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RF and Microwave Oscillator Design

For a listing of recent titles in the Artech House Microwave Library, turn to the back of this book.


Michał Odyniec Editor

Artech House Boston • London www.artechhouse.com

Library of Congress Cataloging-in-Publication Data Odyniec, Michał. RF and microwave oscillator design / Michał Odyniec. p. cm. — (Artech House microwave library) Includes bibliographical references and index. ISBN 1-58053-320-5 (alk. paper) 1. Radio frequency oscillators. 2. Oscillators, Microwave. I. Title. II. Series. TK7872.O7 O34 2002 621.384’12—dc21 2002027960

British Library Cataloguing in Publication Data Odyniec, Michał. RF and microwave oscillator design. — (Artech House microwave library) 1. Radio frequency oscillators 2. Oscillators, Microwave—Design I. Title 621.3’8412 ISBN 1-58053-320-5

Cover design by Gary Ragaglia

 2002 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. International Standard Book Number: 1-58053-320-5 Library of Congress Catalog Card Number: 2002027960 10 9 8 7 6 5 4 3 2 1

Contents

1

Preface

xi

Acknowledgments

xv

Developments of Microwave Oscillator Theory

1

1.1

Introduction

1

1.2

Van der Pol (1927)

2

1.3

J. R. Pierce (1943)

3

1.4

R. Adler (1946)

4

1.5

W. A. Edson and J. A. Mullen (1960)

7

1.6

C. T. Rucker (1969)

8

1.7

K. Kurokawa (1973) References

10 13

2

Methods of Oscillator Design

15

2.1

Introduction

15 v

vi


2.2 2.2.1 2.2.2 2.2.3

Nonlinear Dynamics of a Simple Oscillator Oscillator Equation Phase-Plane Analysis Generalizations of Phase-Plane Analysis

16 16 19 22

2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5

Stability of the Operating Point Introduction Circuit Linearization Counterexample Validity Limits of the Intuitive Criterion Conclusions

23 23 24 25 28 29

2.4 2.4.1 2.4.2

29 29

2.4.4 2.4.5 2.4.6

High-Q Oscillators Steady-State Periodic Oscillations Large Signal Impedance and Corresponding Nonlinear Characteristics Notes on Feedback Representation, High Q, and Small Parameter Large Signal S-Parameters Nonresistive Active Circuit Conclusions

33 36 38 40

2.5 2.5.1 2.5.2

Dynamics of High-Q Oscillators Introduction Oscillation Stability

40 40 41

2.6

Oscillations in the Presence of an External Signal Introduction Circuit Equations Resonance Characteristics

43 43 43 45

Summary References

48 50

Appendix 2A: Nyquist Stability Criterion

52

Appendix 2B: Justification of the Describing Function Method

53

2.4.3

2.6.1 2.6.2 2.6.3 2.7

31

Contents

vii

Appendix 2C: Transformation VoltageCurrent to Amplitude-Phase Equations

56

Appendix 2D: Theorems on Averaging Acknowledgments

58 58

Linearity, Time Variation, and Oscillator Phase Noise

59

3.1

Introduction

59

3.2

General Considerations

61

3.3 3.3.1

Detailed Considerations: Phase Noise Phase Noise of an Ideal Oscillator

64 64

3.4 3.4.1

The Roles of Linearity and Time Variation in Phase Noise Close-In Phase Noise

68 78

3.5 3.5.1 3.5.2

Circuit Examples LC Oscillators Ring Oscillators

79 79 84

3.6

Amplitude Response

88

3.7

Summary References

90 90

Appendix 3A: Notes on Simulation Acknowledgments

92 92

4

High-Frequency Oscillator Circuit Design

93

4.1 4.1.1 4.1.2

Transistor CAD-Oriented Circuit Models Introduction Homojunction and Heterojunction Bipolar Transistor Modeling

95 95

3

96

viii


4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8

4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3

FET Operating and Modeling Transistor I-V and S-Parameter Measurement System Model Extraction Procedure Noise Sources in Semiconductor Devices and Their CAD-Oriented Modeling Transistor Low-Frequency Noise Characterization Modeling of Circuit-CAD–Oriented Noise Sources in HBTs and FETs Oscillator Circuit Design Tools Conventional Linear Theory of Sinusoidal Oscillators Steady-State Analysis of Transistor Oscillators Nonlinear Stability of Free-Running Oscillators Oscillator Phase-Noise Characterization

102 115 118 130 134 150 156 156 167 169 172

4.3.6 4.3.7

Design Rules of Low Phase-Noise FreeRunning Oscillators Phase Noise in One-Port Oscillator Circuit Generalization to Transistor-Oscillator Circuits A Very Useful Design Tool: The Transistor Load-Line Finding the Maximum Added Power of the Transistor by Numerical Calculation Optimization and Localization of the Energy Stored in the Circuit AM/PM Conversion Conclusion

190 193 194

4.4 4.4.1 4.4.2

Practical Examples Breadboard Oscillators Oscillators on MMIC Technology

195 195 200

4.3.1 4.3.2 4.3.3 4.3.4 4.3.5

182 184 186 187 188

Contents

ix

4.4.3 4.4.4

MMIC FET-Based Oscillator Examples MMIC HBT-Based Oscillator Example

202 210

4.5

Conclusion References

212 214

Appendix 4A: HBT and HEMT Nonlinear Models

221

Appendix 4B: Transistor Low-Frequency Noise Characterization

226

Appendix 4C: Numerical Simulations of an Oscillator Benchmark Acknowledgments

235 238

Modern Harmonic-Balance Techniques for Oscillator Analysis and Optimization

245

5.1

Introduction

245

5.2

HB Analysis of Autonomous Quasi-Periodic Regimes in Nonlinear Circuits Autonomous Quasi-Periodic Regimes The Mixed-Mode Newton Iteration Degenerate Solutions and Their Suppression Applications

246 246 248 253 256

5

5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6

Synchronous and Asynchronous Stability Solution Paths in a Harmonic Phasor Space Natural Frequencies of Quasi-Periodic Steady States Nyquist’s Analysis for Time-Periodic Steady States Global Stability Analysis Applications Spurious Oscillations and Related Bifurcation Diagrams

260 261 267 270 274 284 294

x


5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.4.7 5.5 5.5.1 5.5.2 5.6 5.6.1 5.6.2 5.6.3 5.6.4 5.7 5.7.1 5.7.2

CAD-Oriented Oscillator Design Techniques General Optimization Methods Oscillator Optimization by Substitution Methods Design for Oscillation Buildup and SteadyState Stability Computation of the Gradient Applications A Case Study: CAD of a Broadband VCO Oscillator Design for Asynchronous Stability Electromagnetics-Based Optimization of Microwave Oscillators Direct-Newton Optimization Applications Iterative Methods for Large Self-Oscillating Nonlinear Circuit Analysis Inexact-Newton HB for Forced Circuits Computation of the Krylov Subspace Basis Vectors Extension to Large Autonomous Circuits Applications

301 303 312 315 319 320 325 337

343 344 349

351 352 356 360 363

Global Stability Analysis of Large Autonomous Circuits Fundamental Bifurcation Detection for Large Circuits Applications References

366 370 375

About the Authors

377

Index

383

366

Preface There is no need to justify interest in oscillators; they form the heart of all kinds of communication systems and more. Whenever we tune a radio, check the time, use a cellular phone, or even pick up a portable telephone receiver, we rely on a properly working oscillator. Oscillators are intrinsically nonlinear circuits that exhibit a wealth of nonlinear phenomena. Some of these phenomena are desirable from the designer’s point of view; some are not. The very utility of oscillators is based on the nonlinear effect of coexistence of unique and stable periodic oscillations and an unstable quiescent point. Other phenomena, such as the coexistence of several periodic solutions (manifested as spurious oscillations), hysteresis, bifurcation, and the existence of strange attractors, are to be avoided. Because of this wealth of complex behavior, oscillator design remains more an art than a science. At the beginning of oscillator development, theory closely followed practical design. Van der Pol’s theoretical works appeared a few years after the first triode oscillators were invented, and a few years later Krylov and Bogoliubov provided a rigorous justification of the van der Pol approach. The situation changed when triodes, which were well described by van der Pol cubic nonlinearity, were replaced by transistors, which were not. Only a very few of the best designers were able to apply van der Pol methods to microwave oscillators, as is described in Chapter 1. Some designers adapted methods from control theory, which led to successful designs described in Chapters 1 and 2. Still, their methods were little known in the broader design community. A common approach was to make the operating point unstable in the hope that unique and stable oscillations would appear. The xi

xii


circuits and devices were too complicated to attempt much more with standard engineering tools. At the same time, the ideas of van der Pol, Krylov, and Bogoliubov bore fruit in the area of dynamical systems. They merged with the earlier works of Poincare and gave rise to methods of averaging and of integral manifolds. Cartwright and Littlewood’s analysis of the van der Pol equation inspired Smale’s work and his invention of the horseshoe structure. Unfortunately their theory was hardly accessible to engineers, and their examples were limited to triode oscillators and the original cubic van der Pol equation. The situation has changed with the arrival of CAD tools: first, SPICE, and then, harmonic balance and envelope simulators. They made complex circuits and devices tractable and at the same time showed the need for an understanding of nonlinear behavior. We believe that now nonlinear tools and methods have reached the stage at which they can be effectively applied to design. This book aims to make advanced design tools available to practicing engineers. It presents the state of the art in design in language accessible to the hands-on engineer. Since linear design (i.e., design for the unstable operating point) is well covered in the literature, this book focuses on nonlinear design and is structured to guide the reader to its most important aspects. The book consists of five chapters. The first three provide tutorials on oscillator design. The last two present advanced methods in real-life design, including device modeling and circuit-simulation methods. In Chapter 1, Kaneyuki Kurokawa describes the development of nonlinear design, including his pioneering work. He relates the evolution of design methods to development in instrumentation and measurements. Chapter 2 introduces methods of nonlinear analysis, using a simple IC structure as an example. It relates the classic van der Pol approach to the theory of dynamical systems and to the high-Q oscillators described by Kurokawa. Chapter 3 introduces, in detail, phase noise analysis. It uses time domain description to explain noise behavior carefully. Chapter 4 presents the full design of a modern oscillator and focuses on the reproducible performance of the low phase noise microwave transistor oscillator, including a high production volume MMIC VCO. This chapter starts with the nonlinear modeling of active devices; then it discusses linear and nonlinear simulation and noise analysis. The ideas presented in Chapters 1 and 2 are combined with modern simulation tools. Phase noise analysis includes spectral description, as well as the methods developed in Chapter 3. Chapter 5 discusses in detail the harmonic balance methods that proved to be so effective in oscillator analysis and design. It

Preface

xiii

also discusses nonlinear phenomena encountered even in simple oscillators, including the bifurcation of oscillations from a quiescent point and the bifurcation of almost periodic oscillations from existing periodic oscillations. The exposition is aimed at all levels of design expertise; it covers elementary oscillator behavior, as well as advanced methods and models. It is intended for use as both a reference for the practicing designer and as a textbook for senior-level students. We hope that new designers will find here a readable introduction to design, combined with an overview of various methods of differing levels of complexity. At the same time, an experienced designer will find here his favorite methods presented from different viewpoints.

Acknowledgments The inspiration for this book came from the Workshop on Oscillator Design held at International Microwave Symposium in Boston, Massachusetts, in June 2000, which received very favorable feedback. I have been very fortunate to obtain contributions to the workshop and to this book from internationally recognized experts. I would like to thank them all for outstanding contributions. I would also like to acknowledge Jacek Kudrewicz, Ganesh Basawapatna, and Jack Hale, on whose shoulders my personal oscillator knowledge rests. Michał Odyniec El Cerrito, California September 2002

xv

1 Developments of Microwave Oscillator Theory Kaneyuki Kurokawa 1.1 Introduction Oscillators convert direct current (dc) power to radio frequency (RF). They are one of the most fundamental components in RF and microwave systems. However, an oscillator is inherently nonlinear. If it were linear, the oscillation amplitude would grow indefinitely with time. No steady state would be reached. For linear systems, the principle of superposition holds. A complex phenomenon can be decomposed into simpler phenomena; each of these simpler phenomena can be studied independently, and their superposition explains the original complex phenomenon. This type of convenient and powerful method is not applicable for the study of an oscillator because of its nonlinearity. However, if the oscillator is divided into two parts, the linear passive circuit and the nonlinear active device, then the vast amount of knowledge accumulated for linear passive circuits can be advantageously used for the discussion of the linear passive part. In any branch of science, the actual situation is often too complicated to study. Accordingly, a simplified model is constructed, and the behavior of the model is studied. The model usually neglects various factors, which, one hopes, yield only minor effects. After the model is constructed and the equation is given describing the behavior of the model, the direct solution of the equation is sometimes found to be difficult, and approximations 1

2


become necessary. The success of the model and approximations is determined by whether the analysis explains observed phenomena or not. As the techniques of measurement and observation improve, so does the theory, which explains observed phenomena. The history of microwave oscillator theory is the history of the appropriate model construction and approximations. This chapter reviews some of early papers on microwave oscillator theory to determine what kind of models and approximations were used as the observational techniques improved.

1.2 Van der Pol (1927) In his classical paper published in Philosophical Magazine in 1927, van der Pol discussed the phase-locking phenomena observed with beat receivers [1]. A beat receiver is a vacuum tube oscillator that produces audio beat output when a carrier is injected under a near-locking condition. It was used as a sensitive detector of weak signals in wireless telegraphy. A survey of textbooks on wireless communications published around 1920 will show that an enormous number of pages are devoted to the discussion of beat receivers. As late as 1947, the classical textbook [2] by Terman, Radio Engineering, spent one page on beat receivers under the name of oscillating detectors. As a model of the beat receiver, van der Pol presented the following nonlinear differential equation: V¨ + (−␣ + 3␥ V 2 )V˙ + ␻ 20 V = B␻ 21 sin ␻ 1 t

(1.1)

where V stands for the anode voltage of the triode, and the right-hand side represents the injection signal, and −␣ is the negative resistance necessary for oscillation to start. The term 3␥ V 2 represents the nonlinear resistance, without which the oscillation amplitude will grow indefinitely with time. A concrete derivation of (1.1) is presented in [3]. Using (1.1), van der Pol clarified when the phase locking takes place. He also clarified how the RF amplitude varies with the injection signal amplitude and frequency. Although he did not present experimental data, it was obvious that his results agreed very well with observations. The phase locking could be detected by the sudden disappearance of the audio output, and the RF amplitude could be measured by a vacuum tube voltmeter. When solving this nonlinear differential equation, van der Pol used various approximations. Most importantly, he neglected completely the third harmonic, even though the third harmonic


3

amplitude produced by the nonlinear term has the same order of magnitude as that of the fundamental component. The fact that, in spite of this omission, his analysis showed an excellent qualitative agreement with observations profoundly influenced the later development of oscillator theory. The excellent agreement is due to the circuit being detuned at the third harmonic frequency.

1.3 J. R. Pierce (1943) During World War II, researchers at Bell Laboratories and the MIT Radiation Laboratory studied various oscillator circuits for radar applications, including the reflex klystron and magnetron. Summarizing these wide-ranging studies, J. R. Pierce of Bell Laboratories presented a general theory of oscillator behaviors in an internal technical memorandum [4], known as ‘‘mm.’’ This was in 1943. His ‘‘mm’’ was immediately classified as confidential. He neglected harmonics from the beginning. Consequently, in the steady state, the circuit admittance Y plus the electronic admittance Y E must be equal to zero: YE + Y = 0

(1.2)

Note that Y E represents the nonlinear device and that Y represents the linear passive circuit mentioned in the introduction. Pierce assumed that Y E is a function of voltage V alone and that Y is a function of angular frequency ␻ alone. To investigate the stability of each solution satisfying (1.2), he introduced a complex frequency:

␻ = ␻ r + j␻ i

(1.3)

e j␻ t = e j (␻ r + j␻ i )t = e j ␻ r t e −␻ i t

(1.4)

Since

positive ␻ i means that the amplitude is decreasing with time and negative ␻ i means that the amplitude is increasing with time. When V deviates from a steady-state value by dV, the corresponding d␻ is obtained from (1.2) as follows: dY dY E dV + d␻ = 0 dV d␻

(1.5)

4


hence

冤冥

dY E dV dV = kdV d␻ = − dY d␻

(1.6)

If Im (k ) > 0, then the oscillation is stable. This is because when V increases slightly from the steady-state value, dV is positive and d␻ i becomes positive, which means that the amplitude decreases with time, returning toward the original steady-state value. Similarly, when V decreases, the amplitude increases with time and returns to the original steady-state value. If Im (k ) < 0, a slight deviation of V from the steady-state value causes a further deviation. As a result, the oscillation is unstable. By the time Pierce wrote his memorandum, the circuit admittance at microwave frequencies could be accurately measured using a slotted section and the Smith chart. Therefore, the possibility of multiple intersections between the circuit admittance locus and the device line (the negative of the electronic admittance locus) was clearly recognized, and since each intersection satisfies (1.2) and corresponds to a possible oscillation, determining which intersection gives a stable operation was an important subject of study. When the memorandum was declassified 10 years later, W. A. Edson of General Electric (GE) presented the stability criterion in his book, Vacuum Tube Oscillators [5]. J. C. Slater also presented a graphical form of the same criterion in his book, Microwave Electronics [6]. However, Slater did not present the mathematical formula itself, probably because Pierce’s memorandum was yet to be released at the time of his publication. Today, many textbooks on microwaves present Pierce’s stability criterion. However, the appearance is different, since k is first decomposed into the real and imaginary parts and the condition is given in a form in which the expression of the imaginary part is larger than zero. Furthermore, nobody seems to refer to Pierce’s memorandum anymore.

1.4 R. Adler (1946) Independent of these wartime efforts, R. Adler of the Zenith Corporation observed the experimental study of vacuum tube audio oscillators at 1,000 Hz conducted by his colleagues and obtained a differential equation for the oscillator phase to satisfy [7]. It is given by


E ␻ d␣ = − 1 0 sin ␣ + ⌬␻ 0 dt E 2Q

5

(1.7)

where E 1 is the voltage of the impressed signal at ␻ 1 , E is the voltage induced in the grid circuit, Q is the quality factor of the plate load, and ␣ is the phase relative to the impressed signal. ⌬␻ 0 is the difference between the free-running angular frequency ␻ 0 and ␻ 1 . In the steady state, the lefthand side is zero and sin ␣ is not larger than 1, so the locking condition is given by

| |

⌬␻ 0 E1 > 2Q E ␻0

(1.8)

Rewriting this condition, we obtain Adler’s formula for locking range: ⌬␻ max =

␻0 E1 2Q E

(1.9)

This simple and easy-to-understand formula was referred to in many papers. Adler’s formula was, however, misinterpreted in some microwave papers by a factor of 2 as ⌬␻ max =

␻0 2Q

√

P1 P0

In this formula, P 0 is the oscillator output power and P 1 is the injection signal power. The locking range is usually ‘‘verified’’ by locking experiments. The fact that the locking range is proportional to the square root of the injection signal power does not prove the proportionality constant being correct. Norman Kenyon of Bell Laboratories measured circuit parameters of a transistor oscillator at 350 kHz and confirmed that this formula is indeed incorrect by a factor of 2 [8]. The correct formula is given by ⌬␻ max =

␻0 Q ext

√

P1 1 P 0 cos ␸

(1.10)

There are three different Q s in microwaves: the unloaded Q , the loaded Q , and the external Q . Among them, the external Q is the Q to be used in the locking formula. Neglecting the term 1/cos ␸ , (1.10) gives Adler’s

6


formula, since P 0 is E 2 divided by the load resistance R , and P 1 is the injection signal power, which is the available power of the injection signal source, E 12 divided by 4R . ␸ is a new variable. It is the slope angle of the device line shown in Figure 1.1. The more slanting the device line, the wider the locking range. One could use the impedance locus Z (␻ ) as well. Then, the device line is minus the electronic impedance, Z ( A ). In addition to the popular locking range formula, Adler solved (1.7) without using approximations and obtained the closed form solutions

␣ 1 tan = − 2 K ␣ 1 tan = + 2 K

√1 − K

2

K

√K

2

K

−1

⌬␻ 0 (t − t 0 ) tanh 2 ⌬␻ 0 (t − t 0 ) tan 2

√1 − K

√K

K 2

K

−1

2

(K < 1)

(1.11)

(K > 1)

where K = 2Q

⌬␻ 0 E ␻0 E1

(1.12)

These solutions (1.11) show how the oscillator phase changes with time during locking transient (K < 1) as well as under near-locking conditions (K > 1). Note that audio waveforms could be directly observed on the

Figure 1.1 The circuit admittance locus and the device line (minus the electronic admittance) with the angle ␸ .


7

oscilloscope screen, and one could correlate the phase change with observations. Slater presented the same analysis in his book Microwave Electronics even though waveforms could not be observed at microwave frequencies. Long after Slater’s book was published, the spectrum analyzer became standard laboratory equipment and the triangular shape of the spectra envelope under near-locking conditions attracted researchers’ attention. Then, M. Armand of CSF, France, among others, gave an excellent account [9] for the envelope in 1969 by calculating the spectrum intensities using Adler’s solution under near-locking conditions. In terms of sin ␪ =

1 , ⍀ = ⌬␻ 0 cos ␪ K

(1.13)

the result is given by Frequency

Complex Amplitude

␻1

j tan

␻1 + ⍀ ␻1 + n ⍀

冉冉

␪ E 2

冊冊冉

␪ j␪ e E 2 ␪ ␪ 1 − tan2 j tan 2 2 1 − tan2

冊

n−1

e jn␪ E

Except for the first term, the spectrum intensities form a geometric sequence. As a result, on the spectrum analyzer screen with a decibel scale, they become an arithmetic sequence, explaining the triangular shape of the spectra envelope very well. Suddenly, Adler’s solution as a function of time became relevant to microwaves, confirming Slater’s foresight.

1.5 W. A. Edson and J. A. Mullen (1960) W. A. Edson of GE and J. A. Mullen of Raytheon each published a noise paper in the same issue of the Proceedings of the IRE in 1960. Edson’s paper used a conceptual approach [10], while Mullen’s paper was mathematical [11]. Each treated a second-order differential equation, which means that a single resonant circuit is assumed. Both papers studied amplitude modulation (AM) and frequency modulation (FM) noise and presented the output spectrum of oscillators. Note that, by then, the spectrum of oscillator output

8


became directly observable on the spectrum analyzer screen. Consequently, they could discuss oscillator noise in terms of observable quantities.

1.6 C. T. Rucker (1969) To increase the output power of radar, multiple tube oscillators were extensively studied during World War II. Push-pull circuits were thoroughly practical, but the circuits combining more than two tubes became increasingly difficult because of parasitic oscillations, or the so-called mode problem, and no solution of the mode problem was found. In the early days of microwave solid-state oscillator development, a number of power combiners were again proposed. In most of these proposals, individual oscillators were first built and then their output power was combined by means of hybrid circuits or transmission lines. Although the proposed circuits were always reported as stable, the stable operations were rarely reproducible in other laboratories. These circuits were in a near degenerate state and hence inherently unstable. Small changes in the load condition, or in the supply voltage, or even in the ambient temperature bring about sudden changes in the mode of operation. The output power as well as the oscillation frequency literally jumps from one value to another giving an unacceptable performance. This is the same mode problem as suffered by radar pioneers during World War II. This mode problem was solved by the ingenious proposal of a symmetrical oscillator [12] by C. T. Rucker of the Georgia Institute of Technology in 1969. In his oscillator, the power-combing circuit constitutes an integral part of the oscillator circuit. The devices are located physically far apart from each other, yet the behavior of the oscillator becomes identical to that of an oscillator with devices connected in parallel. It consists of five coaxial transmission lines, each about ␭ /4 long, terminated by a device, and arranged radially about a common bias network and a common output network as schematically illustrated in Figure 1.2. A resistor R s is incorporated in each coaxial center conductor to eliminate instabilities commonly observed with multiple oscillator arrangements. The capacitance C between the output coupling disk and each coaxial center conductor provides the necessary coupling to the common load R L . The bypass capacitor in the bias network is located about ␭ /4 away from the hub of the oscillator. Since the relation between one device port and the neighboring device port is identical to the relation between one device port and any other device port, and since the circuit is reciprocal, the impedance matrix [13] of the oscillator circuit looking in from the device ports is given by


9

Figure 1.2 Rucker’s symmetrical oscillator.

Z(␻ ) =

冤

Z 11 Z 12 Z 12 Z 12 Z 12

Z 12 Z 11 Z 12 Z 12 Z 12

Z 12 Z 12 Z 11 Z 12 Z 12

Z 12 Z 12 Z 12 Z 11 Z 12

Z 12 Z 12 Z 12 Z 12 Z 11

冥

(1.14)

The corresponding negative device impedance matrix is given by Z(A) = diag [ Z (A 1 ), Z (A 2 ), Z (A 3 ), Z (A 4 ), Z (A 5 )]

(1.15)

where diag [ ] indicates a diagonal matrix. The oscillation condition is given by Z(␻ )i = Z(A)i

(1.16)

Equation (1.16) can be systematically solved using the eigenvectors of Z(␻ ) and the corresponding eigenvalues. They are respectively given by

10


xn =

1 √5

冤冥 1 exp ( jna ) exp ( j 2na ) exp ( j 3na ) exp ( j 4na )

(1.17)

and

␭ n = Z 11 + Z 12 (e jna + e j 2na + e j 3na + e j 4na )

(1.18)

where n runs from 0 to 4 and a = 2␲ /5. Since

␭ 0 = Z 11 + 4Z 12

(1.19)

␭ 1 = ␭ 2 = ␭ 3 = ␭ 4 = Z 11 − Z 12

(1.20)

one can easily choose the circuit parameters in such a way that ␭ 0 can be equal to Z (A ) but ␭ 1 cannot be equal to Z (A ) in the whole range of A and ␻ . Then, the oscillation will take place in the 0th mode, and all the other modes from n = 1 to 4 are completely suppressed. This is the reason why Rucker’s symmetrical oscillator is free from the mode problem. Once the principle is known, the physical symmetry is not necessary. As long as the circuit is electrically symmetric, the same principle applies. As a result, many practical multiple device oscillators have been successfully proposed. The above solution of the oscillator mode problem may be considered as a triumph of the oscillator circuit theory. However, multiple-device oscillators are no longer being used in practice. This is because, today, transistor power amplifiers are easier to build and maintain.

1.7 K. Kurokawa (1973) My own work on oscillator theory was prompted by the separate measurements of AM and FM noise of phase-locked oscillators conducted by J. Josenhans at Bell Laboratories in 1966 [14]. I presented a graphical interpretation of the phase-locking phenomena extending Slater’s discussion in his book, Microwave Electronics. The phase-locking condition [15] is given by [ Z (␻ 1 ) − Z (A )]I = E

(1.21)


11

where E represents the injection signal voltage at ␻ 1 ; A is the current amplitude; Z (␻ 1 ) is the circuit impedance; and −Z (A ) is the device impedance. For small signal injection, the RF current amplitude A will stay approximately the same as the free-running amplitude A 0 . So, (1.21) gives Z (␻ 1 ) = Z (A ) +

|E | A0

e −j␾

(1.22)

where ␾ is the phase difference between I and E . The formula (1.22) indicates that the distance from Z (A ) to Z (␻ 1 ) is equal to | E | /A 0 under the locking condition. There are, in general, two points on the device line that satisfy this condition as shown by the tails of the solid and dotted arrows in Figure 1.3. However, only one of them, that of the solid arrow, corresponds to a stable locking. I clarified graphically the stability condition of injection locking. Using the graphical discussion, the locking range formula (1.10) was obtained. Furthermore, the AM and FM noise of oscillators under various conditions could be qualitatively discussed. For the analytical study of oscillators, I used the relation

␻ = ␻1 +

1 dA d␾ −j dt A dt

(1.23)

instead of the complex frequency given by (1.3). This is because I was interested in the fluctuations of amplitude A and phase ␾ .

Figure 1.3 The circuit impedance locus and the device line with the injection vector (solid arrow).

12


In the ac circuit theory, the time derivative is everywhere replaced by multiplication by j␻ . This is justifiable since dn dt n

Re [Ae j (␻ t + ␾ ) ] = Re [( j␻ )n Ae j (␻ t + ␾ ) ]

(1.24)

if A and ␾ are constant. When A and ␾ are slowly varying functions of time, to the first-order approximation for n > 1 and exactly for n = 1, the n th time derivative becomes dn dt

n Re [Ae

j (␻ 1 t + ␾ )

] = Re

再冋冉

d␾ j ␻1 + dt

冊

1 dA + A dt

册

n

冎

Ae j (␻ 1 t + ␾ )

(1.25) This means that Re [Z (␻ ) Ae j (␻ 1 t + ␾ ) ] will give approximately the voltage across the impedance Z (␻ ) when ␻ is replaced by ␻ 1 + (d␾ /dt ) − j (1/A )(dA /dt ) everywhere in Z (␻ ). Using (1.23) turned out to be a convenient method of analyzing various oscillator behaviors. Thanks to the orthogonal relation between sine and cosine functions, real-time differential equations of A and ␾ can be obtained. Utilizing these differential equations, topics such as the effects of nonlinear reactance, the variation of circulator output, phase-modulated injection signals, AM and FM noise, noisy injection signals, the effects of nonlinear reactance on noise, and large signal injection could be all discussed in a coherent manner as presented in [15]. The active element in my 1973 paper was a negative resistance device such as a tunnel diode, a Gunn diode, or IMPATT. In the meantime, the preferred active element became a transistor. Note that, interchanging the role of impedance with that of admittance as in the Pierce memorandum, the discussion dual to my paper is directly applicable to regenerative transistor oscillators, since the transistor acts as a negative conductance device in a regenerative oscillator. Similarly, the same discussion is also applicable to Hartley and Colpitts transistor oscillators since the transistor acts again as a negative conductance in the generalized odd mode when the even mode is detuned as is usually the case. Furthermore, imbedding a transistor in a reactive network, a one-port negative conductance device can be created to use as an active device for an oscillator design. The imbedding reactive network must be as simple as possible so that the frequency characteristics of the one-port device can be neglected compared to the frequency characteris-


13

tics of the circuit admittance. Because of the simplicity of the negative conductance model, it becomes straightforward to analyze nonlinear effects, such as the power variation of a phase-locked oscillator and the maximum power oscillation. Now, looking back at all these developments in the past, it seems that what could be observed or measured has profoundly affected what shape the oscillator theory took in the next evolution. Furthermore, history teaches us that the ingenious solution of a difficult problem may turn out to be of little use because of subsequent progress.

References [1]

van der Pol, B., ‘‘Forced Oscillations in a Circuit with Nonlinear Resistance (Reception with Reactive Triode),’’ Phil. Mag., S. 7., Vol. 3, No. 13, January 1927, pp. 65–80.

[2]

Terman, F. E., Radio Engineering, New York: McGraw-Hill, 1947, p. 540.

[3]

van der Pol, B., ‘‘The Nonlinear Theory of Electric Oscillations,’’ Proc. of the Institute of Radio Engineers, Vol. 22, No. 9, September 1934, pp. 1051–1086.

[4]

Pierce, J. R., ‘‘Oscillator Behavior,’’ Memorandum for File, mm-43-140-19, Bell Laboratories, June 19, 1943.

[5]

Edson, W. A., Vacuum Tube Oscillators, New York: John Wiley and Sons, 1953.

[6]

Slater, J. C., Microwave Electronics, New York: Van Nostrand, 1950.

[7]

Adler, R., ‘‘A Study of Locking Phenomena in Oscillators,’’ Proc. IRE, Vol. 34, June 1946, pp. 351–357.

[8]

Kenyon, N. D., ‘‘A Lumped-Circuit Study of Basic Oscillator Behavior,’’ Bell Systems Technical Journal, Vol. 49, February 1970, pp. 255–272.

[9]

Armand, M., ‘‘On the Output Spectrum of Unlocked Driven Oscillators,’’ Proc. IEEE (Lett.), Vol. 57, May 1969, pp. 798–799.

[10]

Edson, W. A., ‘‘Noise in Oscillators,’’ Proc. IRE, Vol. 48, August 1960, pp. 1454–1466.

[11]

Mullen, J. A., ‘‘Background Noise in Nonlinear Oscillators,’’ Proc. IRE, Vol. 48, August 1960, pp. 1467–1473.

[12]

Rucker, C. T., ‘‘A Multiple-Diode High-Average-Power Avalanche-Diode Oscillator,’’ IEEE Trans. on Microwave Theory and Techniques (Corresp.), Vol. MTT-17, December 1969, pp. 1156–1158.

[13]

Kurokawa, K., ‘‘An Analysis of Rucker’s Multi-Device Symmetrical Oscillators,’’ IEEE Trans. on Microwave Theory and Techniques (Corresp.), Vol. MTT-18, November 1970, pp. 967–969.

[14]

Josenhans, J., ‘‘Noise Spectra of Read Diode and Gunn Oscillators,’’ Proc. IEEE (Lett.), Vol. 54, October 1966, pp. 1478–1479.

[15]

Kurokawa, K., ‘‘Injection Locking of Microwave Solid-State Oscillators,’’ Proc. IEEE, Vol. 61, October 1973, pp. 1386–1410.

2 Methods of Oscillator Design Michał Odyniec

2.1 Introduction Since its very beginnings until the 1970s, oscillator design has evolved on three parallel courses. The most common approach focuses on creating conditions under which oscillations can start. It deals with small-signal circuits (i.e., circuits that are linearized about the operating point) and with the selection of structures and components that make the operating point unstable. The second approach, described in Chapter 1, applies van der Pol ideas to microwave oscillator development. It is very effective but little known. The third approach uses van der Pol ideas to characterize nonlinear oscillations. It is even less known in engineering circles than the second. The reason is that the original, cubic nonlinearity of the van der Pol equation has lost relevance to transistor circuits. However, van der Pol ideas gave rise to the theory of nonlinear oscillations and dynamical systems. Among the concepts that can be directly traced to van der Pol equations are averaging, integral manifolds, and Smale’s horseshoe [1–6]. The development of CAD tools in the 1970s transformed design practice [7]. On one hand, it gave designers insight into nonlinear circuits; on the other, it showed the danger of blind applications of the tools. The development of harmonic balance simulators in the 1980s proved particularly useful for oscillator design. 15

16


This chapter shows interrelations between those approaches. We use a simple, commonly used oscillator to illustrate their power and their limits and the ways in which they utilize and enhance the CAD tools. Section 2.2 describes our oscillator with the use of the generalized van der Pol equations and discusses its full dynamics. Section 2.3 presents a rigorous analysis of stability criteria. Sections 2.4–2.7 generalize van der Pol results to high-Q (but otherwise arbitrary) oscillators, discuss the extension of harmonic balance techniques, and combine the high-Q assumption with dynamical analysis.

2.2 Nonlinear Dynamics of a Simple Oscillator 2.2.1 Oscillator Equation This section analyzes one of the simplest and widely used oscillator structures; it is shown in Figure 2.1. The cross-coupled transistors form a nonlinear, locally active circuit while the LC circuit models the resonator. The circuit is simple, yet generic enough to exhibit common oscillator behaviors; its various modifications appear in Chapter 3 and in [8–10]. Here we use the circuit as a modern illustration of the generalized van der Pol theory, which was the starting point of oscillator theory. Let v (t ) denote the voltage across the capacitor and i L (t ) the current through the inductance L . If we neglect the high-frequency transistor parasistics and the reactances of the biasing circuit, the oscillator can be described by a generalized van der Pol equation: L

di L =v dt

C

(2.1)

dv = −i L − i n (v ) dt

The function i n (v ) = i a (v ) + G res v represents the current-voltage characteristics of the oscillator seen from the resonator’s terminals A-B as shown in Figure 2.2. We have included the resonator conductance G res into the nonlinear characteristics for the sake of convenience; it will have no effect on the results. The exact shape of these characteristics is determined by the crosscoupled transistors and the biasing circuit. The bias parameters of V CC = 5V, R1 = R2 = 500 ohm, I dc = 10 mV, and G res = 0.001 mho result in the shape shown in Figure 2.3.


17

Figure 2.1 Simple oscillator.

Let us note that the nonlinear resistance (including the resonator resistance G res ) is ‘‘N-shaped’’ (i.e., its slope, which represents the small-signal conductance, is negative for small voltages and positive for large ones). Thus, the nonlinear circuit is active (i.e., it delivers energy to the resonator for small signals). For large signals it becomes lossy. A cubic polynomial used by van der Pol is the simplest N-shaped nonlinearity. We shall show that practically all of van der Pol results are valid for an N-shaped nonlinear characteristic and therefore applicable for modern oscillator structures.1 The original van der Pol equation is easily obtained from (2.1) by taking a time derivative of its second part and substituting the v /L for di L /dt . That yields 1. Rigorously speaking, we call a function f (v ) ‘‘N-shaped’’ if f (0) = 0, if f (v ) is monotonically decreasing about zero and is monotonically growing for | v | larger than certain constant, and if it is convex for v > 0 and concave for v < 0. Those last conditions mean that its second derivative (if it exists) is positive for v > 0 and negative for v < 0.

18


Figure 2.2 Practical realization of an active circuit (nonlinear resistance).

v¨ +

di n v˙ /C + v /LC = 0 dv

(2.2)

This is a standard form of an oscillator equation with the natural frequency ␻ 20 = 1/LC and the admittance (dissipative force in mechanical di oscillator) n . For (2.2) the N-shape requirement of i n (v ) translates into dv di n (v ) i a (v ) = + G res being negative about zero (thus, G res must be small dv dv enough so that the resonator losses do not quench the oscillations) and monotonically increasing with v. The condition is of course satisfied by van der Pol’s cubic nonlinearity i n (v ) = G (−v + v 3/3) derivative, which results di (v ) = G (−1 + v 2 ) characteristics. in the n dv


19

Figure 2.3 The I-V characteristics that includes G res , seen by the ideal resonator.

2.2.2 Phase-Plane Analysis We apply now the phase plane methods [2, 11–13] to (2.1). We introduce the plane (hence the name of the method) of the possible values of the voltage v and the current i L . In that plane we trace v and i L parameterized by time to obtain the transient and steady-state behaviors. This trajectory tracing can be done numerically as shown in Figure 2.4(a). Superimposed on the trajectories is the reversed characteristic of the nonlinear resistance i L (v ) = −i n (v ). We have put this reversed characteristic in, because it shows the points at which the right side of the second part of (2.1) equals zero. Figure 2.4(b) represents a pair of voltages across the resonator and a pair of currents through the inductance. In each pair one waveform originates at low value of the current and the other one at a high value of the current. In Figure 2.4(a) the same waveforms are represented on the voltage-current plane as trajectories parameterized by time; one of them spirals out, and the other spirals down to the same steady-state solution (which mathematicians

20


Figure 2.4 Dynamics of an LC oscillator ␻ o C = 0.1 mho: (a) phase-plane and (b) waveforms.

call a limit cycle). The steady-state solution is not shown on the plane, but its position can be easily discerned from the behavior of the two trajectories. The power of the phase-plane analysis, however, lies in the fact that we can predict the circuit behavior even without any numerical calculations. Indeed, let us note that the tangent vector to each trajectory equals (dv /dt , di /dt ). Whenever a trajectory crosses the vertical axis v = 0, then di /dt = 0 [because of the first part of (2.1)], and the trajectory is horizontal. Moreover, for v > 0, we have di /dt > 0 and the trajectory goes upward; for v < 0, it goes downward. A similar argument can be applied to the voltage v (t ). On the reversed nonlinear resistor characteristics i L (v ) = −i n (v ), we get dv /dt = 0; thus, every trajectory crosses the characteristic vertically. We already know that it will point upward for v > 0 and downward for v < 0. Moreover, we have dv /dt < 0 above the characteristics and dv /dt > 0 below it. Combining that with the information on the sign of di /dt (positive for positive v and negative for negative v ), we get a complete picture of the behavior of trajectories. Any trajectory passes the positive vertical half axis horizontally directed to the left; then it keeps turning down until it hits the reversed resistive characteristics where it points down directly. Subsequently,


21

it continues turning right until it hits the negative half of the vertical axis where it points horizontally to the right, and then it keeps turning upward to the reversed resistive characteristics where it points up; then it keeps turning left until it hits the positive half of the vertical axis, and so on. The trajectories starting near the origin spiral outward while those originating ‘‘far’’ spiral inward. As another example of a qualitative argument we consider the circuit from Figure 2.1 with small resonator capacitance C . In that case the time derivative of the voltage v˙ = −(i L + i n (v ))/C is large everywhere except in the immediate vicinity of the reversed nonlinear characteristics i L = −i n (v ). Consequently the trajectories either follow the shape of the characteristics or move very fast horizontally as shown in Figure 2.5(a). Thus the corresponding voltage waveforms have parts of slow motion connected by quick jumps as shown in Figure 2.5(b).

Figure 2.5 Relaxation oscillations in the very low-Q resonator ␻ o C = 0.001 mho: (a) phaseplane and (b) waveforms.

22


The above analysis illustrates the way in which the general behavior of the system can be determined even without numerical simulations. Even more can be done [2, 12]: For example, the existence, the number, and the stability of the periodic trajectories can be approximated from the shape of the nonlinear resistance. One does it by constructing an annulus with large external boundary and the internal boundary just about the bias point. If the annulus boundaries meet only entering trajectories, then at least one periodic solution exists [12]. Since for large values of voltage and current the system is dissipative, all trajectories that cross the outer boundary are directed inside the annulus. If the bias point is unstable, then all trajectories that cross the internal boundary are also directed inside the annulus. Consequently, the circuit oscillates. This construction forms the basis of popular oscillator design: ‘‘If we assure that the operating point is unstable, then the circuit oscillates.’’ For an N-shaped symmetric characteristic, we can prove that the oscillations are unique and stable [12, 13]. Moreover, we know [2] that for a second-order equation the only steady-state waveform is either periodic or constant and that for an N-shaped characteristic the system will have an unstable operating point and a stable periodic solution. Also for either large or small values of Q , the shape of the periodic solution and that of the transients can be discerned from the shape of the i n (v ) characteristics. 2.2.3 Generalizations of Phase-Plane Analysis We have analyzed the oscillator model in which the resonator is represented by an LC circuit and the active part by a nonlinear resistance. Within those two assumptions, the results are powerful, intuitive, and rigorous. In particular, we know that the only possible steady states are either constant or periodic and that the orientation of the trajectories is easily estimated from the signs of the right-hand sides of the circuit differential equations. Moreover, for the N-shaped characteristics the oscillations are unique and stable. Since all that we need is the N-shape, our results apply directly not only to cross-coupled transistors but also to IMPATT, Gunn, and tunnel diodes. The theory also allows us to predict the shapes of the voltage and current waveforms and thus the power flow in the oscillator. A thorough analysis of the physical interpretation of second-order classical equations (of van der Pol and Duffing) can be found in [11, 14, 15]. Thanks to the ease of use and the clear physical interpretation of the results, there is a natural tendency to expand the phase plane results to more complex circuits. In the case of high-Q oscillators the generalization can be


23

justified both intuitively and rigorously. Intuitively a high-Q resonator near a resonance can be well modeled by an LC circuit. Moreover, such a resonator varies quickly with frequency so that we can neglect the frequency dependence of the active part and treat it as a nonlinear resistance. The rigorous argument is based on the integral manifolds theory [1–3, 16]. Consequently, the circuit in Figure 2.1 can be treated as a generic oscillator that is simple but includes the main features of any high-Q oscillator circuit. Because of that, we will continue to use it to introduce other methods of nonlinear analysis. The relaxation oscillation behavior that we analyzed for small capacitance was one of the first cases of strange ‘‘chaotic’’ behavior observed and the one that resulted in rigorous analysis of the phenomenon. As long as the system is represented on the plane without an external excitation, the most complicated behavior is periodic. However, if we add a periodic source in parallel with the resonator, then things change. In that case we can imagine the nonlinear characteristics in Figure 2.5(a) as periodically moving up and down. This movement, superimposed with the quick horizontal movement of trajectories caused by the small C, makes the areas near the characteristics extrema stretch and fold onto themselves creating Smale’s horseshoes. A rigorous analysis of a piecewise-linear N-shape has been done by Mark Levi [4]. Another analysis of the classic van der Pol equation from the dynamical system point of view was done by John Guckenheimer [17]. It can serve as a good introduction to the dynamical systems’ language.

2.3 Stability of the Operating Point 2.3.1 Introduction The fundamental question of oscillator design—Will it oscillate?—is usually reduced to: Is the operating point stable?2 The overwhelming majority of oscillator design literature analyzes circuit structures and components values that make the operating point unstable [9, 18–24].3 2. These two problems are not equivalent. In Chapter 5 we will see examples of stable oscillations coexisting with a stable operating point. This situation usually results in hysteresis behavior of the oscillator. 3. Among those, this author’s favorite works are [9] for design philosophy and [20, 23] for practical designs.

