phase. With a convenient choice of the time origin, it is always possible to assign zero offset phase to one of the two signals, for instance u1 . Let u1 ðtÞ ¼ U1 cosðotÞ;

u2 ðtÞ ¼ U2 cosðot þ DjÞ

ð13:77Þ

For the identities (A.92) to (A.94), the sum of the two signals is u1 ðtÞ þ u2 ðtÞ ¼ Ucosðot þ fÞ

ð13:78Þ

where U¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U12 þ U22 þ 2U1 U2 cosðDjÞ;

j ¼ tan1

U2 U1 þ U2 cosðDjÞ

The real quantity U is the amplitude of the resulting sinusoid, which has the same frequency as, and intermediate phase between, u1 , u2 . Note that if the two signals have equal amplitude and 180 relative phase shift ðU1 ¼ U2 ; Df ¼ pÞ, it follows that U ¼ 0, which corresponds to infinite suppression. In the general and non-ideal cases, the ratio U 2 =ðU1 U2 Þ is a good quantifier for how much the two equalfrequency sinusoids cancel each other: U2 U1 U2 ¼ þ þ 2 cosðDfÞ U1 U2 U2 U1

ð13:79Þ

The amplitude matching between the two sinusoids is particularly meaningful if expressed by means of the amplitude imbalance (in dB), while it is convenient to specify the phase in terms of its distance from the ideal case, i.e. using the phase imbalance: U1 DAdB ¼ 20 log10 ð13:80Þ ; Ye ¼ pDj U2 Substituting parameters (13.80) into the quantity (13.79), and expressing the result in logarithmic units, we obtain the suppression function


601

2.0 ∆U = 30 dB

Θe, deg.

1.5 1.0 0.5

40 50

0.0 0.0

(a)

0.1

∆A, dB

0.2

0.3

2

3

20

10

Θe, deg.

15 10

20

5

30 dB 0 (b)

0

Figure 13.18

1

Suppression as function of amplitude and phase imbalance.

DUdB ¼ 10 log10

∆A, dB

U2 U1 U2

h i ¼ 10 log10 10DAdB =20 þ 10DAdB =20 2 cosðYe Þ

ð13:81Þ

For any assigned DUdB , Equation (13.81) describes one curve in the DAdB Ye plane. Figures34 13.18a,b show the result for DUdB ¼ 10; 20; . . . ; 50 dB. More precisely, Figure 13.18a plots the curves relative to small imbalance, while Figure 13.18b displays relatively less ideal cases. The curve relative to a suppression of 30 dB is present in both the graphs, with the thick curve as a reference. Typically, the amplitude imbalance is bound between 0.1–1 dB and the phase imbalance falls within the range 0.1 –5 . The best values are achieved with lower frequency and narrow bandwidths. As an indication, DUdB could be about better than 40 dB in circuits operating around 2 GHz with 5% bandwidth and lower than 20 dB for octave bandwidths allocated above 10 GHz. Therefore, all the balanced mixers present residual levels of the theoretically suppressed mixing products, which is some tens of decibels lower than in an SEM using the same diodes, matching networks, RF/LO frequency and power. Also, the SHM generates residual oRF oLO mixing products. Note also that the graphs in Figure 13.18 apply to any balanced structure: .

The attenuators in Figure 10.21, the phase shifters having the structure of Figures 10.28 and 10.30, and the 90 balanced amplifiers discussed in Section 11.5.4 present non-zero reflection coefficients.

.

The 180 balanced amplifiers discussed in Section 11.5.4 and the push–pull amplifier in Figure 11.25 have residual even-order distortion.

.

The push–push oscillator described in Sections 12.8.4 and 12.9 produces residual fundamental and odd-harmonic spectral components.

34

See the Mathcad file 03_Suppression.mcd.

602

13.3.8


FET mixers

The mixers considered in Sections 13.3.2 to 13.3.5 exploit the nonlinear relation between the diode current and voltage. More precisely, these circuits produce the main mixing products by means of the quadratic term of the diode characteristic: the higher order terms produce undesired spurs. For this consideration, the FET is a good candidate for mixer applications. Equations (9.75), (9.76) and (9.79) show a quadratic relation between the drain current and the gate voltage for any fixed drain–source voltage, provided that the gate–source voltage is higher than the pinch-off. More precisely, given vds , and from the Curtice quadratic model (9.74), the simplified equation set follows:35 8 2 vI < Idss 1 ðvI > Vp Þ ð13:82Þ Ids ðvI Þ ¼ Vp : 0 ðvI Vp Þ where Idss ¼ bVp2 ð1 þ l vds Þtanhða vds Þ The Statz model (9.78) is not exactly quadratic, due to the presence of the factor 1 þ y ðvI Vp Þ in the denominator of that equation. Moreover, the Curtice, Statz and any other model are always approximate, regardless of their accuracy. However, independently of model accuracy, the drain current has a parabolic shape only if vI > Vp ; below that limit the drain current vanishes. The piecewise-defined function (13.82) presents a discontinuity36 in its second derivative for vI ¼ Vp, hence it does not fulfil the conditions for the Taylor series. Nevertheless, by applying curve-fitting algorithms,37 it is possible to approximate the function (13.83) with a polynomial over a finite interval including the point vI ¼ Vp . The fitting function is a polynomial with degree higher than 2: the higher the polynomial, the better the approximation and/or the wider the valid range for a specified error. Therefore, FET mixers present high-order spurs, more or less like diode ones, despite the parabolic equation included in the function (13.82). Figure 13.19 displays the function (13.82) (black solid line), the parabolic segment of the same function extended to the whole range (black dashed line) and the approximating third-order polynomial (grey solid line). The FET is the one considered in Section 9.7.2.2, with v0 ds ¼ 3 V.

13.3.8.1 Single-ended FET mixers Figure 13.20a shows a single-ended FET mixer whose structure resembles that of the diode SEM in Figure 13.10. The directional coupler COUP1 combines the RF and LO signals on the gate of the FET Q1, with some attenuation. The two-port network NT1 matches the transistor input into 50 O at RF and LO frequencies. The inductor Lgg, the capacitor Cgg and the voltage generator Vgg form the gate bias network, which superimposes a DC voltage on the RF and LO voltages. As usual, Lgg (Cgg ) has negligible admittance (impedance) at RF and LO frequencies. Similarly, on the output we have the drain bias network consisting of Vdd , Ldd and Cdd . Again Ldd (Cdd ) presents high (low) impedance at IF. Finally, NT2 matches the drain to the 50 O output impedance at IF. The counterpart of the impedances Z1 , Z2 of the diode SEM is an additional requirement on the FET SEM matching networks: NT1 (NT2 ) has to provide a suitable – usually reactive – impedance at IF (RF, LO) frequencies. The quantitative determination of such additional requirements depends on the performance to optimize, for instance, conversion gain, LO 35 Here, vds is the internal drain–source of the model in Figure 9.40a, and vI is the voltage across the capacitor Cgs of the same model. These voltages differ from the external drain–source and gate–source voltages due to the various drops across Lg , Rg , Ld , Rd , Ls , Rs and RI . 36 The second derivative of the function (13.82) computed for vI > Vp yields the constant d 2 Ids ðvI Þ=dv2I ¼ 2Idss =Vp2 , while the function (13.82) is identically zero, together with all its derivatives – including the second one – for vI Vp. 37 See the Mathcad file 04_Pinch_Off_Polynomial_Curvefit.MCD.

FREQUENCY CONVERTERS 100 80

603

Equation (13.82) I dss (vI / V p - 1)

2

Polynomial approximation

Ids , mA

60 40 20 0 -20 -1.5

VP -1.0

-0.5

0.0

vi , V

Figure 13.19

FET transfer DC characteristic and its polynomial approximation.

power requirement, linearity, etc. An analytical treatment of this topic is possible, but its complexity and length exceed the scope of and the available space in this book. However, the reader has to be aware that the impedance that NT1 (NT2 ) presents to the gate (drain) of Q1 at IF (RF, LO) affects the performance of the circuit in Figure 13.20a. Once this aspect is known, it is relatively easy to make the necessary adjustments by applying nonlinear CAE methods. Of course, the nonlinear model of the FET has to be accurate. Here, we will limit ourselves to presenting some relatively simple considerations, similar to the cases considered in Sections 13.3.2 to 13.3.6. From Equation (13.83) and with the assumptions used in Sections 13.3.2 to 13.3.6, the internal gate–source voltage of Q1 is vI ¼ Vgg þ vRF ðtÞ þ vLO ðtÞ ¼ Vgg þ vR cosðoR t þ fR Þ þ vL cosðoL t þ fL Þ

ð13:83Þ

Again, the peak amplitudes of the RF and LO voltages are proportional to that of the respective generators, with a proportionality constant depending on the directional coupler attenuation, from the matching network and from the transistor model. The latter two passive elements also introduce additional phase shifts, which are embedded within the quantities fR ; fL . Moreover, since fR ; fL are generic terms, we can always assume that vR ; vL are both positive quantities, without loss of generality. Then, the internal gate–source voltage remains above the pinch-off if Vgg vR vL > Vp

ð13:84Þ

Under the condition (13.84), the drain–source current follows the first part of the function (13.83). Substituting expression (13.84) into it yields

2 2 vRF ðtÞ þ vLO ðtÞVgg Idss 1 ¼ 2 vRF ðtÞ þ vLO ðtÞðVp þ Vgg Þ Vp Vp o Idss Idss n ¼ 2 vRF ðtÞvLO ðtÞ þ 2 v2RF ðtÞ þ v2LO ðtÞ þ 2ðVp þ Vgg Þ½vRF ðtÞ þ vLO ðtÞ þ ðVp þ Vgg Þ2 Vp Vp

ids ðtÞ ¼ Idss

ð13:85Þ

604

MICROWAVE AND RF ENGINEERING + COUP1

NT1 1

RF

2

Vdd

Ldd

NT2 Cdd

Cgg

1

2

IF

Q1 L gg

R0 Vgg + (a)

LO

NT1 1

RF

NT2

2

1

2

IF

Q1 NT3 1

LO

2

(b) HYB2 Cdd 1 0°

HYB 1 RF

1

0°

4

0º /90° 90°/180° LO

3

0°

C gg

Q1

L dd L dd

2

Q1

C dd 3

IF

0°/180° 0°

L gg C gg L gg

4

0°

2

R0

+ (c)

Figure 13.20

V gg +

V dd

FET mixers: (a) single ended; (b) single ended with cold FET; (c) singly balanced.

The third equation in (13.85) includes six terms. The terms between the braces produce spectral components at the RF and LO frequencies, at their second harmonic and at zero frequency. The first term generates the principal conversion sum and difference products, whose amplitude is proportional to the product of RF and LO peak amplitude and does not depend on the gate bias voltage: Idss Idss Idss vRF ðtÞvLO ðtÞ ¼ 2 vR vL ½cosðoR þ oL Þt þ jR þ jL þ vR vL ½cosðoR oL Þt þ jR jL Vp2 2Vp 2Vp2 ð13:86Þ The IF output voltage is proportional to the desired one of the two sinusoidal terms in (13.86), in that NT2 and the drain bias network filter out all the remaining terms. Sometimes the filtering is impossible because some of the spurs have a frequency close to the main conversion product.


605

Given the maximum RF and LO amplitudes, it is theoretically possible to choose a gate bias voltage such that the internal gate voltage is always greater than the pinch-off. This condition corresponds to imposing that, for any assigned maximum values of vR and vL , Vgg must fulfil the condition (13.84). In this case, the FET current follows the first segment ðvI > Vp Þ of the function (13.82), consequently the drain current of the mixer equals expression (13.85), and finally the mixer in Figure 13.19a ideally generates second-order mixing products. Unfortunately, high RF and/or LO dynamic ranges, i.e. large values of jvR j; jvL j, determine small negative gate–source bias and high DC drain–source currents. Extracting the DC components from expression (13.85) and by imposing the condition (13.84) on the result, we have the DC drain–source current Idss v2R þ v2L Idss v2R þ v2L 2 2 i ds ¼ 2 þ ðVp þ Vgg Þ > 2 þ ðvR þ vL Þ ð13:87Þ Vp 2 Vp 2 Equation (13.87) gives the maximum DC drain–source current for a specified RF and LO dynamic range. The maximum safe current of the FET could be smaller than the value in (13.87). This circumstance imposes a reduction in the direct current and therefore an increase in Vgg , preventing the mixer from operating in a purely quadratic way. In any case, even if the limit (13.87) is compatible with the transistor safe current, the mixer still generates high-order products, due to the gate–source capacitance modulation. As explained in Section 9.7.2.2, the gate–source capacitance Cgs is not constant, rather it is function of vI , as well as the drain–source current. Thus, large gate–source voltages modulate Cgs with the consequent generation of high-order distortions. An analytical treatment of this phenomenon is too long and complex for this book. Moreover, the most efficient way to investigate the mixer large-signal operation is probably by nonlinear computer analysis, independently of the specific configuration and nonlinear devices.

13.3.8.2 Cold FET mixers Figure 13.20b shows a different type of FET SEM, which has no bias element, known as the cold FET mixer. In this circuit the FETworks like a switch, and the circuit analysis is similar to the one presented in the second part of Section 13.3.4. In the circuit of Figure 13.20b, the two-port networks NT1 , NT2 and NT3 match the transistor into 50 O at RF, IF and LO frequency, respectively. Furthermore, NT1 (NT2 ) presents an open circuit to the drain of Q1 at IF (RF), while the impedance that the gate sees at RF and IF also affects the circuit performance. In the cold FET mixer, the transistor operates in the triode region, working like a switch. If the LO amplitude is high enough, we can assume that if the LO voltage is positive the drain–source channel presents low (ideally zero) resistance. In the opposite case, when the LO voltage is negative, the channel is pinched off, with a resulting high (ideally infinite) drain–source impedance. The IF voltage is then vIF ðtÞ ¼

vRF ðtÞ vLO ðtÞ < 0 ¼ vRF ðtÞ 0 vLO ðtÞ > 0

1 þ swLO ðtÞ 1 vLO ðtÞ > 0 ¼ vRF ðtÞ 0 vLO ðtÞ > 0 2

ð13:88Þ

where the square wave swLO has the Fourier series expansion (13.62). Expression (13.88) is similar to expression (13.61) but with a different multiplying constant (1=2instead of 1) and with the additional term vRF ðtÞ=2. Substituting the series (13.62) into expression (13.88), we get 1 X 1 1 ð1Þk cosf½oR þ ð2k þ 1ÞoL t þ jR þ ð2k þ 1ÞjL g vIF ðtÞ ¼ vRF ðtÞ þ vR 2 p 2k þ 1 k¼0

þ vR

1 X 1 ð1Þk k¼0

p 2k þ 1

cosf½oR ð2k þ 1ÞoL t þ jR ð2k þ 1ÞjL g

ð13:89Þ

606


Therefore, under switching operation, the cold FET SEM produces the same mixing products as the DBM, plus one term proportional to the RF signal. The theoretical conversion gain coming from Equation (13.89) is one-half the one obtained from Equation (13.63): 9.94 dB instead of 3.92 dB. However, Equations (13.88) and (13.89) do not consider the reactive energy accumulated in the OFF state of Q1 , which increases the conversion gain significantly.38 The main advantages of the cold FET mixer over the conventional solution are the absence of any DC power and the availability of two separate electrodes for the RF and LO. This latter circumstance simplifies the design by eliminating the directional coupler and reducing the constraints on NT1 , NT3 : the first works at RF, the second at LO, while the common matching network NT1 in the mixer in Figure 13.20a has to work at RF and LO simultaneously. The main disadvantage of the cold FET mixer is that its conversion gain is always less than 1, due to the passive nature of the circuit.

13.3.8.3 Balanced FET mixers Two or four mixers of the type in Figure 13.20a or 13.20b can be combined together to obtain singly or doubly balanced configurations. Figure 13.20c shows the singly balanced configurations obtained from the mixer in Figure 13.20a; for simplicity, the figure omits the matching networks. A similar arrangement is possible with a cold FET SEM. Assuming that HYB2 is a 180 hybrid, the circuit in Figure 13.20c is the FET counterpart of the SBM in Figure 13.11a or 13.11b, if HYB1 is a 180 or 90 hybrid junction. In both cases, the 180 phase inversion operated by HYB2 in Figure 13.20c plays the role of the diode inversion in Figures 13.11a,b. If HYB2 is of 0 type – as for example a Wilkinson divider – and HYB1 is a 90 type, then the mixer in Figure 13.20c works like the one in Figure 13.11b with inverted D2 : it produces the sum and suppresses the difference frequency. The case of HYB1 of 180 and HYB2 of 0 is a potential SHM. The adjective ‘potential’ is used because the SHM operation requires a third-order nonlinearity, and this is not naturally present in the FET, at least in principle. The FETexhibits third- and higher order nonlinearity mainly if vI swings across Vp . Therefore the design of an FET SHM presents the opposite requirements to the standard FET mixer.

13.3.9

Mixers based on differential pairs

The DP illustrated in Section 11.6.3 offers an interesting mixer application, which is widely used in RFIC, in both bipolar and MOS technology. Here, we will consider the BJT case; however, the operating principle of the circuit is the same for both BJT and FET DP, although with different formulae. Sections 13.3.9.1 to 13.3.9.3 following consider three specific circuits: from the simplest case of one single DP to the Gilbert cell which contains three nested DPs.

13.3.9.1 Singly balanced DP mixer Figure 13.21a shows the basic mixer application of the DP. The circuit presents a differential input ð1Þ

ð2Þ

ð3Þ

½vb vb applied between the bases of Q1 , Q2 , one single-ended input ½vb on the base of Q3 and one ð1Þ

ð2Þ

differential output ½vc vc . In principle, the two inputs can be used interchangeably for the RF and LO, ð3Þ vb

is more convenient for RF, as we will see shortly. although From Equation (11.101) the differential output voltage of the network in Figure 13.21a is " ð1Þ ð2Þ # 1 vb vb ð1Þ ð2Þ ð3Þ vc vc ¼ Rc ic tan h ð13:90Þ 2 nF VT

38 See the SIMetrix file 08_Switching_Cold_FET_MIXER; the mixer described there exhibits a conversion loss of 4.44 dB.


607

Vcc Rc

vb(1)

Rc

vc(1)

Q1 Q2

vb(2)

(2)

vc

ic(3)

vb(3)

Q3 Re

(a)

Vcc (1) (4) ic +ic

ic(2) + i c(5) Rc

Rc

vc(1) (2)

vb(1)

vc

Q1 Q2 Q4

Q5

(2)

vb

(3)

vb

Q6

Q3

(4) vb

Re

Re

(b)

Vcc (5)

ic(2)+ ic

i c(1)+ ic(4) Rc

(1)

vb

Rc

(1)

vc

vc(2)

Q1 Q2 Q4

(2)

Q5

vb

vb(3)

Q3

Re

Re

Q6

vb(6) I0 (c)

Figure 13.21 DP-based mixers: (a) single DP working as SBM; (b) two DPs forming a DBM; (c) Gilbert cell.

608


The Maclaurin series expansion39 of the function (13.90) includes all and only the odd-degree terms ð2Þ vð1Þ c vc ¼ Rc Ic3

1 X

ð1Þ

ð2Þ

a2k þ 1 ½vb vb 2k þ 1

ð13:91Þ

k¼0

with a1 ¼

1 1 ; 2 nF VT

a3 ¼

1 24ðnF VT Þ

3

;

a5 ¼

1 240ðnF VT Þ5

Let the voltage on the base of Q3 be sinusoidal with a superimposed direct current, usually produced by a bias network ð3Þ ð13:92Þ vb ¼ vR0 þ vR cosðoR t þ fR Þ The constant vR0 and the sinusoidal term vR cosðoR t þ fR Þ are the DC and AC components of the RF signal, respectively. The emitter current of Q3 corresponding to the base voltage (13.93) is ieð3Þ ¼

ð3Þ

ð3Þ

ð3Þ

vb vbe vR0 þ vR cosðoR t þ fR Þvbe ¼ Re Re

Assuming that the forward current gain is much greater than 1, as it usually is, the collector current coincides approximately with the emitter one icð3Þ ¼ iR cosðoR t þ fR Þ þ iR0 where iR ¼ vR =Re ;

ð13:93Þ

ð3Þ

iR0 ¼ ½vR vbe =Re

If the voltage applied on the base of Q3 is such as to keep the base–emitter junction always in conduction, ð3Þ ð3Þ ð3Þ i.e. if vR0 vR ?vbe , then the base–emitter voltage vbe is almost constant, and the current ic is the sum of one constant plus one sinusoidal term. Substituting the current (13.93) into the differential output voltage (13.90), we obtain the IF differential output voltage ! " ð1Þ ð2Þ # ð1Þ ð2Þ 1 vb vb 1 vb vb ð1Þ ð2Þ Rc iR cosðoR t þ fR Þtan h ð13:94Þ vc vc ¼ Rc iR0 tan h 2 nF VT 2 nF VT The first term in the function (13.94) simply represents the amplified and distorted input differential LO voltage. It has the same form as Equation (11.101), as a straightforward consequence of iRO being constant. The second term is the product of a sinusoidal quantity for a voltage proportional to the abovementioned one. By applying the series (13.91) to the terms in the function (13.94), it is possible to rewrite the differential output voltage as ð2Þ vð1Þ c vc ¼ Rc iR0

1 X

ð1Þ

ð2Þ

a2k þ 1 ½vb vb 2k þ 1

k¼0

1 X Rc ð1Þ ð2Þ vR cosðoR t þ fR Þ a2k þ 1 ½vb vb 2k þ 1 ð13:95Þ Re k¼0

By hypothesis, the differential voltage between the bases of Q1 and Q2 is sinusoidal,40 and without DC components ð1Þ ð2Þ vb vb ¼ vL cosðoL t þ fL Þ ð13:96Þ ð1Þ

ð2Þ

Hence, the generic factor ½vb vb 2k þ 1 is the sum of the first k þ 1 odd harmonics of that voltage; more precisely, it is the sum of first k þ 1 odd harmonics of the voltage (13.97), all with amplitude þ1 . proportional41 to v2k L 39

See also Equation (11.102). Like the RF applied on the base of Q3 , as always assumed in Sections 13.2.1 to 13.2.8. 41 See identities (A.82) to (A.85), which can be extended to any integer power of the cosine. 40


609

Consequently, the voltage (13.95) contains all the odd harmonics of the differential input and the product between the single-ended and the odd harmonics of the differential input. Therefore, if the single-ended input is the RF one, the circuit in Figure 13.21a works similarly to a 180 SBM, in that it suppresses all the LO even harmonics and all the mixing products having even harmonic order. More than the 180 SBM, the mixer in Figure 13.21a also suppresses all the RF harmonics and all the mixing products having RF order higher than 1. Such a performance implies high RF to IF linearity, which is a peculiar characteristic of the DP-based mixer. Unfortunately, the absence of ð3Þ the RF harmonics is based on the constancy of vbe , which is not true in reality, particularly when the ð3Þ voltage vb approaches zero during its swing. Furthermore, the derivation of Equation (13.95) assumed that the collector current coincided with the emitter one in Q3 : this is rigorously true if and only if the transistor forward current gain is infinite, in the absence of parasitics, and if Q3 constantly operates within its active region.42 No real transistor satisfies any of the first two conditions, while the third one holds true only within a limited range of the RF signal amplitude. ð1Þ ð2Þ For small LO amplitudes, i.e. if vb vb =ð2nF VT Þ ¼ 1,43 then the hyperbolic tangent in the function (13.94) can be approximated by its argument, or equivalently all the higher order terms of the series (13.96) become negligible. The output differential voltage becomes ð1Þ

ð2Þ

ð1Þ

ð2Þ

ð1Þ

ð2Þ

1 vb vb 1 v vb Rc iR cosðoR t þ jR Þ b 2 nF VT 2 nF VT 1 Rc iR0 1 1 Rc ¼ vL cosðoL t þ jL Þ vR cosðoR t þ jR ÞvL cosðoL t þ jL Þ 2 nF VT 2 nF VT Re

vc vc ffi Rc iR0

ð13:97Þ

The second term in the IF voltage (13.97) is the product of the RF and LO voltages, multiplied by the constant Rc =ð2Re nF VT Þ. Therefore, for small LO amplitudes, the mixer in Figure 13.21a behaves approximately as a product detector, with the addition of the LO feedthrough given by the first term of the function (13.58). As the considerations developed in Section 13.3.1 showed, the conversion gain of the mixer in Figure 13.21a, for small LO amplitudes, is proportional to the LO amplitude and equal to 1 vL Rc ðsmall LOÞ ð13:98Þ CGdB ¼ 20 log10 4 nF VT Re with vL =ð2nF VT Þ ¼ 1, by hypothesis. As the LO amplitude increases the multiplying factor of the AC component of the RF voltage, n o ð1Þ ð2Þ tan h vb vb =ð2nF VT Þ ¼ tan h½vL cosðoL t þ fL Þ=ð2nF VT Þ is a periodic non-sinusoidal function, tending to a square wave for high values of vL , as Figure 13.22 shows. The curves of that figure give us the opportunity to define three ranges for the LO amplitude: 1. Small LO, if vL is such that the hyperbolic tangent is approximately equal to its argument, as done to derive Equations (13.97) and (13.98) 1 vL 1 vL tan h cosðoL t þ fL Þ ffi cosðoL t þ fL Þ 2 nF VT 2 nF VT 2. Large LO, if the above function is close to a square wave and 1 vL cosðoL t þ fL Þ ffi swLO ðtÞ tan h 2 nF VT 3. Moderate LO, which is the intermediate case between 1 and 2, although its definition is somewhat arbitrary. 42

That is, the base–emitter junction is forward biased while the collector–base junction is reverse biased. Which is equivalent to assuming that vL ¼ 2nF VT with 2VT ffi 51:5 mV at a room temperature of 27 ˚C and 1 nF 2; see also Section 9.7.2.1. 43

610

MICROWAVE AND RF ENGINEERING tanh[vLcos(ωLOt + φL)/(2 nFVT)]

tanh[(Vb1-Vb2)/(2 nFVT)]

∆Iout

0

SWLO(t)

0 Vb1-Vb2

1 (ωLOt)/(2 π)

1 (ωLOt)/(2 π)

0 Vb1-Vb2= vLcos(ωLOt + φL)

Figure 13.22 RF multiplying factor for different LO amplitudes: lower left, LO voltages with increasing amplitude; upper left, hyperbolic tangent transfer characteristic; upper right waves multiplying the RF sinusoid, with the asymptotic case of the square wave (grey curve). It is possible to arrive at the same conclusion on the large LO operation of our circuit by adopting a simpler model, assuming that Q1 and Q2 work as switches.44 For large vL we can assume that the two transistors work in the active region for a negligible fraction of the LO cycle, when vL cosðoL t þ fL Þ is close to zero. ð1Þ

ð2Þ

ð2Þ

ð1Þ

ð1Þ

ð2Þ

Thus, the voltage vce (vce ) and the current ic (ic ) are negligible when vb vb is positive (negative). Hence, we can write the asymptotic approximate expression of the output voltage as ( ð3Þ ð1Þ ð2Þ vb vb > 0 Rc ic ð1Þ ð2Þ vc vc ¼ ð13:99Þ ¼ ½iR cosðoR t þ fR Þ þ iR0 swLO ðtÞ ð3Þ ð1Þ ð2Þ Rc ic vb vb < 0 However, since the square wave swLO contains all and only the odd harmonics45 of the LO voltage, it follows that the IF voltage (13.99) has the same form as (13.95) but with different values for the coefficients a2kþ1 . Consequently, the mixer in Figure 13.21a does not change the suppressed products when it passes from moderate to strong LO, differently from the DBM of Section 13.3.4. For the conversion gain, the amplitude of the square-wave first harmonic is 4=p. The conversion gain of the mixer in Figure 13.21a, in the switching asymptotic approximation, is totally independent of the LO amplitude and equals 2 Rc ðlarge LOÞ CGdB ð13:100Þ ¼ 20 log10 p Re 44 45

See also [5]. As in expression (13.62).


611

It is reasonable to assume that, for small vL , the conversion gain increases proportionally to vL itself, according to the formula (13.98), and tends asymptotically to the value (13.100).

13.3.9.2 Doubly balanced differential pair mixer Figure 13.21b shows two mixers of the type in Figure 13.21a, combined together in order to obtain a doubly balanced mixer. Applying the same procedure used to derive Equation (11.101), and under the same hypotheses, the collector currents of the six transistors are 8 ð3Þ > ic > ð1Þ > " # > ¼ i c > ð2Þ ð1Þ > > vb vb > > þ1 exp > > > nF VT > > > > > ð3Þ > > i > ð2Þ > " ð1Þ c ð2Þ # ic ¼ > > > > v vb > > þ1 exp b > > nF VT > > > > > > ð6Þ > > i > < icð4Þ ¼ " ð1Þ c ð2Þ # v vb > þ1 exp b > > nF VT > > > > > ð6Þ > > i > > " ð2Þ c ð1Þ # > icð5Þ ¼ > > > v vb > > þ1 exp b > > > nF VT > > > > i > > 1 h vR iR ð3Þ ð3Þ > > vR0 þ cosðoR t þ fR Þ vbe ¼ iR0 þ cosðoR t þ fR Þ > ic ¼ > R 2 2 > e > > > h i > ð6Þ 1 v i > R R ð6Þ > vR0 cosðoR t þ fR Þ vbe ¼ iR0 cosðoR t þ fR Þ : ic ¼ Re 2 2

ð13:101Þ

The last two equations of the system (13.101) imply that Q3 and Q6 work exactly as Q3 in the circuit of Figure 13.21a. Furthermore, the RF voltage is applied differentially between the bases of Q3 and Q6 . After substituting the last two equations of the system (13.101) into the remaining four, the resulting system has four equations in four unknowns Ic1 to Ic4. Extracting from that reduced system the differential output current, we get " ð1Þ ð2Þ # h i v vb ð1Þ ð4Þ ð2Þ ð5Þ ð13:102Þ ic þ ic ic ic ¼ tanh b iR cosðoR t þ fR Þ nF VT The IF output voltage of the mixer in Figure 13.21b is then " ð1Þ ð2Þ # n io h v vb ð1Þ ð2Þ ð1Þ ð4Þ ð2Þ ð5Þ vc vc ¼ Rc ic þ ic ic ic iR cosðoR t þ fR Þ ¼ tanh b nF VT nh Again, the factor tanh

ð1Þ

ð2Þ

vb vb

ð13:103Þ

i

o =ð2nF VT Þ ¼ tanh½vL cosðoLO t þ fL Þ=ð2nF VT Þ contains all the odd

harmonics of that voltage, as seen in the circuit in Figure 13.21a. Hence, the mixer in Figure 13.21b only produces mixing products having unitary RF order (m ¼ 1) and even LO order. This performance, which is better than the diode DBM discussed in Section 13.3.4, justifies well the denomination of doubly

612


balanced, for the mixer in Figure 13.21b. The good suppression properties of that mixer and its good ð3Þ

ð6Þ

ð1Þ

ð2Þ

vb vb to vc vc linearity depend, however, on the same simplifying assumptions as made for the circuit in Figure 13.21a. That is: (a) The circuit presents perfect balance ðQ1 ¼ Q2 ¼ Q4 ¼ Q5

and

Q3 ¼ Q6 Þ.

(b) The sinusoidal voltages on the basis of Q3 and Q4 have the same amplitude and are 180 out of phase. ð3Þ

ð6Þ

(c) vbe ; vbe are perfectly constant. (d) The transistors operate within their active region and have no parasitics. Therefore, real circuits present some RF harmonics and some LO even harmonics, albeit attenuated ones, as they do not satisfy these conditions. Comparing Equations (13.104) and (13.95), we can see that the whole output voltage of the mixer in Figure 13.21b coincides with the opposite of the second term of the one for the mixer in Figure 13.21a. Analyzing the circuit under large LO signals, we obtain an expression similar to (13.100) but with the LO feedthrough term iR0 suppressed due to the doubly balanced structure of the circuit. Consequently, the conversion gain of the mixer in Figure 13.21b with small and large LO amplitudes coincides with the values (13.99) and (13.101), respectively.

13.3.9.3 Gilbert cell Section 13.3.9.2 listed the four conditions required for the mixer in Figure 13.21b to operate as a DMB. As stated, no real circuit fulfils any of them. Nonetheless, the arrangement in Figure 13.21c – also knows as the Gilbert cell – reasonably satisfies condition (b), almost independently of the accuracy of the ð3Þ ð6Þ differential voltages vb and vb . The Gilbert cell achieves that performance by exploiting the properties ð3Þ ð6Þ of the DP consisting of Q3 , Q6 and the current generator46I0 . The collector currents ic and ic are only ð3Þ ð6Þ functions of the difference vb vb : any common-mode voltage has ideally no impact on the two collector currents.47 Therefore, if we assume that the transistors Q3 and Q6 work linearly – as in the mixer of Figure 13.21b – the Gilbert cell output voltage coincides with expression (13.104) with vR , fR related ð3Þ ð6Þ to the voltages vb and vb as follows. ð3Þ ð6Þ It is always possible to express vb and vb as combinations of even and odd mode voltages ð3Þ

ð3Þ

vb ðtÞ ¼

ð6Þ

ð3Þ

ð6Þ

vb ðtÞ þ vb ðtÞ vb ðtÞvb ðtÞ þ ; 2 2 ð3Þ

ð6Þ

vb ðtÞ ¼

ð3Þ

ð6Þ

ð3Þ

ð6Þ

vb ðtÞ þ vb ðtÞ vb ðtÞvb ðtÞ 2 2

ð6Þ

ð3Þ

ð6Þ

The common-mode input voltage ½vb ðtÞ þ vb ðtÞ=2 has an impact on neither ic nor ic , which depend on the differential voltage

ð3Þ ð6Þ vb ðtÞvb ðtÞ.

ð3Þ

Given that

vb ðtÞ ¼ v30 þ v3 cosðoR t þ j3 Þ;

ð6Þ

vb ðtÞ ¼ v60 þ v6 cosðoR t þ j6 Þ

the corresponding differential voltage is ð3Þ

ð6Þ

vb ðtÞvb ðtÞ ¼ v30 v60 þ v3 cosðoR t þ j3 Þv6 cosðoR t þ j6 Þ ¼ v30 v60 þ v3 cosðoR t þ j3 Þv6 cos½ðoR t þ j3 Þ þ ðj6 j3 Þ ¼ v30 v60 þ ½v3 v6 cosðj6 j3 ÞcosðoRF t þ j3 Þ þ v6 sinðj6 j3 ÞsinðoR t þ j3 Þ

46 One implementation of the current generator itself could be as in Figures 11.39a and 13.21a, one transistor (Q3 in those figures) having an emitter resistor (Re ) and a constant base voltage. 47 See also Section 11.6.3.


613

The term v30 v60 is a DC voltage and can be eliminated with a simple series capacitor. The remaining part of the base differential voltage is a combination of the sine and cosine of the same argument. Applying the identities (A.92) to (A.94) with v1 ðtÞ ¼ ½v3 v6 cosðj6 j3 Þ cosðoRF t þ j3 Þ; v2 ðtÞ ¼ ½v6 sinðj6 j3 Þ sinðoR t þ j3 Þ, the differential input voltage becomes ð3Þ

ð6Þ

vb ðtÞvb ðtÞ ¼ v30 v60 þ vR cosðoR t þ f36 Þ

ð13:104Þ

with vR ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v23 þ v26 2cosðj6 j3 Þv3 v6 ;

jR ¼ tan1

sinðj6 j3 Þv6 v3 cosðj6 j3 Þv6

Equation (13.104) states that any amplitude ðv3 6¼ v6 Þ or phase ðf6 f3 6¼ pÞ imbalance between the two base voltages affects the differential amplitude RF vR, which determines the final conversion gain. Nevertheless, the balanced properties of the mixer in Figure 13.21c do not depend on the differential accuracy of the RF voltage. The Gilbert cell mixer also works if the emitter resistors Re are short-circuited. In this case it exhibits lower RF linearity but higher conversion gain. Applying the procedure used in Section 11.6.3 to all three differential pairs ðQ1 ; Q2 ; Q3 Þ, ðQ4 ; Q5 ; Q6 Þ and ðQ5 ; Q6 ; I0 Þ, we obtain the collector currents of the six transistors 8 ð3Þ > i ð1Þ > > " ð2Þ c ð1Þ # > ic ¼ > > v vb > > > þ1 exp b > > nF VT > > > > > > > ð3Þ > > ic > ð2Þ > " # > ¼ i c > ð1Þ ð2Þ > > vb vb > > þ1 exp > > > nF VT > > > > > > > ð6Þ > > ic ð4Þ > > " # i ¼ > c > ð1Þ ð2Þ > vb vb > > > þ1 exp > < nF VT > > ð6Þ > > i > > " ð2Þ c ð1Þ # > icð5Þ ¼ > > > v vb > > þ1 exp b > > > nF VT > > > > > > > I > ð3Þ > > " ð6Þ 0 ð3Þ # ic ¼ > > > v vb > > þ1 exp b > > > nF VT > > > > > > > > I ð6Þ > > " ð3Þ 0 ð6Þ # ic ¼ > > > v v > b > > exp b þ1 : nF VT

ð13:105Þ

614


Eliminating the last two equations from the system (13.105), we obtain 8 I ð1Þ > > )0( " ð6Þ ð3Þ # ) ic ¼ ( " ð2Þ ð1Þ # > > > vb vb vb vb > > > þ1 exp þ1 exp > > nF VT nF VT > > > > > > > > I ð2Þ > > )0( " ð6Þ ð3Þ # ) ic ¼ ( " ð1Þ ð2Þ # > > > v v vb vb > b b > > þ1 exp þ1 exp > > nF VT nF VT < > I > ð4Þ > )0( " ð3Þ ð6Þ # ) > ic ¼ ( " ð1Þ ð2Þ # > > > vb vb vb vb > > þ1 exp þ1 exp > > > nF VT nF VT > > > > > > > I > ð5Þ > )0( " ð3Þ ð6Þ # ) > ic ¼ ( " ð2Þ ð1Þ # > > > vb vb vb vb > > þ1 exp þ1 exp > : nF VT nF VT

which can be rearranged as 8 " ð1Þ ð2Þ # " ð3Þ ð6Þ # > v vb v vb > > exp b exp b > > > 2n V 2nF VT F T > > > icð1Þ ¼ ( " # " ð1Þ ð2Þ #)( " ð3Þ ð6Þ # " ð3Þ ð6Þ #) I0 > ð1Þ ð2Þ > > vb vb vb vb vb vb v vb > > > þ exp exp þ exp b exp > > 2n V 2n V 2n V 2nF VT > F T F T F T > > > > > > > " ð1Þ ð2Þ # " ð3Þ ð6Þ # > > > > v vb v vb > > exp b exp b > > 2n V 2nF VT > F T > ð2Þ > " ð1Þ ð2Þ #)( " ð3Þ ð6Þ # " ð3Þ ð6Þ #) I0 > > ic ¼ ( " ð1Þ ð2Þ # > > vb vb vb vb vb vb v vb > > þ exp exp þ exp b exp > > > 2n V 2n V 2n V 2nF VT F T F T F T > < > " ð1Þ ð2Þ # " ð3Þ ð6Þ # > > > v vb v vb > > exp b exp b > > > 2n V 2nF VT F T > > > icð4Þ ¼ ( " # " ð1Þ ð2Þ #)( " ð3Þ ð6Þ # " ð3Þ ð6Þ #) I0 > ð 1 Þ ð 2 Þ > > vb vb vb vb vb vb v vb > > > exp þ exp exp þ exp b > > 2n V 2n V 2n V 2nF VT > F T F T F T > > > > > > > " ð1Þ ð2Þ # " ð3Þ ð6Þ # > > > > v vb v vb > > exp b exp b > > 2n V 2nF VT > F T > ð5Þ > " ð1Þ ð2Þ #)( " ð3Þ ð6Þ # " ð3Þ ð6Þ #) I0 > > ic ¼ ( " ð1Þ ð2Þ # > > vb vb vb vb vb vb v vb > > þ exp exp þ exp b exp > : 2nF VT 2nF VT 2nF VT 2nF VT


615

The differential output current is thus " ð1Þ ð2Þ # " ð1Þ ð2Þ # " ð3Þ ð6Þ # " ð3Þ ð6Þ # vb vb v vb v vb v vb exp b exp b exp b 2n V 2n V 2n V 2nF VT F T F T F T ð1Þ ð4Þ ð2Þ ð5Þ " ð1Þ ð2Þ # " ð1Þ ð2Þ # " ð3Þ ð6Þ # " ð3Þ ð6Þ # I0 ic þ ic ½ic þ ic ¼ v vb v vb v vb v vb þ exp b þ exp b exp b exp b 2nF VT 2nF VT 2nF VT 2nF VT exp

" ð1Þ ð2Þ # " ð3Þ ð6Þ # vb vb v vb tanh b I0 ¼ tanh 2nF VT 2nF VT

The differential output voltage of the Gilbert cell without emitter resistors is

ð2Þ vð1Þ c vc

¼ Rc

n

icð1Þ

þ icð4Þ

h

icð2Þ

þ icð5Þ

io

" ð1Þ ð2Þ # " ð3Þ ð6Þ # 1 vb vb 1 vb vb ¼ Rc I0 tanh tanh 2 nF VT 2 nF VT

ð13:106Þ

The mixer described by Equation (13.106) generates all the mixing products with odd RF and LO orders, in that the hyperbolic tangent is an odd function, as remarked many times in Sections 11.6.3 and 13.3.9.1 to 13.3.9.3. The Gilbert cell with Re ¼ 0 works similarly to the diode DBM of Section 13.3.4. Moreover, the RF and LO ports are interchangeable, due to the symmetry of Equation (13.106). Observations: (a) If

h i ð1Þ ð2Þ vb vb =ð2nF VT Þ and

h i ð3Þ ð6Þ vb vb =ð2nF VT Þ

are both much smaller than 1, then the two hyperbolic tangents in the function (13.106) approximate their arguments and the function itself simplifies to h i h i ð2Þ ð3Þ ð6Þ 2 ð1Þ ð2Þ vð1Þ vb vb vb vb c vc ffi Rc I0 ð2nF VT Þ Thus the Gilbert cell behaves like an ideal product detector (or voltage multiplier) for small RF and LO. ð3Þ

ð6Þ

(b) If the two differential voltages are both sinusoidal, vb vb ¼ vR cosðoR t þ fR Þ, ð1Þ ð2Þ vb vb ¼ vL cosðoL t þ fL Þ, with small RF and LO amplitude ðvR ; vL ¼ 2nF VT Þ the voltage (13.104) simplifies to 2 ð2Þ vð1Þ c vc ffi ð2nF VT Þ Rc I0 vR vL cosðoRF t þ fR ÞcosðoLO t þ fL Þ

The corresponding conversion gain is ðsmall RF and LOÞ CGdB

ð1Þ

" ¼ 20 log10

Rc I0 vL 8ðnF VT Þ2

# ð13:107Þ

ð2Þ

(c) For large LO, tanhf½vb vb =ð2nF VT Þg ¼ tanh½vL cosðoL tþ fL Þ=ð2nF VT Þ ffi swLO ðtÞ, and small RF,

ð3Þ ð6Þ tanhf½vb vb =ð2nF VT Þg ffi vR cosðoR tþ fR Þ=ð2nF VT Þ,

the IF voltage approximates to

616

MICROWAVE AND RF ENGINEERING ð2Þ vð1Þ c vc ffi

1 X Rc I0 4 ð1Þk cos½ð2k þ 1ÞðoL tþ fL Þ vR cosðoR t þfR Þ 2nF VT p 2k þ 1 k¼0

The conversion gain of the Gilbert cell tends to ðsmall RF; large LOÞ

CGdB

¼ 20 log10

Rc I0 2 nF VT p

ð13:108Þ

(d) Equations (13.108) and (13.98) describe similar circuits under the same operating conditions. They should coincide when the two circuits coincide, i.e. when Re ¼ 0; this is not true, because the quantity (13.98) tends to infinity in that case. Such inconsistency is a consequence of the linear ð3Þ ð6Þ approximation used for Q3 and Q6 with their associated components. If Re ¼ 0 and vb vb is ð3Þ ð6Þ a sinusoid, then vb ; vb are not constant, differently from what was assumed to derive Equation (13.98). Emitter resistors decrease the base–emitter voltage, and thus the gain of the circuit.48 Thus, holding the other parameters equal, the gain (13.98) has to be smaller than the gain (13.108), otherwise Equation (13.98) is not accurate. Moreover, the difference (in dB) between the two values is a parameter that quantifies the effectiveness of the linearization operated by the emitter resistors. (e) The same considerations as in (d) also apply to the large LO gain (13.108) and (13.100). However, both formulae predict that, for large LO, the conversion gain is independent of the LO amplitude. (f) The RF linearity in the mixers in Figures 13.21a–c is a direct manifestation of the linearity ð3Þ ð3Þ ð6Þ ð3Þ ð3Þ ð6Þ between RF voltages vb or vb vb and the currents ic or ic ic . That performance can be 49 investigated with a relatively simple nonlinear circuit analysis of one single DP with two equal resistors in series with the emitters, also known as a long-tail pair. Figure 13.23a plots the differential output versus the differential input for three different emitter resistances, including zero. The graph shows that increasing resistance gives reduced – but more constant – slope. Figure 13.23b confirms and better quantifies this assertion. It plots the derivative of the output with respect to the input voltage – which is the incremental gain – versus the output voltage. A constant difference on the logarithmic y axis indicates a constant relative variation (or a constant difference in dB) of the incremental gain. It can be clearly seen that higher emitter resistance gives lower gain, but less dependence on the output voltage. (g) The mixers in Figure 13.21 operate with balanced voltage. This would not be an issue if the mixer were part of an RFIC, where the amplifiers that interface with the component are most likely DP based and thus differential as well. However, if the application requires a single-ended IF output, a balanced-to-unbalanced adapter is required. It could be a passive device, such as a 180 hybrid, or a balun,50 or an active circuit like the one in Figure 13.24a. It is a normal DP with one of the two output terminals not connected. The transfer characteristic from the differential input to the single-ended output is given by one of the two equations (11.100): the curve shape resembles the hyperbolic tangent. Unfortunately, the characteristic (11.100) is not anti-symmetrical, therefore the circuit in Figure 13.24a does not suppress the even harmonics, differently from the differential-in, differential-out DP. Conversely, if the RF and/or LO source is single ended,

48 The series emitter is a negative feedback configuration, which is the dual of the one considered in Section 11.6.1. As with all negative feedback, it improves the linearity at the expense of the reduced gain. 49 See the SIMetrix file 09_MBC13900_Differential_Pair_Linearity.sxsch, which analyzes a DP employing three MBC13900 transistors (described in Section 9.7) with a total current of Ic3 ffi 2 mA and with different values of the emitter resistance. 50 See Section 13.3.4 for a brief description.

FREQUENCY CONVERTERS (a)

617

Vc1-Vc2

0 Vb1-Vb2 Re =0

(b)

40

80 Ω

100 -d(V -V )/d(V -V ) c1 c2 b1 b2 Re=0

10

40 Ω 80 Ω

1

0

Vc1-Vc2

Figure 13.23 DP with emitter series resistors (long-tail pair): (a) transfer characteristic from the differential input to output voltage; (b) small-signal voltage gain versus the output voltage (logarithmic y axis). then a balun adapter is required. Again, passive 180 hybrids are usable, while Figure 13.24b shows an active, DP-based, realization. The AC voltage on the base of Q2 is constant, due to the capacitor Cin which offers a low RF impedance path to ground. Then, the circuit responds to the differential51 component involved with any input voltage applied on the base of Q1 and transforms it into a balanced output voltage, as required. The circuits in Figure 13.24 can be easily integrated with the mixer on the same chip: this is their main advantage over their passive counterparts. On the other hand, the active balun consumes DC power and has a higher distortion and sometimes more noise than passive ones.

13.3.10

Mixer nonlinearities

This section gives further details on the mixer, considered as a black box. Real mixer working deviates from the ideal model of a product detector, as Sections 13.3.2 to 13.3.9 described. Here we will complete the discussion of non-ideal mixer behaviour and present some further implications of the concepts presented in Sections 13.3.2 to 13.3.9.

13.3.10.1 Nonlinear performances with single tone RF The expressions derived for the various mixer types show that the IF output voltage is the sum of a number – ideally infinite – of spectral components having angular frequency moR þ noL. For the sake of clarity, it is useful to recall that oR ; oL and m; n are the RF and LO angular frequency and index, respectively. Moreover m; n are positive or negative integers, including zero. In our subsequent considerations, the spur at frequency moR þ noL is briefly indicated with its two indexes (m, n) in brackets: for example, the spur (2, 3) is the mixing product at the frequency 2oR þ 3oL. The considerations used to derive 51

See the treatment at the beginning of the present section.

618


Vcc Rc

Rc Cout Vout

Q1 Q2

+ Vin -

I0 (a)

Vcc Rc

Vin

Rc + Vout -

Q1 Q2

Cin I0

(b)

Figure 13.24 Active DP-based balun signal converters: (a) balanced input to single-ended output; (b) single-ended input to balanced output. Equations (13.40) to (13.43) also show that the spur (m, n) results from all the products between the factors jmj

jm j þ 2

v0 RF ; v0 RF

jmj þ 4

; v0 RF

jnj

jnj þ 2

. . . and v0 LO ; v0 LO

jn j þ 4

; v0 LO

. . . where v0 RF ; v0 LO are related to the RF and LO voltages jmj

jnj

as in Equations (13.38). The lowest order factors that produce the spur (m, n) are v0 RF ; v0 LO ; thus for small jmj jnj v0 R v0 L .

vRF ; vLO , the amplitude of the spur (m, n) is proportional to In particular, the main conversion product ðjmj ¼ jnj ¼ 1Þ is obtained from the combinations of infinite integer powers of the RF signal, therefore its amplitude is proportional to the RF one only for small values of the latter. Sections 13.3.4 to 13.3.9 showed that balanced mixers suppress some of the mixing products, although perfect balancing is impossible, and so it is for total suppression.52 The most important consequence of such working is that, for a given combination of RF and LO sinusoidal tones, a wide range of frequency is potentially present at IF. Suitable filters attenuate some unwanted spectral lines, but this could be impossible if some of the spurs are close to the desired IF. If m, n or both are sufficiently high, then the corresponding nonlinear coefficient is also small, and the spur is naturally attenuated. Conversely, for low harmonic order (say jmj; jnj 4) the spur level cannot be negligible. Table 13.1 is a useful tool for mixer spur analysis. It lists the attenuation in dB of all the spurs with given RF, LO 52

See Section 13.3.8.


619

Table 13.1 Typical spurious table for a diode DBM. n!m #

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

–– 13 50 60 65 –– –– –– –– –– ––

11 0 59 46 70 70 –– –– –– –– ––

11 42 50 70 70 70 70 –– –– –– ––

22 22 60 42 70 70 70 70 –– –– ––

47 42 58 63 63 70 70 70 70 –– ––

–– 43 48 54 70 68 70 70 70 70 ––

–– –– 52 63 70 70 70 70 70 70 70

–– –– –– 62 70 70 70 70 70 70 70

–– –– –– –– 60 70 70 70 70 70 70

–– –– –– –– –– 70 70 70 70 70 70

–– –– –– –– –– –– 70 70 70 70 70

maximum order (in our case 0 m; n 10), with respect to the main conversion product. These specific values are just indicative, they do not refer to any real measured or simulated mixer. Tables like this one have to be calculated and/or measured in the given mixer, for each specific RF, LO frequency and power. The resulting numbers include the effects of any filtering and/or matching network used within the mixer. Hence, more accurate indications should present different attenuation values for positive and negative indexes. Spurs always affect any mixer frequency conversion. If their frequency is close to the IF bandwidth, its elimination by filtering becomes cumbersome. For the same reason, RF disturbances having a frequency close to oR could be converted into IF bandwidth as spurs. An example would be useful to clarify the problem. Let us consider the receiver schematic of Figure 13.16 with the following parameters: .

fR ¼ oR =ð2pÞ ¼ 10 GHz

.

fL ¼ oL =ð2pÞ ¼ 11 GHz

.

RF passband 9.2–0.7 GHz

.

IF passband 300–1800 MHz.

Then the frequency product (1, 1) is fR þ fL ¼ 21 GHz; it falls far from the IF bandwidth, and can easily be filtered out. The difference frequency fR fL ¼ 1 GHz is close to the IF centre frequency, and is received. Unfortunately, the product (2, 2) has frequency 2 10 þ 2 11 ¼ 2 GHz, which is close to the IF lowdpass corner and difficult to filter out. Moreover, an interfering RF disturbance located at 10.8 GHz would generate a difference frequency of 200 MHz, which is out of the IF band, but also the (2, 2) spur has frequency 2 10:8 þ 2 11 ¼ 0:4 GHz, which falls inside the IF band and is not filterable. Figure 13.25 shows the simulated53 output spectrum of the SBM mixer in Figure 13.10. The most significant mixer parameters are fR ¼ 10 GHz, fL ¼ 11 GHz, external LO power 18 dBm, directional coupler with 6 dB of coupling factor, for a resulting effective LO power to the diode of 12 dBm, and the RF power is from 6 dBm to þ14 dBm in 10 dB steps. Figure 13.25 also shows the transmission

53 See the SIMetrix file 10_Single_Ended_Mixer.sxsch. The diode parameters are Is ¼ 5 nA; N ¼ 1:1; Rs ¼ 10 O and Cj0 ¼ 1 pF; Vj ¼ 0:5 V; g ¼ 0:5.

620


IF power, dBm

-20 IF filter, 20log10(|s21|)

-40 RF power -6 dBm +4 dBm +14 dBm

-60

-80 0

1

2

3

4

5

IF frequency, GHz

Figure 13.25 power levels.

IF spectrum of the SEM in Figure 13.10, with sinusoidal RF excitation, for three RF

coefficient amplitude of the combination of Z2 and NT2 . It is a frequency-selective IF matching network,54 synthetically labelled as ‘IF filter’. Note that the 2oR 2oL (at 2 GHz) spur amplitude increases by 20 dB when the RF power increases by 10 dB, passing from 6 to þ4 dBm, while this relation does not hold for the RF power increasing by a further 10 dB. The considerations at the beginning of the present section offer a simple explanation for such behaviour. For small v0 R ; v0 L , the mixing products with jmj ¼ jnj ¼ 1ðjmj ¼ jnj ¼ 2Þ have amplitude proportional to v0 R v0 L ðv0 2R v0 2L Þ. Hence the ratio between the two above-mentioned amplitudes is proportional to v0 R v0 L as well. Consequently, the difference between the levels of the main IF product and the spur (2, 2) increases by 1 dB as the RF power decreases by the same amount. At higher RF power, the higher order proportional terms become significant, and the amplitude of the mixing product jmj jnj (m, n) is no longer proportional to v0 R v0 L as Figure 13.25 shows. Moreover, at high RF levels, the main IF power is also no longer proportional to the RF one; in other words, the mixer does not work linearly.55 The visible mixing products in the specific case of Figure 13.25 have equal RF and LO index, m ¼ n ¼ 1; 2; 3; 4. This is typical for IF passband widths greater than one octave. Increasing order spurs have decreasing level, as a combined effect of the IF filter and the natural mixing mechanism. A DBM theoretically suppresses (but does not practically) the even RF and LO harmonics, but the best design solution consists of using LO and IF frequencies that avoid the increase in non-filterable spurs. Figure 13.26 shows a graphical representation56 of the mixer spurs, which can be used in combination with the spurious table to choose the minimum spur RF and LO combination. This type of plot has the name mixer spur chart. The IF of the spur (m, n) has frequency fn;m ¼ jm fR þ n fL j

ð13:109Þ

The absolute sign in Equation (13.109) states that positive and negative IFs are totally equivalent for our considerations.

54

See the SIMetrix file 11_Single_Ended_Mixer_IF_Filter.sxsch. The mixer is inherently nonlinear; the linearity to consider in this context is the one defined in Section 13.3.1. 56 See also the Mathcad file 04_Mixer_Spur_Chart.mcd. 55


621

fI, GHz 15

(3,1) (2,1)

(1,1) (3,0)

(2,0)

(-3,1)

(-3,3)

(1,0)

(-2,1)

(-2,2)

(0,1)

10 (-1,2) (-3,2)

(-2,2) 5 B

(-4,4) (-6,6) A

1.8

(-1,1) fR, GHz

0.3 O

5

Figure 13.26

9.2

10 10.7

15

Spurious response for the mixer considered in Figure 13.25.

Now, for any given fL , Equation (13.109) is a two-segment piecewise-linear relation between the variables fR and fn;m , which is the IF, defined as mfR þ n fL > 0 m fR þ n fL ð13:110Þ fn;m ¼ mfR n fL ðmfR þ n fL Þ 0 The segments (13.110), plotted in Figure 13.26, express the IF obtained for each (m, n) spur as a function of the RF. A segment of the line (13.110) with jmj ¼ jnj ¼ 1 delimits the RF bandwidth on the x axis and the corresponding IF bandwidth on the y axis. Figure 13.26, in particular, shows the case that generates the spectrum of Figure 13.25, with fL ¼ 11 GHz. The RF bandwidth limits, together with their corresponding IF limits, determine a conversion window that is crossed by one of two lines with jmj ¼ jnj ¼ 1: all the other lines crossing the conversion window are spurs falling within the IF band. The rectangle in Figure 13.26 labelled as A is the case of the mixer with the IF spectrum plotted in Figure 13.25, having 9:2 GHz fR 10:7 GHz and

622


300 MHz fI 1800 MHz. Note that the spurs with equal RF and LO index cross the IF window up to orders jmj ¼ jnj 6, in agreement with the simulation result of Figure 13.25. By comparison, the IF window labelled B, having 13:5 GHz fR 15 GHz and 2:4 MHz fI 4 GHz, is free from spurs, despite A and B having the same bandwidth of 1500 MHz.

13.3.10.2 Nonlinear performances with dual tone RF Single tone RF excitation is useful for developing fundamental mixer calculations, which result relatively straightforwardly in that case. Real RF signals are usually more complex than pure sinusoids. However, Fourier transform theory tells us that any practical signal can be approximated by multiple – ideally infinite – sinusoidal terms, each with a suitable frequency, amplitude and phase. The simplest – although significant – case to consider is the dual tone input. Let the RF and LO signals be vRF ðtÞ ¼ vR1 cosðoR1 t þ jR1 Þ þ vR2 cosðoR2 t þ jR2 Þ;

vLO ðtÞ ¼ vL cosðoL t þ jL Þ

ð13:111Þ

The amplitude and phase factors of expressions (13.111) have no superscripts, denoting that no matching network effect has been considered, but that it is not relevant within this context. For the same reason, and for simplicity, we can also assume that fR1 ¼ fR2 ¼ fL ¼ 0. The discussion on the various mixer types shows that the IF voltage is a signal of the type vIF ðtÞ ¼

1 X

ak ½vRF ðtÞ þ vLO ðtÞk

ð13:112Þ

k¼0

Some of the terms resulting from the expansion of expression (13.112) can be attenuated – ideally cancelled – in balanced mixer configurations. Applying the identities (A.80) and (A.81), with a ¼ vR1 ; b ¼ vR2 ; c ¼ vLO and a ¼ oR1 t; b ¼ oR2 t; g ¼ oL t, to the second and third terms of the sum in (13.112) gives the IF voltage that presents spectral products at the following frequencies: .

Second order ðk ¼ 2Þ: zero, 2oR1 ;

2oR2 ;

2oL and oR1 oR2 ;

oR1 oL ;

.

Third order ðk ¼ 3Þ: oR1 ; oR2 ; oL , 3oR1 ; 3oR2 ; 3oL , oR1 2oR2 ; oR1 2oL ; 2oR1 oL , oR2 2oL ; 2oR2 oL and oR1 oR2 oL .

oR2 oL . 2oR1 oR2 ,

More generally, considering the other terms in the sum of (13.112) and deriving the equivalent of the identities (A.80) and (A.81) for a generic integer exponent, we have that the IF voltage includes all the spectral lines having frequency moR1 poR2 noL, and their amplitude for small vR1 ; vR2 ; vLO is p n proportional to vm R1 vR2 vLO . If oR1 oR2 then the mixing products with m ¼ 2; p ¼ 1; n ¼ 1 and m ¼ 1; p ¼ 2; n ¼ 1 produce IFs that are close to oR1 þ oLO oR2 þ oLO . Similarly, the combinations m ¼ 2; p ¼ 1; n ¼ 1 and m ¼ 1; p ¼ 2; n ¼ 1 produce IFs that are close to oR1 oLO oR2 oLO . Therefore, whether the IF desired product is (1, 1) or (1, 1), the mixing products m ¼ 2; p ¼ 1; jnj ¼ 1 and m ¼ 1; p ¼ 2; jnj ¼ 1 fall within the IF bandwidth if the frequencies of the two RF tones are sufficiently close. This behaviour is equivalent to the third-order intermodulation considered in Section 9.5.3 for two-port networks. Also, if a wide-band spectrum is applied to the RF port, each narrow-band portion of that spectrum behaves similarly to a sinusoidal tone: the resulting IF spectrum presents spectral regrowth, similar to Figure 11.34. Figure 13.27 plots the simulated57 output spectrum of the same mixer as in Figure 13.25 but with a dual tone RF excitation. The RF signal is the sum of two tones centred over 10 GHz and spaced by 100 MHz, i.e. 9.95 and 10.05 GHz, respectively. The power of each RF tone is 6, 4, 14 dBm.

57

See the SIMetrix file 12_Single_Ended_Mixer_Dual_Tone_RF.sxsch.


623

0

IF power, dBm

-20 IF filter, 20log10(|s21|)

-40 RF power (each tone) -6 dBm +4 dBm +14 dBm

-60

-80 0

1

2

3

4

5

IF frequency, GHz

Figure 13.27 IF spectrum of the SEM in Figure 13.10, with a dual tone with equal-power RF excitation, for three values of the individual tone power.

The same figure also plots the response of the IF filter, like Figure 13.25. Figure 13.27 shows eight spectral lines placed around 1 GHz, having the following frequencies: .

1110:05 ¼ 0:95 GHz and 119:95 ¼ 1:05 GHz, the main conversion products.

.

11ð2 10:05 9:95Þ ¼ 0:85 GHz and 11 ð10:05 2 9:95Þ ¼ 1:15 GHz, the third-order intermodulation products. By increasing the RF power by 10 dB, from 6 dBm to þ 4 dBm, the third-order intermodulation levels increase by about 30 dB, in agreement with the theory developed in Section 9.5.

.

11ð3 10:052 9:95Þ ¼ 0:75 GHz and 11 ð2 10:05 3 9:95Þ ¼ 1:25 GHz, the fifthorder intermodulation products. They are only visible at the maximum RF power of 14 dBm.

.

No higher order intermodulation product is visible.

This group of eight lines is replicated with some attenuation around 2 GHz, which is the difference between twice the LO frequency and the centre RF. The same thing happens around 3 and 4 GHz, with an attenuation increasing with the LO multiplying factor. This is a combination of the structure of the nonlinear coefficients (13.41), which decays with their order, and of the filtering action of the IF passive network. The group of spectral lines close to DC is clearly due to the even-order terms (13.41). An alternative way to investigate the nonlinear behaviour of the mixer consists of using a single tone RF with swept amplitude. Figure 13.28 plots the IF power versus the RF power: it exhibits the classical compression shape, similar to those in Figures 9.24, 9.27 and 11.16. The RF and IF power at 1 dB compression is I1dB ¼ 9:45 dBm; O1dB ¼ 1:65 dBm, respectively. The conversion loss of the mixer at 1 dB of compression is then 9:45 þ 1:65 ¼ 11:1 dB, with a corresponding small-signal conversion loss of 9:45 þ 1:651 ¼ 10:1 dB. Note also that I1dB is about 2.55 dB lower than the LO power incident on the diode. Considering the direct attenuation of the directional coupler, which is 10log10 ð1106=10 Þ ffi 1:26 dB, the RF power incident on the diode at 1 dB of compression is about 3.8 dB lower than the LO one.

624


IP3

IF power, dBm

single tone power

1 dB CP

0

-20 rd

3 -order IMD

-40

-60

-80 -70

-60

-50

-40

-30

-20

-10

0

10

20

30

RF power, dBm

Figure 13.28

Simulated compression curve of the SEM in Figure 13.10.

Figure 13.28 also includes the third-order intermodulation products, together with their lowlevel extrapolation line. The input and output third-order intercept points are IIP3 ¼ 20:9 dBm; OIP3 ¼ 10:94 dBm, the difference IIP3I1dB ¼ 20:99:45 ¼ 11:45 dB, in reasonable agreement with Equation (9.56), developed for an ideal cubic model.

13.3.10.3 Mixer noise Mixers add not only nonlinear distortion to the converted signal, but also noise. The noise generated by mixers is a complex topic and its treatment requires many pages. Here we will limit ourselves to mentioning its existence, and introduce a few fundamental concepts.58 It is possible to define a mixer noise figure by extending the concepts expressed in Section 9.4.3 to the translated output frequency. Therefore the mixer noise factor is the ratio of the signal to noise ratios at the RF input and at the IF output, assuming that the input noise is the thermal one and the RF bandwidth coincides with the IF one: Fmixer ¼

ðS=NÞRF ðS=NÞIF

ð13:113Þ

The corresponding mixer noise figure, expressed in dB, is NFmixer ¼ 10log10 ðFmixer Þ

ð13:114Þ

The parameters (13.113) and (13.114) can be used like those of a linear two-port network, just by considering that the mixer input and output frequency are different. The output noise power density of any network port cannot be lower than the thermal noise floor at the temperature of the network itself. Mixer IF ports are no exception, therefore, for passive mixers, the noise figure is not less than the conversion loss. The noise figure of diode mixers is typically 1 dB higher than the conversion loss. Transistor mixers can have negative conversion loss (i.e. positive gain), but their noise figure is always positive, and its value is in the range of some decibels.

58

The interested reader can consult [6, 7] for further details.


625

13.4 Frequency multipliers Frequency multipliers are nonlinear two-port networks that produce an output signal having a specific multiple of the input frequency. If the input signal is vIN ðtÞ ¼ vR cosðoR t þ fR Þ

ð13:115Þ

vOUT ðtÞ ¼ vO cos½nðoR t þ fR Þ þ y

ð13:116Þ

the corresponding output signal is

where the integer n is the frequency multiplication factor, and y is an additional phase shift, which depends on the input frequency – and sometimes also on the power – but not on time. The ratio between the output and the input amplitude, usually expressed in dB, is the conversion gain of the multiplier, with a similar definition as for mixers. In the specific and important cases of n ¼ 2 and n ¼ 3, the frequency multiplier is more precisely referred to as a frequency doubler and frequency tripler, respectively. Note that the output signal (13.116) multiplies the frequency and any additional phase of the input signal by the same factor. This means that if the input signal is affected by phase noise, the output signal presents the same phase noise spectrum, but with a level increased by the multiplication factor, or 20log10 ðnÞdB. Frequency multipliers are used to generate high frequencies when the frequency to be generated is too high to be done directly with a fundamental frequency oscillator. Section 12.8.4 also proposed a solution for the same problem, based on the combination of two synchronized oscillators. Other applications for frequency multipliers arise for generating multiple harmonic-related and frequencycoherent signals. The present section describes the basic working principle of some of the most common realizations of frequency multipliers.59 The description of a real circuit, at the end of the section, completes the treatment. Figure 13.29a shows the simplest frequency multiplier, basically consisting of one diode (D1 ) placed between two linear matching networks (NT1 , NT2 ); port 1 (2) is the input (output). That network has the same structure as the SEM in Figure 13.10, with some simplifications: (i) the LO signal is not applied, thus the directional coupler has been eliminated; (ii) the bipoles Z1 , Z2 are not explicitly indicated. However, despite (ii), NT1 matches the input to the diode conjugate impedance at the fundamental frequency, while presenting suitable reactive impedance at the desired harmonic. Similarly, NT2 matches the diode into 50 O at the desired harmonic, and presents suitable reactive impedances to the diode cathode at all the other frequencies. Typically, NT1 (NT2 ) presents a short circuit to the diode at all the harmonic frequencies except the fundamental (desired output harmonic). With these assumptions, the equations of the circuit in Figure 13.29a coincide with those of the SEM in Figure 13.10 without the LO contribution. Thus, if the input voltage has the expression (13.115), then the voltage on the anode of D1 is v0RF ðtÞ ¼ v0R cosðoR t þ j0R Þ v0R =vRF

and difference where the constant ratio matching network and the diode impedance. The diode current is id ðtÞ ¼

1 X

j0R jR

ð13:117Þ

depend on the combined effects of the input

1 X Is Is 0k v ðtÞ ¼ v0k cosk ðoR t þ j0R Þ RF k kV kVk R k!N k!N T T k¼1 k¼1

ð13:118Þ

Now, the generic cosine power cosk ðoR t þ j0R Þ, if k is even (odd), includes all the even (odd) harmonics of the input frequency, from the kth down to zero (one), as in Equations (A.82) to (A.85). Therefore, the kth harmonic results from the combined effects of all the diode nonlinearities with order 59

See [8–10] for more details.

626

MICROWAVE AND RF ENGINEERING NT1 1

IN

NT2

D1

2

1

2

OUT

(a)

NT1A

HYB1 0º

IN

1

2

1

2

vA1

NT2

D1 1

2

OUT

0º 180º 0º

R0

vA2

D2

B

NT1 (b)

NT1 IN

1

NT2

2

1 D1

2

OUT

D2

(c)

+ NT1 IN

1

2

Vdd

Ldd

Cdd

Cgg

NT2 1

2

OUT

Q1 Lgg

(d)

Vgg +

Figure 13.29 Frequency multipliers: (a) with one single diode; (b) with two diodes for even-harmonic generation; (c) with two diodes for odd-harmonic generation; (d) with one single FET. k; k þ 2; k þ 4 . . ., which produce sinusoids with amplitude proportional to v0 kR ; v0 Rk þ 2 ; v0 kR þ 4 . . ., respectively. Thus, the kth harmonic of the diode current is proportional to vkR for small vR , then it changes by laws that change from case to case, but usually tends to an asymptotic limit. Finally, NT2 shorts to ground all the spectral components of the diode current but the one having the desired multiplied outputfrequency. Notethatthecircuit inFigure13.29aispotentiallyabletogenerateallthe harmonics of any order, albeit with decreasing efficiency. The capability to generate multiple harmonics could be a disadvantage in some cases. If the desired harmonic has order k, the circuit also generates all the other ones, including the fundamental. The k1 and k þ 1 harmonics are closest to the desired one, and thus more difficult to filter out. One additional specification of the multipliers is the undesired harmonic rejection. It could be difficult to achieve for high multiplication factors, in that the relative frequency distance between the k and the k1, k þ 1 harmonics decreases with increasing k. The circuits in Figures 13.29b,c alleviate this problem, since in principle they generate only even and odd harmonics, respectively.

FREQUENCY CONVERTERS VA1

VA2

Id1

Id2

627

0

0.25

0.75

1.25

1.75

t/T

Figure 13.30

Waveforms of the circuit in Figure 106.d: top, voltages; bottom, currents.

The circuit in Figure 13.29b is similar to the SBM in Figure 13.11a, with the LO removed, the bipoles ðAÞ ðBÞ Z1 , Z1 and Z2 not explicitly indicated and D2 with inverted polarity. If the input of the circuit in Figure 13.29b is the voltage (13.117) then the voltages on the anodes of the two diodes are v0 vA1 ðtÞ ¼ pRffiffiffi cosðoR t þ j0R Þ; 2

v0 vA2 ðtÞ ¼ pRffiffiffi cosðoR t þ j0R Þ 2

ð13:119Þ

Since vA2 ðtÞ ¼ vA1 ðtÞ, we can set vA2 ðtÞ ¼ vA1 ðtÞ ¼ vA ðtÞ; then the sum of the two diode currents that flow through port 1 of the output matching network is 1 X

Is id1 ðtÞ þ id2 ðtÞ ¼ kVk k!N T k¼0

( ) X 1 vA ðtÞ k vA ðtÞ k Is 1 þ ð1Þk k pffiffiffi þ pffiffiffi ¼ v ðtÞ pffiffiffi k!N k VTk ð 2Þk A 2 2 k¼0

ð13:120Þ

The function (13.120) has only even-order terms, therefore the circuit in Figure 13.29b generates only even harmonics, if perfectly balanced. Assuming the simplified ON/OFF model for the diode current of Section 13.2.2, an alternative way to analyze60 the circuit in Figure 13.29b is possible. Figure 13.30 shows the anode voltages of the two diodes, together with the resulting currents. The diode D1 (D2 ) conducts only when vA1 (vA2 ) is positive, and the resulting current towards port 2 of NT2 is a double half-wave rectified sinusoid 1 v0 id1 ðtÞ þ id2 ðtÞ ¼ pffiffiffi R cosðoR t þ j0R Þ 2 Rs

ð13:121Þ

Rs being the forward resistance of the two diodes. The waveform (13.121) can be expanded into the Fourier series (13.122), which includes only even harmonics, like expression (13.120) ( ) 1 X pffiffiffi v0R ð1Þk 0 ð13:122Þ 1þ2 cos 2kðoR t þ jR Þ id1 ðtÞ þ id2 ðtÞ ¼ p 2 2 Rs k¼1 1ð2kÞ Inverting the polarity of one of the two diodes, the circuit in Figure 13.29b generates only odd harmonics. 60 For a complete nonlinear analysis of the circuit – including some diode parasitics – see the SIMetrix file 14_Schottky_Frequency_Doubler.sxsch.

628


Figure 13.29c shows a multiplier configuration which only generates odd harmonics; it is generally used as a tripler.61 Its structure resembles the SHM of Figure 13.15b with the usual remarks as on the circuits in Figures 13.29a,b. From Equations (13.69) and (13.123) we have that the difference between the two diode currents is id1 ðtÞ þ id2 ðtÞ ¼

1 i X Is h 2Is k k v 1ð1Þ ðtÞ ¼ v2l þ 1 ðtÞ A kVk 2l þ 1 V 2l þ 1 A k!N T T k¼0 l¼0 ð2l þ 1Þ!N

1 X

ð13:123Þ

Thus, the total current through the two diodes has only odd harmonics: a specific one of them produces the desired odd harmonic at the circuit output. The cancellation of the odd and even harmonics operated by the circuits in Figures13.29b and 13.29c, respectively, relies on the perfect balancing of the structure. The two diodes and the networks ðAÞ ðBÞ NT1 , NT1 have to be identical, while the amplitude ratio (phase difference) between the two outputs of HYB1 must be one (zero). Real structures present deviations from the perfect balancing, and therefore produce the unwanted, although attenuated, harmonics: their suppression follows the laws described in Section 13.3.7. Figure 13.29d shows an FET frequency multiplier,62 which is the counterpart of the SEM of Figure 13.20a. FET circuits are generally more suitable for frequency doublers, due to the quadratic nature of the relation between the gate–source voltage and the drain–source current. Section 13.3.8 discusses this point; all the considerations developed for the mixers apply to the multiplier as well, and will be not repeated. If the FET is assumed to be unilateral, the two matching networks can be considered separately. NT1 has simply to match the input generator to the conjugate input impedance of the FET at the fundamental frequency. NT2 can be designed to obtain either the maximum conversion gain or the maximum output power, at the nth harmonic, similar to what was discussed on power amplifiers in Section 11.5. As observed in that section on class F amplifiers, the output load impedance at all the harmonics – not just the nth – influences the multiplier’s performance. However, the unilateral approximation is accurate only for low-frequency circuits. If the FET is not unilateral, the two matching networks interact, and the circuit performances are difficult to predict from analytical considerations. In those cases – which are the majority – experimental and/or nonlinear CAE procedures are the methods of choice. Nowadays, CAE methods are usually preferred. Figure 13.31 shows the layout of an FET frequency doubler, designed to work with a 15 GHz, 0 dBm input signal. The circuit has the base configuration of Figure 13.29d, and employs the same MIC technology as the amplifier in Figure 11.42 and the VCO in Figure 12.28. The two microstrip substrates ðer ¼ 9:8; h ¼ 254 mmÞ PCB1 and PCB2 contain most of the components of NT1 and NT2 , respectively. The microstrip substrates, FET (Q1 ), and three of the four chip capacitors (C2 to C4) are brazed over a metal carrier, which works as the common ground and mechanical support as well. The carrier size is 10 6 mm. NT1 basically consists of the series capacitor C1, a five-element semi-lumped lowdpass filter and a relatively simple matching and bias network. C1 (10 pF chip device) transmits the 15 GHz input signal with negligible attenuation, while isolating the input from the DC gate voltage. The lowdpass filter includes three inductors, realized with high-impedance (narrow-width) lines and two shunt capacitors, with radial stubs (RS1 and RS2 ). Its cut-off frequency is about 16 GHz, so that it presents a reactive impedance to the gate at the output frequency (30 GHz). The rest of NT1 consists of a relatively low impedance line (TL1 ) followed by a stub, short-circuited at RF by the capacitor C3. The bottom electrode of C3 is electrically and mechanically connected to the carrier (which is the common ground plane), and the top electrode shunts the stub terminal while allowing the application of the gate DC voltage. An adjustable resistor voltage divider allows the required gate voltage – which can vary from FET to FET– to 61 See the SIMetrix file 14_Schottky_Frequency_Tripler.sxsch for a nonlinear circuit simulation of a frequency tripler with 1 GHz input frequency, based on the configuration in Figure 13.29c. 62 The interested reader can consult [9, 10] for more in-depth treatments of FET frequency multipliers.


629

carrier Vdd C4

PCB1 C1

RS2

IN

TL1 G

RS1

D Q1

C2

OUT

TL2

C3 Vgg

Figure 13.31 Layout of a microstrip frequency doubler realized following the schematic in Figure 13.25d.

be obtained from a fixed voltage generator Vgg. The capacitor C2 shunts the residual RF on the voltage divider to ground. NT2 is even simpler, and consists of one stub, short-circuited at RF by the capacitor C4 , followed by an edge-coupled bandpass filter, tuned at 30 GHz. The edge-coupled filter performs three important functions: it rejects all the harmonics of the 15 GHz but the second, DC isolates the drain from the output, and presents the required reactive impedance to the drain at the input frequency. The 50 O transmission line TL2 supports this last function by rotating the filter reflection coefficient by a suitable quantity. Figure 13.32 plots the output second-harmonic power versus the input frequency for an input power of 0 dBm of the doubler in Figure 13.31. The measured conversion gain is 2.7 dB and the measured curve is in reasonably good agreement with the simulated one over all the frequency range.

Output power, dBm

5 2.7 0 -5 measured -10

simulated

-15 -20 -25 -30 13

14

15

16

Input frequency, GHz

Figure 13.32

Performances of the circuit in Figure 13.31.

17

630


Bibliography 1. S. A. Maas, Nonlinear Microwave Circuits, Artech House, Norwood, MA, 1988. 2. R. G. Harrison and X. Le Polozec, ‘Nonsquarelaw behavior of diode detectors analyzed by the Ritz–Galerkin method’, IEEE Transactions on Microwave Theory and Techniques, Vol. 42, No. 5, pp. 840–846, 1994. 3. S. A. Maas, Microwave Mixers, Artech House, Norwood, MA, 1986. 4. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 7.2, pp. 420–438. 5. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, section 7.4, pp. 198–200. 6. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, section 7.7, pp. 206–214. 7. S. A. Maas, Microwave Mixers, Artech House, Norwood, MA, 1986, sections 4.4, pp. 113–119, 5.1, pp. 129–144 and 5.2, pp. 145–150. 8. S. A. Maas, Nonlinear Microwave and RF Circuits, Artech House, Norwood, MA, 2003, Chapter 7, pp. 355–391. 9. S. A. Maas, Nonlinear Microwave and RF Circuits, Artech House, Norwood, MA, 2003, Chapter 9, pp. 475–490. 10. E. Camargo, Design of FET Multipliers and Harmonic Oscillators, Artech House, Norwood, MA, 1998, Chapters 3–5, pp. 45–125 and 7, pp. 145–180.

Related files Ansoft files 01_Coupled_Lines_Balun.adsn. Analyzes the coupled-line balun used in the mixer of Figure 13.14c.

Mathcad files 02_Diode_Curves.mcd. Provides some numerical calculations related to the discussion of Section 13.2. 03_Suppression.mcd. Implements the calculations discussed in Section 13.3.7. 04_Pinch_Off_Polynomial_Curvefit.mcd. Calculates the best fit polynomial for the drain–source current when the minimum of the gate–source voltage range is lower than the pinch-off. Such a polynomial is used in Section 13.3.8. 05_Mixer_Spur_Chart.mcd. Plots the spurious response of a mixer with a specified spurious table.

SIMetrix files 06_Schottky_Detector_Power_Sweep.sxsch. Analyzes a diode detector having the differential structure shown in Figure 13.4. Implements a sweep of the RF voltage amplitude. 07_FET_Detector_Power_Sweep.sxsch. Analyzes an FET detector having the differential structure shown in Figure 13.7a. Implements a sweep of the RF voltage amplitude. 08_Switching_Cold_FET_MIXER. Describes a quasi-ideal cold FET mixer under switching operation. The FET has 0.5 O ON and 5 kO OFF channel resistance.


631

09_MBC13900_Differential_Pair_Linearity.sxsch. Analyzes DP with different values of the emitter resistance, and produces the results displayed in Figure 13.23. 10_Single_Ended_Mixer.sxsch. Performs a power swept single RF tone analysis of the mixer in Figure 13.10. 11_Single_Ended_Mixer_IF_Filter.sxsch. Analyzes the IF filter of the circuit in file 10. 12_Single_Ended_Mixer_Dual_Tone_RF.sxsch. Analyzes the same mixer as file 10, but with a dual tone applied to the RF port. 13_Schottky_Frequency_Doubler.sxsch. Analyzes the frequency doubler in Figure 13.29b. 14_Schottky_Frequency_Tripler.sxsch. Analyzes the frequency doubler in Figure 13.29c.

14

Microwave circuit technology 14.1 Introduction This chapter describes some of the most important realization technologies for microwave circuits. Here the term microwave circuit covers any one of the components described in Chapters 7, 8, 10, 11, 12 and 13 up to the complex functional blocks discussed in Chapter 15. In the first and last case we have single function and multi-function blocks, respectively. The present chapter deals with integrated realizations: from the simplest, which is the microwave integrated circuit (MIC), to the most sophisticated one, known as the monolithic microwave integrated circuit (MMIC). The last section presents an overview of the silicon radio frequency integrated circuit (RFIC).

14.2 Hybrid and monolithic integrated circuits The circuits discussed in Chapters 7, 8, 10, 11, 12 and 13 include lumped passive elements, isolated and coupled transmission lines, and semiconductor elements. In the most general case MICs contain all three element types, thus the realization technology must allow the coexistence of all those possible devices. A widely used MIC technology is the hybrid one, in which an insulating substrate – also referred to as the printed circuit board (PCB) – is used to realize the transmission lines and as mechanical support for the whole circuit. Then the other elements, which are discrete devices, are placed on and connected to the substrate elements. Examples of hybrid MICs are shown in Figures 10.4, 11.10, 11.42, 11.45, 12.28 and 13.12. MMICs are a more advanced and expensive technology, consisting of realizing all the circuit elements on the same semiconductor substrate, which works to isolate the transmission lines and doped material for the semiconductor devices. Therefore, the MMIC semiconductor is intrinsic everywhere, except for the regions where semiconductor devices are present. In both MICs and MMICs, the transmission lines are typically microstrips, although coplanar lines are sometimes used.


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14.2.1

High-frequency PCB

The simplest MIC consists of a PCB with discrete elements (resistors, capacitors, inductors, diodes and transistors) in a packaged form and soldered to suitable pads of the PCB itself. In other words, each discrete element, as supplied from the manufacturer, is encapsulated within an isolating house, the package. Metal leads are connected to the element terminals by means of bonding wires, and come out from the package. The latter includes at least as many leads as the number of electrodes the device has, although sometimes one electrode can be connected to more than one terminal,1 and/or some leads are not connected.2 Such a simple technology is a kind of extension of the conventional PCB towards high frequencies. Compared with the standard case, the high-frequency printed circuit technology requires some caution: .

Small tolerance on length, width and gaps on the realized traces. Accuracy is required for the gaps if coupled lines are used.

.

A ground plane covering the whole PCB is required below the traces, unless all the transmission lines are coplanar, which is not frequently the case.

.

The substrate thickness and dielectric permittivity must also present controlled specifications, with tight tolerances.

.

A low dissipation factor ðtan dÞ in the PCB dielectric is also desirable.

.

The size of all the discrete elements has to be much smaller than the wavelength, otherwise the package parasitics become dominant over the intrinsic element performances.

The high-frequency PCB is the technique of choice when low cost is a demanding requirement, the maximum frequency is about 10 GHz, and the relative bandwidth of the circuit is narrower than one octave. The low cost inherent in the high-frequency PCB is related to the simple assembly technique based on conventional tin (Sn) soldering. Its frequency and bandwidth limitation derive from the parasitic elements associated with the packages.3 Figure 14.1 shows one example of a high-frequency PCB, which corresponds to the detector in Figure 13.3a, where the input matching network simply consists of the series capacitor CIN . That layout presents three external ports: the RF input, the DC output and one for the application of the bias current I0 through the inductor Lbias. The input inductor LIN and the output capacitor LOUT have one electrode connected to ground. Such a connection is realized with a rectangular pad, placed on the PCB edge and connected to the back plane by means of a metal ribbon. In more advanced realizations, the ground connections are realized with via holes; this way it is no longer necessary to have the grounded elements placed close to the PCB edge. Furthermore, the series inductance due to a via hole ground is smaller than that of a ribbon connection. The increased flexibility and performance of the via hole technology are counterbalanced by the higher cost associated with the via hole realization itself. 1 This technique is used to reduce the series inductance due to the connection. Typically, the emitter and the source connect to two terminals, normally placed on opposite sides of the package. 2 This happens when a device – for instance, a diode – uses a standard package developed for many device types having different numbers of electrodes, like diodes and transistors. 3 For instance, a surface-mounted capacitor, having a length of 1 mm, presents an associated series of about Ls ffi 0:5 nH. If the capacitance is C ¼ 5 pF, the resulting series resonant frequency is fseries ¼ ð2pÞ1 ðLs CÞ0:5 ffi 3:18 GHz. At that frequency the capacitor behaves approximately like a short circuit; above, it becomes inductive, and its impedance becomes less controllable. On the other hand, the impedance of a 5 pF capacitor becomes lower than 50 O only if the frequency is greater than 637 MHz. Therefore the usable range of our component roughly extends between the above-calculated limits.

MICROWAVE CIRCUIT TECHNOLOGY

Figure 14.1

635

Physical aspect of the detector in Figure 13.3a.

The technology for realizing assemblies of the type in Figure 14.1 is also called surface mount technology (SMT), due to the fact that all the elements but the printed ones are placed on the surface of the PCB. For the same reason, the discrete elements are defined as surface mount devices (SMDs).