24


Because linear design is widely covered in the literature, we shall not repeat its principles here. Instead we shall discuss just one of its aspects, which is widely and usually incorrectly used. To rigorously determine the operating point stability, we linearize the oscillator equations about it. If the eigenvalues of the linearized equations lie in the right half plane, then the point is unstable. In practice, designers prefer to linearize the circuit rather than the equations, which is done by using small-signal (i.e., linearized) models of nonlinear circuit components. The position of the eigenvalues (called characteristic frequencies ) is seldom checked directly. Instead the Nyquist criterion (see Appendix 2A) or some of its simplified forms are used. One of the most widely used criteria [21, 22, 24–35] states that ‘‘when the phase of the transfer function is zero and the magnitude (at the same frequency) is larger than one, then the system is unstable.’’ For microwave circuits the criterion is usually presented in terms of S-parameters and formulated as follows: If the conditions (2.3) hold, arg(S n ) = −arg(S r ),

| Sn Sr | > 1

(2.3)

then the double-reflected signal adds in phase and is bigger than the original one. Consequently, the signal grows and the circuit oscillates. Moreover, after the signal has grown, the gain diminishes so that the conditions (2.3) turn into arg(S n ) = −arg(S r ),

| Sn Sr | = 1

(2.4)

from which the amplitude and frequency of the signal can be determined. This section shows that the criterion (2.3) is not necessarily true and should be replaced by the analysis of the Nyquist plot. The criterion (2.4) is valid and will be discussed in Section 2.4. Although similar in appearance to (2.3), it describes a different physical phenomenon—the steady-state, large-signal oscillation—and is reached at in a very different manner. 2.3.2 Circuit Linearization When interested in ‘‘small’’ signals, we expand the nonlinear characteristics into a Taylor series about the operating point V dc : i n (V dc + v ) = i n (V dc ) + i n′ (V dc ) v + i n″(V dc ) v 2/2 + . . .

(2.5)


25

More precisely, we analyze the voltage and current of the form V dc + v , I dc + i , where I dc = i n (V dc ) and v and i are ‘‘small.’’ Substituting (2.5) back into (2.1), and neglecting all the higher-order terms, we get circuit equations, which are linear (in v, i L ): L

di L =v dt

C

(2.6)

dv = −i L − i a′ (V dc ) v − G res v dt

Note that the ‘‘small signal’’ circuit shown in Figure 2.6(b) is described by (2.6) with g a = −i a′ (V dc ). The circuit losses are determined by the resistors: If G res − g a = i n′ (V dc ) < 0, then the circuit is unstable. This fact can be checked analytically by applying the Laplace transform to (2.6), which yields: sLi L (s ) − Li L 0 = v (s )

(2.7a)

sCv (s ) − Cv 0 = −i L (s ) − i a′ (V dc ) v (s ) − G res v (s ) where v 0 and i L 0 are the initial values of voltage and current. Solving (2.7a) for v (s ), we get v (s ) =

sv 0 2

s + 2⑀ s + ␻ 20

(2.7b)

1 , ⑀ = (G res + i a′ (V dc ))/2C LC The poles of (2.7b) s 1,2 = −⑀ ± √␻ 20 − ⑀ 2 are often called characteristic frequencies of (2.7a); they are equal to eigenvalues of (2.6). Applying the inverse Laplace transform to (2.7b), we obtain voltage waveforms that are sinusoidal (as long as ⑀ < ␻ 0 ) with frequency ␻ ⑀ = √␻ 20 − ⑀ 2 and an exponentially varying amplitude v 0 e −⑀ t. We confirm that if G res + i a′ (V dc ) < 0, then ⑀ < 0, the amplitude grows, and the operating point is unstable.

where ␻ 20 =

2.3.3 Counterexample When G res + i a′ (V dc ) < 0, our circuit is unstable. Let us represent it via S-parameters and apply to it the Nyquist criterion together with the criterion

26


Figure 2.6 Linearization of the simple oscillator: (a) an ‘‘osctest’’ probe from a commercially available simulator separates the resonator from the active circuit and (b) a linearized RLC circuit.

(2.3). We are using the S-parameters for two reasons: (1) S-parameters are commonly used in microwave design and are included in many commercially available software simulation tools and (2) their behavior is counterintuitive. Consequently, the simplified stability criterion (2.3) may easily produce confusing results. The Nyquist plots shown in Figure 2.7 have been calculated for S-parameters with three values of the characteristic impedance Z 0k = 1/Y 0k ,


27

Figure 2.7 Nyquist plots for the simple oscillator for varying Y 0 = 1/Z 0 : (a) G res < g a < Y 01 , (b) G res < Y 02 < g a , and (c) Y 03 < G res < g a .

28


k = 1, 2, 3 such that G res < g a < Y 01 [see Figure 2.7(a)], G res < Y 02 < g a [see Figure 2.7(b)], and Y 03 < G res < g a [see Figure 2.7(c)], where g a = −i a′ (V dc ). Since our circuit is unstable the Nyquist loop, shown in the lefthand plots in Figure 2.7, encircles the point 1 + j 0 for every value of Z 0k , k = 1, 2, 3. However, if we turn to the magnitude-phase plots (shown to the right), then the circuit instability is hard to deduce. Finally the intuitive condition (2.3) obviously fails for G res < Y 02 < g a [Figure 2.7(b)] because we cannot find zero-crossing, and for Y 03 < G res < g a [Figure 2.7(c)] because at the zero-crossing | S n S r | < 1. The S n S r contours in Figure 2.7 clearly show that it is the encirclement of 1 + j 0 that matters (as we know from Nyquist theorem) and not the value of S n S r at the crossing of the real axis. 2.3.4 Validity Limits of the Intuitive Criterion As we have seen above, the intuitive criterion (2.3) may fail. The reason is that it assumes that the Nyquist contour is ‘‘simple,’’ by which we mean that its characteristic is similar to that of a selective filter and has the following properties: • It turns clockwise with ␻ . • It begins and ends at the origin (a bandpass filter has this property). • It crosses the real axis only once for positive ␻ (the negative-␻ -part

of the contour is the mirror image of the positive one). If, as described in Chapter 4, it crosses the real axis more than once, then it should return to the origin (or a ‘‘safe area’’ far to the left from 1 + j 0) between the ‘‘resonance’’ crossings. For a ‘‘simple’’ contour in the above-defined sense, it suffices to find the crossing of real axis [i.e., find the point for which arg(S n ) = −arg(S r )]. If at this point | S n S r | > 1, then, because of the three ‘‘simplicity’’ conditions, the Nyquist loop encircles the point 1 + j 0, and the circuit is unstable. Let us revisit the circuit in Figure 2.6(b). It has S r (␻ ) =

␻ 2 − ␻ 2 + j␻ (G res − Y 0 )/C Y 0 − Y res (␻ ) = − 02 Y 0 + Y res (␻ ) ␻ 0 − ␻ 2 + j␻ (G res + Y 0 )/C

(2.8)

For ‘‘large’’ and ‘‘small’’ frequencies, the contour S r (␻ ) is close to −1 + j *0. Therefore, the contour S n S r (␻ 0 ) always begins and starts at −S n and crosses the real axis at


S n S r (␻ 0 ) =

Y 0 + ga Y 0 − G res Y 0 − ga Y 0 + G res

29

(2.9)

Y 0 + ga Y − G res and S r (␻ 0 ) = 0 vary in the Y 0 − ga Y 0 + G res three cases we get the plots as shown in Figure 2.7. Thus, even in a very simple circuit, such as the one above, an unfortunate choice of Y 0 = 1/Z 0 causes the Nyquist contours to behave nonintuitively and makes the simplified criterion (2.3) useless. Let us note that the admittance/ impedance description is more intuitive than the S-parameter one. That led some authors to the conclusion that the admittance/impedance description is the correct one as opposed to S-parameters [32]. As we have seen above, both descriptions are valid when used with the Nyquist stability criterion. The only advantage of the impedance-admittance representation is that for high-Q resonators it more often produces a ‘‘simple’’ loop. If the resonator has high Q, then it varies in frequency much faster than the active part of the circuit. Consequently, S n can be considered independent of frequency, and the criterion can be rephrased in a form particularly convenient for applications: The system is unstable if the contour S r ( j␻ ) encircles 1/S n . Consequently, it is easy to change parameters of the active circuit so that the encirclement happens. The method of stability circles [36] follows immediately from this formulation. Since the signs of S n =

2.3.5 Conclusions We have used a very simple unstable circuit to show that the widely used intuitive criterion (2.3) may fail and should be replaced by the Nyquist stability criterion. The theory presented here is not new [18, 37, 38], but unfortunately it has often been incorrectly applied. We hope that the example will help to clarify the use of stability criteria.

2.4 High-Q Oscillators 2.4.1 Steady-State Periodic Oscillations This section analyzes the periodic steady state in high-Q4 oscillators. For the sake of simplicity we shall follow our oscillator; however, the methods 4. The rigorous definition of ‘‘high’’ Q is as follows: We require that the nonlinear characteristics of the circuit are bounded and satisfy the Lipschitz condition (i.e., their slopes are bounded). Then, for the theorems of this section and the following sections to hold,

30


presented here are applicable to high-Q but otherwise arbitrary resonators. Also, the nonlinear part of the circuit does not have to be purely resistive. Cavity, DRO, SAW, XTO, or YIG oscillators fall into that category. When interested in the steady-state periodic oscillations, we represent the waveforms by their Fourier series: K

v (t ) =

∑

k = −K

v k e jk␻ t, i (t ) =

K

∑

k = −K

i k e jk␻ t

(2.10)

In principle, the series in (2.10) are infinite; in practice, however, we argue that no physical circuit supports infinitely high frequencies, and we truncate the series at a finite number K . How to choose the value of K is an important design decision. The answer is usually based on the filtering properties of the circuit and often supported by trial and error. Substituting (2.10) into (2.1) we obtain a system of algebraic equations: jk␻ Li k = v k

k = 0, 1, 2, . . . , K

(2.11)

jk␻ Cv k = −i k − i nk (v 0 , v 1 , . . . , v K ) where T

1 i nk (v 0 , v 1 , v 2 , . . . , v K ) = 2T

冕冢∑ K

in

−T

k = −K

冣

v k e jk␻ t e −jk␻ t dt

We consider only the nonnegative indices k = 1, 2, . . . , K because, for the real valued waveforms, the harmonics with the indices of opposite signs are complex conjugate. We have thus reduced a system of differential equations to a (larger) system of algebraic equations from which harmonic coefficients of the oscillations can be found. There are two approaches to solving (2.11). One, called the harmonic balance method, provides solution algorithms for a finite, although possibly large, K . Because of its power and relative simplicity, the there exists a number Q 0 , which depends on the characteristics and their slope boundaries, such that for any Q > Q 0 , the theorems are satisfied. We call the Q that satisfies Q > Q 0 the high Q.


31

harmonic balance became a standard part of circuit simulators. It is discussed in detail in Chapter 5.5 The second approach, which we will discuss here, applies to the resonators with high Q. In that approach we neglect all the harmonics except the first, thereby reducing the system to a single algebraic equation in complex variables. This was the assumption made by van der Pol and by the methods described in Chapter 1. 2.4.2 Large Signal Impedance and Corresponding Nonlinear Characteristics For a high-Q resonator all harmonics except the first are negligible and (2.11) reduces to6 j␻ Li 1 = v 1

(2.12)

j␻ Cv 1 = −i 1 − i n1 (0, v 1 , 0, . . . ) where T

1 i n1 (0, v 1 , 0, 0) = 2T

=

1 ␲

冕

i n (v 1 e j␻ t + v −1 e −j␻ t ) e −j␻ t dt

−T ␲

冕

i n (2 | v 1 | cos ␪ ) cos ␪ d␪

0

For an oscillator with a high-Q but otherwise arbitrary resonator, the first harmonic equation will have the form: Y r (␻ ) v 1 = i n1 (0, v 1 , 0, . . . )

(2.13)

5. As long as the number of equations K is finite, the algorithm is easily generalized to any finite set of frequencies ␻ k that do not need to be harmonically related; therefore, the method also covers quasi-periodic oscillations [39]. 6. It turns out that the assumption can be justified for a general class of oscillators [40, 41]. Since the work is little known and important, not only for justification purposes but also for error estimates, we shall outline its line of reasoning in Appendix 2B.

32


It is easy to see that for

␻ 20 − ␻ 2 , ␻ 20 = 1/LC Y r (␻ ) = jC ␻ (2.13) describes our oscillator. Since we deal with an autonomous system, we can choose an arbitrary time count. We choose it so that the v 1 becomes real and positive (the phase of v 1 becomes zero). To remember that the voltage fundamental is a real number, we introduce a new notation for the voltage amplitude A and for the amplitude of the current fundamental I (A ): A = 2 | v 1 | = 2v 1 2 I (A ) = i n1 (0, v 1 , 0, 0) = ␲

(2.14) ␲

冕

i n (A cos ␪ ) cos ␪ d␪

0

It follows from (2.14) that I (A ) is also real, which was to be expected since we had resistive linearity. Consequently, the oscillator’s voltage and current are close to sinusoidal waveforms v (t ) = A cos ␻ t

(2.15)

i (t ) = I (A ) cos ␻ t where the frequency ␻ and amplitude A are to be found from Y r (␻ ) A = I (A )

(2.16)

Let us note that with Pierce’s notation Y E = I (A )/A (2.16) is equivalent to (1.2). Similarly, in Kurokawa’s notation Y (A ) = I (A )/A , it is equivalent to (1.21) without the injection voltage and with impedances replaced by admittances (Kurokawa [42, 43] analyzed a series resonator while ours is parallel). The function Y (A ) = I (A )/A (or Y E ) can be interpreted as a large signal conductance (admittance in general). The concept of nonlinear characteristics defined by fundamental components of input and output signals has also been used in nonlinear control theory, with Y (A ) = I (A )/A , called a describing function.


33

Equation (2.16) is easy to solve. An ideal resonator’s admittance Y r (␻ ) is purely imaginary while Y (A ) = I (A )/A is real. Thus, we can separate variables and find the resonant frequency and the amplitude from Y r (␻ ) = 0

(2.17)

I (A ) = 0 If the resonator admittance is not purely imaginary, we still can separate the variables into Re(Y r (␻ ))A = I (A ) and Im(Y r (␻ )) = 0 and find the resonance frequency from the second equation and then the amplitude from the first. The variable separation property holds as long as the nonlinear element is resistive. In our circuit the oscillation frequency is found from the first equation to be √1/LC . Equation (2.17) has a simple physical interpretation: The oscillation frequency is determined by the resonant circuit, while the power of oscillation is determined by the active part of the circuit and resonator losses. In our parallel RLC oscillator the steady-state oscillations occur when the combined conductance of resonator’s losses and that of the active circuit is zero and I (A ) = 0. The value of the amplitude for which that happens can be found geometrically as shown in Figure 2.8. Our circuit is highly selective; thus, the power at the fundamental frequency P (A ) = AI (A )/2 well approximates the total power delivered to a high-Q circuit. Consequently, the sign of I (A ) determines whether the circuit is dissipative or oscillatory. Indeed the power is delivered to the circuit when I (A ) < 0 and is dissipated when I (A ) > 0. Consequently, (2.17) determines the amplitude A of a steady-state oscillation and its stability. Steady-state amplitude corresponds to the equilibrium of received and dissipated energy. It is stable because for A < A 0 we have I (A ) < 0; thus, the power is delivered to the system P (A ) = AI (A ) < 0 and the amplitude grows. For A > A 0 we have I (A ) > 0, the power is dissipated P (A ) = AI (A ) > 0, and the amplitude decreases. 2.4.3 Notes on Feedback Representation, High Q, and Small Parameter 2.4.3.1 Feedback Description

Following control theory we will call the method presented above the describing function method. It is particularly well-suited to an input-output description of a system. In fact, when rewritten as

34


Figure 2.8 Nonlinear resistance and the corresponding fundamental I (A ). From I (A ) = 0 we determine the oscillation amplitude.

A = Z r (␻ ) I (A ) (2.16) suggests an input-output representation where the linear characteristic Z r (␻ ) = 1/Y r (␻ ) is interpreted as a ‘‘filter’’ block and I (A ) as a nonlinear block connected in a feedback loop. It is a pretty obvious but seldom-mentioned fact that a dynamical system, in our case (2.1), can have many equivalent feedback representations (2.16) [5]. To illustrate the idea, we introduce an arbitrary parameter 0 ≤ ␣ ≤ 1. For convenience and to be in tune with control theory, we also introduce dimensionless nonlinear characteristics—i n (v ), i a (v ), f (v )—such that i n (v ) = G res i n (v ) = G res (i a (v ) + (1 − ␣ ) v + ␣ v ) = G res ( f (v ) + ␣ v ) In Section 2.4, we chose ␣ = 0, which results in f (v ) = i n (v ).


35

Under the new notation (2.1) reads as L C

di L =v dt dv = −i L − ␣ G res v − G res f (v ) dt

Repeating the fundamental harmonics derivation from Section 2.4.2, we get: A = Z r (␻ ) F (A )

(2.18)

with the dimensionless transfer functions: Z r (␻ ) = −j␻ LG res /(−␻ 2LC + j␻ L + 1) = −j␻ G res /C (␻ 20 − ␻ 2 + j␻␣ G res /C ), F (A ) = I a (A ) + (1 − ␣ ) A Thus, for each 0 ≤ ␣ ≤ 1, (2.18) is a feedback representation of 2.1. Of course, the reasoning and the results of Section 2.4.2 remain valid for any 0 ≤ ␣ ≤ 1, as long as the ‘‘filter’’ Z r (␻ ) remains selective. 2.4.3.2 Small Parameter and High Q

For the describing function method to be valid, the ‘‘filter’’ Z r (␻ ) must be selective (must have high Q). However, the parameter ␣ G res /C is not necessarily small. Only after we normalize frequency to ␻ 0 , we get a smallparameter system: Z r (␻ ) = =

−j (␻ /␻ 0 ) G res /(C␻ 0 ) 1 − (␻ /␻ 0 )2 + j (␻ /␻ 0 ) ␣ G res /(C␻ 0 ) −j (␻ /␻ 0 ) G res /(C␻ 0 ) 1 − (␻ /␻ 0 )2 + j (␻ /␻ 0 ) ␣ /Q

Similarly in (2.2) the parameter 1/C is not necessarily small; if, however, we introduce the dimensionless nonlinear characteristic i n (v ) = i n (v )/G res and a dimensionless time ␶ = ␻ 0 t that is scaled so that the natural frequency equals one, then

36


dv 1 dv d 2v 1 d 2v , = = . d␶ ␻ 0 dt d␶ 2 ␻ 20 dt 2 Consequently, (2.2) reduces to

␻ 20

d 2v d␶

2

+ G res

di n dv ␻ 0 /C + ␻ 20 v = 0 dv d␶

and further to d 2v d␶

2

+ (1/Q )

di n dv +v=0 dv d␶

Thus, a high-Q circuit corresponds to a ‘‘close-to-linear’’ system from the oscillation theory [11, 14, 15, 44] where ⑀ = 1/Q plays the role of the small parameter.7 From now on we will use the small parameter ⑀ together with our original notation t , i n (v ). The reader who prefers the dimensionless notation should treat t as dimensionless time, i n (v ) as the voltage-to-voltage function, and ⑀ as dimensionless ⑀ = 1/Q . The one who prefers physical parameters should treat t , i n (v ) as the original time and current and ⑀ as ⑀ = 1/C (and keep in mind that ⑀ becomes small in dimensionless units). Figure 2.9 compares the value of the oscillations amplitude obtained from (2.16) with phase portraits obtained for the same nonlinear characteristics for two values of Q . The oscillations amplitude obtained from I (A ) = 0 equals to A 0 = 0.8. Phase plane portraits show that for ⑀ = 0.03 the oscillations indeed stabilize at A 0 = 0.8. However, for a lower Q (⑀ = 0.15), they stabilize at about A 0 = 0.84, confirming the fact that the higher the Q is (the smaller ⑀ ), the better the describing function approximation. 2.4.4 Large Signal S-Parameters The above procedure can be easily extended to an S-parameter representation of the circuit. Indeed, for sinusoidal voltage and current (2.15) we define the ‘‘large-signal’’ incident and reflected waves: a = A + Z 0 I (A )/(2√Z 0 ), b = A − Z 0 I (A )/(2√Z 0 )

(2.19)

7. The parameter ⑀ introduced here and used until the end of this chapter is different that the one defined by (2.7b) in Section 2.3.2.


37

Figure 2.9 Comparison of the amplitude obtained via the describing function method with phase-plane portraits: (a) I = I (A ), (b) phase-plane for Q = 33 (⑀ = 0.03), and (c) phase-plane for Q = 6.6 (⑀ = 0.15).

where A and I (A ) are voltage and current fundamentals and we repeat the procedure. On the resonator side we have a = S r (␻ )b and on the active circuit side b = b (a ). These two relationships provide us with the steady-state equation a = S r (␻ ) b (a )

(2.20)

After defining the ‘‘large signal’’ S-parameter (which we easily recognize as an equivalent of the describing function Y (A ) = I (A )/A from the previous section): S n (a ) = b (a )/a, (2.20) is reduced to

38


1 = S r (␻ ) S n (a )

(2.21)

which is equivalent to (2.16). Geometrical interpretation is shown in Figure 2.10. It is as simple as the one presented for large signal admittance. The crossing point of S r (␻ ) = 1/S n (a ) determines the frequency and amplitude of the oscillations. In our circuit the resonator characteristic coincides with the unit circle and the nonlinear resistance with the real axis; their crossing determines the frequency for which Im(S r (␻ )) = 0 and the amplitude of oscillations S n (a ) = 1. Note that (2.21) is exactly (2.4) from Section 2.3 on stability criteria. This equation determines the frequency and the amplitude of oscillations. We see now that when variables are separated then the S r is the same as the one obtained from the Nyquist loop. Consequently, in a high-Q circuit with real valued S n , the frequency at which the loop crosses the real axis is the oscillation frequency (more exactly, as shown in Appendix 2B, the two frequencies are close as long as we have a high-Q resonator). 2.4.5 Nonresistive Active Circuit The real-life active circuits cannot be treated as purely resistive except at very low frequencies. That is, the frequency and amplitude cannot be separated and the describing function equations should be represented as:

Figure 2.10 LS S-parameters.


S r (␻ ) S n (A , ␻ ) = 1

39

(2.22)

However, high-Q resonators are an exception to that rule. Indeed, in their case the variation of S n (A , ␻ ) with frequency is negligible relative to that of the high-Q resonator. As an illustration consider Figure 2.11 in which S r (␻ ) and 1/S n (A , ␻ ) are shown for our circuit with the transistors’ parasitic reactances included. Instead of a single trace S n (A ), we have many: S n (A , ␻ ), ␻ 1 < ␻ < ␻ 2 . However, the resonant part of the circuit varies so quickly with frequency that within the band of interest all the active traces are practically indistinguishable. Consequently, we can treat A and ␻ as separated variables, one determined by S n (A , ␻ ) ≈ S n (A ), and the other by S r (␻ ). Indeed, Figure 2.11 shows the resonator characteristics S r (␻ ) for a narrow frequency band (from 440 to 500 MHz), it coincides with the arc on the unit circle (because we have included the resonator losses into nonlinear circuit). We also plot

Figure 2.11 LS S-parameters for a real-life nonlinear circuit. The resonator characteristics S r (␻ ) coincide with the arc on the unit circle. The thick line shows 1/S n (A , ␻ ) varying with A for a fixed frequency value. It is crossed by very short thin lines that show 1/S n (A , ␻ ) at fixed amplitudes and varying frequency.

40


1/S n (A , ␻ ). The thick line shows 1/S n (A , ␻ ) varying with A for a fixed frequency value. It is crossed by very short thin lines that show 1/S n (A , ␻ ) at fixed amplitudes and varying frequency. The frequency variation is negligible even though S n (A , ␻ ) varies over a much wider frequency band (from 400 to 600 MHz). Consequently, we treat all traces of 1/S n (A , ␻ ) within this band as one. Their crossing with S r (␻ ), which determines the amplitude and frequency, is shown by the marker m 2. 2.4.6 Conclusions We have shown that the oscillations in high-Q oscillators are nearly sinusoidal and that their frequency, amplitude, and power can be determined by a simple graphical construction. The construction has a straightforward physical interpretation; it consists of plotting an active circuit characteristic as depending on the amplitude and the resonator characteristic depending on frequency. Chapter 1 showed how the idea was developed and applied to nonlinear oscillator design. The method is rigorous (i.e., if the graphical method yields amplitude and frequency, we know that the exact solution exists and can estimate the error that we have made.) We have followed Kudrewicz [40] for its justification. A parallel work was done in the 1960s and the early 1970s in control theory. A good overview of this method, called a describing function method, can be found in Mees [5]. Availability of harmonic balance analysis in CAD tools made the describing function particularly easy to implement; at the same time, thanks to its physical insight, it forms a perfect tool to complement the harmonic balance analysis. The describing function concept has recently been applied to BiCMOS oscillator design by Wu and Hajimiri [10].

2.5 Dynamics of High-Q Oscillators 2.5.1 Introduction Let us now consider high-Q oscillators, in which the amplitude and phase vary slowly in time. The slow variation is forced by the high-Q property of the resonator. Chapter 1 shows how the concept arose in works of Adler, Edson, and Kurokawa. Its rigorous justification is based on the method of averaging developed by Krylov, Bogoliubov, and Mitropolskii for the systems with a small parameter [1–3]. Here we use our simple oscillator to show the principles lying behind the method and refer to Appendices 2C and 2D for its justification.


41

2.5.2 Oscillation Stability If the circuit has high Q, then the waveforms are close to sinusoidal, and it is convenient to perform the transformation of variables from voltagecurrent v, i L to amplitude-phase A , ␸ : v (t ) = A cos (␻ 0 t + ␸ ) i L (t ) = −

(2.23)

A␻ 0 sin (␻ 0 t + ␸ ) L

where ␻ 0 = √1/LC is the natural frequency of the circuit. The amplitude of oscillations is described by the following ‘‘averaged’’ equation [1–3]8: A˙ = −⑀ I (A )

(2.24)

where 2 I (A ) = ␲

␲

冕

i n (A cos ␪ ) cos ␪ d␪

0

Mathematical proof of this fact is based on theory of averaging [1–3] (also see Appendix 2C). Intuitively one can argue that the sinusoidal waveform dominates all the others and therefore controls oscillator’s dynamics. Note that I (A ) is the same function that was defined by (2.14) and can easily be computed via a harmonic balanced simulator. Consequently, the constant solutions obtained by the method of averaging are equal to those obtained by the describing function method. The shape of I (A ) determines the oscillation’s buildup, from a small signal to the steady state (which corresponds to the constant amplitude) as shown in Figure 2.12. The insight into the amplitude’s behavior allows us to determine the stability of the constant solution. Indeed, the value of A 0 at which I (A 0 ) = 0 determines the (constant) amplitude of steady-state oscillations. If dI (A 0 )/dA < 0, then the oscillations are stable, because the oscillations with amplitudes smaller than A 0 grow and the ones with A > A 0 decrease. 8. The parameter ⑀ is ‘‘small’’ in a high-Q circuit. As explained in Section 2.4.3 ⑀ = 1/Q in dimensionless variables and ⑀ = 1/C in the original variables.

42


Figure 2.12 Oscillation buildup from a small signal to A 0 such that I (A 0 ) = 0.

We reached the same conclusion from power considerations in Section 2.3 above; however, (2.24) additionally gives us the shape and the rate of the amplitude buildup. Moreover, knowing I (A ), we can calculate the time it takes for the oscillations to build up from an amplitude A 1 = A (t 1 ) to A 2 = A (t 2 ) [45]. Indeed, dividing both sides of (2.24) by ⑀I (A ) and integrating in time, we get t2

t2 − t1 =

1 ⑀

冕 t1

A2

Adt 1 = I (A ) ⑀

冕

dA I (A )

(2.25)

A1

Equation (2.25) has an important design application for minimizing battery usage in portable transceivers. Note that for small A we can linearize I (A ) = I ′(0)A and calculate the start time explicitly:

Methods of Oscillator Design A2

t2 − t1 =

1 ⑀

冕

43

冉冊

1 dA A2 = ln I ′(0)A ⑀ I ′(0) A1

A1

Hence: A (t 2 ) = A (t 1 ) e ⑀ I ′(0)(t 2 − t 1 )

(2.26)

The exponential growth of small A (t ) is clearly visible in Figure 2.12. A general numerical algorithm called the envelope method, which can solve the problem and analyze modulated signals, has been invented independently by Edouard Ngoya and Remy Larcheveque at University of Limoges and by David Sharrit at Hewlett-Packard (now Agilent) and introduced into some commercially available software tools [46, 47]. Here we have shown that for high-Q oscillators a single harmonic can do the job.

2.6 Oscillations in the Presence of an External Signal 2.6.1 Introduction This section extends the results of Sections 2.4 and 2.5 to the oscillations in presence of an external signal. Specifically we show how the large signal characteristic I (A ) can be used to determine the existence and stability of synchronized oscillations. For the sake of clarity we shall use I (A ) simulated for our oscillator. As previously the analysis remains valid for any high-Q circuit with N-shaped nonlinear characteristics. 2.6.2 Circuit Equations Let an external signal s (t ) = B cos (␻ t ) be applied to the oscillator and modeled as a current source in parallel with the resonator. L C

di =v dt dv = −i − i n (v ) − B cos ␻ t dt

(2.27)

44


Repeating the transformation of coordinates from Section 2.5 and Appendix 2C, we end up with averaged equations for amplitude and phase: 1 A˙ = − (I (A ) + B cos ␸ ) C

␸˙ = −␦ +

B sin ␸ CA

If we had used the dimensionless variables, then 1/C would have been replaced by 1/Q . Let us keep going with our universal small parameter ⑀ from Section 2.4.3 A˙ = −⑀ (I (A ) + B cos ␸ )

␸˙ = −␦ + ⑀

(2.28)

B sin ␸ A

where I (A ) is defined by (2.14), B denotes the amplitude of the locking signal, and ␦ = (␻ 2 − ␻ 20 )/2␻ ≈ ␻ − ␻ 0 denotes ‘‘the amount of detuning,’’ in the dimensionless variables ␦ = (␻ 2 − ␻ 20 )/2␻␻ 0 ≈ (␻ − ␻ 0 )/␻ 0 .9 The constant solutions of the equations (2.28) can be found from I (A ) = −B cos ␸

(2.29)

␦ A = ⑀ B sin ␸ which are equivalent to the fundamental harmonic equation I (A )/A − jC

␻ 2 − ␻ 20 = −Be j␸ /A ␻

(2.30)

Equation (2.30) is a special case of the Kurokawa equation (1.22) from Chapter 1: Y (A ) − Y r (␻ ) = Be j␸ /A

(2.31)

9. The difference between our present case (locked oscillations) and the case in Section 2.5 (free oscillations) is that now the frequency is that of the injected signal, and the phase can be uniquely determined with reference to the injected signal; for the free-running oscillations, the phase was undetermined, and we considered only amplitude equations.


45

Let us discuss (2.28) and (2.29) in some detail because they provide a simple graphical method of finding locked oscillations and determining the locking range. Note first that (2.29) possesses the constant solution only if ␦ ≤ ⑀ B /A holds. This condition is a rigorous representation of the fact that the frequency locks only when the synchronizing frequency is close to the natural frequency of the circuit (i.e., the ‘‘amount of detuning’’ is small). Moreover, the locking zone increases with increasing external signal (B ) and decreasing Q (increasing ⑀ ). Section 2.6.3 looks more closely at the behavior of these solutions. 2.6.3 Resonance Characteristics We have seen that the constant solutions of (2.28), and consequently the amplitude and phase of locked oscillations, are determined by (2.29). Let us discuss now the behavior of the solutions of (2.29). They turn out to have a simple geometrical interpretation. Indeed, adding up the squared sides of (2.29) we conclude that the amplitude of locked oscillations satisfies the scalar equation: I (A )2 + A 2(␦ /⑀ )2 = B 2

(2.32)

Equation (2.32) describes the relationship between A , B, and ␦ /⑀ . For fixed B it can be interpreted as a nonlinear resonance characteristic, which represents a variation of an ‘‘output’’ amplitude versus detuning (i.e., A versus ␦ ). The effect of detuning is normalized by ⑀ = 1/Q and increases with increasing Q . The equation can be easily solved graphically and indeed has been for the van der Pol nonlinearity as shown in the Figure 2.13. To this author’s knowledge it has been applied to general circuits only in [48] with the Meissner oscillator as an example. Here we repeat the reasoning of [48] enhanced by three-dimensional plots and illustrated by the oscillator with cross-coupled BJTs. Figure 2.13(d) was derived by van der Pol [49] and is often cited in advanced texts on nonlinear oscillations [14, 15, 44]. It is shown here chiefly for reference purposes and as an illustration of the simple fact that the characteristics obtained for fixed detuning [Figure 2.13(c)] and those obtained for fixed external signal [Figure 2.13(d)] are just two aspects of the threedimensional relationship between A , B, and ␦ /⑀ shown in Figure 2.13(b). Figure 2.14 is of real interest to us. It clearly shows the locking conditions of our oscillator. These conditions are determined by the resonator’s

46


Figure 2.13 Nonlinear resonance characteristics for van der Pol equation: (a) nonlinear characteristics i = G (v − v 3 /3), (b) relationship between oscillations and external signal amplitudes and detuning, given by (2.32), (c) a cross section of (b) for fixed detuning, and (d) contour lines of (b) for fixed B.

quality factor and by the nonlinear characteristics of the active circuit shown in Figure 2.14(a). The relationship between oscillations and external signal amplitudes and detuning, given by (2.32), is shown in three dimensions in Figure 2.14(b). From it we can graphically determine solutions of (2.32) (i.e., the dependence of amplitudes of locked oscillations on the amount of detuning and on the strength of the external signal). The relationship for fixed detuning is shown in Figure 2.14(c); the one for the fixed external signal is shown in Figure 2.14(d). The latter is shown for positive ␦ ; the contours for negative ␦ are the mirror image of those of Figure 2.14(d). They can be interpreted as a nonlinear resonance characteristic. The ‘‘inverted vase’’ shapes of the contours obtained for small B are clearly related to the shape of I (A ). The value of A 0 = 0.8 for which I (A 0 ) = 0 corresponds to the center of the ‘‘vase;’’ the value A 1 ≈ 0.18 for which I (A ) reaches maximum to the ‘‘neck’’ of our ‘‘vase.’’


47

Figure 2.14 Nonlinear resonance characteristics for the oscillator with cross-coupled BJTs: (a) the nonlinear characteristics i n (v ); (b) the relationship between oscillations and external signal amplitudes and detuning, given by (2.32); (c) a cross-section of (b) for fixed detuning; and (d) contour lines of (b) for fixed B.

Figures 2.14(c) and 2.14(d) provide a simple geometrical way to solve (2.32). They clearly show that for small values of B and ␦ , (2.32) has multiple solutions. For large B and ␦ , when A 2␦ 2 dominates I (A 2 ), the plots become monotonic and only one solution exists. So far we do not know which of the graphically found solutions are stable. It turns out that stability can be deduced from Figure 2.14, namely that the stable oscillations correspond to the upper parts of resonance characteristics in Figure 2.14(d), which decrease when ␦ grows. Indeed, the stability of a steady-state solution can be determined from (2.28) linearized about its constant solution A 0 , ␸ 0 : ⌬A˙ = −⑀ I ′(A 0 ) ⌬A + ⑀ B sin (␸ 0 ) ⌬␸ ⌬␸˙ = −⑀

B sin (␸ 0 ) A 20

⌬A + ⑀

B cos (␸ 0 ) ⌬␸ A0

48


Since A 0 , ␸ 0 is a solution of (2.28), we get

␦ A 0 = ⑀ B sin ␸ 0 Consequently,

冋册冋 ⌬A˙ ⌬␸˙

=

−⑀ I ′(A 0 ) −␦ /A 0

册冋册

␦A0 ⌬A = −⑀ I (A 0 )/A 0 ⌬␸

The solution is stable when the trace of the matrix is negative and the determinant positive, that is, when I ′(A 0 ) + I (A 0 )/A 0 > 0

(2.33a)

⑀ 2I ′(A 0 ) I (A 0 )/A 0 + ␦ 2 > 0

(2.33b)

and

Now, from (2.32), the gradient B 2 with respect to (A , ␦ /⑀ ) equals [∂B /∂A , ∂B 2/∂(␦ /⑀ )] = [2I ′(A ) I (A ) + 2A␦ 2/⑀ 2, 2A 2␦ /⑀ ]. Consequently, the condition (2.33b) is satisfied when the gradient points upward (toward growing A ) in Figure 2.14(d). In other words, the condition is satisfied on the downsloped (A decreases when | ␦ | grows) parts of the contours. The first condition (2.33a) is expressed in terms of I (A ) and its derivative. It is shown in Figure 2.15. Since I (A ) is negative for small A and reaches minimum at A 1 ≈ 0.18 and zero at A 0 = 0.8, the expression I ′(A ) + I (A )/A is certainly negative for A < A 1 and positive for A > A 0 ; in between I ′(A ) + I (A )/A , it grows monotonically and crosses zero at A 2 ≈ 0.42, as shown in Figure 2.15. Thus, the locked oscillations are stable if their amplitude is larger than A 2 ≈ 0.42 and the corresponding detuning lies on the sloping down parts (A decreases when ␦ grows) of the contours in Figure 2.14(d). 2

2.7 Summary We have reviewed analysis methods for two classes of nonlinear oscillators: LC and high-Q. The first class relies on phase-plane analysis and provides insight into the global dynamics of the circuit. The second deals with oscilla-


49

Figure 2.15 Determination of the sign of I ′(A ) + I (A )/A : (a) I ′(A ) + I (A )/A is shown in a continuous line and its components I ′(A ), I (A )/A are shown in dotted lines and (b) same plots in zoomed in vertical scale, for reference I (A ) is shown in the thin line.

tions that are close to sinusoidal with the amplitude and phase either constant or slowly varying. Consequently, the behavior of the amplitude and phase determines the oscillations in the circuit. If an approximate solution is found, then the existence of the exact one is proven and the error estimated. The methods presented are all rigorous; references to theorems and to their proofs are included in Appendices 2A–2D. Most of the methods discussed here are not new; some of them can be traced to van der Pol’s intuition from the early 1930s and to the works of Krylov, Bogoliubov, Mitropolski, and Hale from the period 1930–1960. The latter were hardly available to circuit design. For that reason we have chosen a commonly used circuit as an illustration of different approaches. Our approach is related to the works of Pierce, Edson, and Kurokawa described in Chapter 1, which are general and applicable to microwave diode

50


oscillators, although they are not as rigorous as [1–3, 6]. Extensive analysis focused on classical equations can be found in [14, 15, 44]. The linear theory of oscillations is only mentioned in Section 2.3. The rest of that section clarifies a common abuse of stability criterion. We have not discussed phase-noise analysis. Chapter 3 and sections of Chapter 4 are devoted to that important topic. We also did not discuss bifurcation phenomena (either in terms of onset of oscillation or transition to ‘‘chaos’’), which are out of the scope of this chapter. The interested reader may wish to consult [5, 50–52].

References [1]

Bogoliubov, N. N., and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, New York: Gordon and Breach, 1961.

[2]

Hale, J. K., Ordinary Differential Equations, 2nd ed., Huntington, NY: Krieger, 1980.

[3]

Hale, J., Oscillations in Nonlinear Systems, New York: Dover, 1992.

[4]

Levi, M., ‘‘Qualitative Analysis of the Periodically Forced Relaxation Oscillations,’’ Ph.D. thesis, Department of Mathematics, Duke University, 1978.

[5]

Mees, A., Dynamics of Feedback Systems, New York: John Wiley and Sons, 1981.

[6]

Mitropolskii, Yu A., and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics (in Russian), Moscow, Russia: Nauka, 1973.

[7]

Pederson, D. O., ‘‘A Historical Review of Circuit Simulation,’’ IEEE Trans. on Circuits and Systems, Vol. CAS-31, January 1984, pp. 103–111.

[8]

Hajimiri, A., and T. Lee, The Design of Low Noise Oscillators, Boston, MA: Kluwer, 1999.

[9]

Larson, L., RF and Microwave Circuit Design, Norwood, MA: Artech House, 1996.

[10]

Wu, H., and A. Hajimiri, ‘‘Silicon-Based Distributed Voltage-Controlled Oscillators,’’ IEEE Journal of Solid State Circuits, Vol. 36, No. 3, March 2001.

[11]

Andronov, A. A., A. A. Vitt, and S. E. Khaykin, Theory of Oscillations, New York: Dover, 1987.

[12]

Coddington, E. A., and N. Levison, Theory of Ordinary Differential Equations, Malabar, FL: Krieger, 1984.

[13]

Reissig, R., G. Sansone, and R. Conti, Nonlinear Theory of Nonlinear Differential Equations (in Russian), Moscow, Russia: Nauka, 1974.

[14]

Hayashi, C., Nonlinear Oscillations in Physical Systems, New York: McGraw-Hill, 1964.

[15]

Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems, New York: John Wiley and Sons, 1992.

[16]

Odyniec, M., and L. O. Chua, ‘‘Integral Manifolds for Nonlinear Circuits,’’ Intl. Journal of Circuit Theory and Applications, Vol. 12, 1984, pp. 293–328.


51

[17]

Guckenheimer, J., ‘‘Dynamics of the van der Pol Equation,’’ IEEE Trans. on Circuits and Systems, Vol. CAS-27, November 1980.

[18]

Jackson, R. T., ‘‘Criteria for the Onset of Oscillations in Microwave Circuits,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-40, March 1992, pp. 566–568, ‘‘Comments on Criteria . . . ,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-40, September 1992, pp. 1850–1851.

[19]

Khanna, A. P. S., ‘‘Oscillators,’’ Chapter 9 of Microwave Solid State Circuit Design, I. Bahl and P. Bhartia (eds.), New York: John Wiley and Sons, 1988.

[20]

Krauss, H. L., C. W. Bostian, and F. H. Raab, Solid State Radio Engineering, New York: John Wiley and Sons, 1980.

[21]

Martin, J. L. J., and F. J. O. Gonzalez, ‘‘Accurate Linear Oscillator Analysis and Design,’’ Microwave Journal, June 1996.

[22]

Matthys, R. J., Crystal Oscillator Circuits, New York: John Wiley and Sons, 1983, and Malabar, FL: Krieger, 1992.

[23]

Rhea, R., Oscillator Design and Computer Simulation, 2nd ed., Atlanta, GA: Noble Publishing, 2000.

[24]

Vendelin, G. D., U. L. Rohde, and A. M. Pavio, Microwave Circuit Design Using Linear and Nonlinear Techniques, New York: John Wiley and Sons, 1990.

[25]

Abrie, P., Design of RF and Microwave Amplifiers and Oscillators, Norwood, MA: Artech House, 1999.

[26]

Boyles, J. W., ‘‘The Oscillator as a Reflection Amplifier,’’ Microwave Journal, June 1986.

[27]

Camargo, E., Design of FET Frequency Multipliers and Harmonic Oscillators, Norwood, MA: Artech House, 1998.

[28]

Nativ, Z., and Y. Shur, ‘‘Push-Push VCO Design with CAD Tools,’’ Microwave Journal, February 1989.

[29]

Przedpelski, A., ‘‘Simple, Low Cost UHF VTOs,’’ RF Design, May 1993.

[30]

Ratier, N., et al., ‘‘Automatic Formal Derivation of the Oscillation Condition,’’ IEEE Int. Frequency Control Symp., 1997.

[31]

Razban, T., et al., ‘‘A Compact Oscillator,’’ Microwave Journal, February 1994.

[32]

Savaria, S., and P. Champagne, ‘‘Linear Simulators,’’ Microwave Journal, May 1995.

[33]

Schiebold, C., ‘‘Getting Back to the Basics of Oscillator Design,’’ Microwave Journal, May 1998.

[34]

Sweet, A., MIC and MMIC Amplifier and Oscillator Circuit Design, Norwood, MA: Artech House, 1990.

[35]

Warren, D., et al., ‘‘Large and Small Signal Oscillator Analysis,’’ Microwave Journal, May 1989.

[36]

Basawapatna, G. R., and R. B. Stancliff, ‘‘A Unified Approach to the Design of WideBand Microwave Solid State Oscillators,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-27, May 1979, pp. 379–385.

[37]

D’Azzo, J. J., and C. H. Houpis, Feedback Control System Analysis and Synthesis, New York: McGraw-Hill, 1960.

52


[38]

Odyniec, M., ‘‘Stability Criteria Via S-Parameters,’’ EuMC’95, Bologna, Italy, September 1995 (also ‘‘Oscillator Stability Analysis,’’ Microwave Journal, June 1999).

[39]

Kundert, K., Steady State Methods for Simulating Analog and Microwave Circuits, Boston, MA: Kluwer, 1995.

[40]

Kudrewicz, J., ‘‘Frequency Methods’’ (in Polish), WNT, 1972.

[41]

Kudrewicz, J., ‘‘Contribution to the Theory of Weakly Nonlinear Oscillators,’’ Intl. Journal of Circuit Theory and Applications, Vol. 4, 1976, pp. 161–176.

[42]

Kurokawa, K., An Introduction to the Theory of Microwave Circuits, New York: Academic Press, 1969.

[43]

Kurokawa, K., ‘‘Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits,’’ Bell System Technical Journal, July–August 1969, pp. 1937–1955.

[44]

Minorsky, N., Nonlinear Oscillations, Huntington, NY: Krieger, 1974.

[45]

Odyniec, M., and W. Overstreet, ‘‘New Applications of Harmonic Balance Analysis,’’ Intl. Journal on Microwave and Millimeter-Wave CAE, May 1995.