14.2.2

Hybrid MICs

A hybrid technology with higher performances and cost than the high-frequency PCB is the one used for the circuits in Figures 10.4, 11.10, 11.42, 11.45, 12.28 and 13.12. In these cases, the discrete elements are in chip (or die) form, and their terminals connect to the PCB by means of bond wires. This solution increases the circuit performance by reducing the parasitic inductance of the connections4 and reduces the circuit size, due to the elimination of the packages, thus allowing denser assemblies. The bonding wire technology, however, is more expensive than the conventional tin one, in that it involves sophisticated ultrasound-based processes, which need to be performed in a low-powder environment (clean rooms). We can give a better description of MIC technology by referring to the amplifier in Figure 11.42b. It comprises three separate substrates (the three dashed rectangles) brazed on a common metal plate – the carrier – that works as a common ground as well as a mechanical support. More precisely, from the input to the output, there is one PCB for the microstrip elements of the input, the interstage, and the output matching network. The two transistors are placed inside the two slots between the three substrates. An additional feature of the layout in Figure 11.42b is that resistors R1 to R7 are directly realized on the PCB by means of suitable deposited materials of controlled thickness and resistivity, such as nickel–chrome (Ni–Cr) or tantalum (Ta). The capacitors C1 to C4 are square-base parallelepipeds, with a nominal size of 0:5 0:5 0:1 mm. Their two electrodes are placed on the opposite square faces. The top face electrode is a circular metallization having a diameter of 0.2 mm, while the bottom electrode occupies the whole face, which is fully metalized. All the above-mentioned capacitors have one terminal to ground,5 and therefore their 4 Packaged elements include bond wires inside the package itself, between the internal device terminal and the corresponding lead. Then, the leads and the PCB pads present their own additional inductance that adds to that of the internal bond wires. 5 Compare the schematic in Figure 12.42a with the layout in Figure 12.42b.

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bottom electrodes are brazed to the carrier. The non-grounded capacitor terminals connect to the respective PCB pad by means of two gold bond wires, having a diameter6 of 25 mm. The most common material for the bond wires is gold (Au), although silver (Ag) and aluminium (Al) are sometimes preferred. In some cases, the use of two wires, instead of one, increases the reliability of the connection and reduces the associated inductance, although it also increases the assembly time (and thus the cost) and the stress on the capacitors (reducing the overall yield). The two FETs Q1, Q2 are two gallium arsenide (GaAs) parallelepipeds with the same thickness as the capacitors and overall size of 0:6 0:4 0:1 mm. The gate and drain electrodes are squared pads having a size of 60 mm, which are placed on the two opposite edges of the devices. The source is directly grounded to the bottom of the chip – which is fully metal plated – by means of two via holes, symmetrically placed with respect to the gate–drain direction. The source connection to ground is guaranteed by the fact that the bottom plate of the FET is brazed to the carrier. Gate and drain connect to the respective pads through two 25 mm bond wires, like the capacitors. The amplifiers in Figure 11.10 use a different type of source grounding. The transistor is placed over a pad in the PCB, which connects to ground through two via holes. This simplifies the mechanical realization of the PCB, which is a single piece and requires no carrier, but also increases the series inductance of the source ground connection. The amplifiers in Figure 11.10 specifically require a relatively high series inductance to ground, but this is not common, and additional inductance is typically an issue. A technical solution to decrease further the inductance associated with the connection between the device electrodes and the rest of the circuit is the so-called beam lead. The circuits in Figures 10.4b and 13.12 employ such types of devices. The particularity of beam lead devices is that they present one metal (typically Au) lead per electrode, which is an extension of the electrode metallization and grown from that. This solution minimizes the discontinuity between the device electrode and the respective interconnecting element. Furthermore, the inductance of the lead – which has the aspect of a ribbon – is typically smaller than that of the bond wires.

14.2.3

MMICs

In MMICs all the circuit elements – including transistors and diodes – are realized within the same substrate, and the mutual connections are automatically obtained during the realization of the MMIC itself. The MMIC substrate is a semiconductor, typically GaAs, but more advanced materials like indium phosphide (InP) are also in use. The metal continuity between the interconnected terminals of the various MMIC elements guarantees their mutual interconnection. MMIC realization is based on multiple photolithographic steps, each of them essentially being of the same type as for the PCB, although MMIC lithography is much more accurate. Ultraviolet, electron beam and sometimes X-rays are used in order to minimize the light wavelength and associated diffraction effects. Figures 10.5 and 11.37 show two examples of MMIC realizations. Referring to the SPDT in Figure 10.5, it includes eight transistors (Q1A to Q4A and Q1B to Q4B), eight resistors (R1A to R4A and R1B to R4B) and 14 via holes. Eight of those vias connect the source terminals of the transistors that work as shunt elements (Q2A to Q4A and Q2B to Q4B). The remaining six via holes are ground connections for three couples of pads, placed close to the RF ports (P1 to P3). Those elements allow the RF test of the chip on the wafer, as Section 14.5.2 explains in more detail. MMIC technology allows the realization of diodes, transistors, capacitors, resistors, inductors and transmission lines, although the circuit in Figure 10.5 includes only transistors, resistors and transmission lines. The main advantage of MMIC over MIC is the absence of added interconnections: this way, the interconnection parasitics are avoided together with their variability, and the circuit is more compact. An additional advantage deriving from the minimum connection parasitics and associated variability is the 6

More precisely, the wire diameter is d ¼ 1 mil ¼ 25:4 mm.


637

high-performance uniformity of the MMIC in production, from piece to piece. The straightforward connections of the MMIC are also beneficial for the maximum frequency and bandwidth of the circuit. The two main MMIC drawbacks are the high cost and the relatively low quality of its single elements, compared with discrete realizations. The latter inconvenience comes from the need to use the same substrate element for all the elements. Thus the material properties result from a compromise among the needs of the various elements. By comparison, discrete transistors can be realized using the best possible semiconductor for the transistor itself, no matter if a capacitor realized on the same substrate presents low performance and vice versa. In some critical applications, the lower performances of the MMIC elements – transistors in particular – can make the MIC technology a better solution, from the RF performance point of view, despite the higher interconnection parasitics.

14.2.4

Advanced hybrid MICs

The so-called advanced MICs are an intermediate solution between MICs and MMICs. In that case all the passive elements – capacitors, resistors, inductors and transmission lines – are realized over the same dielectric substrate by applying lithographic techniques, similar to the ones used for MMICs. The semiconductor devices, if present, are discrete chip devices. They are wire bonded to their pads present on the substrate, as described for the standard MIC in Section 14.2.2: the respective welding is the only additional assembly procedure required for advanced MICs. The advantages and disadvantages of advanced MICs are intermediate between the ones of standard MICs and MMICs. The dissipation loss in passive elements is better than in MMICs and worse than in MICs, the interconnection parasitics are a minimum as in MMICs if the connections are between nonsemiconductor elements, and the semiconductor device quality is the best possible, as in MICs. The cost of advanced MICs is comparable with, but higher than, standard MICs.

14.2.5

Parasitic elements associated to physical devices

This section presents some qualitative considerations on the parasitic elements associated with the various devices and with their assembly. No physical component is ideal, and this holds true in particular for resistors, capacitors and inductors. Ideal lumped devices only represent first-order approximations of reality, and the accuracy of the model gets worse as the frequency increases. The design of high-frequency circuits has to take into account the behaviour of real devices, thus the parasitic effects have to be considered. In addition, accurate designs, intended for high yield in production, need to consider not only the nominal values of the parasitics, but also their statistical variability, due to variations in the realization processes. Typically, devices having low parasitics also present low variations. Let us begin by considering the SMD used in high-frequency PCBs. Many possible packages are in use in engineering practice; however, the type depicted in Figure 14.2a is very common for two-terminal devices. It is a parallelepiped; two of its opposite faces are metal plated and form the terminals. Such a package is available with different standard denominations, corresponding to different sizes. More precisely, the standard denomination is the union of two numbers that increase with the length and width, respectively. Table 14.1 lists the most common SMD standard sizes. Any device encapsulated inside the package is added by parasitic elements due to the package itself, to the connections between the intrinsic device terminals and the external terminals, and to the metal pads that are necessarily present in the PCB, as Figure 14.2a shows. Figure 14.2b depicts the overall electrical model of the packaged component.7 The capacitances Cp0 and Cp00 are mainly due to the PCB pads, which must have the minimum possible size while allowing the soldering of the device. The capacitance Cp 7 The same problems have been discussed in Section 9.7.1, see in particular Figure 9.34b and the relative comments.

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MICROWAVE AND RF ENGINEERING Table 14.1 Standard sizes of the package in Figure 14.2. Chip size 0201 0402 0603 0805 1206 1210 1217 2010 2020 2045 2512

Figure 14.2

l; mm

w; mm

h; mm

0.60 1.00 1.60 2.00 3.20 3.20 3.00 5.00 5.08 5.00 6.30

0.30 0.50 0.80 1.25 1.60 2.60 4.20 2.60 5.08 11.50 3.10

0.25 0.35 0.50 0.50 0.60 0.60 0.90 0.70 0.90 0.90 0.60

SMD device: (a) physical aspect; (b) electrical model.

models the mutual capacitance between the two leads, and increases by the factor w h l 1 : as the package size increases, w, l and h increase with it, and so it is for Cp, Cp0 and Cp00 . The inductance Ls represents the transit time of the electromagnetic wave inside the device.8 In other words, Ls models the internal connection inductance, increased by the one associated with the finite length of the intrinsic device. The equivalent network in Figure 14.2b is accurate only if the device’s physical length is much shorter than the wavelength, and such a validity limit is approximately reached at the resonant frequency between Cp and Ls . To give some realistic values, in an 0402 package the parasitics could be Lp ffi 0:6 nH;

Cp ffi 0:3 pF;

Cp0 ffi Cp00 ffi 0:05 pF

The resulting maximum usable frequency for the model in Figure 14.2b is fmax ¼

1 pffiffiffiffiffiffiffiffiffiffi ffi 12 GHz 2p Lp Cp

As the geometrical size of the device increase (0603, 0804, etc.), the parasitic values also increase, and the maximum usable frequency decreases proportionally. 8 A more accurate model, although still for a first approximation, considers the device as a piece of transmission line. In discrete lumped approximation, Lp ¼ L l, where L is the inductance per unit length of the transmission line, and the distributed capacitance C contributes to increase Cp0 and Cp00 .


639

d conductor

h ground plane

Figure 14.3

Cross-section of a bond wire running parallel to a ground plane.

Chip devices used in MICs present parasitics of lower entities than SMD elements. The main parasitic element of chip devices is due to the bond wires. Still considering the layout of Figure 11.42b, it is possible to see that the bond wires run almost parallel to the ground plane (carrier). Therefore, we can consider them as transmission lines having the cross-section shown in Figure 14.3. The inductance L and the capacitance C per unit length of that structure are given respectively by9 m m h 1 L ¼ 0 r cosh1 2 ; C ¼ 2pe0 er h ð14:1Þ d 2p 1 2 cosh d Approximating the permittivities of the air by those of a vacuum ðmr ¼ er ¼ 1Þ and considering the typical values of the geometry ðh ¼ 0:2 mm; d ¼ 25 mmÞ, we obtain L ffi 0:7 nH=mm; C ffi 0:016 pF=mm. Now, as Figure 11.42b shows that all the chips present two bond wires, it follows that the effective inductance (respectively, capacitance) is half (double) the value above. Moreover, the length of each bond wire is around 0.2 mm, therefore the series parasitic inductance is Ls ffi 0:2 0:7=2 ¼ 0:07 nH and the capacitance is Cp ffi 0:2 2 0:016 ¼ 4:4 fF. These numbers are about one order of magnitude smaller than those found for the SMD, justifying the MIC realization for a 35 GHz component, like the amplifier in Figure 11.42.

14.3 Basic MMIC elements This section describes the various circuit elements realizable with MMIC technology, illustrating their physical structure and their electrical models. Here, the reference context is the GaAs MMIC; some of the differences between the MMIC and the silicon RFIC will be discussed in Section 14.6. The described MMIC structure is schematic and simplified. It refers to no particular technological process. Nevertheless, there are no conceptual differences between the present descriptions and any real MMIC structure. The GaAs MMIC structure consists of a semiconductor substrate: the back face is fully metal plated, while all the different layers needed for the realization of the circuit elements are deposited on the top side. The realization of a suitable geometry on the various overlapped layers – which could be conducting, insulating, resistive or doping materials – produces the desired circuit element. The layout rules give restrictions on the geometry, in order to keep production compatible with the fabrication process. The factory that produces MMICs takes the name foundry. The MMIC designer can be internal or external to the foundry: in both cases he or she operates following the information about the geometry and electrical models of the various MMIC elements, which are written on a document known as the foundry manual. That document also specifies the layout rules. 9

See [1].

640


d

1

w (a)

METAL1

1 h

substrate, εr ground plane

(b)

1 Lv Rv (c)

Figure 14.4

14.3.1

Via hole: (a) top view; (b) cross-section; (c) electrical model.

Transmission lines

The most used transmission line type in MMIC technology is the microstrip; the coplanar line is used in special cases. The undoped GaAs substrate presents a dielectric permittivity slightly higher than 12, although its exact value depends on the specific process. The loss factor of undoped GaAs substrates is of the order of 0.01 at 10 GHz. The most used thickness for the substrate is 100 mm. The minimum line width and spacing allowed by MMIC processes are both close to 10 mm. However, the current flowing through the line is an additional limiting factor on the minimum usable width. The maximum current is proportional to the line width, with a constant – specified in the foundry manual – that is specific to the particular process. For the maximum line width, it must be considered that wide lines increase the size and consequently the cost of the chip. Furthermore, the microstrip models present in all the CAE tools are quasi-TEM based, thus they become inaccurate if the microstrip effective width10 approaches half of the wavelength, due to the propagation of higher modes.

14.3.2

Via holes

Via holes allow the ground connection of electrodes belonging to components that could be placed in arbitrary positions on the circuit, not just close to the edge as in the layout of Figure 14.1. Figure 14.4a shows the top view of a via hole. It consists of a square pad, which has a typical size w of about 100 mm, and a circular hole – having a typical diameter of 50 mm – is placed at the centre of the pad. 10 The microstrip effective width we f f is defined at the end of Section 3.14. In the example reported there, for a 50 O microstrip line with h ¼ 100 mm, er ¼ 12:5, the effective 1 width is we f f ¼ 265 mm which (neglecting the dispersion) is ffi 198:9 GHz. equal to l=2 at the frequency f ¼ c e0:5 e f f 2we f f


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The walls of the hole are metal plated and connected to the ground plane, so that the pad is grounded, as the cross-section in Figure 14.4b clarifies. Sometimes the pad is octagonal rather than squared, as in the SPDT of Figure 10.5. Furthermore, the hole is not always circular: in the amplifier of Figure 11.37 it is squared. The via hole is not an ideal ground connection. As with all the physical connections it presents some parasitics, which could cause relevant effects at high frequencies. Figure 14.4c shows one possible equivalent circuit for the via hole; the component values are of the order of Lv 0:01 nH for the inductance and Rv 0:05 O for the resistance. The layout rules in the foundry manual specify the pad shape and size, the hole diameter, the minimum possible distance between two vias and, of course, the values of Lv ; Rv . The importance of the minimum distance between vias derives from the risk of breaking the chip, due to the mechanical stress associated with too close perforations. The layers of interest in via hole realizations are: METAL1, which is the one immediately above the substrate surface; and VIA, which is not a physical layer but indicates the point of perforation of the substrate and metal plating the walls of the holes.

14.3.3 Resistors MMIC resistors are realized with materials having controlled thickness and resistivity, like Ni–Cr, Ta or the semiconductor itself with suitable doping. Figures 14.5a,b show the top and side views of an MMIC resistor. The grey rectangle, on the layer RESISTIVE_MATERIAL, is the resistor body, with the two terminals on the layer METAL1. The DC resistance of such an element is R¼r

lr tr wr

ð14:2Þ

where r; tr and wr are the resistivity, thickness and width of the resistive material, and lr is the distance between the two terminals. All these four parameters have a specified nominal value with a tight tolerance, both specified in the foundry manual, which also specifies the maximum current density. Equation (14.2) can be rewritten as R¼

r lr lr ¼ rspeci fic wr tr wr

ð14:3Þ

The parameter rspeci fic ¼ r=tr is the specific resistance of the resistor, normally expressed in O=square, and specified in the foundry model with nominal value and tolerance. Equation (14.3) implies that a squared resistor ðlr ¼ wr Þ presents a resistance equal to rspeci fic , independently of the side. This justifies the name of the unit used for the specific resistance. The electrical model of the MMIC resistor depicted in Figure 14.5c derives from considering that the element is a lossy11 microstrip with step discontinuities at its ends. Normally, lumped reactive elements model the two step discontinuities. Usually, the resistor length12 is much smaller than the wavelength, thus the distributed resistance can be modelled with one single lumped resistor, whose resistance equals the DC resistance (14.3). With these assumptions, the inductors Ls1 and Ls1 , together with the capacitor Cp, model the step discontinuities at the two ends of the resistor. Finally,

The resistance per unit length of the lossy microstrip is clearly Rl ¼ r=ðtr wr Þ ¼ rspeci fic =wr. For strong reasons, wr has to be much smaller than the wavelength, to avoid the lack of model accuracy already discussed in Section 14.3.1 for the microstrip lines. 11 12

642


l′

w′

lr

l0

1

2

wr (a)

METAL1 1

METAL1 2

RESISTIVE_MATERIAL

tr

substrate, εr

h

(b)

ground plane

Z01, εeff1, 1

L s1

L s2

Cp

l r /2

Z01, εeff1,

Rs

lr /2

L s2

L s1

2

Cp (c)

Figure 14.5

MMIC resistor: (a) top view; (b) side view; (c) electrical model.

the two loss-free transmission lines, on the two opposite sides of the lumped resistor, take into account the distributed reactance of the resistor. In the resistor case, the layout rules specify: .

The overlapping distance l0 and the width excess w0 wr of the metal terminals in respect of the resistive material, in order to guarantee the electrical contact between the two.

.

Minimum values for wr and lr .

.

Maximum current as a function of the resistor size. This parameter increases with, but not proportionally to, wr , due to the different heat conduction in the centre and edge of the resistor.

The layers of interest to the resistor realization are METAL1 and RESISTIVE_MATERIAL. The resistor described above is sometimes also realized on the substrate of advanced MIC processes. In that case the resistive material thickness is typically of the order of 1 mm or smaller. The realized resistor takes the name of thin-film resistor(TFR). In contrast, the thick-film resistor also exists, although it is less common and characterized by the higher substrate thickness, which is of the order of some tens of microns. Thin-film resistors usually perform better than thick-film ones, in that the TFR resistance is less dependent on the frequency, due to the lower influence of the skin effect.


643

sl

1

2

wl (a) METAL2 METAL1_2

DIELECTRIC,

ε rc

METAL1 1

hc

2

substrate, ε r

h

(b)

ground plane

Cs Rs

L s1

Ls2

1

2

Z01, εeff1,

l1 Figure 14.6

Cp1

Cp2

Z02, εeff2,

l2

(c)

Planar spiral inductor: (a) top view; (b) cross-section; (c) electrical model.

14.3.4 Inductors The simplest way to realize inductors on a microstrip consists of using short transmission lines with high characteristic impedance, by means of the semi-lumped technique discussed in Section 8.4. More quantitatively, the minimum width – which corresponds to the highest characteristic impedance – for MMIC microstrips is about13 10 mm. From that, and assuming h ¼ 100 mm, er ¼ 12:5, Equations (3.219) and (3.220) give Z0 ¼ 96:43 O, ee f f ¼ 7:435, and then an inductance per unit length of L ¼ 0:876 nH=mm. Inductors in the nanohenry range are thus millimetres long. When higher inductances are required and the transmission line size is not acceptable, it is possible to reduce the inductor size by multiple folding of the line. Clearly the circuit analysis has to consider the bend discontinuities associated with each folding and the mutual coupling between the different segments of the meander line. The transmission lines TL1A to TL3A in the SPDT of Figure 10.5 are examples of the application of that technique. A more compact solution – particularly for high-inductance elements – is the planar spiral inductor illustrated in Figure 14.6. The top view depicted in Figure 14.6a shows the element structure, which 13

See Section 14.3.1.

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consists of a narrow line wound around itself forming concentric squared spirals. Other spiral shapes are also possible, like rectangular, circular or polygonal. Rectangular inductors usually offer the most efficient occupation of the chip area. The right-angled bends of rectangular spirals cause high current densities in the corners, thus higher series resistance and consequent degradation in the quality factor. Circular spirals eliminate the corners and therefore present the best Q factors, although at the expense of a tendential larger area occupation. Polygonal shapes, for instance octagonal, are approximations of the circular shapes that are easier to analyze with CAE programs and to describe in layout CAD programs. The spiral inductor – differently from the folded line – has one terminal trapped inside the inductor itself: it has to pass above or below the spirals to be connected to the rest of the circuit. The cross-section in Figure 14.6b shows the bridge, realized on a suitable additional metal layer (METAL2). That structure is mechanical supported by a dielectric rectangle, slightly wider than the bridge, to avoid accidental short circuits, and realized on the layer DIELECTRIC. Furthermore, a third metal layer (METAL1_2) is required to realize the post to connect the bridge to the layer METAL1. Figure 14.6c shows the electrical model of the spiral inductor. It consists of: . .

The series LR cell ðLs ; Rs Þ that represents the low-frequency behaviour of the element. Two parasitic capacitors to ground Cp1 ; Cp2 modelling the capacitance between the spirals and the ground plane.

.

The capacitor Cs, which takes into account the mutual capacitance between spirals.

.

Two transmission lines, to consider the pieces of microstrip at the inductor terminals and some additional minor effects, thus improving the model accuracy.

The geometric parameters of the inductors are the result of a compromise among different needs: .

A high quality factor implies low series resistance, thus wide strips for the spirals (high wl ). High direct current also requires wide strips.

.

Both the strip width wl and spacing sl must be the minimum allowed by the layout rules, in order to minimize the area occupied and, consequently, also the capacitance to ground Cp1 ; Cp2 .

.

The mutual capacitance between spirals Cs decreases with sl . At the resonant frequency pffiffiffiffiffiffiffiffiffi fs ¼ 1= 2p Ls Cs of Ls with Cs , the two ports 1 and 2 of the inductors are isolated. That frequency is also approximately coincident with the highest limit for model validity. Moreover, at frequencies higher than fr , the inductor performance not only is difficult to predict, but also presents high sensitivity to the manufacturing tolerance. Therefore high fr , thus low Cs, and then high sl , are desirable, in contrast with the previous point.

The layout rules applied to the inductor specify: .

The minimum and maximum for both wl and sl .

.

The margin of the dielectric support on the layer DIELECTRIC with respect to the metal bridge on the layer METAL2, in length and width. The minimum values usually coincide with those of the transmission lines.

.

The size of the posts (layer METAL1_2) connecting the bridge to the first metal layer.

.

The maximum current as a function of wl .

The layers of interest to spiral inductor fabrication are METAL1, METAL2, METAL1_2 and DIELECTRIC.

MICROWAVE CIRCUIT TECHNOLOGY l1

lc lb

w1 wc

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l2

1

w2

2

(a)

METAL2 METAL1_2 METAL1 2

DIELECTRIC, εrc METAL1

1

hc

substrate, εr

h

ground plane

Z01, εeff1, l1

Cs

Rs

Ls1

Ls2

1

(b)

Z02, εeff2, l2 2

Cp1

Cp2

(c)

Figure 14.7 MMIC capacitor: (a) top view; (b) side view; (c) electrical model.

14.3.5 Capacitors MMIC capacitors are parallel-plate devices, with square plates in most cases. The squared shape maximizes14 the cut-off frequency of the first higher order mode, holding the capacitance constant. An additional advantage of the square shape is that the reduced number of design parameters simplifies the modelling, particularly in the scalable case. Figure 14.7 shows the structure and the equivalent circuit of an MMIC capacitor. It is realized on two metals and one dielectric layer: METAL1 for the bottom electrode, METAL2 for the top electrode and DIELECTRIC for the dielectric. The additional metal layer METAL1_2 connects the top electrode to the first-level metal by means of a post, similar to the inductor considered in Section 14.3.4. Also the capacitor dielectric is realized on the same layer as the one used for the bridge support of the inductor. Again, the bottom electrode is slightly wider than the dielectric, which is slightly wider than the top electrode, to avoid short circuits between the two electrodes. 14 Let a ¼ wc =lc be the ratio between width wc and length lc of the capacitor, er the permittivity of the dielectric and hp width. Theffi resulting capacitanceffi is C ¼ e0 er wc lc =hc ¼ e0 er a lc 2 =hc ¼ e0 er wc 2 =ðhc aÞ. Thus, given C, c its ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wc ¼ hc C a=ðe0 er Þ or lc ¼ hc C=ðe0 er aÞ The cut-off frequency of the first non-TEM mode is the one that makes the length between wc and lc equal to a half 2 2 2 2 ¼ 0:25 c2 e1 wavelength, i.e. fNONTEM r =maxðwc ; lc Þ ¼ 0:25 c hc e0 C=maxða; 1=aÞ. Now, fNONTEM is maximum when the highest frequency between a and 1=a is minimum, which occurs when a ¼ 1, as is easy to demonstrate. Hence fNONTEM is the highest possible when wc ¼ lc .

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From an electrical point of view, the structure described in Figure 14.7 can be considered as a section of asymmetrical broadside-coupled lines with some additional discontinuities at the extremes. However, if the capacitor size is much smaller than the wavelength – as it normally is – a simplified model, based on lumped elements and two isolated transmission lines, is sufficiently accurate. Figure 14.7c shows one possibility; the main element is the capacitor Cs, which presents the capacitance Cs ¼ e0 erc

wc lc wc ¼lc w2 ¼ e0 erc c hc hc

ð14:4Þ

Typical values for the relative permittivity and thickness of the dielectric are erc 4; hc 1 mm , respectively. Therefore a capacitor with a size of 100 mm has a capacitance of about 3 pF. The design rules for the capacitor specify: .

The distance between the edges of METAL1 and DIELECTRIC.

.

The distance between the edges of DIELECTRIC and METAL2, which is close if not identical to the previous value.

.

The post size.

.

The minimum and maximum size of the capacitor, and consequently the minimum and maximum realizable capacitance.

.

The maximum voltage between the electrodes allowed by the electrical rigidity of the dielectric. The maximum voltage is about 10 V, corresponding to an electric field intensity of 107 V=m if the dielectric thickness equals 1 mm, as assumed.

The layers of interest in the capacitor fabrication are the same as for the spiral inductor, namely METAL1, METAL2, METAL1_2 and DIELECTRIC.

14.3.6

Semiconductor devices

Figure 14.8 shows the structure of an MMIC FET. The drawings on the left of the dashed line depict the so-called single finger FET, which is the simplest possible one. The single finger FET basically consists of three metal pads, labelled S, G and D, which correspond to the three electrodes, source, gate and drain, respectively. The MMIC substrate is n-doped15 in the region below these three pads: the resulting transistor is a depletion device. Consequently, the drain–source conduction is maximum when the gate–source voltage vgs is zero, decreases with more negative values of vgs , and vanishes when vgs is lower than the pinch-off Vp , as explained in Section 9.7.2.2. All the FET linear and nonlinear characterizations and models described in Section 9.7 also apply to the MMIC FET. In addition, it is possible to include some noise generators in the models of Figures 9.34b and 9.40, in order to model the noise characteristics of the device. Such noise elements could be voltage or current generators, with no, partial or total mutual correlation. The peculiar characteristics of the MMIC FET models are the small values of Ls , Lg and Ld , with respect to similar discrete devices, due to the absence of bond wire connections. There is an approximate proportionality between the length of the FET channel w and the linear and nonlinear model parameters Cgs ; Cgd ; Cds ; gm and the nonlinear current parameters16 b. This allows scalable models to be developed, as Section 14.4.1 will illustrate. For any given w, the gate–source capacitance Cgs decreases with the channel length l, while the remaining parameters above are almost

15 16

The doping is also called implantation, and the doped region is also referred to as the implanted zone. The parameter b is proportional to the saturation current Idss , as in Equation (9.75).

MICROWAVE CIRCUIT TECHNOLOGY Single finger

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Interdigital D G1

S

D

G

h

G D

G3

G4

S

S S1

S

G2

D1

S2

D2

S3

G

S

S1 G1 D1 G2 S2 G3 D2 G4 S3

(a)

S

substrate, εr ground plane (b) METAL0 METAL1 METAL1_2 METAL2 IMPLANT

Figure 14.8 MMIC FET: (a) top view; (b) cross-section.

constant with l. This suggests the realization of transistors with the minimum channel length possible for the available technology. MMIC manufacturers have made appropriate efforts in this direction, approximately halving the channel length every 10 years, passing from a typical value of 1 mm around 1980 to 0.25 mm around 2000. There are two obstacles to the fabrication of a very small channel length: the technological difficulties and the increased intensity of the electric field in the neighbourhood of the gate region. The reason for the second limiting factor can be clarified by considering the effect of Vgs on the channel, as Figure 14.9 shows. The n-doped channel presents the maximum depth when vgs is zero, with the consequent well-known maximum conduction between drain and source. As vgs moves towards more and more negative values, there is a progressive depletion of mobile charges in the region below the gate, with a resulting decrease of channel depth and drain–source conduction. The depleted zone is isolating, while the non-depleted one is conductive. When vgs Vp , the depleted zone depth is equal to the thickness of the doped zone, with a consequent complete isolation of the source from the drain. In this condition, the gate–source voltage falls between the gate and the beginning of the non-depleted portion of the source. The resulting intensity of the electric field is the ratio between that voltage and the amplitude of the depleted zone l 0

648

MICROWAVE AND RF ENGINEERING –vgs+ source

gate

drain

limits of the depleted zone implanted region

vgs = 0 Vp < vgs < 0 vgs ≥ 0

Figure 14.9

Effect of the gate–source voltage on the FET channel. Epinched o f f ¼ vgs l0

ð14:5Þ

The value of Epinched o f f cannot exceed the dielectric rigidity Emax of the material present in the doped and depleted zone. Thus, the absolute value of the maximum applicable vgs is max vgs ¼ Emax l 0 ð14:6Þ Now, l 0 decreases with l, and the electric field configuration in the region between gate and drain is similar to the one between gate and source. In this second case, however, the applied voltage is even higher, due to the presence of a positive drain–source voltage vds . The maximum vds is then maxjðvds Þj ¼ Emax l 0 vgs ð14:7Þ Equations (14.6) and (14.7) are upper limits for the voltages applicable between the FET electrodes: these limits decrease with l 0 and consequently with l,17 as well. Furthermore, small values of l are associated with high values of the parasitic resistances Rg and Ri , which reduce the device gain at high frequencies, similar to Cgs . This issue can be alleviated by fabricating T-shaped gate electrodes:18 the thin part is in contact with the substrate in order to minimize Cgs , while the top side is the wide part, to minimize the overall resistance of the conductor, and thus Rg and Ri . When devices with high saturation currents are required, it is not convenient to increase w above a given limit, in that the FET portion geometrically distant from the input point presents a reduced efficiency, especially at the highest frequencies. This degradation derives from the distributed effects of the gate resistance (Rg and Ri ) and capacitance (Cgs ), which form an RC type of lossy transmission line. This fact is particularly relevant if w is comparable with the wavelength. The input node of the RC transmission line is the access point of the gate, and the output is the opposite extreme, which is approximately an open circuit. Now, the signal travelling from the input towards the output of that RC line is progressively attenuated, therefore the portions of the channel distant from the external access point of the gate are less and less modulated. For that reason, multi-finger transistors with narrower individual channels work better than wide single fingers at higher frequencies. Figure 14.8 shows the structure of a multi-finger at the right of the dashed line. Denoting the number of fingers by nF (in the single finger 17 In order to get an idea of the values involved, let us assume the values Vp 1 V; l 0 l , which are reasonably realistic. The full modulation of the drain current – from zero to Idss – requires the gate–source voltage to swing at least within the interval 0 vgs Vp. The dielectric rigidity needed for vds ¼ 5 V follows from Equation (14.7) and equals 1:2 107 V=m for a channel length of 0.5 mm. 18 That electrode shape is commonly called mushroom.


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FET of Figure 14.8, nF ¼ 1), the total periphery of the FET equals nF w. The main parameters of the transistor models, gm , Cgs , b, Idss , are proportional to the total periphery and approximately independent of nF . Figure 14.8 shows that in a multi-finger19 FET, nF 1 source (S1 to S3) and nF 2 drain (D1 and D2) electrodes interleave with the nF gate ones (G1 to G4), and all the electrodes of the same type are connected together. A metal bridge, realized on the layer METAL2, connects the various source electrodes by means of three metal posts realized on the layer METAL1_2: this interconnecting structure is similar to the one already considered for inductors and capacitors.20 Typically, the most external source pads (S1 and S3 in Figure 14.8) connect to ground by means of via holes, which also work as heat conduction elements, helping to keep the transistor temperature within acceptable limits. For that reason, however, the central portion of the FET is typically hotter than the one close to the via holes: this states an upper limit on the total periphery and to nF . The foundry manual specifies the minimum channel width (typically around 25 mm) and the maximum for vgs and vds . Other parameters, like the channel length, spacing between the various electrodes, sizes of the source bridge and of relative posts, are not modifiable by the MMIC designer. The fabrication of an MMIC FET involves four metal layers: METAL1, METAL2, METAL1_2 and METAL0. The first three layers are the same as for inductors and capacitors, while METAL0 is specific for the gate and requires high precision, due to the small values of l. The additional layer IMPLANT is required to define the doped (or implanted) zone. The regions below the gate metal undergo a different process than the ones below source and drain metal, in that the gate metal–substrate interface has to form a Schottky junction, while the source–substrate and drain–substrate contacts have to be non-rectifying. Additional process-specific layers define the areas for the different treatments. All the electrical models for the FET presented in Section 9.7 are applicable to the MMIC case with no modifications, therefore this section does not deal with them any further. The realization of MMIC diodes is conceptually similar to the FET case. In a first-order approximation, it is possible to consider that a diode derives from an FET, by using the gate–source or the gate–drain junction only. All the linear and nonlinear electrical models of the diode presented in Section 9.7 remain valid, and no further comment is required. Also, the diode fabrication follows the same processes and rules as for the FET.

14.4 Simulation models and layout libraries Sections 14.3.1 to 14.3.5 described the electrical models for the passive linear MMIC elements, while the semiconductor equivalent circuits developed in Section 9.7 hold true for MMIC devices, as Section 14.3.6 pointed out. This section supplies some additional details on the electrical models, in the specific MMIC context. The first point to consider is that all the equivalent circuits – whether linear or nonlinear – have a theoretical foundation. Nevertheless, their values result from a curve fitting to the measured data, which could be [S] parameters, DC curves, results of nonlinear RF tests and combinations of these. Electrical (for the circuit analysis) and graphical (for the fabrication of the required structures on the required layers) data relative to the various realizable elements with a given MMIC process are written in a document known as the process library. Usually, the process library is a collection of files rather than a paper document. Parts of the description contained in the process library are also written in the foundry manual. 19 20

The term finger is used because the gate electrode shape resembles the fingers of a hand. See Sections 14.3.4 and 14.3.5.

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The accuracy, completeness and flexibility of the process library strictly depend on the quality of the characterization that the foundry is able to achieve. From the lowest to the highest level, some solutions are possible, as Sections 14.4.1 to 14.4.5 below show.

14.4.1

Single element models

Knowledge of the scattering parameters is the fundamental requirement of any linear or nonlinear model extraction. The scattering parameters can be used as they are, for the linear circuit analysis. Thus, the lowest possible characterization level of a foundry process consists of fabricating some devices,21 measuring them and storing the scattering parameters obtained in files. If no further characterization is available, the electrical part of the process library is a list of S parameter files – usually in the Touchstone format22 – one per device (and bias condition, if applicable). The graphic part of the library contains the geometry of the required structures on the suitable layers, for the various devices. The number of graphic files is not greater than that for the S parameter files, in that some of the latter could represent the same device with different bias conditions. At this first level of characterization, the only possible circuit analysis is the linear one, with nominal values of the device parameters. A slightly more advanced characterization consists of replacing the S parameter files with equivalent linear circuits, obtained from curve fitting between model and file responses. This gives no appreciable advantage from the circuit analysis point of view and introduces a small degree of approximation.23 On the other hand, the model representation is more compact and is the first step towards more accurate and powerful characterizations.

14.4.2

Scalable models

The second step in the evolution of the process library consists of building scalable models that represent many possible devices of the same type with different geometrical parameters. The variable parameters could be one or more, the corresponding scalable model is one dimensional or multi-dimensional, respectively. The one-dimensional case is simple to explain by referring to one case, like the capacitor in Figure 14.7. Let us assume, for simplicity, a squared device ðwc ¼ lc Þ and that the layout rules allow realizable capacitors with24 25 mm wc 250 mm. Following the procedure of Section 14.4.1, it is possible to fabricate some test square capacitors, for instance with wc ¼ lc ¼ 25; 50; 100; 200; 250 mm, measure the respective scattering parameters and extract the corresponding values for the model of Figure 14.7a. Each of the models is a set of 12 parameter values Z01 , Z02 , ee f f 1 , ee f f 2 , l1 , l2 , Cp1 , Cp2 , Ls1 , Ls2 , Rs and Cs . In other words, a 12-term vector defines each model, and there are as many vectors as the number of different measured capacitors. The vector terms are thus functions of the capacitor side, and the function is known at the discrete points of the variable, corresponding to the measured devices: in our case we have five values from 25 to 250 mm. Now, the model parameters for a generic capacitor side can be obtained by interpolation. The interpolating function can be piecewise linear, a polynomial or a transcendental function. The best solution changes from case to case,25 but allows the replacement of many vectors by one single vector having wc -dependent values. 21 The tested devices could be transmission lines, resistors, capacitors, inductors and FETs with different geometry. The FETs could also have different DC bias conditions. 22 See Section 9.7.1. 23 The model cannot be more accurate than the S parameters it is derived from; rather, it does not give a perfectly identical response, and thus introduces an error, although small. 24 With the typical values in Section 14.3.5, i.e. erc 4; hc 1 mm; the capacitance range resulting from those geometric limits is about 0:19 pF C 19 pF. 25 Frequently used interpolating functions are relatively low-order (2, 3) polynomials, sometimes with additional empirically obtained correction factors. That solution is sufficiently accurate and is relatively easy to implement in the commercially available CAE tools.


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The procedure can be extended to more variables. A typical case is the FET, where the variables could be: total periphery, number of fingers, DC drain–source voltage and DC gate–source voltage, which determine the DC drain current. Many variable parameters involve many measurements with many different devices to realize,26 and multi-dimensional interpolating functions are also difficult to deal with. Nevertheless, the reward for the effort is that the MMIC designer can use the scalable model to increase the number of degrees of freedom of his or her design. It become possible to analyze the circuit performance and sensitivity to one geometric parameter or to a DC bias value of the circuit. The parameters of scalable models can also be used as optimization values to get the best performances from the circuit. It is possible, for instance, to determine the size of the capacitors in one LC filter to tune the frequency bandwidth to the exactly desired value. Another remarkable application of scalable models is the optimization of the DC bias of the transistors, by examining the response of the circuit with the transistor at the various bias values, and choosing the best one.27

14.4.3

Nonlinear models

The circuit analyses performed with linear models do not allow the prediction of the MMIC nonlinear performance, like distortion, power compression, DC to RF efficiency. When those performances are relevant, it is necessary for the foundry to provide nonlinear models for its semiconductor devices. For the electrical nonlinear models themselves, there is no need for additional information to that provided in Section 9.7.2. In the MMIC context, nonlinear model extraction of a device typically involves the extraction of many linear models of the same devices at different DC bias, together with DC curve measurement of the device itself. The remaining part of the task consists of computing the parameters of the nonlinear models from their relations with the parameters of the linear ones and with the DC curves. Very complete and accurate libraries have nonlinear FET and diode models that are scalable with the number of fingers and the periphery of the device, although this requires a remarkable amount of work in terms of fabrication, measurements and computer analysis.

14.4.4

MMIC statistical models

The two main drawbacks of a realized chip are its high cost and the virtual impossibility to change anything inside it in case the resulting performance is not satisfactory. Therefore, the accuracy of the device’s electrical model is as important as the fabrication of the MMIC. For the same reason, it is important to predict by analysis – before realizing the MMIC – the variations of the circuit performance, from piece to piece. This latter aspect involves statistical models, which describe how the device performance changes according to the tolerances inherent in the fabrication process. The statistical analysis is based on models with statistically variable parameters which have a specified nominal value and range of variation. The type of statistical variation could be uniform, within a specified minimum and maximum, or Gaussian with a specified standard deviation. In scalable models, the parameters of the interpolating functions are statistical variables, rather than constant component values. The implementation of this statistical variation is considerably easier with equivalent circuits (parametric or not) than with S parameter files, although it is not impossible in the latter case. This is an additional advantage of the equivalent circuits over their originating S parameter files. It is difficult to retouch the fabricated MMIC if its performance does not meet the specifications, although laser trimming of some parts like transmission lines, inductors and resistors is sometimes possible. Therefore, the MMIC has to be as non-critical as possible by design, and all the possibly high sensitivity to process variations has to be detected and eliminated as soon as possible in the design phase. 26

The number of devices to test increases with a power equal to the number of variable parameters. The definition of the best DC bias depends on the required performance to be optimized. In amplifiers it could be the gain, the input and/or output matching, the noise figure and the total DC power consumption for a specified gain or noise figure. 27

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The statistical analysis is particularly important for MMICs, or more generally for integrated circuits, in that it allows prediction of the variations of the circuit performances over the produced pieces. In other words, the result of the statistical analysis is the prediction of the circuit yield in production. An additional and crucial function of the statistical models is process control. The foundry has to monitor the stability of the fabrication process by realizing, measuring and modelling test devices. The parameters of the resulting models have to fall within tolerances compatible with – and typically better than – the numbers specified in the foundry manual. If this is not so, the cause of the variation needs to be found and removed. The statistical parameters are obtained by measuring and fitting many devices of the same type, geometry and DC bias (if applicable). There is one resulting model with some parameter values for each individual device. The parameter values assume the minimum, mean and maximum values, standard deviation and a statistical distribution. All these characteristics can be extracted and used as a base for a statistical model.

14.4.5

Temperature-dependent models

Circuits have to work with the specified characteristics when the external temperature varies within a given range. The typical working thermal range for military equipment is 54 to þ 85 C. The temperature range for civil applications is narrower; nonetheless an ambient temperature of 20 C is not impossible and the temperature inside a closed car exposed to sunlight in summer can easily exceed þ 70 C. Prediction of the temperature performance of the MMIC by circuit analysis would be beneficial. This would be possible if the relation between the electrical model parameters and the temperature were known and implemented in the model itself, which becomes a temperature-dependent model. Computation of the temperature-dependent functions for the model parameters follows exactly the technique discussed in Section 14.4.2 for scalable models. The only difference is that here the variable parameter is the temperature instead of a geometrical dimension or DC voltage. The three characterizations that give scalable, statistical and temperature-dependent models can be combined together to obtain models with all the three characteristics. The effort involved in the realization and measurement of mathematical models could be really huge, although the result offers a powerful tool to the MMIC designer.

14.5 MMIC production technique MMIC fabrication is the next step after circuit design, once all the simulated performances satisfy the specifications. The circuit layout, containing all the required structures on the various layers, is obtained from the circuit description used for the simulations. In most cases such a process is automatic, in that many CAE tools have automatic layout functions and the graphic libraries are normally linked with the electrical ones. The MMIC fabrication process could be considered as consisting of five distinct phases: substrate preparation, realization of the required structures on the two sides of the substrate, electrical test, cut and selection of the good chips, and packaging (if any). Normally, the foundry buys the semiconductor material from a suitable manufacturer, which supplies that product in the form of a circular disc with a missing and flattened portion, as depicted in Figure 14.10. This basic material, called the wafer, has a diameter of some inches, a typical thickness of 100 mm, is not implanted, and has no deposited metal, although its surface is extremely planar, smoothed and clean. The wafer thickness is also constant within tight tolerances. In most cases, the first operation consists of making the holes – usually with a laser – in the positions of the via holes. Then, the back face of the wafer is metal plated,28 realizing the back ground plane and 28

Typically with gold.

MICROWAVE CIRCUIT TECHNOLOGY chip

653

wafer

horizontal cut lines

vertical cut lines

Figure 14.10

MMIC wafer with the chip reticle.

simultaneously plating the internal walls of the via holes. All these operations are sometimes referred to as the back-end process.

14.5.1 Lithography The operation following the back-end process is the front-end process, which consists of realizing all the structures required for all the elements on the upper side of the wafer. Section 14.3 showed that any circuit elements result from many geometrical figures, realized on different layers. The circuit layout is completely defined as it is for type, space coordinate and orientation of all its devices. Each device placed in a given position of the MMIC requires the realization of its shapes at that point. Now, all the different shapes on the same layer deriving from different devices are realized simultaneously. Thus, for instance, all the geometric figures on METAL1 relative to via holes, transmission lines, resistors, inductors, capacitors and transistors are obtained simultaneously with the same process step. The final MMIC results from the overlap of the various shapes on the respective layers. Lithography is a key technique for defining the geometric figures on the MMIC layers; it can be positive or negative on different layers. Positive lithography differs from negative lithography because of the technique used to form the geometry. In the former case the material is deposited only in the regions where it has to be in the final chip; in the latter case the material is deposited all over the wafer surface and removed where it does not have to be present. Figure 14.11 shows the fundamental phases of a negative photolithography based on a negative photoresist. Beginning with Figure 14.11a, we have a section of the substrate ready to receive the layer on its upper surface. Then the layer is deposited all over the wafer surface, as in Figure 14.11b. Subsequently, a suitable material – the photoresist – is deposited over the previous layer; Figure 14.11c shows the result. Then, Figure 14.11d shows that a mask is superimposed on the photoresist and the material is exposed to the light. The mask covers the entire wafer surface, except the regions destined to form the final structures on the current layer. The fundamental chemical property of the photoresist is that its composition is modified after exposing it to the light. The next phase of the process consists of treating the exposed photoresist to a solvent that melts the non-exposed part only, as in Figure 14.11e. At this point the layer under lithography is protected from the photoresist only in those regions where the final shapes have to be fabricated. A second property of the photoresist is that when exposed it is resistant to the solvent used for etching the layer. Thus, an etching substance in contact with the wafer surface protects the

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(a)

(e)

(b)

(f)

(c)

(g)

light mask photoresist (d) metal substrate

Figure 14.11 Photolithography: (a) naked substrate; (b) substrate after the deposition of the layer under process; (c) after the deposition of the photoresist; (d) exposure to the light; (e) after eliminating the unexposed photoresist; (f) etching of the unprotected material; (g) final aspect of the processed layer, after removing all the photoresist.

material below it, while the material itself is removed wherever the photoresist is absent. Figure 14.11f shows the result. Finally, a third solvent is used to remove all the photoresist, to clean it from above the fabricated structure. This technique is well established and widely used in PCB technology; it is the basis of MMIC realization as well. In the latter case, however, some additional care is required, due to the necessary higher precision: .

The photoresist thickness has to be as small and constant as possible, while presenting no holes or irregularities in the edges of the exposed parts.

.

The light used must have a wavelength as small as possible, in order to minimize the effects of diffraction at the edges of the mask apertures.29 Diffraction produces alternations of illuminated

29

See Figure 1.2.


655

and dark zones, with a distance between two maxima of a wavelength. As a reference, the wavelength of violet light is between 390 and 455 nm. It is too large to ensure sufficient precision in the gate channel. In such critical needs, smaller wavelengths are used, namely ultraviolet (wavelength of the order of 30 nm) or X-rays (wavelength of the order of less than 1 nm). .

Further variations of the lithography eliminate the resist and its associated inaccuracy by directly etching the material, or selectively depositing it.

The last applied layer is a thin isolating layer which covers and protects the whole chip except for the RF pads, DC pads and some sensitive parts of the semiconductor devices.

14.5.2

On-wafer testing

It is possible to test the chips while they are still in the wafer, before cutting and separation. The test on the wafer can be performed at DC (DCOW) and/or RF (RFOW), and the required instruments connect to the chip via probes having elastic metal contacts. For these, there are DC probes for applying bias voltages and RF probes for applying RF stimuli and picking up RF signals. RF probes require the presence of a short path to ground close to the hot RF pad to connect to. For this requirement, MMIC RF pads usually present two side pads connected to ground with via holes, as in the circuits of Figures 10.5 and 11.37. DCOW allows checking for the absence of failures in the circuit, while RFOW can test the effective RF linear and nonlinear performances of the MMIC: scattering parameters, noise figure, output power, linearity, and so on. Conformity of the wafer process is also kept under control. For that reason, test structures populate some portions of the wafer, replacing the real circuits. These test structures form the process control monitor(PCM), and consist of resistors, inductors, capacitors and transistors with different sizes. All the PCM elements are DC and RF tested against suitable specifications. If the yield of the PCM elements is above a specified limit30 the wafer is considered conformal. Clearly, the PCM is present at many points of the wafer, from the centre to the perimeter, in order to check the quality of the fabrication process over the whole wafer surface. Finally, the PCM also includes some standards31 for the calibration of RF cables and probes used for the RFOW.

14.5.3

Cut and selection

The whole wafer surface is populated by chips, with the exception of the PCM and the peripheral regions that are too small for a chip, as Figure 14.10 shows. After the lithography and the electrical test, the wafer is cut and the chips are separated from each other. The cutting tool is a circular diamond saw which can produce only straight cuts, excluding any possibility of zigzags. All the chips of a wafer could be of the same or of different types. The second case frequently occurs in the design phase, when, for cost reasons, one single wafer is used to develop different circuits or multiple versions of the same circuit. In any case, the chips on a row (column) – including the PCM – must have the same height (width), so that the cutting lines are straight. Usually, each chip includes a code, etched on a visible layer, indicating its position (x; y) on the wafer. This makes it possible to measure and identify good and bad chips on the wafer. Coordinate by coordinate, a computer-controlled robot places DC and RF probes on the pads of the chip. Then the test begins, the computer stores its results and compares them with the specifications. The test result is PASS or FAIL, and 30

Typically between 90% and 100%, depending on the test specifications. Open circuit, short circuit, loads and some matched through (50 O) lines. See Section 17.6.2 for more details on the calibration of vector network analyzers. 31

656


the coordinate is marked as relative to a working or non-working chip. After the cut, another computercontrolled machine separates good and bad chips, eliminating the bad and passing the good ones on to visual inspection. The last phase, before packaging (if applicable) and preparation for delivery, consists of a visual control – under the microscope – of the electrically working components. The visual inspection ensures that no mechanical damage is present, in order to minimize the risk of premature electrical failures.

14.6 RFIC RFICs evolved from conventional analogue and digital silicon (Si) integrated circuits, to implement RF functions. Cost32 and RF performance of RFIC technology are both typically lower than for MMICs. However, RFICs allow the realization of not only RF functions but also compact low-frequency analogue and digital circuitry, integrated on the same chip. This way, one single chip can contain the whole architecture of Figures 15.7, 15.12 or 15.13, from the digital input to the RF output and vice versa. Two basic RFIC technologies are currently in use: bipolar and complementary metal oxide semiconductor (BiCMOS) and purely complementary metal oxide semiconductor (CMOS).33 The first one uses bipolar devices for the analogue functions and CMOS transistors for the digital circuitry. This mixed approach combines the advantages of the two types of transistor, each optimized for its specific application. Pure CMOS RFICs, on the other hand, allow more compact solutions, and sophisticated design techniques allow for compensating the lower RF performances of CMOS transistors. The basics of RFIC technology are similar to the MMIC ones, with the following main differences and particularities: 1. The relative permittivity of Si is around 4, while for GaAs it is about 12.5. 2. The conductivity of the intrinsic Si is one order of magnitude higher than in GaAs. Consequently, RFIC passive elements present higher dissipation loss than their MMIC counterparts. 3. RFIC transistors are smaller than MMIC ones. Therefore the RFIC designer is encouraged to use transistor-based circuits34 for parts where MMIC designers use linear passive elements. Typical cases are amplifier bias networks, which mainly consist of transistors (inductors and capacitors) in RFIC (MMIC). For the same reason, RFIC spiral inductors are almost exclusively used for the tank in oscillators35 for the output bias network in power amplifiers.36 4. RFICs have more metal layers – typically around six – than MMICs, allowing denser and more compact interconnections between devices. 5. RFIC technology is usable for frequencies up to 5 GHz, while MMICs work up to frequencies higher than 40 GHz. However, these limits are purely indicative, in that the technology progresses rapidly and constantly, continuously pushing those limits towards higher values.

32 The lower cost of RFICs is mainly due to the lower cost of Si, while the process costs are comparable in the two cases. 33 The qualification ‘complementary’ here denotes the combination of n-channel and p-channel metal oxide field effect transistors on the same substrate. This combination is widely used for digital circuits. 34 See Section 11.6.4. 35 See Sections 12.5.6, 12.8.3 and 12.8.5 for oscillator configurations suitable for RFIC designs. 36 See Section 11.5.


657

Bibliography 1. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 7.2.1, pp. 423–426. 2. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, Chapter 10, pp. 349–400. 3. T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd edition, Cambridge University Press, New York, 2004.

15

RF and microwave architectures 15.1 Introduction The most important application of RF and microwaves is probably in message transmission. Wireless transmissions were the reason for the first studies on RF techniques at the end of the nineteenth century and for the following impressive developments, up to the present day. Wireless systems basically consist of one transmitter, a transmission medium and one receiver. The transmitter produces a quasi-sinusoidal RF signal with some parameters modified according to the message to be transmitted. The transmission medium – which could be the open space between the transmitting and the receiving antennas, a cable or fibre optics – delivers the transmitter output signal to the receiver input, with some attenuation. Finally, the receiver extracts from the RF the information it contains, operating the inverse process of the transmitter. This chapter deals with the most important architectures for RF transmitters and receivers, ignoring propagation media. Section 15.2 presents the basic modulation theory, which is important for describing the working principles of receivers and transmitters. Section 15.3 deals with transmitters, from the basic modulation theory to its implications for the transmitter structures. Section 15.4 describes the most common receiver configurations. Section 15.5 basically describes some transmitting–receiving systems which are typically used for two-way communications. Section 15.6 presents some circuits and subsystems which play important roles in the architectures in Sections 15.3 to 15.5 and find relevant applications also in measurement systems. Section 15.6 completes this chapter with some analyses of real Figure 9.39 and practical architectures.

15.2 Review of modulation theory This section briefly discusses the basic modulation types, introducing the basic concepts to deal with the signals processed by transmitters and receivers. The general form of a modulated signal is vTX ðtÞ ¼ aðtÞ cos½oR t þ fðtÞ


ð15:1Þ

660


The function aðtÞ is the envelope of the signal (15.1) and coincides with the definition used in Sections 11.5.2, 11.5.6.4 and 13.2.2, if its sign is always positive or negative. The functions aðtÞ and fðtÞ produce the amplitude and phase modulation, respectively. Indicating with capital letters the respective Fourier transforms Að f Þ ¼ F ½aðtÞ; Fð f Þ ¼ F ½jðtÞ, we have that Að f Þjj f j> fm ; Fð f Þjj f j> fm ffi 0 with 2pfm ¼ om oR ¼ 2p fR . In other words, the spectrum of both the amplitude and phase modulation signals has a maximum frequency that is much smaller than the RF one. Expression (15.1) does not include the phase noise of the exciter output, which is negligible1 if its contribution is much smaller than intentional phase modulation. Note that jaðtÞj is the envelope of the signal (15.1) if fðtÞ does not present abrupt transitions.

15.2.1

Amplitude modulation

The oldest and probably simplest modulation scheme is amplitude modulation (AM). In AM signals fðtÞ is constant, thus it can be ignored by a convenient choice of the time origin, while aðtÞ is directly related to the message to transmit as aðtÞ ¼ KAM ½mðtÞ þ m0

ð15:2Þ

The multiplying coefficient KAM is a proportionality constant, which determines the RF output power. The mean value of the variable term mðtÞ over time is zero: 1 m ¼ lim T !1 T

T=2 ð

mðtÞ dt T=2

The constant m0 transports no information; however, it can be present on the modulating signal or artificially added to it, in order to simplify receiver operation.2 Substituting the modulating signal (15.2) into the general expression (15.1), and considering fðtÞ as constant, the AM signal assumes the expression vTX;AM ðtÞ ¼ aðtÞ cosðoR tÞ ¼ KAM ½mðtÞ þ m0 cosðoR tÞ

ð15:3Þ

The simplest case to consider is sinusoidal modulation, which occurs when mðtÞ ¼ mp cosðom tÞ. Assuming, without loss of generality, that both m0 and mp are positive, the corresponding transmitted signal is vTX;AM ðtÞ ¼ KAM m0 cosðoR tÞ þ KAM mp cos½ðoR þ om Þt þ KAM mp cos½ðoR om Þt

ð15:4Þ

1 A complete analysis of the phase noise effects on radio links lies outside the purpose of this book. Here we can mention that in phase modulation the phase noise adds to the intentional modulation, degrading the signal to noise ratio. Amplitude modulation is ideally not sensitive to phase noise. Nonetheless, amplitude ripple on the complete chain from the exciter to the receiver can transform the phase noise into amplitude noise. Finally, phase noise produces undesired power on the adjacent channels. This is particularly harmful if the transmitter is close to receivers tuned on adjacent channels. 2 If the modulating signal aðtÞ is always positive or always negative, the demodulation can be performed by means of a simple detector working in the linear region (also known as an envelope detector, see Section 13.2.2). If the sign of aðtÞ is not constant over time, it can become constant by the addition of a suitable constant m0 . The constancy of the sign of aðtÞ implies that the average of aðtÞ itself over time is non-zero. The carrier, which takes a consistent part of the PA output power, then becomes non-zero as well.

RF AND MICROWAVE ARCHITECTURES

Figure 15.1

661

AM signals with sinusoidal modulation and different depth: (a) 50%; (b) 100%; (c) 150%.

Equation (15.4) shows that the AM signal consists of three spectral lines: .

The carrier KAM m0 cosðoR tÞ, having the same frequency as the one of the non-modulated signal, and coinciding with the signal (15.3) itself when no effective modulation is applied ðm ¼ 0Þ.

.

The upper and lower side-tones KAM mp cos½ðoR þ om Þt and KAM mp cos½ðoR om Þt, whose frequencies are the sum and the difference between the carrier and the modulating signal, respectively.

The simple case of sinusoidal AM gives the opportunity for a simple definition of an important parameter. The quantitative parameter to define such performance is the AM modulation depth, usually expressed in percentage units and defined as MD% ¼ 100

maxðaÞminðaÞ % maxðaÞ þ minðaÞ

ð15:5Þ

From the definition (15.5), it follows that the modulation depth of the signal (15.4) is 100mp =m0 %. Figure 15.1 depicts the AM signal for a modulation depth of 50, 100 and 150%. Note that only in the first two cases does the envelope aðtÞ coincide with the one produced by the linear detectors considered in Section 13.2.2. The simple signal (15.4) is useful to reveal two properties of AM signals: 2 1. The power 2 associated with the carrier and with any of the two side-tones is ðKAM m0 Þ =ð2R0 Þ and KAM mp =ð8R Þ, respectively. Therefore the ratio between the total side-tone and the carrier 0 2 power is 0:25 mp =m0 ¼ ðMD% =200Þ2 . Now, the carrier transmits no information,3 hence it is desirable that the carrier level is as small as possible, or, equivalently, that MD% is maximum. 3 The carrier could be helpful to the receiver. Simple AM receivers use a simple diode linear detector as an envelope demodulator, and it works only if the modulation depth does not exceed 100%. More sophisticated solutions, like zero IF receivers, locally reconstruct the carrier and do not need it to be transmitted. In this case, the residuals of the transmitted carrier cause harmful offset and are then totally undesired. See Section 15.4 for further details.

662


δ(f+fR)

δ( f +fR)/2

δ( f – fR)/2 A( f )

M–( f ) –fm2

USB – fm2 – fR

–fm1

fm1

m0 δ( f +fRF)/2 M– ( f +fR)/2

M+ ( f + fRF)/2

fm1 – fR

f

fm2

m0 δ( f – fRF)/2 M+ ( f – fRF)/2

M– ( f – fRF)/2

LSB

– fm1 – fR

(a)

M+( f )

LSB fm2 – fR

USB

– fm2 + fRF – fm1 + fR

fm1 + fR

(b) fm2 + fR

f

0 – fR

Figure 15.2

fR

Spectrum of AM signals: (a) modulating spectrum; (b) modulation spectrum.

2. The maximum instantaneous peak voltage of the signal (15.4) is KAM m0 þ mp with a 2 2 m0 þ mp =ð2R0 Þ. If m0 ¼ mp (respectively, m0 ¼ 0) the maxicorresponding power of KAM mum AM effective power is eight (two) times the total side-tone power. If the modulating signal has a continuous spectrum, as it is more frequent, we need to consider the Fourier transform of the modulating signal. From the definition (B.1) and properties (B.3), (B.6), (B.13), this is VTX;AM ð f Þ ¼ ¼

Að f fR Þ Að f þ fR Þ þ 2 2 KAM KAM m0 m0 M ð f fR Þ þ M ð f þ fR Þ þ dð f fR Þ þ dð f þ fR Þ 2 2 2 2

ð15:6Þ

where the capital letters denote the Fourier transforms of the corresponding lower case time domain functions. The third member of Equation (15.6) includes four terms, which correspond to the three terms of Equation (15.4). The carrier, in particular, is the sum of two Dirac pulses symmetrically placed across the origin: 0:5 m0 dð f fR Þ þ 0:5 m0 dð f þ fR Þ. Figure 15.2 shows a graphical representation of the AM spectrum (15.5) under the hypothesis of a band-limited modulating signal with a maximum frequency much smaller than the carrier, which is how it is in most practical cases. Again, the carrier is present on the output spectrum if and only if the time domain mean of the modulating signal is non-zero.4 The real nature of aðtÞ implies that its spectrum is symmetrical around the origin: Að f Þ ¼ conj½Að f Þ; thus jAð f Þj ¼ jAð f Þj, as in Equation (B.5). Thus the amplitude spectrum in Figure 15.1b consists of four

4

Equivalently, the modulating spectrum presents a Dirac pulse at o ¼ 0.


663

parts with a double symmetry around the origin and around the carrier. Denoting the positive and negative parts of the modulating spectrum as M þ ð f Þ and M ð f Þ, respectively, with5 ( ( Mð f Þ ð f 0Þ 0 ð f 0Þ Mþð fÞ ¼ ; M ð f Þ ¼ 0 ð f < 0Þ Mð f Þ ð f < 0Þ we can rewrite the spectrum (15.6) as dð f fR Þ þ dð f þ fR Þ 2 M þ ð f fRF Þ þ M ð fR f Þ M þ ð fRF f Þ þ M ð f fR Þ þ KAM þ KAM 2 2

VTX;AM ð f Þ ¼ m0

ð15:6Þ

The spectrum (15.7) comprises three terms. The first one is the carrier and consists of two Dirac pulses. The remaining two terms are located outside or inside the region-delimited carrier: they are the lower sideband (LSB) and upper sideband (USB), respectively. Both LSB and USB consist of two symmetrical parts, one with positive and one with negative frequencies. Due to the symmetry between M þ ð f Þ and M ð f Þ, descriptions of microwave circuits consider the positive frequencies only. Note that the spectrum of the AM signal has a total width of 2 fm2 , which is double the maximum frequency of the modulating signal fm2 . Properties 1 and 2, derived for a single tone, hold qualitatively for any modulating signal. They involve demanding requirements on the transmitter output stage. For this reason, the carrier and sometimes one of the two sidebands are suppressed whenever possible, although this implies circuit complication. The suppressed carrier AM is sometimes referred to as double sideband (DSB).

15.2.2

Angular modulation

Besides AM, the other fundamental case to consider is angular modulation, which includes phase and frequency modulation (PM and FM). Generic angular modulation will sometimes be indicated in short by PFM in the rest of this chapter. Equation (15.1) describes angular modulation if aðtÞ ¼ a0 is constant, and fðtÞ is related with the message to transmit: vTX;PFM ðtÞ ¼ a0 cos½oRF t þ fðtÞ

ð15:8Þ

In the PM or FM case the phase or frequency, respectively, varies proportionally to the modulating signal: the relative signals assume the expressions ð15:9Þ vTX;PM ðtÞ ¼ a0 cos oR t þ Kf mPM ðtÞ 8t 1

Consequently, the signal (15.14) becomes vTX;PFM ðtÞ ffi a0 cosðoR tÞ þ

a0 a0 cos½ðoR þ om Þt cos½ðoR om Þt 2 2

ð15:15Þ

6 The choice of sine for PM and cosine for FM is arbitrary. It is always possible to pass from one of the two functions to the other by changing the time origin. The assumption used here leads to simpler expressions. 7 Here, the adjective ‘small’ denotes values that are well below the first zero of the Bessel functions, excluding b ¼ 0, which pertains to all the functions with order greater than 1. From Table 3.5 the smallest of such zeros is p1;0 ffi 2:405. Different considerations suggest consideration of small values of b much smaller than p/2.


665

The angular modulated signal (15.15) presents the same types of terms – carrier and side-tones – as the AM signal given by Equation (15.4). The two cases, however, present different phase relations between lower and upper side-tones. Under the hypothesis of small modulation index, it is possible to derive the PFM spectrum from a generic non-sinusoidal modulating spectrum. Starting from Equation8 (15.9), and generalizing the subscripts to indicate generic angular modulation, we have vTX;PFM ðtÞ ¼ a0 cos½oR t þ KPFM mPFM ðtÞ ¼ a0 cosðoR tÞcos½KPFM mPFM ðtÞa0 sinðoR tÞsin½KPFM mPFM ðtÞ

ð15:16Þ

If the maximum of the quantity jKPFM mPFM ðtÞj – which coincides with the modulation index if mPFM ðtÞ is sinusoidal – is much smaller than p=2, then the approximate expression9 follows from Equation (15.15) vTX;PFM ðtÞ ffi a0 cosðoR tÞa0 KPFM sinðoR tÞmPFM ðtÞ

ð15:17Þ

The same procedure used to derive the spectrum (15.6) gives the spectrum of PFM signals with small modulation index VTX;PFM ðoÞ ffi ja0 KPFM

MPFM ðooR ÞMPFM ðo þ oR Þ dðooR Þ þ dðo þ oR Þ þ a0 2 2

ð15:18Þ

The spectrum (15.17) has the same types of terms – two sidebands and one carrier as in Equation (15.6) – but with different multiplying constants.

15.3 Transmitters As anticipated in the introduction, the transmitter uses the message to be transmitted to modify one or more parameters of the transmitted signal: such a process is known as modulation. The modulation is such that the receiver – which is located at the opposite extreme of the link – can extract the original message from the received signal. RF/microwave transmitters10 basically consist of three cascaded blocks: the exciter, the modulator and the power amplifier (PA). The exciter produces a sinusoidal voltage with a specific frequency that is as stable as possible and at the level required by the modulator. The latter receives two inputs, one from the exciter and one from the message generator, and produces the modulation. Finally, the PA increases the modulated signal up to the required level. Usually, the PA also produces noise and nonlinear distortion products that suitable filters reduce. Sometimes exciter, modulator and PA are not totally separated: the modulator can be part of the exciter and/or of the PA.

15.3.1

Direct modulation transmitters

The present section describes some simple transmitter configurations, capable of generating amplitudeonly or phase-only modulation, to show the implied difficulties and explain the need for more complex solutions. 8

This equation was originally derived for PM, but also applies to FM. Here, limy ! 0 cosðyÞ ffi 1 and limy ! 0 sinðyÞ ffi 0, thus if y p=2 then cosðyÞ ffi 1 and sinðyÞ ffi 0. 10 The separation between RF, microwave and millimetre waves is somewhat arbitrary. A commonly accepted rule defines the frequencies below 2 GHz as RF, the ones between 2 and 20 GHz as microwave, while calling millimetre waves those above 20 GHz. However, in the rest of this chapter we will use the RF denomination, for brevity, each time the precise frequency range has no relevant impact on our considerations, as in many other places in this book. 9

666

MICROWAVE AND RF ENGINEERING ATT1 vOSC(t)

vM1(t)

OSC1

AMP1

vc(t)

vOUT (t)

vM2(t) BP1

(a)

a(t) predistorter vOSC(t)

AMP1

OSC1

vM2(t)

vc(t)

vOUT (t) BP1 (b)

a(t) predistorter MIX1 vOSC(t)

vM1(t)

vIF(t)

R

vOUT (t)

vM2(t)

L

OSC1

BP2

AMP1

BP1

a(t)

(c)

Figure 15.3 Different implementations of AM transmitters: (a) with a variable attenuator; (b) with modulation of the PA: (c) with a mixer.

15.3.1.1 Direct modulation AM transmitters The block diagrams in Figure 15.3 shows three simple implementations of AM transmitters. The circuit in Figure 15.3a exploits a variable attenuator to change the output level of an oscillator over time. In more detail, OSC1 produces the unmodulated or continuous wave (CW) RF signal, The variable attenuator ATT1 modifies the oscillator output voltage according to the voltage (or current) vc applied to its control port. Then the power amplifier AMP1 linearly amplifies the modulated signal. Now, any amplifier generates noise, harmonics and intermodulation between the various spectral components of the amplified signal. Some of these products have a frequency close to the spectrum (15.6) of the signal, and thus cannot be eliminated. Nevertheless, the lowdpass or bandpass filter FILT1 reduces the level of the out-of-band transmitter power below a specified limit. Finally, the present simplified analysis assumes that all the interfacing RF ports of all the components are impedance matched, without significant loss of generality. From a quantitative point of view, the oscillator output voltage is vOSC ðtÞ ¼ vR cosðoR tÞ

ð15:19Þ

n h io ðATTÞ ðATTÞ vm1 ðtÞ ¼ vR s21 ðtÞ cos oR t þ arg s21 ðtÞ

ð15:20Þ

The signal at the attenuator output is

ðATTÞ

where s21 ðtÞ is the transmission coefficient of the variable attenuator ATT1, which varies over time, depending on the control voltage vc ðtÞ.


667

ðATTÞ Section 10.3 showed that s21 ðtÞ (either in linear units or in dB) of variable attenuators depends on the control quantity with a nonlinear law. Conversely, AM requires that the linear magnitude of the ðATTÞ attenuator transmission coefficient s21 is proportional to its control quantity. For this reason, the block in Figure 15.3a includes a predistorter circuit that linearizes the relation between the modulation voltage and the attenuator transmission coefficient. The modulating voltage aðtÞ, the control quantity of the attenuator vc ðtÞ and the magnitude of the ðATTÞ transmission coefficient of the attenuator s21 are related as ðATT Þ ð15:21Þ s21 ¼ h1 ðvc Þ; vc ¼ h2 ðaÞ Then, substitution of the second equation in (15.21) into the first one gives ðATT Þ s21 ¼ h1 ½h2 ðaÞ

ð15:22Þ

ðATT Þ The relation (15.22) is linear between s21 and a if ðATT Þ 1 1 s21 ¼ h1 ½h2 ðaÞ ¼ ka a ) h1 1 fh1 ½h2 ðaÞg ¼ h1 ðka aÞ ) h2 ðaÞ ¼ h1 ðka aÞ

ð15:23Þ

The proportionality constant ka of Equation (15.23) is arbitrary, in that multiplying a by any ka is equivalent to multiplying the linear gain of AMP1 by the same constant: it changes transmitted power, ðATTthe Þ not the modulation characteristics. Thus, ka is such that the maximum of s21 (i.e. the minimum attenuation) is achieved when a reaches its maximum allowed value maxðaÞ, within a convenient margin. This design criterion follows from the consideration that the gain of any controlled two-port network has a maximum. Moreover, attenuators (variable or not) are passive components, thus their gain is upper limited by and smaller than 1. Let us denote the maximum coefficient magnitude of ðATTtransmission Þ ATT1 and the corresponding control quantity by max s21 and maxðvc Þ, respectively. If vc ðATT Þ could hold constant at max sðATT Þ , or decrease. On the other hand, the exceeds maxðvc Þ, then s 21

21

attenuator gain continues to decrease until the control voltage reaches a specified minimum value minðvc Þ. Thus, the attenuation does not change or is not monotonic with vc if the latter falls outside the range of minðvc Þ to maxðvc Þ. In extreme cases, permanent damage to the component could occur. Therefore, additional circuitry ensures that the modulating voltage remains below a specified level.11 Anyway, if the input/output nonlinear transfer characteristic of the predistorter follows the law (15.23) then the transmitter output voltage becomes n h io h i ðAMPÞ ðFILT Þ ðATT Þ ðAMPÞ ðFILT Þ ð15:24Þ vOUT ðtÞ ¼ s21 s21 VRF ka aðtÞ cos oR t þ arg s21 ðtÞ þ arg s21 s21 ðAMPÞ ðFILT Þ ðAMPÞ ðFILT Þ Now, the factors s21 s21 VRF ka and arg s21 s21 are not relevant in that they are constant and correspond to an amplitude scaling and a time translation. The architecture in Figure 15.3a has a number of important limitations: 1. The variable attenuator requires a linearization analogue circuit, which is usually critical to realize. In principle, if the modulating signal assumes only discrete values,12 an SCA13 could be used. This component exhibits any desired characteristic between the digital control word and ðATT Þ The discussion on the maximum allowed modulation voltage assumes that s21 is monotonically increasing with vc , at least for maxðvc Þ vc. In the opposite the considerations remain valid after changing vc for vc. Note ðATTcase, Þ that the linearizing function is the inverse of s21 ¼ h1 ðvc Þ, which has to be monotonic, as the inversion conditions require. 12 As in digital AM. 13 SCA stands for step-controlled attenuator; see Figure 10.23 and the relative description in Section 10.3. 11

668

MICROWAVE AND RF ENGINEERING ðATT Þ s . Unfortunately, in the switching transients from one attenuation to the other, sðATT Þ 21 21 assume uncontrollable values, which degrade the modulated waveform. 2. The power amplifier AMP1 must be linear, because its compression affects the amplitude of the output signal. Some slight compression level can be compensated with a predistorter, which is another critical element. Thus, AMP1 has to generate a maximum output power conveniently smaller than its maximum, with a consequent low efficiency.14 Typically the PA consumes a large portion of the whole transmitter power, therefore a low efficiency in the PA means a low efficiency in the transmitter. 3. The phase variation associated the variation of ATT1 produces undesired PM, represented ðwith ATT Þ by the time-varying term arg s21 of Equation (15.10). The parasitic PM is harmful, in that it widens the emission spectrum, although the AM receiver is usually not sensitive to it. A phaseinvariant variable attenuator15 can significantly reduce such undesired PM, although with additional cost and circuit criticality. In any case, additional parasitic PM comes from the AM–PM conversion of the PA, particularly if it operates under compression.16 Furthermore, the input reflection coefficient of ATT1 is not constant with the attenuation, thus the frequency produced by OSC1 is also dependent on A, due to the pulling.17 4. The transmitter in Figure 15.1a can only generate modulation with limited amplitude variations. Let the attenuator transmission coefficient amplitudes corresponding to the minimum and ðATT Þ ðATT Þ ðATT Þ maximum modulating amplitude be min s21 ; max s21 with min s21 ðATT Þ max sðATT Þ . The quantity 20 log max sðATT Þ =min sðATT Þ is the maximum s 10 21 21 21 21 relative attenuation of the attenuator, and is always a positive finite number.18 Consequently, the ðATT Þ ratio maxðaÞ=minðaÞ also has to be positive. The modulation depth for 20 log10 max s21 = ðATT Þ min s21 g ¼ 10; 20; 30 dB equals MD% ¼ 51:9%; 81:8%; 93:9%, respectively. Therefore, the configuration in Figure 15.1a cannot generate suppressed carrier AM, which presents a modulation depth greater than 100%. 5. The frequency stability of the oscillators described in Chapter 12 is usually insufficient for most transmitter applications.

The block diagram in Figure 15.3b improves the transmitter efficiency by eliminating issue 2. It also improves the performance 4, and simplifies the RF chain, at least in principle. All the other issues remain unchanged. The circuit in Figure 15.3b generates AM signals by varying the bias parameters of AMP1 according to the modulation voltage. The controlled parameters are the DC gate–source and/or drain–source voltage (base–emitter current and/or collector–emitter current) for FET (BJT) amplifiers. This way, the PA always works close to its maximum output power and does not need to be linear, with a consequent high efficiency: typically, class C amplifiers19 are used for this application. Clearly, the output amplitude is not linearly dependent on any combination of the DC control parameters, and a linearization is required, with the same problems as with the variable attenuator. Also, the modulation voltage produces variations in the transmitted signal phase and in the amplifier input reflection coefficient of AMP1 , which 14

See Section 11.5. See Section 10.3. 16 See Sections 9.5.6 and 11.5.7. 17 See Section 12.7, point III. 18 The maximum attenuation of the continuous variable usually ranges from 20 to 60 dB and depends on many factors, such as the technology used (PIN or FET), the maximum working frequency range, the relative bandwidth, the modulation bandwidth, and so on. 19 With a maximum theoretical efficiency higher than 78.5%, as in Figure 11.24 and observation (b) of Section 11.5.5. 15


669

modulates the oscillator frequency oR ¼ 2pfR, by pulling. The scheme in Figure 15.1b can generate a maximum modulation depth of 100%, in that it is possible to obtain a zero output peak amplitude when the drain–source/collector–emitter DC voltage or current vanishes. Figure 15.3c depicts a configuration that potentially removes issue 4. The schematic can also be considered as introducing more modern and flexible solutions. The scheme of Figure 15.3c exploits the mixer multiplication property20 for the generation of AM signals, like the Cartesian modulator described in Section 15.3.3. If the mixer MIX1 behaves like a product detector, its output voltage is vIF ðtÞ ¼ KIF aðtÞ vR cosðoR tÞ

ð15:25Þ

The voltage (15.25) has exactly the same form as Equation (15.3), and the same thing applies to the one resulting from a linear amplification of the voltage (15.25) itself. If MIX1 is non-ideal, the intermediatefrequency (IF) voltage includes multiple mixing products between the harmonics of the modulating and the oscillator voltage, as Section 13.3 illustrates. The filters BP1 and BP2 can be designed to attenuate the latter mixing products strongly. Conversely, the mixing product containing the harmonics of the modulating signal are difficult to eliminate by means of RF filters, and are then more harmful. The nonlinear distortions of the modulating signal must be minimized by keeping the signal amplitude conveniently low. The main drawback of this choice is that with low modulating amplitude, any offset on it or internal to the LO mixer port becomes more relevant, with a consequently increased carrier level. Section 13.3 analyzes the mixer spurs if RF and LO are both sinusoidal, or when one of the two is a dual tone. In transmitter applications, the modulating signal21 has a continuous spectrum, with wide relative bandwidth: thus, the considerations of Section 13.3 need to be extended to this case. This allows applications in both transmitter and receiver analysis. Figure 15.4a presents the amplitude spectrum of a typical modulating signal, denoted as jF ½aðtÞj, by the black solid curve. The corresponding time domain signal aðtÞ can have zero mean value or not; in the second case Að f Þ ¼ F ½aðtÞ presents a Dirac pulse in the origin. Figure 15.4a assumes that the modulation bandwidth is finite, and the lower and upper limits are fm1 and fm2 , respectively, with om2 < 2om1 . Figure 15.4b is the same as Figure 15.4a, but with a modulating spectrum bandwidth equal to one octave: fm2 ¼ 2 fm1 . The shape of jAð f Þj in Figures 15.4a,b is rectangular, for simplicity reasons: the general conclusions derived from the next considerations hold true for any band-limited jAð f Þj. As Section 13.3.10.1 illustrated, the mixing products involving the RF harmonics and the LO fundamental have index ð1; nÞ with n > 1, and are generated by factors proportional to vLO ðtÞvm RF ðtÞ. In our case such factors become proportional to an ðtÞcosðoR tÞ

ð15:26Þ

Passing from the time domain function (15.26) to the corresponding Fourier transform, by applying the Euler exponential form of the cosine and the property (B.6), we have expð joR tÞ þ expð joR tÞ An ð f fR Þ þ An ð f þ fR Þ n F ½a ðtÞcosðoR tÞ ¼ F a ðtÞ ¼ 2 2 n

ð15:27Þ

where An ðoÞ ¼ F ½an ðtÞ. The spectrum (15.27) is determined once An ðoÞ is known, and this can be done for any finite n, by the next applications of the property (B.8). Usually, the most important spurs of the types ð1; nÞ are those with n < 4, as in Table 13.1, therefore the spurious spectra to be computed are A2 ð f Þ ¼ F ½a2 ðtÞ, 20

Mixers ideally behave like multipliers, as Section 13.3, and Section 13.3.1 in particular, explain. Here we assume that the wide-band low-frequency modulating signal is the LO, the sinusoidal high frequency is the RF and the modulated signal is the IF, as in Figure 15.3c. However, RF and LO can be swapped, without affecting the general form of the subsequent considerations. 21

Figure 15.4 AM spectra: (a) amplitude spectrum of the modulating signal having a bandwidth narrower than one octave; (b) as (a) but with one octave bandwidth; (c) amplitude spectrum of the modulated signal.

(c)

(b)

(a)


RF AND MICROWAVE ARCHITECTURES A3 ð f Þ ¼ F ½a2 ðtÞ. From Equation (B.8) we have 8 Að f Þ ¼ F ½aðtÞ > > > > > þð1 > > > > 2 > A ð f Þ ¼ F ½ a ðtÞ ¼ F ½ aðtÞaðtÞ ¼ AðFÞBð f FÞ dF ¼ Að f Þ Að f Þ < 2 1 > > > þð1 > > > > 3 2 > AðFÞA2 ð f FÞ dF ¼ Að f Þ A2 ð f Þ A3 ð f Þ ¼ F ½a ðtÞ ¼ F ½a ðtÞaðtÞ ¼ > > :

671

ð15:28Þ

1

Figures 15.4a,b show jA2 ð f Þj (black dashed curve) and jA3 ð f Þj (grey curve) in the specific simple case of rectangular Að f Þ. From Að f Þ limited within the band ð fm1 ; fm2 Þ, it follows that the band of A2 ð f Þ is ð2 fm1 ; 2 fm2 Þ, while A3 ð f Þ extends over the bands ð2 fm1 fm2 ; 2 fm2 fm1 Þ and ð3fm1 ; 3 fm2 Þ. This assertion holds true for any band-limited Að f Þ, which is relatively easy to demonstrate. Furthermore, Figure 15.4b shows that with one-octave modulation bandwidth, A3 ð f Þ occupies all the frequency range from zero to o3 , and the lower limit of A2 ð f Þ coincides with the upper limit of Að f Þ. If the modulation bandwidth is wider than one octave A2 ð f Þ overlaps with Að f Þ. Hence, if fm2 2 fm1 the second harmonic of the modulating signal cannot even be theoretically filtered from the signal itself. Multiplication by the cosine of Equation (15.26) translates the frequency spectra of the modulating signal and its associated distortion by oR as in Equation (15.27). The left spectral components of Figure 15.4c are relative to the AM spectrum, including the distortion on the modulating signal. The same considerations used to derive Equations (15.28) applied to the higher RF order mixing products ððm; nÞ with 0 < m; n < 4Þ lead to the spectrum shown in Figure 15.4c, centre and right portion. As can be seen, the spectra generated by the RF distortions are well separated from the main product, providing that fR þ 3fm2 2 fR 3m2 . This condition corresponds to fm2 fR =3, and is slightly more stringent than fm2 fR , which is needed in the absence of nonlinear distortions on the modulating signal. The bandpass filter BP1 attenuates the RF harmonics and the relative mixing products. Therefore, the minimum specifications on MIX1 derive from mixing products between the RF and the lowest order LO products ð0Þ vIF ðtÞ ¼ KIF ½aðtÞ0 vR cosðoR tÞ þ KIF aðtÞ vR cosðoR tÞ þ . . . ð0Þ KIF ½aðtÞ0 vR cosðoR tÞ

ð15:29Þ

ð0Þ KIF VR cosðoR tÞ

The first term of the equation ¼ is the mixing product of order ð0Þ (0, 1), and KIF is a constant coefficient representing the respective amplitude. Equation (15.29) can be rewritten as " # ð0Þ KIF þ aðtÞ vR cosðoR tÞ þ . . . ð15:30Þ vIF ðtÞ ¼ KIF KIF ð0Þ

Equation (15.30) points out that the non-ideal mixer adds the constant term KIF =KIF to the effective modulating signal. If the latter has to fall within a specified interval with positive and negative limits, the added quantity has to be smaller than a convenient value. This means that MIX1 must attenuate – ideally suppress – the mixing product (0, 1) by a given amount: in other words, MIX1 is a 180 SBM or DBM.22 The PA AMP1 linearly amplifies the modulated signal, and BP2 reduces the out-of-band unwanted emissions, similar to the other transmitters of Figure 15.3.

15.3.1.2 Angular direct modulated transmitters Figure 15.5 shows the angular modulation corresponding to the AM transmitters in Figure 15.3. 22 See Sections 13.3.3.2 and 13.3.9.1 or 13.3.4, 13.3.9.2 and 13.3.9.3 for more detailed descriptions on some realizations of such mixer types.

672


(a)

(b)

(c)

Figure 15.5 Transmitters with direct angular modulation: (a) PM based; (b) same as (a) but with improved isolation between oscillator and phase shifter; (c) FM based. In principle PM can be generated by adding a variable phase shift to the output signal of an oscillator, as in the arrangement of Figure 15.5a. The oscillator OSC1 could be one of the circuits described in Sections 12.8 and 12.9. The phase shifter PH1 could be a step or continuously variable type. The first case applies only if the modulating signal assumes discrete values, as in digital transmissions. Even in such a restricted application, the main drawback involved with PM generation is related to the switching transients. The switches comprised in the structures of Figures 10.32, 10.33 or 10.36 typically present undefined states in the transients from one stable state to the next. Consequently, when the phase shift passes from one stable value to the next, the transmission coefficient of PH1 varies in a predictably difficult way, producing a parasitic AM and PM on the transmitted signal. Such undesirable behaviour is the exact counterpart to what was observed about ATT1 employed in the transmitter of Figure 15.3a. On the other hand, if PH1 is continuously variable,23 the relation between the phase shift and the control voltage is usually nonlinear, and a predistorter is needed to linearize it, exactly as also observed for ATT1 in Figure 15.3a. Let us momentarily ignore the presence of the network labelled ‘integrator’ and assume ðPH1Þ that the modulating voltage mPM ðtÞ is applied to the predistorter input. Let s21 be the transmission 23

Typically, PH1 is voltage controlled, like the circuit in Figure 10.30b.


673

coefficient of PH1 ; then, by definition of linearization, it is linearly dependent on the modulating voltage. Denoting the proportionality constant by Kf, we have h i ðPH1Þ ¼ Kf mPM ðtÞ arg s21 If the oscillator output voltage is vOSC ðtÞ ¼ vR cosðoR tÞ, the phase shifter output is ðPH1Þ ð15:31Þ vM1 ðtÞ ¼ vR s21 cos oR t þ Kf mPM ðtÞ ðPH1Þ Equation (15.31) coincides with the PM signal (15.9) if the factor v0 s21 is constant. The amplifier AMP1 provides an increase in the level of the modulated voltage up to a specified value, and the bandpass filter BP1 reduces the output harmonics below a specified level. Unfortunately, phase shifters exhibit different attenuation at different phase shifts, as observed at the end of Section 10.4, in point 2. Therefore vM1 is affected by parasitic AM, which is the counterpart of the unwanted PM due to ATT1 in the transmitter of Figure 15.3a. However, the present case is less harmful, in that AM can be consistently reduced if AMP1 operates in compression. For instance, if . .