[46]

Ngoya, E., and R. Larcheveque, ‘‘Envelope Transient,’’ MTTS’1996, San Francisco, CA, June 1996.

[47]

Sharrit, D., ‘‘New Method of Analysis of Communication Systems,’’ Workshop on Nonlinear CAD, MTTS’1996, San Francisco, CA, June 17, 1996.

[48]

Odyniec, M., ‘‘Nonlinear Synchronized LC Oscillators,’’ IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-41, No. 5, May 1993, pp. 774–780.

[49]

van der Pol, B., ‘‘The Nonlinear Theory of Electric Oscillators,’’ Proc. IRE, Vol. 22, No. 9, September 1934.

[50]

DeFeo, O., G. M. Maggio, and M. P. Kennedy, ‘‘The Colpitts Oscillator: Families of Periodic Solutions and Their Bifurcations,’’ Intl. Journal Bifurcation and Chaos, March 2000.

[51]

Palazuelos, E., et al., ‘‘Hysteresis Prediction in Autonomous Microwave Circuits Using Commercial Software, Application to a Ku Band MMIC VCO,’’ IEEE Journal of Solid-State Circuits, Vol. 33, No. 8, August 1998, pp. 1239–1243.

[52]

Ver Hoeye, S., A. Sua´rez, and S. Sancho, ‘‘Analysis of Noise Effects on the Nonlinear Dynamics of Synchronized Oscillators,’’ IEEE Microwave and Wireless Components Letters, Vol. 11, No. 9, September 2001.

Appendix 2A: Nyquist Stability Criterion Consider a feedback system with two linear blocks, one the ‘‘filter’’ with transfer function ‘‘H (s ),’’ the other the ‘‘amplifier’’ G (s ). According to the Nyquist stability criterion [37], the number of the right-hand plane poles of the feedback system equals the number of the right-hand plane poles of H (s ) G (s ) plus the number of the clockwise encirclement of 1 + j 0 by H ( j␻ ) G ( j␻ ). When H (s ) G (s ) has no poles in the right half plane, then the encirclements of 1 + j 0 determine the stability of the system. The above


53

remains valid for any transfer function, in particular when H (s ) G (s ) is replaced by Yn (s ) Zr (s ) or Sr (s ) Sn (s ). For resonators with high Q one can modify the Nyquist criterion as follows: If the resonator has high Q, then the dependence of G on frequency can be neglected and it suffices to check whether H ( j␻ ) encircles 1/G + j 0 [rather than GH ( j␻ ) encircling 1 + j 0]. The Nyquist loop as well as the magnitude and the phase of the transfer function can be easily plotted with commercially available software.

Appendix 2B: Justification of the Describing Function Method Consider an oscillator circuit described by an integral equation ∞

v (t ) =

冕

h (␶ ) f (v (t − ␶ )) d␶

(2B.1)

0

Periodic solutions of (2B.1) can be expanded into a Fourier series: ∞

v (t ) =

∑

k = −∞

v k e jk␻ t, f (v (t )) =

∞

∑

k = −∞

f k (v 0 , v 1 , v 2 , . . . ) e jk␻ t

the coefficients of which can be found from the equation: v k = H (k␻ ) f k (v 0 , v 1 , v 2 , . . . )

(2B.2)

where ∞

H (␻ ) =

冕

h (t ) e jk␻ t dt

−∞

T

1 f k (v 0 , v 1 , v 2 , . . . ) = 2T

冕冢∑ ∞

f

−T

k = −∞

冣

v k e jk␻ t e −jk␻ t dt

where H (␻ ) is the ‘‘filter’’ characteristic and h (t ) is its impulse response; f is the nonlinear element. We use an integral equation to describe the system

54


because in that case we include filters with distributed elements. In particular, a lumped system described by the ordinary differential equation K

∑ ak

k=0

冉冊

k

d v= dt

L

∑ bl

l=0

冉冊 l

d f (v (t )) dt

is a special case of (2B.1) with L

H (␻ ) =

⁄

K

∑ b l ( j␻ ) ∑ a k ( j␻ )k l

l=0

k=0

After neglecting all harmonics except the first, we obtain: v 1 = H (␻ ) f 1 (0, v 1 , 0, . . . )

(2B.3)

Since v 1 is real, we rewrite (2B.3) as: A = H (␻ ) F (A ) where

(2B.4)

2v 1 = A T

冕

1 f 1 (0, v 1 , 0, 0) = 2T

=

1 ␲

f (v 1 e j␻ t + v −1 e −j␻ t ) e −j␻ t dt

−T ␲

冕

f (A cos ␪ ) cos ␪ d␪ = F (A )/2

0

The following theorem is proven in [40]. Theorem (Kudrewicz). If the following conditions hold:

(a) Equation (B2.4) possesses a solution A , ϖ that persists under small perturbations. (b) The active circuit characteristic is bounded and has bounded slope ∃ M > 0 ∃ L > 0 ∀A | F (A ) | < M , | dF (A )/dA | < L


55

(c) The ‘‘filter’’ is selective ∃ ␩ > 0 ∀k ≠ 1 | H (k␻ ) | < ␩ . Then the exact equation also possesses a periodic solution ∞

v (t ) =

∑

k = −∞

v k e jk␻ t, i (t ) =

∞

∑

k = −∞

i k e jk␻ t

(2B.5)

which is close to the approximate one: v˜ (t ) = A cos (ϖ t )

(2B.6)

More exactly:

| v1 − A | < d , | ␻ − ϖ | < ␴ ,

∞

∑ | vk | 2 < ␳ 2

(2B.7)

|k |≠1

where d , ␴ , and ␳ are small numbers that depend on M , L , and ␩ from conditions (b) and (c) above. They estimate the errors in the amplitude of the fundamental (d), frequency (␴ ), and the harmonic distortion ( ␳ ). Outline of the proof. Consider (2B.2) without 1 of k = 1. Thus, we get an equation in ‘‘sequence space’’ l 2 with variables v k , k = 0, 2, 3, . . . and with v 1 = A /2 and ␻ as parameters:

v k = H (k␻ ) f k (v 0 , A /2, v 2 , . . . ), k = 0, 2, 3, . . .

(2B.8)

Since H (␻ ) is selective (i.e., | H (k␻ ) | is small for k = 0, 2, 3, . . . ) and f has a bounded slope, it follows from the theorem of contraction mapping that for any A and ␻ , (2B.8) has a unique and small solution v k = v k (A , ␻ ), k = 0, 2, 3, . . . . Substituting this solution into the equation with k = 1, we get: A = 2H (␻ ) f 1 (v 0 (A , ␻ ), A /2, v 2 (A , ␻ ), . . . )

(2B.9)

Since v k = v k (A , ␻ ) are small for k = 0, 2, 3, . . . , (2B.9) is a small perturbation of (2B.4). Consequently, the existence of a solution A , ϖ of (2B.4) that persists under small perturbations causes the existence of solution of (2B.9), and the two solutions remain close.

56


Appendix 2C: Transformation of Voltage-Current to Amplitude-Phase Equations Our oscillator with the external source is described by: di L =v dt

L

(2C.1)

dv = −i L − i n (v ) − B cos ␻ t dt

C

If the circuit has high Q and the waveforms are close to sinusoidal, then it is convenient to perform the transformation of variables from voltagecurrent v, i L to amplitude-phase A , ␸ v (t ) = A (t ) cos (␻ t + ␸ (t )) i L (t ) =

(2C.2)

A (t ) sin (␻ t + ␸ (t )) L␻

Differentiated in time, (2C.2) reads: dv (t )/dt = A˙ cos (␻ t + ␸ ) − A sin (␻ t + ␸ )␸˙ − A␻ sin (␻ t + ␸ ) Ldi L /dt = (A˙ /␻ ) sin (␻ t + ␸ ) + (A /␻ ) cos (␻ t + ␸ )␸˙ + A cos (␻ t + ␸ ) (2C.3) Consequently, (2C.1) transforms to: (A˙ /␻ ) sin (␻ t + ␸ ) + (A /␻ ) cos (␻ t + ␸ )␸˙ + A cos (␻ t + ␸ ) = A cos (␻ t + ␸ ) CA˙ cos (␻ t + ␸ ) − CA sin (␻ t + ␸ )␸˙ − CA␻ sin (␻ t + ␸ ) = −(A /␻ L ) sin (␻ t + ␸ ) − i n (A cos (␻ t + ␸ )) − B cos (␻ t ) which are linear in A˙ , ␸˙

冋

sin (␻ t + ␸ )/␻

(A /␻ ) cos (␻ t + ␸ )

C cos (␻ t + ␸ )

−CA sin (␻ t + ␸ )

册冋册冋册 A˙ ␸˙

=

0 f

(2C.4)


57

where f = 2AC␦ sin (␻ t + ␸ ) − i n (A cos (␻ t + ␸ )) − B cos (␻ t )

␦ = (␻ 2 − ␻ 20 )/2␻ ≈ ␻ − ␻ 0 Solving for A˙ , ␸˙ , we get 1 A˙ = [2AC␦ sin (␻ t + ␸ ) − i n (A cos (␻ t + ␸ )) C − B cos (␻ t )] cos (␻ t + ␸ )

␸˙ = −

(2C.5)

1 [2AC␦ sin (␻ t + ␸ ) − i n (A cos (␻ t + ␸ )) AC

− B cos (␻ t )] sin (␻ t + ␸ ) after accounting for cos (␻ t ) sin (␻ t + ␸ ) = [sin (2␻ t + ␸ ) + sin (␸ )] cos (␻ t ) cos (␻ t + ␸ ) = [cos (2␻ t + ␸ ) + cos (␸ )] and averaging the terms explicitly dependent on time, we get: A˙ = −I (A )/C − (B /C ) cos ␸

␸˙ = −␦ +

(2C.6)

1 B sin ␸ AC

In case of the free-running oscillator B = 0, we apply the transformation (2C.2) with ␻ = ␻ 0 where ␻ 0 = 1/√LC is the natural frequency of the circuit. Consequently, we get the final equations with B = 0, ␦ = 0: A˙ = −I (A )/C

(2C.7)

␸˙ = 0 Note that (2C.7) yields a constant phase and the same constant amplitude as the one obtained from the describing function method. It provides us with the information of the varying amplitude of Figure 2.12; in particular, it clearly shows the stability of the constant amplitude.

58


Appendix 2D: Theorems on Averaging Many practical problems (including the original van der Pol analysis of oscillators) lead to x˙ = ⑀ X (x , t )

(2D.1)

where X (x , t ) is T-periodic in t . When averaged in time, (2D.1) can be reduced to x˙ = ⑀ X 0 (x )

(2D.2)

T

1 where X 0 (x ) = T

冕

X (x , t ) dt . Equation (2D.2) is called the averaged

0

equation. Two theorems hold: 1. Theorem 1 (transient behavior): If the solutions of (2D.1) and (2D.2) originate from the same initial condition and the parameter ⑀ is ‘‘small,’’ then they remain ‘‘close’’ over the time interval proportional to 1/⑀ . 2. Theorem 2 (steady-state): If (2D.2) possesses a constant regular solution x 0 and the parameter ⑀ is ‘‘small,’’ then (2D.1) possesses a T-periodic solution x (t ) that remains ‘‘close’’ to x 0 . Moreover, if x 0 is asymptotically stable, so is x (t ).

Acknowledgments I gratefully acknowledge the works of Jacek Kudrewicz, Jack Hale, Kaneyuki Kurokawa, and the Kiev school of qualitative analysis of nonlinear systems on which this chapter is based. I wish to dedicate it to Jacek Kudrewicz, who introduced me to oscillator analysis.

3 Linearity, Time Variation, and Oscillator Phase Noise Thomas H. Lee and Ali Hajimiri 3.1 Introduction The theoretical and practical importance of oscillators has motivated the development of numerous treatments of phase noise. The sheer number of publications on this topic underscores the importance attached to it. At the same time, many of these disagree on rather fundamental points, and it may be argued that the abundance of such conflicting research quietly testifies to the inadequacies of many of those treatments. Complicating the search for a suitable theory is that noise in a circuit may undergo frequency translations before ultimately becoming oscillator phase noise. These translations are often attributed to the presence of obvious nonlinearities in practical oscillators. The simplest theories nevertheless simply ignore the nonlinearities altogether and frequently ignore the possibility of time variation as well. Such linear, time-invariant (LTI) theories surprisingly manage to provide important qualitative design insights. However, these theories are understandably limited in their predictive power. Chief among the deficiencies of an LTI theory is that frequency translations are necessarily disallowed, begging the question of how the (nearly) symmetrical sidebands observed in practical oscillators can arise. Despite this complication, and despite the obvious presence of nonlinearities necessary for amplitude stabilization, the noise-to-phase transfer 59

60


function of oscillators nonetheless may be treated as linear. However, a quantitative understanding of the frequency translation process requires abandonment of the principle of time invariance implicitly assumed in most theories of phase noise. In addition to providing a quantitative reconciliation between theory and measurement, the time-varying phase noise model presented here identifies an important symmetry principle, which may be exploited to suppress the upconversion of 1/f noise into close-in phase noise. At the same time, it provides an explicit accommodation of cyclostationary effects, which are significant in many practical oscillators, and of amplitudeto-phase (AM-PM) conversion as well. These insights allow a reinterpretation of why certain topologies, such as the Colpitts oscillator, exhibit good performance. Perhaps more important, the theory informs design, suggesting novel optimizations of well-known oscillators, as well as the invention of new circuit topologies. Tuned LC and ring oscillator circuit examples are presented to reinforce the theoretical considerations developed. Simulation issues and the topic of amplitude noise are considered as well. We first revisit how one evaluates whether a system is linear or timeinvariant. Indeed, we find that we must even take care to define explicitly what is meant by the word ‘‘system.’’ We then identify some very general tradeoffs among key parameters, such as power dissipation, oscillation frequency, resonator Q , and circuit noise power. These tradeoffs are first studied qualitatively in a hypothetical ideal oscillator in which linearity of the noiseto-phase transfer function is assumed, allowing characterization by an impulse response. Although the assumption of linearity is defensible, we shall see that time invariance fails to hold even in this simple case. That is, oscillators are linear, time-varying (LTV) systems, where ‘‘system’’ is defined by the noise-to-phase transfer characteristic. Fortunately, complete characterization by an impulse response depends only on linearity, not time invariance. By studying the impulse response, we discover that periodic time variation leads to frequency translation of device noise to produce the phase noise spectra exhibited by real oscillators. In particular, the upconversion of 1/f noise into close-in phase noise is seen to depend on symmetry properties that are potentially controllable by the designer. Additionally, the same treatment easily subsumes the cyclostationarity of noise generators, and helps explain why class-C operation of active elements within an oscillator can be beneficial. Illustrative circuit examples reinforce key insights of the LTV model. In general, circuit and device noise can perturb both the amplitude and phase of an oscillator’s output. Because amplitude fluctuations are usually greatly attenuated as a result of the amplitude stabilization mechanisms present in every practical oscillator, phase noise generally dominates, at least


61

at frequencies not far removed from the carrier. Thus, even though it is possible to design oscillators in which amplitude noise is significant, we focus primarily on phase noise here. We show later that a simple modification of the theory allows the accommodation of amplitude noise as well, permitting the accurate computation of output spectrum at frequencies well removed from the carrier.

3.2 General Considerations Perhaps the simplest abstraction of an oscillator that still retains some connection to the real world is a combination of a lossy resonator and an energy restoration element. The latter precisely compensates for the tank loss to enable a constant-amplitude oscillation. To simplify matters, assume that the energy restorer is noiseless (see Figure 3.1). The tank resistance is therefore the only noisy element in this model. To gain some useful design insight, first compute the signal energy stored in the tank: E2 =

1 2 CV pk 2

(3.1)

so that the mean-square signal (carrier) voltage is: 2 V sig =

E stored C

(3.2)

where we have assumed a sinusoidal waveform. The total mean-square noise voltage is found by integrating the resistor’s thermal noise density over the noise bandwidth of the RLC resonator:

Figure 3.1 ‘‘Perfectly efficient’’ RLC oscillator.

62

RF and Microwave Oscillator Design ∞

V n2

= 4kTR

冕| |

2 kT Z( f ) 1 = df = 4kTR R 4RC C

(3.3)

0

Combining (3.2) and (3.3), we obtain a noise-to-carrier ratio (the reason for this ‘‘upside-down’’ ratio is simply one of convention): kT N V n2 = = C E 2 stored V sig

(3.4)

Sensibly enough, one therefore needs to maximize the signal levels to minimize the noise-to-carrier ratio. We may bring power consumption and resonator Q explicitly into consideration by noting that Q can be defined generally as proportional to the energy stored, divided by the energy dissipated: Q=

␻ 0 E stored P diss

(3.5)

Therefore, N ␻ 0 kT = C QP diss

(3.6)

The power consumed by this model oscillator is simply equal to P diss , the amount dissipated by the tank loss. The noise-to-carrier ratio is here inversely proportional to the product of resonator Q and the power consumed, and directly proportional to the oscillation frequency. This set of relationships still holds approximately for real oscillators and explains the near obsession of engineers with maximizing resonator Q , for example. Other important design criteria become evident by coupling the foregoing with additional knowledge of practical oscillators. One is that oscillators generally operate in one of two regimes that may be distinguished by their differing dependence of output amplitude on bias current (see Figure 3.2), so that one may write Vsig = I BIAS R

(3.7)


63

Figure 3.2 Oscillator operating regimes.

where R is a constant of proportionality with the dimensions of resistance. This constant, in turn, is proportional to the equivalent parallel tank resistance [1], so that Vsig ⬀ I BIAS R tank

(3.8)

implying that the carrier power may be expressed as P sig ⬀ (I BIAS R tank )2

(3.9)

The noise power has already been computed in terms of the tank capacitance as V n2 =

kT C

(3.10)

but it may also be expressed in terms of the tank inductance: V n2 =

kT = C

kT 1

冉冊

= kT␻ 20 L

(3.11)

␻ 20 L

An alternative expression for the noise-to-carrier ratio in the currentlimited regime is therefore

64


kT␻ 0 L N ⬀ C (I BIAS R tank )2

(3.12)

Assuming operation at a fixed supply voltage, a constraint on power consumption implies an upper bound on the bias current. Of the remaining free parameters, then, only the tank inductance and resistance may be practically varied to minimize the N /C ratio. That is, optimization of such an oscillator corresponds to minimizing L /(R tank )2. In many treatments, maximizing tank inductance is offered as a prescription for optimization. However, we see that a more valid objective is to minimize L /(R tank )2 [2]. Since, in general, the resistance is itself a function of inductance, identifying (and then achieving) this minimum is not always trivial. An additional consideration is that, below a certain minimum inductance, oscillation may cease. Hence, the optimization prescription here presumes oscillation, and in a regime where the output amplitude is proportional to the bias current.

3.3 Detailed Considerations: Phase Noise To augment the qualitative insights of the foregoing analysis, let us now determine the actual output spectrum of the ideal oscillator. 3.3.1 Phase Noise of an Ideal Oscillator Assume that the output in Figure 3.1 is the voltage across the tank, as shown. By postulate, the only source of noise is the white thermal noise of the tank conductance, which we represent as a current source across the tank with a mean-square spectral density of i n2 = 4kTG ⌬f

(3.13)

This current noise becomes voltage noise when multiplied by the effective impedance facing the current source. In computing this impedance, however, it is important to recognize that the energy restoration element must contribute an average effective negative resistance that precisely cancels the positive resistance of the tank. Hence, the net result is that the effective impedance seen by the noise current source is simply that of a perfectly lossless LC network.


65

For a relatively small offset frequency ⌬␻ from the center frequency ␻ 0 , the impedance of an LC tank may be approximated by Z (␻ 0 + ⌬␻ ) ≈ j

␻0L ⌬␻ 2 ␻0

(3.14)

We may write the impedance in a more useful form by incorporating an expression for the unloaded tank Q : Q=

1 R = ␻ 0 L ␻ 0 GL

(3.15)

Solving (3.15) for L and substituting into (3.14) yield: 1

| Z (␻ 0 + ⌬␻ ) | ≈ G

␻0 2Q | ⌬␻ |

(3.16)

Thus, we have traded an explicit dependence on inductance for a dependence on Q and G. Next, multiply the spectral density of the mean-square noise current by the squared magnitude of the tank impedance to obtain the spectral density of the mean-square noise voltage:

冉

v n2 i2 ␻0 2 = n | Z | = 4kTR ⌬f ⌬f 2Q ⌬␻

冊

2

(3.17)

The power spectral density of the output noise is frequency-dependent because of the filtering action of the tank, falling as the inverse-square of the offset frequency. This 1/f 2 behavior simply reflects the fact that the voltage frequency response of an RLC tank rolls off as 1/f to either side of the center frequency, and that power is proportional to the square of voltage. Note also that an increase in tank Q reduces the noise density, when all other parameters are held constant, underscoring once again the value of increasing resonator Q . In our idealized LC model, thermal noise affects both amplitude and phase, and (3.17) includes their combined effect. The equipartition theorem of thermodynamics tells us that, in equilibrium, amplitude and phase noise

66


power are equal. Therefore, the amplitude-limiting mechanism present in any practical oscillator suppresses half the noise given by (3.17). It is traditional to normalize the mean-square noise voltage density to the mean-square carrier voltage, and report the ratio in decibels, thereby explaining the ‘‘upside down’’ ratios presented previously. Performing this normalization yields the following equation for phase noise:

冋冉

2kT ␻0 L {⌬␻ } = 10 log P sig 2Q ⌬␻

冊册 2

(3.18)

The units of phase noise are thus proportional to the log of a density. Specifically, they are commonly expressed as ‘‘decibels below the carrier per hertz,’’ or dBc/Hz, specified at a particular offset frequency ⌬␻ from the carrier frequency ␻ 0 . For example, one might speak of a 2-GHz oscillator’s phase noise as ‘‘−110 dBc/Hz at a 100-kHz offset.’’ It is important to note that the ‘‘per Hz’’ actually applies to the argument of the log, not to the log itself; doubling the measurement bandwidth does not double the decibel quantity. As lacking in rigor ‘‘dBc/Hz’’ is, it is common usage [1]. Equation (3.18) tells us that phase noise (at a given offset) improves as both the carrier power and Q increase, as predicted earlier. These dependencies make sense. Increasing the signal power improves the ratio simply because the thermal noise is fixed, while increasing Q improves the ratio quadratically because the tank’s impedance falls off as 1/Q ⌬␻ . Because many simplifying assumptions have led us to this point, it should not be surprising that there are some significant differences between the spectrum predicted by (3.18) and what one typically measures in practice. For example, although real spectra do possess a region where the observed density is proportional to 1/(⌬␻ )2, the magnitudes are typically quite a bit larger than predicted by (3.18), because there are additional important noise sources besides tank loss. For example, any physical implementation of an energy restorer will be noisy. Furthermore, measured spectra eventually flatten out for large frequency offsets, rather than continuing to drop quadratically. Such a floor may be due to the noise associated with any active elements (such as buffers) placed between the tank and the outside world, or it can even reflect limitations in the measurement instrumentation itself. Even if the output were taken directly from the tank, any resistance in series with either the inductor or capacitor would impose a bound on the amount of filtering provided by the tank at large frequency offsets and thus ultimately


67

produce a noise floor. Finally, there is almost always a 1/(⌬␻ )3 region at small offsets. A modification to (3.18) provides a means to account for these discrepancies:

冋再冉

2FkT L {⌬␻ } = 10 log P sig

␻0 1+ 2Q ⌬␻

冊冎冉 2

1+

⌬␻ 1/f

| ⌬␻ |

3

冊册

(3.19)

These modifications, due to Leeson, consist of a factor F to account for the increased noise in the 1/(⌬␻ )2 region, an additive factor of unity (inside the braces) to account for the noise floor, and a multiplicative factor (the term in the second set of parentheses) to provide a 1/ | ⌬␻ | 3 behavior at sufficiently small offset frequencies [3]. With these modifications, the phase noise spectrum appears as in Figure 3.3. It is important to note that the factor F is an empirical fitting parameter and therefore must be determined from measurements, diminishing the predictive power of the phase noise equation. Furthermore, the model asserts that ⌬␻ 1/f 3, the boundary between the 1/(⌬␻ )2 and 1/ | ⌬␻ | 3 regions, is precisely equal to the 1/f corner of device noise. However, measurements frequently show no such equality, and thus one must generally treat ⌬␻ 1/f 3 as an empirical fitting parameter as well. Also it is not clear what the corner frequency will be in the presence of more than one noise source, each with an individual 1/f noise contribution (and generally differing 1/f corner frequencies). Finally, the frequency at which the noise flattens out is not always equal to half the resonator bandwidth, ␻ 0 /2Q .

Figure 3.3 Phase noise: Leeson versus (3.18).

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Both the ideal oscillator model and the Leeson model suggest that increasing resonator Q and signal power are ways to reduce phase noise. The Leeson model additionally introduces the factor F, but without knowing precisely what it depends on, it is difficult to identify specific ways to reduce it. The same problem exists with ⌬␻ 1/f 3 as well. Finally, blind application of these models has periodically led to earnest but misguided attempts by some designers to use active circuits to boost Q . Sadly, increases in Q through such means are necessarily accompanied by increases in F as well, generally preventing the anticipated improvements in phase noise. Again, the lack of analytical expressions for F can obscure this conclusion, and one continues to encounter various doomed oscillator designs based on the notion of active Q boosting. That neither (3.18) nor (3.19) can make quantitative predictions about phase noise is an indication that at least some of the assumptions used in the derivations are invalid, despite their apparent reasonableness. To develop a theory that does not possess the enumerated deficiencies, we need to revisit, and perhaps revise, these assumptions.

3.4 The Roles of Linearity and Time Variation in Phase Noise The foregoing derivations have all assumed linearity and time invariance. Let us reconsider each of these assumptions in turn. Nonlinearity is clearly a fundamental property of all real oscillators, as its presence is necessary for amplitude limiting. Several phase noise theories have consequently attempted to explain certain observations entirely as a consequence of nonlinear behavior. One of these observations is that a singlefrequency sinusoidal disturbance injected into an oscillator gives rise to two equal-amplitude sidebands, symmetrically disposed about the carrier [4]. Since LTI systems cannot perform frequency translation and nonlinear systems can, nonlinear mixing has often been proposed to explain phase noise. Unfortunately, the amplitude of the sidebands generally must then depend nonlinearly on the amplitude of the injected signal, and this dependency is not generally observed. One must conclude that memoriless nonlinearity cannot explain the discrepancies, despite initial attractiveness as the culprit. As we shall see momentarily, amplitude-control nonlinearities certainly do affect phase noise, but only indirectly, by controlling the detailed shape of the output waveform.


69

An important insight is that disturbances are just that: perturbations superimposed on the main oscillation. They will always be much smaller in magnitude than the carrier in any oscillator worth designing or analyzing. Thus, if a certain amount of injected noise produces a certain amount of phase disturbance, we ought to expect that doubling the injected noise will produce double the disturbance. Linearity would therefore appear to be a reasonable assumption as far as the noise-to-phase transfer function is concerned. It is therefore particularly important to keep in mind that, when assessing linearity, it is essential to identify explicitly the input-output variables. Linear relationships may exist between certain variable pairs at the same time nonlinear ones exist between others. This assumption of linearity is not equivalent to a neglect of the nonlinear behavior of the active devices. It is rather a linearization around the steady-state solution and therefore takes the effect of device nonlinearity into account. There is thus no contradiction here with the prior acknowledgment of nonlinear amplitude control. We see that the word system is actually ill-defined. Most take it to refer to an assemblage of components and their interconnections, but a more useful definition is based on the particular input-output variables chosen. With this definition, the same circuit may possess nonlinear relationships among certain variables and linear ones among others. Time invariance is also not an inherent property of the entire circuit; it is similarly dependent on the variables chosen. We are left only with the assumption of time invariance to reexamine. In the previous derivations, we have extended time invariance to the noise sources themselves, meaning that the measures that characterize noise (e.g., spectral density) are time-invariant (stationary). In contrast with linearity, the assumption of time invariance is less obviously defensible. In fact, it is surprisingly simple to demonstrate that oscillators are fundamentally timevarying systems. Recognizing this truth is the main key to developing a more accurate theory of phase noise [5]. To show that time invariance fails to hold, consider explicitly how an impulse of current affects the waveform of the simplest resonant system, a lossless LC tank (Figure 3.4). Assume that the system is oscillating with some constant amplitude until the impulse occurs; then consider how the system responds to an impulse injected at two different times, as seen in Figure 3.5. If the impulse happens to coincide with a voltage maximum (as in the left plot of Figure 3.5), the amplitude increases abruptly by an amount ⌬V = ⌬Q /C , but because the response to the impulse superposes exactly in phase with the preexisting oscillation, the timing of the zero crossings does not change. On the other hand, an impulse injected at some other time

70


Figure 3.4 LC oscillator excited by current pulse.

Figure 3.5 Impulse responses of LC tank.

generally affects both the amplitude of oscillation and the timing of the zero crossings, as in the right-hand plot. Interpreting the zero-crossing timings as a measure of phase, we see that the amount of phase disturbance for a given injected impulse depends on when the injection occurs; time invariance thus fails to hold. An oscillator is therefore a linear but (periodically) timevarying system. It is especially important to note that it is theoretically possible to leave unchanged the energy of the system (as reflected in a constant tank amplitude), if the impulse injects at a moment near the zero crossing, such that the net work performed by the impulse is zero. For example, a small positive impulse injected when the tank voltage is negative extracts energy from the oscillator, and the same impulse injected when the tank voltage is positive delivers energy to the oscillator. Just before the zero crossing, an instant may be found where such an impulse performs no work at all. Consequently, the amplitude of oscillation cannot change, but the zero crossings will be displaced. Because linearity of noise-to-phase conversion remains a good assumption, the impulse response still completely characterizes that system, even with time variation. Noting that an impulsive input produces a step change in phase, the impulse response may be written as: h ␾ (t , ␶ ) =

⌫(␻ 0 t ) u (t − ␶ ) q max

(3.20)


71

where u (t ) is the unit step function. Dividing by q max , the maximum charge displacement across the capacitor, makes the function ⌫(x ) independent of signal amplitude. ⌫(x ) is called the impulse sensitivity function (ISF), and is a dimensionless, frequency- and amplitude-independent function periodic in 2␲ . As its name suggests, it encodes information about the sensitivity of the oscillator to an impulse injected at phase ␻ 0 t . In the LC oscillator example, ⌫(x ) has its maximum value near the zero crossings of the oscillation, and a zero value at maxima of the oscillation waveform. In general, it is most practical (and most accurate) to determine ⌫(x ) through simulation, but there are also analytical methods (some approximate) that apply in special cases [6, 7]. In any event, to develop a feel for typical shapes of ISFs, consider two representative examples, first for an LC and a ring oscillator in Figure 3.6. Once the ISF has been determined (by whatever means), we may compute the excess phase through use of the superposition integral. This computation is valid here since superposition is linked to linearity, not time invariance: t

∞

␾ (t ) =

冕

−∞

h ␾ (t , ␶ ) i (␶ ) d␶ =

1 q max

冕

⌫(␻ 0 ␶ ) i (␶ ) d␶

(3.21)

−∞

This computation can be visualized with the help of the equivalent block diagram shown in Figure 3.7.

Figure 3.6 Example ISF for (a) an LC oscillator and (b) a ring oscillator.

72


Figure 3.7 The equivalent block diagram of the process.

To cast this equation in a more practically useful form, note that the ISF is periodic and therefore expressible as a Fourier series: ⌫(␻ 0 ␶ ) =

c0 + 2

∞

∑ c n cos (n␻ 0 ␶ + ␪ n )

(3.22)

n=1

where the coefficients c n are real, and ␪ n is the phase of the n th harmonic of the ISF. We will ignore ␪ n in all that follows because we will be assuming that noise components are uncorrelated, so that their relative phase is irrelevant. The value of this decomposition is that, like many functions associated with physical phenomena, the series typically converges rapidly, so that it is often well approximated by just the first few terms of the series. Substituting the Fourier expansion into (3.21), and exchanging summation and integration, one obtains: t

␾ (t ) =

1 q max

冤冕 c0 2

−∞

i (␶ ) d␶ +

∞

∑ cn

n=1

t

冕

−∞

i (␶ ) cos (n␻ 0 ␶ ) d␶

冥

(3.23)

The corresponding sequence of mathematical operations is shown graphically in the left half of Figure 3.8. Note that the block diagram contains elements that are analogous to those of a superheterodyne receiver. The normalized noise current is a broadband ‘‘RF’’ signal, whose Fourier components undergo simultaneous downconversions (multiplications) by ‘‘local oscillator’’ signals at all harmonics of the oscillation frequency. It is important to keep in mind that multiplication is a linear operation if one argument is held constant, as it is here. The relative contributions of these multiplications are determined by the Fourier coefficients of the ISF. Equation (3.23) thus allows us to compute the excess phase caused by an arbitrary noise current


73

Figure 3.8 The equivalent system for ISF decomposition.

injected into the system, once the Fourier coefficients of the ISF have been determined (typically through simulation). Earlier, we noted that signals (noise) injected into a nonlinear system at some frequency may produce spectral components at a different frequency. We now show that a linear, but time-varying system can exhibit qualitatively similar behavior, as implied by the superheterodyne imagery invoked in the preceding paragraph. To demonstrate this property explicitly, consider injecting a sinusoidal current whose frequency is near an integer multiple, m , of the oscillation frequency, so that i (t ) = I m cos [(m␻ 0 + ⌬␻ )t ]

(3.24)

where ⌬␻ 0 I C pos (Vg k , ⌬Vc k ) = I DSS × f g × f d × Correct_Gm × ⌫(V GSN ) (4.34) and the whole expression is written as I C (Vg k , ⌬Vc k ) = I C pos (Vg k , ⌬Vc k ) × ⌫(⌬Vc k ) + I C neg (Vg k , ⌬Vc k ) × (1 − ⌫(⌬Vc k ))

(4.35)

An example of this function is plotted in Figure 4.10. Nonlinear Diode and Capacitance

The I Gk diode and the C Gk capacitance functions are the same as those of the lumped model equations (4.25) and (4.26):

Figure 4.10 I C (Vg k , ⌬Vd k ) function.


冉

115

冊

Vg k

I Gk = I GDS e ␩ gd U T − 1

(4.36)

and

C Gk (Vg k ) = Cg 0

冉

Veff 1− Vb

冊冉 n

冊

1 1 (Vg k − V t ) − (Vg k − V b ) + C b 2d 1 2d 2 (4.37)

with Veff =

1 V + Vb (d 1 − d 2 ) + t 2 2

d1 =

√(Vg k − Vb )

2

+ d2

d2 =

√(Vg k − Vt )

2

+ d2

All the models we have presented have been extracted from measurements. Now, in Section 4.1.4, the pulsed measurement system dedicated to transistor isothermal characterization is described. Then the model-extraction strategies for HBT and FET transistors are explained, and some examples are given. 4.1.4 Transistor I-V and S-Parameter Measurement System Measurement-based models take into account real-world devices, including all the parasitics [31]. They are well-suited for circuit CAD database optimizations and statistical approaches to ensure successful designs of MMICs. The measurement approach requires accurate, repeatable, and safe measurements of the devices. Three families of electrical measurements can be mentioned to proceed to complete extraction models: I-V, S-parameters, and low-frequency noise. 4.1.4.1 Pulse I-V Measurement Setup

To provide suitable models for microwave-circuit CAD, the characterization of microwave devices should be realized with measurement conditions as close as possible to the real-world operating conditions. Temperature time constants of microwave transistors are very high compared with the RF

116


period. Thus, the device temperature does not change during an RF period; thermal drift depends on the average power dissipated inside the device. Because transistor characteristics strongly depend on device temperature, a realistic device characterization must be performed isothermally, at a device temperature as close as possible to its real-world temperature. Starting from the dc quiescent point, short pulses are used to measure the device behavior isothermally. Pulses from 150 ns to a few microseconds can be used, depending on the device. The measurement principle is shown in Figure 4.11 (extracted from [31]) for an FET. VGS 0 , VDS 0 , and I DS 0 correspond to the quiescent point, and VGSi , VDSi , and I DSi are the pulsed point values. The shorter the pulses, the closer the device is to its RF behavior. The pulse duration and duty cycle must be adapted to the DUT, as a compromise between several points: • The pulse width must be large enough for quality of measurement

acquisition.

Figure 4.11 I-V pulse principle. (Source: [31],  1998, IEEE. Reprinted with permission.)


117

• The pulse width must be much smaller than thermal time constant

for isothermal characterization. • The pulse width must be smaller than the trapping time constant, if traps exist. • The pulse duty cycle must be large enough to ensure that the thermal state is driven by the dc quiescent point. • The pulse duty cycle must not be too large to ensure fast and reliable data acquisition with averaging. A duty cycle of 0.1–5% is usually acceptable, but it must be checked for each transistor. This can be done easily with a few measurement points [32]; the pulse durations have to be tested in the same manner. Pulse widths from 300 ns to 600 ns are usually suitable, but pulses as short as 150 ns can sometimes be required for some millimetric HEMTs, due to fast trapping effects. 4.1.4.2 Measurements

Four electrical quantities are measured simultaneously during the pulse, with high accuracy for modeling purposes. This can be done either with a fourchannel GPIB oscilloscope, or with VXI digitizers. The voltage probes are classical high-impedance, low-capacitance active or passive probes. Two configurations allow for the measurement of the currents: differential probes associated with accurate external resistors and Hall probes. All combinations are available for the input and the output of transistors, depending on the current ranges. Our system with its probes provides a specified dc voltage offset accuracy better than 0.25% ± 5 mV up to ± 10V after warm-up and without averaging. Temperature measurements are made with the device inside a thermal enclosure; its temperature should be controlled by software for extensive characterization. 4.1.4.3 Pulsed S-Parameter Measurements

Pulsed S-parameters measurements are performed by superimposing an RF stimulus during the I-V pulses. A vectorial network analyzer (VNA) with short-pulse (≤ 150 ns) capabilities is required for the isothermal measurement of transistors. With these long pulse durations compared with RF periods, and with four identical measurement paths for the transmitted and reflected signals from the DUT to the samplers, the VNA quickly reaches a CW mode during the pulses.

118


Bias tees with a dc bandpass allow I-V pulses to connect the VNA to the I-V setup—see Figure 4.12 (adapted from [31]). A time synchronization between equipment places the RF measurements precisely during the I-V pulses and avoids phase jitter. Depending on device technology and size, RF power levels in the DUT planes between −12 dBm and −21 dBm are suitable for small-signal S-parameter measurements. This level is checked by biasing the device in a nonlinear region (usually at pinch-off for FETs) and by monitoring the trace on the oscilloscope: If the RF level is set too high, the drain current increases by self-biasing during the RF pulse. Moreover, this test is a good way to check the RF pulse position versus the bias pulses. 4.1.5 Model Extraction Procedure From the measurements presented in Section 4.1.4, the whole model extraction can be handled. The base of this work is an optimization procedure of

Figure 4.12 I-V and RF pulse setup. (Adapted from [31].)


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all the parameters included in a homemade software. The optimization algorithm relies on simulated annealing [33, 34] and simulated diffusion [35], which avoid local minima. However, a brute optimization is not desirable for two main reasons: first because a high number of parameters must be optimized to obtain accurate fitting functions and second because some of these parameters must have physical significance. Accordingly, a gradual modeling approach is more advisable. Our software permits this tuning by continuously displaying the measured and calculated curves. Modeling methodologies have been developed for the two types of transistors: bipolar and field effect. 4.1.5.1 Bipolar Transistor Model Extraction

The procedure is divided into four progressive steps: the electrothermal convective model (I CT and diode currents), the extrinsic elements extraction (parasitic capacitances and inductances), the nonlinear capacitances and transcapacitances, and finally the thermal capacitance and resistance [10, 12]. Electrothermal Convective Modeling

The transistor characteristics used for this extraction are the Gummel plot (collector and base currents versus VCE = VCE in logarithm axes) and the input and output characteristics, I CT (VCE ) for constant I B and VBE (VCE ) for constant I B . The parameter extraction steps are the following: • Determination (graphical and numerical) of R E , I SE , ␩ fE , ␩ E , ␤ f ,

and I SFE from the Gummel plot [see Figure 4.3 and (4.3) and (4.6)]; • Numerical optimization of the parameter set of the base collector junction (I SC , ␩ c , I SFC , ␩ fC ), R C , and reverse ␤ R current gain, deduced from the input and output measured characteristic curves; • Numerical optimization of the whole model by taking the previous results as initial guess values. These three steps are repeated for different temperatures, by modifying only the thermal-dependent parameters [see (4.13)–(4.16)]. An example of an HBT convective model is shown in Figure 4.13. Figure 4.13 shows the comparison between the measured and computed input and output characteristics, at two different temperatures: 53°C and 79°C. Extrinsic Elements Extraction

This linear model extraction method combines a direct extraction method [36] with a numerical one. In the HBT linear model (Figure 4.4), the R BC ,

120


Figure 4.13 Comparison between the measured and computed input and output characteristics, at two different temperatures: 53°C and 79°C.

R BE , G m , G d , R C , and R E are fixed from the previous convective extraction (derivatives of nonlinear function at the bias point). The other capacitive, inductive, and R B elements are extracted at the same time, and for some bias points in the region where the transistor operates. The goal of this optimization is to obtain extrinsic elements independent of the bias point and of the frequency. Typically the measured frequency range is from 2 GHz to 40 GHz. An example is given in Figure 4.14 for two bias points: VCE = 2.4V with I C = 100 mA and VCE = 10V with I C = 25 mA. Nonlinear Capacitances and Transcapacitances

The modeling procedure is described as follows: • Extraction of C BC capacitance and C BEC transcapacitance from

S-parameter measurements;


121

Figure 4.14 S-parameters measured and computed from 2 GHz to 40 GHz.

• Fitting the parameters C jC 0 and ⌽C of the junction C jC capacitance,

(4.7) and the parameters C BCd 0 of the diffusion C dC capacitance, and (4.9) from the plot of base-collector capacitance versus base collector voltage; • Fitting the parameters C BEc 0 and C BEc 1 of the transcapacitance

C dEC and (4.12) from the plot of base-emitter capacitance versus base collector voltage. • Extraction of C BE capacitance and C BEC transcapacitance from

S-parameter measurements; • Fitting the parameters C jE 0 and ⌽E , of the junction C jE capacitance,

(4.8) and the parameters C BEc of the diffusion C dE capacitance, and (4.10) from the plot of base-emitter capacitance versus base emitter voltage;

122


• Fitting the parameter C BCe of the transcapacitance C dCE and (4.12)

from the plot of base-collector capacitance versus the base emitter voltage. These steps are repeated versus temperature. By numerical optimization, the coefficients C BEd 0 , C BEd 1 , C BCe 0 , and C BCe 1 are computed. Thermal Model Extraction Thermal resistance determination. The method is based on isothermal measures. The first step is to measure the I CT (VBE ) characteristic for a given VCE and different junction temperatures fixed by the thermal enclosure in which the transistor is placed with very short pulse durations to maintain the transistor to a constant temperature during measurement. The second step is to measure this same curve with dc bias. The superimposition of these two measurements on the same plot allows one to determine the thermal resistance (Figure 4.15): At each intersection of the dc curve and one of the pulsed curve, the dissipated power is given by the dc curve and the temperature is given by the pulsed curves. The plot of the ratio between the dissipated power and the temperature directly gives the thermal resistance value.

Figure 4.15 Determination of correspondence between dissipated power and junction temperature.


123

Thermal time constant determination. In this method, one measures the VBE

voltage evolution during a long pulse in the way that the voltage decreases while the temperature increases. An example of such a measurement is shown in Figure 4.16. Knowing the thermal resistance, the thermal capacitance can thus be deduced from the relation [18, 37]:

␶ th = R th C th

(4.38)

In conclusion, the complete modeling of bipolar transistors necessitates all the described extraction methods and the associated measurement setup. However, depending on the intended application of the transistor—mixer, oscillator, linear, or power amplifier—only a part of this modeling may be achieved. An example of an HBT model is provided in Appendix 4A. Now, Section 4.1.5.2 discusses the FET extraction model. 4.1.5.2 FET Extraction Model

Two different models have been developed for the FET. This section first describes the classical Pi model and then discusses the distributed model essential to accurate noise modeling for nonlinear applications. Pi Model

As for the bipolar transistor, progressive steps are necessary to completely extract the transistor model.

Figure 4.16 V BE voltage pulse measure; determination of thermal constant.

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Extraction of parasitic elements. This is a direct extraction from the measurement of two transistor configurations [38]. A first measurement of the transistor for VDS = 0 is achieved for two VGS bias voltages. For a positive VGS , the gate to the source diode is in conduction region. Then, the Z-parameter computation allows for the determination of the ‘‘series’’ elements: L G , R G , L S , R S , L D , and R D .

Z 11 = R G + R S +

RC + R GS + j␻ (L S + L G ) 3

Z 21 = Z 12 = R S +

RC + j␻ L S 2

(4.39)

Z 22 = R D + R S + R C + j␻ (L S + L G ) where R C is the channel resistance (foundry parameter) and R GS is the incremental resistance of the Schottky diode. For a large negative VGS , the transistor can be considered to be an open circuit, and the capacitances can be obtained from Y-parameter computation: Im (Y 11 ) = j␻ (C pg + 2C b ) Im (Y 21 ) = Im (Y 12 ) = −j␻C b

(4.40)

Im (Y 22 ) = j␻ (C b + C pd ) where C b is the capacitance of residual coupling between electrodes. This extraction method is coupled with an optimization procedure to find all the linear elements of the transistor model, frequency-independent [39]. Extraction of the linear model. This procedure is summarized in Figure 4.17.