.

the compression characteristic of AMP1 is given by Equation (9.44), ðPH1Þ s varies by 3 dB within the allowed range of mPM ðtÞ, which corresponds to an AM 21 modulation depth of 17.1%, and the combination of OSC1 output voltage and AMP1 gain is such that the amplifier compression at the maximum attenuation24 is 1, 2 or 3 dB,

then the amplifier compression at the minimum attenuation becomes 2.1, 4.2 or 5.5 dB and the output power variation is 1.9, 0.8 or 0.5 dB, equivalent to an AM depth of 10.8%, 4.8% or 2.7%, respectively.25 Higher compression levels even further reduce the residual AM, although at the expense of reduced effective gain of the amplifier, and of an increasing level of output harmonics, with the consequently more critical specifications on BP1 . A drawback of this solution could be the parasitic PM introduced by the amplifier, due to its AM–PM conversion.26 Usually AM–PM conversion is particularly strong in the moderate compression range (say around 1dBCP) and decreases under the hard saturation regime, which is then more promising for the present application. An additional drawback of the transmitter in Figure 15.5 is the parasitic FM due to the oscillator pulling. Equation (15.31) assumes that the oscillator frequency is independent of the phase shifter state. This is not true in general, in that the phase shifter reflection coefficient is also dependent on the control voltage, and any change of the latter affects oR , due to the oscillator pulling.27 It is possible to minimize – but not eliminate – the parasitic FM by inserting isolation elements, such as combinations of attenuators, isolators and amplifiers, between OSC1 and PH1 . Figure 15.5b depicts the general structure of this solution. The linear two-port network labelled as BUFF1 consists of cascaded attenuators, amplifiers and isolators. The choice and disposition of which element to use is the result of a trade-off among the different requirements: power consumption, noise, linearity, cost, available technology, and so on. ðBUFF1Þ ffi 1 and However, denoting the scattering parameters of BUFF1 as sðhkBUFF1Þ ðh; k ¼ 1; 2Þ , then s21 ðBUFF1Þ ðBUFF1Þ ðBUFF1Þ s , s and s are as small as possible, particularly the last. For instance, a simple 11 11 12 choice could be an amplifier cascaded with an attenuator having attenuation equal to the amplifier gain. The attenuator will be at the input (output) if linearity (noise) is the most demanding requirement. 24

That is, with the minimum input signal. See the Mathcad file 05_AM_Reduction_By_Compression.mcd. 26 See Section 11.5.7. 27 See Section 12.7, point III. 25

674


The reflection coefficient G1 seen by the oscillator relates to the input one of the phase shifter G2 and ðBUFF1Þ as in Equation (9.2) to shk ðBUFF1Þ

G1 ¼ s11

þ

ðBUFF1Þ ðBUFF1Þ s21 ðBUFF1Þ 1s22 G2

s12

G2

ð15:32Þ

ðBUFF1Þ

With the assumption made on shk , and considering that G2 is usually also small, as the phase shifter is impedance matched, Equation (15.32) simplifies to ðBUFF1Þ

G1 ðmPM Þ ffi s11

ðBUFF1Þ

þ s12

G2 ðmPM Þ

ð15:33Þ

Equation (15.33) states that the oscillator sees any variation of the phase shifter reflection coefficient due to PM as attenuated by the reverse transmission coefficient of BUFF1 , which has amplitude much smaller than 1, as required. Clearly the variation on G1 cannot be zero, as perfectly unilateral ðs12 ¼ 0Þ microwave components do not exist. Figure 15.5c shows the schematic of a direct modulation FM transmitter. Ignoring the presence of the differentiator and assuming the modulation voltage mFM ðtÞ is directly applied to the VCO tuning port, the working principle of this configuration is straightforward. The instant frequency of the VCO monotonically depends on the tuning voltage, and such dependency is linear for relatively small output frequency variations.28 If the VCO tuning linearity is not sufficient for the required frequency variation, then a linearization is required, like the attenuator in Figure 15.3a and the phase shifter in Figures 15.5a,b. Since the instantaneous frequency of the VCO depends linearly on mFM ðtÞ, then the output voltage coincides with Equation (15.11), apart of course from a different cosine amplitude. The amplifier AMP1 increases the power of the modulated signal up to the required value, and the bandpass filter BP1 cleans the harmonic out, exactly as in the schematics of Figures 15.5a,b. An additional function of the amplifier is to reduce the output power variation of the VCO with the tuning range,29 with the same modalities and the same problems as discussed for the transmitter in Figures 15.5a,b. An interesting property of PM (FM) transmitters is that they can also produce FM (PM) by the addiction of a relatively simple low-frequency circuit in the modulation path. Section 15.2.1 shows that PM (FM) with the integral (derivative) of a given modulating signal is equivalent to FM (PM) with the same signal. Therefore, by inserting a linear two-port whose output is the integral (derivative) of the input between the modulation voltage and the internal modulation point of a PM (FM) transmitter, we obtain an FM (PM) one. The integrator in Figures 15.5a,b consists of a simple RC lowdpass filter followed by a linear VCVS. The Fourier transforms of the time domain signals, MPM ð f Þ ¼ F ½mPM ðtÞ, MFM ð f Þ ¼ F ½mFM ðtÞ, are related as MPM ð f Þ ¼

Av MFM ð f Þ 1 þ j2pf R1 C1

ð15:34Þ

If RIN ; CF are such that 2pf R1 C1 1, even at the lowest frequency of the modulation voltage, Equation (12.34) simplifies to MPM ð f Þ ffi

Av 1 MFM ð f Þ R1 C1 j2pf

ð15:35Þ

Now, Av =ðR1 C1 Þ is a constant, while the multiplying factor 1=ðj2pf Þ in the Fourier domain corresponds to the integral in the time domain, by virtue of the property (B.19). Therefore, the voltage mPM ðtÞ in the schematic of Figures 15.5a,b is the integral over time of mFM ðtÞ, as required. The main limitation of the 28 For reference, the VCO described in Section 12.9 presents a modulation sensitivity ranging from 235 to 421 MHz/V when the output frequency varies between 15.1 and 17 GHz. However, the observation in Figure 12.29 reveals that the modulation sensitivity varies less than 10% within any 500 MHz of the output frequency range. 29 The VCO described in Section 12.9 presents an output power variation of 3.8 dB within the whole tuning range.


675

integrator presented in Figures 15.5a,b is that it does not work with input voltages presenting DC or even low-frequency components. In that case, the condition oR1 C1 1 cannot be satisfied. On the other hand, the phase shifter can only produce a finite maximum phase shift. DC components applied to the input of an ideal integrator produce ramp at the output, which continues to increase over time, exceeding any finite limit. In other words, PM transmitters based on the configuration in Figures 15.5a,b cannot constantly shift the oscillator frequency.30 Figure 15.5c presents the dual case of the one in Figures 15.5a,b, MFM ðoÞ ¼

oC1 R1 1 joC1 R1 Av MPM ðoÞ ffi joC1 R1 Av MPM ðoÞ 1 þ joC1 R1

ð15:36Þ

From the property (B.18) of the Fourier transform and from Equation (15.36) it follows that mFM ðtÞ is the derivative of mPM ðtÞ, provided that oC1 R1 1 at the opposite side of the integrator. The limitation of the FM modulator when used for PM is that it cannot produce phase steps, because the derivative of a step becomes infinite at the instance of a discontinuity. More generally, the condition oC1 R1 1 is impossible to satisfy if the modulating signal presents frequencies that are too high, i.e. rapid variations over time.

15.3.2

Polar modulator

The architectures described in Section 15.3.1 presents limited flexibility: the diagrams in Figure 15.3 (15.5) can only produce AM (PFM). Figure 15.6a shows the principle of a transmitter using the polar modulator architecture, which potentially can produce any signal of the type (15.1). From a conceptual point of view, the solution depicted in Figure 15.6a is a combination of the phase modulator of Figure 15.3a with the amplitude modulator of Figure 15.5a. Therefore, the new proposed solution presents all the issues of both these configurations, including the need for linearizing the attenuator and the phase shifter control characteristics. However, it is possible to reduce most of the issues consistently by implementing variable attenuation and phase shift by means of feedback control loops. Figure 15.6b depicts one such realization. The components within the dashed block labelled PLL realize a frequency generator, having a stabilized frequency that can also be varied according to a modulating signal. It should be kept in mind that PM can be obtained from FM by differentiating the modulation signal, as described in Sections 15.2.1 and 15.2.2. The components inside the dashed box labelled ALC realize a precise control of the amplifier output power, in order to generate the AM. The acronyms PLL and ALC stand for phase-locked loop and automatic level control, respectively. Sections 15.6.2 and 15.6.3 illustrate PLL and ALC in more detail. For the description of the polar modulator it is sufficient to mention that: (a) The oscillator OSC1 is a VCO, having output angular frequency oR. OSC2 is a crystal- stabilized oscillator with an output frequency fr ¼ or =ð2pÞ that could be in the range of few megahertz up to about 100 MHz. OSC2 presents frequency stability much better than any RF oscillator of the type presented in Sections 12.5 to 12.9. Furthermore, in some designs, the output frequency of OSC2 can be slightly varied around its nominal value. The component labelled FD1 is a digital element known as a programmable frequency divider: its output frequency equals the input one divided by a frequency division factor NDIV . The phase detector PD1 produces an output voltage

30 A constant frequency shift can be obtained by applying to the phase a sawtooth variation 2p high. Such a periodic PM is equivalent to an indefinite linear, since a phase shift of Df þ k 2p (with integer k) is equivalent to a phase shift of Df. If T is the period of the sawtooth, the periodic phase shift produces a frequency shift D f ¼ T 1 or, in terms of angular frequency, Do ¼ 2p=T. The technique described is known as serrodyne.

676


Figure 15.6 Polar modulator: (a) principle; (b) practical implementation; (c) solution with improved phase accuracy.

proportional to the phase difference between the signals at its two inputs. LF1 is the loop filter and presents a lowdpass response. Without any modulating signal being applied, if the PLL is stable and if the loop gain is sufficient, the phase difference between the two inputs of PD1 is zero. This implies that the frequency of those signals also coincides. Thus oR ¼ 2pfR ¼ NDIV or ¼ 2pNDIV fr , where NDIV is a digitally controlled number, and fr is stable as it comes from a crystal oscillator. By changing NDIV , fr and superimposing the modulating voltage on the PLL tuning voltage, it is possible to generate precise control of the output frequency. In general, not all of these three controls are used, depending on the specific requirements of the system.


677

(b) For ALC, it is also present in the oscillator of Figure 12.27. The short description of Section 12.8.5 is sufficient for our purposes, with the addition of considering a linear instead of a quadratic detector. With that variant, and applying the same considerations of Section 12.8.5, it is possible to find that the output RF amplitude is directly proportional to mAM ðtÞ. The solution in Figure 15.6b is appealing because of its capability to realize precise control of the amplitude and phase. Moreover, the amplitude control actuated directly on the PA potentially offers high and constant efficiency at all amplitude levels. One difficulty involved with the polar modulation consists of controlling the amplifier at low output amplitudes, when the detector becomes quadratic, supplying low voltages. In those ranges the output amplitude becomes less precise and the output power – instead of the RF envelope – becomes proportional to the modulating voltage, with a need for linearization. Furthermore, the main drawback that has so far prevented diffused applications of the polar modulator is related to the parasitic PM associated with the amplifier gain control. Figure 15.6c proposes a slightly different disposition of the same components as Figure 15.6b. This solution includes the transmitter output signal within the PLL control, therefore it can compensate for the parasitic PM introduced by the ALC. However, the solution in Figure 15.6c works only if the frequency divider input signal falls within a given dynamic range, which is typically of the order of 10–20 dB. This implies that the polar modulator of Figure 15.6c is usable if the modulated signal presents limited amplitude variations, and this condition also alleviates the low-level operation of the ALC.

15.3.3

Cartesian modulator

The working principle of the Cartesian architecture is based on the fact that the signal (15.1) can be written as the sum of two 90 out-of-phase sinusoids with constant phase and variable amplitude as vTX ðtÞ ¼ vI ðtÞ cosðoR tÞ þ vQ ðtÞ sinðoR tÞ

ð15:37Þ

with31 vI ðtÞ ¼ aðtÞcos½jðtÞ;

vQ ðtÞ ¼ aðtÞsin½jðtÞ

Equation (15.37) has an interesting mathematical representation: the in-phase (subscript ‘I’) and quadrature (subscript ‘Q’) of the signal (15.37) are the real part and imaginary coefficient of the complex vector aðtÞexp½ jfðtÞ ¼ aðtÞcos½fðtÞ þ jaðtÞsin½fðtÞ ¼ vI ðtÞ þ jvQ ðtÞ

ð15:38Þ

From a physical point of view, the signal (15.37) is the sum of the IF outputs of two identical product detectors, whose RF signals are cosðoR tÞ and sinðoR tÞ, while the LOs are vI ðtÞ and vQ ðtÞ, respectively. Whether we consider an ideal product detector or one of the DBMs presented in Sections 13.3.4 or 13.3.9, the RF and LO ports are interchangeable for present considerations. The choice of what signal to apply to what port depends on the performances of the specific mixer and on the particular requirements of the application: maximum input power, port working bandwidth and linearity. Figure 15.7 shows the principle of a Cartesian-modulation-based transmitter. The 90 hybrid coupler HYB1 distributes the signal coming from the oscillator OSC1 to the RF ports of the mixers MIXI and MIXQ with ideally the same phase and 90 of reciprocal phase shift. The IF outputs of MIXI and MIXQ are proportional to vI ðtÞ cosðoR tÞ and vQ ðtÞ sinðoR tÞ, respectively, apart from the conversion gain factor of the two mixers. The lowdpass filters LPI and LPQ limit the bandwidth of the modulating signals to the minimum values that ensure no appreciable linear distortion on the same signals. Such filtering is needed as vI ðtÞ and vQ ðtÞ are typically produced by a digital processor and then transformed into analogue 31

See the identity (A.98).

678

MICROWAVE AND RF ENGINEERING HYB1 vOSC

0°

vI(t) cos (ω Rt)

R L

90°

OSC1

90°

MIXI MIX Q

R

0°

L

vOUT (t) DIV1

BP1

AMP1

BP2

vQ (t) sin ( ω Rt)

R0 vI LPI vQ LPQ

Figure 15.7

Cartesian-modulator-based transmitter.

voltages by means of digital-to-analogue converters (DACs or D/As). This conversion involves the generation of multiple images of the originating spectrum.32 The power combiner DIV1 sums the two IF outputs and supplies the result to the input of the bandpass filter BP1 which attenuates the spurs produced by the mixers. The two remaining components are the amplifier AMP1, which increases the transmitter power up to the required level, and the bandpass filter BP2, which reduces the out-of-band emissions of AMP1 : noise and harmonics. The main drawback of the Cartesian modulator is the need for the PA to operate linearly, with the associated low efficiency. Apart from that aspect, the two mixers together with the associated circuitry (HYB1 , DIV1 , LPI and LPQ ) mainly limit the performance of the transmitter in Figure 15.7. In detail, the main specifications of this transmitter are as follows: 1. Sideband rejection. If the two modulating signals are two 90 out-of-phase sinusoids having the same amplitude v0 and frequency om vI ðtÞ ¼ v0 cosðom tÞ;

vQ ðtÞ ¼ v0 sinðom tÞ

ð15:39Þ

then from Equation (15.37), the output signal is vTX0 ðtÞ ¼ v0 ½cosðom tÞcosðoR tÞ sinðom tÞsinðoR tÞ ¼ v0 cos½ðoR om Þt

ð15:40Þ

The signal (15.40) is a single side-tone having frequency om lower or higher than the carrier, depending on the sign of vQ . If vI and vQ are signals with a continuous spectrum such that VI ð f Þ ¼ F ½vI ðtÞ ¼ jVQ ð f Þ ¼ jF ½vQ ðtÞ, then the spectrum corresponding to the time domain signal (15.37) has an upper or lower sideband rather than side-tone. However, the signal (15.40) can be used to determine by how much the two mixer paths are amplitude balanced and present the required phase shift. In this regard, it is helpful to reconsider the analysis of the block diagram in Figure 13.9b. In particular, Equation33 (13.36) shows that any amplitude imbalance and phase shift in any of the RF, LO and IF paths affect the conversion product. Therefore, even if the 32

See Section 15.4.6 and [6] for further details. Equation (13.36) neglects the LO amplitude; here this is not possible, in that the system is linearly sensitive to the modulating signals by definition. 33


679

externally applied vI and vQ have equal amplitude and 90 reciprocal phase shift, the converted signal expression differs from Equation (15.39) and assumes the expression v0TX0 ðtÞ ¼ v0 ½aI cosðom tÞcosðoR t þ yI Þ aQ sinðom tÞsinðoR t þ yQ Þ

ð15:41Þ

After some manipulation, it is possible to rewrite the signal (15.41) as v0TX0 ðtÞ ¼

v0 aI v0 aQ cos½ðoR om Þt þ yI cos½ðoR om Þt þ yQ 2 2 v0 aI v0 aQ þ cos½ðoR þ om Þt þ yI cos½ðoR þ om Þt þ yQ 2 2

ð15:42Þ

Each of the two terms on the two lines on the r.h.s. of the equation has the form of Equation (A.87) and their frequencies coincide with the ones in the terms of Equation (15.40). Moreover, their amplitudes are given by Equation (A.93) and equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 vLSB ¼ ½aI aQ cosðyQ yI Þ2 þ ½aQ sinðyQ yI Þ2 2 ð15:43Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 ½aI aQ cosðyQ yI Þ2 þ ½aQ sinðyQ yI Þ2 vUSB ¼ 2 Therefore, a real Cartesian modulator produces the unwanted side-tone (or sideband, in the most general case), attenuated by a number depending on the amplitude and phase balance between its internal paths. The attenuation of the unwanted side-tone, evaluated by comparing the amplitudes (15.43) with the ideal case, is a parameter for characterizing the ideality of the modulator from an internal respect. The amplitudes (15.43) can be measured practically, in order to characterize the signal. Normally, the parameter to evaluate is the ratio – in dB – between the wanted and unwanted side-tones. The comparison can be effected either with þ or 90 of phase shift between vI and vQ , with no significantly different results. Choosing the plus sign, the first (second) amplitude (15.43) is that of the desired (undesired) side-tone, the ratio being 3 2 aI aQ þ þ 2 cosðyQ yI Þ vUSB a a 7 6 Q I 20 log10 ¼ 10 log10 4 aI 5 aQ vLSB þ 2 cosðyQ yI Þ aQ aI

ð15:44Þ

Note that rejection of the unwanted side-tone is a function of the amplitude and phase balance, ðaI =aQ Þ and ðyQ yI Þ respectively. Moreover, for reasonably good balancing, the numerator of the logarithm argument on the right of Equation (15.44) is close to four.34 Therefore, it is possible to rewrite Equation (15.44) as 20 log10

vUSB aI aQ þ 2 cosðyQ yI Þ ffi 10 log10 ð4Þ10 log10 vLSB aQ aI

ð15:45Þ

Equation (15.45) coincides with (13.81), apart from the fixed difference of 10 log10 ð4Þ ffi 6:062 dB, providing it expresses the amplitude imbalance in logarithmic units DAdB ¼ 20 log10 ðaI =aQ Þ and the phase imbalance as Ye ¼ ðyQ yI Þ. For this consideration, it is also possible to use the graphs of Figure 13.18 to determine the unwanted sideband suppression with given unbalanced parameters. The amplitude and phase imbalances to consider are not the only ones due to the hybrid, the mixers and the power combiner. The input lowdpass filters LPI and LPQ 34 For instance, for an amplitude imbalance of 1 dB and a phase imbalance of 10 , the result is 3:97 aI =aQ þ aQ =aI þ 2 cosðyQ yI Þ 4:013.

680

MICROWAVE AND RF ENGINEERING also play a relevant role. Filters present high group delay, particularly at frequencies close to their cut-off, with a consequent rapid variation of their transmission coefficient phase over the frequency. Thus small production tolerances can cause large deviations. Some systems apply digital delays as well as amplitude compensations to the generation of vI and vQ , in order to compensate for the imbalances of the RF hardware. However, such correction is either complicated or not effective if the compensation changes with the frequency of the modulating signal and from piece to piece, as is the case for the impairments introduced by LPI and LPQ . 2. Carrier rejection. From Equations (15.1) and (15.37), and from the comments on Figure 15.2, it follows that the output of an ideal modulator presents a spectral line at the frequency oR (carrier) if and only if the mean value of aðtÞ over time is non-zero. The exact fulfilment of this condition requires that the two mixers present infinite rejection of the mixing product ð1;0Þ, or of the product ð0;1Þ if we swap the oscillator and the modulating signal between the RF and LO ports. None of these products is zero in real cases. The test signals used in the previous point can also be used to characterize the carrier rejection of the transmitter as the ratio – expressed in dB – between the amplitudes of the desired side-tone and the residual carrier. Clearly, the defined carrier suppression increases monotonically with modulating signal level, in that the latter increases the sidetone level. This consideration suggests operating the transmitter with amplitudes of vI and vQ as high as possible. Unfortunately, this is in contrast with the linearity requirements that suggest small modulating signal levels. 3. Linearity. As seen many times in Chapter 13, mixers generate nonlinear distortions. Furthermore, Section 15.3.1.1 showed35 that the most harmful products are the mixing between the oscillator signal and the harmonics of the modulating signal. As a result, the output signal of the arrangement in Figure 15.7 is a reasonable approximation of expression (15.1) until jaðtÞj is smaller than a given value. Clearly an upper limit on jaj implies that vI and vQ also have a maximum allowed absolute value. Section 13.3.10 also illustrated several methods to characterize the mixer distortion,36 which can also be applied to the test of Cartesian transmitters. It is possible to use (i) one single sinusoidal voltage, (ii) two sinusoids with different frequency and (iii) signals with a continuous spectrum. All these signals can be applied to only one or both the modulation inputs, and in the latter case the phase relation between the two inputs is also a parameter. In the simplest case (i), if fm is the frequency of the input, the transmitter output spectrum presents all the spectral lines at the frequencies m fR n fm , where m; n are integers. If fm fR , BP1 and BP2 eliminate all the lines with m „ 2, therefore we can focus our attention on the mixing products with f ¼ fR nfm . Ideal product detectors generate fR fm only, and the amplitudes of those two side-tones depend on the phase and amplitude relations between vI and vQ . In the real case the amplitudes of the lines with f ¼ fR þ n fm and f ¼ fR n fm depend on the combination of three factors: the amplitude and phase relations between vI and vQ , the internal amplitude and phase imbalance of the mixers, and their nonlinear performances. However, by measuring the amplitude of one of the lines with f ¼ fR fm at different amplitudes of the input sinusoid, we can obtain a compression curve like the one in Figure 13.28. By measuring the amplitude of the lines with f ¼ fR 2 fm ð f ¼ fR 3 fm Þ it is possible to obtain the second- (third-)order intercept point of the system.37 The dual tone excitation (ii) with frequencies fm1 and fm2 produces all the spectral lines with f ¼ fR n1 fm1 n2 fm2 , where n1 ; n2 are integers; ideally, only the four lines with f ¼ fRF fm1 fm2 . If fm1 is sufficiently close to fm2 , then the unwanted lines at

35

Particularly in the comments to Figure 15.4. Recall that mixers are inherently nonlinear and all the mixing products come from nonlinear distortions. Nevertheless, here we deal with the linearity (and its opposite, the nonlinearity) in the sense defined in Section 13.3.1. 37 The intercept point measurements based on a single tone are accurate only if the input bandwidth is wide enough to transmit the signal and its harmonics with the same gain. Thus the minimum bandwidth for the second- (third-)order intercept point has a relative extension of 2:1 (3:1). 36


681

a(t) cos [ωR1t + φ(t)] a(t) cos [ωR1t + φ(t)] cos(ωR2) MOD1

MIX1 vOUT(t)

R L

BP1

cos(ωR2) OSC1

Figure 15.8

AMP1

BP2

a(t) cos [(ωRF1 + ωR2) t + φ(t)] or a(t) cos [(ωRF1 - ωR2) t + φ(t)]

Transmitter with frequency conversion.

fRF 2 fm1 fm2 and fRF fm1 2 fm2 are close to the desired one, and allow a measurement of IP3. Finally, the continuous spectrum input (iii) produces output spectra having the shape of Figure 15.4c, where the relative amplitude of the two sidebands depends on the correlation between vI and vQ . The distortion spectra modify the shape of the two sidebands as the nonlinearity effects become relevant. Therefore a comparison between the input modulation spectra and the two sidebands is another way of characterizing the nonlinearity of the transmitter.38 As anticipated at the beginning of this point, the linearity requirement suggests the use of low modulation levels, but this contrasts with the requirement of carrier suppression, as explained in the first point.

15.3.4

Transmitters with frequency translation

In principle, the modulators discussed in Sections 15.3.1 to 15.3.3 can be used to generate modulated carriers with variable selectable frequency. This need arises when many users access the same RF bandwidth with the same modulation scheme, like broadcast radio, mobile phones and radio links. The variable carrier frequency adds criticality to the circuits and tends to reduce the modulator performance in terms of accuracy, sideband and carrier suppression, and linearity. For this reason, in variable-frequency transmitters, it is convenient for the modulator to work at fixed frequency, and for a mixer conversion to translate it into a variable one. Figure 15.8 depicts one possible implementation of the idea; it also shows the expressions for the signal at different points of the system, apart from an amplitude factor. The rectangular box labelled MOD1 is one of the modulators described in Sections 15.3.1 to 15.3.3, typically a Cartesian or polar one. The fixed FM ðoR1 ¼ 2pfR1 Þ carrier mixes with the unmodulated variable one ðoR2 ¼ 2pfR2 Þ generated by the oscillator OSC1. The resulting signal has two main spectral components, each of which is a modulated carrier with frequency fR1 fR2. The bandpass filter BP1 transmits one of these two carriers, rejects the other one and attenuates the mixer spurs as much as possible. The PA AMP1 increases the level of the generated signal up to the required value. The bandpass filter BP2 cleans the harmonics and – more generally – all the unwanted out-of-band emissions of AMP1 . For a given output frequency fR, there are infinite possible combinations of fR1 and fR2 : both, one or none of them could be higher than fR . The choice of the transmitter internal frequencies results from a compromise among some contrasting requirements: 38 The Mathcad file 06_Cartesian_Modulator.mcd implements some calculations on the Cartesian transmitter, assuming a cubic nonlinear characteristic. Amplitude and phase imbalances in the RF and in the baseband filters are also taken into account. The modulating spectra could be single tones, multiple tones, continuous spectra, and combinations of them. The correlation between vI and vQ is adjustable.

682

MICROWAVE AND RF ENGINEERING 1. Cleanliness of the mixer conversion. This requires the use of frequencies as high as possible, in that the smallest are fR1 ; fR2 , the closest are the frequency spurs m fR1 þ n fR2 , and they are more difficult to attenuate. 2. Phase noise. From Equations (12.67) and (12.68) it results that, holding the other parameters constant, the phase noise increases with the oscillator output frequency. Therefore, higher frequency oscillators tend to present higher phase noise. Note that phase noise is equivalent to a disturbance superimposed on the phase modulating signal. 3. Modulation accuracy. The high level of amplitude and phase balance required for highly accurate operation of the Cartesian modulator is easily achieved when its frequency operation is low. For instance, any difference in the physical length of one of its two high-frequency paths produces a phase imbalance proportional to the difference itself, expressed as a fraction of the wavelength. Thus, the phase imbalance increases proportionally with the working frequency. 4. Size. If the transmitter circuits use distributed elements, their size is inversely proportional to the frequency. However, small circuit size typically involves additional criticality. 5. Cost. Normally, the circuit production cost tends to increase with the working frequency.

15.4 Receivers The radio receiver is located at the opposite end of the transmitter in RF links; its input can be connected to a receiving antenna, transmission line or fibre optic transducer. The main task of the receiver consists of extracting the information carried by one of the RF input signals, selectable from the many possible ones. Such a process results from the combinations of three functions: (i) filtering over the frequency; (ii) amplification; and (iii) demodulation. Functions (i) and (ii) are to some extent accessories of (iii), which constitutes the core of the receiver. Section 15.4.1 describes the conceptually simplest receiver structure, consisting of a frequencytuned amplifier followed by an amplifier. The analysis of such an arrangement indicates its weakness and the opportunity to use more complicated solutions, based on frequency conversion. Section 15.4.2 illustrates the superheterodyne receiver, which is the most classical – and in most cases, better performing – architecture. Section 15.4.3 describes a kind of simplified version of the superheterodyne, known as the zero- and low-IF receiver. Section 15.4.4 presents the walking IF receiver, whose complexity and performance as well are a compromise between the above two cases. Section 15.4.5 describes an interesting double conversion superheterodyne receiver, realized on a single integrated circuit,39 namely the MC44S803. Section 15.4.6 completes the receiver descriptions by presenting some solutions that reduce the complexity of the RF circuitry at the expense of the digital part, known as sampling and full digital receiver.

15.4.1

RF tuned receivers

The receiver with the structure in Figure 15.9 is the counterpart of the transmitters described in Section 15.3.1. The proposed configuration performs three basic functions: 1. It selects one of the many signals at its input, by means of the two bandpass filters BP1 and BP2 . The two filters have approximately the same passband width, although their selectivity can be different. 39 The integrated circuit MC44S803 is produced by Freescale Semiconductors. The relative descriptions, data and figures are copyright of Freescale Semiconductor, Inc. 2008. Used with permission.


683

Amplitude demodulator

va(t) vIN(t)

DET1 BP1

AMP1

LP1

DIV1

BP2

Frequency demodulator (P2)

vDIV2 vAMP2 DIV2

AMP2

Figure 15.9

(P1) v DIV1

τ2 TL2 MIXI τ1 TL1

vL

vf (t)

I R

vR

LP2

RF tuned receiver with direct demodulation.

2. It amplifies the receiving signal up to the level required for the demodulator, by means of the amplifier AMP1. 3. It extracts the information sent by the transmitter by a process – known as demodulation – that is the inverse of modulation. The receiver in Figure 15.9 includes both one amplitude and one frequency demodulator, which receive the same signal from the output of BP2 through the two outputs of the power divider DIV1. If only the amplitude of frequency demodulation is needed, the superfluous demodulator is omitted and the remaining one connects directly to BP2 . On the presence of the two bandpass filters, at first glance it seems that all the filtering can be achieved with one single filter, placed at the input or output of the amplifier. Usually, this is not the case. The filter BP1 is necessary in order to prevent strong out-of-band signals intermodulating with the interesting one, particularly in the case where the latter is weak. A typical case of that situation occurs when the receiver is distant from the transmitter, and other transmitters are placed in the neighbourhood of the receiver, which irradiate at close frequencies. On the other hand, BP2 is also needed, because it limits the amount of out-of-band noise, produced by AMP1, that reaches the demodulator. The main drawback of this double cascade filter connection is that the resulting filtering function is less optimal than for one single filter. The analysis of this problem is relatively simple under simplifying hypotheses. If AMP1 presents a negligible input and output impedance mismatch, the forward transmission coefficient from the input of BP1 to the output of BP2 is40 ðBP1 þ AMP1 þ BP2Þ

s21

ðBP1Þ ðAMP1Þ ðBP2Þ s21 s21

¼ s21

If the amplifier gain is flat over a bandwidth sufficiently wider than the passband of the filters, the ðBP1 þ AMP1 þ BP2Þ comes exclusively from the two filters, by the factor frequency dependency of s21 ðBP1Þ ðBP2Þ s21 s21 . It is easy to see that if BP1 (BP2 ) presents an optimum polynomial response41 of order ðBP1Þ ðBP2Þ N1 (N2 ), then s21 s21 is polynomial but no longer optimum. The justification of the first assertion is ðBP1 þ AMP1 þ BP2Þ

If this hypothesis is not valid, the quantity s21 results from the next application of Equation (9.2) to BP1 and AMP1, then to the resulting matrix and BP2. 41 For instance, Bessel, Butterworth, Chebyshev or Cauer, which are optimum under proper additional conditions, as shown in Chapter 7. 40

684

MICROWAVE AND RF ENGINEERING single filter, N=6 two cascaded filters single filter, N=3 0.25

0 →

–20

–40 –0.25 –60

20 log10(|s21|)

20 log10(|s21|)

0.00

–0.50 –80 →

–100 0.75

1.00

–0.75 1.25

Frequency, GHz

Figure 15.10

Response analysis of the filters in the receiver of Figure 15.9.

that the product of two rational functions is a rational function with a numerator (denominator) order equal to the sum of the orders of the two numerators (denominators), except for some cross-simplificaðBP1Þ ðBP2Þ tions that do not occur in this case. Moreover, the response of s21 s21 is non-optimum because no Bessel, Butterworth, Chebyshev or Cauer function is the product of two lower order functions of the same type. In particular, if BP1 and BP2 are Chebyshev filters, and their respective bandpass ripple is RP1 ; RP2 , then the ripple resulting from the cascade42 of BP1 , AMP1 and BP2 is RP1þ 2 ¼ RP1 þ RP2 . The total number of resonators is N1þ2 ¼ N1 þ N2 , where clearly N1 (N2 ) is the order of BP1 (BP2 ). Comparing the response of the input of BP1 to the output of BP2 with that of a single filter having the same passband, order N1þ2 and ripple RP1þ2 , we see that the latter exhibits a higher stopband attenuation. Figure 15.10 shows one possible comparison, with two identical filters43 having a passband of 950–1050 MHz, N1 ¼ N2 ¼ 3 and RP1 ¼ RP2 ¼ 0:25 dB. The thick grey curve is the amplitude response of BP1 or BP2. The thick black curve is the amplitude response of the chain BP1 , AMP1 and BP2 , assuming that the amplifier gain is unitary. The thin black curve is the response of one single Chebyshev filter, with the same passband, order N ¼ 6 and ripple of 0.5 dB. A simple comparison between two black curves reveals that both present the same ripple44 of 0.5 dB, but the single filter presents higher selectivity.45 Figure 15.10 presents the case of ideal Richards response lossy-free filters. The comparison considering dissipation loss and other factors of non-ideality (such as differences between BP1 and 42 In this regard, the presence of AMP1 between BP1 and BP2 is important, because it avoids possible spurious resonances due to the out-of-band mismatched impedances of the two cascaded filters. For further details, see Figure 9.9 and the related comments in Example 9.2. 43 See the Ansoft file 01_Cascaded_Filters.adsn. 44 See the right axis, for close-up detail of the in-band response. 45 At the frequencies of 750 and 1250 MHz, which are at the extremes in the graph in Figure 15.10, the stopband attenuation of the single filter is about 20 dB higher than the combination of BP1 and BP2.


685

BP2 , impedance mismatch, amplifier response) is more complicated, and has to be considered case by case. Moreover, the curves in Figure 15.10 assume the same selectivity for the two filters of the receiver. In the real case BP1 must be more or less selective than BP2 if the immunity to strong out-of-band interfering signals is more or less important than the signal to noise ratio. Finally, as an additional drawback, it must be considered that two filters are typically more bulky and expensive than one single filter having more resonators. After discussing the principal aspects of the operation of the circuitry placed in front of the demodulator, we are now ready to examine the latter in more depth. The solution presented in Figure 15.9 consists of two independent circuits, for amplitude and frequency demodulation, as anticipated in point 3. The first circuit to be described is the AM one; the FM one will follow. The voltage at the receiver input is of the type vIN ðtÞ ¼ aIN ðtÞcos

8t 0Þ (solid line), there is a symmetrical corresponding component derived from NT ð f < 0Þ (dashed line). In other

46 On the contrary, when the noise is comparable with the signal, Equation (15.49) shows that the detector output noise in the presence of an input sinusoidal signal is higher than in its absence. In receiver terminology, the minimum input signal, when the detected voltage is barely distinguishable from the noise, is defined as the tangent sensitivity of the detector.


687

N( f )

(a) 0 – fRF + fm

f fR – fm fR

– fRF

fR + fm

– fRF – fm aIN,1N ( f – fRF – fm)

aIN,0 N ( f – fRF ) aIN,1N ( f – fRF + fm)

(b) 0

f

Figure 15.11 Receiver noise: (a) voltage noise density at the detector input; (b) structure of the voltage noise density at the detector output. words, Vd;n ð f Þ ¼ Vd;n ð f Þ, as a consequence of the real nature of vd;n ðtÞ, as expected by virtue of the property (150.5) and as Figure 15.11 shows. The main consequence of Equation (15.52) is that the output detector noise amplitude is almost flat over a unilateral bandwidth approximately extending from zero to one-half of the RF band.47 Moreover, all six terms of the function (15.52) are uncorrelated noise contributions. Therefore, the total output noise power density is proportional to the quadratic sum of the amplitudes of those terms 2 jNT ð f fR Þj2 þ jNT ð f þ fR Þj2 Pd;n ð f Þ ¼ gAv1 aIN;0 R0 2 gAv1 aIN;m jNT ð f fR fm Þj2 þ jNT ð f fR þ fm Þj2 þ 2 R0 2 gAv1 aIN;m jNT ð f þ fR þ fm Þj2 þ jNT ð f þ fR fm Þj2 þ 2 R0

ð15:53Þ

As can be seen, the voltage noise density is almost flat, so we can put N02 ffi jNT ð f fR Þj2 ffi jNT ð f fR fm Þj2 47 If the RF filters are those considered for Figure 15.10, the detector output noise occupies the bandwidth 0–50 MHz, corresponding to 0–108p rad/s in o.

688


The total noise power corresponding to the density (15.53) then becomes g2 A2v1 2 ðLP1Þ PD;N ffi 2aIN;0 þ a2IN;m N02 NBWII R0 ðLP1Þ

The quantity NBWII detector

ð15:54Þ

is the bilateral noise bandwidth of lowdpass filter LP1, placed at the output of the

ðLP1Þ NBWII

¼

1 ð

2 ðLP1Þ s21 ð f Þ d f ¼ 2NBW ðLP1Þ i2 h ðLP1Þ max s21 1 1

ð15:55Þ

ðLP1Þ

The bilateral noise bandwidth NBWII coincides with twice the unilateral noise bandwidth (9.36). As explained in Section 9.4.5, the noise bandwidth of a filter is the bandwidth of an ideal rectangular filter having the same minimum insertion loss and transmitting the same output total noise power as the original filter, when the input noise density is constant. The noise bandwidth of any filter is wider than the nominal passband width of the filter itself.48 Extracting the power corresponding to the main demodulation product of the function (15.51) and dividing the result by the power noise (15.55), we obtain the signal to noise ratio49 " # A2v1 1 1 ð15:56Þ ðS=N ÞdB ¼ 10 log10 2 2 ðLP1Þ 4a2 IN;m þ 2aIN;0 N0 NBWII If the input amplitude is high enough to make the detector operate in its linear region, the demodulated voltage is proportional to the absolute value of the envelope jaIN ðtÞj. Thus, the envelope detector returns a voltage proportional to the modulating signal – apart from the sign – if the modulation depth does not exceed 100%. In that condition, the detected voltage ideally presents no nonlinear distortion, differently from the quadratic operation. The noise performance analysis of the envelope detector is quite difficult, due to the strongly nonlinear operation of the devices: nonlinear circuit simulations, rather than analytic considerations, are the method of choice.

15.4.1.2 FM demodulator The frequency demodulator of Figure 15.9 exploits the mixer MIX1 to compare the instantaneous frequency of the RF input signal at two different times. For this purpose, the power divider DIV2 splits the input signal and delivers it along two different paths. The two portions of the signal reach the RF and LO inputs of MIX1 with different delays. Two 50 O transmission lines TL1 , TL2 provide the required propagation delay, different for each path. The amplifier AMP2 works under strong compression, in order to guarantee a virtually constant voltage to the input of DIV2 , independently of the input envelope. The last element in the chain is the lowdpass filter LP2. It has a cut-off frequency smaller than the carrier and higher than the highest frequency spectral component of the modulating signal. This way, LP2 transmits the demodulated signal to the output, significantly attenuating the high-frequency products of the mixer.

48 The 3.01 dB attenuation points define the passband of Butterworth and Bessel filters, while the equal-ripple frequency range is the nominal passband width of Chebyshev and Cauer filters. See also Chapter 8. 49 An interesting particular case of the detector application is the direct detection of unmodulated or continuous wave (CW) signals, without interposed ampliers. Setting Av1 ¼ 1; aIN;m ¼ 0 and considering the DC component of the detector output voltage (15.51), the signal to noise ratio becomes n io h ðLP1Þ ðS=N ÞdB;CW ¼ 10 log10 a2IN;0 4N02 NBWII

From this equation it follows that halving the p bandwidth of the output lowdpass filter (or the video ffiffiffi pffiffiffibandwidth) corresponds to increasing the input amplitude by 2, which corresponds to increasing S/N by 10 log10 2 ffi 1:5 dB.

RF AND MICROWAVE ARCHITECTURES Let the voltage at the two outputs of DIV2 be 8t 9 omax , then Dt 2p=omax . It is possible to extract the PM from the received signals by integrating the voltage (15.67) over time. That operation can be performed with the simple circuit in Figure 15.5a or 15.5b, with the same issues and limitations as described in Section 15.3.1.2. Although the architecture of Figure 15.9 is in principle able to receive and demodulate any possible transmitted signal, it suffers from three important drawbacks and limitations: 1. The input filters BP1 and BP2 have a minimum realizable fractional bandwidth. Section 8.7.1 stated that the dissipation loss of bandpass filters is inversely proportional to the product between the fractional passband width and the quality factors of the resonators.51 Without examining this aspect in detail, we can limit ourselves to saying that the fractional passband width of BP1 , BP2 cannot be smaller than the Q factor of their respective resonators, multiplied by some units. Now, Q depends on the realization technology and goes from some tens for lumped elements to some thousands for waveguide resonators. Thus the minimum realizable fractional bandwidth is approximately in the range from less than 1% (waveguide) to about 10% (lumped). Consequently, even in the best case, the receiver in Figure 15.9 cannot separate signals with a relative frequency distance smaller than 1%, which means 10 MHz at 1 GHz. Moreover, that minimum performance can only be achieved by using bulky and expensive waveguide filters. In most cases the unwanted interfering signals consisting of the emissions on adjacent channels are closer. For instance, the frequency spacing between two mobile phone channels is typically smaller than 1 MHz, depending on the standard. 2. The receiver in Figure 15.9 can receive only one specific frequency, which coincides with the centre frequency of BP1 and BP2 . Furthermore, the FM demodulator operates properly if the time delay difference fulfils conditions (15.63) and (15.66), and this can occur at some discrete frequencies only. If multiple selectable channels of different frequency have to be received, BP1 , BP2 and at least one between TL1 and TL2 must be variable. This is quite difficult to realize and implies some degradation in the component performances.52 3. Both the AM and the FM demodulators work until the amplitude of the signal at their inputs is within a specified range. The FM circuit, in particular, used AMP2 to compress the dynamics of the received signal. More sophisticated designs use the information coming from the AM demodulator in order to adjust the receiver gain. If the long-term average detected voltage becomes higher (lower) than a maximum (minimum) threshold, then suitable circuitry provides a decrease (an increase) in the gain. Such a function is known as automatic gain control (AGC), and will be described in Section 15.6.3. Variable attenuators placed along the chain implement the variable-gain actuation. Sometimes the same result is achieved by varying the amplifier biasing.

51

See, in particular, footnote 39 of Section 8.7.1. For more details the interested reader can consult [2]. Continuously variable filters can be realized with yttrium iron garnet (YIG) filters, which employ high-Q resonators based on ferrite devices, or with varactor-tuned circuits like the ones described in [3]. Alternatively, multiple selectable fixed filters placed between two SPnTs implement stepped variable filters; such an arrangement is also known as a filter bank. For the variable-length transmission line, it consists of a true delay variable phase shifter, see Section 10.4. 52

692

MICROWAVE AND RF ENGINEERING DEMODULATOR MIXI

LPI AMPI

A/DI vI

R L

2

RF

IF MIX1

vIN

R

90°

L

BP1 AMP1 BP2

BP3 AMP2

DIV1

2 0.5

(vI +vQ )

HYB1 0° OSC2

arg(vI +jvQ )

0° R0

Digital processor

OSC1 vQ

L R

MIXQ

Figure 15.12

15.4.2

LPQ AMPQ

A/DQ

Superheterodyne receiver.

Superheterodyne receivers

The limitations of the receiver in Section 15.3.1 suggest operation of the main part of the process at a fixed (2) and relatively low (1) frequency. The reception of higher and variable frequency emissions is possible if the receiver converts them into lower and fixed ones, by means of a mixer. Such an architecture is known as superheterodyne or, in short, superhet, and is probably the most widely used one in commercial, professional and military radio receivers. Figure 15.12 shows the block diagram of such a receiver, which is the counterpart of the transmitter in Figure 15.8. The structure in Figure 15.12 has some similarities to and some differences from the one in Figure 15.9. Beginning from the output, the demodulator has a different structure than that in Figure 15.9. This new circuit is particularly suitable for modern receivers that widely employ digital circuits; its description will follow shortly. However, the superhet receiver could also use the same demodulator scheme as in Figure 15.9. The chain from the mixer IF to the demodulator input consists of the bandpass filter BP3, having a centre frequency of oI , and the amplifier AMP2. Compared with the corresponding chain BP1 , AMP1 and BP2 in Figure 15.9, one filter is missing at the output. This is because the noise generated by the input stages (mainly AMP1 and MIX1 ) is the dominating contribution to the total noise. Therefore, the reduction in the noise generated by AMP2 loses importance. Nevertheless, an additional bandpass filter after AMP2 could be present in sophisticated and high-performance circuits. The components between the IF port of the mixer and the demodulator input as a whole form the IF section of the receiver. The chain consisting of the cascade of the bandpass filter BP1, the amplifier AMP1 and the second bandpass filter BP2 is the so-called RF section of the receiver. As a whole, it selectively amplifies the input signal and adds some noise to it. The LO and the received signals mix in the mixer MIX1 and generate a new spectral component that is amplified by the IF section and demodulated. If the input signal to be received has the expression in (15.46), the RF chain operates linearly and its forward transmission coefficient is constant across the input signal bandwidth, then the incident wave at the RF mixer input is n h io ð15:68Þ vRF ðtÞ ¼ sðRF Þ aIN ðtÞcos oR t þ fPM ðtÞ þ fn ðtÞ þ arg sðRFÞ þ nR ðtÞ


693

where: .

sðRFÞ is the forward transmission coefficient of the cascaded BP1 , AMP1 and BP2 . The phase arg sðRFÞ is usually almost constant within the bandwidth of the modulated signal. That term can then be eliminated from the expression by changing the origin of the time axis.

.

nR ðtÞ is the time domain expression for noise resulting from the contributions of the input and output of AMP1 . Ðt jPM ðtÞ ¼ 0 DoðtÞ dt is the instantaneous phase deviation of the input signal. Here, we can use

.

much shorter notation than the one used for the function (15.46) in that we do not deal with the frequency deviation directly. Let the LO voltage applied to the LO port of MIX1 be h i vLO1 ðtÞ ¼ vL1 cos oL1 t þ fðnLO1Þ ðtÞ

ð15:69Þ

where fðnLO1Þ ðtÞ is the phase noise produced by the LO LO1 The IF voltage is h i ð1Þ ðLO1Þ vIF ðtÞ ¼ KIF vL1 sðRFÞ aIN ðtÞcos ðoR oLO1 Þt þ jPM ðtÞjn ðtÞ þ jn ðtÞ h i ð1Þ ðLO1Þ þ KIF vL1 sðRF Þ aIN ðtÞcos ðoR þ oLO1 Þt þ jPM ðtÞ þ jn ðtÞ þ jn ðtÞ h i ð1Þ ðLO1Þ þ KIF vL1 cos oL1 t þ jn ðtÞ nR ðtÞ

ð15:70Þ

The first term of the function represents a spectral component placed across oR oL1 ; it falls inside the passband of the IF channel if oI ¼ joR oL1 j. Therefore the superhet receiver in Figure 15.12 can receive both the input frequencies oR ¼ oL1 oI . One of the two is the desired one, the other is the image frequency, and must be rejected by the RF section. The distance between the received signal and its image is 2oIF . Therefore, the IF in the receiver design results from the trade-off between two opposite requirements. Simplification of the IF and demodulator suggests low values of oI . On the contrary, easy design and realization of BP1 and BP2 require low specified selectivity, and thus high oI . Therefore the received frequency can be changed by changing the frequency of OSC1 , while the reception of a frequency higher or lower than the LO depends on the RF filters. The second term of the function (15.70) occupies the spectrum around oR þ oL1 and is therefore rejected by the IF section. The third term of Equation (15.70) has the spectrum n h i o n h i o ð1Þ ðLO1Þ ð1Þ ðLO1Þ F KIF vL1 cos oL1 1t þ jn ðtÞ nR ðtÞ ¼ KIF vL1 F cosðoL1 tÞcos jn ðtÞ nR ðtÞ n h i o ð1Þ ðLO1Þ KIF vL1 F sinðoL tÞsin jn ðtÞ nR ðtÞ ð15:71Þ ðLO1Þ If the phase noise is small, as is desirable, i.e. jn ðtÞ p=2, then the spectrum (15.71) is approximated as n h i o K ð1Þ v L1 ð1Þ F KIF vL1 cos oL1 t þ fðnLO1Þ ðtÞ nR ðtÞ ffi IF ½NR ðooL1 Þ þ NR ðo þ oL1 Þ 2

ð15:72Þ

From considerations similar to the ones developed in Section 15.2.1,53 we obtain from the function (15.72) that the mixer converts into IF the noise around both the received frequency and the image. This conclusion justifies the presence of the second RF filter BP2, placed between the RF amplifier and the 53

In particular, see the comments on Figure 15.2.

694


mixer. In the absence of BP2 , the output noise of AMP1 in the image band is converted into IF together with the one in the RF band. If the two contributions have the same power,54 the consequence is equivalent to increasing the noise figure of AMP1 by 3.01 dB. The first RF filter BP1 is also required, in order to prevent strong out-of-band signals from saturating the receiver, mainly AMP1. An image reject mixer (IRM) substantially alleviates this problem, making BP2 superfluous. In an ideal case, BP1 would also be unnecessary. Unfortunately, the image rejection of the IRM typically falls within the range of 20–40 dB, when, normally, the required image rejection is in excess of 60 dB. Anyway, the IRM simplifies the RF section at the expense of increased complexity of the mixer circuitry: its use can be convenient in integrated circuit realizations. The receiver of Figure 15.12 can operate with the LO higher or lower than the RF to receive. Both these solutions have advantages and disadvantages. Lower LO frequency is basically easier to generate but also involves mixer spurs with closer frequencies. From the demodulator point of view, the only difference between the two cases is that the frequency of the converted signal is negative, or, equivalently, that the first term of the function (15.70) has a sign inversion in the phase deviation.55 However, that inversion is deterministic and can be easily corrected, so we will assume oL1 < oR in the following considerations, without significant loss of generality. Assuming that the IF section operates in the linear region, it amplifies the signal (noise) corresponding to the first (third) term of the function (15.70) and adds some further noise. If the power divider DIV1 is amplitude and phase balanced, it delivers the same signal to both the RF ports of the mixers MIXI and MIXQ . That signal is a fraction of the IF section output, and assumes the expression h i ðIÞ ðQÞ ðIFÞ ð15:73Þ vRF ðtÞ ¼ vRF ðtÞ ¼ ARF aIN ðtÞcos oI t þ fPM ðtÞfðnLO1Þ ðtÞ þ fn ðtÞ þ nIF ðtÞ ðIF Þ

where ARF is the combined gain from the receiver input to each output of DIV1 , and nIF ðtÞ is the time domain expression for the combined noise at the same outputs. Both the signal (first) and the noise (second) term of the function (15.73) have a spectrum centred across the IF oI ¼ joR oL1 j, which is also equal to the one of LO2 , the second LO. The signal produced by LO2 is h i vLO2 ðtÞ ¼ vL2 cos oI t þ fðLO2Þ ðtÞ ð15:74Þ n where fðnLO2Þ ðtÞ is the phase noise of LO2 . The incident wave (15.74) arrives at the LO ports of MIXI and MIXQ with the same amplitude but with a relative phase shift of p=2. More precisely, the signals that the 90 hybrid HYB1 delivers at the two LO ports of MIXI and MIXQ are h i h i ðI Þ ðQÞ ð15:75Þ vLO ðtÞ ¼ a vL2 cos oI t þ jðnLO2Þ ðtÞ ; vLO ðtÞ ¼ a vL2 sin oI t þ jðnLO2Þ ðtÞ The factor a in expression (15.75) is the amplitude of the transmission coefficients of HYB1 , which pffiffiisffi assumed to be amplitude and phase balanced as with DIV1 . In the ideal loss-free case, it is a ¼ 1 2. The RF voltages (15.73) and LO voltages (15.75) mix into the two above-mentioned mixers and generate the IF voltages h i ðI Þ ðI Þ a ðIF Þ vIF ðtÞ ¼ KIF vL2 ARF aIN ðtÞcos jPM ðtÞjðnLO1Þ ðtÞjðnLO2Þ ðtÞ þ jn ðtÞ 2 h i ðI Þ a ðIF Þ þ KIF vL2 ARF aIN ðtÞcos 2oI t þ jPM ðtÞjðnLO1Þ ðtÞ þ jðnLO2Þ ðtÞ þ jn ðtÞ 2 ðI Þ þ KIF nIF ðtÞa vLO2 cos oI t þ jn;LO2 ðtÞ ð15:76Þ 54 Usually they have similar power, in that the amplifier bandwidth is typically wider than the one of the filters, and easily extends up to the image frequency. 55 If oL1 > oR then, since the cosine is aneven function, the first term of the function (15.70) can be rewritten as cos ðoL1 oRF ÞtfPM ðtÞ þ fðnLO1Þ ðtÞfn ðtÞ , which presents positive frequency, but also a minus sign on fPM ðtÞ.

RF AND MICROWAVE ARCHITECTURES ðQ Þ

h i a ðIF Þ vL2 ARF aIN ðtÞsin jPM ðtÞjðnLO1Þ ðtÞjðnLO2Þ ðtÞ þ jn ðtÞ 2 h i ðQ Þ a ðIF Þ þ KIF vL2 ARF aIN ðtÞsin 2oI t þ jPM ðtÞjðnLO1Þ ðtÞ þ jðnLO2Þ ðtÞ þ jn ðtÞ 2 ðQ Þ þ KIF nIF ðtÞa vLO2 sin oI t þ jn;LO2 ðtÞ

695

ðQÞ

vIF ðtÞ ¼ KIF

ð15:77Þ

Similar to what was observed about Equation (15.70), both the signals (15.76) and (15.77) present a useful baseband component (first term), a high-frequency component (second term, centred on 2oI ) and a baseband noise term (the third one). The lowdpass filters LPI and LPQ reject the high-frequency term and transmit the remaining two at the inputs of the two baseband amplifiers AMPI and AMPQ . The outputs of these amplifiers feed the outputs vI and vQ , respectively. Assuming that the cascades of LPI –AMPI and ðI Þ ðQÞ ðIQÞ LPQ –AMPQ have the same transmission coefficient s21 ¼ s21 ¼ s21 in the band of the respective signals, the demodulator outputs are ð15:78Þ vI ðtÞ ¼ GRX aIN ðtÞcos jI;Q ðtÞ þ nI ðtÞ; vQ ðtÞ ¼ GRX aIN ðtÞsin jI;Q ðtÞ þ nQ ðtÞ where: .

.

.

.

ðIQÞ fI;Q ðtÞ ¼ fPM ðtÞfðnLO1Þ ðtÞfðnLO2Þ ðtÞ þ fn ðtÞ þ arg s21 is the phase deviation with the addition of all the phase noise contributions of the transmitter and of the two LOs. It also includes the phase shift due to the output lowdpass filter; usually it is not proportional to the frequency and therefore introduces some distortion on the demodulated phase. ðLPIQÞ ðI Þ nI ðtÞ ¼ s21 KIF nIF ðtÞa vL2 cos oI t þ fðnLO2Þ ðtÞ is the additive noise that reaches the in-phase output, due to the receiver itself and to all the disturbance emissions in the RF band of interest. ðLPIQÞ ðQÞ nQ ðtÞ ¼ js21 jKIF nIF ðtÞa vLO2 sin oI t þ fðLO2Þ ðtÞ is the equivalent of the above term for the n quadrature output. ðIQÞ

ðI Þ

ðIF Þ

ðIQÞ

ðQ Þ

ðIF Þ

GRX ¼ s21 ða=2ÞKIF vL2 ARF ¼ s21 ða=2ÞKIF vL2 ARF is the overall gain of the receiver, from the input to the outputs vI and vQ , provided that the two mixers MIXI and MIXQ have the same conversion gain.

Usually, the overall gain from the receiver input to the IF outputs of MIXI and MIXQ is high enough to neglect the noise introduced from the baseband filters and amplifier, by virtue of Equation (9.32). Therefore, Equation (15.78) neglects these effects. In the absence of noise, and phase distortion due to LPI and LPQ , the extraction of the AM and PM from the signals (15.78) is straightforward. The functions simplify to vI ðtÞ ¼ GRX aIN ðtÞcos½jPM ðtÞ;

vQ ðtÞ ¼ GRX aIN ðtÞsin½jPM ðtÞ

ð15:79Þ

Squaring and summing the two output voltages, we obtain v2I ðtÞ þ v2Q ðtÞ ¼ G2RX a2IN ðtÞcos2 ½jPM ðtÞ þ G2RX a2IN ðtÞsin2 ½jPM ðtÞ ¼ G2RX a2IN ðtÞ

ð15:80Þ

From Equation (15.80) it follows that aIN ðtÞ ¼

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2I ðtÞ þ v2Q ðtÞ GRX

ð15:81Þ

For the PM, it can be extracted from the ratio of the two signals (15.79) as vQ ðtÞ GRX aIN ðtÞsin½fPM ðtÞ ¼ ) fPM ðtÞ ¼ arg½vI ðtÞ þ jvQ ðtÞ vI ðtÞ GRX aIN ðtÞcos½fPM ðtÞ

ð15:82Þ

696


The function (15.82) uses the argument of the complex number vI ðtÞ þ jvQ ðtÞ instead of the arctangent of the ratio vQ ðtÞ=vI ðtÞ, because the latter is only defined within the range of p=2 to p=2. The quantity fPM ðtÞ is bounded within the limits p and p, thus both the voltages vI ðtÞ; vQ ðtÞ can be positive or negative and their sign denotes the quadrant of fPM ðtÞ. In other words, the function arg½vI ðtÞ þ jvQ ðtÞ is the equivalent of the four-quadrant arctangent of the ratio vQ ðtÞ=vI ðtÞ, which coincides with the standard arctangent when vI ðtÞ > 0. A digital processor can easily perform the numerical operations contained in the functions (15.81) and (15.82), provided that vI ðtÞ; vQ ðtÞ are in digital format. Figure 15.12 shows the principle of the last digital process stage. First, two analogue-to-digital converters (ADCs or A/Ds) convert vI ðtÞ; vQ ðtÞ into binary words, then one digital processor computes the operations (15.81) and (15.82) and returns their results as the demodulated signal in digital form.56 The digital process works if the amplitudes of vI ðtÞ; vQ ðtÞ are sufficiently greater than the A/D resolution but small enough not to saturate it. Therefore, the digital processor not only participates in the demodulation, but also monitors its input digital words, and reduces (increases) the overall gain if they become too great (small). Such is the operation of a digital AGC. The various noise types that affect vI ðtÞ; vQ ðtÞ – that Equations (15.81) and (15.82) neglect – cause errors in the demodulated signals. Expressions (15.81) and (15.82) in the presence of noise become, respectively, v2I ðtÞ þ v2Q ðtÞ ¼ G2RX a2IN ðtÞ þ 2GRX aIN ðtÞ nI ðtÞcos jI;Q ðtÞ þ nQ ðtÞsin jI;Q ðtÞ þ n2I ðtÞ þ n2Q ðtÞ ð15:83Þ vQ ðtÞ GRX aIN ðtÞsin fI;Q ðtÞ þ nQ ðtÞ ¼ vI ðtÞ GRX aIN ðtÞcos fI;Q ðtÞ þ nI ðtÞ

ð15:84Þ

Note that if nIF ðtÞ ¼ 0 the phase noise has no effect on the AM, and the error on the demodulated phase is fn;LO1 ðtÞfn;LO2 ðtÞ þ fn ðtÞ, which is the combination of the phase noise of the two LOs and of the transmitter. If the three noises are uncorrelated the resulting spectrum is the quadratic sum of the three amplitudes; hence the transmitter and the two LOs contribute equally to the total phase noise.

15.4.3

Zero-IF and low-IF receivers

Most of the superheterodyne complexity is related to the presence of the image frequency. One possibility to circumvent this difficulty consists of placing the IF bandwidth across zero: this way, the received frequency coincides with its image. Receivers exploiting this working principle are known as zerointermediate-frequency (ZIF) or direct conversion receivers, and have the block diagram depicted in Figure 15.13. Normally a digital processor of the same type as in Figure 15.12 follows after the two analogue outputs vI , vQ , although Figure 15.13 omits it, for clarity. Such a structure basically consists of the same demodulator as the one used in the RX of Figure 15.12, actually operating at the received – and thus potentially variable – frequency, instead of the fixed IF one. One RF amplifier and one RF filter (AMP1 and BP1 , respectively) complete the RF section of the receiver. In order to explain the reason for the absence of a bandpass filter following AMP1 , we need to develop some considerations about the direct conversion receiver. If oRF ; Do are the received frequency and bandwidth, then oRF is by definition the LO frequency. Thus, the received signal occupies the band ðoRF Do=2; oRF þ Do=2Þ while the corresponding band of the converted signals vI and vQ is 56 The schematic of Figure 15.12 only shows the ADC for simplicity. However, the ADC is not sufficient to execute the analogue-to-digital conversion, in that the ADC take a finite time to operate the conversion itself. Therefore, it is required that the input analogue signal remains constant during that time. For this reason a sampling and hold device is placed between the analogue voltage generator and the ADC input. See Section 15.4.6 for more details.


LPI

MIXI v1

697

v2

R

vI

L

AMPI RF 0°

vIN(t)

90°

BP1

AMP1

DIV1

OSC1 0°

HYB1 v3

L R

v4

MIXQ Figure 15.13

vQ LPQ

AMPQ

Zero-IF receiver.

ð0; Do=2Þ. Therefore, the IF bandwidth of each I and Q channel is half the RF one.57 Let us assume that the RF signal is a pure sinusoid within the RF band: the corresponding voltages at the RF input of the two mixers are v1 ðtÞ ¼ v3 ðtÞ ¼ vR cosðoR tÞ ð1Þ

ð15:85Þ ð3Þ

with oL Do=2 oR oL þ Do=2. For the noise, let Vn ð f Þ ¼ Vn ð f Þ be the voltage noise density at the RF inputs of the two mixers. In order to consider the contribution of the RF amplifiers to the receiver noise, we will neglect the power divider and the mixer noise. Under these assumptions, the S/N at the amplifier output coincides with the one at the RF ports of the mixers, and the low-frequency components of the IF voltages are v2 ðtÞ ¼ KIF vR cos½ðoL oR Þt v4 ðtÞ ¼ KIF vR sin½ðoL oR Þt f Þ ¼ KIF Vn;1 ð f fL Þ þ Vn;1 ð f þ fL Þ ð4Þ Vn ð f Þ ¼ KIF jVn;1 ð f fL Þ þ Vn;1 ð f þ fL Þ ð2Þ Vn ð

ð15:86Þ

The last two equations in (15.86) show that the IF noise at positive frequencies comes from two RF contributions: at the frequencies higher than fL and lower than fL . Inverting all the signs and inequality signs in the above, we obtain the IF noise at negative frequencies. For more clarity, Figure 15.14 shows a graphical representation of the various signal and noise quantities at the RF input (dashed lines) and IF outputs (solid lines) of the two mixers. Now, the two contributions to the IF noise are uncorrelated, therefore the resulting noise power density is the sum of the two densities of each contribution: 57

This assertion has more general validity, and also applies to the Cartesian transmitter, discussed in Section 15.3.3.

698


Figure 15.14

Noise and signal conversion in a zero-IF receiver.

2 Pðn2Þ ð f Þ ¼ Pðn4Þ ð f Þ ¼ KIF

ð1Þ 2 ð1Þ 2 Vn ð f fL Þ þ Vn ð f þ fL Þ 2R0

ð15:87Þ

In many practical cases the amplifier output noise is flat across the received band, but in this case the converted voltage noise in the band ðj f fL j Df =2Þ approximates to ð1Þ 2 V ð f fL Þ ¼ V ð1Þ 2 ð f þ fL Þ ffi j f fL j2 ð fL Þ ¼ 2R0 Pð1Þ ð fL Þ n n n

ð15:88Þ

Substitution of Equation (15.88) into (15.87) gives 2 ð1Þ Pðn2Þ ð f Þ ¼ Pðn4Þ ð f Þ ¼ 2KIF Pn ð f Þ

ð15:89Þ

Equations (15.85) and the first two equations in (15.86) give the signal power at the input and output of the mixers, i.e. P1 ¼ P3 ¼

v2R ; 2R0

2 P2 ¼ P4 ¼ KIF

v2R 2R0

ð15:90Þ

From Equations (15.88) to (15.90) it follows that the ratio between the signal power and the noise power density at the mixer output is 3.01 dB worse than at the input. This worsening implies a degradation of the receiver S/N if the bandwidth after the mixer coincides with the receiving band. This is the case for the superhet receiver, where BP2 is present in order to eliminate the image noise of the RF amplifier. Fortunately, the I and Q channels of ZIF receivers require only half of the bandwidth of the modulated signal. Thus, the S/N at the analogue outputs vI and vQ coincides with the one at the AMP1 output, despite the doubling of the noise power density. The ZIF structure is simpler than a classical superhet. The block diagram in Figure 15.13 has one amplifier, one mixer, one LO and two filters less than the one in Figure 15.12. This is a great advantage for compact and low-cost realizations. In this regard, the two removed filters are particularly relevant for integrated circuit realizations, where RF filters are difficult or impossible to obtain. Furthermore, all the receiver selectivity is realized by the two lowdpass filters LPI and LPQ , operating at low frequency, since their cut-off is slightly higher than Do=2. Finally, the two low-frequency amplifiers AMPI and AMPQ produce most of the receiver gain; AMP1 has the minimum gain needed to make the noise of the rest of receiver negligible. The price to pay for the simplicity of the ZIF receiver consists of some reduced performances and/or more critical operation:


699

(a) The IF bandwidth includes the zero frequency, therefore the residual carrier due either to the transmitters or to the receiving mixers causes an offset voltage, which is harmful to the demodulator. (b) The low-frequency amplifiers AMPI and AMPQ also work at DC and produce offset. (c) The IF relative bandwidth is wider than one octave, which makes the receiver susceptible to evenorder distortion, particularly to second-order harmonics and intermodulation. (d) The demodulator works at variable frequency, which is also higher than for the superhet case, therefore it cannot be optimized as in the superhet itself. (e) The LO signal has the same frequency as the passband of BP1 and AMP1 , so it tends to be present in the antenna, due to leakage from the LO ports of the mixers to the input of AMP1 . This leakage can result in a disturbance to the other receivers, and can generate offset, not differently from the non-suppressed transmitter carrier. The offset problem can be consistently reduced if the transmitting signal is not always present over time, and if the non-operative time is known. One typical case is GSM58 transmission, where the transmitting time slot of each physical channel is 577 ms and the successive seven time slots are idle. The mobile GSM receiver then has 7 577 ¼ 4039 ms of idle time to measure and subtract its offset. When the transmission continues, without idle time, that strategy is not possible. In this case, one frequently used approach is the so-called low-IF architecture. In such receivers, the LO frequency is slightly different from the carrier, with a distance of about 100 kHz, and a numerical process downconverts the low-IF band into another one, which starts from DC. In this case, the offset is removed, in that all the low-frequency amplifiers are AC coupled. The resulting digital hardware is more complicated than the ZIF case, but typically digital circuitry is relatively easy to realize within integrated circuits. Issues (c) to (e) are equally present in the direct conversion and low-IF receiver; in particular, careful design and layout must minimize issue (e).

15.4.4

Walking IF receivers

The walking IF receiver, depicted in Figure 15.15, is a compromise between the classical superheterodyne and the direct conversion architectures. The block diagram of Figure 15.14 is similar to the one in Figure 15.12, with the difference that the second LO signal, which is used for the demodulation, is obtained from the first one by means of a frequency divider. The denomination of walking IF depends on the fact that in the receiver of Figure 15.15, both the IF and the demodulator work at variable frequency. In this context, a frequency divider is a purely digital component, and therefore it is particularly suitable for an integrated circuit realization. Ideally, its input voltage is a square wave with specified low and high levels; the corresponding output voltage is still a square wave with the same low and high levels, but with a frequency that is an integer fraction of the input one. In practice, neither the input nor the output are perfect square waves: rather, they are non-sinusoidal waves with positive and negative half waves that exceed the specified levels for a sufficient time. If fLO1 ¼ oLO1 =ð2pÞ is the frequency of the LO OSC1 , then the fundamental frequency at the output of the frequency divider DIV1 is fL2 ¼ oL2 =ð2pÞ ¼ fL1 =NDIV , where NDIV is an integer. The easiest frequency division factor to realize is NDIV ¼ 2; in this case the divider consists of one single T-type flip-flop.59 Frequency division factors equal to integer powers of 2 are also easily realizable, by cascading multiple flip-flops. That output pseudo-square wave feeds the LO ports of the mixers MIXI and MIXQ , and can be used as it is or lowdpass filtered to obtain 58 The acronym GSM stands for global system for mobile communications, which is the digital standard for mobile communication used in Europe and part of the USA. 59 See [4] for more details.

700

MICROWAVE AND RF ENGINEERING DEMODULATOR LPI

MIXI

AMPI vI

R L

RF

IF 0°

MIX1 vIN

90°

R L

BP1 AMP1 BP2

BP3 AMP2

DIV1

0° HYB1

vQ

L R

MIXQ

LPQ AMPQ

FDIV 1 OSC1

DIV2

Figure 15.15

Walking IF receiver.

a sinusoid. The non-filtered wave is typically used when the mixer operates in switching mode, as in that case the nonlinear elements cyclically switch between ON and OFF at the LO frequency.60 Let oR ¼ 2pfR , oL1 ¼ 2pfL1 and oI ¼ 2pfI be the RF input, the first LO and the first IF angular frequency, respectively. They are related as fI ¼ j fR fL1 j ) fF ¼ fL1 fI

ð15:91Þ

The second LO frequency is equal to the IF one – as required for a coherent demodulation – and the first LO divided by the frequency division factor as well fI ¼ fL2 ¼

fL1 NDIV

Substitution of Equation (15.92) into (15.93) gives 1 fR ¼ 1 fL1 NDIV

ð15:92Þ

ð15:93Þ

Equation (15.93) states the LO frequency to apply for given input frequency range and frequency division factors. The sign þ () applies when the LO frequency is higher (lower) than the RF; both the solutions are possible.61 60

See, for instance, Section 13.3.4. The pros and cons of an LO frequency higher or lower than the RF have been discussed in Section 15.4.2 and will not be repeated here. 61


701

Table 15.1 Comparison of the performances of the three receivers in Sections 15.3.2 to 15.3.4 in the case of a mobile terminal for WCDMA

Number of RF filters Number of LOs First LO frequency (MHz) Second LO frequency (MHz) Immunity to even-order distortion Immunity to offset LO leakage to antenna

Superheterodyne

Walking IF

ZIF

3 2 1530–1590 390 Good Good Low

3 1 1536–1584 384–396 Good Good Low

1 1 1920–1980 –– Poor Poor Medium

The operation of a walking IF receiver can be further clarified by means of an example. Let us assume use of the walking IF architecture in order to realize a WCDMA62 mobile receiver. The receiving frequency range is fR 2 ½1920; 1980 MHz, with a total bandwidth of 60 MHz. Choosing NDIV ¼ 4 and oRF > oLO1 , Equation (15.93) with the þ sign implies that fL 2 ½1536; 1584 MHz. The IF bandwidth must be wide enough to pass all the possible RFs converted by MIX1. From Equation (15.92) the minimum IF bandwidth is fI 2 ½384; 396 MHz, which corresponds to an IF bandwidth of 12 MHz. However, the IF bandwidth resulting from such calculations has to be increased, in order to cover the non-zero bandwidth of the received channels. The second LO signal, generated from the first one, has the same frequency of ½384; 396 MHz as the centre of the walking IF, and is used for the demodulation. Table 15.1 compares the above-described solution with a superheterodyne63 and a direct conversion receiver. Observations: (a) All the circuits of the walking IF receiver operate with a fractional bandwidth much smaller than one octave, with the exception of the demodulator, like a conventional superheterodyne. This makes the receiver immune to the even-order nonlinear distortion products, differently from the ZIF case. (b) The gain of the low-frequency amplifiers is lower than in the ZIF case, due to the presence of AMP2 . This proportionally reduces the output offset. (c) The LO frequency falls out of the passband of the RF section, like a superhet and opposite to a ZIF, with consequent low leakage emission of the LO.

15.4.5

One practical IC-based receiver

Figure 15.16 shows the block diagram of the integrated circuit MC44S803, produced by Freescale Semiconductor.64 The circuit in Figure 15.16 is a single IC receiver for analogue and digital television. The complete receiver needs few additional external parts, namely one RF input balun, two RF filters, the demodulator and a few passive components, including the crystal resonator for the PLL reference oscillator. The MC44S803 is a 3.3 V, low-power (it dissipates as low as 700 mW typical), highperformance, single chip CMOS broadband tuner. It implements a dual conversion superhet with a 62 The acronym WCDMA stands for wide-band code division multiple access and is the standard used for the universal mobile telecommunication service (UMTS). 63 The fixed IF of the considered superheterodyne equals the arithmetic mean of the walking IF extremes fI ¼ ð384 þ 396Þ=2 ¼ 390 MHz, still with fR > fL1 . 64 Data taken from the component data sheet, Document Number: MC44S803, Rev 2.0, 09/2008. Copyright of Freescale Semiconductor, Inc. 2008. Used with permission.

702


Figure 15.16

Block diagram of the Freescale integrated circuit MC44S803.

first IF of 1086 MHz, which is higher than the maximum RF, whose band is 48–861 MHz. In order to explain the advantages of this solution, we need to recall the classical TV receiver structure that was widely used at least up to the 1980s. Such receivers converted the band of 48–861 MHz into an IF of about 40 MHz. This implies that the image frequency is distant from the corresponding image by 80 MHz. A single filter having a passband as wide as the RF input one cannot reject the image. Therefore, old-style TV tuners embodied varactor-tuned bandpass filters having relatively narrow passband width, which had to be aligned with the VCO used as an LO for the conversion. This alignment was a critical aspect of the architecture. The second criticality was the frequency extension of the VCO, which had to cover the bandwidth from 48 þ 40 ¼ 88 MHz to 861 þ 40 ¼ 901 MHz, with a relative extension65 of 901=88 ffi 1024%. That bandwidth is too wide for a single VCO: thus multiple combinations of VCOs were needed to cover the full RF band. Sometimes, multiple tunable filters and/or amplifiers and/or mixers combined with switches served the sub-band66 of each single VCO. However, such a solution is complex from a circuit point of view, and difficult to realize with the IC technique, particularly because of the tunable filters. The solution implemented within the MC44S803 eliminates the above-mentioned inconveniences. The high IF of 1086 MHz implies a distance of 2 1086 ¼ 2172 MHz between the received frequency and its image. This means that at the lowest RF frequency of 48 MHz the image is at 2220 MHz, which is 2220861 ¼ 1359 MHz above the higher limit of the RF band. Therefore a relatively simple filter placed at the input is sufficient to reject the image. If no strong input signal is present at the image frequency, the natural passband behaviour of the input balun, combined with the limited bandwidth of the input amplifier, can provide all the required rejection. Furthermore, the lowdpass response of the RF amplifier limits not only its gain, but also its noise at the image band, making the RF filter superfluous after the amplifier. The final advantage of the solution depicted in Figure 15.16 is the relatively narrow bandwidth of the VCO, which is 1134–1947 MHz, corresponding to a relative extension67 of about 172%. 65 That value results from using a LO frequency constantly higher than the RF. In the opposite case the VCO has to cover the even wider relative bandwidth of 821=8 ¼ 10 262:5%, which is about 10 times wider. 66 The subdivision of the receiving band into sub-bands gives the opportunity to reduce the overall VCO coverage slightly, by using an LO frequency higher (lower) than the RF at the low (high) sub-bands. This way, the relative extension of the VCO frequency becomes 821=88 ffi 933%. 67 The higher frequency, compared with the traditional configuration, is a further advantage of the IC realization, in that it minimizes the size of the VCO tank inductor. However, looking at the same aspect from a relatively old time perspective, we see that higher frequencies are not desirable, due to their inherent technological difficulties.


703

The fixed-frequency conversion from 1086 to the second IF, which can be any value between 30 and 60 MHz depending on the filter used, is quite straightforward. The only comment is that the image of the second conversion – which is from 60 to 120 MHz distant from the second IF itself – suggests a passband width of the first IF filter no wider than 20 MHz. That limit is compatible with the surface acoustic wave (SAW) technology.68 Two PLLs lock the two LOs of the two frequency conversions to a crystal-stabilized oscillator. Note that all the RF and IF stages of the circuit present balanced input and outputs. This is a straightforward consequence of the realization of the amplifiers by means of the DP and of the mixers with Gilbert cells. Indeed, these cells are common building blocks in IC technology, as anticipated in Sections 11.6.3, 12.5.6, 12.8.3 and 13.3.9. From a mechanical point of view, the MC44S803 is available in a Pb-free 64-pin quad lead-less package (QFN) of 9 9 1 mm body size.

15.4.6

Digital receivers

The receivers described in Sections 15.4.2 to 15.4.5 may use digital demodulators which involve the ADC process.69 This section gives some additional details of that process and explores some further possible receiver architectures that exploit similar principles. Figure 15.17a shows the block diagram of a DAC subsystem. Its core element is the ADC itself. As anticipated in Section 15.3.2, the ADC needs a finite time TADC to operate the analogue-to-digital conversion. During that time, the ADC analogue input voltage must be constant, in order to avoid the wrong output of digital words. Analogue signals are by definition continuously variable, therefore an intermediate device is needed to keep the ADC input voltage constant during each conversion. Such a device is known as sample and hold (S/H) and has the principle shown in Figure 15.17a. It consists of an input capacitor ðCÞ to ground with one series switch ðSWÞ placed between the S/H input and output. The same clock signal – consisting of a square wave of period Ts and suitable amplitude – controls both the SW and ADC. The switch is normally open (OFF), with the exception of short time intervals ts Ts at each leading edge of vCLK . In those instants SW becomes ON and charges C to a voltage equal to the input voltage at those times v1 ðkTs Þ, where k is an integer. This requires that the input generator presents a low series impedance for a fast charging of the capacitor. If this is not the case, the unit voltage gain, low output impedance amplifier AMP works as a buffer and provides the required high charge current. Figure 15.18 shows the shape of the continuous wave (v1, solid curve) and of the sampled (v2, dashed line) voltages. Clearly, the time available for the analogue- to-digital conversion is slightly shorter than one clock period, which must then be longer than the minimum analogue-to-digital conversion time: Ts > TADC . Momentarily ignoring the quantization effect due to the ADC, it is possible to derive the relation between the continuous-time and its corresponding sampled signal. In this respect, the block diagram in Figure 15.17a is equivalent to the one in Figure 15.17b. In particular, the S/H operation is functionally equivalent to multiplying the continuous-time signal by a periodic sequence of Dirac pulses, and to filtering the result with a suitable filter. Let the periodic pulse sequence – or the sampling signal – be vs ðtÞ ¼

þ1 X

dðtkTs Þ

ð15:94Þ

k¼1

68 This book does not deal with SAW filters. Briefly, we can say that they exploit conversion of the electromagnetic energy into and back from acoustic waves propagating within special piezoelectric crystals. The interested reader can consult [5] for more details. 69 See [6] for further details.

704

MICROWAVE AND RF ENGINEERING S/H LPF v1

vI N

SW

v3

v4

to the digital processor

C

AMP vCLK

(a) CLK S/H LPF vI N

v1

HLP( f )

v2

HSINC( f )

v3

fA/D (v3 )

v4

to the digital processor

vs = Σ δ (t – k Ts ) (b) sampling mixer BPFRF vI N

LPFIF v1

R

I

v2

to the IF section

L

Σ δ (t – k Ts )

comb generator cos(2 π t / Ts ) (c) local oscillator

Figure 15.17 Sampling functional blocks: (a) ADC with sample and hold; (b) equivalent functional diagram of (a); (c) sampling receiver structure.

The voltage at the output of the multiplier is

v2 ðtÞ ¼ v1 ðtÞ vs ðtÞ ¼

þ1 X

v1 ðkTs ÞdðtkTs Þ

ð15:95Þ

k¼1

The function (15.95) represents a sequence of Dirac pulses located at integer multiples of the sampling time: the area of each pulse equals the input signal at the same time. Figure 15.18 illustrates v2 ðtÞ with thick grey vertical segments, each representing a Dirac pulse having an area equal to the length of the


705

v1, v2, v3

v1

v3

0

t

v1(Ts)

v1(2Ts)

v1(3Ts)

Figure 15.18

Continuous-time and sampled waveforms.

segment. The expression for the S/H output is similar to Equation (15.95), but unitary height rectangular pulses of width Ts replace the Dirac pulses: v3 ðtÞ ¼ Ts

þ1 X

v1 ðkTs ÞrectTs ðtkTs Þ

ð15:96Þ

k¼1

From the functional block diagram, it is possible to model the transformation of the function (15.95) into the function (15.96) with one linear two-port block, indicated as HSINC in Figure 15.17b. Such an element transforms the Dirac pulses coming from the ideal sampler, and having area v1 ðtk=ks Þ, into rectangular pulses with height equal to the same value and width equal to the sampling period 1=fs . In other words, if the input of HSINC is a unitary Dirac pulse dðtÞ, the output is a rectangular pulse, beginning at t ¼ 0, ending at t ¼ 2p=os and with unitary height. Since the frequency response of any linear network is the Fourier transform of its pulse response, this gives HSINC ð f Þ ¼ Ts sincðpTs f ÞexpðjpTs f Þ

ð15:97Þ

where 8 < sinðxÞ x sincðxÞ ¼ : 1

x„0

and fs ¼ 1=Ts

x¼0

The frequency response (15.97) has the well-known amplitude response shown in Figure 15.19. It exhibits a lowdpass characteristic with the maximum gain at f ¼ 0. The amplitude response becomes 2=p ffi 0:637 (corresponding to about 3.92 dB) of its maximum at half of the sampling frequency, and vanishes at integer multiples of the sampling frequency, excluding zero. The factor expðpTs f Þ of Equation (15.95) corresponds to a frequency constant delay of half the sampling period. Let V1 ð f Þ ¼ F ½v1 ðtÞ be the spectrum of the continuous-time input signal; then it is possible to demonstrate that the spectrum of the ideal sampled signal is V2 ð f Þ ¼ F ½v2 ðtÞ ¼ fs

1 X

V1 ð f k fs Þ

ð15:98Þ

k¼1

The spectrum of the S/H output is obtained by weighting the function (15.98) with the filtering function (15.97). That lowdpass action is negligible if the sampling frequency is much greater than

706

MICROWAVE AND RF ENGINEERING 1 sinc(π f/fs )

2/π

0 –5

–4

Figure 15.19

–3

–2

–1

f/fs 1

2

3

4

5

6

Normalized amplitude response of the function (15.97).

the bandwidth of v1 ðtÞ. Under this assumption, the spectrum of the sampled signal contains infinite replicas of the continuous-time signal. The multiple replicas are separable from each other if the sampling frequency is higher than twice the unilateral bandwidth of the continuous-time signal. That is, if f Df ) V1 ð f Þ ¼ 0, then the minimum sampling frequency is fs > 2Df

ð15:99Þ

The inequality (15.99) expresses the condition for the reconstruction of the original signal from the sampled one, as in the famous sampling theorem.70 Figure 15.20a plots the spectrum of the continuoustime (thick black) and sampled (thin grey) spectrum, in the critical case of fs ¼ 2D f . The figure shows that it is possible to separate the fundamental signal spectrum (located at j f j Df ) from the spectral replicas from each other only by means of an ideal brick-wall lowdpass filter, which is not physically realizable. Figure 15.20b shows what happens to the sampled spectrum when fs < 2D f , in violation of the condition (15.99). The multiple replicas overlap and their summation modifies the shape of the fundamental spectrum. Figure 15.20b indicates the overlapping zones, or the aliases, by the grey shaded triangles. Finally, Figure 15.20c shows a more convenient case, which strictly respects the condition (15.99). There is a free band os 2Do on the left and right of the fundamental part of the sampled filter, which is the transition band available for the lowdpass filter that has to remove the spectral replicas. The two cases described in Figures 15.20b,c are sometimes referred to as undersampling and oversampling, respectively. At this point, it must be clarified that, in the present context, the above-mentioned lowdpass filter – sometimes indicated as the reconstruction filter – is not a physical network, rather it is part of the digital demodulation process that follows the ADC. The discussion on the sampled spectrum highlights the fact that the original continuous-time signal can be reconstructed from its sampled version if and only if its bandwidth is limited. It is possible that unwanted out-of-band energy folds in the signal fundamental band, from the effect of the sampling. This must be avoided, by drastically limiting the bandwidth of the signal before sampling, by means of a lowdpass filter. Such a component is indicated as LPF in Figures 15.17a,b. It reduces the level of the outof-band energy ðj f j > fs =2Þ below a specified limit, compatible with an acceptable distortion of the sampled spectrum. 70 Many authors, some independently of each other, demonstrated the sampling theorem, namely Whittaker, Nyquist, Shannon and Kotelnikov. Sometimes, some of them are not mentioned. The interested reader can find the original Shannon paper in [7].


707

V1( f ) , V2( f ) fs (a)

O

f

∆f

fs (b)

O

f

∆f fs (c)

O

f

∆f

fs (d)

O

|VIN ( f )|

V 2( f )

∆f

f

Alias

Figure 15.20 Spectra of sampled signals: (a) undersampled lowdpass signal; (b) critically sampled lowdpass signal; (c) oversampled lowdpass signal; (d) subsampled bandpass signal.

To complete our analysis we need to consider the effects of the ADC. Figure 15.21 plots its transfer characteristic in the simple case of 3 bits (23 ¼ 8 quantization levels). It is clear that the ADC output does not coincide with the input, excluding a finite number of discrete points. That difference can be modelled as either noise or distortion. The added noise – known as quantization noise – is of a different nature to the noise considered so far. Rather, the quantization noise depends on the signal. For our considerations, it is more convenient to consider the ADC effects as distortion, although the characteristic in Figure 15.21 is not continuous, and therefore has no approximating polynomial. However, the ADC input signal contains all the harmonics of the input and of the sampling signal, although the function HSINC ð f Þ limits their energy level at high frequencies. Therefore, the nonlinear distortion associated with the ADC

708

MICROWAVE AND RF ENGINEERING A/D output (v4)

111(6) 110(6) 101(5) 100(4) 011(3) 010(2) 001(1) 000(0)

vmin

Figure 15.21

vmax A/D input (v3)

Transfer characteristic of a (3-bit) ADC.

generates all the spectral components of frequency m f1 n fs, where m; n are integer numbers, and fs =2 < f1 < fs =2 is the generic frequency of the continuous-time signal. The limitation of such distortion levels promotes the use of ADCs with high resolution, i.e. with a high number of bits, typically from 8 to 16, depending on the application.

15.4.6.1 Full digital receiver In principle, the structure in Figure 15.17 can used to build fully digital receivers. The input RF band needs to be limited up to a given Df , by means of a suitable lowdpass or bandpass filter. Then an amplifier must increase the level of the signal up to a convenient value for the ADC. Finally, a numerical process can separate the various signals at the receiver input and demodulate one or more of them. This fully digital receiver architecture is attractive for several reasons: .

Simplicity, particularly in the analogue high-frequency section, which reduces to one filter and one amplifier.

.

Flexibility, since it can receive different channels with different modulations, even simultaneously, and the selection from the many possibilities can be operated by changing the software running in the numerical processor.

.

Cheapness, in that all the digital functions are potentially realizable with low-cost, high-density digital ICs.

Unfortunately, there are two main obstacles to achieving this attractive result, which are the linearity of the ADC and the high data throughput in the numerical process. Real ADC characteristics differ from the staircase shape of Figure 15.21. The deviation is more evident in the low range of the input signal if the step size is small (high resolution or high number of bits), and if the input frequency is high. To our knowledge, the full digital receiver is still in the study phase.


709

15.4.6.2 Sampled receiver Figure 15.17c shows a receiver architecture based on the so-called sampling mixer. This component consists of a mixer – usually a DBM – whose LO signal consists of all the harmonics of a sinusoidal LO. In fact, in the schematic of Figure 15.17c, one comb generator is inserted between the LO and the LO port of the mixer. Comb generators are special frequency multipliers that generate all the harmonics of the input signal. The comb generator output spectrum ideally consists of infinite spectral lines, having the same amplitude and frequency equal to all the integer multiples of the input one. Clearly, such an ideal component cannot exist, at least because infinite spectral lines with finite amplitude have infinite power. Therefore, real comb generators produce spectral lines with decreasing amplitude at high frequency, with negligible amplitude above a given frequency. However, it is possible to demonstrate that the ideal comb generator spectrum has the same shape as the Fourier transform of the sampling signal (15.94). Therefore, the IF signal v2 in the receiver of Figure 15.17c has the same spectral relation with the input v1 as in the receiver of Figures 15.17a,b. As confirmation, it is possible to examine the architecture of Figure 15.17c under a different perspective. Considering the mixer as a multiplier, we have that the IF signal includes the spectral components with frequency equal to the sum and the difference between the RF and the LO. Since the LO frequencies are integer multiples of oLO ¼ os ¼ 2p=Ts , it follows that the IF spectrum has the same form as Equation (15.98), apart from possible different amplitude factors. So far, it looks like the schematic in Figure 15.17c is equivalent to the one in Figure 15.17a, but this is not the case, due to the presence of the input bandpass filter BPFRF instead of the lowdpass filter LPF. Due to BPFRF , the mixer RF signal occupies the bandwidth ð f0 Df =2 to f0 þ Df =2Þ, which does not include the origin. Under this condition, it is possible to demonstrate that the minimum sampling frequency, such that one of the spectral replicas is close to the origin and that produces no aliases, is Df 1þ f 0 fs ¼ 2 f0 1 f0 þ floor 2 Df

ð15:100Þ

where the function floorðxÞ denotes the largest integer not exceeding x. Note that the condition (15.100) coincides with the condition (15.99) when f0 ¼ 0. The minimum sampling frequency (15.100) can be consistently lower than the maximum frequency of the RF signal; for instance, if f0 ¼ 1 109 (1 GHz) and Df ¼ 50 106 (50 MHz), then Equation (15.100) returns f0 > 102:5 106 (102.5 MHz). The condition (15.100) states the possibility for a sampled receiver to operate with a sampling frequency lower than the maximum frequency of the signal to be received. A receiver having the structure of Figure 15.17c and the LO frequency as in Equation (15.100) is sometimes referred to as a subsampled receiver. Figure 15.20d shows one case of a sampled spectrum, obtained by applying the condition (15.100). By exploiting the subsampling, it is possible to build receivers having a simple RF section consisting of one bandpass filter, a sampling mixer and a low-frequency LO. A low-frequency and mainly digital section follows the IF port of the mixer and performs the remaining part of the process: filtering for channel selection, amplification and demodulation. Similar to the full digital receiver, the sampled architecture is also attractive, due to its simple structure and to the limited use of high-frequency analogue components. Compared with the full digital solution, the sampled receiver does not require such a high-frequency ADC. Moreover, subsampled receivers can reduce the numerical processor throughput by transferring some of its tasks to the lowfrequency analogue section. The main limitation on the architecture of Figure 15.17c is that the sampling operation converts into the baseband all the noise located across each spectral line of the sampling signal. This increases the total received noise up to values much higher than the thermal noise background,71 with a consequent drastic limitation of receiver sensitivity. 71

An equivalent noise figure of the order of 30 dB is reported in [7].

710


15.5 Further concepts on RF transmitters and receivers This section covers some aspects of transmitters and receivers that have not been discussed in Sections 15.3 and 15.4 but nonetheless frequently occur in engineering practice. The present treatment is not systematic; rather, it opens some windows on important aspects of the radio equipment technique, although the space available is insufficient to exhaust them. Section 15.5.1 describes the combination of one transmitter and one receiver connected to the same antenna of a cable terminal. Section 15.5.2 presents the CAD analysis of one specific transmitter for radar72 application. Section 15.5.3 explains some receiver performances by means of the analysis of one specific receiver.

15.5.1

Transceivers

In many important applications, such as radar, mobile/cell phones and two-way radios, the same antenna connects to both a transmitter output and a receiver input. The combination of one transmitter and one receiver with a common RF port is referred to as a transmitter–receiver, or receiver–transmitter, or – in short – transceiver. Sometimes, instead of the complete name, the following acronyms are uses: TX for transmitter, RX for receiver, and TX–RX or RX–TX or TRX or RTX for transceiver. Figure 15.22 shows the three basic arrangements for the simultaneous connection of one TX and one RX to the same antenna. More precisely, the configuration in Figure 15.22a can be used when the transmitter and the receiver operate in different time intervals, as in GSM phones. That arrangement is the simplest possible, and allows the design of the output stages of the TX and the input stages of the RX independently of each other. This specific advantage relies on the fact that, at each given time, the SPDT connects only one of the TX and the RX to the antenna. The SPDT isolation between ports 2 and 3 is usually sufficient to make the loading due to the inactive element negligible. The solution of Figure 15.22b is always possible in principle, independently of the TX and RX timing and frequency. In that case, the output circulator transfers the power from the transmitter to the antenna and from the latter to the receiver. It must be considered, however, that the antenna has a finite non-zero reflection coefficient. Therefore, even if the circulator has infinite isolation – and this is never the case – the leakage from the TX to the RX is of the order of the antenna return loss. A typical value for the latter parameter is in the range of 10–20 dB. The resulting TX to RX attenuation is a few decibels less. Thus, the transmitter injects noise in the receiver input. That leakage could be stronger than the signal to receive and can saturate the receiver. Consequently, the possibility to transmit and receive at the same frequency and at the same time with the configuration in Figure 15.22b is more theoretical then practical. Finally, the connection in Figure 15.22c is useful when TX and RX operate on sufficiently different frequency, simultaneously or not. Let fTX and fRX be the transmitting and receiving frequency ranges, and fTX „ fRX by definition. Then, the centre frequency of the output transmitter filter BPFTX (input receiver filter BPFRX ) will be fTX ( fRX ). Thus fRX and fTX fall within the stopband of BPFTX and BPFRX , where both filters present reactive reflection coefficients at both ports. The 50 O transmission line TLTX (TLRX) rotates the output (input) reflection coefficient of BPFTX (BPFRX ) such that it is as close as possible to G ¼ 1 in the frequency range fTX ( fRX ). In other words, looking from the junction node between TLTX and TLRX towards the transmitter we see an open circuit in the RX band, and similarly in the TX band when looking towards the receiver. The assembly of BPFTX , BPFRX , TLTX and TLRX forms a non-contiguous diplexer.73 The acronym ‘radar’ stands for radio detection and ranging. Another possible diplexer type is the contiguous one, which is characterized by the fact that the two filter bands have a common 3.01 dB point, defined as crossover. The configuration in Figure 15.22c is only suitable for noncontiguous bands. However, TX and RX with contiguous bands cannot use that arrangement, due to the poor leakage attenuation between TX and RX, which is about 3 dB in the crossover neighborhood. Moreover, the 3.01 minimum theoretical crossover attenuation of the contiguous diplexer is not acceptable in most cases. For further details, see [8]. 72 73


711

antenna modulating signal

transmitter (TX) 1 SPDT

demodulated signal

2 (a)

receiver (RX)

antenna modulating signal

transmitter (TX)

circulator

demodulated signal

(b)

receiver (RX)

BPF TX modulating signal

antenna

transmitter (TX) TLTX TLRX

demodulated signal

(c)

receiver (RX) BPFRX diplexer

Figure 15.22 Transmitter–receiver configurations: (a) for non-simultaneous transmission and reception; (b) without any constraint on the transmitter and receiver timing and frequency; (c) with transmission and reception at different frequencies. The devices used to connect the same antenna to the TX and the RX – namely, the SPDT of Figure 15.22a, the circulator of Figure 15.22b and the diplexer of Figure 15.22c – take the generic name of duplexer. We are now in a position to consider the effects of the finite TX–RX isolation on receiver performance. Section 10.2 showed that the typical SPDT isolation ranges from 40 or 60 dB for FET or PIN components, respectively.74 This section anticipated that practical values of the circulator isolation are lower than 20 dB. For the diplexer, the TX–RX isolation is a function of the distance between 74 More precisely, the isolation as defined in Section 10.2 is between the common port (connected to the antenna in our case) and the non-selected port. However, the isolation between the selected and non-selected port assumes similar values.

712


Table 15.2 Receiver leakage power for different transmitted power and duplexer isolation. TX power (W) TX power (dBm) RX leakage (dBm)

0.1 20 10 30 70

1 30

10 40

100 50

1000 60

20 20 60

30 10 50

40 0 40

50 10 30

10 50 90

TX–RX isolation (dB)

fTX and fRX and of the filter selectivity as well. Realistic values could fall within the range of 30–90 dB. Summarizing all three cases, we have that the duplexer isolation is bounded within the interval 20–90 dB. Table 15.2 shows the power that arrives at the receiver for different values of the transmitted power and duplexer isolation. How much such leaked power is harmful to the receiver depends on many factors. Moderate levels of injected TX signal degrade the receiver sensitivity. Again, the amount of such degradation depends on the distance between the TX and RF. In the worst case, when the two frequencies differs by less than the IF bandwidth,75 the receiver sensitivity becomes worse than the injected signal. As a reference, it is possible to compare the values in Table 15.2 with the thermal noise power that falls within 1 MHz bandwidth, which equals about 114 dBm at a temperature of 27 C. Furthermore, unwanted TX leakage of the order of 10 dBm or more can also be dangerous for the receiver, in that it could cause permanent damage to the latter. A comparison of the power levels of Table 15.2 with the thermal noise floor suggests that simultaneous transmission and reception on the same antenna and at the same time must be avoided, if possible. However, in the presence of low duplexer isolation, even if the TX has a different frequency from the RX, the TX leakage can still be harmful, due to the out-of-band emission related to the phase noise. It is possible to clarify this point with one example. Let us assume that: .

Both the TX and RX bandwidths are quite narrow, as with the IF bandwidth of the receiver, of course.

.

The distance between the two corresponding frequencies is 30 MHz.

.

The transmitter phase noise is 160 dBc/Hz at 30 MHz offset from the carrier, which represents a good performance.

.

The transmitter power is 0.5 W (þ27 dBm).

.

The duplexer isolation is 25 dB.

The phase noise spectrum of the TX is relatively flat in the narrow IF band of the receiver, and, from the RX point of view, it is no different from the thermal noise. Now, the noise power density injected from the transmitter into the receiver input is 160 þ 2725 ¼ 158 dBm=Hz, which is 16 dB higher than the thermal noise floor, which is 174 dBm/Hz at the room temperature of 27 C, as stated. The equivalent noise power density at the receiver input is then 10log10 ½10ð174 þ NFÞ=10 þ 10158=10 dBm=Hz, where NF is the noise figure of the unperturbed RX. The corresponding equivalent noise figure is NF 0 ¼ 10 log10 ½10NF=10 þ 10ð158 þ 174Þ=10 10log10 ½10NF=10 þ 39:811 dB. If NF ¼ 0; 1; 5; 10; 20 dB 75 In less severe cases, when the distance between TX and RX frequency is consistently larger than the IF bandwidth, the tolerated injected level becomes much higher. In this case it is required that the TX interfering level is small enough not to saturate the input amplifier and the first mixer of the receiver. If the TX–RX frequency distance is much greater than the RF bandwidth of the RX, and if the RX is a superhet, than the second RF filter helps to protect the first mixer from interference. Consequently, the accepted interfering level becomes even higher, being only limited by the linear range of the RF amplifier.


713

then NF 0 ¼ 16:1; 16:1; 16:3; 17; 21:5 dB respectively. Therefore, the noise injection degrades the receiver noise figure at the minimum value of 16.1 dB, if the receiver is totally noise free, and to even worse values in real cases. A similar analysis could be developed for the noise generated by the transmitter amplifiers, even in the absence of a driving input signal. The conclusion also would be similar. The lack of receiver sensitivity due to the presence of unwanted signals is named de-sense. The above considerations suggest that in many cases it is convenient to switch the transmitter off during reception. This minimizes the receiver de-sense while reducing the transceiver power consumption. As a drawback, actuation of that strategy involves consideration of the switch-on transient of the transmitter. It must be activated some time before the transmission begins in order to give time to the transmitter bias and control circuitry to reach their steady state condition. The time difference between the end of reception and the beginning of transmission states a specification for the settling time of all the bias and control circuitry of the transmitter. Clearly the faster it is, the later it is required to activate the TX, with a consequent energy saving. This aspect is particularly important in battery operating equipment, in that it is one of the key factors of the battery charge duration.76

15.5.1.1 TX–RX modules for phased array antennae Figure 15.23 shows an interesting application of TRX modules as part of a so-called phased array antenna.77 This arrangement basically consists of N identical modules, connected between one N-way power divider and one radiating element78 ANTk with k ¼ 1 . . . N. Each module consists of one phase shifter PHk , one transmitter amplifier AMPTXk and one receiving amplifier AMPRXk. One common command actuates all the switches SPDTAk , SPDTAk and SPDTBk , in order to select the transmit (position 1) or the receive path (position 2). In some applications, SPDTBk is replaced by one circulator which exhibits lower insertion loss and higher power handling. The phase shifter PHk can modify the phase of the path from each radiating element to the N-way power divider. In the transmission state, the signal at the input of the power divider is distributed at the TX input of all the modules with ideally identical phase and amplitude. Then each module transfers the input signal to the respective ANTk , after being amplified and phase shifted by fk. Conversely, in the receiving condition, the module delivers the signal received from the respective antenna to the power divider, which sums those N contributions to its common port, still with the same amplitude and phase. The configuration of the different phase shifters determines the angle where the radiated power is directed to or received from, as we will see shortly. One further SPDT connects the divider common port to a modulator (TX) or demodulator (RX), which could include some frequency conversions or not. Three basic variants of the configuration of Figure 15.23 are possible: .

Passive array, when the module simplifies to a phase shifter. In this case, all the required TX and RX amplification is placed after the common port of the power divider.

.

TX- or RX-only array, where AMPRXk or AMPTXk is absent, while straightforward connections to PHk and ANTk replace SPDTAk and SPDTBk .

76 This performance is particularly relevant in mobile/cell phones, where it is specified as ‘conversation time within one battery charge’. 77 The interested reader can find more details on this topic in [9]. This book will limit description of the basic operating principle of the linear phased array. 78 The denomination ‘radiating element’ is a synonym for ‘antenna’. Here, the first is used to denote the single antenna of the array.

714


PH1

SPDTA1 1

AMPTX1

SPDTB1 1

φ1

θ

z

O

2

P

r

ANT 1

2 AMPRX 1

d PH2

SPDTA2 1

AMPTX2

SPDTB2 1

ANT2

φ2

2

2 AMPRX 2

d PH3

SPDTA3 1

AMPTX 3

SPDTB3 1

ANT3

φ3

2

2 AMPRX 3

In TX / Out RX

d PHN

SPDTAN 1

AMPTX N

SPDTBN 1

ANTN

φN

N-way power divider

2

2 AMPRX N

Figure 15.23 Basic structure of a linear phased array antenna. .

Array with different TX and RX angles, where two different, independently set phase shifters are present in the TX and RX path, instead of a single one in the common path. In a subvariant of this approach, SPDTAk is eliminated, and the TX input becomes separate from the RF output. Clearly this requires two different N-way power combiners, one for TX and one for RX.


715

All the radiating elements are identical, geometrically aligned and equally spaced. In other words, all the antennas are disposed along a straight line, and the distance between ANTk1 , ANTk and ANTk , ANTk þ 1 is constant and equal to d. This particular disposition of the single antennae configures a linear array, which is the simplest one. The working principle of the phased array can be understood by analyzing the array itself in transmission. In reception the same conclusions apply, due to the general principle of reciprocity between the same antenna in TX and RX. Let us assume a geometrical reference system with the origin coincident with the position of ANT1 and the z axis orientated perpendicular to the array line. We will compute the electromagnetic field at a point P, placed on a straight line passing through the origin and forming an angle y with the z axis, and located at great distance79 from the array. Due to the long distance of the observation point, the choice of the origin coincident with first radiating element is equivalent to the more intuitive one of the array geometrical centre, and makes it easier to derive formulae. The difference between the two distances from the observation point to the first and the second antenna is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð15:101Þ Dr1;2 ¼ r ½rd sinðyÞ2 þ ½d cosðyÞ2 ¼ r r2 2 d r sinðyÞd 2 In the case of interest, d is comparable with the wavelength, while r is much greater than l, therefore d r and the quantity (15.101) simplifies to80 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 d d 5 d ffi r 1 1 sinðyÞ Dr1;2 ¼ r 41 12 sinðyÞ ¼ d sinðyÞ ð15:102Þ r r r The geometric difference (15.102) corresponds to a phase difference o v1 Dr1;2 ¼ o v1 d sinðyÞ, where o ¼ 2pf is the angular frequency of the signal and v is the propagation velocity of the medium.81 Due to the phase shifters PH1 and PH2, there is an additional phase difference to consider, which is f1 f2 . The total phase shift between the two electromagnetic fields produced by ANT1 and ANT2 at the observation point is thus Dx1;2 ¼ o v1 d sinðyÞ þ f1 f2

ð15:103Þ

Applying the same reasoning to ANT1 and the generic ANTk , where k is any integer ranging from 1 to N, we have the phase difference Dx1;k ¼ k o v1 d sinðyÞ þ f1 fk

ð15:104Þ

The difference between the distances from the various radiating elements and the observation point is sufficient to generate a phase difference, but causes no appreciable amplitude differences, due to the large value of r. Consequently, if the various elements – considered individually – produce the same field intensity82 for any y, then the electric (or magnetic) field at P is proportional to EðPÞ /

N nh o io X exp jDx1;k ¼ exp j k d sinðyÞ þ f1 fk v k¼1 k¼1

N X

ð15:105Þ

where clearly Dx1;1 ¼ 0. 79 Here, ‘great distance’ denotes a distance much greater than the wavelength of the array extension, which equals ðN1Þd. 80 The quantity (15.102) coincides with the limit for r tending to infinity of the quantity (15.101). 81 Typically, the propagation medium is the atmosphere or a vacuum, in terrestrial or space communications respectively, and v ffi c ¼ 2:9979 108 m=s ffi 3 108 m=s. 82 An antenna that produces the same field intensity on all points of a sphere with centre at the antenna itself is called isotropic. For our considerations, it is sufficient to assume an isotropic radiation in the z positive half space.

716

MICROWAVE AND RF ENGINEERING 90°

1

60°

30°

2 π/3

A(θ)=0.5

φ0 =0 θDEG =0°

0 –π/3 –2 π/3

–30°

–60° –90°

Figure 15.24 f ¼ 18 GHz.

Radiation diagrams of a linear phased array antenna with N ¼ 8, d ¼ 8 mm and

Setting the various phase shifters in order to produce a shift proportional to the antenna position, i.e. such that fk ¼ kf0 , the function (15.105) becomes EðPÞ /

n ho io exp jk d sinðyÞ þ f0 v k¼1

N X

ð15:106Þ

The field (15.106) depends only on the variable o v1 d sinðyÞ þ f0 , which means that by changing the value of f0 it is possible to steer the radiation diagram of the array. Figure 15.24 shows the radiation diagram of a linear array with N ¼ 8 antennas, having a reciprocal distance d ¼ 8 mm, and working at the frequency o ¼ 2p 18 109 (18 GHz). The diagram shows that a change in f0 of 2p=3 (corresponding to 120 ) steers the main radiation lobe by about 45 . That diagram is symmetric around the axis y ¼ 0. Note also that the width of the main radiation lobe tends to increase with steering angle.

15.5.1.2 An example of TRX module for active phased array Figure 15.25 shows the simplified electrical diagram (a) and the layout (b) of a transceiver designed for phased array applications.83 For simplicity, neither the electrical diagram nor the layout shows the control and bias circuits for the different RF components of the module. Moreover, the drawing is detailed up to the stage level. The module technology is MMIC based: SPDT1 uses the same chip as in Figure 10.5; most of the amplifiers use chips equal or similar to the one shown in Figure 11.37. The labels in the schematic of Figure 15.25a correspond to the layout in Figure 15.25b.

83

Loris Caporali designed the mechanical structure of this transceiver, for which he is gratefully acknowledged.

RF AND MICROWAVE ARCHITECTURES AMP2

AMP1

PH 1 TX in RX out

AMP3

1

AMP4

CIRC1

φ

717

TX out RX in

2 SPDT1

(a) AMP5

ATT1

120 mm PH1

AMP1

AMP2

AMP3 AMP4

CIRC1

TX in RX out TX out RX in

SPDT1

ATT1

AMP5

screws

(b) carriers enclosure walls chips interconnecting microstrips

Figure 15.25 A 6–18 GHz TRX module for phased array antennas: (a) electrical diagram; (b) layout.

The main design specifications for the module are: .

frequency range minimum (saturated) TX power . minimum TX input power . RX overall linear gain . RX maximum noise figure . phase shifter range .

6–18 GHz þ 30 dBm 0 dBm 10 dB 10 dB 0 to 337.25 in 22.5 steps.

In detail, the TRX module can transmit a signal coming from one RF port to the other one by setting the position of the switch SPDT1 . In position 1, corresponding to the transmitting state, the signal travels from the TX input to the antenna. Conversely, with SPDT1 in position 2, the TRX module transfers the signal from the antenna to the RX output. Clearly, the antenna is connected to the RF port denoted as TX out RX in. The opposite TRX RF port works as the input for the transmission and output for the reception.

718

MICROWAVE AND RF ENGINEERING transmitter active section Antenna PG1 v2 v1 OSC 1

v3

COUP 1

v4 PA1

BPF1

COUP 2

v6

v7

DET1

LF1 v9

COUP 3

v5

CIRC1

v8

DET2

LF2 v10

DET3

to the receiver

LF3 v11

transmitter controller

Figure 15.26 Simplified block diagram of a radar transceiver.

There is a common phase shifter PH1 for the TX and RX paths, realized with four cascaded binary cells of the type84 in Figure 10.30b. The possible phase shift ranges from zero to ð36022:5Þ ¼ 337:5 with a minimum step of 360 =24 ¼ 22:5 . The variable attenuator ATT1 has the schematic as in Figure 10.13d. It produces two attenuation states: the insertion loss in the minimum attenuation state is about 1.5 dB; the attenuation in the other state is 1:5 þ 20 ¼ 21:5 dB. The attenuator driver provides two couples of two values of voltages to the control ports V1 and V2 . The high attenuation is activated in the presence of strong input signals, in order to reduce the RX gain and increase its linearity. An amplification chain, consisting of four cascaded amplifier stages AMP1 to AMP4 , allows the signal power to be increased up to the level required for the transmitting power of þ 30 dBm (1 W). The insertion loss of CIRC1 is about 1.5 dB, therefore the (saturated) output power of AMP4 must be not less than 31.5 dBm (1.4 W). The maximum insertion losses of PH1 and SPDT1 over the frequency and the different possible states are 12 and 3 dB, respectively, for a total of 15 dB. Combining this with the minimum specified input power of 0 dBm, and temporarily assuming linear operation of the transmitter amplifiers, we have that their small-signal gain must be 31:50 þ 12 þ 3 ¼ 46:5 dB. The transmitter amplifier chain works in saturation, and the gain varies over the frequency, temperature and from piece to piece. Thus, the required gain needs at least a 6 dB margin above the previous determination, which ðAMP1...AMP4Þ leads to 20 log10 s21 ¼ 52:5 dB. The receiver must operate linearly, by specification, with an overall gain 10 dB, when the variable ðof CIRC1Þ ðATT1Þ ðAMP5Þ ðSPDT1Þ attenuator ATT1 is set to its minimum attenuation. Therefore 20log10 s21 s21 s21 s21 ðPH1Þ ¼ 10 dB. From the insertion loss given above for PH1, SPDT1 , CIRC1 , and from the fact that the s21 minimum insertion loss of ATT1 is 1.5 dB, we obtain the gain of the receiver amplifier AMP5, i.e. ðAMP5 Þ 20 log10 s21 ¼ 10 þ 1:5 þ 1:5 þ 3 þ 12 ¼ 28 dB. 84 The varactors present in the circuit of Figure 10.30b are capable of generating any continuous variable phase between two given limits, if their reverse bias voltage assumes a suitable continuously variable value. However, in the application of Figure 15.26b the phase shifter driver supplies only two different voltage values per cell, in order to realize the four binary steps 0–22.5 , 0–45 , 0–90 and 0–180 . The directional couplers of the four binary cells are realized with the Lange coupler, described in Section 6.7.3.


719

The circulator CIRC1 allows the signal to travel from the output of AMP4 to TX out RX in, and from the latter to the input of AMP5 , passing through ATT1 . The TRX in Figure 15.25 presents a loop including all the amplifiers, SPDT1 , CIRC1 and ATT1 : if, at some frequencies and state, the loop gain amplitude is higher than 1, then the system becomes unstable.85 This condition must be avoided, in that it produces oscillations that intermodulate with the principal TX and RX signal, and causes the emission of unwanted electromagnetic energy. The stability check starts from the computation of the total gain within the ðAMP1...AMP4Þ ðAMP5Þ ¼ 52:5 þ 29 dB ¼ 81:5 dB. Now, Section 15.4.1 showed loop, which is 20 log10 s21 s21 that the circulator isolation does not exceed 20 dB, while SPDT1 has an isolation of about 40 dB, as in ðCIRC1Þ ðATT1Þ s21 Section 10.2.2. The resulting loop attenuation is approximately equal86 to 20 log10 s21 ðSPDT1;OFF Þ s21 ¼ 10 þ 1:5 þ 40 ¼ 51:5 dB. Thus the total loop gain, which is the sum of gain and attenuation, is well above unity, and the module is unstable. For this reason, and for the ones discussed in Section 15.4.1, the unused amplifiers in either the TX or RX path must be switched off. The inactive amplifier produces attenuation instead of gain, improving the stability margin by some tens, as required. One additional advantage of this solution is the DC energy saving due to the zero current absorbed by the TX (RX) amplifiers in the RX (TX) mode. The main drawback is that the current in TX is different – and typically higher – than the one in RX, hence the power supply has to work with a variable load. This could be harmful to the power supply, particularly if the transition between the two states is fast and abrupt. During the associated transient, the voltage stabilization circuit could suffer and cause the production of spikes that generate parasitic AM and PM. Hence, the power supply design also requires special care, although its description is beyond the scope of this book. Figure 15.26 shows the layout of the TRX module as it looks when the top metal cover is removed. For the sake of clarity, the layout omits the control and bias circuitry – which occupies the apparently empty spaces within each carrier – and the connectors, which are coaxial for the RF ports. The module is slightly longer than 120 mm, due to the connectors, while its width and thickness are both equal to 22 mm. Note that the whole TX and RX amplifier chain is placed between two metal side walls, machined from the module enclosure. They form a rectangular waveguide together with the bottom of the enclosure and the metal cover, having a cross-section of 7.5 7 mm. The corresponding cut-off frequency of the fundamental TE10 mode87 is 20 GHz, which is higher than the maximum working frequency of the module. This ensures no spurious waveguide propagation inside the module up to the maximum frequency of 18 GHz.

15.5.2

CAD analysis of a radar transmitting subassembly

Radar works by transmitting short pulses of RF energy and receiving the echo back from the target. The distance in time between the transmitted and the received pulse gives a measure of the distance to the target. The transmitted pulses have a width and a pulse-to-pulse distance depending on the target type, minimum measurable distance, resolution of the distance itself, and so on. Also, the operating RF depends on the application. Figure 15.26 shows the simplified block diagram of radar. The receiver (not detailed any further) and the transmitter connect at the same antenna by means of the circulator CIRC1. The RF energy produced by the transmitter feeds the antenna, while the received RF energy flows from the antenna to the receiver. 85

See Section 12.4. A more accurate analysis must consider that two mismatched two-ports in cascade behave differently from two matched ones. The amplifiers and the circulator approximately pertain to the latter case, while one of the ports of the switch is necessarily mismatched, unless it is not absorptive, and SPDT1 is not. Thus, from Equation (9.2) it follows that the total isolation could differ from our approximate evaluation. However, since the stability margin is substantially negative ð81:551:5 ¼ 30 dBÞ, the conclusion that the system is unstable is right, despite the approximation involved with the analysis. 87 See Section 3.8. 86

720


The output voltage v1 of the oscillator OSC1 is a sinusoidal CW with a frequency of oR ¼ 2pfR ¼ 2p 109 , corresponding to 1 GHz. The pulse generator PG1 produces a periodic rectangular wave v2 with a pulse width (PW) of 0.5 ms and a pulse repetition interval (PRI) of 5 ms. The resulting duty cycle, defined as d ¼ PW=PRI, equals 10%. The multiplier PD1 uses the rectangular wave to amplitudemodulate v1 with MD ¼ 100%. Omitting the multiplying constant, the multiplier output signal is v3 ¼ v2 v1 . The power amplifier PA1 amplifies the modulated signal, thus the output signal v4 assumes the aspect of a periodic burst of sinusoids, and its envelope88 has the shape of v2 . Note that the described transmitter has the structure of Figure 15.3a, but all the solutions presented in Section 15.2 for AM transmitters are applicable. The directional coupler COUP1 provides, at its coupled port, a signal proportional to the output power of PA1 . Then the detector DET1, which works in the linear region, together with the lowdpass filter LPF1 generate a DC voltage v9 proportional to the envelope of v4 . Similarly, LPF2 eliminates the alternate components from the output of DET2 , and produces the voltage v10 , which is proportional to the envelope of the wave reflected by the bandpass filter BPF1. The transmitter controller checks that v9 is higher than a given threshold; if this condition is missed, an alert will appear, informing that one of the components before COUP1 is about to fail, and consequently the radar is not working properly. The check on v10 is even more important, because if that voltage increases above a certain limit, it means that too much energy is reflected from BPF1 back into the output of PA1 . If this occurs, then the controller switches off the power supply to PA1 , in order to prevent damage to PA1 itself. In this case, the alert message informs that one of the components between the PA and the antenna is broken or disconnected. The directional coupler COUP3 together with detector DET3 and the lowdpass filter LPF3 generate the voltage v11 , which is proportional to the envelope of the radiated RF signal. Thus, the transmitter controller checks the functionality of many RF components (excluding DET1 , DET2 and the circulator) by monitoring v11 . The three directional couplers, detectors and lowdpass filters, together with the associated functions implemented in the controller, form the so-called built-in test equipment (BITE) of the transmitter. The transmitter spectrum occupies a wide frequency band, due to the ON/OFF modulation. Furthermore, and independent of the modulation, the PA produces harmonics that must be minimized. Therefore the signal needs some RF filtering before arriving at the antenna, in order to shape its spectrum and reduce its harmonics. To put some numbers on this, we will assume a PA output peak power89 of 2 kW, corresponding to 63 dBm. Ignoring the harmonics, the output voltage of the power amplifier is þ X ðPA1Þ ðPEAK Þ v4 ðtÞ ¼ s21 v1 ðtÞv2 ðtÞ ¼ v4 cosðoR tÞPW RECTPW ðtk PRI Þ

ð15:107Þ

k¼1 ðPEAK Þ

is the maximum amplitude of the amplifier output waveform, which coincides with The quantity v4 the oscillator peak amplitude multiplied by the gain of PA1 . The modulating signal is periodic, and can be expanded into a Fourier series, which gives v2 ðtÞ ¼ d þ 2

þ1 X sinðkpdÞ k¼1

kp

t cos 2kp PRI

ð15:108Þ

Substituting (15.108) into (15.107) yields ðPEAK Þ

v4 ðtÞ ¼ v4

ðPEAK Þ

d cosðoR tÞ þ v4

þ1 X sinðkpdÞ k¼1

88

kp

cos

oR þ

2pk 2pk t þ cos oR t PRI PRI

See also Figure 15.29. The peak power of ON/OFF modulated signals coincides with the RMS power of a signal with a constant envelope, equal to the maximum of the modulated one. 89


721

0

–20 s21 and spectrum, dB

(BPF1)

(b) 20 log 10(|s21

|)

–40

–60 (a) v4

–80

(c) v5

–100 0.90

0.95

1.00

1.05

1.10

Frequency, GHz

Figure 15.27 Modulation spectra of the transmitter in Figure 15.26: (a) power amplifier output; (b) gain of the RF filter; (c) RF filter output. which can be more elegantly rewritten as ðPEAK Þ

v4 ðtÞ ¼ v4

d

þ1 X k¼1

sin cðkpdÞcos

2pk oR þ t PRI

ð15:109Þ

Equation (15.109) states that the PA output includes a theoretically infinite number of spectral components, symmetrically placed around the carrier, with a frequency spacing of Df ¼ PRI 1 . The carrier frequency is fR and its amplitude is the strongest among all the spectral components of Equation90(15.109). Figure 15.27 shows the amplitude spectrum associated with the signal (15.109). It can be seen that the carrier component is attenuated by 20log10 ðdÞ ¼ 20 dB with respect to the nonmodulated RF power. Furthermore, the sideband amplitude at the limits of the plot is still relatively high, although it decreases with frequency from the carrier. To get a more quantitative idea about the effect of the sideband emission, it is convenient to compute the local maximum spectral amplitude at a given distance from the carrier. For instance, let us consider an offset frequency in the proximity of Df ¼ 100 MHz. The index exactly corresponding to that offset is fRF k100 =PRI ¼ fR 100 106 , thus k100 ¼ 500. The amplitude of these relative spectral lines is sinðk100 pdÞ=ðk100 pÞ ¼ 0. Nevertheless, 0 for frequencies close to 100 MHz, it is possible to find an index k100 ffi k100 such that offset sin k0 pd ffi 1. It is k0 ¼ 505 and corresponds to an offset of Df 0 ¼ 101 MHz. The amplitude 100 100 0 ðPEAK Þ 0 = k100 p , with a relative level of 20 log10 k100 p ffi of the side spectral line amplitude is d v4 64 dB below the carrier. The corresponding power of each side spectral line is 0:1 2000=ð505pÞ ffi 0:126 W. Considering more spectral lines, the total power is clearly higher. However, even one single couple of lines has enough associated power to disturb other radio equipment located in the proximity of the radar antenna, and must be filtered out. Looking at the output spectrum of PA1 in the neighbourhood of the RF harmonics, we can see figures similar to the one in Figure 15.27, just reduced by the harmonic attenuation: typically around 20 dB for the second and third harmonics. 90

It is easy to recognize that the amplitude of the spectral component at the angular frequency oRF 2pk=PRI is ðPEAK Þ

ðPEAK Þ

equal to vRF;5 d sincðkpdÞ, which is maximum if k ¼ 0, when it equals vRF;5 d.

722


Table 15.3 Parameters of BPF1 . k

gk

0 1 2 3 4

1 0.629 0.97 0.629 1

Jk;k þ 1 0.010 967 6 0.010 967 6

Z0L;k ðOÞ

Z0S;k ðOÞ

Ck ðpFÞ

56.61 90.48 77.55 90.48 56.61

428.434 4558.900 4558.900 428.434

2.122 066 2.122 066 2.122 066

Ck ðpFÞ ðoptimizedÞ 2.122 066 2.121 950 2.122 066

Thus, BPF1 is placed between PA1 and CIRC1 in order to reduce the amplitude of the sideband spectral lines. The RF bandpass filter has to provide sufficient attenuation of the harmonics and out-of-band emission (a), while presenting low insertion loss (b) and acceptable linear distortion (c), and must be able to handle the high peak power of 2 kW (d). The specifications for BPF1 result from a compromise among these different requirements. Requirement (a) is best satisfied with high selectivity responses (from the most to the least promising: Cauer, Chebyshev, Butterworth and Bessel), high-order N and high passband ripple (for Cauer and Chebyshev). Unfortunately, the high selectivity is always associated with high variation of the passband group delay and, consequently, with a relevant distortion of the transmitted RF pulse, in contrast with (b). The need for a wide stopband (beyond three times the centre frequency, for the third harmonic) makes the comb-line configuration the almost unique solution. From (d) it follows that a low-order filter with high-Q resonators and low passband ripple91 RPdB must be designed. The need for high-Q resonators is an immediate consequence of the relation between the quality factor and the dissipation loss.92 For different reasons, requirement (d) implies a wide passband width. Indeed, narrow-band filters present high overvoltage values inside the filter and thus a high risk of sparks.93 A reasonable compromise among all the requirements is a three-resonator, Chebyshev comb-line filter with RPdB ¼ 0:01 dB94 and Df ¼ 10 MHz. The filter can be designed using the method illustrated in Section 8.7.3,95 which gives the numbers listed in Table 15.3. The last column of the table presents the tuning capacitances as optimized to compensate the approximation of the synthesis formulae. Note that a very small reduction in the centre capacitance was needed. Without this modification, the filter has a non-equal ripple and a slightly degraded return loss. Figure 15.27 (curve b) shows the transmission coefficient amplitude of the optimized RF filter. Figure 15.28 shows the amplitudes of the transmission and reflection coefficients, together with the group delay of the filter.96 ðBPF1Þ Once the RF filter has been synthesized, and its transmission coefficient s21 ðoÞ is known, it is possible to determine the spectrum of the radiated signal, by applying the filtering function to v5 . The radiated spectrum is derived from Equation (15.109) after modifying the amplitude and phase of

91 Keep in mind that the maximum passband attenuation equals (is higher than) the passband ripple in a loss-free (lossy) filter. 92 Section 8.7 dealt with filter dissipation loss. See [10] for further details. 93 The method described in [11] allows investigation of the filter power-handling capability by means of conventional circuit analysis. 94 Corresponding to a passband return loss of about 26.4 dB. 95 Implemented in the Mathcad file 07_CombLine_Filter_Synthesis.mcd. 96 The three curves can be obtained by a circuit simulation, using the Ansoft file 02_CombLine_Filter_1GHz.adsn or the SIMetrix file 12_CombLine_Filter_AC.sxsch. The latter case is based on AC simulation, which is normally used to compute the AC voltages of some interesting nodes of the network. Some added ideal elements model the equivalent of a directional coupler, in order to compute the reflection and transmission coefficients.

RF AND MICROWAVE ARCHITECTURES 0

10

60

0

40

723

–30 –10

–40 –50

← s21

GD → –20

–60

0

s11 →

–70 –80 0.90

20

Group delay, ns

20 log10(|s21|)

–20

20 log10(|s11|)

–10

0.95

1.00

1.05

–30 1.10

–20

Frequency, GHz

Figure 15.28

Response of BPF1 employed in the schematic of Figure 15.26.

ðBPF1Þ the generic sinusoid at the frequency fR þ k=PRI by the quantities s21 ð fR þ k=PRI Þ and ðBPF1Þ ð fR þ k=PRI Þ , respectively. The result is arg s21 þ1 X k k ðBPF1Þ v5 ðtÞ ¼ v5;k cos 2p fR þ fR þ t þ arg s21 ð15:110Þ PRI PRI k¼1 with v5;k

ðBPF1Þ k ðPEAK Þ v ¼ d sin cðkpdÞ s21 fR þ PRI 4

Different from v4 , the filtered signal v5 could have non-symmetrical spectral amplitudes across the carrier,97 due to the asymmetry of the response of BPF1 . Figure 15.27 shows the filtered amplitude spectra,98 obtained from Equation (15.110); note the significant reduction operated by the filter. The circuit transient analysis99 can be used to investigate other interesting properties of the filtering action operated by BPF1. Figure 15.29 shows the results of that simulation. The three detector output voltages v9 , v10 and v11 are approximately proportional to the envelopes of the PA output, reflected and transmitted waves of BPF1 . More precisely, the voltages v9 to v11 do not exactly follow the envelopes of v6 to v8 , due to the presence of the filters LPF1 to LPF3. They are necessary to eliminate the RF residuals on the detected voltage, but introduce some linear distortion on the respective waveforms, which can be minimized by a careful choice of the transfer function. Our transmitter uses a third-order Bessel response with a 3 dB cut-off frequency of 100 MHz.100 In other words, it is possible that v6;k „ v6;k . See the Mathcad file 08_Pulse_Modulation_RF_Spectrum.mcd. 99 See the SIMetrix file 13_Combline_Filter_Burst_with_Detectors.sxsch, where simple nonlinear transfer characteristics (the output voltage is the absolute value of the input one) model the detector. 100 The frequency and time domain characteristics of the filter can be analyzed with the SIMetrix file 14_VideoFilter_AC_Transient.sxsch. 97 98

724


(a) v9

1.0 0.5 0.0 –0.5 0.0 1.5

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

(b) v10

1.0 0.5 0.0

–0.5 0.0 1.5 (c) v 11

1.0 0.5 0.0 –0.5 0.0

Time, µs

Figure 15.29

Envelope of the transmitter signals.

The most visible effect of the band reduction on the radiated signal is that the envelope of v5 increases its amplitude gradually, rather than abruptly, different from v4 . The reason for this can be easily understood by considering that a reduction in the sideband spectral level is equivalent to a reduction in the harmonic content of the modulating signal. Thus the RF filter operation is somehow equivalent to a lowdpass filtering of the modulating signal. Moreover, the non-flat group delay of the RF filter must be considered: it makes each spectral component propagate with a different velocity within the filter. Therefore, the various spectral components in the RF filter output have a different reciprocal phase shift than they had at the input, with a consequent distortion in the envelope of v5 . The finite rise and fall times of the envelope of v5 imply that the radiated RF burst continues after its ideal end, instead of extinguishing instantaneously. This is a limiting factor on the distance resolution of the radar, although we will not analyze this performance in detail. The envelope of v5 presents a maximum that is slightly higher than that of v4 , despite the passive nature of BPF1 , which is assumed to be loss free. This is not a violation of the energy conservation principle, in that the integral over time of the filter output power (i.e. the output energy) is not higher than the integral of the input one. Moreover, BPF1 exhibits a good passband return loss, no worse than 26.4 dB. Nevertheless, it still produces relevant reflections at the beginning and end of the input RF pulse. This surprising effect has a simple explanation that follows from consideration of the spectrum v6 shown in Figure 15.27. The wave incident on BPF1 presents relevant energy well beyond the passband of BPF1 itself. The out-of-band spectral components are then reflected back to the input, and their combination produces the envelope shown in Figures 15.29c. From the BITE point of view, there are some transients that differ from nominal operation, even if the whole transmitter is perfectly functional. Assuming the leading edge of vDET;1 as a reference, we have that v10 remains high for the first 50 ns, while v11 takes about 100 ns to get close to its maximum. During this time, the controller measures voltages that are similar to a failure condition. Therefore the controller must ignore v9 (v11 ) for the first 50 (100) ns in order to avoid false alerts. Of


725

course, the detector output holding high or low for a longer time means that the RF filter is effectively broken. Thus the exact choice of the ‘don’t care’ time for the BITE follows from a compromise between the risk of undetected failures and the false failure alert rate.

15.5.3

Receiver performance analysis

This section explores some important details on receiver design and performance, and involves many of the concepts presented in Chapters 9, 11, 13 and – of course – 15 above. The descriptions of this section exploit the analysis of one specific superheterodyne receiver having the structure described in Section 15.4.6.2. The analysis is typically performed at the beginning of the design phase, when the receiver components are not yet available. In this situation, the performances of each block must be estimated on the basis of the available knowledge about the possibilities of the existing component technology. The more accurate are those assumptions, the more realistic are the predicted performances. Such high-level analysis is also useful to drive the component specifications, in order to spread the difficulties over the various elements as equally as possible. If a part is defective, the result is that some of the receiver components will have critical specifications, while others will have unnecessary margins. This section also presents some details on the receiver performance, through a reasonable, though not state-of-the-art, receiver. Thus, the numbers assigned to each component parameter are realistic for realizing the component, but do not guarantee that the receiver performance is the best possible one. Figure 15.30a depicts a simplified view of the same receiver as in Figure The demodulator diagram does not show the Q channel, the 90 LO phase splitter and the LO itself. The reason for this is that our analysis covers only the RF behaviour, not the demodulation aspects. From the RF point of view, the 90 relative shift between the LO applied to MIXI and MIXQ is not relevant, so the two analogue voltages vI ; vQ are totally equivalent, and our analysis considers just one of them.

DEMODULATOR MIXI

LPI AMPI vI

R

Out

L

RF, ωRF

IF, ωIF

vOSC2 = vL2 cos(ω L2t)

MIX 1 In

R L

BP1 AMP1 BP2

BP3 AMP2

DIV1

vOSC1 = vL1cos( ωL1t) (a)

Out

In

BP1 AMP1 BP2

MIX 1′

BP3′ AMP2′ DIV1

MIX I,Q′′ LPI,Q′′ AMPI,Q′′ (b)

Figure 15.30 Double conversion superheterodyne receiver: (a) block diagram with a partial view of the demodulator; (b) equivalent block diagram after frequency translation.

726


Table 15.4 Component parameters of the receiver in Figure 15.30. BP1 20 log10 ðjs21 jÞ 2 20 log10 ðjs11 jÞ ¼ 16.4 20 log10 ðjs22 jÞ 0.1 RPdB NF 2 O1dB –– f1 7.5 f2 8.5 N 3

AMP1 BP2 15 10 –– 3 5

––

MIX1 BP3

2 10 4 16.4 10 16.4 0.1 2 –– 7.5 8.5 3

–– 11 0

––

0.1 4 –– 0.95 1.05 5

AMP2 DIV1 25 10 –– 3 10 –– –– ––

MIXI;Q LPI;Q

3.5 5.5 1 20 10 13.5 –– 3.5 –– –– –– ––

–– 6.5 0 –– –– ––

0.2 1 –– 0 0.01 5

AMPI;Q 60 10 –– 5 15 –– –– ––

The frequencies of the example receiver are: .

RF: First IF: . Baseband frequency range: . First LO frequency: . Second LO frequency: .

fR ¼ oR =ð2pÞ ¼ 8 109 fI ¼ oI =ð2pÞ ¼ 1 109 0–10 MHz fL1 ¼ oL1 =ð2pÞ ¼ 7 109 fL2 ¼ oL2 =ð2pÞ ¼ 1 109

The diagram of Figure 15.30b consists of 10 cascaded linear and nonlinear two-port networks. Each of these networks corresponds to one block of Figure 15.30a. The analysis of the receiver in Figure 15.30 is based on the linear relation between the input and output signal, in the sense specified101 in Section 13.3. The IF of each mixer differs from the respective RF by the LO frequency. Thus a bandpass filter centred on oI placed after the IF port is equivalent to a bandpass with the same passband and centre frequency oR ¼ oL oI . The sign in the above expression is positive or negative if the LO frequency is lower or higher than the RF. Therefore, a frequency-dependent s21 ðoÞ transmission coefficient after the IF port is equivalent to the frequency-translated s21 ðo oLO Þ. Similarly, a frequency-dependent response s21 ðoÞ in the RF path is equivalent to a translated response s21 ðo oL Þ in the IF path. Clearly, the translation can be iterated in the case of multiple conversions, providing that linearity and image suppression are ensured for each of them. The advantage of this technique is that both the amplifiers and the mixers can be considered as quasi-linear two-port networks of the type discussed in Section 9.5. In this regard, it is possible to consider the conversion gain of the mixer as a conventional transmission coefficient. The block diagram in Figure 15.30b is derived from the application of the described technique. The element labels with one or two primes indicate one or two frequency translations. Table 15.4 summarizes the parameters assumed for the various receiver blocks. The criterion used for these assumptions is that they have to be reasonably obtainable from real components. In particular, AMP1 is similar to two cascaded amplifiers of the type described in Section 11.4, and MIX1 has approximately the performances of the circuit discussed in Sections 13.3.2 and 13.3.10. Furthermore, the nonlinear components are assumed to be unilateral (s12 ¼ 0 in the linear operation) and with flat response within the working band. As for the filters, a Chebyshev semi-ideal model takes into account the 101 To summarize, if all the unwanted mixing products – including the image – have negligible amplitude, and the RF amplitude does not exceed a specified linearity limit, the RF to IF relation is linear. This linearity implies that if the RF is sinusoidal the IF is sinusoidal as well with a proportionality relation between the two amplitudes. Moreover, the IF signal phase has a constant phase shift with respect to the RF one. Note also that both negligible spur amplitude and linear mixer operation imply small RF amplitude, while the image attenuation relies on the filters.


727

20 log10(|s21|)

0 –10 –20

(a)

BP1 , BP2

–30

LPI,Q

–40 6.0

6.5

7.0

7.5

BP3 8.0

8.5

9.0

9.5

10.0

20 log10(|s21|)

Frequency, GHz 0 –2 –4

(b)

–6 –8 –10

LPI,Q 1

10

BP3

BP1, BP2 100

1000

Offset frequency (f-f0), MHz

Figure 15.31 Amplitude response of the filters used for the receiver in Figure 15.30: (a) frequency translation to the RF; (b) frequency translation to the baseband. dissipation loss.102 The passband response of the filters BP1 to BP3 is obtained from the Richards transform. Under these assumptions, the only frequency-dependent components are the filters. Figure 15.31 shows the amplitude response of the three receiver filters, translated to the input RF (a) and to the output baseband frequency (b). Note that the abscissa of graph (b) is logarithmic. The first aspect to consider is the image attenuation in the first frequency conversion. The mixer MIX1 converts the RF signals having frequency oR ¼ oL oI into the first IF. The first value of 8 GHz is the receiving frequency, while the image is at 6 GHz. The relative attenuation of the image frequency with respect to the desired signal relies on the rejection of the RF section: BP1 , AMP1 and BP2 . The insertion loss of each of these filters at 8 GHz (6 GHz) is 2 dB (32.6 dB), as in Table 15.4 (Figure 15.31a), while the response of AMP1 is flat over the frequency. Hence, the image rejection equals the sum of the relative attenuation of BP1 and BP2 , which equals 2 ð32:62Þ ¼ 61:2 dB. Similarly, an input signal of 8 GHz produces two IF products having frequency oIF ¼ oRF oLO. The first (second) has a frequency of 1 GHz (15 GHz) and is attenuated103 by 4 dB (>60 dB) by BP3. The second mixer works with the LO frequency at the same value as the RF (1 GHz), and thus has no image. The useful final IF signal occupies a bandwidth from 0 to 10 MHz, corresponding to an RF modulating bandwidth of 20 MHz. The unwanted product of the second conversion has a frequency of 2 GHz, which is well above the cut-off frequency of LPI;Q , and thus virtually eliminated. The simplest parameter to evaluate is the small-signal gain amplitude, which under our hypothesis equals the amplitude of the product of the transmission coefficients of the 10 individual components. Thus, the small-signal gain in dB coincides with the sum of the numbers on the second row of Table 15.4, 102 The lumped or distributed parasitic resistance of the filter elements dissipates electromagnetic energy into heat. From the filter overall performance, the dissipation increases the insertion loss by a quantity proportional to the group delay, which increases as the frequency approaches the passband limits. See also [2]. 103 The attenuation of BP3 at 15 times its centre frequency is difficult to predict, in that many second-order effects are negligible in the passband, but become relevant at such higher frequencies. On the other hand, it is unlikely that AMP2 – designed to work at 1 GHz – keeps a constant gain up to 15 GHz. Also the IF output network of MIX1 presents some attenuation at 15 GHz.

728


Table 15.5 Reflection mismatch of the nine couples of two cascaded two-port networks in the receiver of Figure 15.30b. BP1

AMP1 0.83

Ripple

BP2

MIX1 0.83

0.83

BP3

AMP2 0.83

0.83

DIV1

MIXI;Q 0.55

0.55

LPI;Q

AMPI;Q 1.16

1.16

which is 73 dB. However, the real components have finite reflection coefficients, with an associated mismatch ripple, given by Equation (9.2). Normally, the various components are not physically available in the initial stage of the design, and their design is typically not completed.104 Therefore, the various reflection coefficients are not known. All that is available is the expected maximum allowed amplitude of the reflection coefficient in a given band. The phase and the frequency dependence of the reflection coefficient itself could be totally arbitrary. Moreover, such a situation holds in the next stages of the design, in most cases up to production. Consequently, on the basis of the available information, the best way to evaluate the impact of the impedance mismatches is by the application of Equation105(9.3) to each couple of adjacent stages. Table 15.5 lists the maximum possible variation of the gain for each couple of cascaded networks. The worst case occurs when the minima generated by the interactions between all the connected ports fall at the same frequency, and similarly with the maxima. In this worst case the ripple over the band is equal to the sum of the numbers of Table 15.5, which equals 7.56 dB in our case. Such a bad combination is theoretically possible, but unlikely in that it needs phase rotations of 2p in the reflection coefficients, within two frequencies of the received band. This harmful condition is avoided by ensuring that the 50 O lines that connect the various components are shorter than half a wavelength at a frequency coinciding with the receiving bandwidth. A more realistic, although less conservative, prediction considers that the various ripples at each connected couple of ports are uncorrelated. Thus, their combinations generate a total variation equal to the square root of the sum of the squares of the numbers in Table 15.5, or 2.6 dB in our case. The result obtained from the calculation has to be increased by the ripple of the filters. However, the response of BP1 to BP3 is approximately flat within the band of LPI;Q . The dissipation loss in the latter filter cancels the undulation in the response, and replaces it with a monotonic gain decreasing by about 1 dB from 0 to 10 MHz, as Figure 15.31b shows. In the worst case, the maximum and minimum gain obtained from the reflection combinations occurs at the same frequencies of the filter, and the resulting ripple becomes106 8.56 dB (absolute worst case) and 3.6 dB (realistic case). If all the perturbations of the gain flatness are uncorrelated, the resulting ripple is the square root of the quadratic sum of the various terms, in our case 2.8 dB. The noise performance of the receiver can be evaluated by a further application of Equation (9.32) to couples of adjacent blocks. The choice of how to divide the cascade into groups of two networks is arbitrary, and depends only on simplicity and convenience. The final result is independent of the intermediate subdivisions. One possibility consists of grouping, in the first run, all the blocks in couples of adjacent elements: (BP1 , AMP1 ), (BP2 , MIX1 ), (BP3, AMP2 ), (DIV1 , MIXI;Q ) and (LPI;Q , AMPI;Q ). 104 Indeed the component design cannot be completed until the specifications are defined, and such definition requires the higher level design of the receiver to be finalized. 105 In the general case, the resulting transmission coefficients of two cascaded linear two-ports (A) and (B) is given by the term (2,1) of the matrix (9.2), as discussed in Section 9.2. If one of the two connected (or interfacing) ports is ðAÞ ðBÞ ðA;BÞ ðAÞ ðBÞ perfectly matched (s22 ¼ 0, or s11 ¼ 0), the transmission coefficients simplify to s21 ¼ s21 s21 , as in point 1 of Section 9.2. This assumption involves an error which can vary between a minimum and a maximum, depending on the phase of the reflection coefficients of the two interfaced ports: Equation (9.3) gives the resulting variation. 106 A circuit simulation of the worst case, based on a linear frequency-translated representation of the receiver blocks, gives a ripple of about 9 dB. See the Ansoft file 03_Receiver_Ripple.adsn, which uses long 50 O transmission lines placed between the blocks to model a rapidly varying ripple over the frequency.


729

Table 15.6 Accumulated and final gain, noise figure and output 1 dB compression point of the receiver in Figure 15.30. BP1 Gain=NF=O1dB

AMP1

BP2

MIX1

BP3

AMP2

DIV1

MIXI;Q

LPI;Q

13/5/5 11/13/0 21/7/10 9/10/0 2/6.14/6.97 12/7.06/2.54 14/8.25/3.16 73/8.3/15

AMPI;Q

59/6/15 59/6/15 59/6/15

The resulting structure is a cascade of five two-port networks. Multiple reiterations of the same procedure with the new diagram further simplify to three, two and one cascaded blocks. A similar procedure can be used to determine the compression point of the receiver, by the next application of Equation (9.62). Table 15.6 lists the partial results at each step and the final result for both the noise figure (8.3 dB) and output 1 dB compression point (15 dBm). Subtracting the small-signal gain from O1dB, we obtain the input 1 dB compression point, I1dB ¼ O1dBSSG ¼ 1572 ¼ 57 dBm. The input power for 1 dB of compression is a valid indicator of the maximum power applicable to the receiver input. The corresponding third-order input intercept point is given by Equation (9.56), which is IIP3 ¼ 47:4 dBm. Another consequence of Equation (9.56) is the fact that the third-order intermodulation level is 9:6 2 ¼ 19:2 dBc when the input power equals I1dB. Furthermore, the input power for X dB of lower intermodulation level has to be I1dBX=2. For instance, if we want an intermodulation level of 40 dBc, the input power must be I1dBð4019:2Þ=2 ¼ 57:8 dBm. Low input power means a low level of nonlinear distortion, but also increases the importance of the noise. The equivalent receiver input noise power density is related to its noise figure as 10 log10 ðKTÞ þ NF ffi 174 þ 8:4 ¼ 165:6 dBm=Hz, where the numerical value has been computed for a temperature of 27 C. The total input noise power nin is the integral over the frequency of the density weighted by the frequency response of the receiver. Equivalently, expressing the noise power in dBm and using the noise power bandwidth (15.55) we have Nin ¼ 10 log10 ðKT Þ þ NF þ 10log10 ðNBW Þ ¼ 174 þ 8:4 þ 73:1 ¼ 92:5 dBm

ð15:111Þ

where the numerically evaluated noise bandwidth of the receiver is NBW ¼ 20:6 MHz, slightly more than twice the nominal equal-ripple bandwidth of LPI;Q . An input signal power equal to the value in (15.111) corresponds to S/N of 0 dB. The minimum required S/N depends on the quality of the demodulated signal and on the type of the demodulator; however, 10 dB is a typical number in many cases. The minimum receiver input signal for a specified S/N is known as the sensitivity of the receiver.107 Combining this assumption with the –40 dBc for the third-order intermodulation level, we find that the valid input power for our receiver is from 82:5 dBm to 57:8 dBm, with a dynamic range of 82:557:8 ¼ 24:7 dB. Another parameter used to define the receiver dynamics is the spurious free dynamic range (SFDR), defined as the input power that generates intermodulation products of the same power as the noise.108 It is related to the intercept point as 2 SFDR ¼ ðIIP3Nin Þ 3

ð15:112Þ

With our values, SFDR ffi 30 dB. 107 Sometimes, demodulator performances related to S/N are used to specify the sensitivity, instead of S/N itself. For instance, in digital receivers, the parameter most used is the bit error rate (BER), defined as the percentage of wrong bits over the total received bits. 108 See [11].

MICROWAVE AND RF ENGINEERING Nin →

100

0 PI1dB → –20

Small signal gain, dB

50 –40

0 –60 –50

–100 7.8

← s21

7.9

–80

8.0

8.1

Input power for 1dB compression, dBm Input noise power (NBW = 20 MHz),dBm

730

–100 8.2

Frequency, GHz

Figure 15.32

Linear gain, equivalent input noise power and I1dB of the receiver in Figure 15.30.

It is possible to repeat the computation of the linear gain, input noise power and compression power at different frequencies:109 the result is plotted in Figure 15.32. The input compression power over the frequency is particularly of interest, because it is directly related to the possibility that a strong outof-band signal can generate compression. If compression occurs – no matter if it is caused by in-band or out-of-band signals – both the receiver sensitivity and linearity are compromised. A strong saturating input interfering signal not only intermodulates with the desired signal, but also reduces the amplitude of the latter. An extreme case occurs when the strong signal amplitude consistently exceeds the 1 dB compression. In that case, it is possible to approximate the nonlinear characteristic (9.48) with a threesegment piecewise-linear curve 8 vSAT > > v v < > SAT IN > > c1 > > > < vSAT vOUT ðvIN Þ ¼ c1 vIN ð15:113Þ jvIN j > c1 > > > > > vSAT > > : vSAT vIN > c1 If the input signal is the combination of two sinusoids, vIN ðtÞ ¼ v1 cosðo1 tÞ þ v2 cosðo2 tÞ, with v1 v2 , the first term represents the strong interfering signal. It is possible to write the input signal as vIN ðtÞ ¼

v1 v2 v1 v2 cosðo1 tÞ þ cosðo2 tÞ þ cosðo1 tÞ þ cosðo2 tÞ 2 2 2 2

Since o2 ¼ o1 þ ðo2 o1 Þ, the above expression is equivalent to vIN ðtÞ ¼

109

v1 v2 v1 v2 cosðo1 tÞ þ cosf½o1 þ ðo2 o1 Þtg þ cosðo1 tÞ þ cosf½o1 þ ðo2 o1 Þtg 2 2 2 2

See the Mathcad file 09_Receiver_Analysis.mcd.


731

Adding and subtracting the term ðv2 =2Þcosf½o1 ðo2 o1 Þtg, the input signal takes the form of a sum with six terms vIN ðtÞ ¼

v1 v2 v2 cosðo1 tÞ þ cosf½o1 þ ðo2 o1 Þtg þ cosf½o1 ðo2 o1 Þtg 2 2 2 v1 v2 v2 þ cosðo1 tÞ þ cosf½o1 þ ðo2 o1 Þtg cosf½o1 ðo2 o1 Þtg 2 2 2

ð15:114Þ

The terms on the first (second) row of Equation (15.114) represent an AM (approximate FM) signal110 with carrier at the angular frequency o1 and two sidebands with ðo1 o2 Þ offset. If v1 is higher than the saturating limit vSAT =c1 , then the saturating device cancels the AM,111 and the signal (15.114) becomes o vSAT n v2 v2 vOUT ðtÞ ffi v1 cosðo1 tÞ þ cosðo2 tÞ cos½ð2o1 o2 Þt ð15:115Þ v1 2 2 Expression (15.115) is clearly approximate and shows only one of the many intermodulation terms. Comparing the amplitude of small and large signals at the input and output of the saturating two-port, the first decreases by 6.02 dB more than the latter. The difference between the strong and the weak signals at the output of the saturating element is 20log10 ð2Þ ffi 6:02 dB higher than at the input. In addition, the output amplitude of the strong signal is no greater than112 4ðvSAT =pÞ. Therefore, the output amplitude of the small signal does not exceed the limit 2ðvSAT =pÞ ðv2 =v1 Þ. This situation is known as small-signal suppression, and the strong out-of-band signal that saturates the receiver is named the blocking signal or, in short, blocker. Some microwave systems positively exploit the small-signal suppression. In those cases it must be considered as guaranteed if the saturating two-port has a characteristic of the type (9.48) or, better, (15.113). This implies that there is no frequency dependence on the nonlinear transfer function. Such a condition is never perfectly fulfilled, particularly if the signal frequencies fall within the limits of the working bandwidth of the device.

15.6 Special radio functional blocks This section deals with some important building blocks of radio architecture, such as special implementations of the quadrature phase splitter, the PLL, the ALC, the AGC and the successive detection logarithmic video amplifier (SDLVA). Each of these topics could easily be extended to cover a whole chapter at least, particularly the PLL. The approach followed in this section consists of describing the basic working principles of the various functions, with a glance at some design methods.

15.6.1

Quadrature signal generation

Many of the circuits presented in Sections 15.3 to 15.5 use IQ (or quadrature) mixers. Such functional blocks require the application of two signals to the LO ports of two mixers, with equal amplitude and frequency and with a 90 reciprocal phase shift. The schematics in Sections 13.3.6, 15.2 and 15.3 implemented that function with one single LO, which feeds the two mixers through a 90 hybrid coupler having the isolated port terminated on a matched load. Such modelling is valid from an ideal point of view and could also be close to the physical realization of the circuit. However, quadrature hybrids like113 110

See Sections 15.2.1 and 15.2.2. See also Section 15.3.1.2. 112 See Section 9.5.4. 113 See Section 7.7.3. 111

732


branch-line and Lange couplers present a size of the order of a quarter wavelength. To give a concrete idea, assuming a frequency f0 ¼ 2 GHz and an effective permittivity of the propagation medium114 1 ee f f ¼ 4, the size is l=4 ¼ 0:25 c e0:5 e f f f0 ¼ 18:75 mm. Such a size could be reduced by folding the structure, but remains substantially too large for an IC realization. The lumped realization described in Section 7.8 allows smaller dimensions, but the presence of at least four inductors makes the circuits still too large. Other solutions are available for a 90 splitter when size is a demanding requirement. The first possibility is the three-port network depicted in Figure 15.33a. The network consists of a simple first-order RC lowdpass (output vA ) and highpass (output vB ) filter with the same input vIN . The Fourier domain voltage transfer from the input voltage of that network to its two outputs is given respectively by VA 1 ¼ ; VIN 1 þ jot

VB jot ¼ VIN 1 þ jot

ðt ¼ R CÞ

ð15:116Þ

From Equations (15.116) it follows immediately that the relative complex amplitude between the two outputs is VB =VA ¼ jot ¼ j2pf t. Thus, if vIN is a sinusoid, vA and vB are also sinusoids with the same frequency as vIN and with a relative phase shift of argðVB =VA Þ ¼ p=2, as required. The main drawback of this network is that the amplitude of the two outputs is proportional to, rather than constant over, the frequency. Figure 15.34 shows the amplitude response of the network in Figure 15.33a. The two amplitudes coincide at the frequency f ¼ ð2pÞ1 o ¼ ð2pÞ1 t1 ¼ ð2p R C Þ1. The working bandwidth of our network depends on the accepted amplitude imbalance between the two output voltages. For instance, accepting a deviation of 0:5 dB from the perfect balance, or 0:5 20log10 ðjVB =VA jÞ 0:5, we have that the frequency is bound within the limits 0:944=ð2ptÞ f 1:059=ð2ptÞ. The relative bandwidth of the network in Figure 15.33a for an amplitude imbalance of 0:5 dB is then Df = f0 ¼ Do=o0 ¼ 2ð1:0590:944Þ=ð1:059 þ 0:944Þ ffi 11:5%. The network in Figure 15.33b is called a polyphase phase splitter.115 It has exactly the same voltage transfer ratio as the network in Figure 15.33a. The peculiarity of the polyphase network is that it operates with balanced input voltage generator and output loads. Such a characteristic matches well with the needs of bipolar and CMOS ICs, which use DP as building blocks for amplifiers, oscillators and mixers.116 Besides this aspect, the network in Figure 15.33b is less sensitive to component value tolerance than the one in Figure 15.33a. An additional and important property of the polyphase splitter is that it is possible to cascade multiple sections of it, in order to improve the amplitude flatness over a broader frequency range. The phase response of a multi-stage phase splitter is still ideal, like the simple one in Figure 15.33a. Figure 15.33c shows the schematic of a double section polyphase splitter. Inside each section all the resistors and all the capacitors assume the same value, while they change from section to section. The component values can be determined from a specified bandwidth by applying network synthesis techniques or by numerical optimization. A simple set of design equations is R2 C1 ¼ ¼2 R1 C2 R1 C1 ¼ R2 C2 ¼ t ¼

ð15:117Þ 1 o0

ð15:118Þ

where o0 ¼ 2pf0 is the centre working frequency of the network.

114 Which could be either the relative permittivity of a coaxial cable or stripline, or the effective relative permittivity of a microstrip realized on the given substrate. 115 See [13, 14] for further details on that network. 116 See Sections 11.6.3, 12.5.6 and 13.3.9, respectively.

RF AND MICROWAVE ARCHITECTURES R

733

vA

+ C

vIN C

vB

(a)

R R

+ vIN

C R

+ vA –

C R

C R

+ vB –

(b)

C R1

R2

C1

C2

R1

R2

C1

C2

R1

R2

C1

C2

R1

R2

C1

C2

+ vIN

+ vA2 -

+ vB2 –

(c)

Figure 15.33 RC quadrature splitters: (a) simple lowdpass/highpass combination; (b) single section polyphase splitter; (c) double-section polyphase splitter. Once f0 is specified, fixing one among R1 , R2 , C1 , C2 , then Equations (15.117) and (15.118) return the remaining three. Figure 15.34 shows the amplitude response117 of the two-section polyphase p network, ffiffiffi designed with Equations (15.117) and (15.118). Its 0.5 dB bandwidth limits are given by 1=ð2 2ptÞ pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi f 1=ð 2ptÞ, corresponding to a relative bandwidth of one octave: Df = f0 ¼ 2ð 2 0:5Þ= pffiffiffi pffiffiffiffiffiffiffi ð 2 þ 0:5Þ ¼ 2=3 ffi 66%. All the three circuits in Figure 15.33 work as described if the input voltage generator has zero series impedance and the loads are ideal open circuits. For this reason they require buffering amplifiers with low 117

See the Ansoft file 04_Polyphase.adsn.

734


VB

20 log10(|VOUT /VIN |)

–2

VA2 –4

VB2 VA

–6

–8 0.50

0.75

1.00

1.25

1.50

ωτ

Figure 15.34

Amplitude responses of the networks in Figure 15.33.

output (high input) impedance at the input (output). Such buffers are relatively easy to realize by means of DPs. The size of the polyphase splitter plus the three buffers is still much smaller than any distributed or lumped realization of the type discussed in Sections 7.7.3 and 7.8. For the effective output and input impedance of the amplifiers, they are obviously different from zero and infinite, respectively. This nonideality requires an adjustment of the polyphase network values, usually performed with CAE techniques. The result is a narrower working bandwidth with worse amplitude and phase balance than the ideal case. A further common disadvantage of all the networks of Figure 15.33 is the energy dissipation due to the resistors. The buffers have the additional task of restoring the lost RF power. Figure 15.35 shows a purely digital quadrature signal generator118 exploiting two flip-flops of the type ‘D’. Briefly, a ‘D’ flip-flop is a digital device that updates its output (Q) to the value presented on the input (D), in correspondence of the leading edge of the clock (CLK) signal. Then, Q holds its status until a new leading edge of CLK occurs. The other output (Q) is always the logical negation of Q: the first is low when the latter is high and vice versa. The circuit in Figure 15.35a produces the waveforms119 depicted in Figure 15.35b: the outputs vA and vB are ideally square waves with a relative shift of one-quarter of a period and half of the frequency of the input vIN . Therefore, the oscillator connected to the input of our circuit has to generate a double frequency with respect to the one that the mixers need. This could be an obstacle if the circuit working frequency approaches the maximum allowed by the technological process. In other cases, the double frequency operation of the oscillator is a positive advantage, in that it reduces the size of the tank inductor, which typically is one of the largest structures of the IC. For the square wave produced by the circuit of Figure 15.35, it must be observed that: (a) No physical digital circuit produces ideal square waves; rather they generate distorted sinusoids, with harmonic levels that could be acceptable in some applications. (b) The Gilbert cell mixer does not necessarily require a sinusoidal LO. It can also operate with a square-wave LO, working like a switching mixer. 118 119

See [14] for more details. See the SIMetrix file 15_IQ_analog_digital_waveforms.sxsch.


735

vA

D

Q

D

Q

CLK

Q

CLK

Q (a) vB

v IN

v IN

vB

(b)

vA 0

TCLK

Figure 15.35

t

Flip-flop-based quadrature signal generator.

(c) If the waves of the circuit in Figure 15.35a are not usable, it is still possible to lowdpass- or bandpass-filter the square waves, in order to obtain sinusoids from them. However, these filters increase the circuit size and introduce amplitude and phase imbalance, due to the tolerance on the filter components.

15.6.2

PLL

The PLL is one of the key building blocks of modern high-frequency signal generators. It is almost ubiquitous in radio equipment. The PLL discipline involves concepts that are normally marginal for microwave engineers, like control system theory and digital techniques. Nevertheless, the importance of PLLs in modern high-frequency equipment requires an exposition of that topic, even inside a generaloriented book like this one. PLL applications are wide: they extend from frequency synthesis to demodulation and even to motor speed stabilization. This section deals with the application of the PLL to frequency synthesis, presenting some fundamental concepts together with the basic analysis and design principles. The interested reader can find more in-depth considerations in [16–22]. Figure 15.36 shows the diagram of a PLL which includes five basic blocks: phase detector (PD), loop filter (LF), VCO, output power divider (DIV) and frequency divider (FD). The power divider delivers a portion of the VCO output signal to the input of the FD. Thus, DIV can be a Wilkinson power divider, a directional coupler, a hybrid junction, a resistive splitter, and so on. The scheme in Figure 15.36 is used to stabilize the VCO frequency by comparing it with a stable one provided by a suitable crystal oscillator. In essence, if the loop is stable and presents enough loop gain, then the steady state phase difference between the two inputs of the PD is very small, ideally zero. When such a condition occurs, the loop is locked, or the PLL is in the lock state, and the frequency difference

736

MICROWAVE AND RF ENGINEERING LF

PD

Ref

VCO

outPD

ϕ

Out

DIV

vTUNE

:N

(a)

FD Φ(PD) n

Ref ΦREF

Φ (LF) n

OUTPD

Kd

VTUNE

G(j ω)

PD

Φ (VCO) n

δ(f )ω0

LF

Ωvco

Kv

(j ω)

ΦVCO Out

–1

VCO

ΦV

(b) –1

NDIV (FD)

Φn

FD (PD)

(LF)

Φn ΦREF

Kd

G(f ) OUTPD

(VCO)

δ(f )ω0

Φn

Φn –1

(j 2πf )

Kv

Ωvco

VTUNE

ΦVCO

ΦV

(c) N–1 DIV

–1 (FD)

Φn

OUTPD

VTUNE + vc – –

LF

AIvc

RI

C2

C1

(d)

Figure 15.36 The PLL: (a) principle; (b) functional block diagram; (c) signal flow graph equivalent to (b); (d) one possible second-order type II loop filter.

between the two PD inputs is also zero.120 Thus, the VCO steady state frequency equals the reference one multiplied by the frequency division factor. In ideal oscillators, the frequency is constant, while the phase constantly increases over time, unless they are intentionally modulated. Unfortunately, no real oscillator presents that ideal working: the frequency drifts over time, and the phase presents fluctuations known as phase noise, as described in Section 12.7. 120 The frequency difference between the two input signals of the PD is the derivative of the phase difference with respect to time. If the primitive function is zero, the derivative is also zero.


737

This section analyzes the effects of the frequency stability and phase noise, by considering the deviations from ideality in the time domain, or by considering the respective Fourier transforms, following the simplest of the two approaches on a case-by-case base. In the following considerations, lower case letters denote time domain quantities and the corresponding capital letters indicate the respective Fourier-transformed functions; the subscript n denotes a noise term. The PD output quantity – which could be a voltage or a current, depending on the component type – is proportional to the difference between its two input phases. Furthermore, the PD adds some noise to its output error signal. From this and for the linearity of the Fourier transform, it follows that OUTPD ¼ Kd FREF FOUT;FD þ FðnPDÞ

ð15:119Þ

The output voltage of the LF equals the PD output multiplied by a transfer function, and with additional noise: VTUNE ¼ Gð f Þ OUTPD þ VnðLF Þ

ð15:120Þ

The LF is a linear network, typically realized with resistors and capacitors; sometimes linearly controlled generators and/or inductors are also present. Therefore, its transfer function Gð f Þ is a ratio between polynomials of the variable jo ¼ j2pf with real coefficients. It realizes a voltage transfer ratio or a transimpedance, depending on whether the PD output is a voltage or a current. PLL operation requires that Gð f Þ is a lowdpass function, in the sense that Gð f ¼ 0Þ „ 0, jGð f Þj has to decrease with f , and lim f ! 1 Gð f Þ must be finite. From these assumptions it follows that the denominator of Gð f Þ has a degree no lower than the numerator, and the zero-order coefficient of the numerator must be non-zero. For the LF DC response, it could be finite or infinite: the corresponding PLL is of type I or II. Neglecting the phase noise, the relation between the VCO output angular frequency and the tuning voltage can be approximated121 by a first-order function as oV ¼ 2pfV ¼ Kv vTUNE þ o0 ¼ Kv vTUNE þ 2pf0 The parameter Kv is the VCO gain, and is proportional to the modulation sensitivity considered in point 9 of Section 12.8.3, by the factor 2p. The phase is the integral over time of the angular frequency. Passing to the Fourier domain, applying the properties (B.13) and (B.19), and considering the VCO noise, it follows from the relation above that FVCO ¼ Kv

VTUNE þ dð f Þo0 þ FðnVCOÞ j2pf

ð15:121Þ

The FD is typically a digital component, producing an output square wave having a frequency equal to the input one divided by a constant factor NDIV . From the simplest to the most complicated cases, NDIV could be an integer power, a generic integer or a fractional number. Using the regular frequency, it is ðFDÞ

ðFDÞ

ðFDÞ

ðFDÞ

oOUT ¼ oIN =NDIV ¼ 2pfOUT ¼ 2pfIN =NDIV Integrating the first and last members of the above expressions, we pass from the angular frequency to the phase, obtaining ðFDÞ

fOUT ¼

1 ðFDÞ f NDIV IN

121 Figure 12.29 shows one example of output frequency versus tuning voltage of a VCO. It gives an idea about the accuracy of the linear approximation in a given tuning range.

738


Such a relation also subsists between the Fourier domain quantities. From this, and considering that the FD introduces phase noise, like the PD and LF, we get ðFDÞ

FOUT ¼

1 1 ðFDÞ F þ FðnFDÞ ¼ FVCO þ FðnFDÞ NDIV IN NDIV

ð15:122Þ

The relations (15.119) to (15.122) can be represented by the block diagram of Figure 15.36b, which is equivalent to the SFG of Figure 15.36c. The system described by the relations above and the graphical representations is linear. Therefore we can separately calculate and superimpose the effect of the different excitation on the output, which could be the output frequency or the phase noise. The calculations are easily performed by multiple application of Mason’s rule. The result is

FVCO ¼

Kd Kv

Gð f Þ Gð f Þ ðPDÞ 1 dð f Þ ðVCOÞ Gð f Þ ðFDÞ þ Kv FðnLFÞ þ Kd Kv FREF þ Kv F F F j2pf j2pf n jo j2pf n j2pf n Kd Kv Gð f Þ 1þ NDIV j2pf

Rearranging the terms, we obtain FVCO ¼ H ð f Þ FðnINBANDÞ ð f Þ þ F0 ð f Þ þ ½1H ð f ÞFðnVCOÞ ð f Þ

ð15:123Þ

Kd Kv Gð f Þ NDIV j2pf Hð f Þ ¼ Kd Kv Gð f Þ 1þ NDIV j2pf

ð15:124Þ

where:

is the closed-loop gain of the PLL – it is a rational function of the variable jo with real coefficients, as a consequence of the rational nature of GðjoÞ and since Kd , Kv and N are real numbers; " # FðPDÞ ð f Þ FðnLF Þ ð f Þ ð15:125Þ FðnINBANDÞ ð f Þ ¼ NDIV FREF ð f ÞFðnFDÞ ð f Þ þ n þ Kd Kd Gð f Þ is the in-band phase noise of the PLL; and F0 ¼

1H ð f Þ dð f Þ j2pf

Furthermore, the properties are as follows: Kd Kv Gð f Þ Kd Kv Gð f Þ NDIV j2pf NDIV ¼ lim (a) lim H ð f Þ ¼ lim Kd Kv f !0 f !0 Kd Kv Gð f Þ f ! 0 Gð f Þ j2p f þ 1þ NDIV NDIV j2pf and; since by hypothesis lim f ! 0 Gð f Þ ¼ Gð0Þ „ 0, it follows that Kd Kv Gð0Þ N lim H ð f Þ ¼ DIV ¼1 Kd Kv f !0 Gð0Þ NDIV


739

Kd Kv Gð f Þ Kd Kv Gð f Þ NDIV j2pf (b) lim H ð f Þ ¼ lim ¼ lim f !1 f !1 Kd Kv Gð f Þ f ! 1 NDIV j2pf 1þ NDIV j2pf Now, lim f ! 1 Gð f Þ ¼ 0 is finite, due to our initial assumptions on the loop filter, thus lim f ! 1 ½ðKd Kv =NDIV Þ Gð f Þ=ð j2pf Þ ¼ 0 and lim f ! 1 H ð f Þ ¼ 1. (c) The maximum degree of H ð f Þ exceeds that of Gð f Þ by one. It is possible to summarize the properties (a) to (c) by saying that H ð f Þ is a lowdpass rational real function of the variable jo with one order more than Gð f Þ. (d) The PLL order is defined as coincident with the one of H ð f Þ, which equals the denominator degree of H ð f Þ itself, because of (a) and (b). (e) The time domain expression for the effect that the term F0 produces on the PLL output coincides with the inverse Fourier transform of the term itself. The property (B.20) implies j0 ðtÞ ¼ F 1 ½F0 ð f Þ 1H ð f Þ 1H ð f Þ ¼ F 1 dð f Þ ¼ lim f !0 j2pf j2pf ¼ lim

f !0

1 j2p f

1 NDIV 1 ¼ lim Kd Kv Gð f Þ f ! 0 Kd Kv Gð f Þ 1þ NDIV j2p f

The last term of the above equation is zero or finitely non-zero if the LF transfer function has one or no pole at the origin, or the PLL is type II or I. Thus F0 can produce a constant phase offset in the time domain, at most. Therefore, usually PLL models neglect F0 , as we will in the remaining part of this section. The PLL phase noise is derived from Equation (15.123) after extracting the noise contributions. Considering that all the noise terms are uncorrelated, and thus sum quadratically to the output, the PLL phase noise is 2 2 ð15:126Þ jFVCO ð f Þj2 ¼ FðnINBANDÞ ð f Þ jH ð f Þj2 þ j1H ð f Þj2 FðnVCOÞ ð f Þ Again, for the non-correlation between the different in-band noise terms, this results in " 2 ðLF Þ # F ð o Þ 2 ðINBANDÞ 2 ðREF Þ 2 ðFDÞ 2 FðnPDÞ ð f Þ n 2 F NDIV ð f Þ ¼ Fn ð f Þ þ Fn ð f Þ þ þ 2 n Kd2 Kd jGð f Þj2

ð15:127Þ

The relation between the reference and the PLL output frequency is obtained by neglecting the noise contributions of Equation (15.123) and multiplying both members by j2pf. The result is OVCO ¼ NDIV H ð f ÞOREF

ð15:128Þ

Equation (15.127) and property (a) of H ð f Þ imply that if the reference frequency is constant, then the VCO output frequency of a PLL is also constant and equals NDIV times the reference one. Indeed, if oREF ¼ 2pfREF ¼ oR0 ¼ 2pfR0 is constant, it follows from the property (B.13) that OREF ¼ F ½oREF ¼ oR0 dð f Þ. Thus, H ð f ÞOREF ¼ H ð f ÞoR0 dð f Þ is non-zero only if f ¼ 0, therefore OVCO ¼ NDIV H ð f ¼ 0Þ oR0 dð f Þ. Finally, passing to the time domain, oVCO ¼ F 1 ½OVCO ¼ NDIV oR0 F 1 ½dð f Þ ¼ N oR0

ð15:129Þ

740


Equation (15.129) states that if the reference frequency is constant, and the loop is stable,122 then the synthesizer output frequency is constant as well, despite the tendency of the PLL to drift. Moreover, the PLL frequency equals the reference one multiplied by the divider factor. Consequently, it is possible to change the output frequency by changing NDIV . The minimum step between two consecutive frequencies, or the frequency resolution of the PLL, equals the minimum step on the frequency division factor multiplied by the reference frequency.123 The PLL synthesizer has the same relative frequency stability as the reference – typically provided by a highly stable crystal oscillator – independently of N and the inherent VCO stability. From Equations (15.126) and (15.128) it follows that the PLL performances are strongly affected by the closed-loop response. Passing to less abstract considerations, Figure 15.36d shows one possible LF,124 with a second-order transfer function G ð f Þ ¼ AI

1 þ R1 C1 j2pf ðC1 þ C2 Þ j2pf þ R1 C1 C2 ð j2pf Þ2

ð15:130Þ

The PLL with the LF in Figure 15.36d is of type II. Substituting the function (15.130) into the closed-loop function (15.124), after some rearrangements we obtain Hð f Þ ¼

R1 C1 ð j2pf Þ þ 1 NDIV NDIV 3 R1 C1 C2 ð j2pf Þ þ ðC1 þ C2 Þð j2pf Þ2 þ R1 C1 ðj2pf Þ þ 1 Kd Kv AI Kd Kv AI

ð15:131Þ

The function (15.131) is low pass, rational in the variable ð j2pf Þ, with real coefficients, and third order, as expected. The unit gain frequency fm0 of the closed-loop response is such that jH ð fm0 Þj ¼ 1. The unit closed-loop gain is a PLL design parameter which can be used to optimize the noise and/or tuning speed performances, but it cannot be arbitrary. Rather, it must be much smaller125 than the reference frequency. If fm0 approaches the reference frequency, then the tuning voltage presents relevant residuals of the reference frequency, the PLL spectrum presents unwanted lines due to the reference frequency modulation, and the equations developed in this section lose their accuracy. Figure 15.37 plots the closed-loop function126 (15.131) versus the frequency normalized to fm0 , with some particular values of the parameters Kd ; Kv ; AI ; R1 ; C1 ; C2 (black curve). The grey curve in Figure 15.37 is the error function 1H ð f Þ. From Equation (15.127) it follows that the PLL phase noise is a linear combination of the in-band noise – which includes the reference phase noise – and the VCO phase noise. The frequency-dependent weights of the combination are the lowdpass H ð f Þ for the in-band noise and the highpass 1H ð f Þ for the VCO noise. 122 The loop is stable if the closed-loop gain, analyzed in the complete complex plane s ¼ s þ jo, has its poles lying in the left half plane of the variable. Equivalently, it is possible to check the loop stability with the Nyquist criterion already considered in Section 12.4. One possible application of the method consists of breaking the connection between the FD and PD. Then, the gain from the reference of PD to the FD output, evaluated in the complex plane s ¼ s þ jo, must not encircle the point s ¼ 1 þ j0. The sign inversion in Section 12.4 is related to the phase inversion in the PLL loop path. 123 The frequency division factor is typically integer, with a consequent resolution equal to oREF . More sophisticated designs – called N-fractional – allow the implementation of non-integer N, with a smaller than one minimum step. The resulting PLL frequency resolution can be consistently smaller than fREF . 124 The LF in Figure 15.36d is intended to work together with a voltage output PD. If the latter is a current output one, then the voltage-controlled current generator AI can be considered as part of the PD. 125 Typically fm0 =fREF < 1=10, although the true maximum value allowed for fm0 =oREF is strongly dependent on the PD and LF circuit in use. 126 See the Mathcad file 10_PLL.mcd.