To extract the intrinsic model of a transistor (Figure 4.7), successive matrix transformations enable us to eliminate the parasitic elements. The intrinsic elements (C GD , R GD , G D , C DS , C GS , R i , G m , ␶ ) are directly calculated from the Y-parameters following formulas given in [38]. We obtain, for each element, a set of values corresponding to each measurement frequency. An example of extracted parameters is shown in Figure 4.18. Note that the elements must be frequency-independent. Nonlinear convective model extraction. From the I DS (VGS , VDS ) measurements, the fitting parameters of the nonlinear function I DS (VGS , VDS )


125

Figure 4.17 Linear model extraction.

[(4.22)] and those of the drain-source breakdown current source function [(4.28)] I BDS (VGS , VDS ) are extracted by numerical optimization. From the I G (VDS , VGS ) and for low VDS , the fitting parameters of the gate-source and gate-drain diodes I GS (VGS ) and I GD (VGD ) function [(4.23) and (4.24)] are determined, while for high VDS , the fitting parameters of the gate-drain breakdown current source I BGD (VGS , VDS ) function [(4.27)] are computed. The same optimization procedure as for bipolar transistor is employed. The transistor characterization is performed for several temperatures and the thermal dependence of the I G , I D diodes and I DS current source functions [(4.29)] are fitted versus temperature.

126


Figure 4.18 Element extraction verification.

The model consistency is checked by a comparison of the extracted linear elements G D and G m with those calculated from the linear model. Nonlinear capacitances model extraction. This procedure is the same as for the bipolar transistor. Fitting functions are found from the collection of linear capacitance values: C GS and C GD are obtained from linear model extraction as a function of VGS and VDG bias voltages. Thermal circuit extraction. This procedure is based on the measurement of the thermal variation of the Schottky junction threshold voltage. In the first step, the I GS (VGS ) characteristic is measured at several temperatures. Then, for constant I GS current, the VGS voltage is plotted versus temperature. The slope of this curve is a well-known result (between −1 mV/°C and −1.4 mV/°C). In a second step, the same transistor characteristic is measured for several dissipated powers. As for the bipolar transistor, the superimposition of the two characteristics allows for the evaluation of the thermal resistance [27].


127

To evaluate the thermal time constant, drain bias pulses are applied to the transistor. During the bias pulse, a low-level RF signal is applied, and the transistor S 21 parameter is measured (see Figure 4.19). The FET gain exponentially decreases, and the slope at the origin gives the thermal time constant and the thermal capacitance is deduced with the relation (4.38). An example of a Pi model of HEMT is given in Appendix 4A. Section 4.1.5.3 deals with the FET-distributed model, including the low-frequency noise sources. 4.1.5.3 HEMT-Distributed Model Extraction

The distributed model under the gate from source to drain is composed of N identical cells. This number must be chosen in consideration of the function of the gate length L g ; for example, for L g < 0.3 ␮ m, 10 cells are a good compromise between the model accuracy and the complexity of the model extraction [40]. The extraction procedure consists of the same three steps as for the Pi model. The computation of the extrinsic elements, including the drain

Figure 4.19 Thermal gain decrease in the RF measurement window.

128


source capacitance, is performed following the procedure described in Section 4.1.5.2 for the simple Pi model [41]. Convective Model Extraction

The second step is the convective model extraction. Since the nonlinear elements take the same form for all the cells, only one parameter set is to be computed from the I-V characteristic. The procedure is a whole numerical optimization that also computes all the internal node voltages Vg k and ⌬Vc k (k = 1–10). These voltages are only dependent on the external voltages VGS and VDS and on the cell number. Figure 4.20 shows a comparison between the I DS (VGS , VDS ) characteristics computed with a 10-cell model and the measured one. In this example the transistor is a PHEMT with four fingers of 50 ␮ m width by 0.25 ␮ m length. Figure 4.21 shows Vg k and ⌬Vc k voltages (k = 1–10) computed from the model, for external bias voltages VGS = −0.2V and VDS = 3.5V. Capacitances Extraction

The two fringing linear C bs and C bd capacitances and the nonlinear C Gk (Vg k ) capacitance are extracted at the same time. They are extracted from the S-parameter measurements in the same way as for the Pi model.

Figure 4.20 Comparison between the I DS (V GS , V DS ) characteristics, measured and computed with the distributed model (10 cells).


129

Figure 4.21 Internal voltage Vc k and Vg k for fixed external bias voltages.

At last, an example of a comparison between S-parameters measured and computed with the nonlinear distributed model is shown in Figure 4.22. This electrical model, fully extracted from measurements, is the first that is able to handle the distributed low-frequency noise sources. Section

Figure 4.22 Comparison of the S-parameters measured and modeled with the nonlinear distributed model (V GS = −0.2V and V DS = 2.52V).

130


4.1.6 describes and models the main noise sources in the bipolar transistors and FETs. 4.1.6 Noise Sources in Semiconductor Devices and Their CAD-Oriented Modeling An electrical noise signal may be mathematically represented by an infinite sum of pseudosinusoidal components, with random amplitudes and phases. In a CAD-oriented model, a noise source ␦ h (t ) will be represented by:

␦ h (t ) =

√2 ∑ ᑬ (H n e

j␻ n t

)

(4.41)

n

where √ 〈 | H n | 2 〉 is the rms value of the Fourier component, centered at ␻ n , in a 1-Hz bandwidth. The power spectral density of this component will be written as: S ␦ h (␻ n ) = 〈 | H n | 〉 2

(4.42)

4.1.6.1 Noise Sources in Semiconductor Devices

Four main physical noise sources must be considered in the design of low phase-noise oscillators. For a good understanding of the noise origin, let us consider the current flowing in a slice ⌬x of a semiconductor sample. In a one-dimensional model, at an abscissa x , the current I (x ) writes as in relation (4.21) I (x ) = qA (x ) n (x ) v (x )

(4.43)

The current through the slice will fluctuate if at least one of the quantities A (x ), n (x ), and v (x ) (the ensemble average velocity of carriers) fluctuates. Obviously, in a bulk sample, A (x ) cannot fluctuate. Accordingly, the noise sources inside the sample come from the following: • Carrier density fluctuations; • Velocity fluctuations.

Carrier density fluctuations lead to generation recombination noise (G-R) and the associated so-called nonfundamental 1/f noise [42–46].


131

Velocity fluctuations lead to diffusion noise in ohmic semiconductors, the collective approach of the shot noise in semiconductor junctions, and finally to the fundamental 1/f noise through the 1/f random fluctuations of the low-field carrier mobility ␮ 0 and/or the diffusion coefficient D 0 . G-R and 1/f noise are colored noise. They are called low frequency excess noise because their power spectral densities fall, respectively, with a ␶ and 1/␻ , in function of frequency. roll-off of 1 + ␻2 ⭈ ␶2 In nonlinear circuits such as mixers and oscillators, the mixing of the low-frequency noise sources with the carrier signal frequency gives rise to AM and PM noise spectra near the carrier. In conclusion, four main physical noise sources are to be considered in oscillator circuits. Their power spectral densities are described in the following equations: • White noise sources: •

Diffusion noise in ohmic semiconductor regions: For a dc-applied voltage, the power spectral density of the equivalent Norton source of the slice ⌬x is written as: S ␦ i (␻ ) =

4q 2n 0 D 0 A ⌬x

(4.44)

At low field, the conductance of the slice is written as: G 0 = qn 0 ␮ 0

A ⌬x

(4.45)

By taking into account the Einstein relation, one obtains: S ␦ i (␻ ) = 4kTG 0 •

(4.46)

Shot noise in semiconductor junctions: Neglecting the transit time of carriers, the power spectral density of the equivalent Norton noise source of a junction is written as: S ␦ i (␻ ) = 2qI 0 where I 0 is the current flowing in the junction.

(4.47)

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• Low-frequency noise sources: •

Generation recombination noise: In the slice ⌬x , the current due to carrier density fluctuations is written as:

␦ I = qA␦ n (x ) v 0

(4.48)

giving rise to a power spectral density S ␦ i (␻ )GR =

q 2A 2v 20 S ␦ n (␻ )

=

q 2v 20 ⌬x 2

S ⌬N (␻ )

(4.49)

where ⌬N is the total number of fluctuating carriers in the volume of the slice ⌬xA . For G-R noise sources generated by a single trap with a single time constant ␶ , one has S ⌬N (␻ ) = 4⌬N 2 •

␶ 1 + ␶ 2␻ 2

(4.50)

Fundamental 1/f noise: The fluctuating current is written as:

␦ I = qAn 0 ␦ v 0

(4.51)

At low field: ␦ v 0 = ␦␮ 0 E 0 . The resulting power spectral density is S ␦ i (␻ )1/f =

I 02

␮ 20

S ␦␮ 0 (␻ )

(4.52)

where I 0 = qAn 0 ␮ 0 E 0 . 4.1.6.2 Noise Source Behavior in Semiconductor Devices, Driven by RF Time-Varying Signals White Noise Sources

The white noise sources’ behavior under large signals excitations is now well understood. The autocorrelation functions of white noise sources are Dirac delta functions. It is assumed that the noise mechanism follows the RF frequency of the modulating signal [47–50].


133

Let us write a white noise source as:

␦ i (t ) = M (t ) U (t )

(4.53)

where M (t ) is the deterministic modulating function of the applied voltage and U (t ) is a primary white noise source. The pseudosinusoid component of U (t ), centered at the frequency ⍀ is written as: U (t , ⍀) =

√2ᑬ (U˜e

j ⍀t

)

(4.54)

Moreover, in oscillator circuits M (t ) is periodic and can be expanded in Fourier series: k = +N H

M (t ) =

∑

k = −N H

M k e jk␻ 0 t

(4.55)

with M k = M k* Finally, ␦ i (t ) becomes: 1 ␦ i (t ) = √2

k = +N H

∑

k = −N H

˜ e j (k␻ 0 + ⍀)t + U ˜ *e j (k␻ 0 − ⍀)t ) M k (U

(4.56)

This resulting source is called a cyclostationary noise source. This representation may be applied to diffusion and shot noise [47–50]. Low-Frequency Noise Sources

The behavior of the low-frequency noise sources in the presence of timevarying signals is more involved, because their autocorrelation functions are not Dirac delta functions (i.e., their spectra are colored). To explain the modulation mechanism, let us write, for example, the G-R noise current given by the relation (4.48):

␦ i (t ) = q

v ⌬N (t ) ⌬x

(4.57)

where v = ␮ 0 E . Let us suppose that the electric field E is now time-varying. We obtain:

134


␦ i (t ) =

q ␮ E (t ) ⌬N (t ) ⌬x 0

(4.58)

In (4.58), E (t ) contains dc and RF frequency components and ⌬N (t ) only low-frequency noise components. Due to the large time constants involved in the physical mechanism giving rise to the carrier number fluctuations ⌬N (t ), it results that it cannot follow the RF-applied voltage. Nevertheless, ⌬N (t ) is multiplied by E (t ) to give rise to the Norton current source ␦ i (t ), which is so modulated at the RF frequency. The initial low-frequency spectrum of ⌬N (t ) is transposed around the Fourier components of the RF applied voltage [51, 52]. It may be conjectured that the same explanation holds for the fundamental 1/f noise. From (4.51), we obtain:

␦ i (t ) =

I (t ) ␦␮ 0 (t ) ␮0

(4.59)

I (t ) is now a time-varying current, and ␦␮ 0 (t ) is a low-frequency fluctuation. Then ␦ i (t ) contains low-frequency and RF frequency components. To model the noise sources and their modulation coefficients, they must be accurately localized into the semiconductor devices. For example, in an HEMT device, G-R noise sources may originate from the following: • Access resistances; • The supplier layer, under the gate; • The channel itself; • The buffer layer; • The semiconductor surfaces, between the gate drain or gate source.

Actually, the main problem of a CAD-oriented accurate modeling of the noise sources is rightly to find these origins. That is why this domain remains an open topic. 4.1.7 Transistor Low-Frequency Noise Characterization 4.1.7.1 Introduction

This section describes accurate noise measurement techniques used to characterize the noise in FETs and bipolar transistors. The experimental noise


135

measurement setup is presented. Direct determinations of noise generators at the input (gate or base) and the output (drain or collector) are first developed. They are followed by a technique of simultaneous noise measurement of these two generators leading to the correlation between noise sources. In all the cases noise-equivalent circuits are given, and noise calculations are presented in detail. Some remarks are given about the used methodology. The experimental procedure is discussed, and typical results are presented. 4.1.7.2 Noise Representation in Linear Two-Port Devices

In linear two-port networks the noise is taken into account by at least two noise voltage and/or current generators associated with the input and/or the output [53, 54]. This leads to six equivalent noise representations, each of which is naturally associated with a particular matrix of the two-port device. Generally for FETs or bipolar transistors, two noise representations are currently used. In the first, the real network can be seen as equivalent to an ideal noiseless network together with a series voltage noise generator and a shunt current noise generator at the input port (see Figure 4.23). In the following section, noise generators will be presented with the same symbols. In the second representation the real network is associated to an ideal one with shunt current noise generators at each port (see Figure 4.24). For FETs in a common source configuration, the input noise generator is related to the gate current noise, whereas the output one describes the channel noise. For bipolar transistors in common emitter bias, the input noise generator is associated with emitter-base junction and the output noise source with base-collector and emitter-collector carrier diffusion. In general, it should be noted that the noise at each port arises from different contributions of the same internal noise generators and so the

Figure 4.23 Noise representation of a two-port device with noise generators at the input port.

136


Figure 4.24 Noise representation of a two-port device with parallel current noise generators at each port.

equivalent noise generators at the two ports are partially correlated. This remark applies for all the equivalent representations. Classical Noise Analysis

Figure 4.23 allows us to derive the expression of the noise figure of the twoport network when its input is loaded by a source signal generator with a source resistance R S as in Figure 4.25. We have [53] (in the following and for noise measurement purposes we consider ‘‘spot’’ frequency relationships): F( f ) = 1 +

1 [S ( f ) + R S2 S ni ( f ) + 2R S ᑬ (S nv , ni ( f ))] 4k B TR S nv (4.60)

In (4.60) we suppose that R S exhibits only thermal noise; k B is the Boltzmann constant; S nv ( f ) and S ni ( f ) are the spectral densities of the two input noise generators; and S nv , ni ( f ) is their cross-spectrum if these two generators are correlated. On the other hand, the total noise power at the output of the network is proportional to:

Figure 4.25 Noise circuit for noise figure calculation.


137

P out ≈ 4k B TR S + S nv ( f ) + R S2 S ni ( f ) + 2R S ᑬ (S nv , ni ( f )) (4.61) So, if the input load resistance of the device has a low value, S nv ( f ) dominates and the voltage noise generator can be characterized. On the contrary, for a large value of the input load, the current noise generator is evaluated. The knowledge of the noise figure, which is the third noise parameter of the two-port device, allows us to determine the cross-spectrum S nv , ni ( f ). This characterization leads to a global evaluation of the noise but does not allow an investigation of the internal noise generators. In fact, for the determination of S ni ( f ) we must use values of R S larger than the input resistance of the device. This is possible for bipolar transistors in common base or common emitter configurations. Unfortunately, in the case of FETs this method cannot be applied. Noise Measurements for Accurate Characterization

For accurate noise studies the methodology of noise measurements stems from the noise equivalent circuit of Figure 4.24, and the two noise current generators are directly analyzed [55, 56]. Moreover, simultaneous measurements of these generators allow us to obtain their correlation. From the equivalent circuit of the two-port device being tested, the located noise sources can be determined. Section 4.1.7.3 presents these techniques. 4.1.7.3 Experimental Setup

Figure 4.26 reports the noise measurement setup used for the characterization of FETs. (A similar one is used for bipolar transistors.) In this case simultaneous noise measurements at the input (gate or base) and at the output (drain or collector) can be achieved. Vector signal analyzers with two channel inputs allow the measurement of the spectral density of each injected signal, the measurement of the complex cross-spectrum of the two signals, and the coherence between them. The coherence function is defined as: ⌫( f ) = | S VAVB ( f ) | /{S VA ( f ) S VB ( f )} 2

(4.62)

The noise experimental setup requires low-noise amplifiers and vector signal analyzers. They are the main parts of the apparatus. FFT analyzers are advantageously used in low-frequency noise measurements because they are easy to work up. They do not imply calibration by

138


Figure 4.26 General layout for the noise measurement setup.

noise standards and allow direct measurement if the gains of the amplifiers are known. (The FFT analyzer can be used as a network analyzer to determine it.) Actual FFT analyzers work up to 10 MHz (HP 89410A) and can be associated with RF converters for measurements above this limit. Low-noise amplifiers are now available for noise characterization. Tables 4.1 and 4.2 provide examples of the typical characteristics of voltage or transimpedance amplifiers well-suited for noise analysis. The typical input background noise of these amplifiers is reported in Figures 4.27, 4.28, and 4.29. It can be noticed that the cross-spectrum of the input noises of these two amplifiers lead to a negligible coherence function. Figure 4.30 shows the noise experimental setup and the connections between the different parts. 4.1.7.4 FET Noise Measurements Direct Gate Noise Measurements

The gate noise is associated with the low gate currents and involves high impedances. It is measured by the help of ultra-low-noise transimpedance Table 4.1 Typical Characteristics of the 5003 EG&G BROOKDEAL Voltage Amplifier

5003

Input Noise S eA (f )

k

Bandpass

ZE

1,000

0.1 Hz–1 MHz

> 1 M⍀//10 pF Figure 4.27

Re q = S i /4kT White Noise 75⍀


139

Table 4.2 Typical Characteristics of the 5182 EG&G BROOKDEAL Transimpedance Amplifier 5182 k′ Bandpass Input resistance Maximum input current Input noise equivalent current S i /2q Input noise S i′A (f )

S = 10−8 Low Noise S = 10−7

S = 10−6

108 0.5 Hz–10 kHz < 10 k⍀ 100 nA 500 pA

107 0.5 Hz–200 kHz < 100⍀ 10 ␮ A 50 nA

106 0.5 Hz–800 kHz < 1⍀ 1 mA 6.2 ␮ A

Figure 4.28

Figure 4.29

Figure 4.27 Input noise of the EG&G 5003 amplifier.

amplifiers. This type of amplifier allows for a direct determination of the noise current spectral density. In this case the output of the transistor is ac short-circuited, and the configuration of the device is given in Figure 4.31. The noise equivalent circuit of this setup is given in Figure 4.32.

140


Figure 4.28 Input noise of the EG&G 5182 transimpedance amplifier (k ′ = 108).

S i ′ ( f ) is the spectral density of the noise current generator i A′ (t ) A associated with the background noise of the amplifier. For a normal use of this amplifier, only this noise generator is needed to take into account the amplifier noise. S i Gm ( f ) is the spectral density of the noise current generator i Gm (t ) associated with the gate current. G Gm is the low-frequency conductance seen at the input of the amplifier. k ′ is defined as v ′S /i ′e . So we have [57] S iG ( f ) = m

1

| k ′ |2

Sv ′ ( f ) Si ′ ( f ) S

A

(4.63)

Direct Channel Noise Measurements

Channel noise is generally related to higher currents and lower impedances than for gate noise analysis. So, except for critical situations such as very low noise levels near the cut-off voltage or in the saturation range, this noise can be obtained using low-noise voltage amplifiers. In this case the noise


141

Figure 4.29 Input noise of the EG&G 5182 transimpedance amplifier (k ′ = 107).

voltage spectral density is determined. The noise current is then calculated taking into account the values of the channel and load resistances. Channel noise is directly measured as shown in Figure 4.33; the gate circuit is now ac short-circuited. In fact, to avoid oscillations, the transistor is biased with a bias-tee as shown in Figure 4.34. Its bandpass is 10 MHz to 1.5 GHz and for noise measurements in the range 1 Hz to 1 MHz, the transistor load remains R P . The noise equivalent circuit of this setup is given in Figure 4.35. So we have S vS ( f ) = | k |

2

再冉

R P R Dm R P + R Dm

冊

2

冎

[S i D ( f ) + S i P ( f )] + S e A ( f ) m

(4.64) 4k B T as the thermal noise of the load resistance R p , and k as RP the voltage amplification. S i D ( f ) is the spectral density of the channel noise and R Dm is its m dynamic resistance. For correct use of a low-noise voltage amplifier, only with S i P =

142


Figure 4.30 (a) The connections between the different parts of the experimental setup and (b) view of the experimental noise measurement setup.


Figure 4.31 Transistor configuration for gate (input) noise measurements.

Figure 4.32 Noise equivalent circuit for Figure 4.31.

Figure 4.33 Transistor configuration for drain (output) noise measurements.

143

144


Figure 4.34 Transistor bias including the bias-tee.

Figure 4.35 Noise equivalent circuit for Figures 4.33 and 4.34.

the noise voltage generator e A (t ) is necessary to describe the background noise of this amplifier. The measured spectral density of the channel S i D ( f ) is then m

S iD ( f ) = m

冉

1 1 + R P R Dm

冊再 | | 2

S vS ( f ) k

2

冎

− S eA ( f ) −

4k B T RP

(4.65)

In the ohmic range R Dm is easily measurable and R p is chosen as R p >> R Dm . So (4.64) becomes:


S iD ( f ) = m

1

S VS ( f )

(R D m )2

| k |2

S eA ( f )

145

(4.66)

In saturation, if R Dm >> R p , it is not necessary to determine R Dm . Nevertheless, R p must chosen to minimize its noise contribution. In the general case, (4.64) applies. Simultaneous Channel and Gate Noise Measurements

For these measurements the schematic setup configuration of Figure 4.26 is used. To determine the two signals received by the two channels of the analyzer, a general equivalent noise circuit of Figure 4.26 is taken into account. This equivalent circuit is given in Figure 4.36. The intrinsic transistor in a common source configuration is given as an active ␲ two-port circuit ( y 1 , y 2 , y 3 , g m int v GS int ) with additional terminal access resistances (R G , R S , R D ). To these three resistances are associated noise generators i R G , i R S , i R D , whereas, for the intrinsic part, i ch int is the channel noise and i g s and i g d are the noise generators of the admittances y 1 and y 3 . R P and i p are, respectively, the load resistance and its noise generator assuming only thermal noise. For the other noise generators, they can take into account several noise sources, such as 1/f noise, G-R noise, thermal, or shot noise. The voltage amplifier is characterized by its gain k , its input resistance R E , and its input noise generator e A . In a same way k ′, i A′ , and R E′ characterize the transimpedance amplifier. For the evaluation of the spectral densities associated with VS and VS′ we use the spectral densities that characterize the various noise generators and the complex impedances or admittances of the circuit. After several manipulations and transformations in the electrical form of the equivalent circuit (see Appendix 4B), we obtain [see (4B.21b), (4B.28b), and (4B.29b)]:

Figure 4.36 Noise equivalent circuit for simultaneous input-output noise measurements.

146


|

ZS S VS ( f ) = | k | S eA ( f ) + ZS + RP 2

再 +

S ⑀2 ( f ) +

|

|

Z 21 Z 11 + R E′

| | 再再再

Z 21 Z 11 + R E′

S VS′ ( f ) = | k ′

2

2

| 冎

2

|

2

|

|

RP S ep ( f ) + | k | ZS + RP 2

S ⑀ 1 ( f ) − 2ᑬ

再冉

冊

2

S eA′ ( f )

S iA′ ( f ) +

− 2ᑬ

冎

Z 21 (S ⑀*1 ⑀ 2 ( f )) Z 11 + R E′

(4.67) 1

| Z E′ | 2

| 冎 |

冋

S ⑀1 ( f ) +

冎

Z 12 Z 22 + R P

Z 12 Z 12 S (f ) + Z 22 + R P ⑀ 1 ⑀ 2 Z 22 + R P

| |

2

S ⑀2 ( f )

册冎

2

S eq ( f )

(4.68) S V S VS′ ( f ) =

再

冉

冊

kk ′*R P Z 21 Z 12 * S ⑀1 ( f ) S ⑀2 ( f ) * Z + R ′ ′ Z E (Z S + R P ) Z 11 + R E ′ 22 P + S ⑀ 1⑀ 2 ( f ) +

(Z 22

冎

Z *12 Z 21 S ⑀* ⑀ ( f ) + R P )* (Z 11 + R E′ ) 1 2

(4.69)

with S eq ( f ) = S e p ( f ) +

R P2 R E2

S eA ( f )

(4.70)

S ⑀ 1 ( f ) = | z 2 + z 3 | S i GS ( f ) + | z 3 | S i chint ( f ) + | z 2 | S i GD ( f ) 2

2

2

+ S eG ( f ) + S eS ( f )

(4.71)

S ⑀ 2 ( f ) = | z 3 + ␰ | S i GS ( f ) + | z 1 + z 3 | S i chint ( f ) + | ␰ + z 1 | S i GD ( f ) 2

2

+ S eD ( f ) + S eS ( f )

2

(4.72)

S ⑀ 1 ⑀ 2 ( f ) = (z 2 + z 3 )(z 3 − ␰ )*S i GS ( f ) + z 3 (z 1 + z 3 )*S i chint ( f ) − z 2 (z 1 + ␰ )*S i GD ( f ) + S e S ( f )

(4.73)

High-Frequency Oscillator Circuit Design 2

147

S eG ( f ) = R G S iR G ( f )

(4.74)

z 12 z 21 z 22 + R P

(4.75)

Z E′ = z 11 −

2

S eS ( f ) = R S S iR S ( f )

(4.76)

z 12 z 21 z 11 + R E′

(4.77)

Z S = z 22 −

2

S eD ( f ) = R D S iR D ( f )

(4.78)

Remarks

We have presented two methodologies to obtain input (at gate or base) and output (at drain or collector) noise sources: a direct noise measurement of these two noise generators and a simultaneous one that allows us to achieve cross-spectrum and correlation between the input and output noise sources. Obviously these two methodologies must lead to the same results concerning input and output noises. So it is necessary to verify that the measured values obtained at one input of the analyzer are not damaged by the other channel measurements. In particular, the experimental conditions introduced in Appendix 4B must be satisfied. Moreover, for PHEMTs and other structures with large drain current and high output conductance, the load resistance R P must be small enough to avoid inaccurate correlation measurements. On the other hand, the obtained cross-spectrum between input and output noise generators is a complex quantity and (4.69) must be considered as two equations. The dynamic signal analyzer gives the Cartesian or the polar representations of this quantity. Then it appears that the gains of the used amplifiers must be constant in magnitude and phase over the whole frequency range of noise investigations to avoid acute corrections. The above noise calculations apply for FETs or bipolar transistors since the same small signal equivalent circuit and noise generator location can be considered. Nevertheless, the conduction mechanisms and the noise source origins are quite different [53, 54]. Moreover, at high frequencies the behavior of FETs is taken into account as an active transmission line, and it will be shown that the noise sources must be in fact distributed along active RC networks. In all cases, the four relations given by (4.67), (4.68), and the real and imaginary parts of (4.69) can be used as a basis of the noise characterization

148


of two-port semiconductor devices. The four corresponding unknown quantities are the two noise current generators (at the input and the output of the intrinsic part of the device) and their correlation (the real and imaginary part of the cross-spectrum, for example). The other noise sources are located at the parasitic conduction paths. These parasitic elements and their associated noise sources can be predominant for particular biases of the devices and must be determined for an accurate noise characterization. Several contributions [55–63] describe basic investigations into FETs and bipolar transistors. 4.1.7.5 Experimental Procedure

To obtain the noise characterization of the device (i.e., the behavior of all the noise sources versus biases and frequency), two steps must be achieved: The small signal equivalent circuits is determined, and the various noise sources are extracted from the previous noise measurements. Small Signal Equivalent Circuit

From the conduction model of the device, the equivalent circuit of the intrinsic part of the device is determined. This implies the knowledge of the involved characteristic conduction parameters (or their extraction procedure; see [60] for an example). In this procedure the parasitic access resistances are also obtained. Their values can also be extracted from RF scattering parameter measurements. For FETs y 1 and y 2 in Figure 4.36 take into account two equivalent diodes located between gate and source and gate and drain, respectively. These two diodes are in shunt configuration with very low-leakage conductances. The study of the gate current versus various gate and drain biases allows us to obtain these conductances and the characteristic parameters of the diodes [60]. So all the parts of the equivalent circuit are known at each quiescent point of the device. Noise Extraction

In low-frequency noise analysis we expect white noise (thermal noise and shot noise) and excess noise (1/f noise, generation-recombination noise with Lorentzian spectra) [53, 54]. Thermal noise is associated with ohmic paths as parasitic access resistances or leakage conductances. It is the fundamental noise level for channel noise in FETs [53]. Shot noise is the fundamental noise level associated with diode conduction. Hence it is the basic source in bipolar transistors [53]. The 1/f noise is always present when electrical conduction occurs in semiconductor devices, and this noise source can be located at any part of the device [64]. Its magnitude is linked to the fluctuations of the electrical conductivity and to the quality of the technology [65].


149

Generation-recombination noise appears when trapping and detrapping of charge carriers occurs. This is due to particular defects in the device structure or to specific semiconductor materials [55, 65, 66]. In all the cases the noise characterization starts from the identification of these elementary noise sources. Each noise device must be studied versus bias to separate the contribution associated with the intrinsic transistor from the parasitic parts [60]. As expected, the analysis leads to the model of each noise source associated to the device giving its behavior versus bias, frequency, and geometry. Various papers can be considered to make clear this purpose [65–73]. Experimental Results

Some experimental results about GaAs-based PHEMTs are presented, obtained with the previous noise investigation techniques. Figure 4.37 reports the variations of the 1/f noise in the channel as S i D ( f )/I D2 . It can be shown [62] that at a low effective gate voltage the first two behaviors are related to the intrinsic channel, whereas at high gate biases the noise stems from access resistances.

Figure 4.37 Related 1/f channel noise versus effective gate bias showing various behaviors and involving intrinsic channel and access resistances. Here k is a conduction parameter of the device [62].

150


Figure 4.38 shows the 1/f gate noise evolution S i G ( f ) versus the current of diode D 3 located between gate and drain for VGS < 0 (diode D 1 reverse-biased) and 0 < VDS < 3V (diode D 3 forward-biased). The noise follows a quadratic law versus I D 3 as expected for the 1/f noise source. It is related only to D 3 since its magnitude is i -independent. Figure 4.39 reports the coherence function measured for VGS = −0.4V and VDS = 3.75V and the computed values from (4.67), (4.68), and (4.69) after determination of the small-signal equivalent circuit at this bias point and characterization of the noise sources.

4.1.8 Modeling of Circuit-CAD–Oriented Noise Sources in HBTs and FETs This section deals with noise source modeling, which must be inserted in the transistor models:

Figure 4.38 1/f gate noise evolution versus the transistor bias. In this case 1/f noise of D 3 is obtained.


151

Figure 4.39 Example of coherence function measurement and simulation for V GS = −0.4V and V DS = 3.75V.

• In a distributed way for FETs; • As lumped sources in bipolar transistors (at least in first realistic

approach). Due to the strong transistor technology dependence (MESFETs, HEMTs, homojunction bipolar transistors, and HBTs) of the LF noise behavior, a transistor generic model with parametric noise sources expressions is not realistic. Only the noise source locations in the electrical model and their representation by a current or voltage source (described by their pseudosinusoids—see Section 4.1.6) will be defined for each transistor class (FET or bipolar). Finally, an example of a PHEMT LF noise source extraction from measures will be shown. 4.1.8.1 Noise Sources in HBT

The linear Pi HBT model is described in Section 4.1.2 and depicted in Figure 4.3. Strictly speaking, R B is a distributed resistance [74] but it can

152


be taken as a lumped element. The following noise sources have to be considered in HBT [75]: White Noise Sources • Thermal noise sources: They are associated to the resistances R B and

R E . The power spectral density of a thermal noise current source is written as S ␦ i Th =

4kT R

(4.79)

where R can be either R E or R B . T is the resistance temperature. The noise source of R C can be neglected. • Shot noise sources. Van der Ziel [53] has shown that shot-noise in

bipolar transistors can be represented by two uncorrelated current sources: • One in parallel across the intrinsic base-emitter junction: ␦ Ib shot ; • One in parallel across the intrinsic collector-emitter junction: ␦ Ic shot . Their power spectral densities are written as: S ␦ Ib shot = 2qI B

(4.80)

S ␦ Ic shot = 2qI C

(4.81)

Numerical comparisons between the contribution of the noise sources ␦ Ib shot and ␦ Ic shot on the resulting phase noise of several oscillator circuits have been performed. They show that ␦ Ic shot plays the major role, at least in the circuits simulated. Low-Frequency Noise Sources

According to their technology dependence, location determination of the low-frequency noise sources in HBT is more involved. Simultaneous measurements of the low-frequency noise currents at the transistor input and output in function of the bias voltage, for a wide range of the source resistances, allows for discrimination between all the possible locations of the lowfrequency noise sources. Their power spectral density and correlation can


153

be extracted in a function of the base or collector current. Then, for circuitCAD purposes, analytical fitting functions must be found before to include them in HBT nonlinear models. Practically, the main low-frequency noise sources are the following: • First, the low-frequency fluctuations of the base current due to: •

Fluctuations of the surface recombination velocity; Generation-recombination inside the base junction; • 1/f fluctuations of the diffusion coefficient of the minority carriers. • Second, the low-frequency fluctuations of the resistance R B and R E : • Generation-recombination due to the traps; • 1/f fluctuations of the mobility. •

Figure 4.40 shows the resulting HBT model with the noise sources described above: All are uncorrelated. It must be noted that, fortunately, in a transistor many of them can be neglected, according to the technology. 4.1.8.2 LF Noise Model of FET and Associated Extraction

In oscillator circuits based on FET, low-frequency noise sources are practically the main causes of the resulting phase noise. The FET LF noise model is

Figure 4.40 HBT nonlinear model with noise sources.

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established from the distributed nonlinear model seen in Section 4.1.3.3. Generally speaking, each unit cell, includes two noise sources (Figure 4.41). Of course, according to the transistor technology, one of these sources can be neglected. To insert the noise sources in the model, the experimental characterization step is first performed, with the measurement setup described in Section 4.1.7. The two intrinsic power spectral densities Si Gm (gate terminal) and Si Dm (drain terminal) are extracted, in a frequency band from 100 Hz up to 1 MHz for several bias points in the normal operating region of the transistor. Each spectral density depends on three variables: VGS , VDS , and frequency. As an example, Figure 4.42 shows the plot of Si Dm spectral density at a 1-kHz frequency spot, versus VDS and for different VGS . As expected in FET, these measures show that the Si Dm power spectral density is higher than the Si Gm by several orders of magnitude. Moreover, only G-R noise appears in the measures. It may be conjectured that in this example only the ␦ Ic k channel current noise source is to be considered. The second step consists of the extraction of the internal noise sources of the unit cell ␦ Ic 2k as shown in Figure 4.41. Each unitary noise source is supposed to be uncorrelated with the others. Then a relation between all the elementary sources and the measured one at the access port can be written as: N

Si Dm =

∑ | H k | 2 ␦ Ic 2k

k=1

Figure 4.41 LF noise sources of one distributed FET model cell.

(4.82)


155

Figure 4.42 Example of Si Dm spectral density measure.

where H k is the transfer function between the source number k and the external noise current. These transfer functions H k are computed for the 10 cells of the nonlinear model, by linearizing it around the bias point. Now, in the third step, an expression of the unitary noise sources must be found. This calculation is detailed in [76]. The simplest expression relating Si Dm to the local noise source is written as: N

Si Dm = KI C

1 ∑ | H k | 2 ⌬x k

(4.83)

k=1

where K is a fitting constant depending on the material allowing for the adjustment of the model with the measures, ⌬x k the representative length of the cell. This equation is a function of the spot noise frequency and bias point (VGS , VDS ). Figure 4.43 shows an example of Si Dm computed with the distributed model and measured for noise frequency of 1 kHz.

156


Figure 4.43 Comparison between measured Si Dm spectral density and the computed one with the LF noise model.

The correspondence between the two results appears very satisfying, owing to the simplicity of noise current source equations. More accurate modeling can be reached with the counterpart of the more involved fitting equation. In conclusion, this extraction procedure applied to G-R noise sources of the channel can be extended to other noise sources. However, the goal of CAD-oriented modeling is to find a compromise between the model accuracy and ease of use in a CAD environment. To complete the noise model, thermal noise sources of the resistances R g , R s , and R d must be added.

4.2 Oscillator Circuit Design Tools 4.2.1 Conventional Linear Theory of Sinusoidal Oscillators The basic linear operating principles of sinusoidal oscillators will not be discussed here. However, the success obtained in past years by the wellknown negative-conductance concept [77] leads us to carry out a very simple basic calculation to clarify the relationship between the negative-conductance and feedback loop concepts in transistor oscillators.


157

4.2.1.1 Negative-Conductance Concept Applied to a Transistor Oscillator Circuit

The basic oscillator topology is shown in block diagram form in Figure 4.44. In Figure 4.44, the oscillator is represented by the following: • An active amplifying device: the transistor; • An amplitude-limiting and amplitude-stabilizing device; • A frequency-determining element or feedback network.

A very common form of amplitude stabilization is to use the nonlinearity of the active amplifying device. So common applications use the transistor as a self-limiting element. To present the concept of negative conductance, we will use a quasilinear representation of the oscillator circuit as shown in Figure 4.45. This quasi-linear representation allows a very simple analytical calculation. It will be performed in the admittance formalism. Analog (but not identical) results can be obtained with the impedance formalism. We assume that the input and output admittances are reactively tuned to the desired oscillation frequency ␻ 0 , so Y in | ␻ 0 = G in

(4.84)

Y out | ␻ 0 = G out where G in and G out are, respectively, input and output conductances.

Figure 4.44 Functional diagram of microwave transistor oscillators.

158


Figure 4.45 Quasi-linear representation of the oscillation circuit for negative conductance calculation.

The resonator LC is also tuned to the chosen oscillation frequency ␻ 0 . The load resistor is connected in parallel with the conductance of the resonator in order to obtain a total load conductance: G load . In addition, to calculate the input admittance at the oscillation frequency ␻ 0 , the circuit is driven by an external sinusoidal current I 0 (␻ ). The whole admittance observed at the input is written as G tot =

I0 Vin

|

␻0

= G in + N 2(G out + G load ) − NG mo

(4.85)

This conductance can be split into a passive conductance: G p = G in + N 2(G out + G load )

(4.86)

in parallel with a negative conductance due to the transconductance G mo of the transistor: −G active = −NG mo

(4.87)

This active conductance is negative due to the positive feedback introduced by the (out-of-phase) transformer. Near the oscillation frequency ␻ 0 , the total input admittance can be expressed as:


Y (␻ 0 + ⌬␻ ) = G tot (␻ 0 ) +

dG tot dB ⌬␻ + j tot ⌬␻ d␻ d␻

159

(4.88)

dB tot is the total susceptance slope to be seen on the input of this d␻ very simple oscillator circuit.

where

4.2.1.2 Open-Loop Approach

We can redraw the circuit of Figure 4.45 as shown in Figure 4.46, in which the input sinusoidal current source has been replaced by an external voltage E ext (␻ ) controlling the output current source of the transistor modeled once again by its equivalent linear transconductance G mo . The complex open-loop gain is defined as: ˜ ol = Vin G E ext

|

(4.89)

␻

A straightforward calculation shows that at ␻ 0 : ˜ ol = G

NG mo 2

G in + N (G out + G load )

=

G active G passive

Figure 4.46 Open-loop representation of the oscillator circuit.

(4.90)

160


The total input conductance of the circuit is written as: G tot = G passive − G active

(4.91)

Thus, G tot is immediately written as ˜ ol (␻ 0 )) G p (␻ 0 ) G tot (␻ 0 ) = (1 − G

(4.92)

which is the desired result. Thus, it can be seen clearly that when G tot < 0

(4.93)

˜ ol > 1 G

(4.94)

then

So, it comes down to the same thing to study autonomous circuits through the negative conductance concept or with the open-loop gain technique. More generally, when complex admittances are to be handled, the previous relation becomes: ˜ ol (␻ )) Y p (␻ ) Y tot (␻ ) = (1 − G

(4.95)

The study of transistor oscillators with the open-loop gain concept is extremely judicious and will make it easy to study, for example, linear and nonlinear stability, by using commercially available software tools, as will be seen later. 4.2.1.3 Free-Running Oscillator Start Conditions

The circuit is now investigated in the complex-frequency plane: p = ␣ + j␻ . The input admittance becomes: ˜ ol ( p )) Y p ( p ) Y in ( p ) = (1 − G

(4.96)

This linear input admittance is that of the circuit when turning on the power supply. It is obtained by replacing all nonlinear elements of the circuit by their incremental small-signal values at the chosen quiescent point.


161

An instability will appear in the circuit at an angular frequency ␻ 0 , if the current and voltage variables can be written at turn-on as:

␦ X (t ) = ␦ X 0 e ␣ 0 t cos (␻ 0 t + ␸ )

(4.97)

␣0 > 0

(4.98)

with

Since the circuit is autonomous, its equation can be written as: Y in ( p ) ␦ Vin ( p ) = 0

(4.99)

The requirement that ␦ Vin ( p ) ≠ 0 leads to the necessary condition: Y in ( p ) = 0

(4.100)

˜ OL ( p )) Y p ( p ) = 0 Y in ( p ) = (1 − G

(4.101)

besides

But Y p represents a passive admittance; therefore, it cannot contain any zeros with a positive real part (i.e., ␣ 0 > 0). So this clearly implies that (4.102) has to be satisfied. ˜ OL ( p ) = 1 G

(4.102)

The rigorous (4.102) allows for the determination of possible instabilities in the circuit. The zeros of this equation with a positive real part P 0 = ␣ 0 + j␻ 0

(4.103)

P k = ␣ k + j␻ k represent the possible complex oscillation frequencies of the circuit. It must be noted that (4.102) and (4.103) do not guarantee that the oscillation will grow until a stable large signal steady state is reached. They only guarantee that the dc operating point is unstable. This is the first

162


condition to set up oscillations in the circuit. The growth or quench of the oscillation depends on the nonlinear transient behavior of the circuit. ˜ OL ( p ) Referring to the simple but illustrative example of Figure 4.46, G writes: ˜ OL ( p ) = G

NG mo 2

G in + N (G out

1 + G load ) + C p + Lp

=1

(4.104)

and as G p = G in + N 2(G out + G load ) ˜ OL ( p ) = G

NG mo Gp + C p +

1 Lp

=1

(4.105) (4.106)

or equivalently,

冦冋

G p − NG mo + C␣ +

␻ C−

1 L (␣ 2 + ␻ 2 )

␣ =0 L (␣ + ␻ 2 ) 2

册

(4.107)

=0

and, finally, we find two complex conjugate roots ␣ 0 ± j␻ 0 , with

冦

␣0 =

NG mo − G p 2C

␻ 20

␻ 2R

=

冉

NG mo − G p − 2C

冊

2

(4.108)

where

␻ 2R = If NG mo > G p , then ␣ 0 > 0.

1 LC

(4.109)


163

We observe that the start frequency ␻ 0 is nearest to ␻ R when NG mo is nearest to G p and when the selectivity of the tank circuit, which is proportional to 2C , is the highest. Unfortunately, only a few standard simulators can perform this calculation in the complex-frequency plane [78]. ˜ OL ( j␻ ) can be plotted [79]. Then However, the Nyquist diagram G ˜ one can verify that the locus of the G OL ( j␻ ) encircles the critical point: (1,0). A Bode plot can also be drawn [79]. It appears to be a very useful tool. Let us write ˜ OL (␻ ) | e j␸ (␻ ) ˜ OL (␻ ) = | G G

(4.110)

˜ OL | and ␸ (␻ ) are plotted versus frequency. The 20 log | G Three conditions are necessary in a single-frequency parallel tuned oscillator, in order to obtain a growing oscillation from the noise floor of the oscillator circuit.

冦

˜ OL | > 0 20 log | G

␸ (␻ 0 ) = 0 d␸ 0. to take place, is G ˜ OL steady-state ( p ) = G ˜ OL steady-state ( j␻ 0 ) = In the steady state, ␣ = 0 and G 1, so that it can be said that the open-loop gain of an oscillator circuit is always, in the complex plane, ˜ OL ( p ) = 1 G with ␣ > 0 during the transient and ␣ = 0 in the steady state.

(4.112)


165

On the other hand, at real frequencies, one has at turn-on

| G˜ OL small-signal ( j␻ 0 ) | > 1

(4.113)

˜ OL steady-state ( j␻ 0 ) = 1 G

(4.114)

and, in the steady state:

So the transistor works in the steady state with a gain compression: ˜ OL steady-state ( j␻ 0 ) | C compression = 20 log | G

(4.115)

This enables the designer to choose the compression gain at which the transistor will work in the nonlinear steady state. It can be found, at the first order, by a simple linear calculation. Note that for a fixed nonlinear steady state oscillation waveform, the gain compression varies according to the transistor biasing mode: constant voltage, constant current, self-biasing, or feedback biasing. 4.2.1.4 Oscillation Start in Symmetrical Oscillator Circuits

Differential and push-push structures are widely used at RF and microwaves, owing to their easy implementation on a monolithic chip. Moreover, it has been shown that these structures lead to low phase noise, owing to their odd or even symmetry with respect to both loop oscillation and the main low-frequency noise sources in transistors. Figure 4.48 shows a symmetrical structure obtained by cascading two identical active two-port devices. Choosing a chain matrix representation, we obtain in the complexfrequency plane p = ␣ + j␻ :

Figure 4.48 Symmetrical oscillator.