741

20

PLL loop gain, dB

0

H

–20

–40

–60

–80 –3 10

1-H

10

–2

10

–1

10

0

1

10

10

2

10

3

f/fm0

Figure 15.37

PLL closed-loop functions.

Note that, here, the variable f represents the Fourier transform variable of the time-dependent noise fluctuations; it is a kind of ‘frequency of the phase’ and is sometimes referred to as the offset frequency. For offset frequencies much smaller or much greater than fm0 , the closed-loop gain is close to one or zero, respectively, thus 2 2 ð15:132Þ jFVCO ð f fm0 Þj2 FðnINBANDÞ ð f Þ ; jFVCO ð f fm0 Þj2 FðnVCOÞ ðoÞ From relations (15.132) it follows that the PLL synthesizer phase noise is dominated by the in-band (VCO) noise at low (high) offset frequencies. The in-band phase noise power is proportional to the square of the frequency division factor. Holding the in-band noise constant, the low offset frequency phase noise of the PLL increases by 20log10 ðNDIV Þ ¼ 20 log10 ðoVCO =oR0 Þ ¼ 20log10 ð fVCO = fR0 Þ dB. From this consideration it follows that the PLL noise for a given output frequency increases in inverse relation to the reference frequency. On the other hand, Equation (12.68) states that, at small offset frequencies, and holding the resonator quality factor and the active device noise constant, the reference phase noise itself is inversely proportional127 to fR0 . Therefore, if the reference dominates the in-band noise, the PLL phase noise at low offset frequencies is virtually independent of the frequency reference. However, the low-frequency reference prevents use of the wideband PLL, and thus cleaning up the VCO noise. Figure 15.38 shows the intrinsic VCO phase noise (thin black line), the in-band noise (grey line) and the resulting PLL phase noise (thick black line) for a PLL having the closed-loop response as in Figure 15.37 The loop bandwidth is close to the optimum, in that it is close to the intersection between the VCO and the in-band noise. Higher (lower) fm0 will increase the in-band (VCO) contribution to the PLL phase noise in the low (high) offset frequency range, with a resulting overall higher noise. The settling time is another important PLL parameter. It defines how fast the output frequency reaches its steady state value, after the frequency division factor abruptly changes from one value NDIV1 to another NDIV2. It is possible to demonstrate that, in such a transition, the evolution of the PLL is equivalent to that of a hypothetical PLL with constant division factor NDIV2 and with the reference frequency 127 The small offset frequencies are those values such that the frequency-dependent terms of Equation (12.68) are much greater than the constant noise floor. Furthermore, it has to be considered that the low offset frequency noise of the reference is relevant for the PLL noise, due to the lowdpass action of the PLL, as in Equation (15.126).

742

MICROWAVE AND RF ENGINEERING –20

Phase noise, dBc/Hz

–40

VCO

–60 –80

PLL

–100

In-band

–120 –140 –160 –3 10

10

–2

10

–1

10

0

10

1

2

10

10

3

f/fm0

Figure 15.38

PLL phase noise.

1.5 [ωPLL(t)-ωPLL(0)]/∆ωPLL 1.0

0.5

0.0 –1

0

1

2

3

4

5

fm0 t

Figure 15.39

PLL normalized settling time.

abruptly changing from o0R0 to oR0 , provided that o0R0 NDIV2 ¼ oR0 NDIV1 . In other words, the variableNDIV PLL is equivalent to a corresponding variable-oREF PLL if they present the same initial frequency, final frequency and final closed response.128 The variable reference frequency PLL is easier to analyze, in that its output frequency coincides with the step response of the transfer function (15.124), with the initial (final) value of the step of oR0 NDIV1 (oR0 NDIV2 ). Figure 15.39 shows the output frequency, normalized to the step, of a PLL having the close-loop response of Figure 15.37: the variable oPLL ðtÞ is the PLL output frequency, and DoPLL ¼ limt ! 1 oPLL ðtÞoPLL ðt ¼ 0Þ is the difference between the initial and the final 128

The frequency division factor N affects the closed-loop response H ðjoÞ, as in Equation (15.124).


Figure 15.40

743

A practical PLL synthesizer: (a) schematic; (b) photograph (65 25 mm).

steady state value. Note that the output frequency presents damped oscillations at a frequency close to fm0 . This suggests that fast settling PLLs require wide closed-loop bandwidths. The frequency division factor can also be varied over time, in order to apply FM to the PLL.129 Also in this case, the variable-NDIV PLL can be transformed into one equivalent variable-oREF PLL. The consequence is that the closed-loop response lowdpass filters the modulating signal, reducing its highfrequency components. Figure 15.40 shows the block diagram (a) and a photograph (b) of a PLL synthesizer producing an output variable frequency within the range 7–8 GHz, with a step of 1 MHz. The PLL uses a two-stage FD. The first high-frequency divider has a fixed division factor N1 ¼ 8 and delivers an output frequency in the range of 785–1000 MHz. The second divider has a variable factor N2 ¼ 7000–8000. The resulting total division factor is NDIV ¼ N1 N2 ¼ 56 000 to 64 000 with a minimum step of eight. The reference frequency of 125 kHz ensures the required resolution of 8 125 kHz ¼ 1 MHz; the loop bandwidth is 5 kHz.

129 If the modulating digital word that changes N is the integral of the modulating signal, the FM corresponds to PM, as explained in Section 15.2.

744


For the microwave component realization, the circuit employs a VCO similar to a half section of the push–push oscillator of Section 12.9, realized with the same technological process. The element used to deliver the VCO output signal to the FD is a parallel-line directional coupler130 (COUP). The amplifier AMP increases the VCO output power and minimizes the pulling,131 by isolating the oscillator from the external load. Finally, the microstrip semi-lumped lowdpass filter (LPF)132 reduces the output harmonics.

15.6.3

ALC and AGC

This section considers two important RF building blocks known as ALC and AGC. The first is typically used to generate signals with amplitude depending on a control voltage but not on the oscillator power or on the gain of components placed between the oscillator and the output.133 The main AGC application is in receivers, where the demodulator input level has to be constant,134 independently of the attenuation between the transmitter and receiver.135 Both ALC and AGC have the same functional diagram in principle, as depicted in Figure 15.41a. It basically consists of one variable attenuator ATT, placed between the input and output and controlled by a feedback system. The directional coupler COUP connects the attenuator to the AGC/ALC output and to the input of the detector DET. The latter provides a continuous voltage, monotonically increasing with the output power, to a difference amplifier AMP, which drives the attenuator control port through the lowdpass filter LPF2. The other input of AMP is a control voltage vCNTRL , which mainly determines the output power. More generally, the ATT could represent a variable-gain amplifier or a combination of cascaded variable attenuators and/or amplifiers: the equations of the system do not change. The following treatment assumes some simplifying hypotheses in order to make the analysis reasonably easy. Nevertheless, the generality of the results is adequate for most applications. Let us assume that: (a) The ATT is linear from the RF input to the RF output, for any fixed control voltage. The directional coupler is always linear by definition. (b) All the components along the RF path, namely ATT, COUP and DET, are perfectly impedance matched. In other words, all the reflection coefficients at all the ports of all the RF components are zero. (c) The attenuation of the ATT is linear in dB with the control voltage. If it is not, a suitable linearizing circuit is assumed to be inserted between the control voltage and the attenuator control port, although Figure 15.41a does not explicitly show such an element. Alternatively, the linear relation can be assumed, even without linearization, if the attenuation variation range is sufficiently limited. (d) The DC component of the detector output voltage is linear with the input power expressed in logarithmic units. Section 13.2 showed that detectors produce DC voltages proportional to either 130

See also Section 7.7.3.3. See Section 12.7, point 3. 132 See also Section 8.4. The difference from Section 8.4 is that the shunt capacitors in the LPF are realized with shunt open stubs, instead of low characteristic impedance lines. 133 One example of such an application is the polar transmitter block diagram of Figure 15.6c. In that arrangement the ALC compensates the gain ripple of the components placed between the oscillator output and the transmitter output. 134 Here, the expression ‘constant level’ indicates that the signal envelope is bound between a specified minimum and maximum amplitude, which are compatible with demodulator operation. 135 Such attenuation includes transmitted power, antenna gain and receiver gain with associated ripple and variations from piece to piece. 131


745

ATT1 COUP 1

v1(t)

vIN(t)

vOUT(t)

v2(t) v6(t) DET 1

LPF 2

(a)

v3(t)

v5(t) LPF1 AMP2 +

avvin

+ vin –

v4(t)

vCETRL(t) LPF3

COUP –c DIR δ(f)

ATT

Pin,dBM(f)

POUT,dBM(f) –cCOUP δ(f)

–1 ATTdB

(b)

γLOG

kA,0δ(f) γ LOG,0 δ(f)

kA

V3(f)

V6(f)

DET

G1(f) AMP

G2 ( f )

Figure 15.41

aV

–1

vCETRL(f) G3(f)

ALC and AGC: (a) functional diagram; (b) equivalent block diagram.

the input power or envelope, not to any logarithmic quantity. Nonetheless, similar to (c), it is possible to assume that a suitable circuit136 convertsthe detector output voltageinto a linear dB one. Alternatively, RF circuits are available to produce the required dB proportional DC voltage.137 136

Such a device is known as a logarithmic amplifier. They are named successive detection logarithmic video amplifiers (SDLVAs), see Section 15.6.4 for more details. 137

746

MICROWAVE AND RF ENGINEERING Again, in the absence of any of the two solutions above, a simple diode or transistor detector can be approximately considered as linear in dB, within a limited range of input amplitudes. (e) The transfer functions of the filters LP1 to LP3 are low pass in the sense used in Section 15.6.2 for the PLL LF, and the ratio G3 ð f Þ=G2 ð f Þ is low pass in the same sense, as well. (f) One of G1 ð f Þ and G2 ð f Þ has one single138 pole at the origin and also for the product G1 ð f ÞG2 ð f Þ. (g) One at least of G1 ð f Þ and G2 ð f Þ presents transmission zeros at infinite frequency. (h) The AMP is linear and independent of the frequency, so any bandwidth limitation of the amplifier is considered as embedded in one of the three lowdpass filters LP1 to LP3 connected to it.

Further, upper case variables are assumed to be the Fourier transforms of the corresponding lower case ones, as usual. Let the ALC/AGC input voltage be a modulated sinusoid. For hypothesis (a) the output voltage is also sinusoidal and has the same frequency and amplitude proportional to the input. The two voltages are then vIN ðtÞ ¼ aIN ðtÞcos½oR t þ jIN ðtÞ;

vOUT ðtÞ ¼ aOUT ðtÞcos½oR t þ jOUT ðtÞ

ð15:133Þ

From hypothesis (b), the gain from the input to the output is given by the product of the transmission coefficient of the ATT and the direct transmission coefficient of COUP, therefore h i ðATT Þ ðCOUPÞ ðATT Þ ðCOUPÞ jIN ðtÞ ð15:134Þ aOUT ðtÞ ¼ s21 s21 aIN ðtÞ; jOUT ðtÞ ¼ arg s21 s21 ðATT Þ

The coefficient s21 changes – in amplitude and also in phase – with the control voltage. Therefore, the attenuation setting affects not only the amplitude, but also the phase of the output signal. Such side effects – not considered here – could be harmful in applications that do not tolerate PM associated with AM. The instantaneous power of an AM signal at a given instant is the power of a sinusoid having a constant amplitude equal to that of the AM signal at that time. Application of such a definition with logarithmic units, for instance dBm, to the signals (15.133) gives 2 2 a ðt Þ a ðtÞ þ 30; pOUT;dBm ðtÞ ¼ 10 log10 OUT þ 30 ð15:135Þ pIN;dBm ðtÞ ¼ 10log10 IN 2R0 2R0 The Fourier transforms of functions (15.135) are PIN;dBm ð f Þ ¼ F pIN;dBm ðtÞ þ 30 dð f Þ; POUT;dBm ð f Þ ¼ F pOUT;dBm ðtÞ þ 30 dð f Þ

ð15:136Þ

Denoting associated with the transmission coefficients (15.134) as attdB ¼ ðATT Þthe attenuation in dB ðCOUPÞ 20 s21 and cDIR ¼ 20 s21 respectively, quantities (15.135) have the reciprocal relation pOUT;dBm ðtÞ ¼ pIN;dBm ðtÞattdB ðtÞcDIR

ð15:137Þ

Calculating the Fourier transform of functions (15.137), and applying the property (150.13) to the constant term cDIR , we obtain POUT;dBm ð f Þ ¼ PIN;dBm ð f ÞATTdB ð f ÞcDIR dð f Þ

ð15:138Þ

From hypothesis (c), the relation between the attenuation attdB and the control voltage v5 is attdB ðtÞ ¼ kA;0 þ kA v5 ðtÞ

ð15:139Þ

138 A filter with one pole at the origin is marginally stable. Multiple poles at the origin or at any other part of the imaginary axis imply instabilities. For instance, the pulse response of a filter with two poles at the origin is a parabola, therefore the response increases indefinitely over time even with an excitation having a finite time duration.


747

Passing to the Fourier domain, and since kA;0 is constant, we have ATTdB ð f Þ ¼ kA;0 dð f Þ þ kA V5 ð f Þ

ð15:140Þ

Proceeding in a similar way, and considering hypothesis (d), it is possible to derive a relation between the detector DC voltage V4 and PIN;dBm , which is ð15:141Þ V3 ð f Þ ¼ PIN;dBm ð f ÞcCOUP dð f Þ gLOG þ gLOG;0 dð f Þ The Fourier domain input/output relations of the three lowdpass filters LP1 to LP3 are even more straightforward. Combined together, they give V6 ð f Þ ¼ ½G1 ð f ÞV3 ð f ÞG3 ð f ÞVCNTRL ð f ÞaV G2 ð f Þ

ð15:142Þ

Equations (15.138) and (15.140) to (15.142) describe a system corresponding to the block diagram of Figure 15.41b. Applying the same technique used in Section 15.6.2 for the PLL, we obtain the input/ output relation of the ALC/AGC 1 G3 ð f Þ VCNTRL ð f Þ gLOG G2 ð f Þ þ kA;0 HALGC ð f Þ dð f Þ

POUT;dBm ð f Þ ¼ ½1HALGC ð f ÞPIN;dBm ð f Þ þ HALGC ð f Þ gLOG;0 þ cDIR kA;0 þ cCOUP gLOG

ð15:143Þ

where HALGC ð f Þ is the ALC/AGC closed-loop response, defined as HALGC ð f Þ ¼

gLOG aV kA G1 ð f ÞG2 ð f Þ 1 þ gLOG aV kA G1 ð f ÞG2 ð f Þ

ð15:144Þ

From hypotheses (e) to (g) the following observations result. Observations: (a) lim HALGC ð f Þ ¼ lim f !0

gLOG aV kA G1 ð f ÞG2 ð f Þ g aV kA G1 ð f ÞG2 ð f Þ ¼ lim LOG ¼1 f ÞG2 ð f Þ f ! 0 gLOG aV kA G1 ð f ÞG2 ð f Þ

o ! 0 1 þ gLOG aV kA G1 ð

(b) lim ½1HALGC ð f Þ ¼ 0 f !0

(c) lim HALGC ð f Þ ¼ lim f !1

gLOG aV kA G1 ð f ÞG2 ð f Þ ¼ lim g aV kA G1 ð f ÞG2 ð f Þ ¼ 0 f ÞG2 ð f Þ o ! 1 LOG

f ! 1 1 þ gLOG aV kA G1 ð

(d) lim ½1HALGC ð f Þ ¼ 1, as a straightforward consequence of (c). f !1

(e) From (a) to (d), HALGC ð f Þ and 1HALGC ð f Þ present a lowdpass and highpass response, respectively, like the PLL closed-loop response (15.124). The contribution of the terms of the function (15.143) to the ALC/AGC response in the time domain coincides with the inverse Fourier transform of the terms themselves. In particular, for the property (150.20), the third term of the function (15.143) gives the constant gLOG;0 gLOG;0 F 1 cDIR kA;0 þ cCOUP þ kA;0 HALGC ð f Þ dð f Þ ¼ cDIR þ cCOUP gLOG gLOG The effect of such a constant offset on the output power can be neglected, without loss of accuracy, by introducing a proper quantity related to the output power as

748


P0OUT;dBm ð

gLOG;0 f Þ ¼ POUT;dBm ð f Þ cDIR kA;0 þ cCOUP þ kA;0 HALGC ð f Þ dð f Þ ð15:145Þ gLOG

1 Moreover, the effect of the factor g1 LOG G2 ð f ÞG3 ð f Þ in the second term on the r.h.s. of Equation (15.143) operates a linear filtering action on the control voltage. Again, it is possible to eliminate this factor from the ALC/AGC equations by introducing the quantity 0 ð fÞ ¼ VCNTRL

1 G3 ð f Þ VCNTRL ð f Þ gLOG G2 ð f Þ

ð15:146Þ

Substituting quantities (15.145) and (15.146) into Equation (15.143), we obtain 0 P0OUT;dBm ð f Þ ¼ ½1HALGC ð f ÞPIN;dBm ð f Þ þ HALGC ð f ÞVCNTRL ð fÞ

ð15:147Þ

Equation (15.147) is the ALC/AGC counterpart of the PLL equation (15.123). It states that the loop transmits the high-frequency components of PIN;dBm and the low-frequency components of the control voltage, rejecting the other ones. There are two main applications of the loop control of Figure 15.41: .

ALC, where pIN;dBm is almost constant over time,139 and the control voltage is used to superimpose AM on the output signal. Equation (15.147) implies that the instant envelope of the ALC output depends only on the control voltage, not on the input power, providing that the cut-off frequency of 0 HALGC is sufficiently higher than the maximum frequency of VCNTRL . A simplified case of ALC is when it is used to produce a constant and controlled output power. Then, the cut-off frequency of HALGC is not relevant for the ALC response, rather it can be chosen by different considerations, like the accuracy of the control components, noise, cost, etc.

.

0 AGC, typically used in receivers, where vCNTRL is constant over time, while pIN;dBm is variable. Usually PIN;dBm presents both low- and high-frequency components, corresponding to slow and fast variations of pIN;dBm . In general, slow variations are related to variations in the TX to RX distance,140 ripple in the receiver gain141 or in the transmitter power, and gain variations due to the temperature or from piece to piece. Conversely, the modulation generates fast variations on the input envelope. ALC has to eliminate the slow variations without cancelling the modulation. This implies that the highpass cut-off frequency of 1HALGC must be smaller than the lowest frequency component of the AM and higher than the maximum frequency of the slow variations, not related to the modulation.

The simplest AGC/ALC is the one with a simple integrator as a loop filter G1 ð f ÞG2 ð f Þ ¼

1 j2pf t

ð15:148Þ

with the corresponding closed-loop functions

HALGC ð f Þ ¼

1 1þj

f fALGC

;

f fALGC 1HALGC ð joÞ ¼ f 1þj fALGC j

ð15:149Þ

139 The possible variations are due to drift of the generator over the time, ripple in the amplifier chain if the ALC is part of a variable-frequency generator, or variations in the ALC input power from piece to piece. 140 As in mobile phones. 141 The gain ripple in the RX chain causes fluctuations in the amplitude of the demodulator input signal associated with changes in the frequency of the received signal.


749

where fALGC ¼ gLOG aV kA =ð2ptÞ is the 3 dB cut-off frequency of the ALC/AGC response for both the lowdpass and the highpass response. Functions (15.149) have the form of the well-known first-order lowdpass and highpass response with the same cut-off frequency. Note that for any given parameter of the detector and attenuator laws, it is possible to determine the cut-off frequency by setting a proper value in the gain of the AMP. Finally, note also that variable attenuators can only produce finite attenuations. Therefore, the maximum AM depth obtainable with ALC is less than 100%.

15.6.4

SDLVA

Many high-frequency applications require functional blocks supplying a DC voltage proportional to the logarithm of input RF power: demodulators, ALC/AGC and test instruments. Section 13.2 showed that normal detectors produce a voltage proportional to the power or to the peak amplitude – both in linear units – of the input power. Conventional detectors can generate DC voltage linear in dB with the input power if their output is processed by a suitable nonlinear transfer characteristic. Figure 15.42a shows one possible arrangement, where the detector is labelled as DET and the block having the nonlinear transfer characteristic is labelled as LOGAMP, which stands for logarithmic amplifier. If the detector follows the law (13.15) and LOGAMP input and output voltages are related as vOUT ¼ aLOG log10 ðvOUT Þ þ aLOG;0

ð15:150Þ

then the DC output voltage depends on the RF input power as vDC ¼ aLOG log10 g pRF;IN þ aLOG;0 ¼ gLOG pRF;IN;dBm þ gLOG;0

ð15:151Þ

where gLOG ¼ aLOG =10 and gLOG ¼ aLOG log10 ðgÞ þ aLOG;0 3aLOG . The base 10 used for the logarithm in Equation (15.150) is totally arbitrary; any different choice only changes the value aLOG of the logarithmic gain. Note also that the relation (15.151) coincides with the one assumed for the detector used within the ALC/AGC of Figure 15.41. The main drawback of the solution in Figure 15.42a is that the logarithmic amplifier has to operate from DC to a maximum frequency which depends on the application.142 Despite compensation techniques, regardless of their sophistication, the DC operation of components inherently involves offset and drift over time and with temperature. This means that the constant gLOG;0 is subject to change from piece to piece, over time and with temperature. Consequently, offset and drift generate errors in the logarithmic characteristic, which could correspond to relevant errors in the detected input power, due to the dB conversion.143 The solution presented in Figure 15.42b circumvents this obstacle, in that all of its amplifiers work at RF, and are thus DC coupled. The only DC operating circuit is a linear adder, which combines the voltages of the various detectors. The circuit in Figure 15.42b is known as the SDLVA. It consists of N identical cascaded cells, each containing one RF amplifier (AMPk ), one coupler – or more generally, any power-splitting device – (COUPk ) and one detector (DETk ), with k ranging from 1 to N. The output voltages of all the detectors 142 The maximum frequency is related to the high-frequency content of the input RF envelope. The typical bandwidth of a logarithmic amplifier is in the range of some megahertz. This value is similar to the bandwidth of an analogue TV video channel; this justifies the denomination of ‘video’ used for the circuit. 143 Forinstance, an errorof 1 dB corresponds to an error in the absolute power in excess of 20%: 20:56% ffi 100 1100:1 e% 100 1100:1 ffi 25:9%.

750


γ pRF,in

RF IN

DET

AMP1

COUP1

RF IN

AMP2

DET1

vin

DET2

VDC

LOGAMP

AMPN

COUP2

vout

(a)

COUPN

DETN

v1

v2

vN

R

R

R

LP VDC

(b)

vN+1

log(pRF)

vDC VDC

1

γ G0

2

γ G0

3

γ G0

vSAT

v4 4

γ G0

v3

v2

v1

(c)

O

(4)

pSAT

(3)

pSAT

(2)

pSAT

(1)

pSAT

pRF

Figure 15.42 Logarithmic detectors (a) conventional quadratic detector combined with a logarithmic video amplifier; (b) SDLVA block diagram; (c) approximate output characteristics of (b).


751

are summed together by a network schematized in Figure 15.42b as a resistive star network. Finally, the lowdpass filter LP removes the RF residual from the output DC voltage. The SDLVA principle of operation can be explained by considering ideal models for its components, assuming that: (a) All the RF components are impedance matched, i.e. all their reflection coefficients are zero. (b) The amplifier input/output power relation is linear until the output power reaches a given ðOUT Þ saturation level pSAT , then the output power remains constant at that level. This simple model corresponds to the piecewise-linear grey curve of Figure 9.27, and is analytically described by the equation " # 8 ðOUT Þ > pSAT >G p > pIN < > T IN > GT < pOUT ¼ ð15:152Þ " # ðOUT Þ > > pSAT > ðOUT Þ > > pIN : pSAT GT (c) The power gain of the coupler in a direct path from the output of the amplifier k to the input of the amplifier k þ 1 is GD . The power gain from the output of AMPk to the input of the corresponding DETk is GC . If the components COUPk are passive structures, then GD and GC are smaller than 1. (d) The detectors follow the relation (13.15), thus their DC output voltage is proportional to the RF input power, as in the equation, although the case of the envelope detector can be treated in a similar way. (e) The amplifier power gain, reduced by the coupler loss in the direct path, is much greater than 1: GT GD 1. The SDLVA output voltage is the DC component of the voltage at the input of LP, which is vN þ 1 ¼

N 1X vk N k¼1

ð15:153Þ

At very low input power, all the amplifiers operate linearly, and the voltage at each detector output is N N1 v1 ¼ g GC GT pIN ; . . . ; vN1 ¼ g GC GN1 GN2 T D pIN ; vN ¼ g GC GT GD pIN

All the above relations can be synthetically rewritten as vk ¼ g0 Gk0 pIN ðk ¼ 1 . . . N Þ

ð15:154Þ

with g0 ¼ gGC =GD and G0 ¼ ðGT GD Þk . Thus, when all the amplifiers work in their linear region, the SDLVA output voltage is obtained by substituting quantities (15.154) into Equation (15.153) vN þ 1 ¼

N 1X 1 GN 1 1 g0 Gk0 pIN ¼ g0 0 G0 pIN ffi g0 GN0 pIN N k¼1 N G0 1 N

ð15:155Þ

The approximate expression in the fourth member of Equation (15.155) is possible by assumption (e). When the input power increases, the first amplifier to saturate is AMPN , and this occurs when ðN Þ ðOUT Þ pIN ¼ pSAT ¼ pSAT = GNT GN1 . At input power higher than that, the output of DETN remains constant D

752

MICROWAVE AND RF ENGINEERING ðOUT Þ

at the value vN þ 1 ¼ g0 pSAT ¼ vSAT , while relations (15.153) hold true for any k smaller than N. Consequently, the output voltage becomes ! N1 1 X 1 k g G pIN þ vSAT ffi g0 GN1 pIN þ vSAT ð15:156Þ vN þ 1 ¼ 0 N k¼1 0 0 N ðN1Þ

If the input power further increases up to pIN ¼ pSAT also saturates, and the output voltage becomes vN þ 1 ¼

1 N

N 2 X

ðOUT Þ ðN Þ ¼ pSAT = GN1 GN2 ¼ pSAT G0 , then AMPN-1 T D !

g0 Gk0 pIN þ 2vSAT

k¼1

ffi

1 g GN2 pIN þ 2vSAT N 0 0

ð15:157Þ

Comparing Equations (13.155) with (15.157), it can be seen that the slope of the SDLVA output voltage is inversely proportional to the input power, as required by a logarithmic relation.144 Figure 15.42c plots the detector output voltage, the sum and the ideal logarithmic law (grey line) for an ideal SDLVA fulfilling hypotheses (a) to (e) and in the relatively simple case of N ¼ 4. Clearly, if the input power is so high that all the amplifiers saturate, then the output voltage also saturates instead of following the logarithmic characteristic. Also, if the input power is close to zero, the output voltage becomes proportional to the input power in linear units, not in dBm. In particular, if pIN ¼ 0 then vN þ 1 ¼ 0, a different ideally logarithmic characteristic, which should require that vDET ! 1. Hence, the SDLVA approximates the logarithmic characteristic within a certain range of input power that depends on the number of cells and their through gain. A first determination of the input dynamic range identifies it as the interval between the saturation of AMPN and all the amplifiers, i.e. " # ð1Þ pSAT SDLVA INPUT RANGE ¼ 10log ðN Þ ¼ 10 logðG0 Þ þ 10 logðN1Þ pSAT A more accurate consideration gives that the output voltage also approximates the logarithmic law when ðN Þ all the amplifiers are linear and the input power approaches pSAT , as Figure 15.42c also shows. From this, the input dynamic range of an ideal SDLVA is SDLVA INPUT RANGE ¼ 10logðG0 Þ þ 10 logðN Þ

ð15:158Þ

From Equation (15.158) it follows that the input range increases with the number of cells, as expected, and with the gain from the input of one amplifier to the next. However, such a gain also increases the distance between two break points of the piecewise-linear approximation of the logarithmic curve, in that pSAT;k1 ¼ pSAT;k G0 . The error between a curve and its piecewise approximation increases with the distance between the break points. Unfortunately, real amplifiers and detectors present different characteristics different from the ones involved with assumptions (a) to (e). Thus, SDLVA analysis and optimization require computer methods. As an example,145 Figure 15.43 plots the curves of a four-cell SDLVA with amplifiers presenting the cubic-saturated characteristic (9.48) and with detectors having the characteristic plotted in Figure 13.6. The resulting circuit has an input range of about 50 dB, from 50 to 0 dBm.

144 145

Keep in mind that d½loga ðxÞ=dx ¼ loga ðeÞ=x. See the Mathcad file 11_SDLVA.mcd.

RF AND MICROWAVE ARCHITECTURES 2

4

3

1

vDC m Pin,dBm+ q

0

2 Error = [m Pin,dBm + q] –[vDC] → 1

–1

v1 v2

0 –60

–50

–40

–30

Error

Output voltages, V

753

v3 –20

v4 –10

0

–2 10

Input power, dBm

Figure 15.43

Computed curves of SDLVA with four cells.

Bibliography 1. A. B. Kuznetsov, Radio Transmitters, MIR, Moscow, 1981, section 8.1, pp. 248–254. 2. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 2.7.1, pp. 125–129. 3. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, sections 7.2 and 7.3, pp. 420–434 and 439–449. 4. G. Bianchi, Phase-Locked Loop Synthesizer Simulation, McGraw-Hill, New York, 2005, section 2.6, pp. 81–88. 5. J. T. Taylor and Q. Huang, Electrical Filters, Chapter 8.5, ‘Surface acoustic wave filters’ (written by J. H. Hines), pp. 401–418, CRC Press, New York, 1997. 6. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 8.2, pp. 462–482. 7. C. E. Shannon, ‘Communication in the presence of noise’, Proceedings of the IEEE, Vol. 86, No. 2, pp. 447–457, 1998; reprinted from Proceedings of the IRE, Vol. 37, No. 1, pp. 10–21, 1949. 8. T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd edition, Cambridge University Press, New York, 2004, pp. 713–714. 9. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 2.6, pp. 122–125 and section 7.1, pp. 411–420. 10. S. J. Orfanidis, Electromagnetic Waves & Antennas, Chapters 19 and 30, pp. 668–851, published by Rutgers University, Piscataway, NJ, February 2008, available at: http://www.ece.rutgers.edu/ orfanidi/ewa/. 11. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 2.7.1, pp. 125–129. 12. G. Bianchi and R. Sorrentino, Electronic Filter Simulation and Design, McGraw-Hill, New York, 2007, section 7.5, pp. 452–455. 13. T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd edition, Cambridge University Press, New York, 2004, section 12.7, pp. 397–399.

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14. J. Rogers and C. Plett, Radio Frequency Integrated Circuit Design, Artech House, Norwood, MA, 2003, section 7.10.2, pp. 220–224. 15. T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd edition, Cambridge University Press, New York, 2004, section 19.2.6, pp. 700–710. 16. F. M. Gardner, Phaselock Techniques, John Wiley & Sons, Inc., New York, 1966. 17. V. Manassewitsch, Frequency Synthesizers Theory and Design, John Wiley & Sons, Inc., New York, 1976. 18. A. Blanchard, Phase Locked Loops, John Wiley & Sons, Inc., New York, 1976. 19. J. A. Crawford, Frequency Synthesizer Design Handbook, Artech House, New York, 1994. 20. U. L. Rohde, Microwave and Wireless Synthesizers Theory and Design, John Wiley & Sons, Inc., New York, 1997. 21. W. F. Egan, Frequency Synthesis by Phase Lock, John Wiley & Sons, Inc., New York, 2000. 22. G. Bianchi, Phase-Locked Loop Synthesizer Simulation, McGraw-Hill, New York, 2005. 23. R. E. Primek,‘Basic techniques guide the design of AGC systems’, Microwaves & RF, September, pp. 99–106, 1991. 24. R. E. Primek,‘OPAMP filters control responses in AGC systems’, Microwaves & RF, October, pp. 111–121, 1991. 25. ‘Control loop design for GSM mobile phone applications’, White Paper, AVAGO Technologies, May 2006, available at: www.avagotech.com.

Related files Ansoft files 01_Cascaded_Filters.adsn. Analyzes the bandpass filters and the cascaded bandpass filters having the response plotted in Figure 15.10. 02_CombLine_Filter_1GHz.adsn. Analyzes the comb-line filter used in the transmitter of Section 15.4.2. 03_Receiver_Ripple.adsn. Provides a linear analysis of the receiver described in Section 15.5.3. 04_Polyphase.adsn. Analyzes the networks in Figure 15.33 and produces the curves plotted in Figure 15.34.

Mathcad files 05_AM_Reduction_By_Compression.MCD. Analysis of the parasitic AM of the transmitter in Figure 15.5a, taking into account the amplifier compression. 06_Cartesian_Modulator.MCD. Implements some calculations on the Cartesian transmitter with various types of modulating signals and deviations from ideality. 07_CombLine_Filter_Synthesis.MCD. Evaluates the synthesis formulae for comb-line filters. In particular, the file stores the inputs values for the RF filter used in the transmitter of Section 15.4.2. 08_Pulse_Modulation_RF_Spectrum.MCD. Computes the spectra plotted in Figure 15.27. 09_Receiver_Analysis.MCD. Implements the analyses on the receiver discussed in Section 15.5.3. 10_PLL.MCD. Analyzes a type II third-order PLL and produces the graphs shown in Figures 15.37 to 15.39. It also implements some stability analyses. 11_SDLVA.mcd. Analyzes a four-cell SDLVA, producing the curves plotted in Figure 15.43.


755

SIMetrix files 12_CombLine_Filter_AC.sxsch. Analyzes the comb-line filter used in the transmitter of Section 15.4.2. 13_Combline_Filter_Burst_with_Detectors.sxsch. Analyzes the block diagram of Figure 15.27 and produces the curves plotted in Figure 15.29. 14_VideoFilter_AC_Transient.sxsch. Analyzes the step, pulse and frequency response of the detector lowdpass filters of the transmitter in Section 15.4.2. 15_IQ_analog_digital_waveforms.sxsch. Analyzes the network in Figure 15.35a and produces the waveforms plotted in Figure 15.35b.

16

Numerical methods and CAD 16.1 Introduction In the last 30 years or so, the extraordinary increase in computational power and memory storage capabilities of computers made at very reasonable costs and, at the same time, the development of very sophisticated numerical methods have revolutionized the design techniques of microwave and RF circuits and apparatus. The possibility to simulate electromagnetic structures and complex circuits to high accuracy and reliability in modest computing times allows one to perform on a PC virtual experiments that eliminate the cost and time associated with the fabrication and measurements of the prototypes. Modern computer simulators, in constant development, enable accurate analyses of evermore complex systems, from the simple active or passive components to the complete RF link, including the receive and transmit circuits and the effects of the interposed propagation medium. The most sophisticated simulators allow the concurrent analysis of the RF analogue with the low-frequency and digital parts. In engineering applications the term CAD (Computer-Aided Design) indicates any technique that allows the analysis and design to be performed with the aid of a computer. Analogous or equivalent terms are CAE (Computer-Aided Engineering) and EDA (Electronic Design Automation). The CAD of a microwave circuit is a complex procedure that, as schematized in Figure 16.1, can be broken down into the following steps: (a) The starting point is represented by the specifications (or specs), which determine the values of the circuit parameters, such as the VSWR, the frequency range, the gain, the absorbed power, etc. (b) Based on available design data from the literature, handbooks or, simply, on the designer’s experience, a first design is developed. This may consist either of an electromagnetic (3-D or 2-D) structure or an electric circuit whose parameters are given tentative values, subject to subsequent modifications. The first design may also be the result of a synthesis procedure (when available) usually carried out with the aid of a computer. (c) Next, a computer analysis is carried out to evaluate the circuit performance. As we have seen in previous chapters, the term circuit may be used to designate very different physical objects or, to be more precise, different models of physical objects, giving rise to substantially different mathematical problems. When we consider the physical structure, e.g. of a directional coupler in Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-75862-5

758

MICROWAVE AND RF ENGINEERING a. Circuit specification Design data / synthesis b. First design Models

c. Performance analysis Optimization

New design

NO COMPLIANCE?

YES Fabrication

Figure 16.1

Graph illustrating the CAD of microwave circuits.

a rectangular waveguide or a microstrip bandpass filter, the analysis has to be based on the electromagnetic field theory, i.e. on the solution of Maxwell’s equations or equations based on them, such as Helmholtz’s equation. When, on the contrary, the term circuit indicates a device model consisting of transmission lines, capacitors, inductors, etc., its characterization is then based on circuit theory. In the former instance we will speak of electromagnetic or full wave simulation, in the latter we speak of circuit simulation. It is clearly understood that from the computational point of view, the former approach is much more expensive than the latter but allows, at least in principle, for much higher accuracy. Since the circuit simulation is generally based on simplified models of the physical structures, it involves a superior computational speed but cannot account for the fabrication details of the structure, such as metallization thickness, or more subtle phenomena such as EM coupling, package interaction, etc. Note also that, as discussed later on in this chapter, different numerical and analytical approaches are available for the EM simulation of microwave circuits, each having its specific advantages and disadvantages. As an example, consider the comparison between the circuit and full wave simulations of a 10 GHz directional coupler in microstrip technology. Figure 16.2 shows the circuit layout and its electrical schematic. The latter includes the models of a coupled-line section, microstrip bends (with mitred corners for better matching) and microstrip line lengths. Possible and unavoidable coupling between the bends is thus not taken into account. Figure 16.3 shows the circuit (thick grey lines) and the full wave EM simulation (black solid lines) of the microstrip coupler. As a reference, Figure 16.3 also shows the S parameter magnitudes of a directional coupler with ideal purely TEM coupled lines (dashed line).1 1 See the Ansoft file 01_Microstrip_Directional_Coupler.adsn for the circuit analysis of a microstrip and ideal directional coupler.

NUMERICAL METHODS AND CAD 3 3

759

4

4 w = 0.5 mm l = 2 mm

w = 0.5 mm l = 2 mm

w = 0.4 mm s = 0.15 mm I = 3 mm 5 mm

1

2

w = 0.5 mm l = 2 mm

w = 0.5 mm l = 2 mm

Microstrip transmission lines 1

Substrate edge 1

2

RF ports

(a)

(b)

Figure 16.2 Layout of a microstrip directional coupler (er ¼ 9.9, h ¼ 0.508 mm) (a) and its schematic for circuit simulation (b). As can be seen, there is a substantial agreement between the two simulations, but with some notable differences. As already stated, the full wave simulation is usually more reliable, but this is not necessarily true in all cases, since some degree of approximation is always present. Depending on a number of issues, such as the accuracy required, the computational effort and the numerical or analytical algorithm employed, various approaches and models can be chosen for the analysis of microwave circuit performance. Electromagnetic Circuit Ideal

0

–6

10

–8

0

–10

–4 s41

–12

–6 s11 s11

–8

20 log10 (|s31|)

20 log10 (|s21|)

s31 –10

–20

–14

–30

–16 15.0

–40

20 log10(|s11|), 20 log10(|s41|)

s21 –2

s41 –10 5.0

7.5

10.0

12.5

Frequency, GHz

Figure 16.3 Comparison among ideal, circuit and full wave simulations of the coupler in Figure 16.2. Note that the ideal s11 and s41 are not to scale ( < 40 dB).

760

MICROWAVE AND RF ENGINEERING (d) Once the analysis has been performed, one has to check whether the circuit complies with the given specifications. If this is the case, the design is completed and one can proceed with fabrication of the circuit. If not, one needs to change the circuit parameters with the aim of satisfying the specifications. Except for particularly simple or lucky cases, this phase is rather critical, as one has to identify a suitable strategy to modify the circuit parameters to meet the specifications. This strategy normally involves an iterative procedure (i.e. a procedure repeated sequentially a number of times) called optimization, which may require from a few tenths up to several thousands of iterations. It is also to be expected that, in some unfortunate cases, this procedure might not convergeto anysolution, so that the specifications are nevermet. The procedure needs to be stopped in all cases after a predefined number of iterations. At this point, the designer has to choose among various options: change the optimization method, the models adopted, the initial design, and so on.

Within the general framework just described, in this chapter we introduce some fundamental concepts concerning: 1. The EM analysis, with a description of some of the most commonly adopted methods. 2. The circuit analysis, both linear and nonlinear. 3. The basics of optimization. For each analysis technique details are provided on the algorithms employed, the possible applications and relative limitations. We do not consider here the numerical methods for the solution of the numerical problems resulting from the formulations in concept 1 or 2, e.g. the solution of large sets of linear equations or the computation of eigenvalues. Though important, these methods are beyond the scope of this book. One CAD example is the Ansoft file 01_Microstrip_Directional_Coupler.adsn specifically developed for the present chapter. The interested reader can find many more CAD/CAE files in Chapters 6–15.

16.2 EM analysis Maxwell’s equations, as we have seen, represent the monumental synthesis of all EM phenomena. The EM theory consists of the study and solution methods of Maxwell’s equations and constitutes the theoretical foundation of microwave and RF engineering. The latter has the scope of obtaining numerical data useful for the characterization and design of microwave and RF components and systems. There are a huge number of theoretical studies on analytical methods that, starting from Maxwell’s equations, allow one to extract numerical data useful for engineering applications. Such methods have for decades been the only available way to obtain such data using the limited computational power when the present electronic computing capabilities were not available.2 Such capabilities have ultimately made obsolete most of the conventional approaches, which are highly sophisticated from an analytical point of view. The way has thus been paved for the development of less elaborate techniques which lend themselves to electronic computation. Several methods have been developed for the numerical solution of Maxwell’s equations, finalized for the analysis and design of microwave circuits. A numerical method is a mathematical technique to convert an analytical ‘continuum’ (Maxwell’s equations) into a ‘discrete’ one that can be treated numerically (i.e. by a digital computer). In this manner, the infinite degrees of freedom of the analytical 2 It is nonetheless amazing to note the prodigious quantity of information and data produced, particularly at MIT Radiation Laboratory, during the Second World War, then published in the series [1]. These books are still an invaluable reference point today. See, in particular, [2].

NUMERICAL METHODS AND CAD

761

solution are reduced to a finite number of unknowns, which can be found by a digital computer. It must be stressed that the complexity of modern microwave circuits can only be tackled with numerical methods, since analytical methods can be applied only to a limited number of simple cases. The numerical methods for the solution of microwave circuits can be classified according to various criteria. For our purposes, we will limit our attention to the categories of time domain and frequency domain techniques. The former operate on Maxwell’s equations in the form (2.1)–(2.4), the latter operate on the corresponding harmonic version (2.36)–(2.39). In this section we consider just a few examples of numerical methods with the aim of providing the reader with basic information and some application hints. We intend to give an idea of the capabilities of the various methods, without any pretension to exhaust a subject that would require several books to be treated in a satisfactory way.

16.2.1

The method of moments

The moment method or method of moments, often abbreviated as MoM, provides a general mathematical scheme in which almost any approach to the numerical solution of Maxwell’s equations can be included [3, 4]. The MoM is a procedure for transforming a functional equation (where the unknown is a function) into a matrix equation (where the unknown is a set of numerical values). Conversely, any method to approximate a functional equation with a system of equations can be reduced to the MoM. The MoM solution of a differential equation utilizes the concepts of operators and linear spaces. We are going to introduce these concepts here in a simplified form with understandability in mind rather than formal rigour. Let us start from an equation of the type Lð f Þ ¼ g

ð16:1Þ

where L is a generic linear operator (e.g. the Laplace operator), g is a given function (the source or the excitation) and f is the unknown function (the field or the response). We now replace the unknown function with a series expansion X f ¼ a f ð16:2Þ n n n where fn are called basis functions and an are constants, i.e. the series coefficients. A familiar example is the Fourier series expansion of a function f ðxÞ in a limited interval ½a; b. In order for the series (16.2) to be an exact representation of f ðxÞ, the series normally contains an infinite number of terms and the basis functions must constitute what is called a complete set: this means that any well-behaved function can be expressed by such an expansion.3 In practice, the unknown function is approximated with a finite number of terms, however large that may be. Equation (16.2) allows the original problem of finding the unknown f ðxÞ to be replaced by the much easier problem of determining a finite number of constants an . Inserting (16.2) into (16.1) and taking into account the linearity of the operator, we have X a Lð fn Þ ¼ g ð16:3Þ n n In order to compute the unknown coefficients an from (16.3) we need to introduce the concept of inner product of two functions f and g, which is defined on the basis of the following properties: h f ; gi ¼ h f ; gi ha f þ bg; hi ¼ ah f ; hi þ bhg; hi > 0 for f 6¼ 0 h f *; f i ¼ 0 for f ¼ 0

ð16:4Þ

3 By well behaved, we consider a function which is piecewise continuous and square integrable in the domain of definition. The concept of complete set has been already introduced in Sections 3.6 and 5.9.

762


where a and b are scalar constants and denotes the complex conjugate. As can be verified, an example of an inner product is the definite integral ðb h f ; gi ¼ f ðxÞgðxÞ dx a

At this point, we introduce another set of functions wm , called weighting or testing functions, that are used to calculate the inner product with (16.3): P m ¼ 1; 2; 3 . . . ð16:5Þ n an hwm ; Lð fn Þi ¼ hwm ; gi Equation (16.5) constitutes a set of linear equations in an . It can be written in the form ½L½a ¼ ½g with

2 6 ½L ¼ 6 4

ð16:6Þ

hw1 ; Lð f1 Þi . . . hw1 ; Lð fN Þi .. .

.. .

.. .

3 7 7 5

hwN ; Lð f1 Þi . . . hwN ; Lð fN Þi 3 3 2 hw1 ; gi a1 6 6 . 7 .. 7 7 7 ½a ¼ 6 ½g ¼ 6 . 5 4 4 .. 5; 2

ð16:7Þ

hwN ; gi

aN

In the above formulae, expansions (16.3) and (16.5) have been truncated to N terms in such a way that the coefficient matrix ½L is a square matrix. If ½L is non-singular, the expansion coefficients an can formally be obtained as follows: ½a ¼ ½L1 ½g In summary, the approximate solution of (16.1) can be expressed as f ¼ ~f ½a ¼ ~f ½L1 ½g where ~f is the basis function matrix ~f ¼ ½ f1 f1 . . . fN

ð16:8Þ

ð16:9Þ

ð16:10Þ

Note that (16.9) is not the exact solution of (16.1) but only its approximation, whose accuracy depends on the choice of the basis and weighting functions as well as, of course, on the number of terms used in the expansion. The particular case when the weighting functions are the same as the basis functions, thus wn ¼ fn , constitutes Galerkin’s method. This is a very common choice; it has been proved that Galerkin’s method is equivalent to the Rayleigh–Ritz variational method.4 The basis functions can be chosen in a variety of ways. A very common choice is represented by subsectional functions. Such functions are non-zero only in one (or a few) portion(s) of the domain of the unknown function. As an example, Figure 16.4 shows in one dimension the use of triangular subsectional functions represented by the expression 8 1 > > < 1jxxn jðN þ 1Þ for jxxn j < N þ 1 ð16:11Þ Tn ðxÞ ¼ > 1 > :0 for jxxn j > N þ1 4

For variational and perturbational techniques, see for instance the classical book by Harrington [4].


763

f(x)

f(xi+1) f(xi)

x0 = a

Figure 16.4

xi

xi+1

xn = b

x

Linear approximation with triangular subsectional functions.

As can be seen, the superposition of triangular functions yields the piecewise-linear approximation of a generic function. An alternative to the subsectional functions is represented by entire domain functions, defined over the entire domain of the unknown function f. The modal functions encountered in Chapter 3 or the trigonometric functions of the Fourier series are examples of this category. Many commercial software packages today are finalized to the CAD of printed microwave circuits, mostly based on microstrip technology. The key feature of such circuits is that they are planar and the metal thickness can be assumed to be zero with good approximation up to very high frequencies. A great simplification is obtained if the surface current density rather than the EM field is assumed as the unknown, since in this manner the 3-D problem is reduced to a 2-D problem with substantial computational savings. Such an approach is called 2.5-D as it is intermediate between a simplified 2-D model and a rigorous 3-D method capable of accounting for finite metal thickness and non-planar conductors. Such methods are based on the MoM solution of an integral equation (represented by Equation (16.1)) in the surface current density flowing on the metal strips. The current distributions are expanded into triangular subsectional functions called rooftop basis functions. In summary, the MoM reduces the operator problem (16.1) to the linear system of equations (16.9). The system may consist of thousands or even tens of thousands of equations with a usually dense coefficient matrix. From the computational point of view, the challenge is to solve such a system in a numerically efficient way. The interested reader will find there is a vast literature on the subject (see, for instance, [3, 5, 6]).

16.2.2

The finite difference method

The finite difference method is the oldest and most direct method to solve a differential equation by converting it into a set of algebraic equations. Consider for simplicity a 1-D function y ¼ f ðxÞ defined in a finite interval ½a; b of the x axis. The first step, as shown in Figure 16.4, consists of replacing the continuous function f ðxÞ with a discrete set of N þ 1 values fi by dividing the interval into N subintervals of amplitude Dx with fi ¼ f ðxi Þ xi ¼ x0 þ iDx x0 ¼ a ba Dx ¼ N

i ¼ 0; 1; . . . ; N

ð16:12Þ

Next, we need to express the first derivative of f ðxÞ at xi . Consider the Taylor’s series expansion of f ðxÞ at x ¼ xi þ Dx:

764

MICROWAVE AND RF ENGINEERING f ðxi þ DxÞ ¼ f ðxi Þ þ f 0 ðxi ÞDx þ

1 00 f ðxi ÞDx2 þ OðDx3 Þ 2!

ð16:13Þ

where OðDx3 Þ indicates a quantity approaching zero as the third power of Dx. The above equation leads to the following expression for the first derivative of y ¼ f ðxÞ in xi : f 0 ðxi Þ ¼

f ðxi þ DxÞf ðxi Þ 1 00 f ðxi ÞDx þ 0ðDx2 Þ Dx 2!

ð16:14Þ

By neglecting the first- and higher order powers of Dx, (16.14) yields the forward difference approximation of the derivative in xi f ðxi þ DxÞf ðxi Þ Dx

f 0 f ðxi Þ ¼

ð16:15Þ

Considering a negative increment Dx, (16.14) becomes f 0 ðxi Þ ¼

f ðxi Þ f ðxi DxÞ 1 þ f 00 ðxi ÞDx þ OðDx2 Þ Dx 2!

ð16:16Þ

yielding the backward difference f ðxi Þ f ðxi DxÞ ð16:17Þ Dx This expression has the same first-order approximation as (16.15). Notice that the Dx terms in (16.14) and (16.16) have opposite signs. By averaging these equations, one obtains the central difference expression of the first derivative: f 0 b ðxi Þ ¼

f 0 c ðxi Þ ¼

f ðxi þ DxÞ f ðxi DxÞ 2Dx

ð16:18Þ

which is approximated to the second order, i.e. the error approaches zero as Dx2 . For this reason, whenever possible the first derivative is approximated with the central difference (16.18) rather than (16.15) or (16.17). In summary, by using the notation fi ¼ f ðxi Þ, Equations (16.15), (16.17) and (16.18) can be written as fi þ 1 fi Dx fi fi1 ¼ Dx fi þ 1 fi1 ¼ 2Dx

f 0 i; f ¼ f 0 i;b f 0 i;c

ð16:19Þ

Figure 16.5 shows the geometrical representation of the three finite difference approximations: fi; f is the straight line through the points ðxi ; fi Þ and ðxi þ 1 ; fi þ 1 Þ, fi;b is the straight line through ðxi1 ; fi1 Þ and ðxi ; fi Þ, and fi;c is the straight line through ðxi1 ; fi1 Þ and ðxi þ 1 ; fi þ 1 Þ. It is intuitive that, for a given Dx, the central difference provides the best approximation of the tangent in ðxi ; fi Þ. In the solution of EM problems it is often required to compute also the second derivatives of a function, e.g. in Laplace’s equation. In the finite difference approximation, this is obtained by taking the difference between the expansion (16.13) and the corresponding one in xi Dx. One obtains f 00 ðxi Þ ¼

f ðxi þ DxÞ2 f ðxi Þ þ f ðxi DxÞ þ OðDx2 Þ Dx2

ð16:20Þ

Therefore, with a second-order approximation, the second derivative in xi is expressed as f

00

i

¼

fi þ 1 2 fi þ fi1 Dx2

ð16:21Þ

NUMERICAL METHODS AND CAD f(x)

765

f '(xi)

fi+1

' f forward

fi

' fcentral fi–1 f 'backward xi–1

Figure 16.5

xi+1

xi

x

Approximating the first derivative by forward, backward and central differences.

The above results can easily be extended to the 2-D and 3-D cases. In two dimensions, the spatial discretization involves the creation of a rectangular grid or mesh with steps Dx and Dy in the Cartesian plane. In three dimensions we have a parallelepiped mesh with elementary cell Dx, Dy, Dz. As an example, let us apply the above formulae to compute the EM field in two coupled striplines, as in Figure 16.6. The structure is enclosed by metal walls to limit the computational space. The latter is discretized into a 2-D mesh. The generic node ði; jÞ identifies the coordinates ðxi ; yj Þ. The metal enclosure is expected to have a negligible effect on the field distribution as long as it is far away from the metal strips. As discussed in Section 3.16, to determine the even and odd TEM modes of the coupled lines we need to solve the 2-D Laplace’s equation in the transverse potential T: r2t Tðx; yÞ ¼ 0

ð16:22Þ

with boundary conditions T ¼ 0 on the ground planes (metal enclosure) and T ¼ 1 or T ¼ 1 on the metal strips. The symmetry plane (dashed line in Figure 16.6) can be replaced by a magnetic wall (qT=qx ¼ 0) for the even mode, or by an electric wall (T ¼ 0) for the odd mode. In this manner, the number of nodes where the field has to be computed is approximately halved (but the computation has to be repeated with both boundary conditions).

y 30

T=0 ∆y

25 20 15 10

T=0

5 0 0

∆x 10

Figure 16.6

20

30

40

50

x 60

A 2-D discretization of coupled striplines.

766


Using (16.20), with the obvious meaning of the symbols, Laplace’s equation (16.22) in finite difference form becomes Ti þ 1; j 2Ti; j þ Ti1; j Ti; j þ 1 2Ti; j þ Ti; j1 þ ¼0 Dx2 Dy2

ð16:23Þ

If we adopt a square mesh with Dx ¼ Dy, (16.23) simplifies to 1 Ti; j ¼ ðTi; j þ 1 þ Ti1; j þ Ti; j1 þ Ti þ 1; j Þ 4

ð16:24Þ

This expression indicates that in the discrete finite difference formulation, Laplace’s equation implies that the potential at the node ði; jÞ is the average of the potentials at the four adjacent nodes. Applying (16.24) to all internal nodes, we obtain a system of equations in the unknown potential values. The nodal points on the boundary, where the potential is assigned, provide the r.h.s. of the system. According to (16.18), the boundary condition on magnetic walls qT=qx ¼ 0 is imposed by equating the values of the function at points symmetrically located across the boundary. This clearly requires the mesh to be extended one step Dx outside the computational domain. The computation of the potential on the mesh nodes therefore requires the solution of a linear set of equations, whose size is equal to the number of mesh nodes minus those where the potential is assigned (by the boundary conditions). In practical cases this number may easily reach several thousands or even millions. Figure 16.7 shows the computed electric field lines of the even and odd modes of the coupled lines of Figure 16.6. The equipotential lines, orthogonal to the electric field lines, correspond to the magnetic field lines. The computation has been performed using a 60 30 node mesh to discretize one-half of the crosssection and put an electric or a magnetic wall at the symmetry plane to compute the even and odd modes, respectively. The finite difference method can be applied to the solution of Maxwell’s equations in the frequency domain and time domain, and is consequently named FDFD (Finite Difference Frequency Domain) and FDTD (Finite Difference Time Domain) respectively. The former was the first to be applied in practice to the solution of EM problems, in particular for the computation of the modes of waveguides with arbitrary shapes [7]. The formulation of FDTD is equally old [8], but has acquired great popularity only in more recent years, once the required computational power became available.

16.2.3

The FDTD method

The FDTD method was formulated by Yee [8] by discretizing the time domain Maxwell’s equations in rectangular coordinates. In rectangular coordinates, the six components of the homogeneous Maxwell’s equations are5 qHx 1 qEy qEz ¼ m qz qt qy qHy 1 qEz qEx ð16:25Þ ¼ qt qz m qx qHz 1 qEx qEy ¼ qt qx m qy

5 For simplicity, the possible presence of conduction currents J ¼ sE has not been considered in (16.25). Moreover, the time-varying field quantities are written here using italic rather then script fonts as in Chapter 2.


767

Figure 16.7 Electric and magnetic field lines of the even (a) and odd (b) mode of the coupled striplines of Figure 16.6. qEx 1 qHz qHy ¼ qt qz e qy qEy 1 qHx qHz ¼ e qz qt qy qEz 1 qHy qHx ¼ e qx qt qy

ð16:26Þ

Yee used a 3-D rectangular mesh with indexes (i, j, k) to identify the node ðiDx; jDy; kDzÞ and the index n to identify the time instant t ¼ t0 þ nDt. A generic quantity varying with time and in space is then written in the following form: F n ði; j; kÞ ¼ FðiDx; jDy; kDz; t0 þ nDtÞ

ð16:27Þ

768


For a reason that will become apparent shortly, Yee actually used half-integer indexes for nodes located at distances Dx=2 or Dt=2. As a consequence, using the central difference formula the spatial and time derivatives of (16.27) are written as follows: qF n ði; j; kÞ F n ði þ 1=2; j; kÞF n ði1=2; j; kÞ ¼ qx Dx qF n ði; j; kÞ F n ði; j þ 1=2; kÞF n ði; j1=2; kÞ ¼ qy Dy

ð16:28Þ

qF n ði; j; kÞ F n ði; j; k þ 1=2ÞF n ði; j; k1=2Þ ¼ qz Dz qF n ði; j; kÞ F n þ 1=2 ði; j; kÞF n1=2 ði; j; kÞ ¼ qt Dt

These formulae are applied to the EM field components, spatially arranged according to Yee’s cell shown in Figure 16.8. As can be seen, the E-field components are defined along the cell edges, while the magnetic field components are defined at the centres of the cell sides. The resulting rectangular grids, one for the electric field and the other for the H-field, are staggered by one-half of the cell size. Applying (16.28) to the EM field components arranged according to a cubic Yee’s cell with Dx ¼ Dy ¼ Dz ¼ Dl, Maxwell’s equations (16.25) and (16.26) become

x Ey(i+1,j, k +1/2,k+1) Ez(i+1,j,k+1/2)

Ez(i+1,j+1,k+1/2)

Hx(i+1,j,+1/2,k+1/2) Ey(i+1,j+1/2,k)

Hz(i+1/2,j+1/2,k+1) Ex(i+1/2,j+1,k+1)

Ex(i+1/2,j,k+1)

Hy(i+1/2,j+1,k+1/2)

Hy(i+1/2,j,k+1/2) Ey(i+1/2,j,k)

Hz(i+1/2,j+1/2,k)

Ex(i+1/2,j+1,k)

z

Ey(i,j+1/2,k+1) Ez(i,j,k+1/2) (i,j,k)

Figure 16.8

Hx(i,j+1/2,k+1/2) Ey(i,j+1/2,k)

Ez(i,j+1,k+1/2) y

Yee’s cell: spatial distribution of the EM field components according to Yee’s scheme.

NUMERICAL METHODS AND CAD Hxn þ 1=2 ði; j þ 1=2 ; k þ 1=2 Þ ¼ Hxn1=2 ði; j þ 1=2 ; k þ 1=2 Þ þ

769

Dt h n E ði; j þ 1=2 ; k þ 1ÞEyn ði; j þ 1=2 ; kÞ mDl y

þ Ezn ði; j; k þ 1=2 ÞEzn ði; j þ 1; k þ 1=2 Þ Dt h n E ði þ 1; j; k þ 1=2 ÞEzn ði; j; k þ 1=2 Þ Hyn þ 1=2 ði þ 1=2 ; j; k þ 1=2 Þ ¼ Hyn1=2 ði þ 1=2 ; j; k þ 1=2 Þ þ mDl z þ Exn ði þ 1=2 ; j; kÞExn ði þ 1=2 ; j; k þ 1Þ Dt h n E ði þ 1=2 ; j þ 1; kÞExn ði þ 1=2 ; j; kÞ Hzn þ 1=2 ði þ 1=2 ; j þ 1=2 ; kÞ ¼ Hzn1=2 ði þ 1=2 ; j þ 1=2 ; kÞ þ mDl x þ Eyn ði; j þ 1=2 ; kÞEyn ði þ 1; j þ 1=2 ; kÞ

Exn þ 1 ði þ 1=2 ; j; kÞ ¼ Exn ði þ 1=2 ; j; kÞ þ

ð16:29Þ

Dt h n þ 1=2 ði þ 1=2 ; j þ 1=2 ; kÞHzn þ 1=2 ði þ 1=2 ; j1=2 ; kÞ H eDl z

þ Hyn þ 1=2 ði þ 1=2 ; j; k1=2 ÞHyn þ 1=2 ði þ 1=2 ; j; k þ 1=2 Þ Eyn þ 1 ði; j þ 1=2 ; kÞ ¼ Eyn ði; j þ 1=2 ; kÞ þ

Dt h n þ 1=2 ði; j þ 1=2 ; k þ 1=2 ÞHxn þ 1=2 ði; j þ 1=2 ; k1=2 Þ H eDl x

þ Hzn þ 1=2 ði1=2 ; j þ 1=2 ; kÞHzn þ 1=2 ði þ 1=2 ; j þ 1=2 ; kÞ Ezn þ 1 ði; j; k þ 1=2 Þ ¼ Ezn ði; j; k þ 1=2 Þ þ

Dt h n þ 1=2 H ði þ 1=2 ; j; k þ 1=2 ÞHyn þ 1=2 ði1=2 ; j; k þ 1=2 Þ eDl y

þ Hxn þ 1=2 ði; j1=2 ; k þ 1=2 ÞHxn þ 1=2 ði; j þ 1=2 ; k þ 1=2 Þ

ð16:30Þ

For a non-cubic cell, the formulae are more complicated, but the basic features of the method are unchanged. In Yee’s algorithm represented by (16.29) and (16.30) each E component (or H component) is surrounded by four H components (or E components). The reader can easily verify that the same expressions are obtained starting from the integral form (2.6)–(2.7) of Maxwell’s equations. The E and H components are staggered in both space and time. The E field is computed at integer time instants (n) while the H field is computed at half-integer time instants (n þ 1=2). For this reason this has also been called the leap-frog algorithm. In contrast with the FDFD method which yields a system of equations in the unknowns located at the mesh nodes, Yee’s algorithm is a time progressing scheme that does not require the solution of a system of equations. On the contrary, Equations (16.29)–(16.30) allow us to update the EM field distribution progressively in time: 1. Starting from the E-field distribution over the whole E mesh at the n time instant, the H-field components at the n þ 1=2 time instant are computed in all H-mesh nodes using (16.29). 2. From the H-field distribution at the n þ 1=2 time instant, the E-field distribution is updated at the n þ 1 time instant. 3. The above steps are repeated sequentially so as to compute the time evolution of the EM fields starting from a given initial distribution at t ¼ t0 . The method is applied by introducing in the structure a source which excites the EM field from the starting time t ¼ t0 . The time behaviour of the source is chosen so as to be located within the given frequency band where the structure response is to be evaluated. The FDTD algorithm is then launched and is stopped

770


when the transient has died out. The response is evaluated at specific mesh points, e.g. at the ends of the feeding lines. Once the time response has been evaluated, the frequency response can be computed via a fast Fourier transform (FFT) procedure. The details of the procedure for extracting the frequency response of an RF circuit are not reported here but can be found in the literature (see e.g. [9, 10]). The accuracy of the computation depends on the spatial resolution adopted, i.e. on the mesh size Dl. The latter must be much smaller (at least 1/20) than the minimum wavelength in the frequency range considered. For example, if the maximum frequency is 60 GHz and the medium is a vacuum, the mesh size should be of the order of 0.25 mm. This implies that a space region of the order of 1 cm3 contains about 64 000 nodes. This gives an idea of the large number of unknowns even in structures of modest size. For the stability of the algorithm, the spatial resolution Dl also determines the time step Dt. In fact, the time updating algorithm is stable if the time step satisfies the Courant stability condition: 1 Dt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c ð1=DxÞ þ ð1=DyÞ2 þ ð1=DzÞ2

ð16:31Þ

pffiffiffiffiffi where c ¼ 1= me is the velocity pof ffiffiffi light in the medium. In the case of a cubic mesh the stability condition simply reduces to Dt Dl=ðc 3Þ. The FDTD method has numerous advantages that have made it one of the most popular numerical methods in use today for the solution of microwave structures, including antennas. It is indeed extremely flexible, as it can be applied to structures of any geometry, to inhomogeneous and lossy materials, and to transients and time-variable phenomena. Nonlinear elements can also be incorporated into the FTDT scheme [9, 11]. On the other hand, the method obviously has its limitations. While a single impulsive excitation produces a wide-band response in just one shot, the full response can only be computed by applying the FFT after all transients have vanished in the structure. This may require a very large number of time updates, particularly when resonating structures are involved (e.g. filters, resonators, etc.). In the case of open structures, as in other numerical methods it is necessary to limit the computational space by enclosing the structure by suitable absorbing boundary conditions (ABC) capable of simulating the open space (see [9], Ch. 7). Finally, the rectangular mesh implies some difficulties in the discretization of the structures. For example, when small spatial details have to be resolved in one portion of the structure, as often happens in microwave devices, a dense mesh must be adopted so as to yield an exceedingly large number of computational cells. The reader is referred to the literature listed at the end of the chapter for a comprehensive and detailed discussion of the method and the enhancements that have been implemented to it since its first introduction by Yee.

16.2.4

The finite element method

The finite element method (FEM) is probably the most popular numerical method in the engineering sciences. It was in fact originally developed for civil and aeronautical engineering applications [12] and later introduced in the EM computation operating in the frequency domain. Being an important subject that for space limitations cannot be treated here, only a concise and qualitative description is provided below. The interested reader is referred to the bibliography at the end of this chapter, e.g. [13, 14]. The distinctive feature of the FEM is that the frequency domain field-theoretic problem is formulated in terms of an integral equation, usually expressing the minimization of a functional, rather than using the differential form of Maxwell’s equations. While in the FD method the space is discretized into a Cartesian mesh using the discretized Maxwell’s equation, in the FEM, on the contrary, the space is subdivided into volume (or, in two dimensions, surface) ‘elements’ having in general the shape of a tetrahedron (or, in two dimensions, a triangle). Within each element the unknown function, be it a potential or a field component, is approximated by a polynomial, whose coefficients can be expressed in terms of the function at the element vertices or nodes. The minimization of the functional leads to a linear system of equations, where the unknowns are the values of the function at the nodes.


Figure 16.9

771

Example of a waveguide filter (a) and the 3-D FEM mesh employed for its analysis (b).

Compared to the FD method, the FEM has the considerable advantage that it can much more easily fit irregular geometries. This is because the mesh is not rectangular, but tetrahedral, so that it can easily match any 3-D geometry and resolve the small geometrical details only where needed. As an example, Figure 16.9 shows a 3-D FEM mesh employed in the solution of a waveguide structure with coupled resonators loaded with dielectric cubes. In its basic formulation, the method suffers from spurious, thus non-physical, solutions; this problem has been solved by the adoption of specific mathematical expedients. Similar to the FDFD, the FEM requires the solution of very large systems of equations, so the computational effort may become excessive for very large structures. It is nonetheless a very effective method, widely adopted in commercially available software packages.

16.2.5

The mode matching method

Modal analysis is one of the oldest rigorous analytical methods for the solution of resonant cavities and waveguide discontinuities. Its origins date back to the 1940s [15–17]. Starting in the 1960s, along with the spread of electronic computing, the method was developed and extensively employed under the name of mode matching method or, with reference to some of its variants, field matching method. The method, being analytical, has a reduced applicability compared with numerically oriented methods such as the MoM, the FTDT or the FEM. It can only be applied to microwave structures that can be subdivided into simple geometrical shapes, essentially parallelepipeds and cylinders. Although it can

772

MICROWAVE AND RF ENGINEERING y Sc

y b

d

z

Sa

0

Figure 16.10

X

Metal iris in a rectangular waveguide.

be employed also for much more general problems, its formulation is normally limited to waveguide components and discontinuities. Moreover, the method requires a considerable analytical preprocessing. In return, the computational effort is reduced and the computational efficiency very high. The method will be illustrated by the example of a typical problem, such as that depicted in Figure 16.10. A metal diaphragm or iris is located at the section z ¼ 0 of a rectangular waveguide. Sc is the metal surface and Sa is the resulting aperture. The diaphragm is assumed to be thin enough as to be considered of zero thickness.6 Our aim is to compute the equivalent circuit of such a discontinuity, seen as a two-port circuit. Let us designate by the indexes 1 and 2 the two half guides for z < 0 and z > 0, respectively. We suppose that the EM field incident on the diaphragm is the dominant TE10 mode of the rectangular waveguide, while all higher order modes are below cut-off. The incident field will thus produce a reflected TE10 wave for z < 0 and a transmitted TE10 wave for z > 0 in addition to a theoretically infinite number of reflected (for z < 0) and transmitted (for z > 0) higher order modes; such modes, being below cut-off, have exponentially decreasing amplitudes with distance from the iris. The modal analysis or mode matching method is based on the property that waveguide modes form a complete spectrum,7 which means that any EM field within the guide can be expressed as a superposition of waveguide modes, just like any periodic function can be expressed as a Fourier series. On the waveguide cross-sections where discontinuities are present, the unknown EM field is expressed in terms of a modal series: similar to the MoM, the series coefficients become the unknowns of the problem. We use Equation (3.47) to express the EM field of the generic waveguide mode (be it TE or TM) with an additional index m to identify the mode: the most general field in the ith waveguide is expressed by a summation over all its modes:8 ðiÞ

Et ¼

M 1 X

VmðiÞ ðzÞem ðx; yÞ

m¼0 ðiÞ Ht

¼

ð16:32Þ

M1 X

ImðiÞ ðzÞhm ðx; yÞ

m¼0

As in a Fourier series, expansions (16.32) are theoretically valid in the limit of infinite M; in practice one has to truncate the above series to a finite number M of terms, with M large enough to achieve the approximation required. In (16.32) the index m ¼ 0 identifies the dominant mode, while higher order modes correspond to m 1. We recall that the modal vectors satisfy the orthonormalization properties (3.101)–(3.103), i.e. ð ð ep eq dS ¼ hp hq dS ¼ dpq ð16:33Þ S

S

where dpq is the Kronecker delta. The following relation also holds between the modal vectors: 6

If the metal thickness t is not negligible, the iris can be viewed as a waveguide section of length t. See Section 3.6. 8 To simplify the notation with respect to that used in Chapter 3, we omit here the subscript t on the transverse modal vectors em and hm. 7

NUMERICAL METHODS AND CAD hm ¼ ^z em

773 ð16:34Þ

Note that, because of the orthonormalization properties (16.33), the equivalent voltages and currents in (16.32) can be expressed in terms of the tangential fields ð ðiÞ ðiÞ Vm ¼ Et em dS ðS ð16:35Þ ðiÞ ðiÞ Im ¼ Ht hm dS S

The above expressions are obtained by multiplying both sides of (16.32) by the respective modal vectors and integrating over the waveguide cross-section S. Expressions (16.35) allow one to compute the EM field in a waveguide starting from knowledge of the E-field (or H-field) distribution over a waveguide cross-section. Let us now consider the boundary conditions at the discontinuity plane z ¼ 0: the tangential electric field must be continuous across S; it must furthermore vanish on the metal portion Sc of S, while it will be equal to an unknown field E0 on the aperture Sa . Thus, we have ð1Þ

ð2Þ

Et ¼ Et on S ¼ Sa [ Sc ( 0 on Sc ð1Þ Et ¼ E0 on Sa

ð16:36Þ ð16:37Þ

Note that the unknown field E0 can be expressed as if it were the transverse field over a waveguide with cross-section Sa , i.e. P1 X Vap eap ðx; yÞ ð16:38Þ E0 ¼ p¼0

where Vap are unknown coefficients and eap (p ¼ 0, 1. . .) are the modal vectors of such a guide; they therefore satisfy the orthonormalization property over the aperture Sa . In (16.38) we have limited the expansion to the first P modes. We now need to impose the last boundary condition, i.e. the continuity of the tangential H field across the aperture Sa ð1Þ ð2Þ Ht ¼ Ht on Sa ð16:39Þ Now, using expansions (16.32) and (16.38), we can transform the boundary conditions (16.36)–(16.37) and (16.39) into a linear set of equations in the expansion coefficients. Inserting the first of (16.32) into (16.36), we obtain M 1 M 1 X X Vmð1Þ em ðx; yÞ ¼ Vmð2Þ em ðx; yÞ ð16:40Þ m¼0

m¼0 ð1Þ

ð2Þ

For simplicity of notation we have omitted to indicate the coordinate z ¼ 0: the coefficients Vm ; Vm are ð1Þ ð2Þ therefore intended as Vm ð0Þ; Vm ð0Þ. The orthogonality of modal vectors over the waveguide crosssection S implies the identity of the corresponding expansion coefficients at both sides of (16.40): ð1Þ

ð2Þ

Vm ¼ Vm

m ¼ 0; 1; . . . ; M1

ð16:41Þ

This means that the equivalent voltages of all modes are the same on both sides of the iris. As a consequence, from now on we will simply write Vmð1Þ ¼ Vmð2Þ ¼ Vm

ð16:42Þ

We observe that while Equation (16.37) is defined over the entire cross-section S ¼ Sa [ Sc , Equation (16.39) is defined only over the aperture Sa . For this reason we cannot deduce from (16.39) ðiÞ the identity of the coefficients Im of the two magnetic field expansions.

774


Let us now multiply both sides of (16.37) by the generic modal vector em0 ðx; yÞ with m0 ¼ 0; 1; 2; . . . ; M1 and integrate over S ¼ Sa [ Sc : ð M1 P1 X X ð Vm em em0 dS ¼ Van eap em0 dS ð16:43Þ S

m¼0

p¼0

Sa

Taking into account the orthonormalization (16.33), all integrals on the l.h.s. of (16.43) vanish except the one with m ¼ m0 which is unitary. We thus obtain Vm ¼

P1 X Vap dmp

m ¼ 0; 1; 2; . . . ; M1

ð16:44Þ

p¼0

where the coefficient dmp is given by

ð dmp ¼

em eap dS

ð16:45Þ

Sa

Finally, concerning (16.39), after inserting expressions (16.32) for the magnetic field, we multiply both sides by the vector ^z eap0 ðx; yÞ with p0 ¼ 0; 1; 2; . . . ; P1 and integrate over the aperture (not the section!) Sa . By virtue of the orthonormality of the modal functions eap on Sa , all integrals vanish except the one with p ¼ p0 which is unitary. We thus obtain M1 X

Imð1Þ Imð2Þ dmp ¼ 0

p ¼ 0; 1; 2; . . . ; P1

ð16:46Þ

m¼0

where the coefficients dmp coincide with those in (16.45). In fact ð ð ð hm ^z eap dS ¼ ^z em ^z eap dS ¼ em eap dS ¼ dmp Sa

Sa

Sa

Equations (16.44) and (16.46) constitute a linear system of M þ P equations in 3M þ P unknowns. The ð1Þ ð2Þ latter are Vm ; Im ; Im with m ¼ 0; 1; 2; . . . ; M1 and Vap with p ¼ 0; 1; 2; . . . ; P1. Additional relations exist between currents and voltages of the waveguide modes, depending on the boundary conditions at a distance from the discontinuity. Under the assumption that all higher order modes (m > 0) are below cutoff and the iris is far enough from any other discontinuity in the waveguide, the higher order mode amplitudes will decay exponentially with the distance jzj from the iris. For z < 0 there will be only the backward attenuating wave eazm z , while for z > 0 there will be only the forward attenuating wave eazm z . In other words, all higher order modes are ‘matched’, so that currents and voltages are related by the characteristic admittance Ycm : ð1Þ

ð1Þ

Im ¼ Ycm Vm ¼ Ycm Vm ð2Þ

ð2Þ

Im ¼ Ycm Vm ¼ Ycm Vm

m ¼ 0; 1; 2; . . . ; M1

ð16:47Þ

where we have used (16.42). Recalling (3.57), the characteristic admittances of evanescent modes are given by Ycm ¼ jBcm ¼ j

Ycm ¼ jBcm

oe ¼j azm

rffiffiffi e 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ðocm =oÞ2 1

rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi azm e ¼ ¼ j ðocm =oÞ2 1 jom m

for TM modes ð16:48Þ for TE modes

By adding (16.48) to (16.44) and (16.46) we now have 3M P 2 equations in 3M þ P unknowns.

NUMERICAL METHODS AND CAD 1

775

2

jXd

Figure 16.11

Equivalent circuit of the metal iris of Figure 16.10.

With suitable algebraic manipulations, as we will see next, the amplitudes of higher order mode voltages and currents can be eliminated so as to reduce the system to two equations in four unknowns, the latter representing the two voltages and two currents of the TE10 modes in the two half waveguides (z < 0 and z > 0). In this manner we obtain a full characterization of the discontinuity as a two-port network, which can be used to obtain its equivalent circuit. Let us observe first that because of the assumed zero thickness of the diaphragm, the equivalent circuit simply consists of a shunt reactance as in Figure 16.11. Equation (16.42) tells us in fact that ð1Þ ð2Þ V0 ¼ V0 , i.e. that the voltages at the two ports are identical. This automatically implies that the equivalent circuit reduces to a shunt impedance.9 The latter is a pure reactance since no loss has been assumed. It is convenient to write in matrix form the equations we have derived. Let us put 3 3 2 2 3 3 2 2 ð1Þ ð2Þ I1 I1 Va1 V1 7 7 6 6 7 7 6 6 6 ð2Þ 7 ð1Þ 7 6 V2 7 h i 6 6 Va2 7 6 I2 7 h ð2Þ i 6 I2 7 7 7 6 6 ð1Þ 7 7 6 ð16:49Þ ½Vs ¼ 6 . 7; Is ¼ 6 7 6 . 7; Is ¼ 6 . 7; ½Va ¼ 6 6 .. 7 6 ... 7 7 7 6 6 . . 5 5 4 4 . . 5 5 4 4 ð1Þ ð2Þ Va;P1 VM1 IM1 IM1 2

d10

6 6 d21 6 ½D s ¼ 6 . 6 .. 4 dM1;0

d f ¼ ½ d00

d11

...

d1;P1

...

...

...

...

d2;P1 .. .

2 6 ½Bc ¼ diag½Bcm ¼ 4

7 7 7 7 7 5

ð16:50Þ

. . . dM1;P1

dM1;1 d01

3

... Bc1

d0;P1

ð16:51Þ 3

..

7 5

.

ð16:52Þ

BcM1 By keeping apart the coefficients relevant to the dominant mode (m ¼ 0), Equations (16.44) and (16.46) can be written as follows: 9

Using a T network representation as in Figure 4.4, it is immediately seen that ZA ¼ ZB ¼ 0.

776


ð1Þ

I0

df

V0 ¼ d f ½Va

ð16:53Þ

½Vs ¼ ½Ds ½Va

ð16:54Þ

T

þ 2j ½Ds T ½Bc ½Vs ¼ 0

ð16:55Þ

where the superscript T stands for the transposed matrix. We have taken into account that because of (16.47) and (16.48) h i h i ð16:56Þ Isð1Þ ¼ j ½Bc ½Vs ¼ Isð2Þ V0 is the equivalent voltage associated to the fundamental mode and is equal to the voltages V1 ¼ V2 across the reactance Xd . Substituting (16.54) into (16.55), we obtain a system of P equations in the P unknowns Vap . The solution of such a system yields ½Va ¼

1 ½A d f V0 2Xd

ð16:57Þ

where we have expressed the current at port 1 as ð1Þ

I0 ¼

V0 jXd

ð16:58Þ

and ½A is a P P matrix given by 1 ½A ¼ ½Ds T ½Bc ½Ds

ð16:59Þ

Combining (16.59) with (16.55), we finally obtain the expression for the reactance of the iris:10 T Xd ¼ d f ½A d f ð16:60Þ The computation of the field distribution on the aperture Sa and in the waveguides is performed by computing the respective equivalent voltages using (16.57) and (16.54). The iris equations lead us to some further considerations. Observe that the susceptance Bd ¼ 1=Xd can also be expressed in terms of the waveguide modal voltages Vm . Left-multiplying (16.55) by ½Va T we obtain T Bd V0 ½Va T d f 2½Va T ½Ds T ½Bc ½Vs ¼ Bd V02 2½Vs T ½Bc ½Vs ¼ 0 ð16:61Þ In (16.61) we have used (16.53) and (16.54) and the expression for the transpose of the product of two matrices. From (16.61) we obtain Bd ¼

X 2 2 M1 ½Vs T ½Bc ½Vs ¼ 2 Bcm Vm2 2 V0 V0 m¼1

ð16:62Þ

The iris susceptance therefore appears as a linear superposition of the susceptances associated to the higher order modes excited at the discontinuity. Its value depends on the squared amplitudes Vm2 with which such modes are excited, thus on the coefficients dmp in (16.45) representing the projection of the waveguide modes onto the aperture modes and vice versa. Note that Bd does not depend only on the number M of modes retained in the waveguide, but also on the number P of terms employed in the expansion (16.38) of the electric field E0 on the aperture. So far we have not used any information concerning the shape of the iris and its position in the waveguide cross-section, so that the result expressed by (16.62) is fully general. All formulae derived so 10 The computation of the reactance Xd formally requires the matrix inversion in (16.59) but can actually be performed much more efficiently by numerically solving the system (16.57).


777

far can be applied to non-rectangular waveguides and to diaphragms of any shape. Let us now consider instead the case of a capacitive iris, in which the metal diaphragm occupies the upper portion of the rectangular waveguide cross-section, as shown in Figure 16.10. The aperture and the metal surface of the cross-section are defined as Sa f0 y b=2g Sc fb=2 y bg Such a discontinuity involves a cross-sectional variation in the y direction, while no variation occurs along the x axis. Since the incident mode is assumed to be the TE10 mode, whose electric field has only the y component varying as sinðpx=aÞ, the same behaviour will be present on the cross-section z ¼ 0 and along all the guide. The iris will thus excite only modes with the same spatial variation, i.e. the TE1n and TM1n modes. From the mathematical point of view, this is due to the coupling integrals dmp in (16.45) being zero when the modal vectors em and eap have different spatial periods along x, thus when m 6¼ p. Because of these considerations, for a rectangular waveguide of size a b we may consider only the modal vectors having 1 as the first index. We will use the following symbols for TE and TM modes respectively: C1m mp px mpy p px mpy

ðTEÞ ^x þ sin cos ^ ¼ cos sin em y m ¼ 0; 1; . . . ; M1 ð16:63Þ b a b a a b kc;1;m ðTMÞ ¼ em

C1m p px mpy mp px mpy

^x þ ^ cos sin sin cos y m ¼ 1; 2; . . . ; M1 kc;1; m a a b b a b

ð16:64Þ

In order for the normalization condition to be satisfied, the coefficients in the above formulae are given by rffiffiffiffiffiffiffiffi 2dm ð16:65Þ C1m ¼ ab with

( dm ¼

1 for m ¼ 0 2 for m 6¼ 0

ð16:66Þ

Since there is a pair of modes for each m > 0, it follows that the total number of modes in the expansion is 2M1. A further simplification can be made if one considers that the incident field has zero x component of the electric field. Such a component cannot be produced by a discontinuity independent of the y coordinate.11 This implies that the TE and TM modes excited at the discontinuity are combined in such a way that the resulting Ex component vanishes. Each such pair of TE þ TM modes is actually another modal solution deprived of the Ex component. Such modes are therefore TEðxÞ modes or, as they are also called, LSE modes (Longitudinal Section Electric modes). Without going into a discussion about such modes (see for instance [18]), we can simply observe that for each m > 0, (16.63) and (16.64) can be combined in the following manner: em ¼

1 p ðTEÞ mp ðTMÞ

px mpy ^ ¼ C1m sin cos e e y kc1;m a m b m a b

ð16:67Þ

11 One might ask why a similar reasoning does not hold for Hy which is not present in the incident field but is produced by the discontinuity. The difference is due to the metal edge (edge effect), which gives rise to an E field and H field orthogonal to the edge and approaching infinity as r1=2 , r being the distance from the edge. In the present case, since the edge is parallel to x, it produces EM field components along y and z. For a discussion on the edge effect, see [18].

778


so obtaining a new set of modes deprived of the Ex component. In this manner we consider only M þ 1 instead of 2M1 modes. The susceptances (16.48) are to be replaced by those of the LSE modes, which are jBcm ¼ j

o2 meðp=aÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi om ðp=aÞ2 þ ðnp=bÞ2 o2 me

ð16:68Þ

As can be seen, the susceptances are capacitive so that the iris susceptance Bd is also capacitive, hence the name of capacitive iris. In much the same way as for (16.67), the modal vectors of the aperture are expressed as eap ¼ A1p sin with A1p ¼

px ppy ^ cos y a b=2 rffiffiffiffiffiffiffi 4dp ab

ð16:69Þ

ð16:70Þ

The coupling integrals (16.45) are ða ð ðb px mpy 2ppy dmp ¼ em eap dS ¼ C1m A1p sin2 cos dx cos dy a b b Sa 0 b=2 In the computation of the above integrals it is convenient to distinguish three cases: 1 1 m ¼ 2p : d2p;p ¼ abC1;2p A1p ¼ pffiffiffi 8 2 m even 6¼ 2p : d2m;p ¼ 0 pffiffiffi 1 ð2m þ 1Þð1Þm þ p1 2 2 ð2m þ 1Þð1Þm þ p1 C A ¼ ab m odd : d2m þ 1;p ¼ 1;2p 1p p 2p ð2m þ 1Þ2 ð2pÞ2 ð2m þ 1Þ2 ð2pÞ2 In order to compute the iris susceptance using the above formulae, we need to choose the numbers of terms M and P in the field expansions in the waveguide and on the aperture, respectively. One might think that the higher the values of M and P, the better the accuracy. This is not true if M and P are chosen independently. Indeed, the mode matching method exhibits the so-called relative convergence phenomenon. There exists an optimal ratio M/P which provides the best approximated results: the error is higher when M or P is increased so as to deviate from the optimal ratio. By considering that M and P determine the minima spatial periods of the EM field in the guide and on the aperture, respectively, it is understood by intuition that such an optimal ratio is given by M b R ¼ ð16:71Þ P opt d where d is the height of the aperture. In the present case d ¼ b=2, thus R ¼ 2. If we retain P terms of the field expansion on the aperture, we need to retain M ¼ 2P terms in the waveguide. The relative convergence phenomenon is illustrated by Figure 16.12, where the error is plotted as a function of P (number of terms on the aperture) for given M (number of terms in the guide). The case d ¼ b=2 is of special interest as an analytical expression for Bd is available: ! ¥ X bk ð1Þp sin1 ð16:72Þ Bd ¼ 2 tan mp m¼1 The error can thus be calculated as the difference between (16.62) and (16.72). Note that for any given P, once the optimum ratio is exceeded the error increases with M, the solution converging to a value which


779

0.9

susceptance

0.85

P =2 P=3

0.8 P = 11 actual value 0.75 0.7 0.65

P = number of modes in iris

0.6 0

Figure 16.12 method.

5

10 15 20 25 30 35 40 45 number of modes in waveguide (M)

50

Convergence of the numerical solution of the iris of Figure 16.10 with the mode matching

is larger than the exact one. For higher P the error is lower. It is worth noting that with just two terms on the aperture and four in the guide, the error is about 7%. Figure 16.13 shows the distribution of the Ey component (thus the component normal to the iris edge) along the height ( y axis) of the cross-section at z ¼ 0, using M ¼ 40, P ¼ 20. In spite of the very small error in the numerical solution for Bd , the field distribution is locally not very accurate. This is because the procedure for computing Bd with (16.62) has a variational behaviour: an error in the field computation produces an error one order of magnitude smaller in the computation of Bd . Notice in Figure 16.13 the typical ripple of the Fourier series approximation. It is interesting to observe the field increase in the proximity of the metal edge (see footnote 11). The example just illustrated allows us to draw the following conclusions: The mode matching method requires a considerable amount of analytical preprocessing. The formulation presented is relevant to the relatively simple case of a metallic diaphragm in

Ey relative amplitude

.

0.00

M =40 P =20

2.00

4.00

6.00

8.00

10.00

Figure 16.13 Mode matching computation of the Ey distribution at z ¼ 0 for M ¼ 40, P ¼ 20 on the capacitive iris of Figure 16.10, from [19], copyright of John Wiley & Sons.

780

MICROWAVE AND RF ENGINEERING a waveguide. Any other discontinuity requires a specific mathematical development. This is the most serious drawback of this method, like any analytically based technique. .

The method can be applied whenever the analytical expression of the waveguide modes is available. This limits its applicability to simple geometries, such as rectangular or circular waveguides, or to waveguides that can be segmented in such simple geometries (e.g. a ridge waveguide).

.

The method provides very accurate results with small computational efforts (only a few tens of expansion terms are required). The very high numerical efficiency is the greatest advantage of the method. It is therefore amenable to full wave optimization of microwave structures (see Section 16.4).

There are commercial software packages equipped with already developed models for a large number of discontinuities and components. Such models can be used as modules for analyzing a large variety of complicated waveguide circuits and microwave structures.

16.3 Circuit analysis The EM analysis is required for the accurate characterization of the structures used to realize microwave circuits. In many cases, however, a very high accuracy is not required or is computationally too onerous. A circuit approach is much more efficient from the numerical point of view and in many cases can provide very good results, provided that the circuit models employed are adequate to represent the relevant microwave structure. The first commercial simulators appeared on the market at the end of the 1970s. They were based on the nodal description of the circuit. The network to be analyzed was described by a text file with as many lines as network components, each line being of the following type: Component type n1 n2 . . . Parameter value(s) where Component type is a keyword for the component described (resistor, capacitor, etc.), n1 n2 . . . are the nodes where the component is connected, and Parameter value(s) is (are) the values of the component parameter(s) (resistance, for a resistor, or characteristic impedance, propagation velocity and length, for a transmission line, etc). Modern simulators, on the contrary, allow the description of the circuit through its electrical schematic (schematic entry or schematic capture). The basic assumption for the circuit analysis is of course that a circuit description in terms of, for example, admittance or scattering matrix is available for all components. A fundamental distinction is to be made between linear and nonlinear analysis, as we are going to discuss in the next sections.

16.3.1

Linear analysis: the signal flow graph and the admittance matrix methods

The linear analysis is based on the assumption that all circuit elements are linear so that the principle of superposition may be applied. Some simulators deal with nonlinear elements but they are still based on linear models obtained by linearization around the operating point. As a consequence, the amplitude of the excitation does not affect the simulation results. The linear analysis is most often applied in the harmonic regime. In microwave and RF applications, the linear analysis is typically required to yield the scattering parameters versus the frequency. The time response to an arbitrary excitation may be computed by inverse Fourier transforms of the frequency response: this is to be considered as a postprocessing of the linear analysis.


781

SPICE (Simulation Program with Integrated Circuit Emphasis), conceived at University of California, Berkeley, around 1970, is one of the first computer programs developed for circuit analysis. Among several types of analyses that can be performed by SPICE, the AC analysis allows the simultaneous application of several sinusoidal generators having the same frequency but arbitrary amplitudes and phases. As a consequence, all electric circuit variables (voltages at the nodes and current along the branches) are sinusoidal with the same frequency as the excitation. The SPICE AC analysis has some drawbacks when applied to RF networks. The first one is that the scattering parameters of an N-port network cannot be obtained without using some specific tricks:

Figure 16.14 shows a possible way to extract the scattering parameters s11, s21 of a linear two-port network. To do this, some auxiliary elements have been added to the network: .

The input AC voltage generator of unitary amplitude and the series resistance R0.

.

The load resistor at port 2, also of resistance equal to R0.

.

The current-controlled voltage generator H1, controlled by the current absorbed by port 1.

.

The voltage-controlled voltage generator E1, whose output voltage is twice that at port 2.

R0 is assumed to be the normalization resistance. It can be immediately seen that the voltage at the node V2 is equal to s21. The voltage V1 is V1 ¼ 12 R0 I1 ¼ 12 R0

1 Zin;1 R0 ¼ R0 þ Zin;1 Zin;1 þ R0

ð16:73Þ

The last term in (16.73) is just s11. We conclude that the visualization of the magnitudes and phases of s11, s21 is obtained by visualizing magnitudes and phases of the voltages V1 and V2. The procedure illustrated does not allow the simultaneous computation of s11, s21 and s22, s12.

Zin,1

2-port linear network E1 1

2

R0 R0 +

+ vc –

V2 + 2 vc

V1

ic 2 R0 ic H1 + AC 1

Figure 16.14

Computing s11, s21 with the SPICE AC analysis.

782

MICROWAVE AND RF ENGINEERING bL

b L = ΓL a L

ΓL

b2

bg

bs Γg

s22

s11 b1

aL

1

s21

a1

bs = Γg as + bg a2

s12

(a)

as (b)

(c)

Figure 16.15 Signal flow graphs of a two-terminal element (a), a two-port network (b) and a generator (c). The second drawback of the SPICE AC analysis when applied to microwave networks is that only lumped elements are considered, with the only exception of lossless transmission lines. It is therefore difficult to account for the dispersion of characteristic impedances, propagation constants and loss. To remedy such drawbacks, specific algorithms have been devised for RF circuits. The first one is based on representing all components in terms of the scattering matrix. The latter may be computed analytically or available in numerical form as the result of a previous analysis. For example, a two-terminal component connected between two nodes constitutes a two-port network, whose S matrix is given by (4.106). The mathematical relationships expressed by a scattering matrix can be represented by an oriented graph, where the nodes represent the incident and reflected waves ai ; bi at the circuit ports and the paths connecting pairs of nodes are the elements of the scattering matrix. The graph of an N-port network therefore contains 2N nodes (a1, a2, . . . , an), (b1, b2, . . . , bn) and N2 branches. The graph of a two-terminal element with reflection coefficient GL is shown in Figure 16.15a. The two nodes a and b are connected by the branch GL to represent the mathematical relationship b ¼ GL a. Similarly, the graph of a two-port network is shown in Figure 16.15b. Consider now a voltage generator Vg with internal impedance Zg . The relationship between the voltage and the current flowing out of the generator is V ¼ Vg Zg I Replacing V and I with their expressions (4.75) in terms of incident and reflected waves, from simple algebra we obtain a ¼ Gg b þ bg with Gg ¼

Zg Z0 Zg þ Z0

bg ¼

pffiffiffiffiffi Vg Z0 Zg þ Z0

The resulting signal flow graph is shown in Figure 16.15c. Consider now two networks A and B connected at a common port a. Since the incident and reflected waves at a are interchangeable, we have the equalities ðAÞ

ðBÞ

aa ¼ ba ;

ðAÞ

ðBÞ

ba ¼ a a

ð16:74Þ

As a consequence, the graph representing the overall network is obtained by joining the two graphs with the corresponding nodes coinciding as in (16.74). For example, a two-port network fed at port 1 by a source with internal reflection coefficient GS and with a load GL at port 2 is represented by the graph of Figure 11.2b.


783

α β1

Za α1 1 2

1

α3

3

Zc β3 1 2

2 β2 Zb 1 2

Figure 16.16

fictitious 3-port network α2

Connecting three circuits to a common node.

When more than two terminals are connected to the same node a, a dummy M-port network can be introduced so as to reduce the problem to the connection between terminal pairs. This is illustrated in Figure 16.16, where three two-terminal circuits are connected at the common node a. The scattering matrix of the connecting three-port network is given by 2 1 2 14 2 1 ½S ¼ 3 2 2

3 2 2 5 1

ð16:75Þ

In this manner it is possible to draw the signal flow graph of a however complicated network. The scattering parameter shk between the nodes h and k can be evaluated using Mason’s rule [20]: shk

P P P P P1 1 L1;1 þ L2;1 . . . þ P2 1 L1;2 þ L2;2 . . . þ . . . P P P ¼ 1 L1 þ L2 L3 þ . . .

ð16:76Þ

where: .

.

.

.

Pi is the gain of the ith forward path between the nodes h and k. Each gain is the product of the gains of the various branches, oriented from h to k, not crossing any nodes more than once. P L1 is the sum of all first-order loop gains. The loop gain is the product of all gains of the loop, the loop being a closed path where each branch is travelled according to its orientation and no node is crossed more than once. P Li ; i ¼ 2; 3 . . . ; is the sum of the ith-order loops. The ith-order loop is defined as the product of the i first-order non-touching loops, i.e. loops with no common nodes. P Lj;p is the sum of the jth-order loops with no common nodes with the pth path.

The algorithm described reduces the computation of the scattering parameters of a network to a path search in a signal flow graph. The algorithm is robust, but becomes quite onerous when the number of nodes is large. More efficient is an alternative algorithm based on the use of admittance matrices. Consider an Mnode network. Each component is represented by an M M admittance matrix, each port corresponding to a node. The admittance matrix element yhk is non-zero only if the nodes hk are connected. The overall network is obtained by the parallel connection of all elementary networks; the resulting Y matrix is therefore the sum of all elementary Y matrices. Consider for example the network of Figure 16.16. It consists of three impedances Za ; Zb ; Zc and four nodes. As can be readily verified, it results from the parallel connection of the three elementary networks

784


shown in Figure 16.17, whose admittance matrices 2 ðaÞ y11 6 6 6 0 6 ½Ya ¼ 6 6 6 0 6 4 ðaÞ

0

ðaÞ

0 0 y12 0 0 0 0

7 7 0 7 7 7 7 0 7 7 5

0 0 y22 0

6 6 6 0 yðbÞ 11 6 ½Yb ¼ 6 6 60 0 6 4 ðbÞ 0 y21 2

3

ðaÞ

y21 2

are

0 0

6 6 60 0 6 ½Yc ¼ 6 6 60 0 6 4 0 0

0

0

3

7 ðbÞ 7 0 y12 7 7 7 7 0 0 7 7 5 ðbÞ 0 y22 0 0 ðcÞ

y11

ðcÞ

y21

0

ð16:77Þ

3

7 7 0 7 7 7 ðcÞ 7 y12 7 7 5 ðcÞ

y22

Figure 16.17 Breaking down the network of Figure 16.16 into elementary parallel networks. Example illustrating the harmonic balance.


785

where yhk,g (h, k ¼ 1, 2; g ¼ a, b, c) is the hk element of the admittance matrix of the g network. The overall admittance matrix is simply the sum of (16.77): ½Y mm ¼ ½Ya þ ½Yb þ ½Yc 3 2 ðaÞ ðaÞ y11 0 0 y12 7 6 7 6 ðbÞ 7 6 0 yðbÞ 0 y12 7 6 11 7 ¼6 7 6 ðcÞ 7 6 0 0 yðcÞ y12 7 6 11 5 4 ðaÞ ðbÞ ðcÞ ðaÞ ðbÞ ðcÞ y21 y21 y21 y22 þ y22 þ y22

ð16:78Þ

In practice, however, we are normally interested only in the scattering matrix description at the N external ports, with N M. If N ¼ M the external ports coincide with the nodes, and the scattering matrix can be computed from (16.77) using the known relations (see Table 4.1). In general, however, N ¼ MK, K being the number of internal ports. In order to derive the N NY matrix relative to the external ports, observe that since the internal nodes are not connected to the external ports, the corresponding current is zero, thus Ik ¼ 0. Recalling the impedance and admittance matrix definitions given in Sections 4.6.1 and 4.6.2, the open-circuit condition can easily be applied using the Z matrix representation by simply omitting all rows and columns relative to the K open ports. The procedure therefore consists of first computing the M M impedance matrix ½Z ¼ ½Y 1 , then deleting the K ¼ M N rows and columns relative to the internal ports, and finally computing the scattering matrix using (4.83). The algorithm described requires the summation of M M sparse matrices, which is much easier a procedure than the search for the flow graph paths. The matrix admittance analysis is therefore much speedier than that based on scattering parameters. A possible drawback is that, as seen in Section 4.6.2, the admittance matrix might not exist for some elementary networks. This happens when an impedance Zx is shunt connected to ground. This difficulty can be circumvented by replacing Zx with two series impedances Zx =2 with an auxiliary node at their junction. The one-port network is so transformed into a two-port network eliminating the singularity.

16.3.2

Time domain nonlinear analysis

Nonlinear circuits may be viewed as consisting of linear lumped and distributed elements along with circuit elements characterized by nonlinear instantaneous voltage–current relationships.12 For the moment we limit our attention to lumped elements, linear and nonlinear controlled generators and independent sources. The only non-instantaneous relationships are the constitutive relationships of capacitors and inductors, where the time derivatives of the voltages (for capacitors) or currents (for inductors) are involved. The only nonlinear equations are those of the nonlinear controlled generators which are by definition instantaneous. The application of Kirchhoff’s laws yields a differential equation whose order is equal to the number of reactive components contained in the network. The circuit response is obtained by numerical integrations of the differential equations.13 The first computer code implementing the algorithm just described was SPICE, more precisely its module for the TRANSIENT analysis. In RF circuit applications we are normally interested in the harmonic regime response. For example, concerning an amplifier, we are interested in the output signal spectrum when a sinusoidal signal of given power and frequency is applied to the input. This can be obtained via a nonlinear time domain analysis; 12 13

See for example the models described in Section 9.7. See [19].

786


the computation time, however, may be quite long due to the analysis procedure. Starting from the initial values,14 in fact, the voltage computation at the various nodes goes on with the numerical integration from one discrete time instant to the next. The stationary response is obtained after the initial transient has died out.15 Because of the time constants of the bias and feeding networks, the transient has rather long decay rates compared with the signal frequency. The integration time step of the differential equation, on the contrary, must be much shorter than the period of the highest frequency present in the network. This implies that the transient analysis requires a very high number of time steps, thus a considerable computing time. The second drawback of the time domain analysis is, as already mentioned for the SPICE AC analysis, the difficulty of employing dispersive models.

16.3.3

Frequency domain nonlinear analysis

The harmonic balance is a hybrid time domain and frequency domain method for calculating the response of a nonlinear network containing lumped as well distributed linear elements, nonlinear instantaneous elements and independent DC or harmonic sources. In its original formulation the harmonic balance was a ‘single tone’ method, as only a single harmonic excitation was allowed. The method was then extended to periodic excitations, considered as the superposition of harmonics, as well as to excitations containing multiple non-harmonically correlated frequencies, so as to make it possible to compute amplifier intermodulation, mixer conversion products, etc.16 The key point in the harmonic balance is the splitting of the network into a linear and nonlinear portion.17 The linear portion is analyzed in the frequency domain, while the nonlinear portion is analyzed in the time domain. In this manner, the drawbacks associated with the time domain analysis, as mentioned in the previous section, are alleviated. The result of the nonlinear analysis is then Fourier transformed into the frequency domain. To illustrate how the algorithm works, consider the simple example of Figure 16.17, containing only one DC voltage generator, one harmonic generator and only one nonlinear two-terminal element. Connected to port 1 is a cosinusoidal generator having radian frequency o0 and connected to port 2 is a DC generator: V1 ¼ v1 cosðo0 tÞ;

V2 ¼ v0

ð16:79Þ

Since V2 is constant, the network’s stationary response is periodic with T0 ¼ 2p=o0 : voltages at the nodes and currents along the branches can thus be expressed as Fourier series with fundamental frequency o0. The first approximation of the harmonic balance algorithm consists of truncating each series to a finite number N of terms. The higher the value of N, the better the accuracy and the lower the computational speed. A typical choice is N ¼ 8. The admittance matrix of the linear network is computed at the radian frequencies no0 ; n ¼ 0; 1; . . . ; N, using one of the methods described in Section 16.3.1: 32 3 2 3 2 y11 y12 y13 V1 I1 76 7 6 7 6 ð16:80Þ 4 I2 5 ¼ 4 y21 y22 y23 54 V2 5 I3

y31

y32

y33

V3

14 The initial values of the voltages at the various nodes can be either set to zero or precomputed by a DC analysis using the initial values of the independent generators of the network. 15 See for instance the oscillator output voltage of Figure 12.14 where the transient dies out after about 10 ns. 16 See [21], sections 1.4.4, 2.3.4 and 2.5.7.3, and Ch. 3. 17 The linear portion also includes the linear elements of the nonlinear models such as, for instance, Cp, Rs, Ls, in the model of Figure 9.38 or Lg, Rg, Ls, Rs, Ld, Ls, Ri, Cgd, Cds, in the model of Figure 9.40a.


787

The above matrix equation (16.80) is satisfied at all N þ 1 frequencies no0. In particular, the current at port 3 is I3 ¼ y31 V1 þ y32 V2 þ y33 V3

ð16:81Þ

The Fourier series for the voltage V3 and the current I3 truncated to the first N terms are V3 ¼ v3;0 þ

N X v3;n cos no0 t þ fv3;n

ð16:82Þ

n¼1

N X i3;n cos no0 t þ fi3;n I3 ¼ i3;0 þ n¼1

Substituting (16.79) and (16.82) into (16.81), we obtain i3;0 þ

N X

i3;n cos no0 t þ ji3;k ¼ jy31 ðo0 Þjv1 cos o0 t þ jv3;1 þ arg½ y33 ðo0 Þ þ y32 ð0Þv0 n¼1

þ y33 ð0Þv3;0

N X

jy33 ðno0 Þjv3;n cos no0 t þ jv3;n þ arg½y33 ðno0 Þ n¼1

ð16:83Þ Equating harmonic terms of the same frequency no0 at both members of (16.83), we obtain i3;0 ¼ y32 ð0Þv0 þ y33 ð0Þv3;0

ð16:84Þ

i3;1 cos o0 t þ ji3;1 ¼ jy31 ðo0 Þjv1 cosfo0 t þ arg½ y33 ðo0 Þg

þ jy33 ðo0 Þjv3;1 cos o0 t þ jv3;1 þ arg½y33 ðo0 Þ

i3;n cos no0 t þ ji3;n ¼ jy33 ðno0 Þjv3;n cos no0 t þ jv3;n þ arg½ y33 ðno0 Þ ;

ð16:85Þ

n ¼ 2; 3; . . . ; N ð16:86Þ

Equation (16.84) represents the relation between the DC components of the current and the voltages at port 3. Equation (16.85) is a nonlinear equation in the four unknowns i3;1 ; fi3;1 ; v3;1 ; fv3;1 which are the amplitudes and phases of current and voltage at the fundamental frequencyo0. Equations (16.86) are N1 equations in the 4(N 1) unknowns i3;n ; ji3;n ; v3;n ; þ jv3;n ;

n ¼ 2; 3; . . . ; N

Equations (16.84)–(16.86) thus represent (N þ 1) equations in the (4N þ 2) unknown harmonic components of the voltage and current at the port where the nonlinear element is connected: i3;0 ; v3;0

and

i3;k ; ji3;n ; v3;n ; jv3;n

ðn ¼ 1; 2; . . . ; N Þ

The constitutive relation of the nonlinear element can by expanded in a Taylor series: I3 ¼ f ðV3 Þ ¼

¥ X

an V3n

ð16:87Þ

n¼0

If the voltage V3 were purely sinusoidal with radian frequency o0, the nth term of the above expansion would yield integer harmonics up to no0. Since V3 contains N harmonics, the highest resulting harmonic

788


of the nth-degree term will be nNo0. Truncating the series (16.87) to the Nth-order term and substituting (18.82), we obtain " #n N N N X X X i3;n cos n o0 t þ fi3;n ¼ an v3;0 þ v3;n cos no0 t þ fv3;n ð16:88Þ i3;0 þ n¼1

n¼0

n¼1

The l.h.s. of (16.88) contains harmonic terms up to No0 while the r.h.s. contains harmonic terms up to N2o0. Equating terms of the same frequency, we obtain: .

N þ 1 equations relevant to the terms with radian frequencies (appearing in both l.h.s. and r.h.s.) no0 ; n ¼ 0; 1; . . . ; N in the 4N þ 2 unknowns i3;0 ; v3;0 and i3;n ; ji3;n ; v3;n ; jv3;n ; n ¼ 1; 2; . . . ; N.

.

N2 N 1 equations relevant to the terms with radian frequencies (appearing in the r.h.s.) no0 ; n ¼ N þ 1; N þ 2; . . . ; N 2 in the 2N þ 1 unknowns v3;0 and v3;n þ jv3;n ; n ¼ 1; 2; . . . ; N.

Equation (16.88) represents the relationship between the voltage and current harmonic components in the nonlinear element resulting from its constitutive relation. It corresponds to a system of N 2 nonlinear equations, where the unknowns are the harmonic components (amplitudes and phases) of voltages and currents. Combining (16.83) with (16.88), we have N 2 þ N þ 1 equations with 2N þ 2 unknowns. The harmonic balance analysis of the network consists of ascribing suitable values to the unknown parameters of the harmonic components so as to approximately satisfy (16.83) and (16.88) simultaneously. The solution cannot be exact, but the harmonic balance algorithm can most often identify an approximate solution such that the error is smaller than a prescribed value. While it is very difficult to determine beforehand the convergence conditions of the harmonic balance algorithm, it is nevertheless possible to provide some general criteria. For example, the convergence is slower when dealing with nonlinear elements with step discontinuities (with discontinuous derivative) or when the saturation rate is high. This happens for example when the input power to an amplifier is much higher than that at 1 dB compression. A typical case of non-convergence occurs when the circuit tends to oscillate. In the unfortunate cases of non-convergence, one may try to modify some parameters, such as: .

Increasing or reducing the number N of harmonics.

.

Reducing the maximum error in the solution of the system (16.83), (16.88).

.

Reducing the power applied.

As a final remark, let us observe that the harmonic balance is based on the analysis of the linear network up to the frequency of the highest harmonic. By considering the first eight harmonics in a circuit operating at 10 GHz fundamental frequency, it is necessary to dispose of models of the linear network up to 80 GHz.

16.4 Optimization As we saw in the introduction to this chapter, optimization is a crucial phase of the microwave and RF circuit design as it is normally necessary to refine or modify an approximate design so as to satisfy the given specifications. Because of the vastness of the subject we cannot but provide some general basic concepts. Quite often the CAD user does not even realize the complexity of the problem, as he or she blindly confers to the optimizer the task of trimming the circuit to match the specifications. Though superficial, some knowledge of the basics of optimization and of the advantages and limitations of the various approaches is important for a reasoned choice among them.


16.4.1

789

Definitions and basic concepts

The circuit optimization consists of the minimization of a scalar function UðFÞ, called the objective function or error function, which represents the distance between the performances obtained and those required. F is the vector of the circuit parameters (e.g. the component values), which can be considered as the design parameters. In the case of a microstrip filter, for instance, UðFÞ may depend on the difference between its gain and the specified gain, while F, depending on the filter model adopted, may contain the values of the LC elements of the filter or the characteristic impedances of the line sections, or the dimensions of the metal strips, etc. In general, the circuit parameters are subject to some constraints. These can consist of inequalities (e.g. gðFÞ 0) or equalities (e.g. hðFÞ ¼ 0). For example, for technological reasons the width of a microstrip line is to be contained between given limits. The space region R where all such constraints are satisfied constitutes the feasible region. That is, R fFjgðFÞ 0; hðFÞ ¼ 0g The region R is said to be closed or open depending on whether the equality is contemplated or not. The optimization procedure consists of the search for Fmin inside R; the minimum is said to be a global minimum when Umin UðFmin Þ < UðFÞ for any feasible F non-coincident with Fmin . Usually, optimization methods do not find a global minimum, but just a local minimum, such that UðFmin Þ ¼ min UðFÞ F2Rl

where Rl is the region in the proximity of Fmin. Figure 16.18 describes in a 2-D space some typical conditions occurring in optimization problems. The figure shows the feasible region, the constant-U lines,

Figure 16.18

Map of a 2-D optimization problem.

790


a global and a local minimum, a saddle point and a narrow valley. A saddle point is characterized by the fact that the function has a maximum along certain directions and a minimum along others.

16.4.2

Objective function

There are several ways to define the objective function to be minimized in the optimization procedure. Let us designate by C the independent variable, e.g. the frequency or the time, on which the circuit response FðF; CÞ depends, and by SðCÞ the desired response. We can define a weighted error function as eðF; CÞ ¼ wðCÞ½FðF; CÞSðCÞ

ð16:89Þ

where wðCÞ is a weighting function which, depending on the value of C, enhances or reduces the difference between the actual and the desired responses. In the cases when C is a discrete variable, an error vector is defined as follows: e2 ðFÞ . . . en ðFÞ T

eðFÞ ¼ ½ e1 ðFÞ

ð16:90Þ

where ei ðFÞ ¼ eðF; Ci Þ and n is the number of discrete C values. For any value of C, the function (16.89) or (16.90) provides the weighted deviation of the response from the specification. In order to have only one error parameter to be minimized, for continuous variables we use the pth norm of (16.89) kekp ¼

(ð c2

)1=p p eðF; CÞ dC

ð16:91Þ

c1

where p is an integer greater than or equal to 1. In the case of discrete variables, the integral is replaced by a summation over the Ci values: kekp ¼

nX o1=p eðj; c Þp i

i

ð16:92Þ

In the case p ¼ 2, the norm is called Euclidean since it measures the length of the error vector (16.90) in the multi-dimensional space. Two important objective functions are based on (16.90). Using the least pth approximation the objective function is written as ð c2 eðj; cÞp dc UðjÞ ¼ ð16:93Þ c1

The case p ¼ 2 corresponds to the well-known least squares approximation, where the error is the sum of the squared errors. In the discrete case we have UðjÞ ¼

n X

½ei ðjÞ2 ¼ ½eðjÞT ½eðjÞ

ð16:94Þ

i¼1

In many practical problems one requires the circuit response to be contained within a range of values that are both upper and lower bounded. Consider, for example, a filter gain specified in terms of maximum ripple, see Figure 16.19. In such cases the minimax approximation can be employed, which is based on the following objective function: UðFÞ ¼ max ½wu ðCÞfFðF; CÞSu ðCÞg; wl ðCÞfSl ðCÞFðF; CÞg ½Cl ;Cu

ð16:95Þ

where wu and wl are the weighting functions (supposedly greater than 0) relative to the upper bound Su and to the lower bound Sl, respectively. The minimization of (16.95) constitutes the minimax approximation. It aims at minimizing, with the respective weights, the distances between the response and the upper and


Figure 16.19

791

The minimax optimization criterion, upper and lower bounds.

lower bounds. If such bounds are not specified, but just a desired value SðCÞ ¼ Su ðCÞ ¼ Sl ðCÞ is given, (16.95) reduces to ð16:96Þ UðfÞ ¼ max jwðcÞfFðf; cÞSðcÞgj The problem arising from the discontinuity of the derivatives of the objective function can be circumvented by a suitable reformulation of the objective function.

16.4.3 Constraints It is apparent that the geometrical parameters of a microwave circuit or the components of its electrical model or the physical parameters of a device cannot assume fully arbitrary values but are necessarily subject to some limitations due to various types of constraints. A microstrip line, for example, may not be too wide in order not to allow the onset of higher order modes, nor can it be too narrow so as to compromise its electrical continuity. In practice, therefore, some constraints need to be imposed during the optimization procedure to avoid the solution converging towards non-physical or unrealizable values. The constraints can be expressed in the form of inequalities imposed on the circuit parameters, thus on the elements of F, so constraining Fi to be contained within a prescribed interval: jli ji jui ;

i ¼ 1; 2 . . .

ð16:97Þ

where the equality signs are to be eliminated in the case of an open feasible region. Then (16.97) can be applied by fixing the value of Fi to the upper or lower bound whenever the optimization procedure attempts to move it outside the prescribed interval. This procedure has the disadvantage of leading to a non-continuous objective function. As an alternative, the constraint (16.97) can be imposed analytically by defining a new objective function in a new variable F0 i such that Fi remains in the prescribed interval, as in the following example: Fi ¼ Fli þ ðFui Fli Þ sin2 F0 i

16.4.4

Optimization methods

Once the objective function and, through the constraints, the feasible region have been defined, one needs to search for the minimum of the objective function. The search methods are classified into two categories, direct search methods and gradient methods. The former consist of sequentially comput-

792


ing the objective function at various points of the feasible region, recording the minimum computed value and following a suitable strategy to look for the next point. Many strategies can be applied, including a random search. For a discussion on search methods the interested reader is referred to [23–25]. Gradient methods use the gradient of the objective function to direct the search for the minimum. The gradient in fact identifies the direction of maximum variation of the function. In a 3-D representation of a two-variable function, the opposite of the gradient is the direction of quickest descent. The most immediate application of this concept is implemented in the steepest descent method. From the first attempt point F1, the search for the optimum proceeds iteratively in the direction of the gradient according to the formula fi þ 1 ¼ fi þ li si

ð16:98Þ

In this expression, the quantity rUðji Þ si ¼ rUðji Þ identifies the search direction and li is a suitable parameter that determines the distance at which the optimum is sought. The use of the gradient thus allows one to reduce the search to just one direction. The iterative procedure is stopped when, from two successive iterations i and i þ 1, one of the following criteria is verified: (a) The relative difference between the values of the objective function is less than a prescribed value: j½UðFi þ 1 ÞUðFi Þ=UðFi Þj e1 . (b) The difference between the independent variable vectors Fi þ 1 and Fi is, in absolute value, less than a prescribed quantity, jFi þ 1 Fi j e2 . (c) The derivatives of the objective function are, in absolute value, less than a prescribed quantity, jqUðFÞ=qFk j e3 . The method is apparently very efficient, as it reduces the problem to a 1-D search. Nevertheless, since it intrinsically converges to a local minimum, it is normally not very useful, unless previous information is available about the localization of the global minimum. Another drawback of the steepest descent method is that it may not converge at all in the presence of a narrow valley (see Figure 16.18), since the direction of the gradient is not that of the minimum. A number of modifications have therefore been developed to the basic method in order to avoid such difficulties. Among these methods there are the second-order methods which make use of the second derivatives of the objective function.

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8. K. S. Yee, ‘Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media’, IEEE Transactions on Antennas and Propagation, Vol. AP-14, pp. 302–307, 1966. 9. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, NJ, 1995. 10. T. Weiland, ‘Time-domain electromagnetic field computation with finite difference methods’, International Journal of Numerical Modelling, Vol. 9, pp. 259–319, 1966. 11. J. A. Pereda, F. Alimenti, P. Mezzanotte, L. Roselli and R. Sorrentino, ‘A new algorithm for the incorporation of arbitrary linear lumped networks into FDTD simulators’, IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 6, pp. 943–949, 1999. 12. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, John Wiley & Sons Australia Ltd. 13. T. Itoh, G. Pelosi and P. P. Silvester, Finite Element Software for Microwave Engineering, John Wiley & Sons, Inc., New York, 1996. 14. J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, Inc., New York, 2002. 15. W. C. Hahn, ‘A new method for the calculation of cavity resonators’, Journal of Applied Physics, Vol. 12, pp. 62–68, 1941. 16. J. R. Whinnery and H. Jamieson, ‘Equivalent circuits for discontinuities in transmission lines’, Proceedings of the IRE, Vol. 32, pp. 98–116, 1944. 17. G. Conciauro, M. Guglielmi and R. Sorrentino, Advanced Mode-Matching Techniques, John Wiley & Sons, Inc., New York, 2000. 18. R. E. Collin, Field Theory of Guided Waves, 2nd edition, IEEE Press, Piscataway, NJ, 1991. 19. R. Sorrentino, M. Mongiardo, F. Alessandri and G. Schiavon, ‘An investigation of the numerical properties of the mode-matching technique’, International Journal of Numerical Modelling, Vol. 4, pp. 19–43, 1991. 20. A. Vladimirescu, The SPICE Book, John Wiley & Sons, Inc., New York, 1993. 21. R. W. Kruse, ‘Microwave design using standard SPICE’, Microwave Journal, November, pp. 164–171, 1988. 22. S. A. Maas, Nonlinear Microwave and RF Circuits, 2nd edition, Artech House, Norwood, MA, 2003. 23. J. A. Dobrowolski, Introduction to Computer Methods for Microwave Circuit Analysis and Design, Artech House, Norwood, MA, 1991. 24. K.C. Gupta, R. Garg and R. Chadha, Computer-Aided Design of Microwave Circuits, Artech House, Norwood, MA, 1981. 25. IEEE Transactions on Microwave Theory and Techniques, Special Issue, Vol. 52, No. 1, Part 2, 2004.

17

Measurement instrumentation and techniques 17.1 Introduction A test and measurement system basically consists of one or more instruments and one device under test (DUT). This chapter will describe the main microwave measurement instruments, as well as discuss some interactions between the instrument and the DUT. The basic working principles and internal structure of the main RF/microwave (RF/mW) test instruments will be illustrated, with some considerations of the obtainable performance. Some modern instruments are digital extensions of more dated analogue counterparts. When this applies, the description begins with the analogue – albeit obsolete – version of the instrument, for a better understanding of the working principles. We will apply this concept in particular to the spectrum analyzer and the scalar network analyzer.1

17.2 Power meters The power meter is one of the simplest RF/mW measurement instruments. One of its simplest realizations is a bloomer, which measures the RF power through temperature measurements. Figure 17.1 shows a block diagram of the instrument. It is normally composed of two parts, the power sensor and measurement unit, each with its own case. The sensor is connected to the measurement unit through a multi-wire flexible cable that transports only direct currents. Two twisted and shielded wires transmit the sensor output voltages to the measurement unit, and some additional wires deliver the DC voltages – generated inside the measurement unit – to the sensor circuitry. The sensing part of the sensor is a matched load,2 absorbing the whole signal power. The load is thermally isolated from the environment, or, in other words, the thermal resistance Rc;a between the load and environment is quite high and known with precision. Let pRF be the power of the signal to be 1

Michele Ancis is gratefully acknowledged for his contribution to the English of this chapter. Usually, the sensor impedance is 50 O for coaxial input. Waveguide power sensors are matched to the impedance of their waveguide. 2


796


DISPLAY TRAN1 RF INPUT

R0

s1 T1

TRAN2 R0

s2 T2

AMP1 + + vin,1 av,1vin,1 -

vOUT AMP3 + + vin,3 a v,3 vin,3 –

AMP2 + + vin,2 av,2 vin,2 –

Power sensor

voffset

+ Controller

Measurement unit

Figure 17.1

Schematic of an RF/mW power meter.

measured, T1 the temperature reached by the load as an effect of the dissipated power and T2 the environment temperature. The three are related by T1 ¼ Rc;a pRF þ T2

ð17:1Þ

Now, assuming the environment temperature to be constant, one could measure the RF power by measuring twice the load temperature, with and without the applied signal. In fact, from Equation (17.1) it follows that with no RF applied the sensor temperature equals that of the environment. In practice, however, it is difficult to operate in constant temperature conditions, therefore the meter has a built-in temperature sensor that allows the ambient temperature contribution to be subtracted from the final measurement. Temperature measurements are normally done with a thermoelectric transducer, which provides a voltage proportional to the temperature. In Figure 17.1 two transducers are present, namely s1 and s2. The output voltages of the hot (s1) and cold (s2) transducers are ð17:2Þ v1 ¼ k1 T1 þ k0;1 ¼ k1 Rc;a pRF þ T2 þ k0;1 ; v2 ¼ k2 T2 þ k0;2 where k1 ; k2 and k0;1 ; k0;2 are the constants (slope and offset) describing the supposedly linear characteristic of the two transducers. The parameters of the two thermoelectric transducers are nominally identical, although never exactly so because of component tolerances. The transducers’ output voltages are amplified by the amplifiers AMP1 and AMP2 and then fed to the differential amplifier AMP3, whose output is ð17:3Þ v3 ¼ av;1 v1 av;2 v2 ¼ av;1 k1 Rc;a pRF þ av;1 k1 av;2 k2 T2 þ av;1 k0;1 av;2 k0;2 Now if the gains of AMP1 and AMP2 compensate for their respective transducer slopes, so that av;1 k1 ¼ av;2 k2 , then the voltage v3 becomes independent of the environment temperature v3 ¼ av;1 k1 Rc;a pRF þ av;1 k0;1 av;2 k0;2

ð17:4Þ

The differential voltage v3 at the output of the transducer given by Equation (17.4) is proportional to the signal RF power, plus an offset av;1 k0;1 av;2 k0;2 due to the mismatch between the two transducers and in the respective amplifiers. This offset is, however, independent of ambient temperature3 and can be

3 In a more detailed analysis it must be considered that AMP1 and AMP2 also present their own offset, usually depending on the temperature and variable over time (drift).

MEASUREMENT INSTRUMENTATION AND TECHNIQUES

797

eliminated by zeroing the instrument before applying the signal. The RF power is therefore measured in the following steps: 1. Zeroing. Without RF signal input, the meter measures the offset at the output of AMP3 vout;0 ¼ av;3 av;1 k1 Rc;a 0 þ av;1 k0;1 av;2 k0;2 ¼ av;3 av;1 k0;1 av;2 k0;2 ð17:5Þ This voltage is memorized and subtracted from the amplifier output. This operation can be either manual or automatic. In the former case the offset is regulated with a knob until the meter reads zero when the signal is not applied. In the latter case, in modern automatic instruments, the voltage vout;0 is converted into digital form, memorized and then automatically subtracted by the instrument. 2. Measurement. The output of AMP3, after subtracting the offset, is vdisp ¼ vout vout;0 ¼ av;3 av;1 k1 Rc;a pRF ) pRF ¼

1 vdisp av;3 av;1 k1 Rc;a

ð17:6Þ

The resulting voltage is sent to a voltmeter whose scale is arranged to read powers directly. In principle, av;1 , av;3 , k1 and Rc;a are known quantities. However, to accommodate the unavoidable tolerances in construction and aging effects, the value of the constant 1=ðav;3 av;1 k1 Rc;a Þ can be determined before the measurement through the calibration process. 3. Calibration. An oscillator4 providing a known output power is also contained in the measurement unit. The oscillator can be set ON or OFF and its signal is accessible through a connector in the front panel of the instrument. After zeroing and prior to measuring, the sensor is connected to the oscillator output, the oscillator is turned on and the gain av;3 is regulated until the known oscillator power is displayed. Gain adjustment is also automatic in modern instruments. Often the gain of AMP3 is selectable from a certain set of values, in order to accommodate measuring different power ranges. In modern instruments, a digital display giving the measure either in W or dBm replaces the analogue voltmeter. The reading of the instrument is based on the output voltage of the thermal sensor TRAN1. Now, if pRF varies rapidly over time, then neither the load temperature T1 nor any of the internal instrument voltages will be able to follow these variations,5 due to thermal inertia. Therefore the measure gives the average power of the signal. The instrument described above measures RMS power, typically over an integration time of about 1 second. It is possible to replace the thermoelectric power sensor with the double detector arrangements of Figure 13.4 or 13.7a. If the detectors work in their quadratic region, the resulting DC output voltage is proportional to the square of the peak RF voltage and therefore to the power. Detectors supply higher output voltages and exhibit faster responses6 than bloomers. Therefore, if the measuring unit has analogous speed characteristics,7 the crystal-based power meter can be used to measure both RMS and peak power. The amplifiers contained in the sensor and meter must have narrow bandwidth for the bolometer-based instrument and wide bandwidth for the diode-based one. The peak value is obtained by sampling v3 and numerically computing the value. Bolometers are more precise, while crystal sensors allow for lower power ranges to be measured. Generally, constructors provide measuring units compatible with both sensor types. 4

The oscillator frequency is of the order of a few tens of megahertz, while its power is 1 mW (0 dBm). Amplifiers AMP1 to AMP3 have to amplify extremely weak DC voltages, and must present very low offset, drift and noise. Such characteristics are typical of narrow-band devices. 6 The video bandwidth of a diode detector is of the order of some megahertz, as calculated in Section 13.2.1. 7 A suitable measuring unit has to be able to operate at narrow bandwidth, in order to work with bolometers and detectors to measure RMS power, and at wide bandwidth, to measure peak power (only with detector sensors). 5

798


17.3 Frequency meters Microwave/RF meters are based on the electronic counter. In its easiest realization a frequency meter is made of an AND gate, an oscillator providing the counting time reference, a counter and a display. The maximum measurable frequency for such an instrument is around 1 GHz. This limit can be extended to around 10 GHz by using digital frequency dividers and up to 40 GHz using frequency conversion techniques.

17.3.1

RF digital frequency meter

The arrangement in Figure 17.2a is normally used for frequencies up to hundreds of megahertz. A detailed description of the digital frequency meter is beyond the scope of this book, but we will illustrate its working principle here to explain RF/mW frequency meters. The signal to be measured is fed to a

DISPLAY AMP

n1

IN

AND n2

Counter

Reset

n3

(a)

Time base

INPUT2

Prescaler

AMP

DISPLAY 2

n1

AND n2

INPUT1

Counter

1

Reset

n3

Time base

Figure 17.2

(b)

Block diagrams of digital frequency meters: (a) without prescaler; (b) with prescaler.


799

limiting amplifier, so that the resulting output resembles a square wave,8 regardless of the input waveform.9 Indicating by fx the frequency of the signal to be measured, a square-wave logic signal with the same frequency fx is present at the amplifier output (node n1 in Figure 17.2a). At the other AND gate10 input (n3 ) a square-wave logic signal11 with a stable and known frequency fB is applied, sometimes called the time base. The AND output differs from zero only when both its inputs are high. Thus when the time-base signal is high, i.e. during TCOUNT ¼ 1=ð2 fB Þ, a number of pulses NPULSES will have gone through the gate, given by NPULSES ¼ fx TCOUNT ¼

1 fx 2 fB

ð17:7Þ

The quantity (17.7) is proportional to the frequency fx ; it is counted and displayed at the end of the operation: that is, when the time-base signal goes from high to low. The information displayed is then frozen while the time-base signal stays low. The next low-to-high transition resets the counter and allows for a new measurement. From Equation (17.7) it follows that if fB is 0.5, 5, 50 Hz, and so on, the pulse number counted each cycle is respectively the signal frequency in units, tenths, hundredths, and so on, of hertz. Now, the counter can only allow for an integer number of pulses, therefore two frequencies fx1 , fx2 will be distinguishable only if the pulses counted differ for at least unity, i.e. if j fx1 fx2 j > 1 ) j fx1 fx2 j ¼ Dfmin ¼ 2 fB 2 fB

ð17:8Þ

Therefore the frequency resolution is equal to twice the time-base frequency; this implies in turn that precise measurements require long measure times. The digital frequency meter in Figure 17.2a has a maximum input frequency in the hundreds of megahertz, a limit that tends to be pushed back by advancements in technology. For higher frequencies it is necessary to convert them, bringing them within the operational limit of the digital meter. This can be done by the use of two basic techniques: frequency division and frequency conversion.

17.3.2

Microwave digital frequency meter

Figure 17.2b shows an application of the frequency division technique. The scheme in Figure 17.2b is derived from that in Figure 17.2a by adding one SPDT and one high-frequency divider, usually called a prescaler. If a signal is fed to INPUT1 and the switch is set to position 1, the circuit works exactly as the one in Figure 17.2a. If the signal is applied to INPUT2 and the SPDT is in position 2, the digital frequency meter will measure a square wave whose frequency is fxp ¼

fx P

ð17:9Þ

Again, fx is the frequency to be measured, i.e. the frequency of the signal applied to input 2, and P is the prescaler division factor, which is integer, normally an integer power of 2 ðP ¼ 2; 4; 8 . . .Þ. If the input

8 Clearly, the high and low levels of the square wave must be compatible with the digital logic used in the frequency meter circuitry (TTL, CMOS, ECL, etc.). 9 Sometimes, the squaring operation is performed by a specific mixed analogue–digital component, known as a Schmitt trigger. 10 The AND gate is a digital component with two inputs. The output is at the high logic state when both the inputs are high, otherwise it is zero. 11 Usually a highly stable crystal resonator oscillator generates the time-base signal.

800


signal is within the dynamic range12 of the prescaler, the output will be a square wave with logic values compatible with the rest of the digital circuitry of the frequency meter. The instrument in Figure 17.2b allows for a frequency extension by a factor of P. As for the frequency resolution, note that every cycle on n1 corresponds to P cycles of the input signal. Consequently, the frequency resolution originally given by Equation (17.8) is also multiplied by P. Therefore, holding the time base constant, the maximum measurable frequency increases with P, but at the same time the frequency resolution is worsened by the same factor. The maximum input frequency in modern prescalers is in the gigahertz range, and technological development promises to push this limit upwards.

17.3.3

Frequency conversion frequency meters

A more complicated as well as better performing13 technique is that of frequency conversion. Figure 17.3a depicts a block diagram of a microwave frequency meter. Such an instrument consists of a conventional digital frequency meter, a frequency mixer with a tunable local oscillator, and a digital processing unit. The counter maximum measurable frequency fx;max is in the hundreds of megahertz range; if necessary this frequency can be reached by using a prescaler14 as in Figure 17.2b. The microwave signal is applied to the mixer RF port, while the LO is fed by a signal whose frequency is controlled by the processor. The lowdpass filter LPF, having a cut-off frequency slightly higher than fx;max , filters the mixer IF signal. Then the saturating amplifier AMP produces an output signal similar to a square wave, with high and low levels compatible with the logic gate AND. The LO signal is generated starting from a conversion oscillator, whose frequency fL is very stable and close to fx;max . This signal is then fed to a comb generator, which generates all harmonic frequencies of its input. Its signal can be written as vLO ðtÞ ¼

N X k¼1

vk cosðk 2p fL t þ fk Þ

ð17:10Þ

where the harmonic amplitudes vk have nearly the same value, although decreasing with k. We therefore have a ‘comb’ of spectral lines equally spaced in fLO at the output of the comb generator.15 The tunable filter16 is a passband with a bandwidth much narrower than fLO and a central frequency which is tunable at the different LO harmonics. Once tuned to a certain harmonic, the filter has low attenuation for that frequency and high attenuation for the other harmonics. In this manner, the processor can select which frequency is to be applied to the LO mixer port, hence the reason for the name preselector taken by the filter. The measurement process is composed of two steps: tuning and then measuring. During the tuning phase, the processor tunes the filter to the different harmonics k fL starting from a minimum frequency kmin fL. For every k set, the IF mixer port has a signal whose frequency is fI ¼ fin k fL

ð17:11Þ

where fin is the frequency to be measured. If and only if fI fx;max the digital counter can measure fI and the detector’s output goes high, informing the processor that the search is over. In this case, the processor has all the data to calculate fin using Equation (17.11), knowing k (set by the processor itself) and measuring fI by means of the counter.

12 The range of the working input prescaler input typically falls within the interval 30 to 0 dBm, with a resulting dynamic range of about 30 dB. 13 Particularly at the maximum measurable frequency, which can be higher than 40 GHz. 14 Omitted in Figure 15 The comb generator is also used in the sampling receiver, discussed in Section 15.3.6. 16 Such a filter is typically a yttrium iron garnet (YIG) device. In special cases it could be realized with varactortuned resonators or switched filter banks.


801

Conversion oscillator n6

DISPLAY Comb generator n4

Processor

BPF n5 IN

DET

AMP

n1

L

n2

R

Counter

MIX AND

Reset

n3

Time base

(a)

DISPLAY Conversion oscillator n6

Processor

Comb generator n4 IN

DET

AMP

n1

L

n2

R

MIX

Counter

LPF AND

Reset

n3

Time base

Figure 17.3 preselector.

(b)

Frequency conversion microwave frequency meters: (a) with preselector; (b) without

802


If on the other hand fI > fx;max , neither the counter nor the detector measure anything and the processor then increments k by one unit. The search time can vary from hundreds of milliseconds to hundreds of microseconds, depending on the type of filter used.17 To speed up the search phase, some instruments allow the search field to be narrowed; in this case the search time can be shortened if one approximately knows the frequency to be measured. As explained in Section 13.3,18 the IF signal contains all modulation products jm fR n fL j, for integer m and n. For the case at hand, at the LO port adjacent frequencies to the selected one are present, although attenuated. This implies that, in general, spurious signals are present at the amplifier input and can cause false measurements. To minimize this effect, the amplifier is equipped with an AGC,19 whose role is to keep spurious levels under the value tolerated by the counter. Finally, note that, contrary to frequency division, frequency conversion does not reduce the resolution of the frequency meter.

17.3.4

Frequency conversion frequency meter without preselector

The preselector is one of the most expensive parts of the instrument in Figure 17.3a. Figure 17.3b shows a possible simplified architecture, without a tunable filter. Here the preselector has been removed and the processing unit changes the frequency of the conversion oscillator instead of choosing its harmonic. All harmonic frequencies at the output of the comb generator are present simultaneously at the LO port of the mixer. Often the mixer and comb generator are realized with a single component called a harmonic mixer or sampling mixer. The instrument in Figure 17.3b can be considered as a sampling receiver,20 with a special demodulator, consisting of a frequency counter. The frequency fL of the conversion oscillator is selectable from at least three values fL1 ; fL2 ; fL3 , all around 2 fx;max . Figure 17.4 shows the signal spectrum applied to the LO and three possible input spectra of the signal to be measured. For any value of fL set by the processor, the frequency fin of the signal will always be found between two consecutive harmonics of the LO; let us suppose these to be k1 and k. At IF two main spectral lines are present, fI;LOW ¼ fin ðk1Þ fL , fI;HIGH ¼ k fL fin . As for the position of the frequency to be measured between the two harmonics, three cases are possible: (a) The signal frequency is near to the ðk1Þth harmonic, the frequency fI;LOW is sufficiently below fx;max and can therefore be measured by the counter, whereas fIF;HIGH is high enough not to influence the measure. (b) The signal frequency is midway between the kth and (k 1)th harmonic. Both fI;LOW and fIF;HIGH are close to fx;max and therefore influence the measure. Now, the counter does not provide a stable value when two signals with similar frequency and amplitude are present at its input. (c) Specially to (a), the signal frequency is near to the kth harmonic, the frequency fIF;HIGH is sufficiently below fx;max and can therefore be measured by the counter, whereas fI;LOW is high enough for fx;max not to influence the measure. In condition (b), the processor receives an unstable signal from the counter and modifies fL until it reaches one of the conditions (a) or (c), which are not distinguishable from the processor. However, whichever one of conditions (a) or (c) occurs, if the frequency measured by the counter is fI then the unknown frequency is 17

The YIG filter is the slowest device to tune, the filter bank is faster. Particularly in Section 13.3.10.1. 19 See Section 15.6.3. 20 See Section 15.4.6. 18

MEASUREMENT INSTRUMENTATION AND TECHNIQUES (k–1) f LO

k fLO fin,a fin,b

803

(k+1) fLO

fin,c

f

Figure 17.4 preselector.

Spectrum of the mixer input signals in a frequency conversion frequency meter without fin ¼

k fL þ fI ¼ ðk þ 1Þ fL fI

11 kþ fL fI 2

ð17:12Þ

The input frequency is known once the sign and the k index in Equation (17.12) have been determined. This can be obtained by using a new LO frequency f 0 L slightly shifted from fL . The new f 0 L reading differs from the old one by the quantity 11 f 0 I fI ¼ k þ ð17:13Þ ð fL f 0 L Þ 2 where all terms fI , f 0 I , fL and f 0 L are known to the processor: the first two are measured, the latter two are set by the processor itself. Only one of the two relations (17.13) is satisfied and determines the valid sign to use. The same sign also applies to relations (17.12) and selects the valid one. With two successive readings it is therefore possible to determine k and the sign in Equations (17.12) and (17.13). Thus, two valid readings – none of them in condition (b) – of different frequencies of the conversion oscillator give the frequency measure. In the most general case, three different fL frequencies are needed in order for two of them to be valid. The method described so far, based on Equations (17.12) and (17.13), needs fin to be stable over a long enough time to carry out the three readings; the frequency meter in Figure 17.3b therefore has a longer acquisition time than that in Figure 17.3a and needs a more stable input frequency.

17.4 Spectrum analyzers The spectrum analyzer (SA), as the name suggests, gives a visual representation of the spectral content of the input signal. Nearly all microwave spectrum analyzers give amplitude but not phase information.

17.4.1

Panoramic receiver

The simplest microwave spectrum analyzer is the so-called panoramic receiver, whose block diagram is depicted in Figure 17.5. The input signal is scanned by a tunable bandpass filter BPF, controlled by a sawtooth signal which simultaneously moves the electronic beam of a cathode ray tube (CRT) along the x axis. The beam is moved along the y axis by a voltage proportional to the amplitude of the input signal, in linear or in logarithmic units. In the second case a logarithmic converter is placed between the detector DET and the vertical amplifier AMPV input, or an SDLVA is used instead of a conventional detector. Figure 17.6 plots the voltage of the ramp generator: it is periodic with period T and grows linearly between t1 and t2 , returning to the initial value between t2 and t3 . The tunable bandpass filter central frequency varies linearly with the applied voltage. The tunable bandpass filter is typically a YIG device for frequency in the range of 2–40 GHz and a varactor-tuned resonator component for frequencies below 1 GHz. YIG filters are current tuned, and present a quite linear relation between drive current and centre frequency. On the contrary, varactor-tuned filters are voltage tuned and their centre frequency is marked nonlinearly depending on the tuning voltage. Therefore, a detailed panoramic receiver block diagram should include either a voltage-to-current converter (YIG) or a linearizer (varactor).

804

MICROWAVE AND RF ENGINEERING BPF

LPF

DET

IN

AMPH

AMPV

Ramp generator

+ v1 –

+ a2v2

+ a1 v1

+ v2 –

CRT

Figure 17.5

Panoramic receiver.

Let us consider a single period of the ramp; the filter control voltage is tt1 vtuning ðtÞ ¼ vstart þ vstop vstart t2 t1

ð17:14Þ

with a corresponding filter centre frequency of tt1 f0 ðtÞ ¼ f0;start þ f0;stop f0;start t2 t1

ð17:15Þ

The filter therefore scans the input signal spectrum in the range ½f0;start ; f0;stop . At the same time, the beam’s abscissa travels from left to right at constant speed. Let Sin ð f Þ be the signal spectrum and H ðvTUNE ; f Þ the frequency response of the tunable filter at the tuning voltage vTUNE . Both of those functions are Fourier transforms of real signals, hence for the property (B.5) it follows that jSin ð f Þj ¼ jSin ðf Þj and jH ðvT ; f Þj ¼ jH ðvT ; f Þj. Assuming that the detector works in the quadratic region, its output voltage at a generic time instant t 2 ½t1 ; t2 is proportional to the power of the signal around the frequency. Precisely, it results in ð þ1 vDET ðtÞ ¼ 2 ð17:16Þ jSin ð f Þj2 jH ½vTUNE ðtÞ; f j2 d f 0

The multiplying factor of 2 takes into account the bilateral frequency nature of the functions Sin ð f Þ and H ðvT ; f Þ.

v

vSTOP

vSTART O

t1

t2

t3

T

Figure 17.6

Scanning ramp.

t

MEASUREMENT INSTRUMENTATION AND TECHNIQUES To a first approximation, we can consider the response of BPF as rectangular, i.e. 8 Dv > > ð v Þ < 1 f f j > 0 TUNE j 2 < H vtuning ; f ¼ > Dv > > :0 j f f0 ðvTUNE Þj 2

805

ð17:17Þ

where Df and f0 ðvTUNE Þ are, respectively, the passband width and the voltage-dependent centre frequency – given by Equation (17.15) – of the filter. The parameter Df ¼ Do=2p is the resolution bandwidth(RBW) of the instrument. Real filters have no rectangular response, thus instrument manufacturers use the 3 dB bandwidth to specify the RBW. Substitution of the response (17.17) into expression (17.16) gives the approximated visualized spectrum f0 ½vTUNE ððtÞ þ D f =2

jSin ð f Þj2 d f

vDET ðtÞ ffi 2

ð17:18Þ

f0 ½vTUNE ðtÞD f =2

Equation (17.18) states that, at the instant t, the detector output voltage is proportional to the power of the signal in the tunable filter band at the same instant. The y axis value of the light point on the screen is therefore proportional to the power of the signal around the frequency on the abscissa. During the interval ½t2 ; t3 the ramp voltage goes back to its initial voltage to start a new cycle. During this time, called the retrace time, theCRT beam is turned off. By acting upon the minimum and maximum ramp voltages vstart ; vstop it is possible to change the start and stop frequencies f0;start ; f0;stop accordingly. The two horizontal (AMPH) and vertical (AMPV) amplifiers, in Figure 17.5 below, have gain and offset such that at t1 (respectively t2 ) the light point is at the left (right) edge of the screen, and the inferior (superior) edge of the screen corresponds to a fixed minimum (maximum) power of the input signal. Such a minimum and maximum could be either linear power units (e.g. W) or logarithmic units (e.g. dBm), depending on the detector type. The panoramic receiver has two major drawbacks: a low-frequency resolution and a limited sensitivity. The former is due to the tunable filter’s instantaneous bandwidth RBW ¼ Df ¼ Do=2p, which is of the order of 50 MHz. If therefore two signal components differ by less than RBW, they will be visualized as a single spectral line with a value that is the sum of the power of each component. Figure 17.7 shows the effect of the finite RBW on the visualization of an equal-amplitude two-tone input spectrum21. The four curves have a different ratio between RBW and the tone spacing. The panoramic receiver has a sensitivity slightly lower than that of the detector, because of the tunable filter’s insertion loss, and ranges from 50 to 60 dBm. This value can be improved by reducing the passband VBW of the postdetector video filter LPF, obtaining 1.5 dB for each halving22 of the bandwidth. Now, it must be considered that the spectrum visualized by the panoramic receiver does not coincide with the square magnitude of the input spectrum, as ideally required. The reason for this difference is due to the limited observation time of each portion of the spectrum: the time-varying filter explores different portions of the frequency range at different times. Moreover, the transient response of any filter produces output RF energy even after the input signal is extinguished, and the duration is inversely proportional to

21

See the Mathcad file 02_Resolution_Bandwidth_Bessel.MCD. Halving the video bandwidth corresponds to halving the pffiffiffitotal noise power transmitted from DET to the input of AMPV. This increments the resulting signal to noise ratio of 2 in linear units and corresponds to increasing the output DC voltage by the same amount. If DET is quadratic, such equivalent voltage increasing can be by increasing pffiffiobtained ffi the input power by the same amount, which corresponds in dB to the number 10 log10 ð 2Þ ffi 1:5051499. 22

806

MICROWAVE AND RF ENGINEERING f2

Amplitude

f1

f2 –f1

(f2 –f1)/40

∆ f= 2 (f2–f1) (f2 –f1)/2

Frequency

Figure 17.7 Visualization of a two-tone spectrum for different values of the RBW.

the filter bandwidth. Hence, the sweep time t2 t1 must be no smaller than the quantity t2 t1 >

17.4.2

f0;stop f0;start minðRBW; VBW Þ

ð17:19Þ

Superheterodyne spectrum analyzer

A better performing instrument is the superheterodyne spectrum analyzer, whose block diagram is shown in Figure 17.8. It can be thought as an extension of the panoramic receiver, in which the scan process is performed by converting a variable frequency to a fixed one. This way, one can allow, at least in theory, for an analysis bandwidth which is both tunable and narrow, as required. The ramp voltage is the same as in Figure 17.6 and has the same analytical expression as Equation (17.14). The central instantaneous tunable

BPRF

MIX

BPIF

AMP

IN

DET

LPF

R L

OSC

Ramp generator

AMPV

+ v1 –

AMPH

+ a2v2

+ a1v1

CRT

Figure 17.8

Superheterodyne spectrum analyzer.

+ v2 –


807

filter frequency is again as in Equation (17.15), whereas the instantaneous frequency of the tunable oscillator is distant from the variable filter one by the IF frequency fI :23 fL ðtÞ ¼ f0 ðtÞ fI

ð17:20Þ

This situation is often rendered by saying that the tunable oscillator ‘tracks’ the tunable filter, also called the preselector. The tunable input RF filter BPRF eliminates the image frequency, so that at each instant t the IF signal is a replica of the input with its centre frequency translated from f0 to fI . As a secondary effect of this frequency conversion, the amplitude of the converted signal is reduced by the conversion loss of the mixer and of BPRF, which also vary with f0 . Moreover, some conversion spurs24 are present, although reducing the input power below a given limit, by means of a suitable attenuator, can limit their amplitude below a specified value. At the output of the IF filter BPIF the signal spectrum is A Sin ½ f f0 ðtÞ þ fI HIF ð f Þ, where HIF ð f Þ is the frequency response of the filter and A is a constant less than unity accounting for the losses in the tunable filter, in the IF filter and in the mixer.25 Approximating the filter response with a rectangle of width RBW around fI at the output of the IF amplifier, we have 8 RBW > >1 j f f0 ðtÞj < < 2 ð17:21Þ G A½ f0 ðtÞ Sin ½ f f0 ðtÞ þ fI RBW > > :0 j f f0 ðtÞj 2 where G is the gain of the IF amplifier AMP, and the frequency-dependent loss of the mixer and preselector is explicitly indicated by the factor A. In the superheterodyne analyzer the detector works over the bandwidth of the IF channel, whereas in the panoramic receiver it has to work over the whole input signal bandwidth. At time t, the detector outputs a voltage proportional to the power of the input signal in an RBW band centred on the analysis frequency f0 ðtÞ ¼ o0 ðtÞ=ð2pÞ. From the video filter LPF onward, the superheterodyne receiver is identical to the panoramic one already described. The frequency conversion technique allows an improvement in both sensitivity and frequency resolution, but at the expense of a higher cost. The minimum measurable signal for a superheterodyne spectrum analyzer is smin;dBm ¼ 114 þ 10 log10 ðNBWMHz Þ þ NF0

ð17:22Þ

where: .

NF0 is the noise figure resulting from cascading all the different blocks from BPRF to AMP.26

.

The thermal noise density of 114 dBm/MHz is for a resistor at 290 K.

.

NBWMHz RBWMHz is the noise bandwidth27 in MHz of the IF channel, which is slightly wider than the RBW.

Typically, NF0 amounts to 20–30 dB, so Equation (17.22) implies that the minimum signal for NBW ¼ 1 MHz is in the range of 94 to 84 dBm. Moreover, from Equation (17.22) it follows that that the sensitivity improves by 3.01 dB for every halving of the IF bandwidth.28 In the superheterodyne 23

See Section 15.4.2. See Section 13.3.10. 25 All three elements are passive components. 26 Section 15.5.3 shows how to calculate the noise figure resulting from a superheterodyne receiver chain. 27 See Section 9.4.5. 28 By comparison, in a panoramic receiver, the sensitivity improves by 1.5 dB for each halving of the video bandwidth. 24

808

MICROWAVE AND RF ENGINEERING 20 RBW = 500 kHz RBW = 100 kHz

Amplitude, dB

0

–20

–40

–60

–80 –50

Figure 17.9

–25

0 Offset frequency, MHz

25

50

Visualization of a single tone input signal with two different RBW values.

analyzer IF and video bandwidths can be set independently, although the default setting is often the same value for both. The RBW of a superheterodyne analyzer is far narrower than that of a panoramic one, and can reach 1 kHz (1 Hz) with the use of an analogue (digital) filter. Figure 17.9 shows two readings of the same sinusoid of 1 mW (0 dBm) at 10 GHz. The analysis bandwidth is SPAN ¼ f0;stop f0;start ¼ 100 MHz. On the x-axis it is indicated as the offset frequency from the centre (10 GHz). The first curve (black) refers to a measurement using RBW ¼ 500 kHz, while the second (grey) is a measure with RBW ¼ 100 kHz. Note here how the noise level of the grey line is reduced by around 6 dB, as predicted by Equation (17.22), and the spectral line appears narrower in the measure with a reduced bandwidth. As in the case of the panoramic analyzer, the minimum sweep time is given by Equation (17.19), but in the superheterodyne analyzer the bandwidth to consider is narrower between BPRF and BPIF. The latter coincides with the RBW and typically dominates, being much narrower than the first. In modern spectrum analyzers the blocks shown in Figure 17.8 are, mostly, digital. Thus: .

The tunable oscillator is not a simple free-running VCO but a PLL-synthesized generator29 whose frequency is tracked to a very stable quartz reference and is changed by a digital control word.

.

The ramp generator outputs digital words such that the synthesized oscillator varies its frequency linearly with time; this way, a staircase discrete waveform approximates the continuous ramp.

.

An analogue-to-digital converter (A/D) is present after the video filter.

.

A computer unit with a synthetic display replaces the horizontal amplifier, the vertical amplifier and the CRT. The staircase generator sends the instant frequency (x) to the computer, while the A/D provides the corresponding detected voltage (y), and the computer then presents the corresponding points on the display.

.

Sometimes a second A/D is present after the IF filter, followed by a digital signal processor (DSP), in order to realize very narrow-band (below 1 Hz) filters.

29

See Section 15.6.2.

MEASUREMENT INSTRUMENTATION AND TECHNIQUES .

809

Modern spectrum analyzer also include IQ demodulators, of the type described in Section 15.3, for sophisticated measurements like the phase spectrum.

Some relatively economic analyzers use subharmonic mixers for the frequency conversion, to reduce the maximum oscillator frequency by a factor equal to the maximum harmonic order used. This factor is around 10, so that an analyzer with 20 GHz maximum input frequency uses a tunable oscillator of 2 GHz maximum frequency. Another advantage of using the harmonic mixer is to allow the oscillator a reduced tuning bandwidth. Let us say, for the sake of clarity, that the oscillator has an output frequency of 1 to 2 GHz: the second harmonic will cover the band 2–4 GHz, the third 3–6 GHz, and so on. Using signal harmonics it is therefore possible to achieve a span of more than a decade, starting from an oscillator that has only an octave tuning bandwidth. The main drawback of using the harmonic mixer is the conversion loss dependence on the harmonic being used: as the input frequency rises, so does the harmonic to be used and the conversion loss of the mixer; the instrument noise figure, and then its sensitivity, get worse for higher frequencies. Another possibility for the analyzer is the zero-span modality, where the ramp generator (or its digital counterpart) is disconnected from the oscillator, whose frequency is set to a fixed value. The start of the ramp is triggered by an external signal or by the detector: the ramp starts after the detected voltage passes a specified limit, with an adjustable delay between the trigger event and the ramp start. The sweep time is settable, too. In zero-span mode, the analyzer works as a narrow-band tunable oscilloscope, allowing visualization of the signal envelope around a certain frequency.

17.5 Wide-band sampling oscilloscopes Figure 17.10a shows the internal structure of a digital sampling oscilloscope. This instrument acquires a certain discrete number of samples of the input signal, converts them into digital form and outputs them on the screen. The transformation of a time-continuous signal into its time-discrete counterpart is called sampling, which is uniform if the time interval between samples stays constant. The equivalent process on the y axis, consisting of taking discrete amplitude values from continuously varying ones, is called quantization. Section 15.4.6 explores the processes of sampling and quantization. Nevertheless, some additional considerations need to be developed, in order to focus the effects of these two processes on waveform visualization. The sampling operation is accomplished through two elements: a clock oscillator – which provides the timing signal – and a sample and hold (S/H). Let sIN ðtÞ and fCLK be the input signal and the clock frequency respectively. The time interval between two consecutive samples is then dt ¼ 1= fCLK . The result is that the S/H output signal (node n2 ) is a staircase approximation of sIN ðtÞ. Figure 17.11 shows the resulting waveform at the output of S/H. Then, the A/D transforms the staircase waveform into a numerical sequence. From the A/D point of view, the S/H output signal results as a sequence of samples spaced by dt on the time axis and constant along the interval between two adjacent samples. Let sk ¼ sIN ðk dtÞ be the sample at the kth sampling interval; then, the A/D has a time slightly smaller than30dt to convert sk into a corresponding N-bit31 digital word DWk . After the conversion, DWk is stored in the memory before a new conversion starts and a new word DWk þ 1 arrives. After the acquisition of a fixed sample number NSAMPLE , the digital processor reads those NSAMPLE couples of numbers ðk; DWk Þ and sends them to the display as ðx; yÞ coordinates of the signal representation.

30 The effective time is dt decreased by the time needed for SW to charge the S/H capacitance, and by a margin to take before the next sample arrives. 31 Typically N ranges between 8 and 12. Moreover, higher sampling frequencies correspond to a lower number of bits and then to a lower amplitude resolution.

810


DISPLAY

Processor S/H LPF s I N (t)

SW

n2

A/D

Memory

C

(a)

CLK

DISPLAY

Processor S/H LPF IN

SW

n2

R L

A/D

Memory

C

Comb generator Sampling Mixer SQ OSC

Figure 17.10

(b)

Digital sampling scope: (a) direct sampling; (b) subsampling.

The stored and displayed waveform consists of a discrete number of points, each with a discrete amplitude. The quantization error depends on the number of bits N of the A/D. The ratio between the maximum and minimum signal is 2N , so the dynamic range in dB is therefore DynamicADC ¼ 20 log10 2N ¼ 20 log10 ð2Þ N ffi 6:02 N

ð17:23Þ

The quantity (17.23) is directly proportional to the bit number. The effect of the A/D conversion can be equated to adding the quantization noise to the input signal.

Input, sampled, and linearly interpolated signals


s4

s6

s12

s5

s1

s11

s7

s2

s8

s3 0

811

s13

s14

s9

s10 1

2

t /T

Figure 17.11 Sampling scope waveforms (two periods). Input signal (thick black curve), S/H output staircase (thin black line) and linear interpolation between samples (grey line).

The effect of sampling depends on signal bandwidth. The sampling theorem32 states that a timecontinuous signal can be reconstructed starting from its sampled version, if the sampling frequency is greater than twice the unilateral signal bandwidth. Indicating the input signal spectrum as the Fourier transform of the input time domain signal, þð1

SIN ð f Þ ¼ F ½sIN ðtÞ ¼

sIN ðtÞexpðj2pftÞ dt 1

If SIN ð f Þ is band limited, with unilateral bandwidth BW, which means that Sin ðj f j > BW Þ ¼ 0, and providing that fCLK > 2 BW, then it subsists the identity sin ðtÞ ¼

þ1 X

þ1 X sin½pð fCLK tkÞ sin ts;k sk sinc½pð fCLK tkÞ ¼ pð fCLK tkÞ k¼1 k¼1

ð17:24Þ

Equation (17.24) tells us that the original waveform can be reconstructed by starting from its samples and using them as the weight of an infinite sum of sinc½pð fCLK tkÞ ¼ sin½pð fCLK tkÞ=½pð fCLK tkÞ interpolating functions, shifted in time. These functions are one at the time instant corresponding to their sampling and zero at all other sampling times: sk t ¼ ts;k sk sinc½pð fCLK tkÞ ¼ 0 t ¼ ts;j 8j 6¼ k Equation (17.24) is rigorously valid only for band-limited signals and implies an infinite sum of terms. None of these conditions are realizable in the real world. On the one hand, an infinite sum of non-zero terms implies an infinitely long time signal;33 on the other hand, truncating the sum (17.24) implies 32 A demonstration of the theorem, also known as the Whittaker, Nyquist, Shannon or Kotelnikov theorem, can be found in [6]. 33 This means that there is no finite time interval ðtstart ; vstop Þ, such that t 2 = ðtstart ; vstop Þ implies that the signal is identically zero sin ðtÞ 0.

812


considering a time-limited signal and therefore one with infinite bandwidth. The LPF limits the effective bandwidth at the input of S/H and then brings the operating conditions of the sampling oscilloscope close to those dictated by the sampling theorem. This way, the scope displays sin without its higher frequency components or, if the signal is periodic, without its higher order harmonics. Limiting the bandwidth of the signal would require an ideal LPF with cut-off frequency just below half of the sampling frequency. Realworld filters, however, have finite selectivity and phase nonlinearity in the passband. Thus in practice the cut-off frequency of the filter must be lower than the theoretical value fCLK =2; moreover, phase distortion in the passband reduces the scope’s useful band even more. The maximum allowed cut-off frequency for the LPF depends on the available filter technology. To give an idea of this, an ideal loss-free seventh-order Chebyshev filter with a passband ripple of 0.5 dB presents 40 dB of attenuation at a frequency which is 1.44 times the cut-off, as Figure 7.5 shows. Consequently, the maximum usable input frequency for a sampling oscilloscope is about fCLK =3. Typical numbers could be: sampling frequency34 fCLK ¼ 4 109 samples=s, input lowdpass cut-off frequency fT ¼ 1:3 GHz and a maximum input signal frequency fMAX ¼ 1 GHz. Figure 17.11 plots two periods of the periodic signal 1 2 3 sin ðtÞ ¼ A1 sin 2p t þ A2 sin 2p t þ j2 þ A3 sin 2p t þ j3 ð17:25Þ T T T

Input signal and interpolating sinc functions

The signal (17.25) is band limited by definition, with a maximum frequency of 3=T. Figure 17.11 also shows the first 14 samples of sIN ðtÞ corresponding to a clock frequency fCLK ¼ 7:2=T > 2 3=T. The grey line staircase waveform represents the S/H output as mentioned before. The thin black line joins the samples through a piecewise-linear function, which is seen to approximate the target function quite poorly, unless one takes many more samples. Such a solution is, however, impractical because it implies using a higher sample frequency, and then demands more performance from the S/H and A/D blocks, and larger memory to accommodate the generated samples. A more efficient solution is to apply Equation (17.24) from the sampling theorem, substituting the (inferior) superior sum limit, i.e. (1) þ 1 with (0) NSAMPLE . Figure 17.12 shows a single period of sin ðtÞ (thick black curve), seven samples of it (s1 to s7 ) and the

s4 s5

s4 sinc[ π fCLK(t – t s,4)]

s6 s7

s1 s2 sin (t)

s3

0

1

t/T

Figure 17.12

34

used.

Interpolation of a sampled waveform (one period).

The unit of sampling frequency is samples/second; more frequently its multiples, k (103), M (106), G (109) are


813

piecewise-linear interpolation (grey line), the seven terms of the sum (17.24) corresponding to the seven sample values. The term in the sum (17.24) corresponding to the fourth term is slightly thicker than the others and is labelled with its expression to highlight it. The sampling frequency of digital oscilloscopes can reach values as high as 10 Gsamples/s, with a trace memory depth of 100 Mbytes. The maximum useful frequency for the input signal is around 4 GHz; since up to 108 samples, spaced at 0.1 ns each, can be saved, it follows that up to 10 ms of signal can be acquired. For a given sampling frequency, the maximum input signal frequency can be made higher by subsampling. Figure 17.10b shows a block diagram of a subsampling digital scope. The working principle is similar to that of the stroboscope: the periodic input signal is sampled with a frequency slightly lower than an integer multiple of the fundamental frequency. The arrangement in Figure 17.10b is an extension of that in Figure 17.10a, with the addition of a mixer (MIX) and harmonic generator (OSC with the comb generator). The sinusoidal signal produced by OSC is fed to a hard limiter basically consisting of a saturated amplifier. The output of such a device approximates a square wave of frequency fCLK which works as the timing signal for S/H and A/D, similar to the conventional sampling oscilloscope. Furthermore, the oscillator output also feeds the mixer LO port through a harmonic generator. As discussed in Sections 15.4.6, 17.3.4 and 17.4.2, the combination of a mixer with a harmonic generator feeding the LO is often referred to as a harmonic mixer. With respect to the cases considered in the above-mentioned sections, the harmonic generator used in the subsampling scope must satisfy two additional requirements: the harmonics must have nearly constant amplitude and a precise phase relation. The first requirement is also useful for minimizing sensitivity variations in band for the frequency meter and spectrum analyzer as well; the second is not essential for the former, but it is mandatory for the subsampling scope. It is known35 that an infinite sum of constant amplitude and phase harmonics corresponds to a Dirac pulse train of the same fundamental frequency: 1 1 X fCLK X m 1 vH ðtÞ ¼ cosð2pk fCLK tÞ ¼ d t ð17:26Þ 2 f 2 CLK m¼0 k¼1 A real harmonic generator outputs a finite number of harmonics, with decreasing amplitudes and not perfectly in phase with each other. Thus, the signal is a train of finite width pulses, not perfectly symmetric. As an example, Figure 17.13 shows four cases of the signal (17.26) with 10 and 20 terms and different additional phase shifts. The thick curves (a, grey; b, black) are expression (17.26) truncated at the first 10 and 20 terms, respectively. The thin black curves (c, solid; d dashed) both have the first 20 terms of the sum (17.26), but also include additional small and large perturbation phases at the argument of each term of the sum. It is possible to see that the main lobe of the signal gets narrower as the number of harmonics increases, following Pulse_Width ðNumber_O f _ Harmonics fCLK Þ1

ð17:27Þ

An ideal sampling mixer includes a harmonic generator that produces harmonics having constant amplitude and no reciprocal phase shift. In the ideal case, the IF signal is the product of the RF input and a sampling signal, which has finite amplitude at sample times ts;k ¼ k= fCLK and is zero elsewhere. The resulting IF output is proportional to the input at sample times and zero elsewhere. We will consider nonidealities in the components subsequently.

35

From Equations (15.94), (15.95), (15.98) and (B.13), by setting v1 ðtÞ ¼ 1; fs ¼ fCLK , we get " # þ1 þ1 1 X X X dðtk=fCLK Þ and V2 ð f Þ ¼ F ½v2 ðtÞ ¼ F dðtk=fCLK Þ ¼ fCLK dð f k fCLK Þ v2 ðtÞ ¼ k¼1

k¼1

k¼1

814


(a) 10 harmonics, in phase (b) 20 harmonics, in phase (c) 20 harmonics, slightly out of phase (d) 20 harmonics, totally out of phase

20

vh (t)

10

0

–10

–20 0.0

0 .5

1 .0

fCLK t – 0.5

Figure 17.13

Possible output waveforms of a harmonic generator.

Consider a periodic signal sIN ðtÞ at the oscilloscope input, having period T0 ¼ 1=f0 ¼ 2p=o0 , with sIN ðtÞ ¼

Nin X k¼1

Ak cos ðko0 t þ jk Þ

ð17:28Þ

The waveform (17.28) can be reconstructed starting from samples taken at a rate which is slightly less than an integer multiple of the fundamental frequency. The sampling frequency is therefore fCLK ¼

1 o0 1 ¼ TCLK 2p M þ e

ð17:29Þ

where M is a positive integer and e 1. In order to set the oscillator to the value in (17.29), it is necessary for the processor to know the fundamental frequency of the input, which can be either measured with a frequency meter or input manually.36 Once the right value has been set, reconstruction takes the form shown in Figure 17.14. Here M ¼ 1 has been chosen for the sake of clarity: any other positive value for M would have produced the same results. The signal samples are marked with progressive numbers. If we put the origin of the t axis at the first sample, then the second sampling occurs at t2 ¼ ðM þ eÞT0 . Since the signal has period T0 , the second sample is s2 ¼ sIN ðt2 Þ ¼ sIN ½ðM þ eÞT0 ¼ sIN ðeT0 Þ. In the same way the third sample is taken at t3 ¼ 2ðM þ eÞT0 and is s3 ¼ sIN ð2eT0 Þ. Generalizing, the kth sample is taken at tk ¼ ðk1ÞðM þ eÞT0 and is sk ¼ sIN ½ðk1ÞeT0 . The sample sequence at the mixer output therefore reproduces the input signal, but with an expanded time scale whose expansion factor is known and can be accounted for by the processor when displayed. The bottom curve in Figure 17.14 shows a waveform period obtained from 16 samples. 1 ¼ ðM þ eÞ f0 1 ¼ ðM þ eÞT0 seconds from the next, which is a value Each sample is spaced fCLK very close to M times the period of the input signal, T0 . Thus, increasing M (i.e. reducing the subsampling frequency) reduces the frequency of the samples coming to S/H and ADC. On the other hand, increasing M implies increasing the number of input signal periods necessary to reconstruct a 36 In particular, the input frequency is known if a computer-controlled generator produces the input signal. This happens when testing the response of a device to a known periodic signal.


815

T S3 S3

S5 S6

S2

S7

S1

sIN(t) S8

S9

S10

S2 S21

S10

S1

reconstructed waveform (expanded)

t

Figure 17.14

Waveforms of a subsampling oscilloscope.

single period on the scope display. In the same way, when e approaches zero the samples on the reconstructed waveform become more and more dense, again at the expense of an increased acquisition time. The number of samples necessary to reconstruct a period of the input signal is 1/e, while the acquisition time is Tacquisition ¼

Mþe M T0 ffi T0 e e

ð17:30Þ

The output of the sampling mixer is a train of narrow pulses whose area is proportional to the amplitude of the input signal. This pulse train is lowdpass filtered and fed to the S/H and A/D stages. These two blocks are operated with the same clock driving the sampling mixer. The signal fed to the LPF is a sequence of 1 pulses with a width much smaller than T0 and reciprocally spaced by fCLK ¼ ðM þ eÞ T0. The filter eliminates the components whose frequency is above fCLK =2, outputting a signal similar to the input, time stretched, and with an amplitude reduced by the ratio between width PW and the period of the sampling pulse TCLK PW 1 1 ðM þ eÞ T0 Mþe

ð17:31Þ

The amplitude of such a signal can be amplified, while the choice of pulse width and subsampling frequency is a trade-off among maximum input signal frequency, maximum A/D conversion frequency, instrument sensitivity and reconstructed waveform resolution. We have supposed, so far, the sampling mixer to be ideal. However, every physical component will necessarily deviate from ideality. The first non-ideality to consider is the frequency-dependent mixer conversion loss. The input signal (17.28) is transformed at the mixer’s IF port by Nin X o0 jk sIF ðtÞ ¼ ak Ak cos k tþ ð17:32Þ Mþe Mþe k¼1

816


In expression (17.32), the frequencies are reduced by a factor ðM þ eÞ, and the amplitude is multiplied by a factor ak due to the mixer’s conversion loss, which varies with frequency. The distortion introduced by this mechanism can be corrected by the following procedure: 1. The processor knows the fundamental input frequency and therefore its harmonics, as well as the mixer’s conversion loss as a function of frequency. 2. The mixer output is sampled, converted into digital form and stored in memory. Starting from a signal period, the processor calculates its Fourier transform and therefore the amplitude ak Ak of the signal harmonic frequencies. The processor knows the coefficients ak , thus it can extract the values of the original amplitudes Ak ; then the effect of this mixer non-ideality is overcome. A second kind of non-ideality resides in the harmonic generator and has already been described in the present section. From the point of view of the RF/mW periodic signal, the first requirement is for the width of the pulse generated by the harmonic generator to be far smaller than the period of the signal. We want therefore that Pulse_Width ¼ ðNumber_O f _Harmonics fCLK Þ1 f0 1 hence, the number of harmonics has to be much greater than the ratio f0 = fCLK . In this respect, typical numbers could be f0 ¼ 40 GHz, fCLK ¼ 100 MHz, and therefore a number of harmonics well in excess of 400. Now, a great harmonics number implies a wide spread of the LO energy. If we suppose the amplitude of the harmonics to be constant and produced without losses,37 then each one will have a power equal to the Nth fraction of the oscillator power, for N harmonics. That is, with the numbers introduced, every harmonic is reduced by 10log10 ð400Þ ffi 52 with respect to the oscillator’s fundamental one. The mixer conversion loss is therefore of the order of 20–30 dB. It is in any case crucial for the harmonic generator’s output not to be as in Figure 17.13, curve (d). In this case, the IF mixer output, close to the sampling instants, would present two pulses of opposite amplitude that are very closely spaced, which would then be cancelled by the LPF at S/H input, thus producing a constant signal of very small amplitude. Instruments adopting the working principle of the one in Figure 17.10b have been produced with an input maximum frequency of 40 GHz and 1 picosecond time resolution.

17.6 Network analyzers The network analyzer (NWA) is probably the most used microwave measurement instrument. It consists of the combination of a variable-frequency generator, which applies a sinusoidal stimulus to the DUT input, and one or more receivers that measure the response at the output ports of the DUT. The NWA measurement assumes that the DUT is linear: under this condition it returns the scattering parameters of the device. If the DUT exhibits nonlinear behaviour at the test power used, such as harmonics and/or power compression, then the NWA measurement gives the wrong results. A first NWA classification is based of the kind of measurement performed by the instrument. The term scalar network analyzer (SNWA) refers to an instrument capable of measuring only the magnitude, vector network analyzer (VNWA) if it can also provide phase information. A special kind of network analyzer is the noise figure meter (NFM), which measure noise density produced by a device and is used to measure the gain and noise figure of two-port devices.

37 This assumption is quite optimistic, in that harmonic generators convert into various harmonics a total power of about 10% of the input.


17.6.1

817

Scalar analyzers

The simplest SNWA implementation is analogue, and the block diagram in Figure 17.15 shows its principle. The ramp generator outputs a waveform as in Figure 17.6, which moves the light point on the CRT screen, through the horizontal amplifier AMPH, from left to right. At the same time, the frequency of the VCO sweeps linearly between its start and stop values. The detector output voltage, amplified by the vertical amplifier AMPV, governs the vertical position of the CRT light point. Such amplifiers are most often logarithmic, so that the vertical coordinate of the trace is proportional to the detected power expressed in dBm. As seen for the spectrum analyzer, the cathode ray is turned off during retrace. Usually the ramp generator, horizontal and vertical amplifiers and the CRT are contained in a single block called the amplitude analyzer. Present on the front panel are a connector for the ramp generator output and one for the detector input. The instrument shown in Figure 17.15 can measure the amplitude of transmission |s21| and reflection |s11| coefficients of a two-port DUT. Connecting the DUT output to the detector input as in connection b in Figure 17.15, the detector output, and therefore the trace vertical amplitude, are proportional to the square amplitude of the DUT ðDUT Þ 2 transmission coefficient s21 ð f Þ , ðCOUPÞ 2 ðDUT Þ 2 ð f Þ s21 ð f Þ gDET ð f Þ VDET ð f Þ ¼ PVCO ð f Þ s21

Amplitude analyzer AMPV Ramp generator AMPH + v1 –

+ a1v1 + a2v2 CRT

VCO

1

COUP

2

1

DUT

2

b

3

a b R0

Figure 17.15

Analogue SNWA.