166


冋册冋册冋 V1( p ) I1( p )

=

A

B

V2( p )

C

D

−I 2 ( p )

册

(4.116)

册

(4.117)

and

冋册冋册冋 V 1′( p ) I 1′( p )

=

A

B

V 2′( p )

C

D

−I 2′( p )

The connecting wires impose: I 1 = −I 2′ I 2 = −I 1′

(4.118)

V 1 = V 2′ V 2 = −V 1′ Then the equation of the circuit is written:

再冋

A−1

B

C

D−1

册冋

A+1

B

C

D+1

册冎冋册

= 0 (4.119)

册冎

= 0 (4.120)

V1( p ) I1( p )

The oscillation start conditions become: det

再冋

A−1

B

C

D−1

册冎再冋 det

A+1

B

C

D+1

which is the desired result. By inspection of the determinants, two possible modes of oscillation are obtained, namely, the odd and even modes. For the out-of-phase mode: V 1 = −V 1′. For the in-phase mode: V 1 = V 1′. To eliminate the undesired mode, the architecture of the active twoport devices must be carefully designed: • The use of the out-of-phase mode leads to differential and push-

push structures, with good phase-noise characteristics and reproducibility (see example at the end of the chapter). • The in-phase mode is generally used to combine the output power of two (or more) identical oscillators.


167

4.2.1.5 Oscillation Start Conditions in Multitransistor Circuits

The governing equation that describes an autonomous linear circuit can be formulated as a matrix equation in the following form: ›

[H ( p )] ␦ X ( p ) = 0

(4.121)

›

In order to have solutions ␦ X ( p ) different from zero, the system determinant must be equal to zero. The roots of det [H ( p )] = ⌬( p ) = 0 must be calculated. If the designer has no dedicated tools to find these zeros, the determinant ⌬( j␻ ) = 0 may be plotted in the complex plane and the encirclement of the critical point must be studied. Unfortunately, the resulting figure is generally too complicated to offer any particular interest. This is due to a lack of determinant normalization. In that case the system should rather be handled by plotting the normalized determinant function (NDF) [79, 80] or by computing the system’s eigenvalues [81]. This method will prove extremely effective even in the case of nonlinear analysis, as will be seen with the local stability of the oscillator steady state. After verifying the oscillation start conditions, the nonlinear steady state simulation will ensure that the circuit will oscillate properly. 4.2.2 Steady-State Analysis of Transistor Oscillators 4.2.2.1 Simulation Techniques

The oscillator steady state can be calculated by two general-purpose simulation techniques: • Time domain integration (TDI); • Harmonic balance (HB).

The most commonly used time domain simulation technique for steadystate autonomous circuits is the shooting method [82]. However, the HB method appears as the most widely used technique for RF, microwave, and millimeter-wave applications and it is well-suited for tackling efficiently the analysis of multitransistor circuits with distributed components, with reasonable computer simulation time. The latest generations of HB circuit simulators rely primarily on the modified nodal analysis (MNA) formulation to describe electronic circuits. This formulation proves efficient and accurate when HB applications are to

168


be extended to the analysis of modulated signals and noise in nonlinear circuits. Other network formulating equations are possible in order to further reduce the system matrix size to be computed, but results may be insufficiently accurate to be used with the analysis of modulated signals and noise in nonlinear circuits. Finally, the computing time of nonlinear circuits has been dramatically reduced nowadays thanks to methods using Krylov subspaces to solve the network matrices. The use of Krylov subspace methods in circuit simulation has been proposed by Freund and Golub [83]. It significantly reduces the computer simulation time, which currently makes RF, microwave, and millimeter-wave circuit analysis by the HB method and its associated techniques, particularly efficient and reliable. For surveys and further references, we refer the reader to [84–90] for details on the principles and implementation of the HB method. 4.2.2.2 Simulation Results

The main output characteristics given by a circuit simulator are the following: • Oscillation frequency; • Node voltages (i.e., complex voltages at harmonic frequencies, or

time-domain waveforms); • Mesh currents; • Average power dissipated in the circuit elements. • Other characteristics required by the designer. 4.2.2.3 Conditions for a Successful Steady-State Simulation

Special care is needed in choosing the number of spectral components as well as the number of samples processed by HB simulators. As for the number of harmonics, it should obviously not be too small, to obtain accurate amplitudes and phases for the first few harmonic frequencies. Conversely, if the number of significant harmonics is too high, it can also lead to erroneous results owing to the limited frequency range validity of active and passive component models used by foundries. Some publications indicate simulations performed at millimeter waves with a number of harmonics higher than 10 without any indication of the frequency range validity of the models used. Harmful interactions at frequencies beyond which passive and active models are not valid must be avoided. These frequencies must be shortcircuited on the active device ports. With this precaution, a high number of harmonics can be used.


169

As for oversampling, a high number of samples are required to reach an accurate solution: Even if the simulated circuits are driven by quasisinusoidal voltages (variables), the resulting nonlinear currents (functions) are strongly nonlinear (e.g., exponential functions used in diodes and bipolar transistor models). To check the stability of the simulated steady-state solution, the nonlinear local stability analysis of the circuit must now be undertaken. 4.2.3 Nonlinear Stability of Free-Running Oscillators Two kinds of stability are to be studied in nonlinear oscillator circuits: global and nonlinear local. 4.2.3.1 Global Stability

A possible bifurcation of a stable oscillation regime toward another stable regime can be detected by means of the variation of one of the parameters of the circuit, such as the dc bias voltage or the value of an element of the circuit. The most commonly encountered bifurcations are described as follows: • Amplitude jump and hysteresis phenomena (direct bifurcation D-

type); • A sudden appearance of a divided-by-two solution of the fundamental frequency ␻ 0 (indirect bifurcation or I-type); • Appearance of spurious frequencies ␻ = ␣ ⭈ ␻ 0 with ␣ ∈ ᑬ and nonrational (Hopf bifurcation H-type). All these phenomena, which are very spectacular to observe experimentally, can now be subjected to a detailed theoretical study with commercial simulators [91–96]. Such analyses, however, require a lot of computer time and should be reserved—in our opinion—to verifying and diagnosing oscillator circuits suffering from working disorders. Nevertheless, it is important for a designer to know these tools so that he or she can make a diagnosis if necessary. Moreover, such tools will soon be implemented into many commercial simulators, thus enabling nonspecialists to use them. 4.2.3.2 Nonlinear Local Stability of an Oscillator

The aim of this analysis is to verify that no spurious oscillation can occur in the circuit under the desired stable oscillation frequency ␻ 0 , calculated

170


previously. The nonlinear stability study can be carried out by a perturbation analysis of the periodic steady state. The principle of the analysis [78, 81, 95, 97, 98] consists of introducing a small-signal complex perturbation of the form: e (␣ + j␻ )t

(4.122)

This small signal is superimposed on the oscillation steady-state signal obtained previously. Thus, the state variables controlling the nonlinear elements will be written in the steady state as k = +N

x ss (t ) =

∑

k = −N

X k e jk␻ 0 t

with X −k = X k*

(4.123)

where N is the number of harmonics components and ␻ 0 is the fundamental angular frequency. Since the perturbing signal is small, higher order terms in ␻ may be neglected, and so, in the presence of this small-signal perturbation, the previous state variable expression becomes: x (t ) = x ss (t ) + ␦ x (t )

(4.124)

with the resulting perturbation k = +N

␦ x (t ) =

∑

k = −N

⌬X k e ␣ t e j (k␻ 0 + ␻ )t

(4.125)

A nonlinear element y (t ) controlled by x (t ) will be written by neglecting higher order terms in ␻ . y (t ) = y ss (t ) + ␦ y (t )

(4.126)

with k = +N

␦ y (t ) =

∑

k = −N

⌬Y k e ␣ t e j (k␻ 0 + ␻ )t

(4.127)


171

Finally, for the whole circuit, the resulting perturbation vectors of state variables ␦ x›(t ) and nonlinear elements ␦ y›(t ) are related by the Jacobian matrix [U ] of nonlinear elements. The terms in the Jacobian matrix [84] are simply the Fourier components of the derivative of the vector of nonlinear › › elements y (t ) with respect to the vector of state variables x (t ). This matrix is evaluated from the steady-state circuit analysis. By taking into account the nonlinear HB system of equations [78, 95, 98], the following perturbation equation in the complex-frequency plane can be written in matrix form: ›

{[1] − [A y ] [U ]} ⌬X = 0

(4.128)

where [1] is the identity matrix. [A y ] is the matrix of the linear circuit at frequencies: −j␣ + (␻ ± k␻ 0 )). [U ] is the Jacobian matrix of nonlinear elements. The above equation is the basis of the local stability analysis of a nonlinear circuit in the presence of a periodic or quasi-periodic steady-state signal [78]. Several processing methods can be undertaken from this equation depending on the available simulators. The first method, both natural and clever, yet not so easy to implement, is to find the complex eigenvalues of this equation and extract the complex frequencies solutions for which ␣ > 0. Then we immediately obtain frequencies at which instability can occur with either (␣ > 0; ␻ = 0 or ␻ = imaginary) or (␣ > 0; ␻ ≠ 0 and real). This extremely efficient and rigorous method permits one to handle local stability analysis of oscillators in the presence of either periodic (CW) or quasi-periodic (modulated) steady-state solutions [78]. The calculation in the complex-frequency plane can be avoided by plotting the determinant of this equation in the j␻ plane (i.e., by examining the resulting Nyquist locus). Unfortunately, as in the linear case discussed previously, the investigation of this diagram proves difficult for nonexpert designers who will find it very complicated and/or impossible to use. Once again, this is due to a lack of normalization of the resulting diagram. Another interesting computer approach to be performed consists of calculating the eigenvalues of the perturbed system in the j␻ plane [81]. By appropriate eigenvalue ordering, the complete Nyquist plot in the j␻ plane can be obtained, allowing a check of the clockwise encirclement of the

172


critical point. This method ensures the normalization of the determinant characteristic through a simple Nyquist locus. The detailed implementation of this rigorous method in a commercial simulator will be found in [81]. More recently a simplified method was proposed [99]: It consists of injecting a small signal current perturbation I ( j ⍀) into one node of the circuit in presence of the large signal steady state. The impedance Z observed at the particular node where the current perturbation has been injected can be calculated. The frequency ⍀ has to be swept in a wide frequency range. The result is then subjected to system identification methods, which permit to progress from Z ( j␻ ) to Z ( p ).

写 M

Z ( j␻ ) —————> Z ( p ) = identification

( p − zi )

i=1

( p − pi )

(4.129)

For the small poles p i which have a positive real part, the circuit is unstable and unwanted oscillations might occur. This very attractive method presents a drawback; in fact, strictly speaking, the impedance observed on a particular node of the circuit may well not contain poles with positive real parts located elsewhere in the circuits. See, for example, circuits with unilateral elements. This drawback becomes obvious when the characteristic matrix of the circuit is investigated by inspection. So, as indicated by the authors of the method, the location of the small-signal current perturbation is crucial. It has to be a ‘‘nonisolated node.’’ In complex circuits combining an oscillator with buffers the input port of a transistor participating to the oscillation is a good choice. 4.2.4 Oscillator Phase-Noise Characterization One of the fundamental characteristics of an oscillator is its phase noise (PM) spectrum. This characteristic must be precisely assessed. The amplitude noise (AM) spectrum is not an essential characteristic owing to its low level (in general between −10 dB and −30 dB) as compared to the phase noise level. Moreover, one of the common oscillator applications is to be a local oscillator in receiver systems. In such applications, AM noise rejection can be achieved with balanced configurations. However, this characteristic can prove significant in some applications, where the transmitter must have a very low AM noise (e.g., in collision warning/avoidance systems in automotive applications).


173

4.2.4.1 Output Signal Spectrum

Phase and amplitude noise generation being a stochastic process, it is necessary to know: • The statistical properties of noise sources; • The nonlinear interaction between these random sources and the

circuit elements, which generates random fluctuations on amplitude and phase in a free-running oscillator circuit. To define the amplitude and phase fluctuations in terms of spectral power density or autocorrelation function, the stationarity of the random phenomena must be checked. Numerous controversial publications have been issued in the past years addressing the validity of the phase noise representation in free-running oscillators as a stationary process [100]. This problem has now been solved [101, 102], but designers should also have some knowledge of the hypotheses relevant to the simulation tools of which they make use. Strictly speaking, the oscillator phase noise is a nonstationary process, whereas amplitude noise is a stationary process. According to Rutman [101], the pure oscillator output signal can be expressed simply by: V (t ) = V 0 sin (2␲ f 0 t )

(4.130)

where V 0 and f 0 are nominal amplitude and frequency, respectively. The actual output signal can be expressed as: V (t ) = (V 0 + ⑀ (t )) sin (2␲ f 0 t + ␸ (t ))

(4.131)

where ⑀ (t ) and ␸ (t ) are amplitude and phase fluctuations, respectively. It must be pointed out that a frequency variation implies a related phase variation: More precisely, the instantaneous frequency is the time rate of change of phase divided by 2␲ . By definition, the instantaneous frequency can be written as: f (t ) = or else:

1 d␸ (t ) 1 d [2␲ f 0 t + ␸ (t )] = f 0 + 2␲ dt 2␲ dt

(4.132)

174


f (t ) = f 0 + ⌬ f (t )

(4.133)

with ⌬ f (t ) =

1 d␸ 2␲ dt

(4.134)

The frequency noise ⌬ f (t ) is a stationary process as can be justified by physical considerations; then, ␸ (t ), which is denoted by t

␸ (t ) = ␸ 0 +

冕

2␲ ⌬ f (␪ ) d␪

(4.135)

0

is, in general, a nonstationary process. It must be noted that a free-running oscillator does not have a reference phase. This assertion is best visualized by turning off then turning on a freerunning oscillator: Although the circuit goes back to the same oscillation frequency every time, there is no reason why its phase, once it has been turned on, should be related to the one observed at turn-off. As a matter of fact, an oscillator gradually loses memory of its original phase with time. The conventional example of phase noise generation is that of a theoretically ideal oscillator, with an internal white-noise source that generates a socalled phase diffusion process similar to the mathematically ideal Brownian motion. As was pointed out by Kuva¨s [103], phase diffusion is a slow process. So the linearization of equations involving the phase in stabilities is valid as long as short-term stability is concerned. It is theoretically demonstrated that stationary models can be adopted for the representation of phase fluctuations in free-running oscillators. On the other hand, formalisms related to cyclostationary random processes can be used to describe all nonlinear noise sources. To verify these assumptions, a series of numerical calculations have been carried out on many different oscillator circuits using reliable active and passive device models. In the benchmark, transistors were described with linear and/or cyclostationary white noise and/or colored noise sources. Circuit simulations have been successively carried out using a time domain Monte Carlo (TDMC) method developed by Bolcato [104] and a frequency domain method [105] developed in our laboratory [88]. Results obtained with the two methods have been successfully compared.


175

The TDMC method can be regarded as the most rigorous method for noise calculation in lumped nonlinear circuits and may be used as a reference for comparison purposes. The method analyzes the noise excitation and the deterministic signal together, without stationary approximation. However, a drawback of this TDMC method is its high amount of simulation time. The higher the numerical precision required, the more CPU time will be needed. The frequency-domain analysis is based upon the assumption of the stationarity of the phase noise process. It is derived from the HB method by a linearization of the nonlinear network equation around the steady-state solution to construct a linear time-varying model for noise analysis [106]. The nonlinear elements are represented by means of conversion matrixes directly derived from the HB Jacobian evaluated at the steady state, and on the representation of linear [107] and cyclostationary noise sources by their correlation matrices. Note that in our method primary noise sources can be located in any node of the circuit, which is not the case for all the methods of oscillator phase-noise calculation. In addition, the aforementioned method and the TDMC technique permit accurate handling of the slow and fast dynamics of the circuit, which is crucial for oscillator circuits with low-frequency noise sources (i.e., generation-recombination and 1/f noise sources). The circuit example described in Appendix 4C verifies all these properties. The required computing time to simulate noise spectra with the frequency-domain method is extremely reduced as compared to the oscillation steady-state calculation. In conclusion, despite the theoretical problems that may arise concerning the stationarity of phase noise, ␸ (t ) may be assumed as a stationary process: Its one-sided spectral density S ␸ ( f ) and its double-sided spectral density L ( f ) can be defined from the Fourier transform of the autocorrelation function R ␸ (␶ ). The concept of S ␸ ( f ) [101, 108] is recognized today as the established fundamental quantity for the characterization of phase noise in free-running oscillators. 4.2.4.2 Oscillator Phase Noise Calculation in the Frequency Domain by Means of Conversion Matrices Formalism

The output waveform for an oscillator can be expressed as:

冉

V (t ) = V 0 1 +

冊

⌬v (t ) cos (␻ 0 t + ␸ 0 + ⌬␸ (t )) V0

(4.136)

176


where V 0 is the peak amplitude of the carrier ␻ 0 . ␸ 0 is the phase of the carrier frequency at t = 0. ⌬v (t ) is the amplitude noise modulation. ⌬␸ (t ) is the phase-noise modulation. The AM and PM noise modulations can be expressed as pseudosinusoids at the noise frequency ⍀. ⌬v (t ) = ⌬V 0 cos (⍀t + ␾ A )

(4.137)

⌬␸ (t ) = ⌬␸ 0 cos (⍀t + ␾ ␾ )

(4.138)

where 〈 ⌬v 2(t ) 〉 and 〈 ⌬␸ 2(t ) 〉 denote the associated spectral densities. By introducing (4.137) and (4.138) into (4.136) and assuming that the amplitude of these modulation indexes are very small in comparison with the carrier signal, the voltage V (t ) contains three components at the following frequencies: • Carrier frequency: ␻ 0 ; • Lower sideband frequency: ␻ 0 − ⍀; • Upper sideband frequency: ␻ 0 + ⍀.

These components are evaluated with the software tool based on the linear time-varying approach and lead to the following expression:

冋

V (t ) = V 0 cos (␻ 0 t + ␸ 0 ) +

V⌬ cos [(␻ 0 − ⍀)t + ␸ ⌬ ] V0

+

V⌺ cos [(␻ 0 − ⍀)t + ␸ ⌺ ] V0

册

(4.139)

According to the conventional definition of AM and PM modulations, we can define AM noise at an offset frequency ⍀ from the carrier as:

冋册 N C

at ⍀ dBc

= 10 log

P ssb (AM ) by hertz P carrier

(4.140)


177

PM noise at an offset frequency ⍀ from the carrier as: L at ⍀ = 10 log dBc

P ssb (PM ) by hertz P carrier

(4.141)

The above-mentioned definitions are the ratio of the single sideband noise power spectral density (in a 1-Hz bandwidth at ⍀ offset from the carrier) to the total carrier power. Let us write in complex notation: ⌬V 0 e j␾ A ⌬V˜ = V0 V0 ⌬␸ 0 e j␾ ␾ = ⌬␸˜ V⌬ e j␸ ⌬ = V˜⌬

(4.142)

V ⌺ e j␸ ⌺ = V˜ ⌺ By identifying (4.136) with (4.139) and by taking into account (4.137), (4.138), and (4.142), we obtain the following expression for the amplitude and phase noise spectra 2 | V˜⌬* e j␻ 0 + V˜ ⌺ e −j␻ 0 | 2 P ssb (AM ) | ⌬V˜ | = = P carrier V 02 V 02

(4.143)

| V˜⌬* e j␻ 0 − V˜ ⌺ e −j␻ 0 | 2 P ssb (PM ) 2 ˜ = | ⌬␸ | = P carrier V 02

(4.144)

and finally the PM-AM complex correlation C=

⌬V˜ ⌬␸˜ * V 02

=j

1

(|V⌺ | V 02

2

− | V⌬ | 2 + 2j Im (V⌬* V ⌺ e 2j␸ 0 )) (4.145)

The PM-AM noise spectra and the AM-PM noise correlation spectrum of free-running oscillators can be calculated using the above-mentioned equations. Note that there may be some confusion between two quantities that permit the ‘‘measurement’’ of oscillator phase noise.

178


Let us write the oscillator signal as: V (t ) = V 0 cos (␻ 0 t + ⌬␸ (t )) = V 0 cos (␾ (t ))

(4.146)

where ⌬␸ (t ) and cos (␾ (t )) are fluctuating quantities. Remember that the standard definition of phase noise is S [⌬␸ ]⍀ : that is, the spectral density of phase fluctuations with ⌬␸ (t ) calculated at ⍀ offset from carrier [108]. However, in some publications, phase noise is expressed as S [cos (␾ )]⍀ , where the spectral density of cos [␾ (t )] is calculated at ⍀ offset from carrier. By invoking the relation between the autocorrelation function of ⌬␸ (t ) and of cos [␾ (t )], the relation between S [⌬␸ ]⍀ and S [cos (␾ )]⍀ may be found. Nevertheless, the two ‘‘measures’’ must not be confused. As an example, the asymptotic behavior of phase noise near the carrier, in an oscillator with white noise sources, may be calculated. By neglecting the low-frequency dynamics of the circuit, one finds that: • S [⌬␸ ]⍀ varies as K /⍀2. • The associated S [cos (␾ )]⍀ varies as

K1 K 22

+ ⍀2

S [⌬␸ ]⍀ is measured by passing the signal through a phase detector, whereas S [cos (␾ )]⍀ is directly observable on a spectrum analyzer, if the AM noise is negligible. The numerical processing of the AM and PM noise spectra with a simulator software package is described in the following equation. 4.2.4.3 Description of the Numerical Method

Applying Kirchoff’s current law in the time domain to all the nodes of a general autonomous circuit, we can obtain the well-known conventional nonlinear equation i(v(t)) +

d q(v(t)) + y(t) * v(t) + i G = 0 dt

(4.147)

where t is the time; v(t) is the vector of node voltage waveforms; i G is the vector of independent sources; i(t) represents all the nonlinear current sources and q(t) all the nonlinear charges; and y(t) represents the impulse response matrix of the linear circuit.


179

Steady State

If we expand all the previous variables in truncated complex Fourier series, and we call Y the Fourier transform of impulse response y(t), we obtain the steady state in the frequency domain: I(V 0 ) + j ⍀Q(V 0 ) + YV 0 + I G = 0

(4.148)

where I(V 0 ) denotes the current-voltage dependence of the nonlinear conductances and Q(V 0 ) describes the dependence of the charge stored in the nonlinear capacitors on the voltages. ⍀ is a diagonal matrix representing the frequency domain differentiation operator. V 0 denotes a nontrivial solution of (4.148) and characterizes the oscillator steady-state node voltages. Noise Analysis

Let us assume that the noise sources do not perturb the large-signal steady state of the circuit. From this postulate, we can apply the frequency conversion analysis to determine noise voltage and current anywhere in the circuit. Let us consider the time domain steady-state solution v 0 (t ) of (4.147) corresponding to the phasor representation V 0 in the frequency domain and superimpose a small perturbation to it: v(t) = v 0 (t) + ␦ v(t)

(4.149)

Thus, the system of the nonlinear equations can be linearized around the steady state d i(t) dv

|

v 0 (t )

␦ v(t) +

冉 |

d d q(t) dt dv

v 0 (t )

冊

␦ v(t) + y(t) * ␦ v(t) = b(t) (4.150)

where b(t) describes all the possible forms of the modulated noises sources. It can be expressed as: b(t) = m(t) n(t)

(4.151)

For the sake of clarity, let us express the primary noise source n(t) [47] in a developed form:

180


n(t) =

冦冤

√2 ᑬ

k = +NH

∑

+

冥冤

N k e j (k␻ 0 + ⍀p )t + ᑬ

k=1

k = +NH

∑

−

冥

N k e j (k␻ 0 − ⍀p )t

k=1

冧

+ ᑬ [N 0 e j ⍀t ]

(4.152)

+

where √2 N k represents the complex amplitude of the pseudosinusoid relative to the frequency k␻ 0 + ⍀p with ⍀p 0.0 then I DST a = I DSS × f g × f d then I DST a = 0

224


with V GSN = 1 +

V DSN = V DSP

V GS (t − ␶ ) − V␸ VP V DS V (t − ␶ ) 1 + w GS VP

冉

冊

V P = V P 0 + pV DS + V␸ fg =

冉

1 1 V GSN − (1 − e −mV GSN ) m 1 1 − (1 − e −m ) m

冊

2

f d = (1 − e −V DSN (1 + aV DSN + bV DSN ) ) correct_ gm = 1 + ␤ gm × (V DS − V dm ) × (1 + tanh (␣ gm (V GS (t − ␶ ) − V gm ))) I DS = I DST a × correct_ gm Table 4A.5 Parameters of the PHEMT I DS Current Source V DSP

V␸

a

b

m

P

I DSS

0.5865

2.971

0.9854

0

32.57

6.172 × 10−2

0.8337

␤ gm

␣ gm

w

V dm

␶ (pS)

V gm

V P0

0.1088

0.2732

1.113

26.86

0.257

4.555

0.4958

4A.2.2 Gate-Drain Diode

冉

I GD = I GDS e

V GD ␩ gd U T

−1

冊


225

Table 4A.6 Parameters of the I GD Current Source I GDS

␩ gd

5.966 × 10−12

1.912

4A.2.3 Gate-Source Diode

冉

I GS = I GSS e

V GS ␩ gs U T

冊

−1

Table 4A.7 Parameters of the PHEMT I GS Current Source I GSS

␩ gs

9.939 × 10−11

2.1832

4A.2.4 Gate-Drain Capacitance

C GD (V GD ) = Cg 0d

冉

V′ 1 − eff V bd

冊冉 n′

冊

1 1 (V GD − V td ) − (V GD − V bd ) + C bd 2d 1′ 2d 2′

with V ef′f =

V + V bd 1 (d ′ − d 2′ ) + td 2 1 2

d 1′ =

√(V GD − V td )

2

+ d ′2

d 2′ =

√(V GD − V bd )

+ d ′2

2

Table 4A.8 Parameters of the PHEMT Gate-Drain Capacitance Cg 0d

V bd

1.354 × 10−14 1.887

V td

d′

C bd

n′

−0.5914

0.2543

3.371 × 10−14 7.842

226


4A.2.5 Gate-Source Capacitance

C GS (V GS ) = Cg 0s

冉

V eff 1− V bs

冊冉 n

冊

1 1 (V GS − V ts ) − (V − V bs ) + C bs 2d 1 2d 2 GS

with V eff =

1 V + V bs (d − d 2 ) + ts 2 1 2

d1 =

√(V GS − V ts )

2

+ d2

d2 =

√(V GS − V bs )

+ d2

2

Table 4A.9 Parameters of the PHEMT Gate-Source Capacitance Cg 0s

V bs

18.85 × 10−14 1.176

V ts

d

C bs

n

−0.7791

−0.1419

0

−0.1086

4A.2.6 Extrinsic Elements Table 4A.10 PHEMT Extrinsic Elements L G (pH)

L D (pH)

L S (pH)

R G (⍀)

R D (⍀)

R S (⍀)

C pg (fF)

C pd (fF)

30.85

16.5

16.35

1.95

2.575

2.575

48.98

20.5

R i (⍀)

R GD (⍀)

C DS (fF)

0.366

15.65

56.5

Appendix 4B: Transistor Low-Frequency Noise Characterization 4B.1 Equivalent Circuit for Noise Calculations From Figure 4B.1, the equivalent circuit of the intrinsic transistor is given in Figure 4B.2 and the admittance matrix { y int } is directly obtained.


227

Figure 4B.1 Noise equivalent circuit for simultaneous input-output noise measurements.

Figure 4B.2 Equivalent circuit of the intrinsic transistor.

In order to take into account the parasitic access resistances the impedance matrix of this intrinsic part is calculated. We have, respectively: {Y int } =

再

{ Z int } =

再

y1 + y3

−y 3

g m − y3

y2 + y3

冎

(4B.1)

冎

(4B.2)

and z2 + z3

z3

z3 − ␰

z1 + z3

with (for n = 1, 2, 3) zn =

1 y det { y int } n

(4B.3)

and

␰=

g m int det { y int }

(4B.4)

228


So we have the following equivalent circuit (see Figure 4B.3). Now the same transformation is done with the associated noise sources. For the intrinsic transistor the steps of the calculation are summarized in Figures 4B.4–4B.6.

Figure 4B.3 T representation of the intrinsic transistor.

Figure 4B.4 Circuit transformations for noise calculations.




229

And we obtain in complex notations: J 1 = i gs − i gd

(4B.5a)

J 2 = i ch int + i gd

(4B.5b)

and

再冎 v1

v2

= { Z int }

再冎 i 1′ i 2′

(4B.6)

with: i 1′ = i 1 + J 1 i 2′ = i 2 + J 2 So we have: v 1 − (z 2 + z 3 ) J 1 − z 3 J 2 = (z 2 + z 3 ) i 1 + z 3 i 2

(4B.7)

v 2 − (z 3 − k ) J 1 − (z 1 + z 3 ) J 2 = (−␰ + z 3 ) i 1 + (z 1 + z 3 ) i 2 Let us put: E 1 = (z 2 + z 3 ) J 1 + z 3 J 2

(4B.8)

E 2 = (z 3 − ␰ ) J 1 + (z 1 + z 3 ) J 2 the equivalent circuit of the intrinsic part is given in Figure 4B.7. This representation is advantageously used to obtain the amplifier output signal. Now the parasitic access resistances and their associated noise

Figure 4B.7 Equivalent noise circuit in the T representation.

230


generators are introduced. From Figures 4B.1 and 4B.7 the total noise representation is given in Figures 4B.8 and 4B.9 with: eg = R G iR G es = R S iR S

(4B.9)

ed = R D iR D The following relations are used: { Z ext } =

再

Z 11

Z 12

Z 21

Z 22

冎再 =

z2 + z3 + RG + RS z3 − k + RS

z3 + RS

v 1 − E 1 − e g − e s = Z 11 i 1 + Z 12 i 2 v 2 − E 2 − e d − e s = Z 21 i 1 + Z 22 i 2

Figure 4B.8 Noise representation of the real transistor.

Figure 4B.9 Noise representation of the real transistor.

冎

z1 + z3 + RS + RD (4B.10) (4B.11)


231

⑀ 1 = E1 + eg + es

(4B.12)

⑀ 2 = E2 + ed + es 4B.2 Noise Current Calculation at the Input of the Transimpedance Amplifier Figure 4B.10 gives the equivalent form of the circuit presented in Figure 4B.1. To compute the input noise current i E′ of the transimpedance amplifier the part at the right-hand of the dotted line b ′b ″ is replaced by a Thevenin generator (e eq , R eq ) with: e eq =

ep R E − eA R P RP + RE

R eq =

RP RE RP + RE

(4B.13)

As shown elsewhere [119], the following experimental conditions apply: R E ≥ 1M⍀; R p < 100⍀. So we have: e eq ≈ e p − e A

Rp RE

R eq ≈ R p

(4B.14)

and V 2 = e eq − R p i 2

(4B.15)

From (4B.11), (4B.12), and (4B.15) we have: i2 =

e eq − ⑀ 2 − Z 21 i 1 R P + Z 22

v1 = ⑀ 1 +

(4B.16)

冉

冊

Z 12 Z Z (e ′ − ⑀ 2 ) + Z 11 − 12 21 i 1 Z 22 + R P eq Z 22 + R P

Figure 4B.10 Equivalent representation of the noise measurement setup of Figure 4B.1.

232


Let us put e ′ = ⑀1 +

Z 12 (e − ⑀ 2 ) Z 22 + R P eq

Z E′ = Z 11 −

(4B.17)

Z 12 Z 21 Z 22 + R P

(4B.18)

we have: v 1 = e ′ + Z E′ i 1

(4B.19)

So Figure 4B.8 can be reduced to the circuit given in Figure 4B.9 and iE′ = iA′

e′ Z E′ + ′ ′ ′ Z E + R E Z E + R E′

(4B.20)

The experimental condition R E′ > Z ′S and v E = e A + e s′ = e A +

ZS RP ep + e ZS + RP ZS + RP

(4B.28a)

and the spectral density given by the analyzer is: S VS ( f ) = | k | S VE ( f ) 2

= |k |

再 +

2

再

S ⑀2 ( f ) +

|

|

ZS S eA ( f ) + ZS + RP

|

Z 21 Z 11 + R E′

Z 21 Z 11 + R E′

|

2

| 冎

2

|

2

冎

|

| 冊

RP S ep ( f ) + | k | ZS + RP

S ⑀ 1 ( f ) − 2RP

2

再冉

S eA′ ( f )

2

冎

Z 21 (S ⑀*1 ⑀ 2 ( f )) Z 11 + R E′

(4B.28b)

4B.4 Calculation of the Cross-Spectrum It must be noticed that for the used dynamic signal analyzer (HP 89410A) the measured cross-spectrum of the two channels A and B is related to the synthetic form: (channel A)(channel B)*


235

So the measured cross-spectrum is calculated with the help of (4B.21) and (4B.28) as: S V S VS′ ( f ) = (kV E (k ′iE′ )*)( f S V S VS′ ( f ) =

再

(4B.29a)

)

冉

冊冎

kk ′*R P Z 21 Z 12 * − S ⑀1 ( f ) S ⑀2 ( f ) Z 22 + R P Z E′ * (Z S + R P ) Z 11 + R E′ ′ + S ⑀ 1⑀ 2 ( f ) +

(Z 22

Z *12 Z 21 S ⑀* ⑀ ( f ) + R P )* (Z 11 + R E′ ) 1 2

(4B.29b)

Appendix 4C: Numerical Simulations of an Oscillator Benchmark Table 4C.1 Oscillator Element Values L chock1

L chock2

C dec1

C dec2

V gs0

V ds0

L in

R load

10 nH

5 nH

10 pF

5 pF

−0.4 V

3.0 V

12 nH

50 ⍀

Lr

Rr

Cr

9.3 pH

1e 3 ⍀

26 pF

N1

N2

N3

N4

N5

N6

1

6.4

6.18

1

1.37

1

Numerical values: See Section 4A.2. Note: Harmonic frequencies higher than nine are short-circuited across the intrinsic input and output port of the transistor. 4C.1 Small Signal (AC) Open-Loop Simulation The transistor is linearized around its bias point. The transconductance is now controlled by an external independent voltage source: E ext (␻ ).

236


The complex open-loop gain is defined as: ˜ ol = G

Vgs E ext

|

␻

(4C.1)

where Vgs is taken across the gate capacitance. See Figures 4C.1 through 4C.4. 4C.2 Negative Conductance Concept The input admittance is calculated by introducing an external sinusoidal current I 0 (␻ ) (AC) across the extrinsic transistor input port. 4C.3 Steady-State Simulation The steady-state simulation is performed after elimination of the spurious frequency. Figures 4C.5 and 4C.6 show the linear admittance seen at the input port versus frequency and the oscillator load cycle.

Figure 4C.1 Simulated oscillator circuit.


237

Figure 4C.2 Nonlinear PHEMT model.

4C.4 Phase-Noise Simulation K Power spectral density: S ␦ i 1/f = f Z c /c : Short circuit for harmonic frequencies higher than 9. Table 4C.2 Oscillator Element Values K

L lf

R plf

C lf

4e-09

20 ␮ H

0.1⍀

0.5 ␮ F

Figure 4C.7 shows the noisy transistor used for the simulation of the oscillator phase noise (Figure 4C.8) for two unloaded quality factor values. The low-frequency filter across the intrinsic output port of the transistor is activated. 4C.5 Relation Between Steady-State Accuracy and Phase-Noise Results Figure 4C.9 shows the influence of a low-frequency circuit on the phase noise simulation. Accurate phase-noise calculation is directly related to the

238


Figure 4C.3 Simulated linear open-loop circuit.

large steady-state accuracy (Figure 4C.10). It is only a numerical problem that is easily solved by increasing oversampling of the steady-state simulation.

Acknowledgments Many special thanks go to R. Que´re´, M. Valenza, and J. C. Vildeuil for a number of helpful technical discussions, support, and constructive suggestions. Our warmest thanks go to M. Delagnes for his painstaking care in correcting our English and for spending his valuable time. It is a pleasure to thank Mrs. M. C. Lerouge and Mrs. H. Rivie`re for the countless hours they devoted to solving logistic problems.


239

Figure 4C.4 Numerical results of open-loop gain G˜ ol (␻ ) versus frequency: (a) 450–550 MHz and (b) 10.14–10.36 GHz. (Note: There are two possible oscillation start frequencies.)

Our colleagues at IRCOM and many others throughout the industry deserve our deep gratitude for their valuable advice and technical discussions. It is a pleasure to recognize the stimulation of our past Ph.D. students for their searching questions and enthusiasm. We also wish to acknowledge the support of the French research center: CNRS and of the universities of Limoges and Montpellier for many of the results included in this chapter. The sponsorship of CEE, RNRT, CNES, and DGA was gratefully appreciated. Finally, J. C. Nallatamby, J. Obregon, and M. Prigent would like to thank Agathe, Denise, Maı¨lys, Pierrette, Soizic, Sylvain, and The´o for their support, patience, and encouragement.

240

Figure 4C.4 (Continued.)



241

Figure 4C.5 (a) Linear admittance seen at the input port versus frequency: 450–550 MHz, and (b) linear admittance seen at the input port versus frequency: 10.21–10.27 GHz.

242


Figure 4C.6 Oscillator load-line (see Section 4.3.2.1).

Figure 4C.7 P-HEMT transistor with 1/f noise source, thermal noise sources (Tamb ) and low-frequency filter.


243

Figure 4C.8 Simulated phase noise for two unloaded Qs and without low-frequency filter.

Figure 4C.9 Influence of low-frequency dynamic on phase noise (see Section 4.2.4.3).

244


Figure 4C.10 Curve (1): accurate result of phase noise; and curve (2) and curve (3): inaccurate result of phase noise near the carrier due to an inaccurate steady-state analysis (see Section 4.2.4.3).

5 Modern Harmonic-Balance Techniques for Oscillator Analysis and Optimization Vittorio Rizzoli, Andrea Neri, Alessandra Costanzo, and Franco Mastri 5.1 Introduction This chapter discusses the analysis and design of autonomous (self-oscillating) nonlinear circuits making use of the harmonic-balance (HB) principle. Attention is focused on the piecewise HB technique based on the decomposition of the nonlinear circuit into a linear and a nonlinear multiport subnetwork interconnected through a number of common ports [1]. It is noteworthy that this approach—historically the first HB method proposed in the technical literature [2]—is better suited than the nodal HB approach proposed much later [3] for truly general-purpose applications encompassing the RF as well as the microwave and millimeter-wave ranges. Indeed, the piecewise technique represents the only possible choice in all those cases where the voltage concept is only significant on a local basis but becomes physically meaningless on a global basis, so that a common reference node and a consistent set of node voltages do not exist. Possible examples are a system composed of a number of integrated circuits coupled by electromagnetic fields (e.g., a beam-forming network [4]), or even a high-clock-rate integrated circuit where electromagnetic propagation in the ground plane is not negligible. In all cases, the piecewise HB method describes the linear subnetwork as a multiport in the 245

246


frequency domain, so that the interaction of the nonlinear devices with the embedding circuit always takes place through a limited number of local ports [1], irrespective of how the passive circuit as a whole is analyzed—from elementary lumped circuit theory to electromagnetic simulation. Starting from this viewpoint, the chapter develops a self-consistent set of analysis and design algorithms for self-oscillating circuits explicitly devised for generality and computational efficiency—from elementary single-device topologies up to entire front ends for heterodyne communication systems, including the (single or multiple) local oscillator function.

5.2 HB Analysis of Autonomous Quasi-Periodic Regimes in Nonlinear Circuits This section discusses the basic HB techniques for autonomous circuit analysis. Most numerical approaches to the analysis of autonomous nonlinear circuits are normally restricted to the time-periodic oscillator case. This limitation is very reductive, since some of the most interesting and practically important applications of nonlinear circuits involve multitone autonomous operation, typical examples being represented by RF front ends for telecommunications systems. In this section we introduce an HB approach to the treatment of autonomous quasi-periodic regimes in nonlinear microwave circuits, which is fully compatible with the needs of a general-purpose microwave CAD environment. We use a rigorous and absolutely general piecewise HB formulation, whereby the problem is reduced to a well-posed nonlinear algebraic system to be solved by a specialized (mixed-mode ) Newton iteration. This allows the simultaneous determination of the steady-state harmonics and of the unknown fundamental frequencies of the quasi-periodic regime. Unrivaled speed and robustness are guaranteed by a general algorithm for the exact computation of the Jacobian matrix. Section 5.2.1 is devoted to a formal definition of autonomous multitone regimes. Section 5.2.2 introduces the fundamental analysis algorithms that will be systematically used throughout the chapter. Section 5.2.3 provides a preliminary treatment of the degenerate solution problem. 5.2.1 Autonomous Quasi-Periodic Regimes This section introduces the autonomy concept for general quasi-periodic regimes. Let us consider a nonlinear microwave circuit operating in a quasiperiodic electrical regime generated by the intermodulation of F timeperiodic signals of incommensurable fundamental angular frequencies ␻ i . Any signal a (t ) supported by the circuit may be represented in the form

Modern Harmonic-Balance Techniques for Oscillator Analysis

247

冣

(5.1)

a (t ) =

∑

k∈S

∑

A k exp ( j ⍀ k t ) =

k∈S

冢

F

A k exp j ∑ k i ␻ i t i=1

where ⍀ k is a generic intermodulation (IM) product of the fundamentals— that is, F

⍀k =

∑ ki ␻i

(5.2)

i=1

In (5.1) and (5.2), k i is an integer harmonic number and k is the F-vector of harmonic numbers k1

k=

冤冥 k2

...