DET

+ v2 –

ð17:33Þ

818


where VDET ð f Þ is the DC output voltage of the detector DET, assumed to work in the quadratic region, ðCOUPÞ 2 PVCO ð f Þ is the VCO output power, s21 ð f Þ is the transmission coefficient square amplitude of the directional coupler COUP direct path, and gDET ð f Þ is the detector sensitivity. The voltage (17.33) is the square magnitude of the DUT transmission coefficient multiplied by a frequency-dependent constant ðCOUPÞ 2 ð f Þ gDET ð f Þ ð17:34Þ K21 ð f Þ ¼ PVCO ð f Þ s21 If generator power, detector characteristic and transmission loss of the coupler are constant over the ðCOUPÞ 2 frequency – or K21 ð f Þ is at least constant – then the detected voltage is proportional to s21 ð f Þ . In reality, however, the mentioned parameters are not constant and there is therefore a frequency-dependent factor between the detected voltage and transmission coefficient ðDUT Þ 2 VDET ð f Þ ¼ K21 ð f Þs21 ð f Þ ð17:35Þ The factor (17.34) can be measured if a device with known transmission coefficient is available. The simplest way to measure K21 ð f Þ is to connect the detector directly to port 2 of the directional coupler. This ðDUT Þ 2 is equivalent to measuring a DUT with s21 ð f Þ ¼ 1; the resulting trace amplitude corresponds to 0 dB attenuation. Since kS21 is frequency dependent, the curve is not a horizontal line; moreover, since the instrument is analogue, there is no means of storing the trace. The procedure followed in this case38 consists of marking the trace on the screen, and then reading DUT gain or attenuation as the distance between the displayed and previously marked curve. This operation is called normalization and sometimes – erroneously – calibration. The reflection coefficient measurement is similar to what was discussed on the transmission coefficient. In this case, the test set is as in configuration a in Figure 17.15. Port 2 of the DUT is terminated on a matched load and DET is connected to the coupled port (3) of COUP. The detected voltage is proportional to the DUT reflected power at its port 1, and thus to its reflection coefficient square ðDUT Þ 2 amplitude s11 ð f Þ , ðDUT Þ 2 VDET ð f Þ ¼ K11 ð f Þs11 ð f Þ

ð17:36Þ

ðCOUPÞ 2 ð f Þ gDET ð f Þ K11 ð f Þ ¼ PVCO ð f Þ s32

ð17:37Þ

with

ðCOUPÞ

where s32 ð f Þ is the coupling coefficient of COUP (from port 2 to port 3). Again, the detected output is proportional to the square of the reflection coefficient, through the frequency-dependent factor (17.37). ðDUT Þ ðDUT Þ can be performed in the same way as previously described for s21 , but The normalization of s11 this time using a device whose reflection coefficient is known, like a short or open circuit, to be connected to the coupler’s port 2 instead of the DUT. ðDUT Þ 2 Swapping the ports allows the inverse transmission coefficient s12 ð f Þ and output reflection ðDUT Þ 2 coefficient s22 ð f Þ to be measured. Similarly, the measurement of the N 2 coefficients ðDUT Þ 2 shk ð f Þ ðh; k ¼ 1 . . . N Þ of N-port devices is possible – one at a time. The transmission coefficient

38

Analogue NWAs are nowadays completely obsolete.


819

DISPLAY

Processor

DET3

AMP 3 ADC3

DET2

AMP 2 ADC2

1

VCO

COUP2

3

3

2

2

1

DET1 1

DUT

Memory

AMP 1

2

ADC1

COUP 1

Amplitude analyzer

Figure 17.16

Digital SNWA.

ðDUT Þ 2 shk ð f Þ

ðh 6¼ kÞ is measured by applying the stimulus at port k, the detector at port h, and ðDUT Þ 2 terminating the remaining N2 ports on a matched load. The reflection coefficient skk ð f Þ is measured by applying the stimulus at port k, the detector at the coupled port of the directional coupler, and terminating the remaining N1 ports on a matched load. A first extension of the instrument described is in the use of a double channel amplitude analyzer, i.e. one with two electron beams, two vertical deflecting plates and two vertical amplifiers. This way, it is possible simultaneously to visualize traces of different measurements: transmission and reflection coefficients at the same time, for instance, or two transmission coefficients of a three-port device. As seen for the superheterodyne spectrum analyzer, the analogue instrument in Figure 17.15 has been replaced nearly everywhere nowadays by its digital version, having the structure in Figure 17.16. One digitally controlled oscillator – typically PLL synthesized – replaces the VCO, and a command sequence – sent by the processor to the generator through a digital interface – replaces the ramp generator. This produces a frequency sweep with discrete steps rather than a continuous one. The detected voltages are digitally stored in memory for successive display after eventually having been elaborated by the processor. Usually the digital amplitude analyzer possesses at least three channels, connected to three different detectors. Each channel consists of a detector, postdetector DC amplifier (typically logarithmic) and ADC. The three channels in the configuration of Figure 17.16 comprise detector, amplifier and ADC with the same corresponding index. In subsequent considerations, the measurement channels will be denoted with the name of their respective detector. The channel output is a digital word proportional to the RF power (typically in dB units) at the detector input. The three channels are normally used39 to measure the generator’s output power (DET3), reflection (DET2) and transmission coefficient (DET1). 39 This employment is the most standard one. However, the three detector channels can be used independently to display any arbitrary combination of their three measured parameters.

820


Differently from the analogue instrument, the digital one can store and mathematically manipulate previously measured traces. In the standard configuration in Figure 17.16, two traces are presented on the display: DET1/DET3 and DET2/DET3. These ratios are independent of the generator’s output power, so that the measure is insensitive to its power drift. The storage and manipulation capabilities allow for an easier normalization procedure with respect to the analogue instrument: .

DET1 is directly connected to port 1 of COUP1. This is equivalent to measuring a device with ðDUT Þ 2 s21 ð f Þ ¼ 1. The output of ADC1, [K21 ð f Þ], is stored. The displayed curve is the ratio of ADC1, with DUT inserted, to the stored trace, which is seen from Equation (17.35) to be ðDUT Þ 2 s21 ð f Þ . If the channels have logarithmic amplifiers, the displayed curve is the difference –

.

.

rather than the ratio – between the measured and the stored values. A short or open circuit [GðDUT Þ ¼ 1] is connected to port 2 of COUP1 and the ADC2 output [K11 ð f Þ] is stored. The displayed curve is the ratio (or the difference in the logarithmic channel case) of the ADC2 output word, with the DUT inserted, to the stored trace, which equals ðDUT Þ 2 s11 ð f Þ , as in Equation (17.36). Passing from the square magnitude to the linear magnitude (if needed) requires a simple arithmetic operation in the processor. If the channels are logarithmic, the elaboration is even simpler, and only impacts on the multiplying factor, which must be halved.40

The main advantage of the digital versus the analogue instrument is that the ratio between the actual and stored measure is displayed, a constant attenuation or gain resulting in a perfectly horizontal line. Both the analogue and digital scalar analyzers suffer the following measurement limitations: 1. Dynamic range. The DC voltage of the detector is proportional to the RF power at its input and therefore to the square of RF input voltage. Thus 10 dB of variation in the latter causes a 20 dB increase in the detected voltage. For this range expansion effect, the detected voltage diminishes quite rapidly as the RF amplitude becomes smaller. On the other hand, detectors exhibit a quadratic characteristic up to a maximum input power41 of 10 mW ( þ 10 dBm). Consequently, the SNWA dynamic range rarely exceeds 60 dB. ðDUT Þ 2. Mismatch error. The insertion loss measured with the DUT connected coincides with s21 ð f Þ if and only if both generator output and detector input are exactly impedance matched. In practice this is only approximately true. As an example, if both generator and detector have a 20 dB return loss (corresponding to a reflection coefficient linear amplitude of r ¼ 0:1),from Equation (9.3) it ðDUT Þ follows that the peak-to-peak error on s21 ð f Þ is approximately maximum 2 2 20log10 1 þ 0:1 = 10:1 ffi 0:17 dB, or 0:085 dB. 3. Directivity error. Consider again the simplified instrument in Figure 17.15, measuring reflection (path a). If the directivity of COUP were infinite and the DUT were a matched load, the detected voltage would be zero and the screen would display the floor noise. However, since the directivity has a finite value, the detected voltage is non-zero despite the DUT being a matched load. The 40

i h ðDUT Þ ðDUT Þ 2 That is, 20log10 s11 ð f Þ ¼ 10log10 s11 ð f Þ .

41 Such a performance is obtained with specifically designed components. General purpose elements, like the one in Figure 13.6, is quadratic up to an input power of 105 W, corresponding to 20 dBm. Clearly, it is possible to increase the maximum input power by interposing an attenuator between the detector and DUT, which also improves the detector input matching, but at the expense of the sensitivity: the total dynamic range does not change.


821

coupler directivity depends on its realization and on the frequency band; a value of 30 dB is a very good result. The SNWA cannot measure return loss values close to its coupler’s directivity. Because of the mismatch and directivity errors, the zeroing procedure is called normalization instead of calibration. The latter is more properly reserved for the VNWA which, measuring both amplitude and phase of the parameters, can correct mismatch and directivity, at least to some degree.

17.6.2

Vector analyzers

The three measurement channels of the SNWA in Figure 17.16 are three simple broadband receivers. Therefore, their output is affected by the total noise in the whole possible test bandwidth covered by the instrument: this is a limiting factor for the receiver sensitivity, which impacts on the maximum measurable DUT attenuation. Furthermore, the simple structure of the SNWA receivers cannot measure the phase. The VNWA overcomes such limitations by replacing the three detector-based receivers with superheterodyne ones. Compared with the SNWA, the VNWA can measure not just the amplitude, but also the phase of the DUT scattering parameters. Additional advantages of the VNWA are the higher dynamic range, due to the use of superheterodyne instead of simple detector receivers. Furthermore, the vector measurement gives the possibility to the instrument to characterize its own errors and remove them from the measurement. The main drawback of the VNWA is its higher complexity and cost than for the SNWA. Figure 17.17 shows the simplified block diagram of a VNWA. The three receivers – sometimes also referred to as channels – are labelled CH1 to CH3, each of them ðCHk ; k ¼ 1; 2; 3Þ consisting of four cascaded components: one mixer MIXk , one bandpass filter BPk , one amplifier AMPk and one analogueto-digital converter ADCk. Similar to the SNWA in Figure 17.16, the three receivers in Figure 17.17 measure the DUT stimulus power (CH3), reflected power (CH1) and transmitted power (CH2). Two tunable oscillators are necessary to implement such a solution, one to feed the DUT (VCO1 in Figure 17.17); the second oscillator (VCO2) provides the LO signal for the frequency conversion in the three measurement channels, through the three-way power divider42 DIV. Furthermore, VCO1 and VCO2 are phase locked to the same reference, therefore their frequencies are stable, precise, coherent and present a constant difference. The processor controls each oscillator: VCO1 outputs the test frequency fTEST ¼ oTEST =ð2pÞ and VCO2 generates a frequency oTEST oIF having a fixed offset from the first, which coincides with the centre frequency of the bandpass filters BP1 to BP3. The typical value of the offset frequency fI ¼ oI =ð2pÞ is around 10 MHz. The DUT is fed by a system composed of VCO1, three directional couplers (COUP1 to COUP3) and the absorptive SPDT (SW). The VNWA depicted in Figure 17.17 has two RF test ports, P1 and P2, which coincide with port 1 of COUP1 and COUP2, respectively. Connected to the coupled branch (port 3) of both these couplers are three nominally identical channels. The three channels convert the microwave signal to a relatively low frequency, in order to allow the phase measurement and improve amplitude accuracy. CH3 (reference) connected to COUP3 measures the output power of VCO1. CH1 (CH2), connected to COUP1 (COUP2), measures the power reflected from 42 More precisely, the schematic in Figure 17.17 shows a three-way resistive power splitter. It presents higher attenuation than a conventional – and ideally loss-free – Wilkinson divider. Conversely, the resistive splitter is physically smaller and works over a wider bandwidth, from zero to a frequency such that the component size is much smaller than the wavelength. The design equation for the N-way star resistive splitter is Rk ¼ R0 ðn1Þ=ðN þ 1Þ ðk ¼ 1 . . . N þ 1Þ, where N is the number of ways and, as usual, R0 is the normalization resistance. In our case N ¼ 3, thus R1 ¼ R2 ¼ R3 ¼ R4 ¼ 50 2=4 ¼ 25 O. With this value, the splitter is perfectly matched to all of its ports skk ¼ 0 ðk ¼ 1 . . . N þ 1Þ, while the transmission coefficient between any port couple is skk ¼ N 1 ðh; k ¼ 1; . . . ; N þ 1; h 6¼ nÞ. In comparison, the transmission coefficients of a perfectly matched and lossfree divider – assuming port 1 as input – are jsk1 j ¼ N 0:5 ðk ¼ 1 . . . N þ 1Þ.

822


VCO1

DISPLAY

VCO2 1

RI R2 2

R3 3

R4

DIV

4

BP3

MIX3

AMP3

CH3

L

ADC3

R

2

3

COUP 3 1

BP1

MIX1

AMP1

L

2 SW

1 1

Processor

ADC1

R

1

COUP1 2

CH1

COUP2

3

2

BP2

MIX2

CH2

L

3 R

P1

AMP2

ADC2

P2

Memory 1

DUT

2

Figure 17.17 VNWA. port 1 (2) of the DUT. Note that the power reflected from one of the DUT ports is proportional to the transmitted one, if the stimulus is applied to the other port. Setting SW in position 1 (2), the instrument feeds the DUT at port 1 (2) and therefore can measure s11 ; s21 (s22 ; s12 ). Each parameter couple, corresponding to a switch position, can be measured with a single frequency sweep. The instrument is therefore capable of measuring all four parameters with two frequency sweeps (one for each SW position) without requiring a change in the DUT connection. This possibility is peculiar to the VNWA. To describe the VNWA working in more detail, it is convenient to consider SW in position 1. For position 2 the same considerations apply, swapping the ports of the DUTand CH1 with CH2. Furthermore, we will initially assume that the passive components and the stimulus generator are ideal. In particular, the initial assumptions are that the test ports of the instruments are perfectly impedance matched, and COUP1 and COUP2 present infinite directivity, so that the signals at the RF ports of MIX1 and MIX2 are proportional to the reflected and transmitted wave of the DUT. The reference signal measured by CH3 is the stimulus generated by VCO1, attenuated and phase shifted by the coupled branch (ports 1, 3) of COUP3. The signal measured by channel 1 (2) is the reflected wave from port 1 (2) of the DUT, attenuated and phase shifted by the coupling coefficient of COUP1 (COUP2).


823

h i ðDUT Þ ðDUT Þ ðDUT Þ ðh; k ¼ 1; 2Þ be the complex and frequency-dependent Let shk ¼ rhk exp jjhk scattering coefficients of the DUT from port k to port h. The signals at the coupled ports of COUP1 to COUP3 are h i ðDUT Þ ðDUT Þ vRF;1 ðtÞ ¼ v0 g1 r11 cos oTEST t þ y1 þ f11 h i ðDUT Þ ðDUT Þ ð17:38Þ vRF;2 ðtÞ ¼ v0 g2 r21 cos oTEST t þ y2 þ f21 vRF;3 ðtÞ ¼ v0 g3 cos½oTEST t þ y3 where: .

The amplitude v0 depends on the output power of VCO1.

.

The amplitudes gk and the phases yk ðk ¼ 1; 2; 3Þ depend on the transmission coefficients of the components connecting VCO1 to the RF ports of the three mixers.

.

The above-mentioned quantities v0 , gk and yk are frequency dependent, although not explicity indicated in Equations (17.38).

The analysis of the arrangement in Figure 13.9b shows that the IF signals conserve the amplitude and phase relation of the RF inputs. Applying the same considerations to the VNWA receivers, and neglecting the quantization, the ADC outputs corresponding to the signals (17.38) are h i ðDUT Þ ðDUT Þ vADC;1 ðtÞ ¼ a1 v0 g1 r11 cos oIF t þ y1 þ b1 þ f11 h i ðDUT Þ ðDUT Þ ð17:39Þ vADC;2 ðtÞ ¼ a2 v0 g2 r21 cos oIF t þ y2 þ b2 þ f21 vADC;3 ðtÞ ¼ a3 v0 g3 cos½oIF t þ y3 þ b3 The frequency-dependent amplitude and phase factors ak and bk ðk ¼ 1; 2; 3Þ result from the mixer conversion gain, the filter loss, the amplifier gain and the ADC input/output relation. They depend on the test frequency and on the components of the three channels – which are nominally identical – but not on the DUT, provided that the receivers work linearly. With purely numerical manipulations of the ADC output words (17.39), the VNWA processor can extract the amplitude and phase of the test channels (CH1, CH2) relative to the reference channel (CH3). Figure 17.18 shows the block diagram of the required numerical process. It consists of one part dedicated to the amplitude (nodes labelled from DW1 to DW10) and one dedicated to the phase (DW11 to DW21). The blocks labelled x2 produce an output digital word proportional to the square of the input one: for simplicity, and without any significant loss of generality, the relative proportionality constant will be is the average of the input one over considered as unitary. The output digital word of the blocks labelled x one period 2p=oIF of it. The amplitude processing section also includes two two-input, one-output blocks, whose output x2 =x1 is the ratio between the two inputs x1 ; x2 . Two additional – and optional – nonlinear transfer characteristics f ðxÞ are present. From the above descriptions and from Equations (17.39) it follows that the digital sequences at the nodes DW1 to DW3 are h i h i ðDUT Þ 2 ðDUT Þ DW1 ðtÞ ¼ a1 v0 g1 r11 cos2 oIF t þ y1 þ b1 þ f11 h i h i ðDUT Þ 2 ðDUT Þ cos2 oIF t þ y2 þ b2 þ f21 DW2 ðtÞ ¼ a2 v0 g2 r21

ð17:40Þ

DW3 ðtÞ ¼ ða3 v0 g3 Þ2 cos2 ½oIF t þ y3 þ b3 During each elaboration, the test frequency is held constant, therefore the phase shifts present inside the cosine arguments of the functions (17.40) are constant over time as well. The digital words DW4 to DW6

824


ADC3

x

2

DW3

DW6

AMPLITUDE PROCESS

x

x1 ADC1

x

2

DW1

DW7 x2/x1

DW4 x

x1 ADC2

x

2

DW2

x2 .x1

DW9

f(x)

DW10

DW8 x2/x1

DW5 x

x1

f(x)

x2

x2

DW12

DW16 x

x1 arg (x2 + jx1) x2

x2

DW20

DW11 x1

x2 . x1

DW13 x DW17

x2

x1

x2 . x1

DW14

DW18 x

x1

x2

x1

arg (x2 + jx1) x2

x2 . x1

DW21

DW15

x2

x DW19 PHASE PROCESS

Figure 17.18

Block diagram of the VNWA processor.

are the average over one IF period of the words DW1 to DW3, respectively, 2p=o ð IF

DW4 ¼

DW1 ðtÞ dt ¼ 0 2p=o ð IF

DW5 ¼


DW6 ¼

DW3 ðtÞ dt ¼ 0

h i ðDUT Þ 2 a1 v0 g1 r11 2 h i2 ðDUT Þ a2 v0 g2 r21 2 ða3 v0 g3 Þ2 2

ð17:41Þ


825

Clearly, the processor computes a discrete sum of samples, rather than a true integral, and the sampling frequency is higher than 2 fI , in order to fulfil the conditions of the sampling theorem. The digital words DW7 and DW8 immediately follow from functions (17.41) DW4 a1 g1 2 h ðDUT Þ i2 ¼ r11 DW6 a g 3 3 2 h i2 DW5 a2 g2 ðDUT Þ DW8 ¼ ¼ r21 DW6 a3 g3

DW7 ¼

ð17:42Þ

The quantities DW7 and DW8 are then proportional to the square amplitude of the DUT reflection and transmission coefficient, respectively. The relative proportionality constants depend on the VNWA circuitry (directional couplers and receivers) but not on the power of VCO1, as the factor v0 disappears from functions (17.42). Passing from the square amplitude (17.42) to the linear or the logarithmic one involves two relatively straightforward computations, schematized by the blocks f ðxÞ. For elaboration of the phase, the diagram in Figure 17.18 includes three additional types of blocks. The first, labelled p=2, delays the input signal by a quarter of a period 1=ð4 fI Þ, which corresponds to a phase shift of p=2, as the label suggests. Then, there are two multipliers ðx1 x2 Þ whose output is the product of the two inputs. The last type of block is a trigonometric function, which returnsp a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number [argðx1 jx2 Þ] such that its cosine and sine are equal to the inputs x1 and x2 , normalized to x21 þ x22 , respectively. With these assumptions, the digital word at the p=2 phase shifter output is DW11 ðtÞ ¼ a3 v0 g3 sin½oIF t þ y3 þ b3

ð17:43Þ

Then DW12 ðtÞ DW13 ðtÞ DW14 ðtÞ DW15 ðtÞ

¼ vADC;1 ðtÞvADC;3 ðtÞ ¼ vADC;1 ðtÞDW11 ðtÞ ¼ vADC;2 ðtÞvADC;3 ðtÞ ¼ vADC;2 ðtÞDW11 ðtÞ

ð17:44Þ

Expanding the trigonometric products in functions (17.44) and computing the mean value over one period of the resulting sequences, we have 2p=o ð IF

ðDUT Þ

DW12 ðtÞ dt ¼

DW16 ¼

a1 a3 v20 g1 g3 r11 2

0 2p=o ð IF

DW17 ¼

ðDUT Þ


DW18 ¼

a1 a3 v20 g1 g3 r11 2

ðDUT Þ

a2 a3 v20 g2 g3 r21 DW14 ðtÞ dt ¼ 2

0 2p=o ð IF

DW19 ¼

h i ðDUT Þ cos y1 y3 þ b1 b3 þ j11

h i ðDUT Þ cos y2 y3 þ b2 b3 þ j21

ðDUT Þ

DW15 ðtÞ dt ¼

a2 a3 v20 g2 g3 r21 2

h i ðDUT Þ sin y1 y3 þ b1 b3 þ j11 ð17:45Þ

h i ðDUT Þ sin y2 y3 þ b2 b3 þ j21

0

The trigonometric operators then return ðDUT Þ

DW20 ¼ arg½DW16 jDW17 ¼ y1 y3 þ b1 b3 þ f11 ðDUT Þ DW21 ¼ arg½DW18 jDW19 ¼ y2 y3 þ b2 b3 þ f21

ð17:46Þ

The values in (17.46) are the phases of the DUT reflection and transmission coefficients, shifted by quantities due to the instrument’s circuitry.

826


The amplitude process is the digital equivalent of applying three detectors at the three amplifier outputs to measure their amplitudes. More precisely, the block x2 is the digital equivalent of the quadratic behaves like a digital LPF. detector, while x Elaboration of the digital phase is the exact counterpart of the IQ mixer demodulator used for the receivers in Sections 15.4.2 to 15.4.4. In the present case, the signal at the output of AMP3 would work as the LO for the IQ mixer, while the outputs of AMP1 and AMP2 are the signal to be demodulated. The multiplier blocks can be seen as numerical implementations of the product detector, while the p=2 phase works like the hybrid coupler in the analogue IQ mixer. Other purely digital methods are possible, like the one based on the discrete Fourier transform(DFT) of the signals from the three ADCs. The required amplitude and phase of the three channels are simply extracted from the DFT coefficient corresponding to the IF. Regardless of the implemented method, the instrument in Figure 17.17 measures the amplitude and phase at the same time. Removing the initial ideality assumptions on VCO1, COUP1 to COUP3 and SW, we have that ðDUT Þ ðDUT Þ and s21 , due to the non-zero directivity of quantities (17.42) and (17.46) include error terms on s11 COUP1 and the imperfect impedance matching of the test ports, and the non-flat stimulus power over the frequency. Using the amplitude process only, which returns quantities (17.42), the VNWA can work in scalar mode. The measured transmission and reflection coefficient amplitudes are affected by the same mismatch and directivity errors as seen for the SNWA. Also, the normalization procedure is the same as in the digital SNWA. However, the VNWA uses for the amplitude measure a superheterodyne receiver having a far greater sensitivity than the detector, which results in a dynamic range around 100 dB. Apart from the phase measurement capability, the instrument in Figure 17.17 can thus be considered as a high dynamic range SNWA. A further advantage of the instrument in Figure 17.17 with respect to that in Figure 17.16 is the possibility to measure s12 ; s22 by switching SW instead of changing the DUT connections, as already mentioned. Furthermore, the VNWA is by definition capable of measuring the phase of the scattering parameters, since thisinformation isavailable conversion as well. In fact, the phase differences after the frequency ðDUT Þ ðDUT Þ between vADC;1 ; vADC;3 and vADC;2 ; vADC;3 are the sum of f11 and f21 plus some constant 43 terms due to the instrument’s components. However, all these constant terms can be measured during a normalization procedure, similar to the ðDUT Þ

ðDUT Þ

one described in Section 17.6.1, and subtracted from the measure in order to obtain f11 and f21 . The VNWA normalization is similar to the one for the SNWA. Both require two standards with known ðSTDÞ

reflection and transmission coefficients. The simplest choice consists of using an open circuit (s11

¼ 1)

ðSTDÞ for s11 and a through connection (s21

¼ 1) for s21. The only peculiarity of the VNWA normalization is that it works for both the amplitude and phase of the parameters. Therefore, after subtracting the amplitude in dB (phase) obtained with the standard from that obtained with the DUT, the result is the amplitude in dB (phase) of the DUT, apart from the directivity and mismatch error, already introduced in Section 17.6.1. The measured parameter is represented in either polar (amplitude and phase) or Cartesian (real and imaginary parts) form. For each frequency sweep an Np 3 matrix is written into memory, where Np is the number of points – typically of the order of some hundreds – in which the frequency band has been divided. The first column contains the measurement frequencies, the second one the amplitude (or the real part) and the third one the phase (or the imaginary part). Thus, when

43 Here, the word ‘constant’ indicates that they do not depend on the DUT. However, these terms are still variable with the test frequency.


827

normalization is applied, what is displayed is the ratio of two complex numbers – or vectors – measured with and without the DUT. The possibility to measure phase allows for the correction of mismatch and directivity errors through the calibration procedure. Calibration consists of seven steps, usually guided by the processor, whereas the operator is asked to realize the necessary connections: 1. The processor puts SW in position 1 and three reflection calibration standards, whose reflection coefficients are known, are measured. By measuring the first two standards (short and open circuit, both with high reflection) and checking against the nominal values, the transmission in amplitude and phase from P1 to the RF port of ADC1 can be measured. With the third standard (matched load) the instrument measures the isolation of COUP1 in amplitude and phase. This contribution will then be subtracted from the measure, correcting the directivity error. 2. The processor puts SW in position 2 and the same operations as described in step 1 are performed for P2, ADC2 and COUP2. 3. After steps 1 and 2 are performed, the VNWA is calibrated for reflection measurements on both ports. The coupling coefficients of COUP1 and COUP2, together with the respective measurement channels, are accounted for, and the directivity errors are compensated. 4. The processor switches SW back to position 1 and a known44 transmission calibration standard, with low reflection coefficients, is connected between the two NWA test ports. The transmission (amplitude and phase) of the path from VCO1 to the ADC2 output (which includes the standard) is measured and then checked against the transmission coefficient’s nominal value. Under this condition, moreover, the reflection coefficient of P2 is measured (being P1 calibrated for reflection measurements). With this information, a normalization of the measured transmission coefficient with respect to the calibration standard (as in the scalar analyzer) is possible, as is a correction of load mismatch at P2. 5. The processor puts SW to 2 and repeats step 4 for P2. 6. After performing steps 1 to 5, the VNWA measurement takes into account all instrument non-idealities: power flatness of the source, gain/loss of internal components, finite directivity of the couplers and port mismatch. The last two can be corrected only by means of amplitude and phase measurement of the different signals, and are therefore peculiar to the VNWA. 7. An accessory correction, which can be omitted in most cases, is the isolation. The residual transmission coefficient (due to internal crosstalk, typically of the order of 80 dB) is measured when the two ports are terminated on a matched load. The signal obtained is subtracted from the measurement. The resulting noise floor of the instruments improves by some tens of decibels, allowing measurements of DUT attenuations in excess of 100 dB. In more quantitative terms, Equations (17.42) and (17.46), respectively, have been derived from (17.39) by assuming infinite directivity for COUP1, COUP2 and perfect impedance matching for P1 and P2. After considering the above-mentioned non-idealities, Equations (17.42) and (17.46) would include further

44 Typically, the transmission standard is a short piece of impedance-matched coaxial cable. Again, the normalization impedance, and therefore the characteristic impedance of the transmission standard, is 50 O in most cases.

828


sinusoidal error terms. Furthermore, it is possible to derive the equations for DW7, DW8 and DW20 , DW21 , for SW in position 2. The resulting expressions have the same form as (17.42) and (17.46), after swapping the index 1 and 2. In both the switch positions, use of phasors simplifies the measurement error analysis: with this formalism, Equations (17.39) become h i ðDUT Þ vADC;1 ðtÞ ¼ Re A1 v0 G1 s11 expðjoIF tÞ h i ðDUT Þ vADC;2 ðtÞ ¼ Re A2 v0 G2 s21 expðjoIF tÞ

ð17:47Þ

vADC;3 ðtÞ ¼ Re½A3 v0 G3 expðjoIF tÞ with Ak ¼ ak expðjbk Þ, Gk ¼ gk expðjyk Þ and k ¼ 1; 2; 3. As we have seen, the VNWA processor extracts the DUT parameters as the ratio between phasors ðDUT Þ

ðDUT Þ

ðDUT Þ

ðDUT Þ

A1 v0 G1 s11 =ðA3 v0 G3 Þ ¼ A1 G1 s11 =ðA3 G3 Þ and A2 v0 G2 s21 =ðA3 v0 G3 Þ ¼ A2 G2 s21 =ðA3 G3 Þ. More precisely, the quantities (17.42) and (17.46) are the amplitude and phase of the above-written ðDUT Þ

ðDUT Þ

complex numbers. Hence, A3 v0 G3 , A1 v0 G1 s11 and A2 v0 G2 s21 can be considered as the incident wave, the reflected wave and the transmitted wave, respectively, as affected by the various VNWA components. Such quantities, together with the one coming from the non-idealities of the instruments, can be represented by the graphs in Figure 17.19. The figure presents one SFG per position of SW: (1) and (2). Each branch of both the graphs represents either one of the VNWA error terms or a scattering parameter of the DUT. The branch weights are frequency-dependent complex numbers, equal to the corresponding error term or DUT scattering parameter: 1. Forward directivity e11 : produces a non-zero output on b1 , even in the absence of reflections from ðDUT Þ the DUT (s11 ¼ 0). 2. Forward transmission tracking e21 : models the frequency dependence of the stimulus power applied to the DUT input (P1) and the frequency response of the test receiver CH2. The latter should be modelled separately with a specific weight on the branch ending on b2 . Using a single non-unitary weight on one single branch leads to simpler equations and involves no additional approximation. 3. Forward reflection tracking e12 : models the non-flat coupling coefficient of the test coupler COUP1 and the frequency response of the test channel CH1. 4. NForward source matching e22 : represents the imperfect impedance matching of the test port P1. 5. Forward isolation eX : produces an output on b2 even in the absence of transmission in the DUT ðDUT Þ ¼ 0). The physical cause of the isolation is the crosstalk between the components (s21 connected to the two test ports of the instrument. 6. Forward load matching eL : takes into account the finite reflection coefficient P2 to port 2 of the DUT. These six forward error terms refer to the VNWA operation with SW in position 1. There are six other reverse error terms for SW in position 2; they are marked with a prime. The definition of each reverse error term is derived from the one of the corresponding forward error, after swapping the index 1 and 2 in the DUT and test ports.


829

ex

e22

e11

b1

(DUT)

s21

e21

a1

(DUT)

s11

b2

(DUT)

eL

s22,D

(DUT)

s12

e12

(1)

(DUT)

e12 ′

s21

eL′

b1

(DUT)

s11

e22 ′

(DUT)

s22,D

(DUT)

e21′

s12

b2

e11′

a2

(2)

ex′

Figure 17.19 Figure 17.17.

Error model for the VNWA for the two different positions of the switch SW in

ðDUT Þ

ðM Þ

Denoting the DUT scattering parameters as shk , the measurement results shk are ðh; k ¼ 1; 2Þ 8 " # ðDUT Þ > s11 eL > ðM Þ ðDUT Þ ðDUT Þ > þ s s12 > s11 ¼ e11 þ e21 e12 > ðDUT Þ ðDUT Þ 21 > > 1e22 s11 1eL s22 > > > > ðDUT Þ > > ðM Þ s21 > > s21 ¼ eX þ e21 > > e eL 22 ðDUT Þ ðDUT Þ > > 1 s s12 > ðDUT Þ ðDUT Þ 21 > < 1e22 s11 1eL s22 ð17:48Þ ðDUT Þ > s12 > > sð M Þ ¼ e 0 þ e > X 12 > 12 > e0 22 e0 L > ðDUT Þ ðDUT Þ > 1 s s12 > > ð DUT Þ ðDUT Þ 21 > 0 0 > 1e 22 s22 1e L s11 > > " # > > ðDUT Þ > > ðM Þ s22 e0 L > ðDUT Þ ðDUT Þ 0 0 0 > s ¼ e þ e e þ s s > 11 21 12 12 : 22 ðDUT Þ ðDUT Þ 21 1e0 22 s22 1e0 L s11 The transmission and reflection tracking affect the measurements of the DUT reflection coefficients by their products e21 e12 and e0 21 e0 12 , not separately. Steps 1 and 2 consist of determining the reflection error parameters e11 , e22 , e21 e12 and e0 11 , e0 22 , e0 21 e0 12 by measuring – at both the test ports – the reflection calibration standards. The measurement of two reciprocally isolated one-port devices is equivalent to ðDUT Þ ðDUT Þ measuring one two-port without transmission between its two ports, i.e. s21 ¼ s12 ¼ 0. If the DUT has no transmission, there is no interest in the second and third equations in (17.48), while the remaining

830


two simplify to

8 ðDUT Þ > s11 > ðM Þ > ¼ e þ e e s > 11 21 12 11 > ðDUT Þ > < 1e22 s11 > > > > ðM Þ > > s22 ¼ e0 11 þ e0 21 e0 12 :

ðDUT Þ

s22

ð17:49Þ

ðDUT Þ

1e0 22 s22

Let the reflection coefficients of the three calibration standards be GðSTD;pÞ and the corresponding ðM;pÞ ðM;pÞ measurement results at the two test ports be s11 and s22 , respectively, with p ¼ 1; 2; 3. Combining the known standard reflection coefficients with their resulting measurements, from each of the two equations (17.49) a system of three equations with three unknowns follows: 8 GðDUT;1Þ > ðM;1Þ > > > s11 ¼ e11 þ e21 e12 > > 1e22 GðDUT;1Þ > > > > > < GðDUT;2Þ ðM;2Þ ð17:50Þ s11 ¼ e11 þ e21 e12 > 1e22 GðDUT;2Þ > > > > > > > > GðDUT;3Þ > ðM;3Þ > : s11 ¼ e11 þ e21 e12 1e22 GðDUT;3Þ 8 GðDUT;1Þ > ðM;1Þ > > s22 ¼ e0 11 þ e0 21 e0 12 > > > 1e0 22 GðDUT;1Þ > > > > > < GðDUT;2Þ ðM;2Þ ð17:51Þ s22 ¼ e0 11 þ e0 21 e0 12 > 1e0 22 GðDUT;2Þ > > > > > > > > GðDUT;3Þ > > sðM;3Þ ¼ e0 11 þ e0 21 e0 12 : 22 1e0 22 GðDUT;3Þ Ideally, GðSTD;1Þ ffi 1 (open circuit), GðSTD;2Þ ffi 1 (short circuit) and GðSTD;3Þ ffi 0 (matched load). Each of the two systems (17.50) and (17.51) has three equations in the three unknowns e11 , e22 , e21 e12 and e0 11 , e0 22 , e0 21 e0 12 , respectively. The solutions of these systems are 8 NAa þ NBa þ NCa > e ¼ > > > 11 DAa þ DBa > > > > > < NAb NBb e21 e12 ¼ ð17:52Þ > fDAb þ DBb g2 > > > > > > > NAc þ NBc > : e22 ¼ DAb þ DBb and

8 NA0 a þ NB0 a þ NC 0 a 0 > > ¼ > e 11 > DA0 a þ DB0 a > > > > > < NA0 b NB0 b e0 21 e0 12 ¼ > fDA0 b þ DB0 b g2 > > > > > > > NA0 c þ NB0 c > : e0 22 ¼ DA0 b þ DB0 b

ð17:53Þ

MEASUREMENT INSTRUMENTATION AND TECHNIQUES where

831

ðM;1Þ ðM;2Þ ðSTD;1Þ s11 G GðSTD;2Þ GðSTD;3Þ

NAa ¼ s11

ðM;2Þ ðM;3Þ ðSTD;2Þ s11 G GðSTD;3Þ GðSTD;1Þ

NBa ¼ s11

ðM;3Þ ðM;1Þ NCa ¼ s11 s11 GðSTD;3Þ GðSTD;1Þ GðSTD;2Þ h i ðM;1Þ ðM;2Þ ðSTD;3Þ G DAa ¼ s11 s11 GðSTD;2Þ GðSTD;1Þ h i ðM;2Þ ðM;3Þ ðSTD;1Þ G DBa ¼ s11 s11 GðSTD;2Þ GðSTD;3Þ h ih ih i ðM;1Þ ðM;2Þ ðM;2Þ ðM;3Þ ðM;3Þ ðM;1Þ s11 s11 s11 s11 NAb ¼ s11 s11 NBb ¼ GðSTD;1Þ GðSTD;2Þ GðSTD;2Þ GðSTD;3Þ GðSTD;3Þ GðSTD;1Þ h i ðM;1Þ ðM;2Þ ðSTD;3Þ DAb ¼ s11 s11 G GðSTD;2Þ GðSTD;1Þ h i ðM;2Þ ðM;3Þ ðSTD;1Þ G DBb ¼ s11 s11 GðSTD;2Þ GðSTD;3Þ h i ðM;1Þ ðM;3Þ ðSTD;1Þ G NAc ¼ s11 s11 GðSTD;2Þ h i ðM;1Þ ðM;2Þ ðSTD;3Þ G GðSTD;1Þ NBc ¼ s11 s11 The terms of Equations (17.53) are derived from the corresponding ones of Equations (17.52) after ðM;pÞ ðM;pÞ replacing s11 with s22 ðp ¼ 1; 2; 3Þ. The optional calibration step 7 consists of terminating both the test ports with matched loads,45 such ðDUT Þ

ðSTD;4Þ

that s21 ¼ s21 ¼ 0, and measuring the resulting forward and reverse transmission coefficients. From Equations (17.48) they result in ( ðM;4Þ s21 ¼ eX ð17:54Þ ðM;4Þ s12 ¼ e0 X The omission of step 7 is equivalent to assuming eX ¼ e0 X ¼ 0. The final calibration steps 4 and 5 consist of measuring the four scattering parameters of the ðSTD;5Þ

transmission standard. Let shk

ðM;5Þ

and shk

be the scattering parameters of the transmission standard ðM;5Þ

and the corresponding measured values, respectively, with h; k ¼ 1; 2. Substitution of the quantities shk ðSTD;5Þ shk

0

and into the system (17.48) gives a system of four equations with four unknowns, eL , e21 , e L and e0 21 , which are the remaining four error terms to be identified. The simplest transmission standard is the ðSTD;5Þ

direct connection of the two test ports to each other, corresponding to s11 ðSTD;5Þ

s12

¼ 1. Equations (17.48) with those calibration standards give 8 ðM;5Þ > s11 ¼ e11 þ e21 e12 eL > > > > > 1 > ðM;5Þ > > < s21 ¼ eX þ e21 1e e 22 L > 1 ð M;5 Þ > 0 > s ¼ e X þ e12 > > > 12 1e0 22 e0 L > > > : ðM;5Þ s22 ¼ e0 11 þ e0 21 e0 12 e0 L

ðSTD;5Þ

¼ s22

ðSTD;5Þ

¼ 0 and s21

¼

ð17:55Þ

45 The impedance of the termination is not critical, in that what matters is the absence of transmission in the standard. However, the matched terminations minimize the radiation from the test ports, and thus the associated external crosstalk, which is not part of the isolation.

832


From the first and fourth equations of (17.55), we obtain the forward and reverse load matching (eL and e0 L , respectively), while the second and third equations give the forward and reverse transmission tracking (e21 and e0 21 , respectively): 8 ðM;5Þ > s11 e11 > > ¼ e > L > e21 e12 > > > h i > > ð M;5Þ > > < e21 ¼ s21 eX ð1e22 eL Þ ð17:56Þ h i > ðM;5Þ 0 0 0 0 > > e 1e ¼ s e ð e Þ 21 X 22 L 12 > > > > > > ð M;5 Þ > > s e0 11 > : e0 L ¼ 22 0 e 21 e0 12 The direct connection of the VNWA test ports is possible if the respective test connectors are of the same type and opposite gender. If this condition does not occur, the procedure is slightly more complicated, but the same principle applies. ðDUT Þ gives another four equations with unknowns eL , e0 L , In any case, the measurement of a known s21 0 e21 , e 21 . That system can be solved to find the desired four error terms. Observations: (a) The full 12-term calibration requires knowledge of the same number of complex quantities. From the physical point of view, this implies the measurement of amplitude and phase, which is only possible in vector analyzers. (b) The normalization procedure described in Section 17.6.1 is derived from a simplified version of the SFG in Figure 17.19(1), after assuming e11 ¼ e22 ¼ eL ¼ eX ¼ 0, and considering the amplitude only of the parameters. (c) Equations (17.52) and (17.53) give the products e21 e12 and e0 21 e0 12 , respectively, while Equations (17.56) return the values of e21 and e0 21 only. Hence, e12 and e0 12 result as a consequence. (d) Calibration reduces but does not completely cancel measurement errors. Its effectiveness depends on several factors, like the precision of the standards and the accuracy of their scattering parameter characterization, the time stability of the standards characteristics, the precision and intrinsic (hardware) quality of the instrument, before calibration, and the time stability of the measurement circuitry. For these reasons VNWA performance is specified for a particular calibration kit, at a specified temperature and humidity. The equivalent source and load return loss, after calibration, is around 30 dB, whereas the equivalent directivity is around 40 dB. That is, 10 dB better than the scalar NWA. Lastly, to give a quantitative idea, an equivalent return loss of 30 dB in the test port corresponds to an error of 0.004 dB, 0.06 in the transmission coefficient. Once the 12 error terms e11 , e21 , e12 , e22 , eL , eX and e0 11 , e0 21 , e0 12 , e0 22 , e0 L , e0 X are known, by means of the calibration, they can be removed, or de-embedded, from the measure, in order to obtain a virtually result free of error. Indicating the values that the VNWA measures in hardware by 3 2 b1 ðMÞ b1 ðM Þ ¼ s ¼ s 6 11 a1 12 a2 7 7 6 4 ðM Þ b2 ðMÞ b2 5 s21 ¼ s22 ¼ a1 a2 the corrected scattering parameters of the DUT can be derived from the inversion of the system (17.48), which gives


ðDUT Þ

ðM Þ ðM Þ ð mÞ s e0 s e s e0 1 þ 22e0 12 e0 2111 e0 22 eL 21e12 e21X e120 12 e0 21X ¼ ðM Þ ð mÞ ðM Þ ðmÞ s e s e0 s e s e0 1 þ 11e12 e2111 e22 1 þ 22e0 12 e0 2111 e0 22 eL e0 L 21e12 e21X e120 12 e0 21X

ðDUT Þ

ðe0 22 eL Þ ¼ ðM Þ ðM Þ ðmÞ ðM Þ s e s e0 s e s e0 1 þ 11e12 e2111 e22 1 þ 22e0 12 e0 2111 e0 22 eL e0 L 21e12 e21X e120 12 e0 21X

ðDUT Þ

ðe22 e0 L Þ ¼ ðM Þ ðM Þ ðM Þ ðM Þ s e s e0 s e s e0 1 þ 11e12 e2111 e22 1 þ 22e0 12 e0 2111 e0 22 eL e0 L 21e12 e21X e120 12 e0 21X

ðDUT Þ

ðM Þ ðM Þ ðM Þ s e s e0 s e 1 þ 11e12 e2111 ES e0 L e120 12 e0 21X 21e12 e21X ¼ ðM Þ ðM Þ ðM Þ ðM Þ s e s e0 s e s e0 1 þ 11e12 e2111 e22 1 þ 22e0 12 e0 2111 e0 22 eL e0 L 21e12 e21X e120 12 e0 21X

ðM Þ

s11

s11 e11 e12 e21

1þ

ðM Þ

s21

s21 eX e12 e21

ðM Þ

s12

833

s12 e0 X e0 12 e0 21

ð17:57Þ

ðM Þ

s22 e0 11 e0 12 e0 21

1þ

ð17:58Þ

ðM Þ

s11 e11 e12 e21

ð17:59Þ

ðM Þ

s22

s22 e0 11 e0 12 e0 21

ð17:60Þ

The calibrated measurement is ideally error free. However, in practice, the calibration procedure returns the error terms with uncertainties due to imperfect knowledge of the calibration standard scattering parameters. Such errors affect the results of Equations (17.57) to (17.60), with a consequent error in the final measurement. Nevertheless, the calibrated VNWA offers substantially better performances than the scalar NWA in terms of the final error, as anticipated. ðDUT Þ ðDUT Þ ðDUT Þ ðDUT Þ Note that for each DUT scattering parameter s11 , s21 , s12 or s22 , the calibrated ðM Þ

ðM Þ

ðM Þ

ðM Þ

measurement requires the measurement of all four parameters s11 , s21 , s12 and s22 .

17.6.3

Noise figure meters

Figure 17.20 shows the principle of a noise figure meter. Essentially, it is a noise generator (NG), sometimes called a noise source, and a narrow-band heterodyne power meter. The latter includes the local oscillator OSC, the mixer MIX, the IF amplifier AMP, the IF filter BPF2, the detector DET and the postdetection lowdpass filter LPF. In some applications, an additional bandpass filter BPF1 is placed between the DUT output and the RF input of MIX. In this case, the selective power meter is a superheterodyne receiver with a quadratic amplitude demodulator. The instrument measures the gain and noise figure of the DUT by measuring the noise at device output when NG is on and off. If the oscillator frequency is fTEST ¼ oTEST =ð2pÞ, BPF1 is not present, and the centre frequency of BPF2 is fI ¼ oI =ð2pÞ; the DC output of LPF is proportional to the total DUT output power at fTEST þ fI and fTEST fI . In this case, typical values for the centre frequency and bandwidth of BPF2 could be fI ¼ 100 106 and DfI ¼ 1 106 . ðDET Þ Let PDIN;0 be the power noise density at the input of DET, when the RF port of MIX is terminated with a matched resistance: it includes the thermal noise of the input termination and the detector noise, ðDET Þ

referred to its input. The total noise power density at the detector input PDIN noise density

ðMIX Þ PDRF

is related to the power

at the RF port of the mixer as ðDET Þ

PDIN

ðÞ

ðþÞ

ðDET Þ

¼ PDIN þ PDIN þ PDIN;0

ð17:61Þ

834


DISPLAY

Processor OSC

L

NG

MIX

ADC

R

AMP

BPF2

DET

Memory

LPF

BPF1 A

B

DUT IN

OUT

Figure 17.20

Noise figure meter.

where: . .

h i ðÞ ðMIX Þ PDIN ¼ PDRF ð fTEST fI ÞkT Gð fTEST fI Þ h i ðþÞ ðMIX Þ PDIN ¼ PDRF ð fTEST þ fI ÞkT Gð fTEST þ fI Þ

.

Gð f Þ is the total conversion gain – in linear units and evaluated at the RF input frequency f – from the RF input of MIX to the output of BPF.

.

k and T are the Boltzmann constant and the absolute temperature, hence kT is the thermal noise power density;46 usually T ¼ 290:15 K, thus kT ¼ 3:987 1021 mW=Hz, corresponding to 143.994 dBm/Hz. ðÞ

ðþÞ

If the DUT bandwidth is far greater than 2oIF , then PDIN ffi PDIN and h i ðÞ ðþÞ ðÞ ðMIX Þ PDIN þ PDIN ¼ PDIN ffi 2 PDRF ð fTEST ÞkT Gð fTEST Þ

ð17:62Þ

The quantity (17.62) is approximately equal to twice the DUT output power around the measurement frequency fTEST , amplified by the instrument components, from MIX to BPF2. In this case the instrument performs a bilateral noise figure measurement at fTEST . On the other hand, if the DUT is a narrow-band ðÞ ðþÞ device, then the two contributions PDIN and PDIN need to be separated by connecting the anti-image filter BPF1, having centre frequency oT . Moreover, OSC needs to be tuned at fTEST þ fI or fTEST fI . The latter test set performs the so-called unilateral noise figure measurement. We will describe below the most common bilateral measure. In any case, the considerations here can be easily extended to the other case, keeping a correction factor of 2.

46

See also Section 9.4.1.


835

Noise figure and gain are usually expressed in dB, but for ease of treatment we will use linear units; instead of noise figure NF, we will use the noise factorF, being47NF ¼ 10 log10 ðF Þ. The total noise power at the detector input following is ðDET Þ

PIN

1 ð

ffi

h i ðÞ ðDET Þ PDIN ð f Þ þ PDIN;0 ð f Þ d f

ð17:63Þ

1 ðÞ

ðDET Þ

Normally, PDIN PDIN;0 , provided that the gain of AMP is high enough to make its noise dominant over the detector and the thermal noise. Moreover, Gð f Þ is negligible if f falls sufficiently outside one of the two intervals ð fTEST fI DfI =2; fTEST fI þ DfI =2Þ or ð fTEST þ fI DfI =2; fTEST þ fI þ D fI =2Þ. Since DfI fTEST , the terms of Equation (17.61) are approximately constant with the frequency, and the integral (17.63) simplifies to h i ðDET Þ ðÞ ðDET Þ ðBPF2Þ ffi PDIN ð fTEST Þ þ PDIN;0 ð fTEST Þ NBWII ð17:64Þ PIN ðBPF2Þ

where NBWII is the bilateral noise bandwidth48 of the filter BPF2. The detector output voltage is proportional to the power (17.64), by the constant g. LPF is a lowdpass ðLPF Þ ðLPF Þ filter with corner frequency fc NBW ðBPF2Þ ; sometimes fc is lower than 1 Hz. Thus the DET output voltage – filtered by LPF – is a DC value proportional to the power (17.64) h i ðLPF Þ ðÞ ðDET Þ vOUT ffi g PDIN ð fTEST Þ þ PDIN;0 ð fTEST Þ NBW ðBPF2Þ ð17:65Þ Note that all the factors of Equations (17.64) and (17.65) are constants, although the original signal is noise, which inherently fluctuates over the time. The lowdpass averaging effect of LPF guarantees that the voltage at the ADC input is effectively constant. The noise power density (17.62) is related to the gain GðDUT Þ ð fTEST Þ and the noise factor ðDUT Þ ð fTEST Þ of the DUT at the radian frequency fTEST as F nh i o ðÞ ð17:66Þ PDIN ð fTEST Þ ¼ 2 PDðNGÞ ð fTEST ÞkT þ kT F ðDUT Þ ð fTEST Þ GðDUT Þ ð fTEST Þ2kT where PDðNGÞ is the noise power density at the output of the noise generator NG. In most cases, NG assumes the schematic depicted in Figure 17.21. The DC generator Id;0 in combination with the inductor L reverse biases the diode D at a voltage close to the breakdown.49 The breakdown process produces significant random fluctuations in the diode direct current. In other words, the diode current is the sum of the constant Id;0 plus a random term, which is noise and presents relevant energy up to microwave frequency. Noise generators employ special diodes specifically designed, produced and selected to achieve the maximum and flat over frequency noise. The noise current cannot flow through the DC generator, due to the presence of L that exhibits high impedance in the relevant range of high frequencies. Rather, the noise flows in the direction of the load, through the capacitor C, whose reactance is negligible at high frequency, as is typical in RF/mW 47

See also Section 9.4.3, in particular Equation (9.30). The noise bandwidth of a bandpass filter is defined in Section 9.4.5. Here, for the reader’s convenience, we can define the bilateral noise bandwidth of BPF2 by the quantity 48

ðBPF2Þ

NBWII

1 ð h i2 ðBPF2Þ 2 H ¼ maxH ðBPF2Þ ð f Þ ð f Þ d f 1

where H ðBPF2Þ ðf Þ is the transfer function of the filter. If the output (input) of AMP (DET) is matched with the filter ðBPF2Þ ðf Þ of BPF2. If the filter has a real pulse impedance, then H ðBPF2Þ ðf Þ coincides with the transmission coefficient s21 response H ðBPF2Þ ðf Þ ¼ conj½H ðBPF2Þ ðf Þ, the bilateral noise bandwidth equals twice the unilateral one. 49 See Section 9.6.2.

836


MN L

C D

Id,0

Figure 17.21

1

2

OUT

Noise generator.

bias networks. The matching network MN maximizes the noise power transfer from the diode to the output of NG, if needed. Finally, note that the current limitation – inherent in the current-biased device – prevents diode avalanche destruction, by keeping the diode power Id;0 VBREAK below the safe limit. If the DUT is a very low-noise device, then a more precise noise source is preferred, realized with a metallic resistor whose temperature is varied from two different values T1 and T2 . In these two temperatures the resistor produces the noise densities kT1 and kT2 , respectively. The measure in this case is more precise but also more time consuming as the time necessary for thermal equilibrium of the source to be reached must be accounted for. From the external point of view, NG behaves as a purely passive matched load when the direct current is zero, and produces an excess of noise, relatively flat over the frequency. The two above-mentioned conditions are defined as OFF and ON states and the corresponding measurements are defined as cold and hot, respectively. In both conditions the output port of NG is ideally perfectly matched. In the OFF or ON state, the output noise density at the test frequency of the noise generator is 1 ðOFFÞ PDðNGÞ ðoT Þ ¼ kT ð17:67Þ 1 þ enrðoT Þ ðONÞ where the frequency-dependent real number enr is the excess noise ratio of the noise source, which is assumed to be a known term. Substituting the values in (17.67) into (17.66), and the result into (17.65), we obtain the LPF output voltage in the cold and hot measures n o h i ðLPF;OFF Þ ðDET Þ g NBW ðBPF2Þ ffi 2kT GðDUT Þ F ðDUT Þ 1 þ PDIN;0 ð17:68Þ vOUT ðLPF;ON Þ

vOUT

ðDUT Þ

ðLPF;OFF Þ

¼ vOUT

ðDUT Þ

þ 2kT enr g GðDUT Þ NBW ðBPF2Þ

ð17:69Þ

ðDET Þ PDIN;0

,F , and enr are functions of the test frequency, although not where the parameters G explicitly indicated in these equations. Equations (17.68) and (17.69) can be considered as a system of two equations with five unknowns ðDET Þ GðDUT Þ , F ðDUT Þ , PDIN;0 , g NBW ðBPF2Þ and enr. The first two unknowns are the DUT parameters to be measured, the remaining three are constructive parameters of the instrument and can be either given by ðDET Þ the manufacturer or known via calibration processes. Let us temporarily assume that PDIN;0 , NBW ðBPF2Þ and enr are known. The DUT gain follows from Equation (17.69) as GðDUT Þ ¼

ðLPF;ON Þ

ðLPF;OFF Þ

vOUT vOUT 2kT enr g NBW ðBPF2Þ

ð17:70Þ

Once the gain is known, by Equation (17.70), it is easy to obtain the noise factor of the DUT, by substituting Equation (17.70) into (17.68), and extracting F ðDUT Þ from the result (" ) # ðLPF;OFF Þ vOUT 1 1 ðDET Þ F ðDUT Þ ¼ PD ð17:71Þ 1 IN;0 2kT g NBW ðBPF2Þ GðDUT Þ


837

Equations (17.70) and (17.71) return the required gain and noise factor of the DUT from two DC voltages measured with the noise generator ON and OFF. We are now in a position to remove the temporary ðDET Þ assumption about knowledge of the parameters PDIN;0 , g NBW ðBPF2Þ and enr. Two of these three parameters can be determined by measuring a known calibration DUT, by means of Equations (17.68) and (17.69). For instance, if enr is known the calibration is a simple direct connection of the two test ports ðDUT Þ ðDUT Þ GCAL ¼ FCAL ¼ 1; from Equations (17.68) and (17.69) it follows that g NBW ðBPF2Þ ¼

ðLPF;ON Þ

ðLPF;OFF Þ

vOUT;CAL vOUT;CAL ; 2kT enr

ðDET Þ

PDIN;0 ¼

ðLPF;OFF Þ

vOUT g NBW ðBPF2Þ

ð17:72Þ

If the excess noise ratio is also unknown, the calibration procedure is more complicated and requires the measurement of at least two known two-port networks: for instance, two matched attenuators with different attenuations.50 The test procedure described above requires a noise source to measure both the gain and the output noise of the DUT. However, if the power gain of the DUT is known from a previous measurement,51 the noise stimulus becomes superfluous. If GðDUT Þ is known, then F ðDUT Þ is derived from Equation (17.68), which basically expresses the DUToutput noise as modified by the instrument parameters. The extraction of the noise factor of a DUT from its gain and its output noise power density is called cold noise figure measurement.

17.7 Special test instruments Sections 17.2 to 17.6 described standard types of RF/mW test instruments, which are specifically designed to measure the performances of RF/mW devices, components, subsystems or signals. The present section offers a panoramic view of non-standard RF/mW instruments. These instruments are peculiar because of their operation, like the IFM described in Section 17.7.1, or because of their application, like the material testing arrangements of Section 17.7.2. Section 17.7.3 describes complex test benches, in which the high-frequency test is only part of the DUT measurement.

17.7.1

IFM

The acronym IFM stands for instantaneous frequency meter, and defines an analogue frequency demodulator having a structure similar to the one in Figure 15.9. In principle, such equipment is able to deliver an output voltage depending on the frequency of the input signal. If the relation between output voltage and input frequency is monotonic, then it can be linearized by means of analogue or digital circuits, as described similarly in Sections 10.3 and 15.3.1. Figure 17.22a shows the simplest possible IFM configuration, consisting of one power divider (DIV), two transmission lines (TL1 and TL2), one mixer (MIX) and one lowdpass filter (LP). The power divider delivers signals – both deriving from the input one – to the RF and LO ports of MIX, through TL1 and TL2, which have different lengths. The lowdpass filter ideally eliminates the high-frequency components from the output voltage. Let the input signal of the circuit in Figure 17.22a be vRF ðtÞ ¼ v0 cos ðo0 tÞ

ð17:73Þ

If TL1 and TL2 are matched to the impedances of the output ports of DIV and to the RF and LO ports of MIX, respectively, then the RF and LO input voltage will be

50 51

The noise figure of a matched attenuator coincides with its attenuation, both expressed in dB. For instance, the DUT power gain could have been previously measured with an NWA.

838


TL2 RF

vDC ( t )

I

DIV

MIX

(a)

R

LP

TL1

TL2 I-Q MIXER

RF

DIV1

MIX1 vR2

TL1

HYB 0°

R

vL2

I

MIX2

DIV2 vR1

90° 90°

R

0°

vL1

I

R0

LPI

(b) vI ( t )

ADCI

DW I DWOUT

ROM vQ ( t )

ADCQ

DW Q

LPQ HYB2,3 R

L

0°

0°

90° 90° DET 1,3

DET2,4

+ vin – (c)

Av vin

AMP1,2 I

Figure 17.22 IFM: (a) simplified structure; (b) complete block diagram; (c) alternative circuit arrangement for the mixers in (a) and (b).

MEASUREMENT INSTRUMENTATION AND TECHNIQUES vR ðtÞ ¼ aR v0 cos½o0 ðtt1 Þ;

vL ðtÞ ¼ aL v0 cos½o0 ðtt2 Þ

839 ð17:74Þ

where: .

aR and aL are the amplitude attenuations from the input of DIV to the RF and LO ports of MIX, including the dissipation loss of DIV, TL1 and TL2.

.

t1 and t2 are the propagation delay from the input of DIV to the RF and LO ports of MIX, due to the internal delay between the input and the two outputs of DIV and to the electrical length of the two transmission lines.

If MIX operates as a product detector, the IF output voltage is vI ðtÞ ¼ kIF vR ðtÞvL ðtÞ kIF aR aL v20 kIF aR aL v20 ¼ cos½o0 ðt1 t2 Þ þ cos½2o0 t þ o0 ðt1 þ t2 Þ 2 2

ð17:75Þ

The voltage (17.75) is the sum of a constant term plus a term having twice the input frequency. The cut-off frequency of LP is by definition much smaller than the one in the second term above, therefore the output voltage of the circuit in Figure 17.22a contains the constant term only vDC ¼ v0 0 cosðo0 DtÞ

ð17:76Þ

where: .

v0 0 ¼ aLP ðkIF aR aL v20 Þ=2.

.

aLP is the DC attenuation of LP.

.

Dt ¼ t1 t2 .

Figure 17.23 plots the function (17.76), showing that such a relation is periodic with the input frequency f0 ¼ o0 =ð2pÞ, and monotonically increases within the range p þ 2kp o0 Dt 2p þ 2kp

ð17:77Þ

Although kcould be any positive, zero or negative integer, IFM typically works with k ¼ 0 which makes both the limits (17.77) positive. Equation (17.76) implies that the sensitivity of the output voltage to the input frequency increases with the difference Dt between the two delays from the input to the RF and LO ports of the mixer. On the other hand, from Equation (17.77) it follows that the amplitude of the monotonic range of input frequencies equals o0 2 ½p=Dt; 2p=Dt. It decreases with Dt, which has to be dimensioned according to a convenient compromise between the two performances. Moreover, the voltage (17.76) depends not only on the frequency, but also on the amplitude of the input signal. Therefore, an amplitude stabilization element is normally required at the input of the circuit in Figure 17.22a: for instance, a saturating amplifier. The test set in Figure 17.22a can also be used to measure the phase noise of an oscillator, if the output of that DUT is connected to the input of the instrument. In this application, if the nominal angular frequency of the oscillator is fOSC ¼ oOSC =ð2pÞ, the delay difference Dt is an odd multiple of quarter wavelength at the DUT frequency. This ensures that the two signals at the RF and LO inputs of the mixer are in quadrature. The oscillator angular frequency, including the phase noise, is o0 ðtÞ ¼ oOSC þ

djn ðtÞ dt

Considering a time-variable frequency, Equations (17.74) for the mixer input signals become

ð17:78Þ

840


Normalized output

2

arg=(– vI – j v Q ) vDC=vI –vQ

0

–2

–4

arg=(– vQ + j vI)

–2

–1

0

1

(ω0∆τ – 1.5 π)π

Figure 17.23

2

–1

Output voltages of the circuits in Figures 17.22a,b.

2tt 3 ð1 vR ðtÞ ¼ aR v0 cos4 o0 ðxÞ dx5;

2tt 3 ð2 vL ðtÞ ¼ aL v0 cos4 o0 ðxÞ dx5

0

ð17:79Þ

0

Again, the initial time of the integrals is somehow arbitrary; different choices correspond to different constant phase constants or to different choices of the time origin. Substitution of Equation (17.78) into (17.79) gives 8tt 9 8tt 9 = = < ð1 < ð2 djn ðxÞ djn ðxÞ vR ðtÞ ¼ aR v0 cos oOSC þ oOSC þ dx ; vL ðtÞ ¼ aL v0 cos dx : ; : ; dx dx 0

0

Expanding the expressions for vR ðtÞ; vL ðtÞ, and computing the IF signal of the mixer assuming that for the constant v0 0 the same value was used for Equation (17.76), we obtain the low-frequency component vDC ðtÞ ¼ v0 0 cos½oOSC ðt2 t1 Þjn ðtt1 Þ þ jn ðtt2 Þ ¼ v0 0 cosðoOSC DtÞ cos½jn ðtt1 Þjn ðtt2 Þ þ v0 0 sinðoOSC DtÞ sin½jn ðtt1 Þjn ðtt2 Þ If Dt is such that oOSC Dt is an odd multiple of p=2, then cosðoOSC DtÞ ¼ 0 and sinðoOSC DtÞ ¼ 1. Moreover, if the phase noise is small – as is desirable – then cos½fn ðtt1 Þfn ðtt2 Þ 0 and sin½fn ðtt1 Þfn ðtt2 Þ fn ðtt1 Þfn ðtt2 Þ. Under all these assumptions, vDC ðtÞ v0 0 ½jn ðtt1 Þjn ðtt2 Þ

ð17:80Þ

The ambiguity in the sign of expression (17.80) arises because we assumed that oOSC Dt is an odd integer multiple of p=2, without specifying the multiplicity factor. However, since the relevant quantity is the spectrum amplitude, that sign is not relevant. The magnitude of the Fourier transform of the voltage (17.80) is jVDC ð f Þj 2jv0 0 j jsinðp f DtÞj jFn ð f Þj

ð17:81Þ

Now, the variable f in Equation (17.81) is the offset frequency, which is much smaller than the oscillation frequency: f fOSC . Normally, Dt is of the order of some periods of the DUT signal, therefore pf Dt p=2 and the sine of Equation (17.81) can be approximated by its argument, so

MEASUREMENT INSTRUMENTATION AND TECHNIQUES jVDC ð f Þj 2jv0 0 j jpf Dtj jFn ð f Þj

841 ð17:82Þ

According to Equations (17.81) and (17.82), the output voltage of the setup in Figure 17.22a is proportional to the oscillator phase noise multiplied by a constant, which is proportional to the offset frequency, for small values of it. Moreover, the proportionality factor jsinðp f DtÞj presents a first null when the offset frequency is f ¼ 1=ðDtÞ. Therefore, the sensitivity of the instrument increases with Dt, but the maximum measurable offset frequency decreases with Dt. Clearly, the cut-off frequency of LP has to be greater than the maximum offset frequency of interest. The arrangement52 in Figure 17.22b is an improvement on the one in Figure 17.22a. Neglecting the digital circuitry – consisting of ADCI, ADCQ and PROM – the main modification consists of replacing the single mixer with an IQ one. Such a modification involves of course the duplication of the output lowdpass filter. Indeed, the diagram in Figure 17.22b has two filters, one per mixer output, namely LPI and LPQ. Under the same assumptions used for the diagram of Figure 17.22a, the input voltages at the mixer inputs are vR1 ðtÞ ¼ aR1 v0 cos½o0 ðtt1 Þ;

vR2 ðtÞ ¼ aR2 v0 cos½o0 ðtt1 Þ

ð17:83Þ

vL1 ðtÞ ¼ aL1 v0 cos½o0 ðtt2 Þ;

vL2 ðtÞ ¼ aL2 v0 sin½o0 ðtt2 Þ

ð17:84Þ

where aR1 ; aR2 ; aL1 ; aL2 are the attenuations from the input of the instrument to the RF, LO ports of the two mixers. The two filter output voltages are then vI ¼ v0 0 cosðo0 DtÞ;

vQ ¼ v00 0 sinðo0 DtÞ

ð17:85Þ

with v0 0 ¼ aLP

kIF aR2 aL2 v20 ; 2

v00 0 ¼ aLP

kIF aR1 aL1 v20 : 2

Assuming the same conversion loss for the two mixers, the same DC loss for the two lowdpass filters and the same path loss for the outputs of DIV1, DIV2 and HYB, then v0 0 ¼ v00 0 . Under these conditions, it is possible to extract information about the input frequency from a simple operation on the quantities (17.85). Both the voltages vI ; vQ and their respective opposites could be used as the real part or the imaginary coefficient of a complex number to be used to extract the frequency to be measured. This generates eight possibilities, as Table 17.1 shows. However, half of the combinations can be eliminated because they identify two opposite numbers, argðx þ jyÞ ¼ argðxjyÞ þ p. Similarly, half of the remaining four combinations are equivalent in that argðx þ jyÞ ¼ argðxjyÞ. The two remaining main possibilities are ðaÞ

ð17:86Þ

ðbÞ

ð17:87Þ

vOUT ¼ argðvQ þ j vI Þ vOUT ¼ argðvI j vQ Þ

Expressions (17.86) and (17.87) are real functions of the two real variables vI ; vQ , despite the presence of the imaginary unit. Figure 17.23 plots these two functions, together with functions (17.85). The function (17.76) is also presented for comparison. Note that functions (17.86) and (17.87) are monotonic increasing within any of the respective intervals p p ðaÞ þ 2kp o0 Dt þ 2ðk þ 1Þp ð17:88Þ 2 2

52 The Ansoft file 01_IFM.adsn provides a simplified analysis of the system. It also includes a description of a threesection power divider and hybrid coupler, for better wide-band operation of the system.

842


Table 17.1 Possible combinations of IFM mixer output voltages for extracting the input frequency. ReðzÞ ¼ ImðzÞ ¼

1

2

3

4

5

6

7

8

vI vQ

vI vQ

vI vQ

vI vQ

vQ vI

vQ vI

vQ vI

vQ vI

ðbÞ

2kp o0 Dt 2ðk þ 1Þp

ð17:89Þ

Both functions (17.86) and (17.87) present a monotonic range of amplitude o0 Dt ¼ 2p. The difference between the two cases is the position of the monotonic range along the axis x ¼ o0 Dt. By comparison, the functions (17.86) and (17.87) not only present a wider monotonic interval than the function (17.76), but also are linear with the input frequency in the respective monotonic intervals. Furthermore, the function (17.76) approximates the function (17.86) for small values of the variable ðo0 Dt1:5pÞp1 . Multiplication of both the real part and the imaginary coefficient of a complex number by any real constant leaves the argument of the complex number itself unchanged. Consequently, functions (17.86) and (17.87) are independent by any real scaling factor simultaneously applied to vI ; vQ . Therefore, the output of the configuration in Figure 17.22b is ideally independent of the amplitude of the RF input signal, differently from the one in Figure 17.22a. In real cases, however, the mixers operate as expected only if their RF and LO amplitudes fall within specified limits; this reintroduces the necessity for an amplitude levelling device at the input of the system in Figure 17.22b. From an implementation point of view, a simple way to compute the two functions (17.86) and (17.87) is to convert the voltages vI ; vQ into digital ones by means of the two converters ADCI, ADCQ. The two obtained digital words DWI ; DWQ form the address of a read-only memory(ROM), which realizes a look-up table. The ROM data at the address corresponding to a given combination of vI ; vQ return the discrete53 value of the input frequency through one of the functions (17.86) or (17.87). Figure 17.22c shows an alternative circuit that can replace each of the mixers in Figure 17.22b. Such mixer replacement basically consists of a 90 hybrid, two detectors and one differential amplifier. The analysis of the system in Figure 17.22b modified by replacing each of the two mixers with the circuit in Figure 17.22c can be performed by the same method used for the original arrangement, and gives conceptually identical results. Compared with mixer IFM, detector IFM can operate with lower input power and a wider acceptable input range. On the contrary, mixer IFM produces analogue output voltages vI ; vQ with higher amplitude, simplifying the design of the digital circuitry. The simplified analysis of the arrangements in Figure 17.22 assumes no impedance mismatch between the interfacing ports of the various components. This hypothesis is too optimistic for real cases. All the unavoidable mismatches cause amplitude and phase ripple, as described in Section 9.2. Consequently, functions (17.76), (17.86) and (17.87) present superimposed ripple, which can partially be corrected by properly modifying the ROM data. The mismatched IFM calibration is possible by applying known frequencies at the input – and correcting the ROM data at the corresponding address can do this – such that the output coincides with the nominal value. The instruments in Figure 17.22 can be considered as purely analogue frequency meters, besides the postprocessing digital network of the system in Figure 17.22b. The advantage of such analogue over digital frequency meters illustrated in Section 17.3 is their shorter measurement time. Typically the bandwidth of the output lowdpass filters is of the order of some megahertz, thus the output voltages need a few microseconds to reach their steady state value. In comparison, the measurement time of a digital frequency meter is of the order of some milliseconds. The main drawback of the IFM is derived from its analogue nature: the result is less precise, the performances vary over time (aging), from piece to piece, and with temperature. 53

The precision of the output number increases with the number of bits of the ROM.


17.7.2

843

Complex test benches

The complex configurations presented in Chapter 15 often involve combinations of many interacting functions: not only purely high frequency and analogue, but also low frequency and/or digital. For instance, the transmitter in Figure 15.7 has two low-frequency analogue inputs and one RF output, while the receiver in Figure 15.12 has one RF input and two digital outputs. The need for testing such a complex assembly drives the technique towards the development of sophisticated test equipment that embodies all the functions described in Sections 17.2 to 17.6. Particularly demanding is the problem of integrated circuit (IC) testing, in that parts or all of the transmitter and receiver architectures are nowadays realized with IC technology. Moreover, the integration level gets higher and higher as the technology improves, so that it is not uncommon for one single IC to realize a complete transmitter/receiver, from the digital input/ output to the RF output/input, including the antenna switch. One interesting piece of test equipment, specifically designed for IC testing, is the Verigy 93000.54 It basically consists of one power supply, one computer, one cooling unit and a configurable plurality of test functions. The power supply delivers the DC voltages to all the other modules of the apparatus. The computer drives all the test functions and processes the data resulting from each elementary measurement, allowing the user to program complex sequences of measurements. The cooling unit keeps the temperature of the system at an approximately constant value, within a tight tolerance, in order to minimize the performance variations in the various elements of the system. The hardware test functions of the system are realized by means of different types of cards: .

Purely digital. which are capable of applying stimuli to the DUT, receiving and processing the output coming from the DUT. Within the present context, a digital stimulus is an arbitrary and programmable sequence of digital words.

.

Low-frequency analogue. which basically consists of combinations of D/As and A/Ds together with the respective signal conditioning circuitry. The system computer can apply arbitrary sequences of digital words to the inputs of these cards, which transform the digital sequence into the corresponding analogue arbitrary waveform to apply to the DUT. On the other hand, the analogue cards transform the analogue waveform coming from the DUT into a sequence of digital words that can be analyzed by the computer. Many cards of this type are available, with different numbers of channels, speed and precision. As a trend, high speed is associated with relatively low precision and vice versa.

.

RF. the most important one for our description. There are two types of RF cards, namely source and front end. The first is a PLL generator with ALC,55 producing an output signal with highly stable frequency and amplitude. The sources also include the circuitry required to modulate their output signal. Therefore, the source can also produce arbitrary amplitude and phase-modulated signals. The front end applies the signal coming from the source to the DUT, receives the signal from the DUT and converts it to low frequency, compatible with the speed accepted and analogue cards. Figure 17.24 shows the configuration of the RF section of the 93000 test system. It consists of up to four sources (S1 to S4) and one front-end card. One of the four sources is used to generate the LO signal for the two mixers MIX1 and MIX2 of the front-end card. By setting the various switches it is possible to apply a single (only S4 connected to the three-way power splitter), double (S4 and S1 connected to the splitter) and triple (S4, S1 and S2 connected to the splitter) tone stimulus to the DUT, through the test ports 3A and/or 3B. Two variable attenuators are present in the path from the source to each of those test ports, in order to adjust the power applied to the DUT in a range of 0–65 dB. The test ports 3A and 3B can also be isolated from the source, by means of two absorptive

54 All the descriptions in this section and all the figures are published with the authorization of Verigy Ltd, Singapore, www.verigy.com. 55 See Section 15.6.2 for a description of the PLL and Section 15.6.3 for the ALC.

S3

S2

S1

S4

RF

Figure 17.24

MT

40 MHz

40 MHz

RF

MT

RF

MT

RF

MT

MIX2

MIX1

to a-term on s2

30 dB

LNA

35 dB

30 dB

LNA

30 dB

30 dB

35 dB

30 dB

30 dB

35 dB

to a-term on s2

LNA

b-term on s1

LNA

b-term on s2

Block diagram of the complete RF test section of the Verigy 93000 system.

ADC pogo pin

D2

ADC pogo pin

D1

RF

MT

35 dB

35 dB

35 dB

35 dB

3D

3C

2D

2C

1D

1C

3B

3A

2B

2A

1B

1A

RF3

RF2

RF1



845

SPSTs, while keeping the DUT port matched into 50 O. Alternatively, the RF stimulus can be routed to the ports 1A, 1B, 1C, 1D, 3C, 3D, which are similar to 3A, 3B, or 2A, 2B, 2C, 2D, that produce lower power, due to the absence of the last amplifier in the chain. The ports 1B, 1D can also eliminate the last amplifier and attenuator from their path. The RF signal coming from the DUT can be applied to the ports 2A, 2B, 2C and 2D. Therefore, the ports 3A, 3B, 3C, 3D work only to apply a stimulus to the DUT (TX mode), the ports 2A, 2B, 2C, 2D work only in RX mode, and the ports 1A, 1B, 1C, 1D are mainly used for the scattering parameter measurement, as we will see shortly. The DUT output signal applied to any of the ports 2A, 2B, 2C, 2D can pass through one variable attenuator, one or two low-noise amplifiers (LNAs) and arrive at the RF port of a mixer: MIX1 for 2A, 2B and MIX2 for 2C, 2D. The LO port of the mixers is connected to one of the sources S1 to S3. The resulting IF signal has a frequency smaller than 1 GHz. It can be delivered as it is to an external user (which could be an analogue card), or converted into digital form (A/D D1 and D2 for MIX1 and MIX2, respectively) and delivered to a digital card. A lowdpass filter with 40 MHz cut-off frequency can be selected to limit the bandwidth of the above-mentioned signals. Two directional couplers are present in one of the selectable paths of the ports 1B, 2A, 2B, 1D, 2C, 2D. The coupled ports of the directional coupler can be routed to the RF ports of the respective mixers, allowing the computer to measure the wave reflected back from the DUT. Therefore, the arrangement of Figure 17.24 can work as an arbitrary multi-tone modulated stimulus, spectrum analyzer and vector network analyzer. Additionally, the signals coming from the source can be routed to the RF port of the mixers (loop-back mode), for diagnostic and calibration purposes. All the V93000 cards have the aspect shown in the photograph of Figure 17.25. One PCB includes all the circuits and has an aluminium cover, which works as an electric shield and for liquid-cooled heat transfer. The size of each board is about 200 300 mm. The V93000 system can include many boards of different types, depending on the type of test to be performed. Finally, Figure 17.26 shows the aspect of the complete system, which is about 2 metres high.

Figure 17.25

Physical aspects of the cards forming the block diagram of Figure 17.24.

846


Figure 17.26

17.7.3

The Verigy V93000 test system.

Test instruments for non-electrical quantities

This section gives a short overview of the application of RF/mW test instruments to the measurement of non-electrical properties. Figure 17.27 shows the first example of such possibilities. A network analyzer measures the transmission coefficient of a coaxial resonator somehow coupled to the test ports P1 and P2 of the NWA. The response s21 of that measurement has the same aspect as the one depicted in Figure 5.3, although that figure shows the normalized impedance rather than the transmission coefficient. Focusing our attention on the amplitude response, there are two fundamental parameters to consider: the resonant frequency o0 and the half-power bandwidth Do. The measurement principle is that the material under test(MUT) occupies a fixed and known part of the resonator dielectric, so that it affects the resonator performance. In more detail, the resonator dielectric is air, except for the part consisting of the MUT. Therefore the MUT dielectric permittivity is higher than the rest of the dielectric, and shifts o0 towards lower frequencies.56 Similarly, the MUT conductivity decreases the resonator quality factor, and therefore increases Do. Inserting materials with known characteristics in place of the MUT, and measuring the resulting o0 , Do, it is possible to obtain the calibration curves of the instrument. Sometimes, these curves are obtained from calculations. In normal operation, a computer sets the NWA measurement frequency band and acquires the s21 curve from the NWA itself. Then the computer extracts the parameters o0 , Do from the response curve.

56

For instance, if the resonator is a l=4 line short-circuited at one end and open-circuited at the other, then pffiffiffiffi o0 ¼ 0:5p c l= er , where c is the speed of light in a vacuum, l is the resonator length (including the lengthening due to the end effect in the open-circuited side) and er is the relative permittivity. Now, if the MUT is absent, all the dielectric is air, then er ffi 1 and o0 0 ffi 0:5 p c l. On the contrary, if the whole resonator is filled with the MUT, which qffiffiffiffiffiffiffiffiffiffiffiffiffi ðMUTÞ ðMUT Þ < o0 0 . In the actual case, the dielectric is a combination of has er > 1, it follows that o00 0 ¼ 0:5 p c l= er air and the MUT, and the resulting effective permittivity is intermediate between the two values. Thus the resonant ðMUT Þ : frequency ranges between o0 0 and o00 0 , and decreases with er


847

Material under test (MUT) Coaxial resonator

P1

P2

Figure 17.27

NWA

Computer

Test set for the measurement of material characteristics.

Resonant frequency, bandwidth and calibration curves give the required complex dielectric permittivity ðMUT Þ of the MUT ec ¼ eðMUT Þ jsðMUT Þ =o. ðMUT Þ can be used to derive non-electrical properties of the MUT, such as: Knowledge of ec .

eðMUT Þ is an indicator of material density. If the chemical composition of the MUT is known, the dielectric constant can be used to determine the density or the weight of the MUT.

.

sðMUT Þ increases with the content of conducting elements in the MUT; in particular it can be used to detect the water content, especially if o0 is close to a water absorption peak.

The interested reader can find more details on this technique in [13–15]. An instrument derived from the IFM in Figure 17.22b can be used to measure distances,57 as Figure 17.28 shows. In fact, the instrument described in Section 17.7.1 exploits the interference between a sinusoidal signal and its delayed replica to measure the signal frequency. This is possible because the delay Dt between the two signals is known. On the contrary, the arrangement in Figure 17.28 uses an oscillator with a stable and known frequency as input, while the delay difference between the two paths from OSC to the input of DIV2 and HYB has to be determined. A further difference between the instruments in Figures 17.22b and 17.28 is that, in the latter, the path from the input to HYB includes one transmitting (ANTTX) and one receiving (ANTRX) antenna. Assuming that ANTTX and ANTRX are placed on the same plane and directed towards a reflecting plane obstacle, the delay Dt includes one term that is proportional to the distance between the two antennas and the reflecting plane. Developing the equations of the arrangement in Figure 17.28 with the same methods and hypotheses used for the one in Figure 17.22b, we obtain the same results. In particular the final output digital word is given by Equation (17.86) or (17.87), depending on the choice made for the output function. Consequently, if the oscillator frequency is o0 , the unambiguous measurable delay is derived from the intervals (17.86) or (17.87) and equal respectively to p p þ 2kp o0 DtðaÞ þ 2ðk þ 1Þp ð17:90Þ 2 2 2kp o0 DtðbÞ 2ðk þ 1Þp

57

More details can be found in [16].

ð17:91Þ

848


d ANTTX

∆τ I-Q MIXER OSC

MIX1 0º

R

DIV1

δ

I

90º DIV2

MIX2

ANTRX

90º

R

R0 0º

I

Reflecting plane

LPI vI(t)

vQ(t)

Figure 17.28

ADCI

ADCQ

DWI DWOUT DWQ

ROM

Altimeter derived from the IFM in Figure 17.22b.

Again, high values of o0 imply high sensitivity to the delay, but also smaller unambiguous range and vice versa Another factor to consider in the choice of o0 is the sensitivity of the instrument to the roughness of the reflecting surface. One possible application of the instrument in Figure 17.28 is the measurement of the height of a car from the ground. In this case, the oscillator frequency has to be such that any ground protuberance results are smaller than the wavelength. Assuming that the maximum protuberance size is 20 mm, and that is one-third of the wavelength, we have o0 =ð2pÞ ¼ 5 GHz and an unambiguous measurable range of Dtmax Dtmin ¼ 2p=o0 ¼ 0:2 109 s. Finally, the desired distance d between the two antennas and the reflecting plane is related to Dt through a trigonometric function, which also depends on the distance between the two antennas and on their radiation diagram. In the simplest case, where the two antennas are isotropic and have no other transmission contribution than the one due to the reflection from the plane, this is 2 Dt ¼ c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi d d2 þ 2

ð17:92Þ

where c is the speed of light in a vacuum. If d d, the amplitude of the unambiguous range becomes dmax dmin ¼ c

Dtmax Dtmin c 1c l ¼p ¼ ¼ 2 o0 2 f0 2

ð17:93Þ

In the above example f0 ¼ 5 GHz and the unambiguous distance range given by Equation (17.95) is dmax dmin ¼ 30 mm.


849

Bibliography 1. Agilent Application Note 200-1, ‘Fundamentals of microwave frequency counters’, Santa Clara, CA, 1997. 2. C. Raucher, V. Janssen and R. Minihold, Grundlagen der Spektrumanalyse, Rhode & Schwarz, Munich, 2008. 3. Rohde & Schwarz Application Note 1EF16, ‘Phase noise measurements with spectrum analyzers of the FSE family’, Munich, October 1995. 4. Rohde & Schwarz Application Note 1MA15, ‘Measurement of frequency settling time of synthesizers and transmitters’, Munich, February 1999. 5. Rohde & Schwarz Application Note 1EF48, ‘Power measurement on pulsed signals with spectrum analyzers’, Munich, January 2003. 6. C. E. Shannon, ‘Communication in the presence of noise’, Proceedings of the IEEE, Vol. 86, No. 2, pp. 447–457, 1998; reprinted from Proceedings of the IRE, Vol. 37, No. 1, pp. 10–21, 1949. 7. Agilent 71500A/70820A Microwave Transition Analyzer, Brochure, Santa Clara, CA, 1997. 8. Agilent Application Note 1287-1, ‘Understanding the fundamental principles of vector network analysis’, Santa Clara, CA, 1997. 9. Agilent Application Note 1287-2, ‘Exploring the architectures of network analyzers’, Santa Clara, CA, 2000. 10. Agilent Application Note 1287-3, ‘Applying error correction to network analyzer measurements’, Santa Clara, CA, 2002. 11. M. Hiebel, Fundamentals of Vector Network Analysis, Rhode & Schwarz, Munich, 2007. 12. Noise Com Application Note 121, ‘Improving noise figure measurements’, Parsippany, NJ. 13. G. Bianchi, M. Dionigi, D. Fioretto and R. Sorrentino, ‘A resonant cavity sensor for moisture monitoring in wet powder’, Asia-Pacific Microwave Conference, Yokohama, December 1998. 14. G. Bianchi, M. Dionigi, D. Fioretto and R. Sorrentino, ‘A microwave system for moisture monitoring in wet powder for industrial application’, IEEE MTT Symposium, Anaheim, CA, 1999, pp. 1603–1606. 15. E. Fratticcioli, A. Ocera, M. Dionigi, F. Orfei and G. Bianchi, ‘A low-cost complete system for complex permittivity measurement using resonant probes’, EuMA Proceedings, December 2005. 16. G. P. Merlino, ‘Computer-aided design of a microwave device for real-time control of Formula 1 cars’ trim’, Masters Dissertation, University of Perugia, Faculty of Engineering, 1991–1992 (in Italian).