(5.3)

kF

The vector k spans a finite subset S of the k-space (containing the origin), which will be conventionally named the signal spectrum. The Fourier coefficient A k will be named the harmonic of a (t ) at ⍀ k [or the kth harmonic of a (t )]. Since we want to deal with real signals, S must be symmetrical with respect to the origin, and A −k = A k*. We shall also denote by S + the subset of S such that ⍀ k ≥ 0 for k ∈ S +. In general, the circuit will be excited by dc (bias) sources for which k = 0, and by a number of free periodic generators of frequencies ⍀ b (h) (1 ≤ h ≤ B ) where b (h ) ∈ S +. Let us consider the excitation matrix B = [b(1) | b(2) | . . . | b(B )]

(5.4)

of dimensions F × B. We define the rank of B, namely, R , as the maximum size of the nonsingular square submatrixes of B (R ≤ F ), and introduce the following definitions: 1. If R = F, the electrical regime is nonautonomous or completely forced. 2. If R < F, the circuit is autonomous of order M = F − R , or M-autonomous. The physical explanation of these definitions is as follows. First of all, for an autonomous regime of order M , we can always define a new set of

248


fundamentals that are linear combinations (with rational coefficients) of the original ones, in such a way that M rows, say, the first M ones, of the excitation matrix (5.4) become zero. We shall assume henceforward that the ␻ i ’s have been selected in this way. In such conditions, the first M of the ␻ i ’s will be referred to as the free fundamentals. If we now introduce the quantities zi = ␻i t

(5.5)

(1 ≤ i ≤ F ) the generic signal (5.1) takes on the form a (t ) =

冢

F

∑ A k exp j ∑ k i z i

k∈S

i=1

冣

(5.6)

and may thus be viewed as a function defined on an F -dimensional normalized time space Z, which is 2␲ -periodic in each dimension z i . The meaning of an autonomy of order M is now clear: The electrical regime has no forcing term (other than dc), and thus no phase reference, in M dimensions (out of F ) of the normalized time space. As a consequence, the phase of the electrical regime with respect to z i is indeterminate for 1 ≤ i ≤ M . Each arbitrary choice of the phase in the i th dimension corresponds to one of an ∞M family of equivalent electrical regimes having the same amplitude spectrum, as shown by (5.6). For M = F = 1, this implies the well-known invariance of the time-domain waveforms with respect to a shift of the time origin. 5.2.2 The Mixed-Mode Newton Iteration Throughout this chapter we shall assume that the nonlinear subnetwork may be described by a set of generalized parametric equations of the form [1]

冋冋

册册

v(t ) = u x(t ),

d nx dx , . . . , n , x D (t ) dt dt

i(t ) = w x(t ),

d nx dx , . . . , n , x D (t ) dt dt

(5.7)

where v(t ) and i(t ) are vectors of voltages and currents at the common ports, x(t ) is a vector of state variables (SV), and x D (t ) is a vector of time-


249

delayed state variables [i.e., x Di (t ) = x i (t − ␶ i ), ␶ i = constant]; u and w are vector-valued memoryless nonlinear functions of their arguments. Note that (5.7) is equivalent to a set of implicit integro-differential equations relating voltages and currents at the device ports and is general enough to accommodate the vast majority of nonlinear device models of common use. However, some kinds of models such as spatially distributed nonlinear models described by partial differential equations, may not fall directly within the reach of (5.7) and may require some sort of approximation (such as the well-known ‘‘slicing’’ technique) for the methods discussed in this chapter to apply. The linear subnetwork has the usual frequency-domain representation Y( j␻ )V(␻ ) + I(␻ ) + N(␻ ) = 0

(5.8)

where V(␻ ) and I(␻ ) are vectors of voltage and current phasors; Y( j␻ ) is the linear subnetwork admittance matrix; and N(␻ ) is a vector of Norton equivalent current sources at the linear subnetwork ports. The set of complex HB errors at a generic IM product ⍀ k (k ∈ S + ) has the expression E k = Y( j ⍀ k )U k + W k + N(⍀ k )

(5.9)

where U k and W k are vectors of kth harmonics of u(t ) and w(t ). All vectors in (5.7)–(5.9) have a same size n d equal to the number of interconnection ports (device ports ). To avoid the use of negative frequencies, the nonlinear solving system is formulated in terms of vectors E and X H of real and imaginary parts of the HB errors (5.9) and of the SV harmonics, respectively. The size of these vectors is N T = n d (2N + 1), where N is the number of positive IM products to be taken into account in the analysis. To allow the existence of M-autonomous electrical regimes with prescribed free fundamentals, the circuit should make available M-independent degrees of freedom under the form of tuning parameters, which usually represent bias voltages and/or free parameters of the linear subnetwork. The solving system is thus written in the form E(X H , T, V␻ ) = 0

(5.10)

where T and V␻ are the M-vectors of the tuning parameters and of the free fundamentals, respectively. If T and V␻ are treated as unknowns, (5.10) is apparently a nonlinear system of N T real equations in N T + 2M real unknowns. However, as it was shown in the previous section, the phases of the electrical regime with respect to the free fundamentals are not determined

250


by (5.10), and may thus be arbitrarily selected. In the following this will be accomplished by setting to zero the phases of M selected SV-harmonics at M intermodulation products representing linearly independent combinations of the free fundamentals (reference harmonics ). In the harmonic state vector X H the real parts of the reference harmonics are replaced by their magnitudes and the imaginary parts are suppressed, thus generating a reduced state vector X that will be used throughout the chapter for autonomous HB analysis. The vector X only contains N T − M real quantities to be determined, and (5.10) is thus replaced by the system of N T equations in N T + M unknowns E(X, T, V␻ ) = 0

(5.11)

Two kinds of situations are most often encountered in practice: 1. V␻ is fixed and X, T must be determined (tuning problem ). 2. T is fixed and X, V␻ must be determined (analysis problem ). Both cases result in well-posed nonlinear systems of N T real equations in as many unknowns. Each system can be solved by a norm-reducing Newton iteration [5], with a set of unknowns simultaneously including the reduced state vector and the tuning parameters or the free fundamentals. Because of this hybrid set of unknowns, this will be referred to as a mixedmode Newton iteration as opposed to the purely harmonic Newton used for nonautonomous circuits [1]. Of course, mixed problems where M elements of T ∪ V␻ are fixed and the remaining M must be found, may also occur and can be solved in a similar way. For an efficient implementation of the Newton algorithm, it is of paramount importance that the Jacobian matrix of the HB errors with respect to the unknowns be computed exactly, rather than by numerical perturbations [1]. This simultaneously reduces the Jacobian evaluation cost and the number of Newton iterations required to solve (5.11), due to the increased accuracy of the derivatives. From (5.9) for k, s ∈ S + we get ∂W k ∂U k ∂E k = Y( j ⍀ k ) + ∂ Re [X s ] ∂ Re [X s ] ∂ Re [X s ] ∂W k ∂U k ∂E k = Y( j ⍀ k ) + ∂ Im [X s ] ∂ Im [X s ] ∂ Im [X s ] ∂N(⍀ k ) ∂E k ∂Y( j ⍀ k ) = Uk + ∂T ∂T ∂T

冋

∂Y( j␻ ) ∂N(␻ ) ∂E k = ki Uk + ∂␻ i ∂␻ ∂␻

册|

␻ = ⍀k

(5.12)

+ Y( j ⍀ k )

∂U k ∂W k + ∂␻ i ∂␻ i


251

The derivatives of Y and N with respect to both circuit parameters and frequency can be computed by adjoint-network techniques. The derivatives of the voltage and currents harmonics U k and W k are found in the following way [1]. We first introduce the Fourier expansions ∂u = ∂y m

p ∈ Sd

∑

C m , p exp ( j ⍀ p t )

∂u = ∂x D

∑

p ∈ Sd

C p exp ( j ⍀ p t )

∂w = ∂y m

p ∈ Sd

∑

D m , p exp ( j ⍀ p t )

∂w = ∂x D

∑

D p exp ( j ⍀ p t )

p ∈ Sd

D

(5.13)

D

where y 0 = x, y m = d mx /dt m (1 ≤ m ≤ n ), and S d is named the derivatives spectrum. S d is usually larger than the signal spectrum in order to make available all the harmonics that are required for the subsequent calculations. Note that (5.13) is real, so that C m , −p = C m* , p and the like. The derivatives of U k and W k with respect to the real and imaginary parts of the SV harmonics are expressed by [1] ∂U k = ∂ Re [X s ] ∂U k = ∂ Im [X s ] ∂W k = ∂ Re [X s ] ∂W k = ∂ Im [X s ]

n

∑ ( j ⍀ s )m[Fm , k − s + (−1)m Fm , k + s ]

m=0 n

∑ j ( j ⍀ s )m[Fm , k − s − (−1)m Fm , k + s ]

(5.14)

m=0 n

∑ ( j ⍀ s )m[G m , k − s + (−1)m G m , k + s ]

m=0 n

∑ j ( j ⍀ s )m[G m , k − s − (−1)m G m , k + s ]

m=0

where the F’s and G’s are simple linear combinations of the Fourier coefficients of (5.13): D

Fm , k − s = C m , k − s + ␦ 0m C k − s exp (−j ⍀ s TD ) D

G m , k − s = D m , k − s + ␦ 0m D k − s exp (−j ⍀ s TD )

(5.15)

252


In (5.15) ␦ is Kronecker’s symbol and TD is the diagonal matrix of the time delays. By definition, the amplitude of a generic reference harmonic, say, X R , is one of the entries of X, while its phase ⌽R is arbitrarily fixed. For this case we have ∂U k ∂U k ∂U k + sin ⌽R = cos ⌽R ∂ Re [X ] ∂ Im [X R ] ∂|XR | R

(5.16)

∂W k ∂W k ∂W k + sin ⌽R = cos ⌽R ∂ Re [X ] ∂ Im [X R ] ∂|XR | R Finally, the derivatives with respect to the free fundamentals are given by [6] n

∂U s =j ∑m ∂␻ i m=1

冤∑

k∈S

冤∑ ∑ 冤∑ 冤∑

−j

k∈S

k∈S

−j

k∈S

D

冥冥冥冥

k i Cs − k TD exp (−j ⍀ k TD )X k

n

∂W s =j m ∂␻ i m=1

k i ( j ⍀ k )m − 1C m , s − k X k

k i ( j ⍀ k )m − 1D m , s − k X k D

(5.17)

k i Ds − k TD exp (−j ⍀ k TD )X k

The computation of the derivatives of Y usually represents a major contribution to the overall analysis cost. In particular, finding ∂Y /∂␻ is more expensive than finding ∂Y /∂T, since the former involves all the reactive circuit components, and the latter only those belonging to the tuning parameters. Thus, in general tuning is a faster process than analysis. Much more so if all the tuning parameters are bias voltages, since in this case the third of (5.12) reduces to ∂E k 0 ∂N(0) = ␦k ∂T ∂T

(5.18)

The admittance then remains constant throughout the iteration and its derivatives are not required. These facts can be effectively exploited in


253

many important applications, such as the computation of the frequencyvoltage characteristic of VCOs [6]. Multitone autonomous problems can be treated by the mixed-mode Newton iteration introduced in this section in a most efficient way. This method is considerably faster than the continuation method with artificial embedding [7], because it directly leads to the solution with just one Newton loop, without requiring the generation of a solution path in the state space. To avoid the degenerate (nonoscillatory) solutions of (5.11), a suitable starting point for the Newton algorithm can always be found in a straightforward way by carrying out a preliminary quasi-Newton iteration, as discussed in Section 5.2.3. Alternatively, the analysis may be carried out by the substitution algorithm, which will be discussed in Section 5.4.2. Note that in many cases the initial-point problem does not exist, since the multitone analysis can be started from the results of an oscillator optimization (see Section 5.4). Generally speaking, the pattern of possible solutions of the system (5.11) may be extremely complicated, especially for M > 1. A general solution may require a global stability analysis of the parameterized circuit based on the principles of bifurcation analysis, as discussed in Section 5.3. 5.2.3 Degenerate Solutions and Their Suppression Let us consider a nonlinear circuit for which the HB system (5.11) admits a one-autonomous solution (X ss , T, V␻ ), where X ss is the steady-state reduced state vector, T is the only tuning parameter, and V␻ ≡ ␻ 1 . We can partition X ss in the following way:

冋册 (1)

X ss =

X ss

(R )

(5.19)

X ss

(1)

where X ss contains the magnitude of the reference harmonic and the real and imaginary parts of the steady-state harmonics for which the harmonic vector (5.3) has the form k1

冤冥 0

k=

... 0

(k 1 ≠ 0)

(5.20)

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Since the fundamental frequency ␻ 1 is autonomous, by definition the circuit does not contain any free sources at ␻ 1 nor at any of its harmonics. As an immediate consequence, the system (5.11) always admits at least one further solution of the form X ss =

冋册 0

(R )

(5.21)

X ss

that will be called a degenerate solution. From a physical viewpoint, the electrical regime described by (5.21) simply consists of the circuit response to the free sources (both dc and time-periodic) exciting the circuit. On the other hand, the electrical regime described by (5.19) consists of such responses superimposed on the free oscillation of fundamental frequency ␻ 1 . If the circuit only contains dc sources, a degenerate solution will also be called a dc or stationary solution or a bias point. The degenerate solution concept can be extended to the case M > 1 in an obvious way. If the system (5.11) is solved by a mixed-mode Newton iteration, whether or not the iteration will converge to an oscillatory regime depends on the choice of the initial values of X, T, and ␻ 1 , namely, X in , Tin , and ␻ in . It is thus clear that a systematic treatment of the oscillator analysis and tuning problems by the HB technique requires that a straightforward (and possibly cheap) procedure for finding such initial values be established. This result is pursued in this section. For the sake of formal simplicity, the discussion will be restricted to the oneautonomous case (V␻ ≡ ␻ 1 ). Let us consider the analysis problem first (fixed T, unknown ␻ 1 ). Reasonable starting values ␻ in and X in can be found in a straightforward way under very general conditions by the following algorithm. The degenerate solution (5.21) is first determined by ordinary HB analysis for forced circuits. The steady state is then perturbed by introducing in the circuit a sinusoidal current source of frequency ␻ in with an impressed current of suitable amplitude, connected in parallel to the load branch. When ␻ in is far away from the autonomous frequency of oscillation ␻ 1 , the circuit response to the perturbing source is also small and results in the appearance of a small load voltage component at frequency ␻ in , proportional to the source amplitude. On the other hand, when ␻ in falls within the locking bandwidth of the free oscillation, the oscillation is locked and the output voltage at ␻ in is close to the free oscillating value. Thus, by sweeping ␻ in across a suitable range, a fast increase of the output voltage at ␻ in is observed as ␻ in approaches ␻ 1 . The state associated with the voltage peak can be safely chosen as a starting point for the mixed-mode Newton iteration.


255

A less general but computationally faster algorithm can be used when the only free generators contained in the circuit are bias sources. In this case we may find the bias point and draw a Nyquist stability plot by ordinary linear circuit analysis (see Section 5.3.4). For ␻ ≥ 0, Nyquist’s plot crosses the real axis at (at least) one point at the left of the origin of the complex plane. The associated frequency value is chosen as ␻ in . To find an acceptable X in , the amplitude of the reference harmonic is then fixed to some relatively large value V in , usually on the order of the corresponding bias voltage, and the objective function F OB = || E(X, T, ␻ in ) ||

(5.22)

(where || • || denotes the Euclidean norm) is minimized with respect to the remaining entries of X by means of a standard optimization algorithm, such as a quasi-Newton method. In this procedure, all harmonics are started from zero except for the one being held fixed. The purpose of this calculation is not to reach the harmonic balance (which would be impossible), but just to move the state vector closer to the oscillatory solution in the state space. Experience shows (see example in Section 5.2.4) that a small number of iterations (usually on the order of 20) are sufficient to reach a point X in from which the mixed-mode Newton iteration can be safely started. The process is very cheap when the problem size (i.e., the number of scalar unknowns) is small-to-moderate: As an example, for the circuit of Figure 5.4 (27 unknowns with N = 4), the generation of X in in 20 iterations typically requires 0.05 CPU seconds on a 800-MHz PC. For large circuits such as entire front ends (many thousands of unknowns) carrying out a quasi-Newton iteration may be impractical (as is the use of continuation methods), because each iteration may be extremely expensive—so that their number must be minimized. In such cases, the alternative algorithm discussed in Section 5.6.3 may be preferable. The case of a tuning problem is easier to handle, because the frequency of oscillation is assigned. In this case a suitable initial state vector X in may be simply arrived at by minimizing (5.22) by the same procedure discussed above starting from a ‘‘typical’’ value of the tuning parameter, such as 50⍀ for a characteristic impedance or ␭ /4 at ␻ in for a transmission line length. As a final point, it is worth mentioning that the HB system (5.11) may be modified in a number of ways so that any degenerate solutions will be a priori suppressed. As an example, two of the HB equations may be replaced with the Kurokawa conditions for oscillation [8]. Another possibility is to modify (5.11) as follows:

256


E(X, T, V␻ )

|| X (1) ||

=0

(5.23)

where X (1) contains the real and imaginary parts of the SV harmonics for which the harmonic vector (5.3) takes on the form (5.20). The left-hand side of (5.23) approaches a nonzero limit as the oscillation tends to vanish (i.e., as X (1) → 0), so that the algorithm used to solve (5.23) is prevented from approaching a degenerate solution. Note, however, that these techniques normally do not improve the convergence of the Newton iteration, so that the above discussed search for a suitable starting point still remains necessary. For this reason, (5.23) will not be normally used in the rest of the chapter. 5.2.4 Applications To illustrate the analysis of a quasi-periodic steady-state regime by the mixedmode Newton iteration, we consider a self-oscillating mixer whose schematic topology is given in Figure 5.1. The circuit consists of a reflection-type DRO using a 150-␮ m FET as the active device. The nominal LO and radio frequencies are 8 GHz and 8.51 GHz, respectively. The required capacitive source feedback and inductive gate feedback are realized by an open stub and a loaded microstrip line coupled to a dielectric resonator, respectively. The same line is also used to input the RF signal. At the output, a lumped IF filter/matching circuit is fed through another microstrip line whose purpose is

Figure 5.1 Schematic topology of a dielectric-resonator tuned self-oscillating mixer.


257

to provide the optimum load reactance to the FET drain at the LO frequency. In turn, the IF matching circuit is designed in such a way as to load the FET drain by a real impedance of about 150⍀ at the nominal intermediate frequency of 510 MHz. Note that for the sake of the present example the circuit topology is regarded as a priori known. In reality, it is obtained by optimizing the LO alone (i.e., after suppressing the IF section lying beyond point L in Figure 5.1) for maximum drain current at a fundamental frequency of 8 GHz by the techniques discussed in Section 5.4, with eight harmonics taken into account. The circuit is first analyzed in the absence of RF signal (M = 1, F = 1) by the methods discussed in this section, and the drain voltage spectrum shown in Figure 5.2 is obtained. The observed frequency shift with respect to the nominal LO frequency is about +675 kHz, and is due to the small additional reactive loading of the FET drain introduced by the IF filter. An RF signal with available power −10 dBm at 8.51 GHz is then injected into the gate through the input microstrip, and the circuit is analyzed once again with M = 1 and F = 2, starting from the results of the previous step. The resulting drain voltage spectrum is shown in Figure 5.3. The additional LO frequency shift due to the injection of the RF signal is about −8 kHz. The transducer conversion gain of the self-oscillating mixer is about −3 dB without RF input matching. Because of the superlinear rate

Figure 5.2 Drain voltage spectrum of the local oscillator in the absence of RF signal.

258


Figure 5.3 Drain voltage spectrum of the self-oscillating mixer for an available RF input power of −10 dBm.

of convergence of the Newton iteration in the vicinity of the solution, finding the frequency of oscillation with very high accuracy requires a limited numerical effort. In both cases the unknown frequency is determined with a relative error smaller than 10−9, which requires 12 and 9 Newton iterations, respectively. This allows an excellent numerical control of the small frequency shifts that are typical of dielectric-resonator tuned oscillators and selfoscillating mixers. On an 800-MHz PC, the analysis of Figure 5.2 takes about 0.3 CPU second, 58% of which is spent for linear subnetwork computations. In turn, the multitone analysis of Figure 5.3 takes about 1.4 seconds, 65% of which is spent for linear subnetwork computations. It is thus clear that the nonlinear analysis algorithm is extremely efficient. For comparison, a time-domain analysis would be at least two orders of magnitude slower due to the high Q of the dielectric resonator (around 2,700 in the present case). To examine more in depth the degenerate solution problem, we now investigate the convergence properties of the mixed-mode Newton iteration as a function of the starting point for the VCO circuit shown in Figure 5.4. The reference harmonic is chosen as the fundamental component of the FET drain voltage. Figure 5.5 shows the boundary of the region of convergence of the Newton iteration in the (V in , f in ) plane when the analysis is carried out


Figure 5.4 Schematic topology of a VCO.

Figure 5.5 Region of convergence of the Newton iteration for the VCO.

259

260


by performing all the steps of the algorithms discussed in Section 5.2.3 ( f in = ␻ in /2␲ ). For reference, the exact frequency of oscillation f 1 = ␻ 1 /2␲ , the lowest value of f in obtained from the Nyquist analysis, the value determined by the method of the injected current, and the drain bias voltage V DS 0 (see Figure 5.4) are also reported in Figure 5.5. By letting V in = V DS 0 , the desired result is achieved without trials, since the initial values actually used by the algorithm lie well within the region of convergence. As a matter of fact, the tolerance of the analysis procedure with respect to the initial values is loose, since convergence is obtained for 0.83 f 0 ≤ f in ≤ 1.3 f 0 , and for 0.75 V DS 0 ≤ V in ≤ 1.33 V DS 0 in any combination. Qualitatively similar results and an equally reliable behavior of the analysis algorithm were obtained for several different oscillator topologies. It is thus clear that the method proposed in Section 5.2.3 offers consistent advantages over other approaches, which require either the generation of a solution path [7], or a considerable number of initial trials [8].

5.3 Synchronous and Asynchronous Stability This section discusses the numerical implementation of the fundamental principles of bifurcation theory in conjunction with HB-based nonlinear analysis algorithms, as a general-purpose tool for the synchronous and asynchronous stability analysis of nonlinear circuits. Section 5.4 shows that the bifurcation theory can be used not only as an analysis tool for determining the stability properties of a known circuit but also as a support to optimization techniques for including stability constraints among the design goals. The stability analysis methods discussed in this section are applicable to all the circuit topologies that fall within the reach of the HB technique. This is much more general than the classic dynamic analysis based on time-domain methods [9], which is normally limited to very low-order nonlinearities. In exchange for this, the frequency-domain description of the linear subnetwork limits the available waveforms to quasi-periodic regimes, which in turn allows only the fundamental bifurcations to be detected in this way. Some complex aspects of nonlinear circuit dynamics, especially those involving continuous spectra such as chaos, cannot be exactly described by HB. The important point, however, is that with few exceptions the methods discussed in this section are normally sufficient for ordinary engineering


261

purposes, and thus represent a precious complement to the ordinary HB CAD environment. Section 5.3.1 discusses the representation of solution paths in the state space (i.e., loci of steady states having similar spectra) for a parameterized nonlinear circuit. It is shown that HB-based analysis techniques are best suited for producing neat nonredundant descriptions, especially in the case of autonomous circuits for which the steady states are not uniquely defined. Section 5.3.1 also introduces a generalized continuation algorithm for the automatic construction of solution paths in the state space. Section 5.3.2 discusses the determination of the natural frequencies of quasi-periodic steady states, which provides the basic numerical tool for the stability analysis of general nonlinear circuits operating in steady-state conditions. The generation of Nyquist stability plots for time-periodic steady states is developed as a local stability analysis tool in Section 5.3.3. Both forced and autonomous states are considered for the sake of clarity, and the autonomy-related peculiarities are examined in detail. Section 5.3.4 pursues global stability analysis by searching known solution paths for the fundamental bifurcations. A peculiar feature of HB is that unlike time-domain analysis of nonlinear circuits, this method allows the determination of unstable steady states, which is of great help in the interpretation of complex stability patterns through bifurcation analysis. Section 5.3.5 uses the methods discussed earlier to solve a few elementary stability problems. Finally, Section 5.3.6 specifically addresses the important and difficult problem of spurious generation in oscillators as a relevant application example of the proposed methodology. 5.3.1 Solution Paths in a Harmonic Phasor Space 5.3.1.1 State-Space Representation

According to the discussion developed in Section 5.2, after a suitable finite spectrum of N positive frequencies plus dc has been defined, the real timedependent state vector x(t ) may be replaced by the vector X H of real and imaginary parts of the SV harmonics. The circuit states may then be represented making use of two different state spaces: 1. The ordinary phase space P, defined as an n d -dimensional real Euclidean space for which the coordinates are the instantaneous values of the SV [i.e., the entries of x(t )]; 2. The harmonic state space H, defined as an N T -dimensional real Euclidean space [N T = n d (2N + 1)] for which the coordinates are the entries of the harmonic state vector X H .

262


Let us now assume that the circuit supports an M-autonomous electrical regime of the kind introduced in Section 5.2.1. The phases of the autonomous oscillations are not determined by the HB equations and can thus be arbitrarily selected. This means that adding an arbitrary real quantity ⌬⌽i to the phase of each autonomous oscillation changes a solution into another solution of the HB equations. Two such M-autonomous steady states have the same amplitude spectrum, while the phases of a generic intermodulation product (5.2) differ by the quantity [6] M

⌬⌽ k =

∑ k i ⌬⌽i

(5.24)

i=1

where k is the harmonic vector (5.3). Thus, in the H -space the family of M-autonomous regimes related by (5.24) with 0 ≤ ⌬⌽i < 2␲ is represented by an M-dimensional manifold (e.g., a closed surface for M = 2 or a closed line for M = 1). In most practical cases only the amplitude spectra of the oscillations are of interest, so that the states belonging to the family (5.24) may be considered equivalent to each other. Also, the actual values that the ⌬⌽i ’s take on in a real circuit will depend on the oscillation buildup process (e.g., on the particular noise waveform by which the process is started right after the excitations are turned on), so that the occurrence of a specific state of the same family cannot be predicted or distinguished from the other ones. This justifies the convention of taking one (arbitrarily chosen) state of the family as representative of the whole family (5.24). In the following we shall adopt a special case of the convention introduced in Section 5.2.2—that is, we shall arbitrarily set to zero the phase of the harmonic of a suitably selected SV, say, x n i (t ), at each autonomous fundamental frequency ␻ i . This uniquely defines the reduced state vector X (see Section 5.2.2). The domain of X is a subspace RH of the harmonic state space H. We shall conventionally call RH the autonomous state space. The entire family of quasi-periodic regimes (5.24) is represented by a single isolated point in the state space RH, so that the use of RH is particularly well-suited for obtaining simple pictorial representations of the complex bifurcation diagrams generated by the occurrence of instabilities in parameterized self-oscillating circuits. 5.3.1.2 Numerical Construction of a Solution Path

We now assume that a given nonlinear circuit is parameterized by a single real parameter P, representing any physical or electrical quantity affecting the circuit performance. Throughout this section we shall assume that P is the only independent variable available in the circuit. Since by assumption the


263

tuning parameters T are not available, the HB solving system (5.11) describing the autonomous quasi-periodic steady states supported by the circuit may be cast in the form E(X, V␻ ; P ) = 0

(5.25)

where the free parameter has been put into evidence for later convenience. Equation (5.25) is a real system of N T real equations in as many unknowns (the entries of X, V␻ ). On the other hand, if the steady state of interest is forced, V␻ is a priori known, but all phases are uniquely determined, so that (5.25) is replaced by E(X H ; P ) = 0

(5.26)

where the unknowns are now the entries of X H . We are thus in a position to treat autonomous and forced steady states in a unified way by introducing an auxiliary state vector of dimension N T , defined by

XA =

冦

冋册 X V␻

for autonomous steady states

(5.27)

X H for nonautonomous (forced) steady states

so that the HB solving system may be cast in the unified form E(X A ; P ) = 0

(5.28)

By the implicit function theorem, (5.28) implicitly defines an equation of the form X A = X A (P )

(5.29)

representing a curve in the RH -space for autonomous steady states, or in the H -space for nonautonomous steady states. Such a curve will be named a solution path. As we will show in the rest of this chapter, the numerical construction of solution paths is an essential part of HB-based stability analysis of nonlinear circuits and thus deserves special attention. Building a solution path requires the computation of a continuous sequence of solutions of (5.28), which may represent a formidable task from the numerical viewpoint. It is thus of primary importance that the adopted nonlinear analysis

264


strategy provides the best available numerical efficiency. For this reason, the mixed-mode Newton iteration discussed in Section 5.2.2 represents an excellent choice for this task. A peculiar advantage of this approach is that any steady state can be used as a good starting point in the search for its next neighbors, so that the quadratic convergence of the Newton iteration in the vicinity of the solution can be most efficiently exploited in the construction of the solution path. Although this analysis technique provides unsurpassed computational efficiency, it leads to numerical ill-conditioning when regular turning points are encountered on the solution path. A regular turning point is defined as a regular point of the curve where the parameter reaches a relative minimum or maximum, so that

| |

DP =0 D XA

(5.30)

where the differential operator D denotes the derivative taken along the solution path. If we now differentiate (5.28) with respect to P, taking (5.29) into account, we get J A (X A ; P )

D X A ∂E + DP ∂P

|

=0

(5.31)

X A = constant

where J A is the Jacobian matrix J A (X A ; P ) =

∂E ∂X A

|

(5.32) P = constant

From (5.31) we then obtain ∂E D XA = −[ J A (X A ; P )]−1 DP ∂P

|

(5.33) X A = constant

Due to (5.30), at a regular turning point || D X A /DP || becomes infinite. On the other hand, the partial derivatives ∂E /∂P appearing on the righthand side of (5.33) are not taken along the solution path and are thus regular at the turning point, which implies that the Jacobian matrix (5.32) must be singular. Solving (5.28) with respect to X A by a Newton iteration is then


265

impossible at a turning point and is severely ill-conditioned in the close vicinity of it. In practice, if the solution path is generated by stepping the parameter and solving (5.28) with regards to X A for each value of P, the construction will stop before reaching the turning point due to failure of the Newton loop to achieve convergence. For a systematic and effective implementation of stability analysis in a general-purpose CAD environment (see Section 5.3.4) it is essential that solution paths can be efficiently generated with uniform accuracy and without requiring any manual intervention, irrespective of the presence of turning points. To reach this result, two kinds of actions must be taken: 1. In the vicinity of a turning point the solving system (5.28) must be replaced by an equivalent system free of numerical ill-conditioning. 2. A quantitative criterion must be implemented allowing the proximity of the singularity to be detected and the replacement to be carried out automatically. Regarding the first action necessary, the singularity of the Jacobian matrix (5.32) at a turning point originates from the fact that in its neighborhood the correspondence between the circuit state X A and the parameter P is not one-to-one. The correct behavior of the Newton-iteration based HB analysis may thus be restored by exchanging the roles of P and of one of the entries of the reduced state vector X, say, X P , in the vicinity of the turning point. We will denote by XA′ the vector generated by suppressing X P from the state vector X A , and by Z the modified state vector Z=

冋册 XA′ P

(5.34)

In the neighborhood of a turning point, the HB system (5.28) may be rewritten in the form E(Z; X P ) = 0

(5.35)

and the solution path may be generated by stepping X P and solving (5.35) with regards to Z by a Newton iteration for each value of X P . In principle the real or imaginary part (or the magnitude) of any SV harmonic is eligible as X P , provided that the Jacobian matrix

266


JZ =

∂E ∂Z

|

(5.36) X P = constant

is not singular at the turning point. This has the meaning of restoring the one-to-one correspondence between the points of the solution path and the auxiliary parameter X P . The implementation of the second condition requires a quantitative definition of the concept of neighborhood of a turning point. To ensure a uniform performance of the algorithm in general-purpose applications, such a definition should be as far as possible independent of the particular problem under consideration. Since the determinant of the Jacobian matrix (5.32) may change wildly as a function of the circuit topology and of the spectrum adopted in the HB analysis, it is not possible to establish in general whether the determinant is ‘‘close enough to zero.’’ In order to develop a ‘‘universal’’ switching criterion, we make use of the condition number of the Jacobian matrix, here defined as the ratio between the maximum and minimum magnitudes of the singular values, that is, [10]

␬ ( JA ) =

| S ( J A ) | max | S ( J A ) | min

(5.37)

where S(•) denotes a generic singular value of the matrix indicated in brackets. Note that ␬ = 1 for an identity matrix, while ␬ → ∞ for a singular matrix. Thus, ␬ ( J A ) is a normalized measure of the degree of ill-conditioning of the matrix J A . In the construction of the solution path we then use the formulation (5.28) of the HB equations when ␬ ( J A ) ≤ ␬ T and the formulation (5.35) when ␬ ( J A ) > ␬ T , where ␬ T is an empirically established threshold value. Excellent results have been obtained for a variety of circuits and spectra by taking ␬ T = 104 (see Section 5.3.5). The switching is operated automatically by the program at any time the condition number crosses the threshold. The derivatives of the parameter P and of the auxiliary parameter X P are continuously monitored during the generation of the solution path, so that the search can be restarted in the correct direction after a switching takes place. A similar criterion is also used to check that the auxiliary parameter X P has been properly chosen. The choice is considered acceptable if ␬ ( J Z ) ≤ ␬ T . If this condition is violated, the entries of the state vector X are tested in sequence until an acceptable choice is obtained.


267

5.3.2 Natural Frequencies of Quasi-Periodic Steady States We denote by x ss (t ) a generic quasi-periodic steady state of the form (5.1) supported by a nonlinear circuit and assume that x ss (t ) is perturbed by a small self-excited signal of the form exp [(␴ + j␻ )t ] where ␴ and ␻ are real. Since by assumption the perturbation is small, only IM products that are first-order with respect to the perturbation need be retained in the analysis, so that the perturbed state may be represented in the form x(t ) = x ss (t ) +

∑ ⌬X k exp {[␴ + j (␻ + ⍀ k )] t }

(5.38)

k∈S

In (5.38) each frequency of the form ␻ + ⍀ k will be conventionally named a sideband, and ⌬X k is an n d -vector containing the (complex) phasors of the perturbations superimposed on the kth steady-state harmonics. Note that the complex form of the HB equations will be retained throughout this section because of the transient nature of the analysis to be carried out. In transient conditions the advantage of avoiding negative frequencies is lost, and the higher simplicity of the complex formulation prevails. We now replace (5.38) into (5.7) and linearize the nonlinear subnetwork equations in the neighborhood of x ss (t ). We obtain a set of equations of the form ⌬Vk = ⌬I k =

∑ P k, s ⌬X s

(5.39)

s∈S

∑ Q k, s ⌬X s

s∈S

where ⌬Vk , and ⌬I k are n d -vectors containing the phasors of the perturbations superimposed on the kth steady-state voltage and current harmonics, respectively. Starting from (5.7) and making use of the expansions (5.13), we can readily find explicit expressions for the matrices P k, s and Q k, s : n

P k, s =

∑ [␴ + j (␻ + ⍀ s )]m C m , k − s + C k − s exp {−[␴ + j (␻ + ⍀ s )]TD }

m=0 n

Q k, s =

D

∑ [␴ + j (␻ + ⍀ s )]m D m , k − s + D k − s exp {−[␴ + j (␻ + ⍀ s )]TD } D

m=0

(5.40)

268


Note that the Fourier coefficients C and D are independent of (␴ + j␻ ). In addition, the perturbation must obviously satisfy the linear subnetwork equations at all sidebands. Such equations may be written in the form Y[␴ + j (␻ + ⍀ k )]⌬Vk + ⌬I k = 0

(5.41)

(∀k ∈ S) where Y( j␻ ) is the linear subnetwork admittance matrix. Combining (5.41) with (5.39) yields the homogeneous linear solving system

冦

∑ {Y[␴ + j (␻ + ⍀ k )]P k, s + Q k, s }⌬Xs = 0

s∈S

(5.42)

(∀k ∈ S)

which can be rewritten in the compact matrix notation R(␴ + j␻ )⌬X = 0

(5.43)

where the complex N T -vector ⌬X is the direct sum of the sideband perturbation vectors ⌬X s for all s ∈ S, and the submatrices of R(␴ + j␻ ) are given by R k, s = Y[␴ + j (␻ + ⍀ k )]P k, s + Q k, s

(5.44)

(∀k, s ∈ S) ⌬X may be interpreted as the spectrum of the perturbation superimposed on the circuit state. Note that (5.43) involves the sidebands only and is thus a complex linear system of N T equations in as many unknowns. The condition for the (autonomous) existence of the perturbation (i.e., ⌬X ≠ 0) is then D (␴ + j␻ ) = det [R(␴ + j␻ )] = 0

(5.45)

The solutions of (5.45) are the natural frequencies of the quasi-periodic steady state. A steady state is stable if all of its natural frequencies have negative real parts. A steady state having one (at least) positive real natural frequency (but no complex natural frequency with positive real part) will be said synchronously unstable. A steady state having one (at least) complex natural frequency with a positive real part will be said asynchronously unstable.


269

A stability analysis essentially consists of a search for the solutions of (5.45) having positive real parts, and for the physical conditions resulting in sign reversals of such real parts (bifurcations ). Considering that ⍀ −k = −⍀ k and that C m , −p = C m* , p (and the like), from (5.44) and (5.40), we get R k, s (␴ − j␻ ) = Y[␴ + j (−␻ + ⍀ k )]

冦冦∑ n

⭈

D

m=0 n

+

冧冧

∑ [␴ + j (−␻ + ⍀ s )]m C m , k − s + C k − s exp {−[␴ + j (−␻ + ⍀ s )]TD } D

[␴ + j (␻ + ⍀ s )]m D m , k − s + D k − s exp {−[␴ + j (−␻ + ⍀ s )]TD }

m=0

= Y[␴ + j (−␻ − ⍀ − k )]

冦∑ n

⭈

[␴ + j (−␻ − ⍀ − s )]m (C m , s − k )*

m=0 D

+ (C s − k )* exp {−[␴ + j (−␻ − ⍀ − s )]TD }

冦∑

冧

n

+

[␴ + j (−␻ − ⍀ − s )]m (D m , s − k )*

m=0 D

+ (D s − k )* exp {−[␴ + j (−␻ − ⍀ − s )]TD }

冧 (5.46)

Furthermore, assuming that the linear subnetwork is passive, its admittance matrix is positive real, so that (5.46) yields R k, s (␴ − j␻ ) = {Y[␴ + j (−␻ + ⍀ − k )]}*

冦∑ 冦∑ n

⭈

D

[␴ + j (␻ + ⍀ − s )]m C m , s − k + C s − k exp {−[␴ + j (␻ + ⍀ − s )]TD }

m=0 n

+

D

冧冧

[␴ + j (␻ + ⍀ − s )]m D m , s − k + D s − k exp {−[␴ + j (␻ + ⍀ − s )]TD }

m=0

*

*

= [R − k, − s (␴ + j␻ )]* (5.47)

270


Since the spectrum is symmetric with respect to the origin, changing the sign of both k and s in (5.43) is equivalent to reversing the order of the intermodulation products ⍀ k , which leaves the determinant unchanged. In conclusion, we have D (␴ − j␻ ) = [D (␴ + j␻ )]*

(5.48)

so that the natural frequencies either are real or occur in complex conjugate pairs. In particular, (5.48) implies that D (␴ ) is real. For reasons to be explained in the following section, the particular case ␴ + j␻ = 0 deserves some special attention. If we let ␴ + j␻ = 0 in (5.38) and make use of (5.1), the perturbed state takes on the expression x(t ) = x ss (t ) +

∑ ⌬X k exp ( j ⍀ k t ) = ∑ (X k + ⌬X k ) exp ( j ⍀ k t )

k∈S

k∈S

(5.49) Let us assume that the unperturbed steady state is not a bifurcation. If such a state is forced, its harmonics X k are uniquely determined by the HB system. Equation (5.49) then implies that the only possible solution of (5.42) is ⌬X k = 0 ∀k ∈ S, which in turn requires D (0) ≠ 0. However, the situation is different for an autonomous steady state. Indeed, in this case the phases of the autonomous oscillations are indeterminate, as was discussed in Section 5.3.1, which means that (5.42) has infinitely many solutions defined by ⌬X k = [exp ( j ⌬⌽ k ) − 1]X k

(5.50)

where ⌬⌽ k is given by (5.24). As a consequence, the determinant of the linear solving system (5.42) must vanish, so that for an autonomous steady state D (0) = 0. 5.3.3 Nyquist’s Analysis for Time-Periodic Steady States If the unperturbed steady state is time-periodic with fundamental angular frequency ␻ 1 , an important property of the determinant (5.45) is of considerable help in the stability analysis. Recalling that the coefficients C and D are independent of (␴ + j␻ ), by inspection of (5.40), we get


271

R k , s [␴ + j (␻ + ␻ 1 ) = Y[␴ + j (␻ + ␻ 1 + k␻ 1 )]

冦∑ n

⭈

[␴ + j (␻ + ␻ 1 + s␻ 1 )]m C m , k − s

m=0 D

冧

+ C k − s exp {−[␴ + j (␻ + ␻ 1 + s␻ 1 )]TD }

冦∑ n

+

[␴ + j (␻ + ␻ 1 + s␻ 1 )]m D m , k − s

m=0 D

+ D k − s exp {−[␴ + j (␻ + ␻ 1 + s␻ 1 )]TD }

冧

= R k + 1, s + 1 (␴ + j␻ ) (5.51) Increasing ␻ by ␻ 1 thus results in a shift of the submatrixes of R along the principal diagonal in such a way that R k , s is replaced by R k + 1, s + 1 . If the number of positive harmonics N is infinite, this shift leaves the matrix unchanged, so that D (␴ + j␻ ) is a periodic function of ␻ of period ␻ 1 . This behavior will be retained with some approximation if N is finite but sufficiently large. Due to such periodicity, each solution ␴ + j␻ of (5.45) is actually representative of a numerable family of natural frequencies expressed by ␴ + j␻ + jh␻ 1 , where h is an arbitrary integer. Combining this property with (5.48), we find that two solutions of the form ␴ ± j␻ 1 /2 represent the same natural frequency from a mathematical viewpoint and that D (␴ ± j␻ 1 /2) is real. For practical purposes, the most relevant consequence of the periodicity property is that Nyquist’s analysis becomes easily applicable to (5.45). In order to apply Nyquist’s criterion [11], the complex frequency ␴ + j␻ must be swept clockwise along a closed contour enclosing the entire right-hand half of the complex plane. This contour may consist of the j -axis (running from −j ∞ to +j ∞) ideally closed by a half-circle of infinite radius. The relevant information is then the number of turns around the origin made by the plot of D ( j␻ ) as ␻ is swept from −∞ to +∞. This number is normally infinite for infinite N, because D (␴ + j␻ ) then has an essential singularity at infinity owing to (5.40). This difficulty may be nicely overcome thanks to the periodicity of D ( j␻ ), since the plot can be simply built by sweeping ␻ across the finite range [−␻ 1 /2 ≤ ␻ ≤ ␻ 1 /2], so that the number of turns

272


within each period automatically becomes finite. In particular, the essential singularity at infinity produces a finite number of counterclockwise turns of D (␴ + j␻ ) around the origin in each interval of length ␻ 1 , in a way similar to a complex exponential function. Other properties of (5.42) allow for a simple formulation of the stability equation. If the linear subnetwork is passive, Y(␴ + j␻ ) is positive real and has no poles in the RHP. In addition, if circuit losses are properly taken into account, Y(␴ + j␻ ) is equally free from j -axis poles. The same is then true for D (␴ + j␻ ) due to (5.44) and (5.40), so that the Nyquist plot becomes a bounded closed curve. The possible occurrence of zeros of D (␴ + j␻ ) on the imaginary axis deserves some special attention. In a generic situation, that is, if we exclude the occurrence of particular cases (such as double zeros) requiring special circumstances to be verified, the condition D ( j␻ C ) = 0

(5.52)

(␻ C real) implies that the real part of one (at least) natural frequency undergoes a sign reversal as some circuit parameter is swept across a critical value for which (5.52) holds. In turn, this implies that the circuit stability portrait undergoes a qualitative change at criticality, such as turning from stable to unstable or the like. The only exception to this general rule is given by autonomous steady states, for which the condition D (0) = 0 always holds, as shown in Section 5.3.2. The origin of the complex plane thus automatically lies on the Nyquist plot of an autonomous steady state without implying a stability change. The steady states for which (5.52) holds (with the additional condition ␻ C ≠ 0 in the autonomous case) are called bifurcations. This name is intuitively explained by the fact that new branches of the solution path are originated (or terminated) at bifurcations, as discussed in Section 5.3.4. Under the assumption that the steady state under consideration is not a bifurcation, D ( j␻ ) has no j -axis zeros if the steady state is forced, and has a simple zero at ␻ = 0 if the steady state is autonomous. We are now in a position to state Nyquist’s stability criterion for a generic periodic steady state. If the steady state is forced, the equation takes on the form [12] NZ = NC + N∞

(5.53)

where: N Z = number of zeros of D(␴ + j␻ ) lying in the rectangular domain [␴ > 0, −␻ 1 /2 ≤ ␻ ≤ ␻ 1 /2];


273

N C = number of clockwise turns around the origin made by D( j␻ ) as ␻ is swept across the range [−␻ 1 /2 ≤ ␻ ≤ ␻ 1 /2], owing to the zeros of D(␴ + j␻ ) lying in the rectangular domain [␴ > 0, −␻ 1 /2 ≤ ␻ ≤ ␻ 1 /2]; N ∞ = number of counterclockwise turns around the origin made by D( j␻ ) as ␻ is swept across the range [−␻ 1 /2 ≤ ␻ ≤ ␻ 1 /2], owing to the essential singularity of D(␴ + j␻ ) at infinity. If the steady state is autonomous, the origin lies on the plot, so that (5.53) is replaced by

NZ = NC + N∞ −

1 2

(5.54)

In (5.53) and (5.54), N C represents the total phase change of D ( j␻ ) around the plot divided by 2␲ and may be easily obtained by inspection of the Nyquist plot, while N ∞ is unknown. However, (5.44) and (5.40) clearly show that the order of the essential singularity at infinity only depends on the size of the submatrixes R k , s and on the maximum order n of the derivatives appearing in the device equations (5.7). Thus, N ∞ is normally a constant for all the periodic steady states of a given circuit, and can be computed once for all by carrying out a Nyquist stability analysis for a state having a known N Z . As an example, such a state can be found by passivating the circuit (i.e., turning off the bias sources of all active devices), and reducing the amplitude(s) of the RF source(s) in such a way that the circuit behaves linearly. Since the Nyquist plot may be quite complicated for large circuits (i.e., N ∞ may be large), it is usually convenient to apply Nyquist’s criterion to the auxiliary function

冋

F (␴ + j␻ ) = exp −

册

2␲ N ∞ (␴ + j␻ ) D (␴ + j␻ ) ␻1

(5.55)

F (␴ + j␻ ) has obviously the same zeros as D (␴ + j␻ ), but for it N ∞ = 0, so that N C may be evaluated much more easily. When (5.55) is used, (5.53) and (5.54) are replaced by

274


NZ = NC NZ = NC −

(5.56) 1 2

(5.57)

for forced and autonomous periodic steady states, respectively. As a final point, we note that the application of the above technique to autonomous steady states may be somewhat delicate from the numerical standpoint. This happens because the computation of D ( j␻ ) is carried out by a complex numerical procedure involving several FFTs and a matrix decomposition. Due to the unavoidable numerical errors, this procedure may return for D (0) = F (0) a small but substantially random real number instead of zero. The automatic recognition of this numerical zero may then be difficult, because the order of magnitude of | D ( j␻ ) | varies wildly as a function of ␻ , of the number of harmonics, and of the circuit topology. A possible contrivance for circumventing this problem is to replace (5.55) in the autonomous case by

冋

F (␴ + j␻ ) = exp −

册

2␲ N ∞ (␴ + j␻ ) D (␴ + j␻ ) − D (0) (5.58) ␻1

The resulting Nyquist plot exactly contains the origin (to machine accuracy) without being significantly perturbed for ␻ ≠ 0 because D (0) is real and small. Generally speaking, it has been found that the aliasing errors introduced by the FFT give a major contribution to the uncertainty on the computed value of D (0), so that the accuracy increases together with the number of steady-state harmonics taken into account in the HB analysis. 5.3.4 Global Stability Analysis The techniques discussed in the previous sections allow stability to be examined in a local sense. However, a global stability analysis can be produced making use of bifurcation theory. To illustrate this procedure, we assume that the nonlinear circuit is parameterized by a single free parameter P ranging in some interval P 1 ≤ P ≤ P 2 . We also assume that a periodic solution path (F = 1) has been determined in the state space by sweeping P across the range of interest following the method discussed in Section 5.3.1. 5.3.4.1 Fundamental Bifurcations of a Periodic Solution Path

According to the definition introduced in Section 5.3.3, if the real part of one (at least) natural frequency changes sign as the parameter is swept across


275

some critical value, say, P C , the state X A (P C ) is a bifurcation of the solution path [9]. At a bifurcation (5.52) is satisfied for some critical value ␻ = ␻ C . Because of (5.48), the Nyquist stability plot consists of two identical halves having a mirror symmetry with respect to the real axis. We can thus assume without loss of generality that ␻ C satisfies the inequality 0 ≤ ␻ C ≤ ␻ 1 /2. For a generic circuit parameterized by a single real parameter, one of the following cases may then occur. 1. At criticality (P = P C ) one real natural frequency changes sign, or equivalently the steady state has one zero natural frequency, and (5.52) is satisfied with ␻ C = 0. Note, however, that according to the results established in Section 5.3.3, for an autonomous steady state D (0) is always zero, which requires the forced and autonomous cases to be treated separately. (a) In the forced case the real quantity D (0) changes sign at criticality, and D ( j␻ ) has a simple zero at ␻ = 0. The Nyquist stability plot makes a simple crossing of the origin as P is swept across P C , so that the number of turns around the origin changes by one. (b) In the autonomous case, D ( j␻ ) has a double zero in the origin of the complex plane at criticality, so that both D (0) = 0 and dD ( j␻ )/d␻ = 0 at ␻ = 0. Now, from (5.48) we can easily infer that for P ≠ P C the derivative dD ( j␻ )/d␻ is purely imaginary at ␻ = 0. We can thus conclude that dD ( j␻ )/d␻ changes from positive imaginary to negative imaginary (or conversely) as P is swept across the critical value. The number of turns of D ( j␻ ) around the origin thus changes by one [from k + 1/2 to k − 1/2 or conversely (k integer)] in spite of the fact that D (0) = 0 for any P. At criticality the Nyquist plot has a cusp in the origin, and is tangent to the real axis because dD ( j␻ )/d␻ = 0. If either condition 1(a) or 1(b) holds, we say that the steady state undergoes a direct-type or D-type bifurcation, normally resulting in the appearance of synchronous instability [9]. The following argument shows how the occurrence of a D-type bifurcation affects the solution path in the H -space (forced case) or in the RH -space (autonomous case). According to the discussion of Section 5.3.1, the HB solving system and the solution path equation may be formulated in a unified way by (5.28) and (5.29), respectively. If the steady state has a zero natural frequency at P = P C , a self-

276


sustained (small) synchronous perturbation of the form (5.38) with ␴ + j␻ = 0 may exist in the circuit. If we denote by ⌬X A the associated perturbation on the auxiliary state vector X A , the perturbed state vector must still satisfy (5.28), that is, E[X A (P C ) + ⌬X A ; P C ] = 0

(5.59)

Since || ⌬X A || +3 dBm). This task is relatively easy from a numerical viewpoint, so that the optimization converges smoothly in only three iterations. The fundamental output power and the tuning parameters T and U are plotted in Figure 5.31 as a function of the number of iterations. The oscillatory regime generated in this way (corresponding to point I in Figure 5.32) is then checked for stability. With the circuit parameterized by T = V P , the bifurcation diagram given in Figure 5.32 is obtained. H 1 and H 2 are supercritical Hopf bifurcations of the dc solution path. According to the results established in Section 5.3.4, the dc states lying between H 1 and H 2 are unstable, and all the states belonging to the periodic solution path are stable. Thus, I represents an acceptable starting point. At I the phase noise computed by (5.105) at 10 kHz offset is −42 dBc/Hz, and the matrix SS d (␻ 1 ) has one eigenvalue of magnitude 1.18. Starting from I, a single-frequency optimization is then carried out at center-band (␻ 1 /2␲ = 5 GHz) with nominal specifications on output power and phase noise. This optimization converges in about 200 iterations and takes about 30 seconds on an 800-MHz PC. Figure 5.33 plots the fundamental output power versus the number of iterations, showing that 150 iterations are sufficient to meet the design goal. Figure 5.34 shows the evolution of the phase noise at 10 kHz offset from the carrier, of the function (5.104) with w = 0.1, and of the largest eigenvalue of SS d (␻ 1 ) (in magnitude) along this optimization.