Related files Ansoft files 01_IFM.adsn. Analyzes the test set in Figure 17.22b. It also includes a description of a three-section power divider and hybrid coupler, for a better wide-band operation of the system.

Mathcad files 02_Resolution_Bandwidth_Bessel.MCD. Computes the noise bandwidth of a Bessel bandpass filter. Also analyzes the effect of the same filter (used in the IF of an SA) on dual tone visualization.

Appendix A

Useful relations from vector analysis and trigonometric function identities A.1 Symbol definitions Vectors are in bold, scalars in regular font. r F ¼ F(r) A ¼ A(r) A Au ^ u rF rA rA r2 F; r2 A @F=@u V S ^ n ^sH H H‘ A dl ¼ H‘ A ^s dl ^ ‘ A dS ¼ ‘ A n dS

position vector scalar field vector field magnitude of A component of A in the oriented u direction unit vector in the oriented u direction (direction vector) scalar product cross product gradient of F divergence of A curl of A Laplacian of F and A partial derivative of F along the oriented u direction volume surface (also the contour of the volume V) direction vector normal to an oriented surface S direction vector tangent to an oriented line l circulation of the vector A computed along the closed, oriented line l flux of the vector A computed through the oriented surface S


852

APPENDIX A

A.2 Common use algebraic identities aðA þ BÞ ¼ aA þ aB

ðA:1Þ

aðA BÞ ¼ ðaAÞ B ¼ A ðaBÞ

ðA:2Þ

aðA BÞ ¼ ðaAÞ B ¼ A ðaBÞ

ðA:3Þ

AþB ¼ BþA

ðA:4Þ

AB ¼ BA

ðA:5Þ

A B ¼ B A

ðA:6Þ

A A ¼ A2

ðA:7Þ

û ^¼1 u

ðA:8Þ

^ ¼ Au Au

ðA:9Þ

A ðB þ CÞ ¼ A B þ A C

ðA:10Þ

A ðB þ CÞ ¼ A B þ A C

ðA:11Þ

AB C ¼ CA BþBC A

ðA:12Þ

A ðB CÞ ¼ BðA CÞCðA BÞ

ðA:13Þ

^ ðA u ^ Þ ¼ AAu u ^ u

ðA:14Þ

A.3 Parallelism, orthogonality, coplanar conditions If A, B and C are non-zero vectors, the following hold:

AB¼0

A and B are parallel

(A.15)

AB ¼ 0

A and B are orthogonal

(A.16)

AB C ¼ 0

A, B and C are coplanar

(A.17)

^ as normal unit vector A is orthogonal to the surface S having n

(A.18)

^ as normal unit vector A is tangent to the surface S having n

(A.19)

Â¼0 n Â ¼ 0 n

APPENDIX A

853

A.4 Differential identities ^ rF ¼ u

@F @u

ðA:20Þ

rðF þ GÞ ¼ rF þ rG

ðA:21Þ

rðFGÞ ¼ FrG þ GrF

ðA:22Þ

r ðA þ BÞ ¼ r A þ r B

ðA:23Þ

r ðFAÞ ¼ Fr A þ A rF

ðA:24Þ

r ðA BÞ ¼ B r AA r B

ðA:25Þ

r ðA þ BÞ ¼ r A þ r B

ðA:26Þ

r ðFAÞ ¼ Fr AA r F

ðA:27Þ

r ðrF Þ ¼ 0

ðA:28Þ

rr A ¼ 0

ðA:29Þ

r2 F ¼ r rF

ðA:30Þ

r r A ¼ rr Ar2 A

ðA:31Þ

r2 ðFAÞ ¼ Ar2 F ðif A is constantÞ

ðA:32Þ

Equations (A.28) and (A.29) show that the gradient of a scalar field is an irrotational vector, while the curl of a vector field is a solenoidal vector, respectively.

A.5 Vector operations in common coordinate systems Rectangular coordinates In a rectangular coordinate system vectors are represented using their components along the x, y and z axes. We thus have rF ¼

@F @F @F ^x þ ^ ^z yþ @x @y @z

ðA:33Þ

@Ax @Ay @Az þ þ @x @y @z

ðA:34Þ

rA ¼

@Az @Ay @Ax @Az @Ay @Ax þ ^y þ ^z r A ¼ ^x @y @z @z @x @x @y

ðA:35Þ

854

APPENDIX A r2 F ¼

@2F @2F @2F þ 2 þ 2 @x2 @y @z

r2 A ¼ ^xr2 Ax þ ^yr2 Ay þ ^zr2 Az

ðA:36Þ ðA:37Þ

Cylindrical coordinates In a cylindrical coordinate system vectors are represented using their components computed in the ^ and ^z: directions of the fundamental vectors ^r, u ^ þ Az^z A ¼ Ar ^r þ Af u @F 1 @F @F ^r þ ^þ ^z u @r r @f @z

ðA:39Þ

1 @ðrAr Þ 1 @Af @Az þ þ r @r r @f @z

ðA:40Þ

rF ¼

rA ¼

1 @Az @Af @Ar @Az 1 @ðrAf Þ @Ar ^ þu þ ^z r A ¼ ^r r @f r @z @z @r @f @r r2 F ¼

ðA:38Þ

1@ @F 1 @2F @2F r þ 2 2þ 2 r @r @r r @f @z

Ar 2 @Af A 2 @A ^ r2 Ay 2y 2 r þ ^zr2 Az r2 A ¼ ^r r2 Ar 2 2 þu r r r @f r @f

ðA:41Þ

ðA:42Þ

ðA:43Þ

Spherical coordinates In a spherical coordinate system vectors are represented using their components computed in the ^: directions of the fundamental vectors ^r, ^h and u

rF ¼

rA ¼

^ A ¼ Ar^r þ Ay ^h þ Af u

ðA:44Þ

@F 1 @F ^ 1 @F ^r þ ^ hþ u @r r @y r sin y @f

ðA:45Þ

1 @ðr2 Ar Þ 1 @ðsinyAy Þ 1 @Af þ þ r2 @r r sin y @y r sin y @f

r A ¼ ^r

@ðsin yAj Þ @Ay 1 @y r sin y @j 1 1 @Ar @ðrAj Þ 1 @ðrAy Þ @Ar ^ ^ þu þh r sin y @j r @r @y @r

r2 F ¼

ðA:46Þ

1 @ 1 @ @F 1 @2F 2 @F r þ sin y þ 2 2 2 2 r @r @r r sin y @y @y r sin y @f2

ðA:47Þ

ðA:48Þ

APPENDIX A

855

2 1 @Ay 1 @Aj r2 A ¼ ^r r2 Ar 2 Ar þ Ay þ þ r tan y sin y @j @y 1 1 @Ar cosy @Aj A þ 2 2 þ ^h r2 Ay 2 y @y r sin2 y sin2 y @j 1 1 2 @Ar cos y @Ay ^ r2 A j 2 A 2 þu j r sin2 y sin y @j sin2 y @j

ðA:49Þ

Relations between rectangular and cylindrical coordinates ^r ¼ ^x cos y þ ^y sin y

ðA:50Þ

^h ¼ ^x sin y þ ^ y cos y

ðA:51Þ

^x ¼ ^r cos y^h sin y

ðA:52Þ

^y ¼ ^r sin y þ ^ h cos y

ðA:53Þ

Relations between rectangular and spherical coordinates ^r ¼ ^x sin y cos f þ ^y sin y sin f þ ^z cos y

ðA:54Þ

^h ¼ ^x cos y cos f þ ^y cos y sin f þ ^z sin y

ðA:55Þ

^ ¼ ^x sin f^ u y cos f

ðA:56Þ

^x ¼ ^r sin ycos f þ ^h cos ycos f^ u sin j

ðA:57Þ

^ cos f ^y ¼ ^r sin y sin f þ ^h cos ysin f þ u

ðA:58Þ

^z ¼ ^r cos y^ h sin f

ðA:59Þ

Integral identities The following formulae allow for the transformation of a volume integral over V into a surface integral over the enclosing surface S, and of a surface integral over oriented surface S into a linear integral along its ^ is the unit vector normal to S contour ‘. Both S and ‘ can be the union of two or more separate regions. n and directed outwards from the volume V. We assume that in the region where integrals are defined (boundary included) both F and A are continuous and differentiable as many times as required. Divergence (or Gauss’s) theorem

I

ð

^ dS An

r A dV ¼ V

S

ðA:60Þ

856

APPENDIX A

Stokes’ theorem

ð

I r A dS ¼

Curl formula

ðA:61Þ

^ A dS n

ðA:62Þ

F^ ndS

ðA:63Þ

I

ð r A dV ¼ V

Gradient formula

A ^sd‘

‘

S

S

ð

I rFdV ¼

V

S

Integral of a conservative field computed along a line directed from P to Q ðQ rF ^sd‘ ¼ FQ FP

ðA:64Þ

P

Circulation of a conservative field

I ‘

First Green’s formula

ð

rF ^sd‘ ¼ 0

Gr2 F þ rF rG dV ¼

V

Second Green’s formula

ð

Gr2 FF r2 G dV ¼

V

ðA:65Þ I G S

@F dS @n

I @F @G G F dS @n @n S

ðA:66Þ

ðA:67Þ

A.6 Basic trigonometric identities Addition and subtraction formulae sinða bÞ ¼ sinðaÞcosðbÞ cosðaÞsinðbÞ

ðA:68Þ

cosða bÞ ¼ cosðaÞcosðbÞ sinðaÞsinðbÞ

ðA:69Þ

1 1 sinðaÞsinðbÞ ¼ cosðabÞ cosða þ bÞ 2 2

ðA:70Þ

1 1 sinðaÞcosðbÞ ¼ sinðabÞ þ sinða þ bÞ 2 2

ðA:71Þ

1 1 cosðaÞcosðbÞ ¼ cosðabÞ þ cosða þ bÞ 2 2

ðA:72Þ

Factor formulae

APPENDIX A

857

Euler identities eja ¼ cosðaÞ þ jsinðaÞ

ðA:73Þ

cosðaÞ ¼

eja þ eja 2

ðA:74Þ

sinðaÞ ¼

eja eja 2j

ðA:75Þ

A.7 Powers of sinusoidal combinations From the subsequent applications of Equations (A.68) to (A.72), it is possible to expand the factor ½a cosðaÞ þ b cosðbÞk ; k ¼ 2; 3; 4; 5, as follows: ½a cosðaÞ þ b cosðbÞ2 1 ¼ a2 ½1 þ cosð2aÞ 2 þ ab ½cosðabÞ þ cosða þ bÞ þ

ðA:76Þ

1 2 b ½1 þ cosð2bÞ 2

½a cosðaÞ þ b cosðbÞ3 1 ¼ a3 ½3cosðaÞ þ cosð3aÞ 4 3 þ a2 b ½cosð2abÞ þ cosð2a þ bÞ þ 2cosðbÞ 4 3 þ ab2 ½cosða2bÞ þ cosða þ 2bÞ þ 2cosðaÞ 4 1 þ b3 ½3cosðbÞ þ cosð3bÞ 4

ðA:77Þ

½a cosðaÞ þ b cosðbÞ4 1 ¼ a4 ½3 þ 4cosð2aÞ þ cosð4aÞ 8 1 þ a3 b ½cosð3abÞ þ cosð3a þ bÞ þ 3cosðabÞ þ 3cosða þ bÞ 2 1 þ a2 b2 ½cosð2a2bÞ þ cosð2a þ 2bÞ þ 2cosð2aÞ þ 2cosð2bÞ þ 2 2 3 þ ab3 ½cosða3bÞ þ cosða þ 3bÞ þ 3cosðabÞ þ 3cosða þ bÞ 4 1 þ b4 ½3 þ 4cosð2bÞ þ cosð4bÞ 8

ðA:78Þ

858

APPENDIX A

½a cosðaÞ þ b cosðbÞ5 1 ¼ a5 ½10cosðaÞ þ 5cosð3aÞ þ cosð5aÞ 16 5 þ a4 b ½cosð4abÞ þ cosð4a þ bÞ þ 4cosð2abÞ þ 4cosð2a þ bÞ þ 6cosðbÞ 16 5 þ a3 b2 ½cosð3a2bÞ þ cosð3a þ 2bÞ þ 3cosða2bÞ þ 3cosða þ 2bÞ þ 6cosðaÞ þ 2cosð3aÞ ðA:79Þ 8 5 þ a2 b3 ½cosð2a3bÞ þ cosð2a þ 3bÞ þ 3cosð2abÞ þ 3cosð2a þ bÞ þ 6cosðbÞ þ 2cosð3bÞ 8 5 þ ab4 ½cosða4bÞ þ cosða þ 4bÞ þ 4cosða2bÞ þ 4cosða þ 2bÞ þ 6cosðaÞ 16 1 þ b5 ½10cosðbÞ þ 5cosð3bÞ þ cosð5bÞ 16 Similarly, the factor ½acosðaÞ þ bcosðb þ ccosðgÞÞk ; k ¼ 2; 3; can be written as the sum of simple sinusoidal terms as ½a cosðaÞ þ b cosðbÞ þ c cosðgÞ2 a2 b2 c2 þ þ 2 2 2 a2 b2 c2 þ cosð2aÞ þ cosð2bÞ þ cosð2gÞ 2 2 2 þ ab ½cosðabÞ þ cosða þ bÞ þ ac ½cosðagÞ þ cosða þ gÞ þ bc ½cosðbgÞ þ cosðb þ gÞ

¼

½a cosðaÞ þ b cosðbÞ þ c cosðgÞ3 3 3 3 ¼ a a þ 2b2 þ 2c2 cosðaÞ þ b b þ 2a2 þ 2c2 cosðbÞ þ c c þ 2a2 þ b2 cosðgÞ 4 4 4 a3 b3 c3 þ cosð3aÞ þ cosð3bÞ þ cosð3gÞ 4 4 4 3 2 3 þ ab ½cosða2bÞ þ cosða þ 2bÞ þ a2 b ½cosð2abÞ þ cosð2a þ bÞ 4 4 3 2 3 2 þ ac ½cosða2gÞ þ cosða þ 2gÞ þ a c ½cosð2agÞ þ cosð2a þ gÞ 4 4 3 2 3 þ bc ½cosðb2gÞ þ cosðb þ 2gÞ þ b2 c ½cosð2bgÞ þ cosð2b þ gÞ 4 4 3 þ abc ½cosðabgÞ þ cosða þ b þ gÞ þ cosða þ bgÞ þ cosðab þ gÞ 2

ðA:80Þ

ðA:81Þ

Putting a ¼ 1; b ¼ 0, Equations (A.76) to (A.79) simplify to 1 1 cos2 ðaÞ ¼ þ cosð2aÞ 2 2

ðA:82Þ

3 1 cos3 ðaÞ ¼ cosðaÞ þ cosð3aÞ 4 4

ðA:83Þ

3 1 1 cos4 ðaÞ ¼ þ cosð2aÞ þ cosð4aÞ 8 2 8

ðA:84Þ

APPENDIX A 5 5 1 cos5 ðaÞ ¼ cosðaÞ þ cosð3aÞ þ cosð5aÞ 8 16 16 More generally, a cosine power can be written in the form n X n cos½ðn2kÞa cosn ðaÞ ¼ 2n k

859 ðA:85Þ

ðA:86Þ

k¼0

where n is a positive integer and

n! n ¼ k k!ðnkÞ!

is the kth coefficient of the Newton binomial with degree n.

A.8 Composition of two sinusoids having the same frequency The combination of two sinusoidal signals having the same frequency but different amplitude and phase is relatively common in RF/microwave circuits and subsystems. This section presents some useful mathematical manipulations for that case. Let the two sinusoidal functions be v1 and v2, with v1 ðtÞ ¼ a1 cosðo0 t þ j1 Þ;

v2 ðtÞ ¼ a2 cosðo0 t þ j2 Þ

ðA:87Þ

Adding and subtracting the phase f1 from the cosine argument of v2, and summing the two functions (A.87), gives ðA:88Þ v1 ðtÞ þ v2 ðtÞ ¼ a1 cosðo0 t þ f1 Þ þ a2 cos½o0 t þ f1 þ ðf2 f1 Þ Expanding the second cosine, we get v1 ðtÞ þ v2 ðtÞ ¼ a1 cosðo0 t þ f1 Þ þ a2 cosðo0 t þ f1 Þcosðf2 f1 Þa2 sinðo0 t þ f1 Þsinðf2 f1 Þ Grouping together the terms containing the factor cosðo0 t þ f1 Þ, we obtain v1 ðtÞ þ v2 ðtÞ ¼ ½a1 þ a2 cosðf2 f1 Þcosðo0 t þ f1 Þa2 sinðo0 t þ f1 Þsinðf2 f1 Þ ðA:89Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi After dividing both sides by the factor ½a1 þ a2 cosðf2 f1 Þ2 þ a2 sin2 ðf2 f1 Þ, Equation (A.89) becomes v1 ðtÞ þ v2 ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a cosðo0 t þ f1 Þb sinðo0 t þ f1 Þ ½a1 þ a2 cosðf2 f1 Þ2 þ ½a2 sinðf2 f1 Þ2

ðA:90Þ

with

a1 þ a2 cosðj2 j1 Þ sinðj2 j1 Þ ffi ; b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½a1 þ a2 cosðj2 j1 Þ þ ½a2 sinðj2 j1 Þ ½a1 þ a2 cosðj2 j1 Þ2 þ ½a2 sinðj2 j1 Þ2

The quantities A, B are real constants, with absolute value not greater than 1, and the sum of their square is unitary 1 a 1; 1 b 1; a2 þ b2 ¼ 1 Therefore, it is possible to assume that a ¼ cosðyÞ;

b ¼ sinðyÞ

ðA:91Þ

860

APPENDIX A

Substituting quantities (A.91) into the function (A.90), we have v1 ðtÞ þ v2 ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cosðyÞcosðo0 t þ f1 ÞsinðyÞsinðo0 t þ f1 Þ ½a1 þ a2 cosðf2 f1 Þ2 þ ½a2 sinðf2 f1 Þ2 Finally,

with a1;2 ¼

v1 ðtÞ þ v2 ðtÞ ¼ a1;2 cosðo0 t þ f1 þ yÞ

ðA:92Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a1 þ a2 cosðf2 f1 Þ2 þ ½a2 sinðf2 f1 Þ2

ðA:93Þ

y ¼ tan1

a2 sinðf2 f1 Þ a1 þ a2 cosðf2 f1 Þ

ðA:94Þ

Note that the arctangent in expression (A.94) has to consider the quadrant where the numerator and the denominator of its argument are located. Thus expression (A.94) is equivalent to y ¼ arg½a1 þ a2 cosðf2 f1 Þ þ ja2 sinðf2 f1 Þ

ðA:95Þ

On the contrary, given a1;2 and y, it is possible to compute the two sinusoids that generate the function (A.92) by inverting (A.93) and (A.94): 8
> jtj < < Dt 2 dðtÞ ¼ lim ½rectDt ðtÞ ¼ lim ðB:9Þ Dt Dt ! 0 Dt ! 0 > > :0 jtj > 2 The subtended area of the rectangular pulse is unitary, independently of its width, and this holds true for the Dirac pulse. The Fourier transforms of time domain pulses and inverse transforms of frequency domain pulses are Dt=2 ð

F ½rectDt ðtÞ ¼ Dt=2

1 sinðpf DtÞ expðj2pftÞ dt ¼ ¼ sincðpf DtÞ Dt pf Dt

F ½dðtÞ ¼ lim fF ½rectDt ðtÞg ¼ 1 Dt ! 0

F rectD f ð f Þ ¼ 1

Dð f =2

D f =2

1 sinðpDftÞ expðj2p ftÞ d f ¼ Df pD ft

F 1 ½dð f Þ ¼ lim

Df !0

F rectD f ð f Þ ¼ 1

ðB:10Þ

ðB:11Þ

ðB:12Þ

ðB:13Þ

APPENDIX B

863

9. Fourier transforms of important particular functions. From properties 3 and 5, Equations (B.6) and (B.13), and Euler’s formulae, we have expðj2pf0 tÞ þ expðj2pf0 tÞ dð f f0 Þ þ dð f þ f0 Þ F ½cosð2p f0 tÞ ¼ F ¼ ðB:14Þ 2 2 F ½sinðo0 tÞ ¼ F

expðj2pf0 tÞexpðj2pf0 tÞ dð f f0 Þdð f þ f0 Þ ¼ 2j 2j

ðB:15Þ

From Equation (B.6) and Euler’s formula, it follows that 1 1 F ½aðtÞ cosð2p f0 tÞ ¼ Að f f0 Þ þ Að f þ f0 Þ 2 2 F ½aðtÞsinð2pf0 tÞ ¼ jAð f f0 Þ þ jAð f þ f0 Þ 10. Differentiation and integration over time. daðtÞ ¼ j2pf F ½aðtÞ F dt 2 F4

ðt

ðB:16Þ ðB:17Þ

ðB:18Þ

3 aðtÞ dt5 ¼

1

11. Impulse response in the time domain. 2 61 F 1 ½dð f Þ H ð f Þ ¼ lim 4 D f ! 0 Df

Dð f =2

D f =2

1 F ½aðtÞ j2p f

ðB:19Þ

3 7 H ð f Þ expðj2pftÞ d f 5 ¼ H ð0Þ

ðB:20Þ

Appendix C Orthogonality of the eigenvectors in ideal waveguides We demonstrate here the orthogonality properties of the TM and TE modes in an ideal waveguide. We first recall that the transverse potentials satisfy the eigenvalue equation r2t T ðx; yÞ þ kt2 Tðx; yÞ ¼ 0

ðC:1Þ

To distinguish between TM and TE modes, we will use the notation ( p) to designate TM modes and [ p] for TE modes, p being the mode index. The boundary conditions associated with (C.1) are therefore Tð pÞ ðx; yÞ ¼ 0 @T½ p ðx; yÞ ¼0 @n

on C

ðC:2Þ

Applying to the cross-section S of the waveguide, with contour C, the second Green’s formula (A.67) in two dimensions, with F ¼ Tð pÞ and G ¼ TðqÞ , we have ð þ @Tð pÞ @TðqÞ TðqÞ r2t Tð pÞ Tð pÞ r2t TðqÞ dS ¼ TðqÞ Tð pÞ dl ðC:3Þ @n @n S C From the boundary condition (C.2) the r.h.s. of (C.3) vanishes. Using (C.1), we then obtain ð 2 2 Þ Tð pÞ TðqÞ dS ¼ 0 if ktð2 pÞ 6¼ ktðqÞ ðktð2 pÞ ktðqÞ

ðC:4Þ

S

The functions TðpÞ and TðqÞ are therefore orthogonal over the cross-section S of the waveguide, provided that the modes have different eigenvalues, i.e. are non-degenerate. In a similar manner we can demonstrate that, for a pair of TE modes, ð 2 2 ðkt½2 p kt½q Þ T½ p T½q dS ¼ 0 if kt½2 p 6¼ kt½q ðC:5Þ S


866

APPENDIX C

Let us now prove the following orthogonality between the field vectors of non-degenerate TM modes: ð

ð pÞ

ðqÞ

et ðx; yÞ et ðx; yÞ dS ¼ 0

ðC:6Þ

S ð pÞ

Using et ðx; yÞ ¼ rt Tð pÞ ðx; yÞ, the above equation is equivalent to ð rt Tð pÞ rt TðqÞ dS ¼ 0

ðC:7Þ

S

From the first Green’s formula (A.66) in two dimensions, with F ¼ Tð pÞ and G ¼ TðqÞ , we get ð

S

TðqÞ r2t Tð pÞ þ rt Tð pÞ rt TðqÞ dS ¼

þ C

TðqÞ

@Tð pÞ dl @n

ðC:8Þ

Using Equations (C.1) and (C.2), (C.8) becomes ð ð 2 Tð pÞ TðqÞ dS þ rt TðpÞ rt TðqÞ dS ¼ 0 kðqÞ S

ðC:9Þ

S

2 2 Because of the orthogonality (C.4), for non-degenerate modes (ktðpÞ 6¼ ktðqÞ ) we obtain (C.7). With the same procedure, but using the boundary condition for TE modes, we obtain ð rt T½ p rt T½q dS ¼ 0 ðC:10Þ S

This formula can be transformed as follows: ð

ð S

rt T½ p rt T½q dS ¼

S

^z rt T½ p ^z rt T½q dS ¼ 0

ðC:11Þ

Recalling that, for TE modes, et ðx; yÞ ¼ ^z rt Tðx; yÞ, we deduce for non-degenerate TE modes 2 (kt½2 p 6¼ kt½q ) that ð

½ p

½q


ðC:12Þ

S

Finally, let us prove the following orthogonality between TM and TE modes: ð

ð pÞ

½q


ðC:13Þ

S

Expressing the field vectors in terms of the respective potentials, the above equation is equivalent to ð S

rt Tð pÞ ^z rt T½q dS ¼ 0

ðC:14Þ

To prove (C.14) we compute the following expression: rt Tð pÞ^z rt T½q ¼ rt Tð pÞ ^z rt T½q þ Tð pÞ rt ð^z rt T½q Þ

ðC:15Þ

APPENDIX C

867

The last term on the r.h.s. of (C.15) is zero because r ^z rt T½q ¼ r r ð^zT½q Þ ¼ 0 The integrand in (C.14) can thus be replaced with the l.h.s. of (C.15). Applying the divergence theorem (A.60) in two dimensions, we then obtain ð

ð S

rt Tð pÞ ^z rt T½q dS ¼

S

rt Tð pÞ^z rt T½q dS ¼

þ c

^ dl ¼ 0 Tð pÞ^z rt T½q n

ðC:16Þ

^ is the unit vector normal to the contour C. Note that since Tð pÞ is zero on C. In (C.16) n ^ ¼ rt T½q n ^ ^z ¼ rt T½q ^t, ^t being the unit vector tangent to the contour. ^z rt T½q n

Appendix D

Standard rectangular waveguides and coaxial cables Standard rectangular waveguides The inner dimensions of the rectangular waveguide are: a broader side, b narrower side. Standard designation WR

1

US Mil.

WR975 RG204 (a) WR770 RG205 (a) WR650 RG69 (b) RG103 (a) WR510 WR430 RG104 (b) RG105 (a) WR340 RG112 (b) RG113 (a) WR284 RG48 (b) RG75 (a) WR229 RG340 (c) RG341 (a) WR187 RG49 (b) RG95 (a) WR159 RG343 (c) RG344 (a) WR137 RG50 (b) RG106 (a) WR112 RG51 (b) RG68 (a)

fmin fmax

fc

WG6

0.75–1.12 0.96–1.45 1.12–1.70

WG8

UK Mil.

a

Frequencies (GHz) IEC

inch

b mm

inch

mm

0.61 9.75 0.77 7.7 0.91 6.5

247.65 195.58 165.1

4.875 3.85 3.25

123.825 97.79 82.55

1.45–2.20 1.70–2.60

1.16 5.1 1.37 4.3

129.54 109.22

2.55 2.15

64.77 54.61

WG9A

2.20–3.30

1.74 3.4

86.36

1.7

43.18

WG10

2.60–3.95

2.08 2.84

72.136 1.34

34.036

WG11A R40

3.30–4.90

2.58 2.29

58.166 1.145

29.083

WG12

R48

3.95–5.85

3.15 1.872

47.549 0.872

22.149

WG13

R58

4.90–7.05

3.71 1.59

40.386 0.795

20.193

WG14

R70

5.850–8.200

4.30 1.372

34.849 0.622

15.799

WG15

R84

7.050–10.000

5.26 1.122

28.499 0.497

12.624

(Continued ) Microwave and RF Engineering Roberto Sorrentino and Giovanni Bianchi © 2010 John Wiley & Sons, Ltd. ISBN: 978-0-470-75862-5

870

APPENDIX D

(Continued ) Standard designation WR

US Mil.1

UK Mil. IEC

WR90

RG52 (b) RG67 (a) RG346 (c) RG347 (a) RG91 (b) RG349 (a) RG352 (c) RG351 (a) RG53 (b) RG121 (a) RG354 (c) RG96 (s) RG271 (c) RG97 (s)

WG16

WR75 WR62 WR51 WR42 WR34 WR28 WR22 WR19 WR15 WR12 WR10 WR8 WR7 WR4 WR3 1

RG98 (s) RG99 (s) RG138 (s) RG136 (s) RG137 RG139 (s)

a

Frequencies (GHz) fmin fmax

R100

8.20–12.40

fc

inch

b mm

inch

6.56 0.9

22.86

0.4

mm 10.16

WG17

10.0–15.0

7.87 0.75

19.05

0.375

9.525

WG18

12.40–18.00

9.49 0.622

15.799 0.311

7.899

WG19

15.00–22.00

11.58 0.51

12.954 0.255

6.477

WG20

18.00–26.5

14.06 0.42

10.668 0.17

4.318

WG22

20.0–33.0 26.50–40.00

17.37 0.34 21.09 0.28

8.636 0.17 7.112 0.14

4.318 3.556

5.69 4.775 3.759 3.099 2.54 2.032 1.651 1.092 0.864

2.845 2.388 1.88 1.549 1.27 1.016 0.826 0.546 0.432

WG23 WG24 WG25 WG26 WG27 WG28

33.00–50.00 40.00–60.00 50.00–75.00 60.00–90.00 75.00–110.0 90.00–140.0 110.0–170.0 170.0–260.0 220.0–325.0

26.36 31.41 39.90 48.41 59.06 73.82 90.85 137.34 173.69

0.224 0.188 0.148 0.122 0.1 0.08 0.065 0.043 0.034

0.112 0.094 0.074 0.061 0.05 0.04 0.0325 0.0215 0.017

(a) ¼ aluminium, (b) ¼ brass, (c) ¼ copper, (s) ¼ silver.

Standard coaxial cables Table D.1 Flexible cables 2a Denomination

Dielectric type

RG-6/U RG-8/U RG-11/U RG-58/U RG-59/U RG-174/U RG-178/U RG-179/U RG-213/U RG-214/U RG-223 RG-316/U

PE PE PE PE PE PE PTFE PTFE PE PTFE PE (foam) PTFE

2b

2c

Z0 ðOÞ

inch

mm

inch

mm

inch

mm

75 50 75 50 75 50 50 75 50 50 50 50

0.185 0.285 0.285 0.016 0.242 0.1 0.033 0.063 0.285 0.285 0.285 0.06

4.7 7.2 7.2 2.9 3.7 2.5 0.84 1.6 7.2 7.2 7.2 1.5

0.039 0.085 0.064 0.035 0.032 0.019 7 0.0039 7 0.0039 7 0.0012 7 0.0012 0.108 7 0.0067

1 2.17 1.63 0.9 0.81 0.48 7 0.1 7 0.1 7 0.0296 7 0.0296 2.74 7 0.17

0.270 0.405 0.412 0.195 2.42 0.1 0.071 0.098 0.405 0.425 0.405 0.102

8.4 10.3 10.5 5 6.1 2.54 1.8 2.5 10.3 10.8 10.3 2.6

APPENDIX D

871

Table D.2 Semi-rigid cables 2c Denomination

Dielectric type

UT-020 UT-031 UT-034 UT-047 UT-056 UT-085 UT-120 UT-141 UT-215 UT-250 UT-325 UT-390

PTFE PTFE PTFE PTFE PTFE PTFE PTFE PTFE PTFE PTFE PTFE PTFE

Z0 ðOÞ

inch

mm

50 50 50 50 50 50 50 50 50 50 50 50

0.020 0.031 0.034 0.047 0.056 0.085 0.141 0.141 0.215 0.250 0.325 0.390

0.508 0.787 0.864 1.194 1.422 2.159 3.581 3.581 5.461 6.350 8.255 9.906

fmax ðGHzÞ 270 175 115 110 61 61 40 34 21 19 14 12

Notes: 1. PE ¼ polyethylene; PTFE ¼ polytetrafluoroethylene. 2. The quantity 2b is the internal diameter of the outer conductor, which coincides with the outer diameter of the dielectric. Conversely, 2c is the overall external diameter of the cable which includes the thickness of the outer conductor and the external dielectric jacket, if present. See Figure D.1. 3. RG flexible cables have an external dielectric jacket, while semi-rigid UT cables do not. Therefore, in the latter case, the overall diameter is the dielectric diameter plus twice the thickness of the outer conductor. 4. The inner conductor of the cables RG-178/U, RG-179/U, RG-213/U, RG-214/U and RG-316/U consists of seven twisted cylindrical conductors. The resulting section is not exactly circular; rather it has the aspect shown in Figure D.2. 5. The maximum frequency of a coaxial cable coincides with the cut-off frequency of the first nonTEM mode (TE). The values listed for the semi-rigid cables are indicative. Their exact values depend on the effective internal diameter 2a of the outer conductor, which is not specified, and varies from manufacturer to manufacturer. 6. The value of the inner diameter 2b for the semi-rigid cables also depends on 2a and thus on the thickness c a of the outer conductor. However, the ratio a=b is such that the characteristic impedance of the cable Z0 ¼ 60 e0:5 lnðb=aÞ is equal to the specified value. r 7. Table D.2 lists 50 O cables only. Other characteristic impedances are possible. More generally, the denomination UT-D-Z0 indicates a semi-rigid coaxial cable with an overall external diameter, expressed in tenths of inches, equal to D, and with characteristic impedance equal to Z0 ohms. For example, UT-085-75 indicates a cable with an external diameter 2c ¼ 0:085 inches and Z0 ¼ 75 O. The value of Z0 affects the inner conductor diameter and, consequently, the maximum frequency.

872

APPENDIX D

2b

2c

2a

Inner conductor Outer conductor External dielectric sheath

Figure D.1 Section of a cylindrical coaxial cable.

1

3

2

4

6

5

7

Circumscribed circle Circle with the same area as the sum of the seven conductor sections

Figure D.2 Section of the inner conductor of the cables RG-178/U, RG-179/U, RG-213/U, RG-214/U and RG-316/U.

Appendix E

Symbols for electrical diagrams High-frequency port

+

voltage generator

High-frequency terminal DC or low-frequency port

current generator

marked node solder dot

+ vin -

ground

+ vin -

+

voltage controlled voltage source (VCVS)

E•vin

gm•vin

voltage controlled current source (VCCS)

inductor resistor

i

+

H•i

current controlled voltage source (CCVS)

capacitor generic bipole

i

F•i

current controlled current source (CCCS)

diode

BJT, npn

varactor

BJT, pnp


874

APPENDIX E

4-terminal transmission line

FET, n-channel

2-terminals shunt short-circuit stub

FET, p-channel

2-terminals shunt open-circuit stub

MOSFET, n-channel

series short-circuit stub

MOSFET, p-channel

series open-circuit stub

2-terminal transmission line

coupled lines

microstrip

coupled microstrips

termination

amplifier

attenuator

oscillator

variable attenuator

phase shifter

lowdpass filter

VCO

R

I L

R

I L

mixer

sampling mixer

APPENDIX E

highpass filter

bandpass filter

A/D

analog to digital converter

bandstop filter

D/A

digital to analog converter

all-pass filter

Wilkinson divider

coupled-lines directional couplert

0° 90° 90° 0°

0° 180° 0° 0°

detector

hybrid junction, 90

hybrid junction, 180

directional coupler

875

Appendix F

List of acronyms A/D ABC AC ADC AGC ALC AM AM–PM ASCII BER BiCMOS BITE BJT BPF BRF CAD CAE CAM CCCS CCIR CCVS CMOS CPW CRT CW D/A DAC DBM DC

Analogue-to-Digital converter Absorbing Boundary Conditions Alternating Current Analogue-to- Digital Converter Automatic Gain Control Automatic Level Control Amplitude Modulation Amplitude Modulation to Phase Modulation American Standard Code for Information Interchange Bit Error Rate Bipolar Complementary Metal Oxide Semiconductor Built-In Test Equipment Bipolar Junction Transistor Bandpass Filter Band-Reject Filter Computer-Aided Design Computer-Aided Engineering Computer-Aided Manufacturing Current-Controlled Current Source Consultative Committee on International Radio Current-Controlled Voltage Source Complementary Metal Oxide Semiconductor Coplanar Waveguide Cathode Ray Tube Continuous Wave Digital-to-Analogue converter Digital-to-Analogue Converter Doubly Balanced Mixer Direct Current


878

APPENDIX F

DCOW DFT DP DR DRO DSB DSP DUT EBG ECL EDA EHF EM EMC ENR enr EURAMIG FDFD FDTD FEM FET FFT FM GAM GPS GSM HEMT HPF I1dB IC IEM IF IFM IIP2 IIP3 IL IM2 IM3 IMD IMPATT IP2 IP3 IRM JFET LC l.h.s. LINC LMDS LNA LO

Direct Current (test) On Wafer Discrete Fourier Transform Differential Pair Dielectric Resonator Dielectric Resonator Oscillator Double Sideband Digital Signal Processor Device Under Test Electronic Band Gap Emitter Coupled Logic Electronic Design Automation Extra High Frequency Electromagnetic Electromagnetic Compatibility Excess Noise Ratio (in dB) Excess Noise Ratio (in linear units) European Radio And Microwave Interest Group Finite Difference Frequency Domain Finite Difference Time Domain Finite Element Method Field Effect Transistor Fast Fourier Transform Frequency Modulation Generalized Admittance Matrix Global Positioning Satellite/Global Positioning System Global System for Mobile Communications/Generalized Scattering Matrix Heterojunction Electron Mobile Transistor Highpass Filter Input 1 dB compression point Integrated Circuit Image-Enhanced Mixer Intermediate Frequency Instantaneous Frequency Meter Input second-order Intercept Point Input third-order Intercept Point Insertion Loss Second-order Intermodulation Product Third-order Intermodulation Product Intermodulation (or Intermodulation Distortion) Impact Avalanche Transit Time Second-order Intercept Point Third-order Intercept Point Image Reject Mixer Junction Field Effect Transistor Inductor Capacitor Left Hand Side Linear Amplification with Nonlinear Components Local Multipoint Distribution Systems Low-Noise Amplifier Local Oscillator

APPENDIX F LPF LSB LSE MBFN MEMS MESFET MIC MKS MLS MMDS MMIC mmW MoM MOS MOSFET MSG MUT mW NBW NFM NG NWA O1dB OIP2 OIP3 PA PAE PCB PCM PCS pec PFM PHEMT PIN PLL PM pmc PPW PRI PRR PW radar RBW RC RF RFIC RFID RFOW r.h.s. RL

Lowdpass Filter Lower Sideband Longitudinal Section Electric Multi-Beam Forming Network Microelectromechanical Systems Metal Epitaxial Field Effect Transistor Microwave Integrated Circuit Metre Kilogram Second Microwave Landing System Multipoint Multichannel Distribution Systems Monolithic Microwave Integrated Circuit Millimetre Wave Moment Method/Method of Moments Metal Oxide Semiconductor Metal Oxide Semiconductor Field Effect Transistor Maximum Stable Gain Material Under Test Microwave Noise Bandwidth Noise Figure Meter Noise Generator Network Analyzer Output 1 dB compression point Output second-order Intercept Point Output third-order Intercept Point Power Amplifier Power-Added Efficiency Printed Circuit Board Process Control Monitor/Pulse Code Modulation Personal Communication Systems Perfect Electric Conductor Phase and Frequency Modulation, angular modulation Pseudo-morphic Heterojunction Electron Mobile Transistor P-doped Intrinsic N-doped Phase-Locked Loop Phase Modulation Perfect Magnetic Conductor Parallel-Plate Waveguide Pulse Repetition Interval Pulse Repetition Rate Pulse Width Radio Detection And Ranging Resolution Bandwidth Resistor Capacitor Radio Frequency Radio Frequency Integrated Circuit Radio Frequency Identification Radio Frequency (test) On Wafer Right Hand Side Resistor Inductor

879

880 RL RLC RMS ROM RTX RX RX–TX SA SAW SBF SBM SCA SDLVA SEM SFDR SFG S/H SHF SHM SMD SMT S/N SNWA SPDT SPICE SPnT SPST SSA SSB SSG TE TEM TFR TM TRX TTD TTL TWA TX TX–RX UHF UMTS USB UWB VBW VCCS VCO VCVS VHF VLSI VNWA

APPENDIX F Return Loss Resistor Inductor Capacitor Root Mean Square Read-Only Memory Receiver–Transmitter, transceiver Receiver Receiver–Transmitter, transceiver Spectrum Analyzer Surface Acoustic Wave Stopband Filter Singly Balanced Mixer Step-Controlled Attenuator Successive Detection Logarithmic Video Amplifier Single-Ended Mixer/Scanning Electron Microscope Spurious Free Dynamic Range Signal Flow Graph Sample and Hold Super High Frequency Subharmonically Pumped Mixer Surface Mount(ed) Device Surface Mount Technology Signal to Noise Ratio Scalar Network Analyzer Single Pole, Double Throw Simulation Program with Integrated Circuit Emphasis Single Pole, n Throw Single Pole, Single Throw Solid State Amplifier Single Sideband Small-Signal Gain Transverse Electric Transverse Electromagnetic Thin-Film Resistor Transverse Magnetic Transmitter–Receiver, transceiver True Time Delay (phase shifter) Transistor–Transistor Logic Travelling Wave Amplifier Transmitter Transmitter–Receiver, transceiver Ultra High Frequency Universal Mobile Telecommunications Standard Upper Sideband Ultra Wide Band Video Bandwidth Voltage-Controlled Current Source Voltage-Controlled Oscillator Voltage-Controlled Voltage Source Very High Frequency Very Large-Scale Integration Vector Network Analyzer

APPENDIX F VSWR WCDMA WLANs YIG ZIF

Voltage Standing Wave Ratio Wide-band Code Division Multiple Access Wireless Local Area Networks Yttrium Iron Garnet Zero Intermediate Frequency

881

Index 0 dB coupler, 212 1 dB compression point, 326, 328, 331, 334, 366, 457, 474–5 180 hybrid (see also 180 hybrid junction, hybrid in phase opposition, hybrid T, magic T), 209, 224, 454, 468, 588, 594, 606, 616–7 90 hybrid junction (see 90 hybrid coupler) 396, 401, 404, 677, 731 ABCD matrix, 111–2, 117–20, 122, 265, 267, 289, 306–8 absorbing boundary conditions, 770 absorptive switch, 365 admittance inverter, 264, 267 admittance parameters, 111, 114, 344 advanced MIC, 637, 642 alternating current, 877 amplitude analyzer, 817, 819 amplitude modulation, 329, 660, 877 amplitude modulation to phase modulation, 329, 877 amplitude modulation to phase modulation conversion (see AM-PM conversion) AM-PM conversion, 329, 482, 668, 673 analogue to digital converter, 395, 696, 808, 877 analysis, 3, 37, 76, 85–7, 91, 142, 167, 199, 207, 210, 214, 289, 296, 307, 313, 316, 329, 331, 338, 342, 346, 348–50, 356, 389, 402, 416, 423, 433, 438, 450, 454, 470, 474–5, 483, 489, 493, 496–7, 512, 516–8, 520, 524, 526, 528, 530–2, 534, 544, 546, 565, 570, 572–3, 575, 592, 594, 598, 600, 605, 614, 618, 643, 649–52, 660, 666, 669, 678, 682–3, 685, 688, 707, 710, 713, 719, 722–3, 725–6, 735, 744, 752, 757–60, 771–2, 780–2, 785–6, 788, 806–8, 823, 828, 842, 851 angular modulation (see also frequency modulation, phase modulation, FM, and PM), 663, 664, 665, 671 anti-resonance, 133, 139–40 anti-resonant circuit (see also parallel resonant circuit), 160, 503–5, 507–8, 512, 516, 530 approximation problem, 239 attenuation vector, 23, 26 automatic gain control, 691, 877 automatic level control, 675, 877 available power gain, 417, 419

back-end process, 653 backward difference, 764 balun, 591, 593, 616–8, 701–2 bandpass filter, 257–60, 266, 270, 274, 277, 299–300, 315, 470, 472, 541, 555, 629, 666, 671, 674, 678, 681–3, 692, 696, 875 band-pass filter (see bandpass filter) band-reject filter (see bandstop filter) bandstop filter, 238, 254, 260–1, 270 band-stop filter (see bandstop filter) Barkhausen criterion, 514 basis functions, 761, 762 beam lead, 589–90, 636 BiCMOS, 656, 877 bilateral noise figure measurement, 834 bilinear transform, 97, 100 binary attenuator, 397, 398 binomial transformer, 168, 171–2, 175–6 bipolar and complementary metal oxide semiconductor (see BiCMOS) BiCMOS, 656, 877 bipolar device (see also bipolar transistor, BJT), 415, 451, 519, 523, 656 bipolar junction transistor, 334–5, 338–40, 345, 347–8, 356, 358–9, 440–1, 448, 459, 489, 493, 495, 530, 557, 561, 606, 668, 873 bipolar junction transistor (see BJT) bipolar transistor (see also bipolar device, BJT), 338, 347–8, 358–9, 492, 551–2, 877 bit error rate, 729, 877 BJT (see also bipolar transistor, bipolar device), 334, 338–40, 345, 356, 358–9, 440–1, 448, 459, 489, 493, 495, 530, 555, 557, 561, 606, 668, 873, 877 Blass matrix, 224–7 blocker (see blocking signal) blocking signal, 731 branch-line coupler, 214–7, 222, 404, 732 built in test equipment, 720, 724–5, 877 Butler matrix, 224–5 Butterworth filter, 240–2 CAD, 641, 710, 719, 757–8, 760, 763, 788, 877 CAE, 343–4, 416, 472, 512, 572, 577, 603, 628, 640, 644, 650, 652, 734, 757, 760, 877


884

INDEX

calibration, 342, 430, 655, 797, 818, 821, 827–33, 836–7, 842, 845, 846–7 canonical (network, form, filter), 247–8, 256, 270, 272–3, 522 capacitive iris, 777–9 carrier, 336–7, 339, 341–2, 356, 375, 358, 474, 477, 482, 498–9, 538, 540, 542–3, 549, 557–8, 628–9, 635–6, 639, 660–3, 665, 668–9, 678, 680–1, 685–6, 688, 699, 712, 717, 719, 721, 723, 731 carrier lifetime, 370–1, 376 cathode ray tube, 803–6, 808, 817, 877 Cauer filter, 244, 247–8, 688 central difference, 764–5, 768 chain matrix (see ABCD matrix) change of reference impedance, 100 characteristic impedance, 50–4, 74–84, 87–8, 95–7, 100, 103–8, 113–4, 117–9, 123, 125–7, 137, 167–8, 172, 175–6, 179, 181, 193–5, 197, 203, 209, 218, 221–2, 238, 250–2, 264–5, 269–70, 273, 281, 283–4, 296, 299, 374, 378, 387, 402, 404, 435–6, 438, 446, 485, 497, 499, 544–6, 590, 593, 643, 744, 780, 782, 789, 827, 871 Chebysheff (see Chebyshev) Chebyshev filter, 301, 240, 242–5, 247–8, 315, 402, 688, 809, 812 Chebyshev polynomials, 168, 172–5, 240–1, 244 Chebyshev approximation, 240 Chebyshev, response, 168, 171, 241, 245, 247, 251, 284, 722 Chebyshev, transformer, 172, 174, 176–7 circuit, 2–4, 20, 50, 52, 82, 84, 86–7 circuit theory, 1–2, 19, 50, 52, 91, 94, 131, 237, 514, 758 circuit element, 2–5, 19, 39, 91, 146, 167, 195, 220, 249, 263–5, 407, 487, 497, 499, 530, 546, 633, 636, 639, 653, 780, 785 circuit model, 3, 86, 94, 119, 123, 284, 305, 403, 512, 780 circular polarization, 13 Circulator, 188, 222, 233, 710–1, 713, 719–20 class A, 440, 442, 459–3, 466–7, 470, 472–3, 477 class AB, 461, 468–9, 473 class B, 459, 461–2, 467–70, 472–3, 478 class C, 459, 462, 466–7, 473, 477–9 class D, 470, 473 class E, 462, 470–1, 473 class F, 462, 472–3 cleaning filters, 262 cold FET, 390, 573, 604–6 cold FET mixer, 605–6 cold noise figure measurement, 837 cold switching, 366 collector efficiency, 444 comb generator, 704, 706, 800–2, 810, 813 combiner, 188, 204–5, 678–9, 714 combline (see comb-line) comb-line filter, 281–4, 722

commensurate filters, 264, 278, 299 commensurate response, 263 complementary metal oxide semiconductor, 340, 656, 701, 732, 799, 877 complete set of of basis functions, 761 complete set of of orthonormal functions, 59 complete set of of vector functions, 92 computer aided design (see CAD) computer aided engineering (see CAE) computer aided manufacturing, 877 computer analysis, 605, 651, 757 Computer-Aided Design (see CAD) Computer-Aided Engineering (see CAE) condition of separability, 43, 143 conduction angle, 460–1, 465–8, 472 conductivity, 11, 15, 17, 19, 35–6, 52, 59–61, 144–6, 188, 275, 335, 656, 846 conformal mapping, 77, 80, 84 conjugate matching, 107, 426–7, 435, 496 constant gain circles, 428 constant noise circles, 431 constant Q circles (see also constant Q curves), 103 constant Q curves (see also constant Q circles), 102 constitutive parameters, 11 constitutive relations, 10–1, 14, 19, 785 consultative committee on international radio, 4, 877 continuity equation, 10, 14 continuously variable attenuator, 390–1, 393–4, 397–8 conversion gain, 579, 590, 593, 598–9, 602, 606, 609–13, 615–6, 625, 628–9, 677, 695, 726, 823, 834 conversion loss, 579, 590, 597, 599–600, 606, 623–4, 807, 809, 815–6, 841 coupled port, 127, 194, 212, 451, 555, 720, 818–9, 823 coupled-line coupler, 217 coupling, 84, 88, 127, 142, 147, 198, 212, 214, 218–9, 225, 277, 281, 347, 352, 376–7, 380, 476, 488, 497, 508, 528, 537, 556, 581, 619, 643, 758, 777–8, 818, 822, 827 coupling coefficient, 127, 225, 476, 537, 556, 818, 822, 827 Courant stability condition, 770 cross over (see crossover) crossover, 212, 224–5, 710 crossover distortion, 470 current controlled current source, 339, 345, 442, 873, 877 current controlled voltage source, 347, 448, 873, 877 Curtice model, 356, 359, 377, 602 cutoff (see cut-off) cut-off frequency, 39, 46–8, 56, 58, 63–5, 72, 76, 183, 195–6, 199, 239–40, 243–4, 252, 255–7, 300, 373, 402, 450, 484, 487, 590, 628, 645, 688, 719, 723, 727, 748–9, 800, 812, 839, 841, 845, 871 cyclotron frequency, 231–2

INDEX D/A, 407, 481, 875, 877 DAC, 703, 877 damping factor, 135–6, 144 DBM, 590–4, 600, 606–7, 610–1, 615, 619–20, 671, 877 DC input power, 443–4 DCOW, 655, 878 degenerate modes, 57–8, 70, 866 delay line, 193, 229 denormalization, 239, 252, 255, 257, 259, 263 de-normalization (see denormalization) depletion FET, 340 de-sense, 713 detector video bandwidth, 569–70, 797, 805, 807, 877 detector video resistance, 569 device under test, 342–4, 795, 816–8, 820–1, 823, 825–9, 832–7, 839–40, 843, 845, 878 DFT, 826, 878 dielectric permittivity, 10, 15, 87, 634, 640, 846, 847 dielectric resonator, 142, 146–7, 157, 547–8, 878 dielectric resonator oscillator, 547, 878 dielectric rigidity, 61, 648 dielectric strength (see dielectric rigidity ) differential form, 9–10, 15, 770 differential pair (see DP) diffraction, 2–3, 636, 654 digital attenuator, 193, 399 digital signal processor (see DSP) digital to analog converter (see DAC, D/A) digital to analogue converter (see DAC, D/A) diode quad (see diode ring) diode ring, 590 diplexer, 710–1 Dirac pulse, 538, 662–3, 669, 703–5, 813, 862 direct current [test] on wafer (see DCOW) direct port, 127, 194, 212 direct search methods, 791 directional coupler, 84, 127, 188, 204, 211–4, 218, 224–5, 441, 475, 536, 555, 581, 602, 606, 623, 625, 718, 720, 722, 735, 744, 757–60, 818–9, 821, 825, 845, 875 directivity, 199, 212–3, 227, 820–2, 826–8, 832 Dirichlet condition, 44 discontinuity, 91, 93, 142, 146, 168–9, 176–7, 188, 195–7, 203, 253, 274, 323, 602, 636, 675, 772–7, 780, 791 discrete Fourier transform (see DFT) dispersion, 11, 15, 55–6, 79, 82, 177, 275, 334, 407, 581, 597, 640, 782 dispersion diagram, 48, 56, 81 dispersive, 11, 15, 40, 81, 167, 786 distributed amplifier (see TWA) divider, 188, 204–9, 221, 224, 294–8, 448, 457, 475–6, 480, 516, 528, 567–8, 580, 598, 606, 628–9, 675, 677, 683, 688, 694, 697, 699, 713, 735, 740, 743, 798–9, 821, 837, 841, 875

885

Doherty amplifier, 477–8 dominant mode, 48, 65, 81, 83–4, 93, 145, 155, 190, 198, 202–3, 232, 772, 775 doping, 334–6, 339–41, 639, 641 double side band (see DSB) doubly balanced mixer (see DBM) downconversion, 579, 598 DP, 489–94, 502, 528, 530, 555–6, 606–7, 609, 612, 616–8, 703, 732, 878 drain efficiency, 444 DSB, 663, 878 DSP, 808, 878 dual network, 247, 256, 273, 594–5 dual-mode filter (see dual mode filter) duplexer, 711–2 dynamic effective permittivity, 82 E plane, 40, 65, 196–7, 210 EDA (see electronic design automation) edge coupled filter (see edge-coupled filter) edge-coupled filter, 277–81, 577, 629 effective permittivity, 79–82, 84, 140–1, 439, 732, 846 efficiency (see collector efficiency, drain efficiency, power added efficiency) EHF, 4, 878 eigenfunction, 46, 760 eigensolution, 44–5, 48 eigenvalue, 44–6, 58, 62–3, 69–71, 143, 149–52, 865 eigenvector, 148–54, 865 electrical length, 94, 119, 123, 125, 165–6, 169, 176–8, 193, 206, 229, 263, 265, 267, 271, 275, 277–9, 281, 283–4, 387, 435–6, 438–9, 497, 537, 545–6, 839 electro magnetic (see electromagnetic) electromagnetic, 1–2, 6, 9, 11, 25, 91, 203, 237, 365, 367, 380, 382, 638, 703, 715, 719, 727, 754, 757–8, 878 electro-magnetic (see electromagnetic) electromagnetic compat ibility, 6, 878 electromagnetic simulation, 203 electronic design automation, 757, 878 elliptic filters (see Cauer filters) elliptically polarized, 28 EMC (see electromagnetic compatibility) end effect, 140, 279, 846 energy velocity, 55–7 enhancement FET, 340 ENR, 878 enr, 836–7, 878 envelope, 450–1, 479, 505, 561, 660–1, 677, 665, 688, 720, 723–4, 744–5, 748–9, 809 envelope detector (see also linear detector), 561, 570, 572, 577, 751 envelope elimination and restoration, 479 E-plane (see E plane) equal-ripple (See Chebyshev)

886

INDEX

equiripple (See Chebyshev) equi-ripple (See Chebyshev) equivalent circuit, 86, 92–3, 128, 138, 157, 161, 179, 190, 196–7, 251, 281, 283–4, 308, 353–4, 357–9, 370, 382–3, 478, 485, 490, 508, 521–5, 529, 531, 547, 641, 645, 649, 651, 772, 775 error function, 740, 789–90 european radio and microwave interest group, 6, 878 excess noise ratio (see enr, ENR) excitation, 115, 157, 159, 195–6, 199, 207–8, 226–7, 229, 305–6, 316–9, 321, 326–7, 332, 348, 350, 370, 372, 441, 457–61, 467, 493–4, 505–6, 513, 528, 562, 565, 620, 622, 680, 738, 746, 761, 770, 780–1, 786 external quality factor, 136 extra high frequency, 878 Fano, 163, 183, 424, 428–9, 568 Fano’s inequality, 164 fast Fourier transform (see FFT) FDFD, 766, 769, 771, 878 FDTD, 766, 769–70, 878 feedback, 435, 450, 483–5, 495–6, 503, 512–8, 523, 525–8, 530, 534, 540, 543, 547, 550, 555, 557, 616, 675, 744 feed-back (see feedback) feedforward, 475, 477 feed-forward (see feedforward) FEM, 770–1, 878 ferrite, 109, 230–1 FET, 190, 339–40, 345–6, 353–6, 359, 363, 367, 375–9, 382, 390–3, 408, 415–6, 440, 449, 459, 485, 487, 489, 493, 495–6, 502, 519, 523, 530, 555, 557, 561, 573, 575–7, 602–3, 605–6, 628, 646, 648–51, 668, 711, 874, 878 FET nonlinear models (see Curtice model, Statz model) FET periphery, 649 FFT, 770, 878 field effect transistor (see FET) field matching, 771 fin line, 39–40 finite difference frequency domain (see FDFD) finite difference time domain (see FDTD) finite element method (see FEM) finline (see fin line) first design, 757 FM, 663–5, 673–5, 681, 685, 688, 691, 731, 878 forward difference, 764 foundry, 639, 650–2 foundry manual, 639–41, 649, 652 Fourier series, 13, 92, 369, 463, 469, 535, 592, 605, 627, 720, 761, 763, 770, 772, 778, 786–7 Fourier transform, 303, 538, 569, 622, 660, 662, 669, 674–5, 686, 705, 709, 737, 739, 741, 746–7, 780, 804, 811, 816, 840, 861–3 free oscillation, 42, 91, 131, 133–5

frequency conversion, 204, 619, 682, 703, 713, 727, 798–800, 802, 807, 809, 821, 826 frequency divider, 675, 699, 735, 743, 798–9 frequency division, 700, 799, 802 frequency division factor, 675, 699–700, 736, 740–3 frequency domain, 14, 21, 195, 343, 509, 770, 786, 862 frequency doubler, 625, 628 frequency modulation (see FM) frequency multiplier, 316, 321, 561, 625, 628, 709 frequency resolution, 740, 799, 800, 805 frequency tripler, 625 front-end process, 653 full wave simulation, 758–9 fundamental mode, 48, 61, 63, 67, 71–2, 75, 78, 82, 93–4, 155, 203, 719, 776 gain ripple, 494, 744, 748 Galerkin’s method, 762 GAM, 155, 878 generalized admittance matrix (see GAM) generalized Chebysheff filters, 244 generalized scattering matrix (see GSM) generalized transmission lines, 41, 49–50, 91, 93–4, 105, 131 global minimum, 792 global positioning satellite (see GPS) global system for mobile communications (see GSM) GPS, 5–6, 878 gradient methods, 791–2 ground redefinition, 524 group delay, 265, 275, 402, 476, 680, 722, 724, 727 group velocity, 55–7 GSM, 155, 489, 699, 878 gyromagnetic ratio, 231 H plane, 65, 196–7, 210 harmonic balance, 784, 786, 788 harmonic distortion, 317 Helmholtz’s equation (see also wave equation), 21, 38, 41–3, 57, 141, 758 HEMT, 356, 367, 375, 878 heterojunction electron mobile transistor (see HEMT) HFET, 367, 375 high-pass filter, 256–7, 410, 875, 878 hot switching, 366 H-plane (see H plane), 210 hybrid, 204, 209, 216, 224, 396, 401, 404, 407, 413, 454, 456, 585–6, 588–9, 594, 598, 677, 679, 694, 732 hybrid coupler, 438, 454–8, 468, 584, 590, 600, 606, 616–7, 826, 841–2 hybrid in phase opposition, 458 hybrid junction, 194–5, 204, 209–12, 216, 219, 221, 584, 593, 598, 606, 735, 875 hybrid MIC, 497, 633, 637 hybrid mode, 43, 82 hybrid ring, 209, 221

INDEX hybrid T (see magic T) hybrid technology, 487, 633, 635 IC, 378, 494, 497, 552, 701–3, 732, 734, 843, 878 IEM, 599–600, 878 IF, 577, 579–84, 586, 588–90, 592–4, 596–600, 602–5, 608–11, 615–24, 669, 677–8, 682, 689, 692–4, 696–7, 699–703, 709, 712, 726–7, 800, 802, 807–8, 813, 815–6, 823–4, 826, 833, 839–40, 845, 877 IFM, 837, 839, 842, 847 image enhanced mixer (see IEM) image frequency, 597, 599–600, 693–4, 696, 702, 727, 807 image reject mixer (see IRM) IMD, 320, 878 immittance inverter, 267, 269 impedance inverter, 264, 267–271, 273–4, 277–8, 280, 283 impedance matrix (see also Z matrix), 109–10, 112, 114–5, 117–8, 122, 124–5, 190, 250, 785 impedance parameters (see also Z parameters), 109, 117 impedances normalized, 95 implantation, 335, 646 impressed current, 199, 509–511 incident wave, 2, 28, 30, 32, 34, 36, 44, 47–8, 53, 73, 91, 95, 105, 107–8, 113, 145, 194, 238, 302, 316, 396, 417–8, 454, 485, 692, 694 inner product, 761–2 input 1 dB compression point, 325, 332, 729, 878 input second-order intercept point, 878 input third-order intercept point, 328, 333, 624, 878 insertion loss, 187, 199, 206, 222–3, 238, 244, 295, 315, 365–6, 371, 374, 377–80, 382, 385–8, 393, 397–9, 409, 412, 452, 581, 688, 713, 718, 722, 727, 805, 820, 878 integral form, 10, 14–5, 769 integrated circuit (see also IC, MIC, MMIC, RFIC), 39, 190, 289, 340, 415, 494, 652, 656, 682, 694, 698–9, 701, 878 intermediate frequency (see IF) intermodulation, 317, 319–21, 326, 332–3, 348, 381, 429, 468, 477, 482, 622–4, 666, 699, 729, 731, 786 intermodulation distortion (see intermodulation, IMD) intrinsic impedance, 26, 36, 49, 60, 146 intrinsic Q (see unloaded Q) IQ mixer, 598, 731, 826 IRM, 597, 694, 878 isolated port, 127, 219, 600, 731 isolation, 187, 206, 212–3, 295, 365–6, 371, 374, 377–81, 383, 385–9, 392, 398, 498, 543, 558, 647, 673, 710–12, 719, 827–8 Isolator, 68, 188, 232–3, 543, 673 istantaneous frequency meter (see IFM) iterative procedure, 760, 792

887

JFET, 340–2, 356, 878 junction diode, 334, 336, 368, 370, 549 junction field effect transistor (see JFET) Kotelnikov theorem (see sampling theorem) Kronecker delta, 58, 772 Kurokawa stability condition, 423, 481 Larmor frequency (see cyclotron frequency) LC, 157, 221–2, 247, 258–60, 270, 408, 463, 485, 546, 651, 789, 878 leap-frog algorithm, 769 least pth approximation, 790 least squares approximation, 790 Leontovic boundary condition, 20 Leontovic condition, 60, 145 LINC, 480–1, 878 linear amplification with nonlinear components (see LINC) linear approximation, 370, 737 linear detector (see also envelope detector), 561, 661 linear polarization, 13 LMDS, 878 LNA (see also low-noise amplifier), 430–1, 458, 481, 489, 499, 845, 878 LO, 577, 579–82, 584–90, 592–4, 596–606, 608–13, 615–20, 622–3, 625, 669, 671, 677–8, 680, 688, 692–4, 696, 699–702, 709, 725–7, 731, 734, 800, 802, 813, 816, 821, 826, 837, 839, 841–2, 878 LO harmonic index (see LO index) LO harmonic order (see LO index) LO index, 620, 622 load line, 443, 446, 448–9, 565 load pulling, 451, 453 loaded Q (see also quality factor), 136 loaded quality factor (see also loaded Q), 136, 470–1 loaded-line phase shifter, 403 local minimum, 789–90, 792 Local Multipoint Distribution Systems (see LMDS) local oscillator (see LO) lock state, 735 Longitudinal Section Electric (see LSE) Longitudinal Section Electric modes, 777–8 long-tail pair, 616 loop filter, 676, 739, 748 Lorentz gauge, 37 low-IF, 682, 696, 699 low-loss transmission lines, 54 low-noise amplifier (see also LNA), 429–30, 497, 499 low-pass filter, 238–9, 243, 245, 248–52, 255, 257, 269, 299–300, 373, 448, 479, 481, 487, 555, 573, 590, 628, 674, 679, 688, 695, 698, 706, 709, 720, 744, 751, 800, 833, 837, 845, 874, 879 LSB, 663, 879 LSE, 777–8, 879

888

INDEX

Maclaurin series, 369, 492–3, 562–3, 581, 583, 608 magic T, 210–1 magnetic conductor, 20, 76, 140–2, 146, 157 magnetic vector potential, 37, 41 main filter, 262 majority carriers, 336–7 Mason’s rule, 303, 418, 513, 738, 783 matched termination, 187, 831 material under test (see MUT) maximally flat, 168, 171–2, 176, 240 maximally flat filter (see Butterworth filter) maximum stable gain, 428, 879 MBFN, 224, 227, 879 MEMS, 180, 363, 367, 379–83, 385, 403, 408, 410, 879 MESFET, 334, 340, 345, 355–6, 358, 367, 375–7, 381, 393, 432, 441, 448, 451, 493, 879 metal epitaxial field effect transistor (see MESFET) metal oxide semiconductor (see MOS) metal oxide semiconductor field effect transistor (see MOSFET) method of moments (see MoM) method of separation of variables, 20, 22, 43, 141 MIC, 628, 633–7, 639, 642, 879 micro electromecanical semiconductor (see MEMS) microstrip, 39–40, 74, 78–87, 91, 140–2, 146–7, 157, 177–8, 187, 192, 197–8, 203, 205, 210, 212, 214, 216, 219, 221–2, 228–9, 251–2, 264, 274–5, 277–9, 281, 372, 379–80, 438, 440, 446, 497–9, 545–7, 549, 557–8, 588–9, 628, 633, 635, 640–1, 643–4, 732, 744, 758, 763, 789, 791, 874 microwave circuit, 3, 91, 93–4, 109, 112–3, 123, 163, 197, 211, 423, 633, 663, 757–61, 763, 780, 791, 859 microwave integrated circuit (see MIC), 190, 497, 633 microwave landing system (see MLS) milimetre wave, 1, 381, 383, 497, 594, 879 minimax approximation, 790 minority carriers, 336 mixer noise figure, 624 mixer spur, 584, 618, 620, 669, 681, 694 mixer spur chart, 620 mixing products, 584, 586–8, 592–3, 596, 601–2, 606, 609, 611, 615, 618, 620, 622, 669, 671, 680, 726 MKS, 9, 879 MLS, 6, 879 MMDS, 6, 879 MMIC, 378, 482, 485, 633, 636–7, 639–41, 643, 645–7, 649, 651–6, 879 modal analysis, 771–2 mode jumping, 521, 525, 531, 536, 546, 549 mode matching, 771–2, 778–9 modulation, 339, 355, 450, 479, 579, 581, 592, 605, 648, 659–60, 663, 665–8, 671, 674–5, 677, 680–3, 688, 708, 720, 737, 740, 748, 802 modulation bandwidth, 668–9, 671 modulation depth, 661, 668–9, 673 modulation index, 664–5

modulation sensitivity, 552, 558, 579, 674 MoM, 761, 763, 771–2, 879 moment method (see MoM) monolitic microwave integrated circuits (see MMIC) MOS (see MOSFET) MOSFET, 334, 339–42, 356–8, 367, 375–6, 393, 493, 874, 879 multi-beam forming network (see also MBFN), 223–4 multi-finger FET, 648–9 multipoint multichannel distribution systems (see MMDS) mushroom gate, 648 MUT, 846–7, 879 narrow valley, 790, 792 NBW (see noise bandwidth) negative feedback, 483–4, 616 network analyzer (see also NWA), 342, 430–1, 452, 816, 846 Neumann condition, 45 NFM (see also noise figure meter), 816, 879 noise bandwidth, 314–5, 688–9, 729, 807, 835, 879 noise circles (see constant noise circles) noise factor, 306, 310–4, 429–32, 540, 624, 835–7 noise figure, 3, 306, 311, 313, 415–6, 424, 429–33, 435, 438, 440, 458, 497–9, 501, 624, 651, 655, 686, 694, 709, 712–3, 729, 807, 809, 816, 833–5, 837 noise figure meter, 816, 833, 879 nonlinear distortion, 363, 375, 377, 398, 412, 454, 468, 470, 476, 582, 624, 665, 669, 671, 680, 686, 690, 701, 707, 729 normalization, 100, 102, 152, 165, 191, 243, 392, 452, 579, 818, 820–1, 826–7, 832 normalization condition, 50–1, 58 normalized impedance matrix, 114 notch filter (see bandstop filter) numerical method, 760, 770 NWA (see also network analyzer), 816, 818, 827, 832–3, 837, 846, 879 Nyquist criterion, 513–4, 740 Nyquist theorem (see sampling theorem) objective function, 789–92 offset frequency, 538, 541, 721, 741, 808, 821, 840–1 Ohm’s law, 11, 15, 18, 60, 190 operating power gain, 417, 428, 444 operator, 24, 42, 761, 763, 825, 827 optical tunnel effect, 3 optimization, 253–4, 271, 274, 282, 344, 352, 354, 387, 407, 410, 415–6, 440, 444, 483, 488, 651, 732, 752, 760, 780, 788–91 orthogonality properties, 57, 865 orthonormalization, 58, 201, 772–4, 777, 781 oscillator phase noise, 541 oscillator pulling, 536–7, 543, 555, 558, 668–9, 673, 744

INDEX oscillator pushing, 537, 543 oscillator startup, 507, 513, 517–8, 555, 557 oscillator steady state, 507, 508–9, 511, 517–8, 533–4, 539, 555, 557, 736, 741 output 1 dB compression point, 325, 332, 457, 729, 879 output second-order intercept point, 879 output third-order intercept point, 328, 624, 879 oversampling, 706 PA (see also power amplifier), 440, 444, 459, 473–6, 479, 481–2, 495, 660, 665, 668, 671, 677–8, 720–1, 723, 879 PAE, 444, 470, 879 panoramic receiver, 803, 805–7 parallel resonance, 132–3, 137, 139, 275, 371 parallel resonant circuit, 147, 161, 503, 546 parallel-coupled filter (see edge-coupled filter) parallel-plate waveguide (see also PPW), 74, 82, 879 passband, 3, 163, 174, 238–44, 252, 255, 257–9, 262–3, 270, 273–5, 278–9, 281, 284, 299–301, 310, 315, 402, 547, 597, 619–20, 682–4, 688, 691, 693, 699, 701–3, 722, 724, 726–7, 800, 805, 812 passband filter, 129, 257, 538, 691, 698 passband ripple, 241–4, 252, 254–5, 260, 262, 271, 274, 300, 315, 410, 722, 812 PCM, 655, 879 PCS, 4, 879 pec (see also perfect electric conductor), 76, 879 perfect electric conductor, 17, 20, 44, 60, 146, 154, 879 perfect magnetic conductor, 20, 76, 141, 146, 157, 879 phase detector, 675, 735 phase modulation (see also PM), 581, 592, 660, 663, 879 phase noise, 537–43, 547, 549, 555, 579, 625, 660, 682, 685, 690, 693–6, 712, 736–41, 839–41 phase vector, 23, 26–7, 30, 36 phase velocity, 22, 25, 53–6, 58, 79, 81, 87, 137, 165, 167, 195, 252, 403–4 phased array, 193, 713, 715–6 phased array antenna, 224, 401, 713 phase-invariant attenuator, 400 phase-locked loop (see PLL) PHEMT, 367, 375, 878 piecewise-linear approximation, 330, 752, 763 PIN, 190, 367–73, 375–79, 381, 388, 390–1, 393, 399, 406, 408, 668, 711, 879 planar circuit, 39–40, 76, 82, 140–2 planar circuit, 40, 76, 140–2 planar waveguide model, 82, 140 plane wave, 21–7, 30, 36, 46–7, 49, 55, 76, 146 PLL, 543, 675–7, 701, 703, 731, 735, 737–43, 746–7, 808, 819, 843, 879 PLL closed-loop-gain, 738, 740–1 PLL frequency resolution, 740 PLL in-band noise, 739–41

889

PM (see also phase modulation), 663–5, 668, 672–5, 677, 691, 695, 719, 743, 746, 879 pmc (see perfect magnetic conductor) polarization angle, 34 polyphase phase splitter, 732–4 positive feedback, 483–4, 503, 512–3, 517, 526 power added efficiency (see PAE) power amplifier (see also PA), 415, 440, 454, 479–80, 565, 628, 656, 665–6, 668, 720, 879 power measurement unit, 795 power sensor, 795, 797 Poynting’s theorem, 17–8, 60, 127 PPW (see also parallel-plate waveguide), 74, 76, 879 predistortion, 473–5 prescaler, 799–800 preselector, 800, 802, 807 PRI, 720, 879 printed circuit, 39–40, 76, 82, 139, 197, 214, 224, 227, 277, 281, 299, 347, 457, 497, 546, 589, 633–4 printed circuit, 39, 76, 80, 139, 197, 214, 224, 227, 277, 281, 299, 347, 457, 497, 546, 633–4, 879 process library, 649–50 programmable frequency divider, 675 propagation vector, 23, 25, 30, 34, 36 prototype, 239, 245–7, 249, 252, 254–62, 266, 269–71, 274, 278, 282, 299–301, 410, 757 pseudo-morfic heterojunction electron mobile transistor (see HEMT) pull-in voltage, 381 push-pull, 469, 553, 601 push-push, 553–5, 557, 601, 741 PW, 717, 815, 879 Q (see also quality factor), 41, 102–5, 134, 136, 144–6, 494, 505–8, 512–3, 518, 528, 531, 540, 543, 546–7, 691, 722 Q factor (see Q) quadratic detector (see also quadratic-law detector), 561, 566, 571, 575, 677, 826 quadratic-law detectors (see also quadratic detector), 573 quadrature mixer (see IQ mixer) quality factor (see also Q), 40, 102, 134–5, 144, 275, 505–6, 530, 543, 547, 644, 722, 741, 846 quantization, 703, 707, 809–10, 823 quasi-static model, 78 quasi-TEM mode, 78, 81, 83, 85 quiescent current radar, 193, 223, 237, 537, 710, 719–21, 724, 879 radio frequency [test] on wafer (see RFOW) radio frequency integrated circuit (see RFIC) rat-race, 209–10, 588 RBW, 805, 807–8, 879 read-only memory, 394, 481, 842, 880

890

INDEX

receiver, 233, 390, 489, 561, 577, 579, 597, 619, 659–60, 665, 668–9, 682–3, 685–6, 689, 691–2, 694–9, 701, 702–3, 708–11, 713, 718–9, 725, 728–30, 744, 748, 800, 802, 807, 816, 821, 823, 825–6, 828, 833, 843 receiver sensitivity, 709, 729 receiver-transmitter (see transmitter-receiver) reciprocity, 19–20, 112, 116, 126, 187, 200, 204, 211, 217, 230–31, 715 reconstruction filter, 706 reference impedance, 100, 114, 117, 119, 122–4, 126, 165, 167–8, 176, 188, 191, 206, 209, 213, 257, 271, 274, 314, 363, 484 reference prototype, 255, 257, 259–62, 270 reflected wave, 28, 30, 34, 44, 53, 61, 93, 105–6, 113, 145, 169, 208, 212, 216, 290, 302, 316–7, 404, 517, 578, 600, 782, 822, 828 reflection coefficient, 31, 33–4, 53–54, 94–7, 99–102, 104–5, 107–8, 114–6, 118–9, 123–4, 127, 146, 163–4, 166–9, 171–2, 174–6, 178, 188, 195, 216, 239, 246–7, 257, 260, 271, 291–2, 294, 296, 299, 301, 344, 346, 365, 396, 401, 404, 419–20, 422, 424, 426–9, 431–2, 444, 452, 454, 458, 484, 517–8, 532–4, 536, 547, 575–6, 629, 668, 673–4, 710, 722, 728, 744, 751, 782, 818–20, 826–30 reflection phase shifter, 193, 401, 404 reflective switch, 365, 388, 393 refractive index, 29 regeneration, 513 relative convergence, 778 resolution bandwidth (see RBW) resonance isolator, 232 resonant circuit (see also resonator), 122, 132–3, 136, 470–1, 508, 523, 528, 536, 546, 550, 556 resonant condition, 131–3, 137, 139 resonant mode, 19, 131, 139, 142–7, 152, 155–6, 160–1 resonator (see also resonant circuit), 19, 131, 134–6, 138–42, 144–7, 149, 157–9, 187, 258–9, 270, 275, 277, 281, 383, 472, 503, 505–12, 516–8, 520, 527, 530–2, 534, 540, 542–3, 546–7, 549, 553, 558, 684, 691, 698, 722, 741, 770–1, 799, 800, 803, 846 retrace, 817 retrace time, 805 return loss, 163–4, 167, 176, 183, 187, 193, 199, 216, 238, 240, 244, 254, 257–8, 260, 271, 274–5, 278, 284, 293–8, 301, 365, 371, 379, 387–8, 396, 399, 402, 411–2, 569, 710, 717, 722, 724, 820–1, 832, 880 RF circuit, 91, 745, 757, 770, 782, 785, 788 RF harmonic index (see RF index) RF index, 584, 617, 620, 622 RFIC, 482, 528, 606, 616, 633, 639, 656, 879 RFID, 6, 879 RFOW, 87, 655, 879 Richards transformation, 261–4, 266, 270, 299 ROM (see read-only memory)

rooftop basis functions, 763 rotary attenuator, 190 Rotman lens, 224, 227–30 Rotman lens, 224, 227–9 S/H (see also sampling and hold), 703, 705, 809, 812–16, 880 S/N (see also signal to noise ratio), 309–10, 312–3, 688, 697–8, 729, 880 SA (see spectrum analyzer) saddle point, 790 sample and hold (see also S/H), 696, 703, 809, 880 sampled receiver, 709 sampling, 682, 703–4, 706–7, 709, 797, 809, 811–6, 825 sampling mixer, 709, 802, 815 Sampling theorem, 706, 811–2, 825 SAW, 703, 880 SBM, 584–9, 600, 606, 609, 619, 627, 671, 880 SCA, 390, 394, 397, 399–400, 667, 880 scalar network analyzer (see also SNWA), 795, 816 scattering matrix, 112–5, 117, 122–6, 155, 191, 193, 206–8, 210–1, 213–4, 216–8, 232–3, 289, 293, 302, 395, 421, 423, 454, 457–8, 469, 593–4, 780, 782–3, 785 schematic capture, 780 schematic entry, 780 Schottky diode, 370, 562 Schottky junction, 338, 649 SDLVA, 731, 745, 749, 751–2, 803, 880 second-order intercept point, 328, 878 SEM, 581, 584, 586, 594, 602, 605–6, 625, 628 sensitivity, 178, 275, 371, 381, 416, 429, 431, 453, 480, 489, 556, 568–9, 575, 576–7, 597, 644, 651, 686, 691, 709, 712–3, 729–30, 737, 805, 807, 809, 813, 815, 818, 820–1, 826, 839, 841, 848 separability condition, 23–4, 46, 62 series resonance, 133, 137–9 series resonant circuit, 503, 508, 546, 597 serrodyne, 675 SFDR, 729, 880 Shannon theorem (see sampling theorem) SHF, 4, 880 SHM (see also subharmonic mixer, subharmonically pumped mixer), 594, 596–7 601, 606, 628, 880 sideband, 537–8, 543, 663, 678–9, 681, 721–2, 724, 731 signal to noise ratio (see also S/N), 309, 624, 660, 685, 688–9, 805, 880 simultaneous conjugate matching, 426 single finger FET, 646, 648 single-ended mixer (see SEM), 581 singly balanced mixer (see SBM) skin depth, 16–7, 35–6, 145 small signal suppression SMD, 222, 499, 635, 637, 639, 830

INDEX SMT, 635, 830 SNWA (see also scalar network analyzer), 816–7, 820–1, 826, 880 solid state amplifier (see SSA) SP2T (see also SPDT), 408–10 SPDT (see also SP2T), 366, 372, 374, 378–9, 386–9, 393, 398, 400, 408–9, 411, 430–1, 451–2, 636, 641, 643, 710–11, 713, 799, 821, 880 spectral regrowth, 481–2, 622 spectrum analyzer, 451, 803, 806–9, 813, 817, 819, 845, 880 SPICE, 352, 781–2, 785–6 SPnT, 363, 365–6, 370–1, 377, 386, 399, 408, 691, 880 SPST, 363, 365–7, 371–2, 375, 377–8, 385–6, 390, 392–3, 399, 452, 845, 880 spurious free dynamic range (see SFDR) square-law detector (see quadratic detector) SSA, 415, 880 stability circle, 422–3, 431–2, 434, 481 Statz model, 357, 377, 602 steepest descent, 792 step controlled attenuator (see SCA) stopband, 3, 238, 240, 242–4, 251, 254–5, 257, 260–2, 281, 284, 299–300, 684, 710, 722 stripline, 39–40, 46, 76–8, 80, 85, 197, 251, 264, 279, 281, 446, 732, 765 stub, 91, 122, 136–8, 163, 165, 178–180, 188–9, 215–6, 259, 263–7, 270–1, 273–5, 277, 281, 299, 305, 386, 403, 408, 435–6, 440, 497–9, 544–7, 589–90, 628, 744, 874 subharmonic mixer (see also subharmonically pumped mixer, SHM), 594, 809 subharmonically pumped mixer (see also subharmonic mixer, SHM), 584, 594 subsampled receiver, 709 subsectional functions, 762–3 successive detection logarithmic video amplifier (see SDLVA) superhet (see also superheterodyne), 692–3, 698–9, 701, 712 superheterodyne (see also superhet), 597, 682, 692, 696, 699, 701, 725, 806–8, 819, 821, 826 surface acoustic wave (see SAW) surface charge, 16 surface current density, 16–7, 60, 65, 763 surface mount device (see SMD) surface mount technology (see SMT) surface wave, 30 symmetrical transformer, 171 synthesis, 167–8, 171, 191, 218, 225–7, 237–9, 242, 245–50, 252, 254–7, 259, 261–2, 266, 270, 273–4, 278, 282, 316, 389, 410, 412, 530, 722, 732, 757 tangent of delta, 15 tangential impedance, 31–3

891

Taylor series, 317, 323, 510, 539, 566, 602, 787 TE, 26–7, 30, 41–3, 45–50, 57–9, 61–3, 68, 70–1, 73, 82, 143–4, 152, 200, 772, 774, 777, 865–6, 871, 880 telegrapher’s equations, 50, 52, 87–8, 94 TEM, 25–7, 39, 41–2, 45–51, 55, 57, 59, 62, 72, 74–8, 80–1, 83, 85, 87–8, 125, 196, 198, 203, 252, 279, 281, 284, 545, 758, 871, 880 tensor permeability, 12 tensor permittivity, 12 testing functions, 762 TFR, 558, 642, 880 thin-film resistor (see TFR), 558, 642, 880 third-order intercept point, 328, 332–3, 878 time base, 799–800 time domain, 11, 14, 22, 53, 195, 303, 509, 516, 538, 586, 662, 669, 674, 678, 685, 693–4, 723, 737, 739, 747, 811, 861–3 time domain (simulation) techniques, 761, 766, 785–6 time period, 13, 18, 22 TM, 26–27, 30, 41–50, 57, 59, 61–4, 70–1, 73, 82, 141, 143–4, 152–3, 200, 772, 774, 777, 865–6, 880 total reflection, 3, 28, 30, 146, 195 Touchstone file, 343–4 Touchstone format, 343, 650 transceiver (see also transmitter-receiver), 710, 713, 716, 880 transducer power gain, 418–9, 428, 449 transfer switch, 388–9 transimpedance, 109, 737 transmission matrix (see ABCD matrix) transmission phase shifter, 193 transmitted wave, 28, 30–1, 34, 36, 93, 723, 822, 828 transmitter, 233, 561, 577, 659–60, 663, 665–6, 668–9, 672–5, 677–8, 680–3, 685–6, 689–90, 692, 695–7, 699, 710–3, 718–9, 720, 723–4, 744, 748, 843 transmitter-receiver (see also transceiver), 366, 710 traveling wave amplifier (see TWA) true delay phase shifter (see TTD) true-phase phase shifter, 193, 401, 404, 410, 412 true-time-delay phase shifer (see TTD), 193 TTD, 193–4, 401, 410 tunable filter, 555, 702, 800, 802, 804–5, 807 TWA, 485–8, 497, 882 Two-hole coupler, 214 UHF, 4, 880 undersampling, 706 uniform plane waves, 25, 27, 30, 36, 46–7, 49, 76 unilateral noise figure measurement, 834 unloaded Q, 136, 281 upconversion, 555, 559, 579, 597, 599 up-conversion (see upconversion) USB, 663, 880

892

INDEX

varactor, 380, 401, 403, 405–6, 549–52, 555, 557–8, 691, 702, 718, 800, 803, 873 varactor diode (see varactor) VBW (see video bandwidth), 569 VCO, 549–50, 552–5, 557–8, 628, 674–5, 702, 735–7, 739–41, 744, 808, 817–9, 874, 880 VCO gain, 737 vector network analyzer (see also VNWA), 451, 655, 845 via hole, 275, 277, 379, 438–9, 487, 499, 634, 636, 640–1, 649, 652–3, 655 video amplifier, 565 video bandwidth, 569, 688, 797, 805, 807–8, 880 VNWA (see also vector network analyzer), 816, 821–3, 825–8, 832–3, 880 VSWR, 95–6, 99–100, 102, 105, 757, 878 wafer, 636, 652–3, 655 walking IF, 682, 699, 701 wave equation (see also Helmholtz’s equation), 21–2, 24 wave impedance, 25–7, 48–51, 73–4, 188, 195, 203

wavelength, 1–2, 4, 6, 22, 39, 44, 47, 52, 58, 78, 96, 97, 99, 101, 103, 137, 139, 163, 165, 167, 188, 195, 201, 207, 224, 250–1, 261, 269, 281, 373, 386, 403, 543, 547, 589, 593–4, 634, 636, 638, 640–1, 646, 648, 654–5, 682, 715, 728, 732, 770, 821, 839, 848 weighting functions, 762 Wilkinson combiner, 188, 205 Wilkinson power divider, 188, 204–6, 209, 294–5, 457, 480, 606, 735, 821, 875 Wilkinson splitter, 205 wireless, 1, 237, 659 Wittaker theorem (see sampling theorem) YIG, 549, 691, 800, 802–3, 881 Z matrix (see also impedance matrix), 109–10, 114, 117, 125–6 Z parameters (see also impedance parameters), 109–11 zero IF (see also ZIF), 661, 696 zero-span, 809 ZIF (see also zero IF), 696, 698–9, 701, 878

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