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Figure 5.31 Performance of the optimization used to generate a starting-point oscillatory regime.

Figure 5.32 Bifurcation diagram for the VCO, after generation of the starting-point oscillatory regime (represented by point I) by the substitution algorithm.


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Figure 5.33 Fundamental output power versus number of iterations during the singlefrequency optimization.

Figure 5.34 Phase noise, function (5.104), and magnitude of the largest eigenvalue of SS d (␻ 1 ) for the VCO, during the single-frequency optimization.

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Both the phase noise and the condition number of the Jacobian decrease and remain finite, and the magnitude of the eigenvalue remains larger than 1, so that the final point F is expected to be stable and to fulfill the oscillation buildup condition. This is confirmed by the bifurcation diagram shown in Figure 5.35. Oscillation buildup is guaranteed since the dc state corresponding to point F is unstable. Figure 5.36 shows the VCO tuning characteristic and output power as a function of frequency, resulting from the singlefrequency optimization. As could be expected, the available tuning band is far narrower than the specification, and the linearity is relatively poor. For the broadband design, R = 9 values of the fundamental frequency of oscillation, uniformly spaced across the nominal band, are selected. Before starting the final optimization, a sequence of broadbanding steps is carried out as discussed in Section 5.4.1. The purpose of these computations is to find a circuit topology that supports an oscillatory regime across the entire nominal band. At each intermediate step, the band is defined as the set of all the fundamental frequencies falling inside the band of oscillation obtained at the previous step. A broadband optimization limited to this portion of the nominal band is then carried out, with specifications on the fundamental output power (P out > +10 dBm) and on the tuning characteristic only. The specification on the tuning characteristic is always of the form (5.91), but the requested tuning sensitivity (MHz/V) is gradually increased from step to step, to generate a broader band of oscillation for the same varactor tuning

Figure 5.35 Bifurcation diagram for the VCO after a single-frequency optimization at 5 GHz, leading to the oscillatory steady state represented by point F.


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Figure 5.36 VCO tuning characteristic and output power after the single-frequency optimization at 5 GHz.

range at the end of each optimization. This approach works nicely, and produces the results plotted in Figures 5.37 and 5.38, showing that the band of oscillation is gradually increased at each step until it reaches the nominal value. The overall cost of the broadbanding steps is about 70 CPU seconds. The broadband design may now be carried out with nominal specifications across the entire tuning band. In order to specify the tuning linearity, a goal of the form (5.91) is added with ⑀ = 0.01. The parameters A and B in (5.91) are left free, as discussed in Section 5.4.1. In addition, the tuning parameter range is limited by a specification of the form (5.92) with T L = −25V and T U = 0. The broadband optimization converges in about 350 seconds on an 800-MHz PC. The final tuning characteristic and the percent deviation from the best-fit straight line are plotted in Figure 5.39 across the prescribed band. Figures 5.40 and 5.41 show the output power and the bifurcation diagram for the VCO parameterized by T. The oscillatory regime appears to be stable across the entire nominal tuning range (corresponding to the shaded area). Also, all the dc states within the shaded area lie between the supercritical Hopf bifurcations H 1 and H 2 and are thus unstable because of two complex-conjugate natural frequencies with positive real parts (see Section 5.3.4). This guarantees oscillation buildup from the dc bias point at any frequency of the tuning band.

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Figure 5.37 Evolution of the VCO tuning characteristic throughout the broadbanding process.

Figure 5.38 Evolution of the VCO output power throughout the broadbanding process.

As a final performance check, we now apply a digital signal consisting of an NRZ periodic sequence of 512 bits with a bit rate of 100 Kbps to the tuning input of the VCO biased for oscillation at center band. Each bit in the sequence is treated as a statistically independent random variable that may take on the values ± 1V, corresponding to frequency deviations of ± 25 MHz. The circuit is then analyzed by advanced techniques for digitally


335

Figure 5.39 VCO tuning characteristic and percent deviation from linearity after the broadband optimization.

Figure 5.40 Fundamental output power after the broadband optimization.

modulated autonomous circuits [20]. Figure 5.42 compares the actual signal spectrum at the VCO output with the theoretical spectrum of a binary CPFSK signal generated by an ideal modulator fed by an infinite sequence of bits. These results show that the optimized VCO can provide an excellent performance as a linear frequency modulator.

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Figure 5.41 Bifurcation diagram for the VCO parameterized by the varactor bias voltage.

Figure 5.42 Normalized output spectrum of the VCO used as a frequency modulator, compared with the spectrum of a theoretical CPFSK signal (f = frequency offset from the carrier, T = bit interval).


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5.4.7 Oscillator Design for Asynchronous Stability Oscillator design for asynchronous stability is a difficult task. According to the discussion of Section 5.3.6, a spurious oscillation starts to build up when a Hopf bifurcation is encountered on the periodic solution path of the circuit parametrized by an arbitrary free parameter. The techniques for the automatic detection of such bifurcations discussed in Section 5.3 are well suited for analysis purposes but are not fast enough to be used inside an optimization loop. Also, it is difficult to express in the form of a minimization constraint the requirement that no Hopf bifurcations exist on the periodic solution path. Thus, the spurious-free design of nonlinear microwave circuits requires a specific treatment. An interactive solution to this problem is developed in this section. The rationale of the approach is as follows. The substitution principle introduced in Section 5.4.2 is extended to develop an approximate but accurate technique for efficiently locating the Hopf bifurcations of the periodic solution path. This technique provides the basis of a predictor-corrector scheme that can efficiently generate Hopf bifurcations loci in an arbitrary twodimensional parameter space. In turn, this enables the designer to locate regions of the space of designable circuit parameters where spurious-free operation is likely to take place. A conventional optimization with the variable ranges constrained to such regions then leads to a spurious-free design. If necessary, the process can be iterated to find the best tradeoff between the stability requirement and the electrical specifications. The design of a stable varactor-tuned broadband VCO will be discussed for illustrative purposes. 5.4.7.1 Efficient Construction of Hopf Bifurcations Loci

To approximately detect a Hopf bifurcation of the periodic solution path, the following technique can be used. In the state space RH, a quasi-periodic solution path bifurcates from the periodic solution path at a Hopf bifurcation. Each steady state belonging to such a path is two-autonomous, with lines at all the intermodulation products of the two free fundamentals ␻ 1 and ␻ 2 , the latter representing the spurious oscillation. In such conditions, we have two reference harmonics, X R 1 and X R 2 , whose amplitudes will be denoted by A 1 and A 2 , and two tuning parameters, T1 and T 2 (see Section 5.2.2). In turn, the reduced state vector X (of dimension N T − 2) contains the real and imaginary parts of all the SV harmonics (except for X R 1 and X R 2 ) and the amplitudes A 1 and A 2 . For our present purposes, we parameterize the circuit by the tuning parameters T1 and T 2 , which thus play the role of independent variables. We further suppress the entry A 2 from X, and denote

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by X S the (N T − 3)-vector generated in this way. The autonomous system (5.11) may then be cast in the form E(X S , A 2 , ␻ 1 , ␻ 2 ; T1 , T 2 ) = 0

(5.112)

According to the substitution principle introduced in Section 5.4.2, we now interchange the roles of the amplitude A 2 and of the free parameters T1 and T 2 , in the sense that we treat the former as a known quantity, and the latter as unknowns to be determined. Equation (5.112) is thus changed into the system of N T equations in N T + 1 unknowns E(X S , ␻ 1 , ␻ 2 , T1 , T 2 ; A 2 ) = 0

(5.113)

By the implicit function theorem, for any fixed value of A 2 , (5.113) implicitly defines an equation of the form F (T1 , T 2 ; A 2 ) = 0

(5.114)

representing a curve parametrized by A 2 in the T1 − T 2 space. When the circuit state approaches the Hopf bifurcation on the quasi-periodic solution path, then simultaneously ␻ 2 → ␻ 2C and A 2 → 0, where ␻ 2C is the value of the spurious angular frequency at criticality. Thus, in principle, the Hopf bifurcation could be located by a single Newton iteration by solving (5.113) for X S , ␻ 1 , and ␻ 2C , and either T1 or T 2 with A 2 set to zero. This is practically impossible, however, because the Jacobian matrix of (5.113) is singular at A 2 = 0, so that the method can only be used with A 2 set to some near-zero but finite value A m . The tradeoff is that instead of finding the Hopf bifurcation, we find a state of the quasi-periodic solution path very close to it. Nevertheless, very accurate estimates of the bifurcating values of the quantities of interest may be normally found in this way. As an example, if A 2 is an FET drain voltage it has been found that letting A 2 = 10−5V still ensures excellent convergence, and allows ␻ 2C to be evaluated with a relative error typically better than 10−4. By letting A 2 = A m , (5.114) then becomes the (approximate) equation of a Hopf bifurcation locus in the twodimensional parameter space T1 − T 2 . The approximate technique discussed earlier can only perform a local search for the Hopf bifurcation, in the sense that the Newton iteration for (5.113) will only converge if the starting point is close enough to the bifurcation. As such, this method is not a substitute for the global search algorithm discussed in Section 5.3.4, which can locate all the Hopf bifurca-


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tions of a parametrized circuit even if no starting-point information is available. However, the local algorithm is ideal for efficiently generating Hopf bifurcations loci by continuation [7] in a multidimensional parameter space, starting from any of the bifurcations located by the global algorithm. This technique is discussed in detail next. A similar method is obviously usable in the simpler case of Hopf bifurcations of dc states. At each point of the locus, one of the two parameters actually plays the role of the independent variable and is suitably stepped to generate the next point. The system (5.113) is then solved either for the set of unknowns Y 1 = [X S , ␻ 1 , ␻ 2 , T1 ] (if T 2 acts as the free parameter), or for Y 2 = [X S , ␻ 1 , ␻ 2 , T 2 ] (if the reverse is true). The decision as to which parameter should be stepped is taken on the basis of the derivative DT i /DT j (i , j = 1,2), which can be evaluated from (5.113) as ∂E DT i = −L[ J(Yi )]−1 DT j ∂T j

|

(5.115) Yi = constant

where the symbol D indicates that the derivative is taken along the locus, and L is the row matrix [0 0 0 . . . 1]. The Jacobian matrix J(Yi ) is defined by J(Yi ) =

∂E ∂Yi

|

(5.116) T j = constant

In practice, T j is chosen as the independent parameter if the magnitude of (5.115) does not exceed a specified threshold. Otherwise, the roles of the two parameters are interchanged. In this way the algorithm can automatically overcome any turning point that may be encountered on the locus (5.114). Once the independent parameter has been selected (say, T j ), the next point of the locus can be efficiently found by a simple predictor-corrector scheme. In the predictor step, the increment of Yi corresponding to an increment ␦ T j of the free parameter is estimated by application of the implicit function theorem to (5.113):

␦ Yi ≈ −␦ T j [ J(Yi )]−1

∂E ∂T j

|

(5.117) Yi = constant

The corrector step is just the solution of (5.113) by a Newton iteration starting from the point defined by (5.117). Note that the predictor step is

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virtually costless (except at those points where a parameter switching takes place), since the factorized Jacobian matrix is automatically available after performing the corrector step for the previous point. The generation of a two-dimensional locus by this technique typically takes a few tens of seconds on an 800-MHz PC. 5.4.7.2 Spurious-Free Design

For the sake of clarity, the design procedure will now be illustrated by a case study. We consider once again the VCO design problem discussed in Section 5.4.6 (also see Section 5.3.6). The VCO topology is given in Figure 5.30, and the design specifications are the same, with the added requirement that the VCO should not generate any spurious oscillation. The purpose of the reactance-compensating network introduced on the gate (see Figure 5.30) is to provide the frequency dependence of the feedback reactance required for the linearization of the tuning characteristic. The resistor R is added for out-of-band stabilization. As in many broadband circuits, multiple resonances might occur owing to the relative complexity of the circuit topology, leading to the possible buildup of spurious oscillations. The detection and elimination of such spurious tones is an important aspect of the design problem. A global stability analysis of the circuit under consideration has been carried out in Section 5.3.6. The main result is that, indeed, the VCO exhibits one spurious periodic solution path (H 4 S 2 H 3 in Figure 5.20), which is in connection with the existence of a Hopf bifurcation S 1 on the nominal periodic solution path H 1 S 1 H 2 (see Figure 5.20). What is perhaps more important, the VCO exhibits a hysteresis cycle and is thus bistable in the voltage range V 1 ≤ V P ≤ V 2 , where V P is the intrinsic varactor bias voltage (see Figure 5.20). This means that if the drain bias is turned on with the varactor biased within this range, either the nominal or the spurious oscillation will build up with equal probability, depending on the noise waveforms that actually excite the oscillation start-up. The VCO behavior as described above is obviously not acceptable, and a further design step must be carried out to suppress any kind of asynchronous instability. For this purpose, a number of two-dimensional Hopf bifurcations loci may be built to visualize the dependence of the spurious generation mechanism on selected circuit parameters. The most interesting result is given in Figure 5.43. In Figure 5.43 one of the parameters is the varactor bias voltage, which implicitly carries the information on the VCO tuning range, and the other one is the resistance R (see Figure 5.30). Figure 5.43 simultaneously shows the loci of the four primary Hopf bifurcations H 1 , H 2 , H 3 , and H 4 , and of the two secondary Hopf bifurcations S 1 and S 2 .


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Figure 5.43 Primary and secondary Hopf bifurcations loci for the broadband VCO.

The generation of these loci takes about 250 seconds on an 800-MHz PC. Figure 5.43 provides some interesting design information. Both the secondary Hopf bifurcations and the spurious periodic branch are seen to completely disappear when R exceeds a threshold value R T ≈ 135⍀. The preselected value R 0 = 50⍀ falls well within the region where spurious generation takes place. Figure 5.44 shows an expanded view of the primary Hopf bifurcations loci. The spurious solution is found to disappear for R T 1 < R < R T 2 , with R T 1 ≈ 120⍀ and R T 2 ≈ 374⍀. Of course, simply setting R to a value within that range would considerably degrade the remaining aspects of the circuit performance. We thus carry out a second design step consisting of a new optimization starting from the final point of the previous one, but with the added constraint R T 1 < R < R T 2 . The final optimization takes about 140 seconds on an 800-MHz PC and meets all the specifications. The resulting bifurcation diagram given in Figure 5.45 describes a very well-behaved tunable oscillator, completely free of instability phenomena of any kind. The conclusion that can be drawn from the results presented in this section is that the stability analysis techniques discussed in Section 5.3 and the oscillator optimization methods presented in Sections 5.4.1 and 5.4.2 can be effectively combined to include the requirement that the oscillator be spurious-free. The proposed approach is an interactive technique based on the alternate use of optimization and numerically generated Hopf bifurcations

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Figure 5.44 Primary Hopf bifurcations loci for the broadband VCO.

Figure 5.45 Bifurcation diagram for the VCO, after final broadband optimization.


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loci. The latter are found to represent a very powerful engineering tool, providing a synthetic overview of the influence of selected parameters on the entire oscillator bifurcation pattern. This allows the spurious generation process to be understood by the circuit designer, and to be prevented by intelligent interventions that would be fatally beyond the reach of any optimization method. For typical microwave oscillators the generation of twoparameter Hopf bifurcations loci by the substitution algorithm is usually much faster than a conventional optimization, so that this design technique is numerically efficient as well.

5.5 Electromagnetics-Based Optimization of Microwave Oscillators The optimization of microwave oscillators based on standard HB circuit analysis techniques has been treated in detail in Section 5.4. However, in many cases the circuit layout and/or the frequency range of operation may be such that a meaningful simulation of the linear part of the circuit can only be obtained by electromagnetic analysis based on the numerical solution of Maxwell’s equations. A direct optimization may then become practically impossible, due to the large number of CPU-time intensive electromagnetic simulations that are required to compute the objective function and its gradient. The key to electromagnetic-based oscillator design is thus the development of clever algorithms allowing the overall number of electromagnetic analyses to be effectively minimized. An important family of methods whereby this result may be achieved relies upon the space-mapping concept [21–23]. This approach has been widely demonstrated in the optimization of both linear and forced nonlinear microwave circuits [21–24]. The method presented in this section is conceptually related to space-mapping but takes advantage of the peculiarities of the oscillator design problem to improve the numerical conditioning of the design process. Space-mapping establishes a map between the electromagnetic model (layout) parameters and those of an ordinary circuit model and solves the optimization problem by inverting the map, in such a way that all optimizations are performed in the circuit model domain [21]. A significant difficulty encountered in the application of this principle to nonlinear circuits [24] is that the map is normally evaluated by numerical optimization with respect to the circuit model parameters, and is thus rather fuzzy. In particular, it is not necessarily single-valued and may thus not be locally invertible. With the technique discussed here, a preliminary optimization of the circuit model

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is carried out, as with the space-mapping, but its results are used to directly convert the oscillator optimization problem into the solution of a nonlinear system. The latter is generated by adding to the ordinary set (5.78) of HB equations a suitable set of additional equations introducing the design goals. All such equations are then simultaneously solved by a norm-reducing Newton iteration, so that no further optimizations are required during the search for the layout parameters. For this reason, the method will be referred to as direct-Newton optimization. This approach is equivalent to dealing with a sharply defined, locally invertible map and leads to a well-conditioned and considerably more efficient nonlinear design process. The approach is demonstrated by the design of an MMIC oscillator, showing an order-of-magnitude reduction of the total number of electromagnetic simulations with respect to an ordinary optimization. It is also shown that even for simple microstrip topologies the electromagnetic-based design may lead to significantly different circuit performance with respect to a conventional circuit model-based approach. 5.5.1 Direct-Newton Optimization This section assumes that the linear subnetwork is described by its layout [possibly including the active device(s) metallizations], and is analyzed as a whole by electromagnetic methods. For the sake of simplicity, we shall also assume that the tuning parameter T is a bias voltage, and that the remaining designable parameters (i.e., the entries of the vector P) are layout geometrical parameters, for example, as shown in Figure 5.46. The number of such parameters will be denoted by n L . In the solving system (5.78), the HB errors then depend on P exclusively through the linear subnetwork admittance parameters. For later convenience, this dependence will be explicitly put into evidence by introducing a real vector Y(P) containing the real and imaginary parts of the linear subnetwork admittance parameters at all harmonics of interest. The HB system (5.78) is then rewritten in the form E[X T ; ␻ 1 , Y(P)] = 0

(5.118)

The computation of Y(P) for a given layout requires a sequence of N electromagnetic simulations, where N is the number of positive frequencies of the spectrum. At dc, only the off-chip bias resistors and (if any) the lumped resistors integrated on-chip are considered, while the rest of the layout is approximated by a set of short circuits. The dc admittance parameters may thus be computed by ordinary circuit algebra. As usual, (5.118) is a


345

Figure 5.46 Schematic circuit layout for a two-port microstrip oscillator.

system of N T = n d (2N + 1) equations in as many unknowns, and the free fundamental ␻ 1 is fixed as a design specification. To develop a unified vector notation, the design goals are formulated by imposing that the network functions of interest are greater than or equal to a set of specified values. This set is denoted by S, and the number of specifications (i.e., the dimension of S ) is denoted by n S . However, in this case we do not solve the system (5.118) prior to each objective function evaluation, so that the network functions depend directly on both X T and P. The design goals can thus be synthetically cast in the form F[X T ; ␻ 1 , Y(P)] ≥ S

(5.119)

where F in an n S -vector of network functions. In the following we shall assume that the number of design parameters is such that n S ≤ n L . As a preliminary step of the design procedure, a simplified circuit model consisting of interconnected components is adopted for the linear subnetwork. The corresponding set of designable parameters (of size n C ) is denoted by P C , and the associated state vector obtained by solving (5.118) by X TC . The number n C of designable circuit parameters is completely arbitrary and is not related to n L . The oscillator is then optimized with

346


respect to P C for the set of specifications (5.119) (with X T and P replaced by X TC and P C ) by the methods discussed in Section 5.4. Now let a solution of the problem be defined by the vectors X TCopt and P Copt , satisfying both (5.118) and (5.119), and the corresponding set of network function values be denoted by S opt . The primary purpose of this initial step is to provide the designer with a general feeling of the problem conditioning, and to generate some basic information to be utilized in the subsequent layout design. Such information consists of a reasonable initial guess (obtained from P Copt ) for the circuit layout, and a feasibility check of the design goals through the identification of a realizable set (S opt ) of network function values. The key idea underlying direct-Newton optimization is to make use of the information generated by the preliminary step to allow the formulation of the layout design task as a system-solving problem rather than an optimization. Indeed, since by definition S opt ≥ S, any solution of the nonlinear system of (N T + n S ) equations in (N T + n L ) unknowns

再

E[X T ; ␻ 1 , Y(P)] = 0 F[X T ; ␻ 1 , Y(P)] = S opt

(5.120)

satisfies all the specifications and thus defines an acceptable design. Since system solving is much more efficient than optimization, this normally leads to a dramatic reduction of the required number of iterations and thus of the overall number of expensive electromagnetic analyses. Furthermore, assuming that one of the network functions specified in the second of (5.120) is the oscillator output power, (5.120) can only be satisfied by a nondegenerate (oscillatory) solution of the HB equations. Thus, the degenerate (dc) solution(s) of (5.118) are not solutions of (5.120). Note that the second of (5.120) is a more stringent specification than (5.119), which implies that some degrees of freedom might be lost if the design process were based on a conventional solution of (5.120) rather than on (5.118) and (5.119). In order to overcome this problem, such degrees of freedom are restored by dynamically updating the specifications during the solution of (5.120) by a norm-reducing Newton method, in the way discussed below. The solving system (5.120) can now be rewritten in the compact form W(Z) = 0 where

(5.121)


W= Z=

冋冋册

E[X T ; ␻ 1 , Y(P)] F[X T ; ␻ 1 , Y(P)] − S opt

347

册

(5.122)

冥

(5.123)

XT P

The Jacobian matrix of (5.121) has the form

J(Z) =

∂W = ∂Z

冤

∂E ∂X T

∂E ∂Y ∂Y ∂P

∂F ∂X T

∂F ∂Y ∂Y ∂P

The Jacobian matrix ∂E/∂X T of the HB system can be computed analytically by the algorithms discussed in Section 5.2.2. The dependence of the HB errors on the Y-parameters is explicitly given by (5.9), and the network functions are normally expressed in terms of the entries of X T and Y by simple algebraic equations. Thus, the only delicate step in the computation of (5.123) is the evaluation of ∂Y/∂P by numerical perturbation based on electromagnetic analysis. To avoid numerical discontinuities that might cripple the accuracy of the derivatives, the grid used in the electromagnetic simulation must be kept constant in the perturbation process. Thus, the parameter to be perturbed, say, P k (1 ≤ k ≤ n L ), is simply changed by one grid step ⌬, and the corresponding change of Y is computed as

␦ Y = Y(P + Uk ⌬) − Y(P)

(5.124)

where Uk is an n L -vector whose k th entry is equal to 1, while the remaining ones are zero. The k th column of ∂Y/∂P is then approximated by ␦ Y/⌬. Now let the generic (i th) iterate in the unknown vector Z be denoted by Z (i ). By perturbing (5.121) in the neighborhood of Z (i ) and imposing that the perturbed vector W equals 0, we obtain the update formula J(Z (i ) ) [Z (i + 1) − Z (i ) ] = −W[Z (i ) ] X TCopt Z (0) = P Copt

冦冋册

(5.125)

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The Jacobian matrix J(Z) has dimensions (N T + n S ) × (N T + n L ), and is thus normally rectangular because n S ≤ n L . This means that only n S layout parameters are determined by the update formula (5.125), while the remaining n F = n L − n S are free. We may now partition Z (i ) in the form

冤冥冋 (i )

XT

Z

(i )

(i )

=

⌬

=

(i ) PS (i ) PF

(i )

ZS

(i )

PF

册

(5.126)

(i )

where the subvectors P S , P F have dimensions n S , n F , respectively. Similarly, the Jacobian matrix may be partitioned in the form J[Z ] = [ J S | J F ] (i )

(i )

(i )

(5.127)

(i )

(i )

where J S is square of dimensions (N T + n S ) × (N T + n S ), and J F is rectangular of dimensions (N T + n S ) × (N T + n F ). By replacing (5.126) and (5.127) into (5.125) and adopting a norm-reduction scheme [1], we finally obtain the explicit update formula

再

(i + 1)

PF

= PF + A

(i )

(i )

ZS

= ZS − ␣

(i )

(i )

(i + 1)

[ J S ]−1 {W[Z ] + J F A } (i )

(i )

(i ) (i )

(5.128)

where A (i ) is an arbitrary n F -vector and ␣ (i ) is a scalar parameter. In a conventional norm-reducing iteration, ␣ (i ) would be determined by a onedimensional search in such a way as to minimize the norm of the error vector. With direct-Newton optimization, we take advantage of the onedimensional search step to allow individual network function values to depart from S opt , provided that the corresponding original specifications (5.119) are satisfied. To do so, a reduced network function vector F R is dynamically generated by suppressing from F all the network functions satisfying (5.119) with the ‘‘greater than’’ sign. The quantity to be minimized by the onedimensional search is then defined as

√|| E ||

2

+ || F R ||

2

(5.129)

At the end of the search, the network functions not belonging to F R will take on a set of values S v satisfying (5.119). Such values are replaced


349

into S opt before starting the next Newton iteration. Finally, the degrees of freedom introduced by A (i ) may be used to fulfill a number of layout constraints, such as the requirement that the chip dimensions do not exceed prescribed bounds. Such constraints normally have mathematical formulations of the kind

∑ ak P k ≤ b

(5.130)

k

where the a ’s and b ’s are constants. Equations (5.128) through (5.130) can be immediately converted into bounds on the entries of A (i ) and on ␣ (i ), so that this kind of specification can be addressed in a most straightforward and efficient way. The same degrees of freedom may also be of help for improving the convergence of the algorithm. 5.5.2 Applications As an illustrative example, we consider the design of a microstrip oscillator with the layout shown in Figure 5.46. This topology was purposefully chosen to emphasize the fact that proximity effects may significantly modify the circuit performance with respect to the prediction of ordinary circuit models even in a simple stub layout. The FET is a five-finger 300-␮ m device manufactured by Alenia-Marconi Systems, for which a very accurate largesignal model is available [25]. The gate, source, and drain pads are incorporated in the layout, but the electrode fingers are not, as shown in Figure 5.46, so that only the intrinsic FET model and the small parasitics introduced by the fingers need be added to the layout in order to analyze the circuit. The intrinsic model inclusive of the electrode fingers is obtained by the identification procedure discussed in [25]. The starting information consists of data obtained from measurements by de-embedding the Y-matrix of the FET electrode metallizations (except for the fingers), computed by electromagnetic analysis. This procedure avoids the need for including the fine details of the FET electrode layout in the electromagnetic analysis of the oscillator. The grid size can thus be increased from about 0.1 ␮ m to 2 ␮ m with no appreciable loss of accuracy. The oscillator design specifications are listed in Table 5.2 (first row). The chip size should not exceed 2 mm × 2 mm on 100-␮ m thick GaAs substrate. The bias circuit is off-chip. Figure 5.47 shows the circuit model adopted to approximately describe the oscillator layout. The circuit model parameters are the lengths and widths of all microstrip sections, so that

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Table 5.2 Results of Electromagnetic Optimization

Network Functions

Number of Linear Subnetwork Frequency of Fundamental Spectral Analyses Oscillation Output Power Purity

Specification CM design 206 EM analysis of 1 CM design EM design by 18 system solving

Drain Efficiency

10 GHz ≥ 17 dB ≥ 16 dBc ≥ 20% 10 GHz 17.5 dB 18.4 dBc 24% The Newton iteration converges to a degenerate solution. 10 GHz 17.6 dB 16.8 dBc 22.5%

Figure 5.47 Circuit-model corresponding to the oscillator layout of Figure 5.46.

n C = 8. Four harmonics plus the fundamental and dc are used in each HB analysis, so that N = 5. The preliminary optimization with respect to the circuit model parameters converges in about 50 seconds on an 800-MHz PC and meets all the design specs. The electromagnetic-based analysis of the starting point generated by circuit model optimization is found to converge to


351

a degenerate solution. We next carry out the electromagnetic-based design by the procedure discussed in the previous section. In this step only the lengths of the microstrip lines are included in the set of designable parameters, for a total of n L = 4 unknowns. The solution of (5.120) requires only six Newton iterations, and a total of 18 electromagnetic analyses. The overall CPU time is about 8 hours on an 800-MHz PC. Figure 5.48 shows the evolution of the oscillator performance with respect to the number of Newton iterations. The final oscillator performance is again given in Table 5.2 (fourth row), and meets all the design goals. For comparison a direct electromagnetic optimization, starting from the same initial point, would take a total of 206 electromagnetic analyses and more than 4 days of CPU time. Note that the final performance achieved by electromagnetic-based optimization is slightly inferior to the results of the circuit model-based optimization due to the high level of idealization of the circuit description, which normally makes the circuit model-based results somewhat optimistic.

5.6 Iterative Methods for Large Self-Oscillating Nonlinear Circuit Analysis The Newton-iteration–based HB technique discussed in Section 5.2 is generally acknowledged as the principal method for the analysis of nonlinear microwave circuits working in steady-state regime under multitone excitation. HB simulators relying upon this method are robust and well-behaved and can normally reach convergence starting from zero harmonics even at very high drive levels [1]. The only outstanding drawback of this simulation approach lies in the huge demand of computer resources when the problem size becomes large. This is due to the fact that the storage of the Jacobian matrix requires N T2 words, and its factorization time is O (N T3 ), where N T is the number of scalar unknowns. These difficulties may be partly overcome making use of an artificially sparse Jacobian matrix coupled with sparse-matrix solvers [1, 3], but this unavoidably cripples the power-handling capabilities of the analysis algorithm. Thus, the need exists for an HB technique that can tackle large-size problems (say, N T ≥ 10,000) on ordinary PCs (or at most work stations), while fully retaining the convergence properties of traditional HB. An algorithm of this kind, named the inexact-Newton harmonic balance (INHB) is outlined in this section. The INHB provides the same robustness and accuracy as ordinary HB techniques with a dramatic reduction of memory storage and CPU time and thus opens the way to the HB treatment of very large size simulation problems.

352


Figure 5.48 (a) Drain efficiency, (b) output power, and (c) spectral purity of the oscillator as a function of number of direct-Newton optimization iterations.

5.6.1 Inexact-Newton HB for Forced Circuits In the case of a forced circuit (M = 0), all fundamental frequencies are a priori known and there is obviously no need for tuning parameters, so that the HB system (5.10) becomes a nonlinear system of N T equations in N T unknowns of the simple form


353

Figure 5.48 (Continued.)

E(X H ) = 0

(5.131)

In the following we shall assume that the nonlinear map E: R N T → R N T is continuously differentiable. Given an approximation X Hi to the exact solution X H , the corresponding Newton update n i is defined as the solution of Newton’s equation J(X Hi )n i = −E(X Hi )

(5.132)

where J(X H ) is the Jacobian matrix of E with respect to X H . The Newton iteration is then defined by X Hi + 1 = X Hi + n i

(5.133)

This solution technique requires the Jacobian matrix to be stored and factorized at each step and is thus not well suited for large values of N T because of memory and CPU time requirements. Both difficulties can be simultaneously overcome by resorting to an inexact Newton method [26]. An inexact Newton update is defined as a vector s i satisfying the condition

|| J(X Hi )s i + E(X Hi ) || ≤ f i || E(X Hi ) || (0 ≤ f i < 1)

(5.134)

354


where f i is named the forcing term [26]. The inexact Newton iteration is then defined by X Hi + 1 = X Hi + s i

(5.135)

Note that for f i = 0, s i reduces to n i , so that the forcing term provides a normalized measure of the maximum allowed deviation between the exact and inexact Newton updates. The iteration is terminated when the relative error on each element of E(X Hi ) drops below a prescribed threshold. The vector appearing on the left-hand side of (5.134), namely, r i = J(X Hi )s i + E(X Hi )

(5.136)

is called the residual associated with s i . (5.136) is zero for s i = n i , so that the residual provides a measure of the actual deviation between the exact and inexact Newton updates. Inexact Newton methods have a number of interesting features that make them an ideal choice for solving large-size nonlinear systems. They can be globalized by suitable techniques such as backtracking [26], and their efficiency can be greatly improved by appropriately updating the forcing term at each step [27]. Above all, they do not require a large linear system to be exactly solved at each step, since s i must satisfy (5.134), but is otherwise arbitrary. Instead, for a given f i the inexact update can be iteratively refined starting from an arbitrary initial guess (0th order approximation), until (5.134) is met. For application to HB analysis, best results have been obtained making use of the GMRES iterative solver to be discussed below [28]. Let us introduce a suitable approximation C i of J(X Hi ). The 0thorder approximation to n i is defined by −1

(0)

s i = −C i E(X Hi )

(5.137)

where C i is named the preconditioner [29]. A set of real N T -vectors defined by the following recursive relation is then computed and stored: (1)

−1

(0)

K i = r i = [1 N T − J(X Hi ) C i ] E(X Hi ) (q )

Ki

−1

(q − 1)

= J(X Hi ) C i K i

(q > 1)

(5.138)

Modern Harmonic-Balance Techniques for Oscillator Analysis (0)

355

(0)

where r i is the residual associated with s i , and 1 N T is the identity matrix of order N T . The vector space spanned by the vectors K i(q ) with 1 ≤ q ≤ Q is called a Krylov subspace of dimension Q [28]. To explain the role of the vectors (5.138), we now derive a recursive formula for the generation of a sequence of iterates s i(q ) starting from the 0th order approximation (5.137). If we let (0)

di = ni − si

(5.139)

and replace (5.139) into (5.132), making use of (5.136), we obtain (0)

J(X Hi ) d i + r i = 0

(5.140)

A 0th order approximation to d i is then (0)

−1 (0)

d i = −C i r i

(5.141)

Equation (5.141) may now be used together with (5.139) to generate the first-order approximation (1)

(0)

(0)

(0)

−1 (0)

−1

(0)

(1)

si = si + di = si − Ci ri = si − Ci Ki

(5.142)

The procedure leading from the 0th-order approximation (5.137) to the first-order approximation (5.142) can now be iterated in a straightforward way. At the Q th step we obtain the Q th-order approximation (Q )

si

(0)

−1

= si − Ci

Q

∑ Ki

(q )

(5.143)

q=1

The residual associated with (5.143) is (Q ) ri

=

(Q ) J(X Hi ) s i

+ E(X Hi ) =

(0) ri

Q

+

(q + 1)

∑ Ki

(5.144)

q=1

The Q th-order residual r i is thus a linear combination of the first Q + 1 vectors (5.138) (i.e., is a vector belonging to the Krylov subspace of dimension Q + 1). Finding the vectors (5.138) is computationally expensive. In order to get from this effort the best possible result, we search the Krylov

356


subspace for the residual having the minimum Euclidean norm. This means that we replace (5.143) by (Q ) si

=

(0) si

+

Q

∑ ␣ q Ki

−1 Ci

(q )

(5.145)

q=1

where the ␣ q are unknown coefficients to be determined in such a way as to minimize the Euclidean norm of the associated residual, that is, (Q )

ri

Q

(0)

= ri +

(q + 1)

∑ ␣ q Ki

(5.146)

q=1

In practice, the basis vectors (5.138) are first combined to generate an orthonormal basis for the Krylov subspace (Arnoldi’s orthogonalization [28]), then a standard least-squares method is applied to minimize || r i(Q ) || . If the resulting norm satisfies (5.134), s i(Q ) is taken as the inexact Newton update. Otherwise, Q is increased and a higher-order approximation is generated. Convergence is guaranteed to be achieved for sufficiently large Q , because it can be shown that [28] (Q )

lim s i

Q→∞

= ni

(5.147)

From the numerical viewpoint, however, it is not practical to increase Q beyond some upper bound Q max , because the memory occupation of the basis vectors would otherwise become too large. Thus, if (5.134) is not max ) instead satisfied for Q = Q max , the algorithm is restarted by taking s (Q i of (5.137) as the new 0th order approximation, and the whole procedure is repeated until (5.134) is verified. In such case the algorithm is more properly referred to as Q max -GMRES [28]. Unfortunately, in this case a convergence theorem such as (5.147) is no longer available, so that convergence to n i is not guaranteed. Nevertheless, it has been found that Q max ≈ 50 is normally sufficient to provide good convergence properties in HB applications, even at high drive levels. 5.6.2 Computation of the Krylov Subspace Basis Vectors With the INHB, the CPU time required to exactly solve Newton’s equation is essentially replaced by the time required to compute the basis vectors


357

(5.138). As a matter of fact, the subsequent minimization process has negligible cost for large values of N T , since Q 0. The initial values of A and ␻ 1 can be determined by application of the method of Section 5.2.3 to a small part of the given large circuit containing the oscillating device(s) (e.g., the local oscillator of a large front end). The oscillatory regime X H and ␻ 1 , generated in this way is used as the starting point for (5.159) with T and U , reset to their original values. The subsequent solution of (5.159) by the INHB thus determines the actual oscillatory regime (including the free fundamental frequency) of the given circuit. In this way, the problem can be solved at the cost of just two inexact Newton iterations, and the use of expensive continuation methods [7] can be avoided. The iterative solution strategy for (5.165) is similar to the one already discussed for (5.159). The Jacobian matrix of (5.165) has dimensions (N T + 2) × (N T + 2) and the same form as (5.160). R is now the 2 × N T matrix R=

冋

册

0

0

...

0

sin (␾ R )

−cos (␾ R )

0

...

0

0

0

0

...

0

2 Re [X R ]

2 Im [X R ]

0

...

0

0 (5.166)

and the column vector ∂E/∂␻ 1 is replaced by the N T × 2 matrix

冋

∂E ∂T

∂E ∂U

册

(5.167)

In the multiplication of the Jacobian matrix times a vector, the overhead due to the increased number of equations of (5.165) with respect to (5.131) is now 2N T + 4 flops, which is still negligible for N T large. 5.6.4 Applications As a representative example, let us consider a typical single-conversion receiver front end, whose functional diagram in terms of interconnected blocks is given in Figure 5.49. The circuit basically consists of two doubly balanced

364


Figure 5.49 Schematic topology of a microwave front end.

mixers arranged in an image-rejection configuration, a local oscillator, coupling networks, amplifiers, and filters. The band of operation is 935–960 MHz with a fixed IF of 90 MHz. The passband of the RF preselection filter coincides with the front-end band, and its attenuation at 900 MHz is about 30 dB. The circuit-level description of the front end is very detailed, and includes many (linear) parasitic components. The total number of device ports is n d = 178, and the total number of nodes is 1,502. The front end is analyzed as a single circuit, so that interblock couplings that may exist for various reasons such as imperfect isolation or proximity effects may be accounted for without difficulty. The reverse isolation of the local oscillator amplifier is about 15 dB. For illustrative purposes, a simple transistor local oscillator is designed for 0-dBm output power at 857.5 MHz when loaded by an ideal 50-⍀ resistance. The active device is biased at V BE = 0.6V and V CE = 3V. After connecting the local oscillator to the front end, we want to evaluate the front-end performance in terms of conversion gain and local oscillator frequency drift in the following cases: (1) in normal operating conditions, with a −85-dBm RF signal at 947.5 MHz (corresponding to center band) received by the antenna and (2) in the presence of a strong 915-MHz interfering tone superimposed on the useful signal. The frequencies taken into account in the HB analysis are given by pf LO + qf INT ± f RF , where f LO , f INT , and


365

f RF are the local oscillator, interferer, and radio frequency, respectively, and | p | + | q | ≤ 4 ( p , q integers). This results in N = 62 and N T = 22,250, corresponding to 187,750 nodal unknowns. The reference harmonic is selected as the collector voltage harmonic of the oscillating transistor at the fundamental local oscillator frequency, with ␾ R = 0. The first analysis is based on (5.165), with A , ␻ 1 held fixed to the values obtained from the analysis of the ideally loaded oscillator (A = 2.5V, ␻ 1 = 2␲ ⭈ 947.5 MHz). The remaining harmonics are initialized to zero, and the interferer power is set to P INT = −50 dBm. The tuning parameters T and U are selected as the base and collector bias voltages V BE and V CE . The analysis converges in about 250 seconds on an 800-MHz PC. The final values of the tuning parameters are V BE ≈ 0.65V, V CE ≈ 3.21V. In the second step, the bias voltages are reset to their nominal values, and (5.159) is solved for X H , ␻ 1 starting from the results of the previous run. The analysis converges in about 890 seconds with the results shown in Figure 5.50. The effects of the interferer power are then investigated. The analysis is carried out by continuation with respect to P INT in the range −50 dBm ÷ +5 dBm (2.5-dB steps). Only (5.159) needs to be solved in this simulation, because the results of each step provide sufficient starting-point information to prevent the Newton iteration from converging to the degenerate solution at the subsequent point. The front-end conversion gain and the local oscillator frequency shift are

Figure 5.50 Front-end conversion gain and LO frequency shift.

366


plotted against P INT in Figure 5.50. The average CPU time for this run is about 1,900 seconds per power point. The results show that a strong interferer may shift the local oscillator frequency by more than one GSM channel if the local oscillator is not sufficiently buffered.

5.7 Global Stability Analysis of Large Autonomous Circuits In the previous section, the HB technique coupled with Krylov-subspace methods has been introduced as a numerical tool for the analysis of nonlinear microwave circuits containing large numbers of devices and/or supporting steady states with discrete spectra including large numbers of lines. However, the extension of these techniques to global stability analysis for general circuits (particularly high-frequency distributed circuits) is not straightforward. Generally speaking, the stability analysis methods discussed in Section 5.3 reduce the search for the fundamental bifurcations of a parameterized circuit to the manipulation of a determinantal equation of the form (5.45) through Nyquist’s analysis. When the number of unknowns is very large, Krylovsubspace methods become indispensable to waive the need for storing and factorizing the Jacobian matrix. Unfortunately, these methods do not provide an efficient way of computing the determinant of a large complex matrix, which prevents the use of Nyquist’s analysis. The exact location of the fundamental bifurcations by the algorithms of Section 5.3.4 thus becomes impossible. To circumvent this problem, this section resorts to the technique for the approximate detection of Hopf bifurcations that was introduced in Section 5.4.7.1. This method is combined with the Krylov-subspace method discussed in Section 5.6.3 to produce nonlinear systems that approximately locate the Hopf bifurcations and can be solved by the inexact Newton iteration coupled with the GMRES method, irrespective of their size. This approach may be applied to Hopf bifurcations of both autonomous and forced (including dc) solution paths, and may be extended to I-type bifurcations. Regular turning points may be located by a slight variation of the switching-parameter algorithm developed in Section 5.3.1.2. 5.7.1 Fundamental Bifurcation Detection for Large Circuits 5.7.1.1 Hopf Bifurcations of a One-Autonomous Solution Path

Let the circuit be parametrized by a single parameter P. We assume that a Hopf bifurcation occurs on a one-autonomous solution path (M = 1,


367

F arbitrary), and consider a point of the quasi-periodic solution path bifurcating from criticality. On this path the electrical regime is two-autonomous (M = 2), with two unknown free fundamentals ␻ 1 and ␻ 2 . We properly select two reference harmonics X R 1 and X R 2 , whose phases are held fixed to some arbitrary values ␾ R 1 and ␾ R 2 . The amplitude of X R 2 is denoted by A 2 . Making use of the substitution principle, we interchange the roles of P and A 2 and treat the former as a problem unknown, and the latter as a free parameter. As in Section 5.6.3, we want to formulate the solving system is such a way as to allow its solution by the INHB, and in particular the straightforward use of the algorithms developed in Section 5.6.2 in the Jacobian-vector multiplications. We thus retain the entire harmonic vector X H as unknown, and fix the values of ␾ R 1 , ␾ R 2 , and A 2 , by auxiliary equations. The resulting HB system is a straightforward extension of (5.165), namely,

冦

E(X H , ␻ 1 , ␻ 2 , P ) = 0 sin (␾ R 1 ) Re [X R 1 ] − cos (␾ R 1 ) Im [X R 1 ] = 0 sin (␾ R 2 ) Re [X R 2 ] − cos (␾ R 2 ) Im [X R 2 ] = 0

(5.168)

(Re [X R 2 ])2 + (Im [X R 2 ])2 − A 22 = 0

Equation (5.168) is a system of N T + 3 equations in as many unknowns X H , ␻ 1 , ␻ 2 , and P, and can be efficiently solved by the INHB algorithm in a way similar to (5.165). At this stage we can proceed as in Section 5.4.7. When the circuit state approaches the Hopf bifurcation on the bifurcated solution path, then simultaneously ␻ 2 → ␻ 2C , P → P C , and A 2 → 0, where ␻ 2C and P C , are the bifurcating (critical) values of ␻ 2 , and P, respectively. The Jacobian matrix of (5.168) is singular at A 2 = 0, however, so that the bifurcation can only be approximately located by solving (5.168) with A 2 set to some near-zero but finite value A m . This means that instead of finding the Hopf bifurcation itself, we find a state of the two-autonomous solution path located very close to it. Nevertheless, very accurate estimates of the bifurcating values of the quantities of interest may be normally found in this way. This is particularly true for ␻ 2C and P C , since Hopf’s theorem [9] shows that for the states belonging to the bifurcated branch, the conditions ∂␻ 2 =0 ∂A 2

(5.169)

∂P =0 ∂A 2

(5.170)

368


hold at criticality. Thus, the bifurcation can be accurately located even with relatively large values of A m . The starting point for (5.168) is usually a state belonging to the one-autonomous solution path on which the Hopf bifurcation occurs. If necessary, a few trials may be carried out in order to find a point close enough to the bifurcation to ensure convergence of the inexact Newton iteration. 5.7.1.2 Hopf Bifurcations of a Forced Solution Path

Hopf bifurcations of a forced (including a dc) solution path (M = 0, F arbitrary) can be treated in a similar way. In this case the electrical regime on the bifurcated path is one-autonomous (M = 1), with one unknown free fundamental ␻ 1 , and one reference harmonic X R . By fixing the phase and amplitude of X R to ␾ R and A m , respectively, we obtain the HB system for a state of the bifurcated solution path close the Hopf bifurcation

冦

E(X H , ␻ 1 , P ) = 0 sin (␾ R ) Re [X R ] − cos (␾ R ) Im [X R ] = 0 2

2

(Re [X R ]) + (Im [X R ]) −

A 2m

(5.171)

=0

Equation (5.171) is a system of N T + 2 equations in as many unknowns X H , ␻ 1 , and P, and can be efficiently solved by the INHB algorithm in a way similar to (5.168). Equation (5.171) approximately yields the Hopf bifurcation if A m is sufficiently small. Nevertheless, the bifurcating values of ␻ 1 and P may be accurately found in this way even with relatively large values of A m , due to relationships similar to (5.169) and (5.170). 5.7.1.3 Period-Doubling Bifurcations of a Periodic Solution Path

The above algorithm can also be used to approximately find period-doubling (I-type) bifurcations of a periodic one-autonomous solution path (M = 1, F = 0). In this case the bifurcated solution path is still periodic and oneautonomous, with a free fundamental equal to ␻ 1 /2 at criticality. The reference harmonic X R must be chosen as a harmonic directly related to the output power at the halved frequency. The solving system is formally similar to (5.171), except that the spectrum is extended to include the odd multiples of the halved frequency. However, a more efficient formulation is


冦

冋

E XH ,

册

369

␻ 1 (P ) + ␦␻ , P = 0 2

sin (␾ R ) Re [X R ] − cos (␾ R ) Im [X R ] = 0

(5.172)

(Re [X R ])2 + (Im [X R ])2 − A 2m = 0

where the unknowns are X H , ␦␻ , P, and ␻ 1 (P ) is the free fundamental computed as a function of P along the original solution path. If the original periodic solution path is forced (M = 0, F = 1), the solution path bifurcating from an I-type bifurcation is still periodic and has a fixed fundamental frequency ␻ 1 /2. In addition, since the frequency division process is coherent [9], the phase of the states belonging to the bifurcated path is determined by the excitation phase, and is thus a problem unknown (i.e., is not arbitrary). A state of the bifurcated branch close to the bifurcation may thus be found by solving with the INHB technique the system of N T + 1 equations in as many unknowns

冦

冉

E XH , P ;

␻1 2

冊

=0 (5.173)

(Re [X R ])2 + (Im [X R ])2 − A 2m = 0

5.7.1.4 D-Type Bifurcations of a One-Autonomous Solution Path

Direct-type bifurcations occurring on a one-autonomous solution path may be detected by the switching-parameter algorithm introduced in Section 5.3.1.2. If N T is large, the HB system to be solved in order to generate the solution path is (5.159), which is rewritten here for convenience:

再

E(X H , ␻ 1 ; P ) = 0 sin (␾ R ) Re [X R ] − cos (␾ R ) Im [X R ] = 0

If we now differentiate (5.173) with respect to P along the solution path, we obtain

J H␻

冤冥 D XH DP D␻ 1 DP

=−

冤冥 ∂E ∂P 0

(5.174)

370


where J H␻ is the Jacobian matrix defined by (5.160), and the operator D denotes the derivatives taken along the solution path. Equation (5.174) may be solved by the inexact Newton algorithm discussed in Section 5.6.2 to compute the derivative D X H /DP. At a D-type bifurcation we have

| |

DP =0 D XH

(5.175)

| |

(5.176)

so that the quantity

␦=

D XH DP

can be used as a control variable for parameter switching. The switching is automatically performed when ␦ rises above some empirically defined threshold and is suppressed in the opposite condition. A convenient choice for the switching parameter is the amplitude A of the reference harmonic. This means that, after a switching takes place, the nonlinear system (5.159) is replaced by (5.171) with A m = A . The amplitude is continuously swept until (5.176) drops again below threshold. 5.7.2 Applications Let us consider the double-ring mixer with active baluns schematically depicted in Figure 5.51. The circuit contains eight diodes and 12 FETs for a total of n d = 32 device ports and 67 nodes. All the FETs have the same nominal bias point, V GS = −0.5V, V DS = 4.5V. The circuit operates at a nominal local oscillator power level of +13 dBm at 7 GHz, and a nominal RF power of −10 dBm at 6.5 GHz. In such conditions the conversion gain is about −3.6 dB. Owing to its complex topology (see Figure 5.51), the circuit provides a variety of feedback paths that might result in the onset of unwanted spurious oscillations. In order to check the possible existence of such anomalous behaviors, we thus carry out a search for Hopf bifurcations of the nominal two-tone regime, from which spurious tones might start to build up. Since the nominal regime is completely forced (M = 0, F = 2), the system to be solved is (5.171), and the adopted spectrum is the set of harmonic vectors of the following form:


371

Figure 5.51 Schematic topology of a monolithic double-ring mixer.

冦

冤冥 k1

k=

k2

±1 |k1| + |k2| ≤ 8

(5.177)

Note that to limit the problem size the third harmonic number is limited to ±1, which is allowed by the low RF power. With this spectrum, the total number of positive frequencies is N = 218, and the total number of scalar unknowns is N T = 13,922. The reference harmonic is chosen as the fundamental component of the drain voltage for one of the FETs of the output balun (see Figure 5.51), and the drain bias voltage is chosen as the tuning parameter (P = V DS ). With A m set to 0.1V, the system (5.171) is solved by the INHB technique, and one Hopf bifurcation H is found at P C ≈ 2.139V, ␻ 1 /2␲ ≈ 16.4 GHz. The solution takes about 770 seconds on an 800-MHz PC. Note that the adopted value of A m belongs to a parameter range where DP /DA may be considered zero for all practical purposes. Indeed, the accuracy of the above quoted estimate of P C turns out to be better than 4 ⭈ 10−3. Starting from H, the quasi-periodic one-autonomous solution path (M = 1, F = 2) shown

372


in Figure 5.52 is built by ordinary continuation. The path shows that at the nominal drain voltage of 4.5V the mixer does, indeed, generate a spurious oscillation, with the resulting three-tone spectrum shown in Figure 5.53. Figure 5.54 shows the degradation of the mixer conversion gain as a function of the spurious oscillation buildup with increasing drain voltage. The circuit performance resulting from the above analysis is obviously not acceptable, so that some measure should be taken in order to suppress the spurious oscillation. A possible solution is to introduce in the circuit a simple ‘‘trap’’ represented by the resonant circuit shown in the inlet of Figure 5.51. The purpose of the trap is to ground the drain of one of the selfoscillating FETs, while avoiding perturbing the output circuit thanks to the infinite impedance of the parallel LC resonator tuned at the intermediate frequency. To gain a thorough insight in the effects of the trap, the system (5.171) is solved by the INHB technique with A m = 0.1V and several values of the capacitance C (see Figure 5.51). The locus of the Hopf bifurcation H generated in this way is shown in Figure 5.55 in the two-dimensional parameter space C − P. The computation of the locus takes about 1,425 CPU seconds per point. Figure 5.55 shows that the spurious oscillation is suppressed for C > 1 pF and V DS set to the nominal value of 4.5V. Indeed, with C = 1.2 pF the mixer is found to behave correctly at any drain bias with the canonical spectrum shown in Figure 5.56. The conversion gain computed in such conditions is again given by the upper curve in Figure 5.54.

Figure 5.52 Bifurcation diagram for the double-ring mixer.


373

Figure 5.53 Output spectrum of the double-ring mixer in the presence of the spurious oscillation.

Figure 5.54 Conversion gain of the double-ring mixer.

374


Figure 5.55 Hopf bifurcation locus for the double-ring mixer.

Figure 5.56 Output spectrum of the double-ring mixer after connecting the trap.


375

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About the Authors Michał Odyniec received an M.S. in applied mathematics and a Ph.D. in electrical engineering from the Technical University of Warsaw in 1972 and 1977, respectively. Subsequently, he became an assistant professor at Technical University, where he worked on nonlinear circuits and systems. From 1981 to 1984 he continued the research as a visiting assistant professor at the University of California at Berkeley. From 1985 to 1989 he worked for Microsource, Inc., where he was responsible for the design of microwave oscillators. From 1989 to 2001, Dr. Odyniec worked for Hewlett-Packard (later Agilent Technologies) in the areas of nonlinear CAD, fast-switching synthesizers, and 60-GHz radio. In 2001 he joined Signature Bioscience as a principal engineer, where he is responsible for the development of oscillatorbased sensors. He has published more than 60 technical papers. He is a senior member of IEEE, a member of SIAM, a member of the IMS Technical Program Committee, a member of the Technical Committee IEEE MTT-1 on CAD, and a recipient of the Myril B. Reed best paper award. Marc Camiade received a Dpl. Eng. in physics and electronic engineering from the Institut National des Sciences Applique´es in Toulouse, France, in 1981. He joined Thomson-CSF in 1982 as a design engineer of hybrid circuits; in this capacity he participated in a variety of microwave and millimeter-wave circuits. He later became the application group manager in charge of new product development based on MIC and MMIC components. He joined United Monolithic Semiconductors in 1996 and is currently in charge of the development of components for defense and automotive 377

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applications. He is now mainly involved in all the functions for radar front end from the L band to the W band. Alessandra Costanzo received a Dr. Ing. in electronic engineering from the University of Bologna, Italy, in 1987. In the same year she obtained a research grant issued by the Department of Electronics and Telettra S.P.A. In 1989 she joined the University of Bologna as a research associate. Since 2001, Dr. Costanzo has served as an associate professor of electromagnetic fields at the University of Bologna. Her teaching and research activities have focused on several topics, including electrical and thermal characterization and modeling of nonlinear devices and simulation and design of active microwave integrated circuits. She has also been devoted to the development of software tools for the broadband design of autonomous circuits and systems for electrical, stability, and noise performance. Most recently, Dr. Costanzo has worked on the development of algorithms for the analysis of self-oscillating circuits and systems excited by digitally modulated signals and for the broadband design of self-oscillating integrated antennas based on electromagnetic analysis. From 1995 to 1997, Dr. Costanzo was a member of the Technical Program Committee of the European Microwave Conference. She is also a member of the IEEE. Ali Hajimiri received a B.S. (honors) in electronics engineering from the Sharif University of Technology in 1994, and an M.S. and a Ph.D. in electrical engineering from Stanford University in 1996 and 1998, respectively. Dr. Hajimiri worked as a design engineer for Philips on a BiCMOS chipset for GSM cellular units. He has also worked for Sun Microsystems and Lucent Technologies (Bell-Labs). He is currently an assistant professor at the California Institute of Technology. Dr. Hajimiri’s group does research on integrated circuits and their applications from high-speed and RF to lowfrequency, high-precision circuits. His research group is involved in both the theoretical analysis of the problems in integrated circuits and the practical implementations of new systems in very large-scale integrated circuits. Dr. Hajimiri is a coauthor of a book, The Design of Low Noise Oscillators (Kluwer, 1999), and the author of more than 25 refereed papers. Kaneyuki Kurokawa received a B.S. in electrical engineering in 1951 and a Ph.D. in electrical engineering in 1958, both from the University of Tokyo. In 1957, he became an assistant professor at the University of Tokyo. From

About the Authors

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1959 to 1961, he worked on parametric amplifiers at Bell Laboratories, Murray Hill, New Jersey, while on a leave of absence from the university. In 1963, he joined Bell Laboratories as a member of the technical staff and was later promoted to a supervisor. At Bell, he developed microwave-balanced transistor amplifiers, millimeter-wave path-length modulators, and the theory of microwave solid-state oscillators and initiated the development of optical fiber transmission systems. In 1975, after the delivery of six optical transmitter modules from Murray Hill to Holmdel for the Atlanta Experiment, Dr. Kurokawa left Bell to join Fujitsu. At Fujitsu, he directed the efforts to develop optical fiber systems, array and simulation processors, Josephson junction ICs, and silicon-on-insulator technology. From 1985 to 1989, he was in charge of Fujitsu’s Atsugi Laboratories, which successfully developed HEMT devices and distributed feedback lasers. Dr. Kurokawa became the director of Fujitsu Laboratories in 1979, the managing director in 1985, the vice president in 1992, and a Fujitsu fellow in 1994. From 1986 to 1989, he also served as a visiting professor at the Institute of Industrial Science of the University of Tokyo. In 2000, he retired from Fujitsu. Dr. Kurokawa is the author of An Introduction to the Theory of Microwave Circuits (Academic Press, 1969). In addition, he is an IEEE life fellow and an honorary member of the Institute of Electronics, Information, and Communication Engineers. He received the Certificate of Appreciation from the International SolidState Circuits Conference in 1965 and the IEEE MTT-S Pioneer Award in 1996. Thomas H. Lee received a B.S., an M.S., and a Ph.D. in electrical engineering from the Massachusetts Institute of Technology in 1983, 1985, and 1990, respectively. In 1990 he joined Analog Devices, where he was primarily engaged in the design of high-speed clock recovery devices. In 1992, he joined Rambus, Inc., in Mountain View, California, where he developed high-speed analog circuitry for 500-Mbps CMOS DRAMs. Dr. Lee has also contributed to the development of PLLs in StrongARM, Alpha, and K6/K7 microprocessors. Since 1994, he has been an assistant professor of electrical engineering at Stanford University where his research focus has been on gigahertz-speed wireline and wireless integrated circuits built in conventional silicon technologies, particularly CMOS. Dr. Lee has twice received the best paper award at the International Solid-State Circuits Conference, was coauthor of a best student paper at the ISSCC, won a best paper award at the CICC, and was recently awarded a Packard Foundation fellowship.

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In addition, he is a distinguished lecturer of the IEEE Solid-State Circuits Society and was recently named a distinguished microwave lecturer. He holds 12 U.S. patents, is the author of a textbook, The Design of CMOS Radio-Frequency Integrated Circuits (Cambridge Press, 1998), and is a coauthor of three additional books on RF circuit design: The Design of Low Noise Oscillators (Kluwer, 1999); The Design and Implementation of LowPower CMOS Radio Receivers (Kluwer, 1999); and Multi-GHz Frequency Synthesis & Division (Kluwer, 2001). Dr. Lee is also a cofounder of Matrix Semiconductor. Franco Mastri received a Laurea (cum laude) in electronic engineering from the University of Bologna, Italy, in 1985. In 1987 and 1988 he obtained research grants issued by the Fondazione G. Marconi, Pontecchio Marconi, Bologna, and Selenia S.p.A., Rome, to carry out a study on the application of nonlinear CAD techniques in MIC and MMIC design. In 1990 he joined the Department of Electrical Engineering at the University of Bologna as a researcher. His main research interests are in the field of nonlinear circuits, with a special emphasis on numerical methods for circuit simulation. Jean-Christophe Nallatamby received a DEA in microwave and optical communications in 1988 and a Ph.D. in electronics in 1992, both from the University of Limoges, France. He is now a lecturer in the GEII department at Brive of the IUT of Limoges. His research interests are in nonlinear noise analysis of nonlinear microwave circuits, the design of the low-phase noise oscillator, and the noise characterization of microwave devices. Andrea Neri received an M.S. in electronic engineering from the University of Bologna in 1981. In 1983 and 1984 he obtained research grants issued by the Fondazione G. Marconi, Pontecchio Marconi, Bologna, and Selenia S.p.A., Rome, to work on dielectric resonators and their applications in MIC. In 1985, he joined the Fondazione U. Bordoni, Rome, where he is currently involved in research on nonlinear microwave circuit design. In 1987 he received a Ph.D. in electronics and computer science. Since 1994 Dr. Neri has been a lecturer on microwaves at the University of Bologna. His main fields of interest are microwave oscillator design, stability analysis, and numerical techniques for the circuit-level simulation of communication subsystems. Juan Obregon received an MSEE from C.N.A.M., Paris, France, in 1967 and a Ph.D. in electrical engineering from the University of Limoges, France,

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in 1980. He has worked for THOMSON-CSF, where he developed lownoise parametric amplifiers for radar receivers at L and C band, and the Radiotechnique Compelec (RTC), where he worked on solid state oscillators. In 1969 he received a patent for the first multidiode oscillator combiner consisting of N active diodes coupled to a common cavity working in the TM010 mode. Dr. Obregon became a professor at the University of Limoges in 1981, where he founded a research team devoted to the modeling, simulation, and optimization of nonlinear circuits at IRCOM, the research institute in microwave and optical communications of the University of Limoges and CNRS. He has performed extensive research in the framework of national and European programs in the area of simulation tools dedicated to CAD of nonlinear microwave circuits and taught short courses at several universities in Brazil, France, and Spain. Dr. Obregon has authored and coauthored numerous publications and holds 10 patents. He was awarded the European Microwave Prize in 1985 for his work on millimeter wave DROs. Since 1981 he has been a consultant to microwave industrial laboratories. He is now professor emeritus at the University of Limoges. Michel Prigent received a Ph.D. from the University of Limoges, France, in 1987. He is a lecturer at the University of Limoges. His field of interest is the design of microwave and millimeter-wave oscillator circuits. He is also involved in characterization and modeling of nonlinear active components (i.e., FET, PHEMT, and HBT) with a particular emphasis on low-frequency noise measurement and modeling for the use in MMIC CAD. Dominique Rigaud studied physics and electronics at the University of Montpellier, France, where he received the Doctorat en Sciences Physiques in 1973. His dissertation concentrated on the electrical noise in silicon field effect transistors. Dr. Rigaud has worked at the Centre d’Etudes d’Electronique des Solides at Montpellier, the Institut Universitaire de Technologie in Nimes, France, and the Universite´ des Sciences et de la Technologie d’Oran in Algeria, where he was the director of the Department of Electrical Engineering. He is currently a professor of electrical engineering at the University of Montpellier at the Centre d’Electronique de Montpellier. Dr. Rigaud’s research interests include electrical noise studies and modeling in field effect devices, especially in III-V field effect transistors and in Si MOSTs. He is involved in fundamental studies of noise behavior of scaled submicrometer MOS devices and in the quality and reliability evaluation of field effect transistor technologies. Dr. Rigaud is also involved in low-noise

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signal amplification and electrical noise measurements. He has published more than 100 technical papers in the above areas. Vittorio Rizzoli obtained his degrees in electronic engineering from the University of Bologna, Italy. From 1971 to 1973 he held a research grant issued by Fondazione Ugo Bordoni and joined the Centro Onde Millimetriche in Pontecchio Marconi, Bologna, where he was involved in the development of IF circuitry for a millimeter-wave circular-waveguide communications system. Rizzoli has worked for the Stanford Park Division of Hewlett-Packard in Palo Alto, California, in microwave transistor modeling and medium-power amplifier design; and also for the University of Bologna, Italy, first as an associate professor of circuit theory, and then as a full professor of electromagnetic fields and circuits. His teaching and research activities have focused on several topics, including the theory of electromagnetic propagation in optical fibers and the simulation and design of passive and active microwave integrated circuits. More recently, he has been engaged in the development of algorithms and software tools for the CAD of very large nonlinear circuits. Dr. Rizzoli has authored or coauthored more than 160 technical papers. He is a member of the editorial boards of the IEEE Transactions on Microwave Theory and Techniques, the IEEE’s Microwave and Wireless Components Letters, and John Wiley’s International Journal of RF and Microwave Computer-Aided Engineering. He is also a member of the paper review board of Electronics Letters. Dr. Rizzoli has been on the Technical Program Committee of the European Microwave Conference. He has also served as the distinguished microwave lecturer of the IEEE MTT-S for Region 8, lecturing in Europe, the United States, and the Middle East on simulation and design of nonlinear microwave circuits. In 1994 he was elected Fellow of the IEEE. Since 1993 he has been a member of the Technical Committee IEEE MTT-1 on CAD. In addition, since 1999, he has been a TPC member of the IEEE MTT-S International Microwave Symposium.

Index 38.25 GHz VCO, 205–8 defined, 205–6 frequency-tuning characteristic, 206 fully integrated, frequency tuning characteristic, 209 fully integrated, phase noise, 209 phase noise, 206 photograph, 207 See also VCOs

constant, stability, 57 control dynamics, 90 fluctuations, 60 impulse sensitivity function, 88 stabilization, 60, 157 steady-state oscillations, 41 thermal noise and, 65 Amplitude-limiting mechanism, 66 Amplitude modulation (AM) noise, 7 modulations, 176 rejection, 172 simulation methods, 181 spectra processing, 178–82 spectrum, 172 Amplitude response, 88–90 overdamped, 89 underdamped, 89 Amplitude-to-phase (AM-PM) conversion, 60, 193–94 Arnoldi’s orthogonalization, 356 Asynchronous instability, 268 Asynchronous stability, 337–43 Hopf bifurcations loci construction, 337–40 interactive solution, 337 oscillator design for, 337–43 spurious-free design, 340–43 See also Stability

Abrupt-junction law, 327 AC circuit theory, 12 Active circuits nonresistive, 38–40 practical realization of, 18 real-life, 38 See also Circuit(s) Admittance input, 192 linear, 240 linear input, 160 linear subnetwork, 359 load, 188 matrix, 226, 269 resonator, 33 total input, 158–59 Alder, R. (1946), 4–7 Amplitude comparison, 37

383

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Autonomous circuits extension to, 360–63 global stability analysis of, 366–74 See also Circuit(s) Autonomous quasi-periodic regimes, 246–48 applications, 256–60 degenerate solutions and, 253–56 HB analysis of, 246–60 mixed-mode Newton iteration and, 248–53 Autonomous state space, 262 Averaged equation, 58 Averaging theorems, 58 Bifurcations, 169 of dc solution path, 279–82, 296 defined, 272 detection for large circuits, 366–70 diagram for broadband VCO, 296 diagram for double-ring mixer, 372 diagram for multiple-cavity DRO, 326–27 diagrams, 279 direct-type, 275, 280 Hopf, 277, 279, 280, 281, 294, 298 inverse-type, 276 numerical detection of, 277–79 period-doubling, 278, 368–69 of periodic solution path, 274–77 pitchfork, 276, 280–81 saddle-node, 276, 280 of solution paths, 275 stability exchange at, 282–84 states lying close to, 278 theory, 260 transcritical, 276, 280–81 Bipolar transistor model extraction, 119–23 electrothermal convective modeling, 119 extrinsic elements extraction, 119–20 nonlinear capacitances/ transcapacitances, 120–22 thermal model extraction, 122–23 See also Model extraction procedure

Bipolar transistors breakdown phenomenon, 98–99 electrothermal model illustration, 102 electrothermal nonlinear model, 100–102 Gummel plot, 97 heterojunction, 96–102 homojunction, 96–102 ideal, 97 Kirk effect, 98 linear model illustration, 103 parasitic elements, 97–102 thermal effects, 100 Black-box model, 95 Block-diagonal matrix, 360 Bode plot, 163 Breadboard oscillators, 195–200 circuit, 196, 198 defined, 296 high-Q sapphire resonator, 199, 200 illustrated, 198 phase-noise variation, 198 Breakdown voltage, 98 Broadband optimization preliminary steps, 310 starting point, 310 See also Optimization Broadband VCOs, 295–99, 325–36 bifurcation diagram, 296, 330, 332, 336 bifurcation diagram (after final optimization), 342 case study, 325–36 evolution of output power, 334 evolution of tuning characteristic, 334 fundamental output power, 335 nominal mode, 298 normalized output spectrum, 336 Nyquist plot, 297 output power vs. number of iterations, 331 performance check, 334–36 phase noise function, 331 primary Hopf bifurcation loci for, 342 schematic topology, 295, 328 spurious mode, 298

Index starting-point oscillatory regime, 329, 330 tuning characteristic, 332, 333 See also VCOs CAD-oriented oscillator design, 301–43 applications, 320–25 asynchronous stability, 337–43 case study, 325–36 computation of gradient, 319–20 general optimization methods, 303–12 optimization by substitution methods, 312–15 oscillation buildup, 318–19 robustness evaluation, 321 steady-state synchronous stability, 315–17 See also Oscillator design CAD tools harmonic balance analysis, 40 oscillator design and, 15–16 Capacitance(s) base-collector junction, 99–100 base emitter junction, 100 depletion-layer, 327 diffusion, 100 extraction, 128–30 HBT base-collector, 223 HBT base-emitter, 222–23 nonlinear, 108, 120–22, 126, 327 nonlinear equations, 99 PHEMT gate-drain, 225 PHEMT gate-source, 226 resonator, 21 tank, 63 Capture range defined, 321 establishing, 321 Cavity oscillators, 30 Channel noise, 140–45 1/f, 148, 149 defined, 140 measurements, 140–45 See also Noise; Noise measurements Characteristic frequencies, 24, 25 Circuit design tools, 156–82 free-running oscillators, 169–72

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nonlinear stability, 169–72 phase-noise characterization, 172–82 sinusoidal oscillators, 156–67 steady-state analysis, 167–69 transistor oscillators, 167–69 Circuit examples, 79–88 LC oscillators, 79–88 ring oscillators, 84–88 Circuit(s) active, 18, 38–40 admittance locus, 6 analysis, 191 autonomous, 360–63, 366–74 energy optimization, 190–93 equations, 43–45 feedback, 196 high-frequency, design, 93–214 high-Q, 23, 38 hybrid, 8 with idealized feedback lossless circuit, 191 impedance, 11 impedance locus, 11 impedance matrix, 8 linearization, 24–25 losses, 25 lossless feedback, 191–93 multitransistor, 167 noise, 136 noise evolution into phase noise, 75 nonlinear, 17 one-port, 184–86 parameters, 5 power location in, 186 push-pull, 8 simulated, 236 transistor-oscillator, 186–87 Clapp oscillator, 80 Close-in phase noise, 78–79 CMOS biCMOS, 40 differential, 87 single-ended, 87 Colpitts oscillator, 79 Computer-aided design. See CAD tools

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Conductance active, 158 load, optimization of, 185–86 negative, 157–59 tank, 64 total input, 160 Continuously tunable oscillators, 308–10 Convective model extraction, 128 Correlation matrix global, 180 noise, 317 output noise, 180 total, 180 Cross-spectrum calculation, 234–35 Current-voltage characteristics, 16 dc solution path, 279–82 defined, 280 for DRO, 292 D-type bifurcation, 280 Hopf bifurcation, 280, 281, 296 pitchfork bifurcation, 280–81 saddle-node bifurcation, 280 transcritical bifurcation, 280–81 See also Solution paths Degenerate solutions, 253–56, 306 defined, 254 stationary, 254 Derivatives spectrum, 251 Describing function, 32 Describing function method defined, 33 justification of, 53–55 outline of proof, 55 theorem (Kudrewicz), 54–55 Dielectric resonator oscillator (DRO), 30, 208, 287–94 dc solution path, 292 determinant of Jacobian matrix for, 288 free-running microstrip, 287 multiple-cavity, 323–25 Nyquist plots, 289, 291, 292–94 periodic solution path, 292 schematic topology, 288 stable dc state, 289 stable periodic state (point S), 293–94

tuning conditions of, 326–27 unstable dc state, 291 unstable periodic state (point U), 292–93 Dimensionless nonlinear characteristics, 34–35 Direct gate noise measurements, 138–40 Direct-Newton optimization, 344–49 Jacobian matrix, 347 one dimensional search step, 348 oscillator performance as function of, 352 preliminary step, 345–46 update formula, 347 See also Optimization Direct-type bifurcation, 275, 280 dc-type solution paths, 280 one-autonomous solution path, 369–70 periodic solution path, 275 stability exchange, 283 See also Bifurcations Discrete tuning oscillators, 310–12 Double-ring mixer, 370–74 bifurcation diagram, 372 conversion gain, 373 Hopf bifurcation loci, 374 output spectrum, 373 output spectrum (after connecting trap), 374 schematic topology, 371 Doubling oscillators, 197 Drain noise measurement, 143 Edson, W.A. (1960), 7–8 Eigenvalue ordering, 171 Electromagnetics-based optimization, 343–51 applications, 349–51 circuit model corresponding to oscillator layout, 350 direct-Newton, 344–49 results, 350 space-mapping, 343 See also Optimization Electrothermal convective modeling, 119 nonlinear model, 100–102

Index Envelope method, 43 Equivalent circuit(s) experimental setup at input of voltage amplifier, 234 for intrinsic transistor, 227 for noise calculation, 226–31 for simultaneous input-output noise measurements, 227 Escapement defined, 77 transfer and, 77–78 Extrinsic elements extraction, 119–20 Feedback circuits, 196 FET extraction model, 123–27 extraction of linear model, 124 extraction of parasitic elements, 124 nonlinear capacitances model extraction, 126 nonlinear convective model extraction, 124–26 Pi model, 123 thermal circuit extraction, 126–27 See also Model extraction procedure FET MMIC-based oscillator, 202–10 defined, 202–3 quarter-wave structure, 201, 202 VCOs, 203 See also MMIC-based oscillators FET noise measurements, 138–48 FET nonlinear distributed model, 111–15 channel current source expressions, 113–14 defined, 112 nonlinear diode and capacitance, 114–15 topology, 113 Field-effect transistors (FETs), 94 in common source configuration, 135 GaAs, 109–11 metal oxide semiconductor (MOSFETs), 103 metal semiconductor (MESFETs), 103 modeling, 102–15 noise measurements, 138–48 nonlinear distributed model, 111–15 operating, 102–15

387

output characteristics, 104 Pi model of, 105 Forced solution path, 368 Free fundamentals, 248 Free-running oscillators design rules, 182–95 nonlinear stability, 169–72 optimization, 303–8 start conditions, 160–65 Frequency modulation (FM) noise, 7 GaAs FETs, 109–11 breakdown effects, 109 thermal effects, 110–11 trapping effects, 109–10 See also Field-effect transistors (FETs) Gate noise, 138–40 1/f, 150 defined, 138 measurements, 138–40 measurement transistor configuration, 143 See also Noise; Noise measurements Global correlation matrix, 180 Global stability, 169 Global stability analysis, 274–84 bifurcations of dc solution path, 279–82 bifurcations of periodic solution path, 274–77 of large autonomous circuits, 366–74 numerical detection of bifurcations, 277–79 stability exchange at bifurcations, 282–84 GMRES iterative solver, 354 GMRES method, 366 Harmonic balance (HB), 167, 245–374 circuit simulators, 167 inexact-Newton (INHB), 351–56 introduction, 245–46 piecewise technique, 245 principle, 245 See also HB analysis; HB errors Harmonic balance method, 30 Harmonic state space, 261

388


HB analysis of autonomous quasi-periodic regimes, 246 availability in CAD tools, 40 HB errors at generic IM product, 249 Jacobian matrix of, 250 HBT MMIC-based oscillator, 210–12 communication applications, 210 defined, 210 measured frequency tuning characteristic, 213 measured phase noise, 213 phase noise analysis, 210–11 shot noise, 211 simulated phase noise, 212 See also MMIC-based oscillators HBT model base-collector capacitances and transcapacitance, 223 base-emitter capacitances and transcapacitance, 222–23 extrinsic elements, 223 nonlinear current sources and gain, 222 See also Nonlinear models HBT noise sources, 151–53 low-frequency noise, 152–53 nonlinear model, 153 white noise, 152 HEMT-distributed model extraction, 127–30 capacitances extraction, 128–30 convective model extraction, 128 See also Model extraction procedure HEMT nonlinear model, 223–26 current source, 223–24 extrinsic elements, 226 gate-drain capacitance, 225 gate-drain diode, 224–25 gate-source capacitance, 226 gate-source diode, 225 See also Nonlinear models Heterojunction bipolar transistors (HBTs), 96–102 model extraction strategies, 115 nonlinear model, 99–100

See also Bipolar transistors; HBT noise sources HF circuit design, 93–214 conclusion, 212–14 knowledge fields, 212–14 overview, 93–95 practical examples, 195–212 rules, 182–95 tools, 156–82 transistor CAD-oriented models, 95–156 High-electron mobility transistors (HEMTs), 103, 104–8 defined, 104 DX traps in, 110 extrinsic elements, 105–6 intrinsic elements, 106–7 linear elements, 107 nonlinear capacitances, 108 Pi model of FET, 105 pseudomorphic (PHEMT), 105, 149, 204, 225–26 transconductance in, 107 See also HEMT nonlinear model High-Q oscillators, 22, 29–43 conclusions, 40 dynamics, 40–43 feedback description, 33–35 frequency, 33 large signal impedance, 31–33 large signal S-parameters, 36–38 nonlinear resistance, 34 nonresistive active circuit, 38–40 small parameter and high Q, 35–36 stability, 41–43 steady-state periodic oscillations, 29–31 underdamped amplitude response, 89 High-Q sapphire resonator oscillator, 199–200 illustrated, 299 phase noise spectrum, 200 Homojunction bipolar transistors, 96–102 Hopf bifurcation of dc solution path, 280, 281, 282, 296, 314 defined, 277 of forced solution path, 368

Index global search, 338–39 local search, 338 location of, 300–301 of one-autonomous solution path, 366–68 in oscillator performance, 294 of periodic solution path, 277 primary, 340, 341 quasi-periodic solution path from, 298 search for, 279 secondary, 340, 341 stability exchange, 283 See also Bifurcations Hopf bifurcation loci, 303 for broadband VCO, 342 efficient construction of, 337–40 equation, 338 numerically generated, 341–43 Ideal oscillator amplitude response, 88 phase noise, 64–68 transistor off in, 77 Impedance matrix, 8, 9 Impulse sensitivity function (ISF), 71 amplitude, 88 approximation by triangles, 86 coupled with noise equations, 87 decomposition, equivalent system for, 73 determination, 71 example, 71 as function of waveform, 76 n th harmonic of, 72 periodic, 72 shot noise, 81, 82 value, 86 Inductance, tank, 63 Inexact-Newton HB (INHB), 351–56 CPU time, 356 defined, 351 for forced circuits, 352–56 maximum allowed deviation, 354 preconditioner, 354 Inexact Newton update, 353–54 Injection locking, 11 Instantaneous frequency, 173

389

Intrinsic transistor, 227, 228 Intuitive criterion, 28–29 Inverse-type bifurcation, 276, 283 Iterative methods, 351–66 applications, 363–66 extension to large autonomous circuits, 360–63 INHB, 351–56 Krylov subspace basis vectors, 356–60 I-V characteristics, 19 pulse measurement, 115–17 Jacobian matrix, 264, 265 approximation, 360 condition number of, 286 definition, 339 determinant of, 266, 288 direct-Newton optimization, 347 factorization of, 319–20 of forced system, 361 HB errors, 250 magnitude of determinant of, 287 Nyquist plot of, 289 partitioning, 348 singular, 264 Jitter, 87 Kirchoff’s current law, 178 Kirk effect, 98 Krylov subspace basis vectors, 356–60 of dimension Q, 355 Kurokawa, K. (1973), 10–13 Laplace transform, 25 Large autonomous circuits applications, 370–74 extension to, 360–63 fundamental bifurcation detection, 366–70 global stability analysis, 366–74 See also Circuit(s) Large-state S-parameters, 36–38 illustrated, 38 for real-life nonlinear circuit, 39 See also S-parameters

390


LC oscillators circuit examples, 79–84 excited by current pulse, 70 phase-plane, 20 waveforms, 20 Leeson model, 68 Linear, time-invariant (LTI) models, 76 theories, 59 Linear, time-varying (LTV) model, 76 phase-noise predications with, 81 systems, 60 theory, 81 Linearity assumption, 69 of noise-to-phase conversion, 70 phase noise role, 68–79 Linear model extraction, 124 defined, 124 illustrated, 125 Load-line, 187–88 defined, 188 illustrated, 241 nonoptimum, 189 optimum, 189 Local stability, 169–72 analysis, 171 nonlinear, 169–72 See also Stability Locking range formula, 5 Lorentzian spectrum, 78 Lossless feedback circuits, 191–93 illustrated, 191 stored energy calculation, 191–93 Low-frequency noise analysis, 148 characterization, 134–50 model of FET and associated extraction, 153–56 sources, 132, 133–34 sources (HBT), 152–53 sources of one distributed FET model cell, 154 spectrum, 134 See also Noise

Low-frequency noise characterization, 226–35 cross-spectrum calculation, 234–35 equivalent circuit for noise calculations, 226–31 noise current calculation, 231–32 noise voltage calculation, 232–34 Low-phase free-running oscillator design, 182–95 Low-Q oscillators, overdamped response, 89 Maximum added power, 188–90 circuit schematic, 189 procedure, 189–90 Maximum allowed deviation, 354 Measurements, 117 channel noise, 140–45 direct gate noise, 138–40 drain noise, 143 FET noise, 138–48 I-V pulse, 115–17 pulsed S-parameter, 117–18 pulse I-V, 115–17 simultaneous input-output noise, 227 See also Noise measurements Metal oxide semiconductor FETs (MOSFETs), 103 Metal semiconductor FETs (MESFETs), 103 Microwave Electronics, 4, 7, 10 Microwave front end conversion gain and LO frequency shift, 365 schematic topology, 364 Microwave oscillators, 1–13 early papers, 2–13 electromagnetics-based optimization, 343–51 two-port, 284–87 two-port (expanded scale), 286 Mixed-mode Newton iteration, 246, 248–53 convergence properties, 258 defined, 250 for solution paths, 264 starting point, 254 See also Newton iteration

Index Mixed-mode state vectors defined, 304 Newton method for, 308 ‘‘mm,’’ 3 MMIC-based oscillators, 200–212 advantages, 220 bias point, 201 communication systems, 201 FET example, 202–10 generic multifunction for frequency generation, 203 HBT example, 210–12 load impedance, 201 main architectures, 201–2 multifunctions, 202 open-loop gain, 201 oscillation frequency sensitivity, 201 radar applications for automotive, 200 Model extraction procedure, 118–30 bipolar transistor, 119–23 FET, 123–27 HEMT-distributed, 127–30 Mode problem, 8, 10 Modified nodal analysis (MNA), 167 Mullen, J.A. (1960), 7–8 Multiple-cavity DRO, 323–25 bifurcation diagrams, 326–27 defined, 323 illustrated, 324 optimized phase noise of, 328 output power comparison, 324–25 resonator diameter/coupling, 324 See also Dielectric resonator oscillator (DRO) Multiple-stub reactance-compensating network, 326–27 Multitransistor circuits, 167 Negative conductance active, 158 calculation circuit, 158 concept, 157–59, 237 Network function, 304–5 Newton algorithm, 250 Newton iteration convergence of, 258, 259 mixed-mode, 246, 248–53

391

optimization, 307, 312 replaced by optimization algorithm, 306 solving system by, 322 starting point, 308 Noise AM, 7, 172 analysis, 179–82 channel, 140–45 circuit, 75, 136 classical, analysis, 136–37 correlation matrix, 317 drain, 143 extraction, 148–49 FM, 7 gate, 138–40 low-frequency, 132, 133–34 near the carrier, 74 phase, 59–88 PM, 176, 177, 178–82 representation in linear two-port devices, 135–37 shot, 81, 82, 148, 211 simulation, 112 sources in semiconductor devices, 130–34 voltage, 64 white, 64, 131, 132–33 Noise calculations circuit transformations for, 228, 233 current, 231–32 equivalent circuit, 226–31 voltage, 232–34 Noise measurements for accurate characterization, 137 direct channel, 140–45 direct gate, 138–40 drain, 143 experimental procedure, 148–50 experimental results, 149–50 experimental setup, 137–38 experimental setup connections, 142 experimental setup illustrated, 142 FET, 138–48 simultaneous input-output, 145

392


Noise sources (circuit-CAD-oriented), 150–56 G-R, 256 HBT, 151–53 LF noise model of FET, 153–56 Noise sources (semiconductor devices), 130–34 behavior, 132–34 G-R, 134 low-frequency noise, 131, 133–34 types of, 131–32 white noise, 131, 132–33 Noise-to-phase transfer characteristic, 60 Nonlinear analysis, 302 Nonlinear capacitances, 108, 120–22 equations, 99 model extraction, 126 See also Capacitance(s) Nonlinear characteristics dimensionless, 34–35 high-Q oscillators, 31–33 resonance, 46, 47 Nonlinear circuits, 17 Nonlinear convective model extraction, 124–26 Nonlinear dynamics, 16–23 oscillator equation, 16–19 phase-plane analysis, 19–23 Nonlinearity, 1, 68 Nonlinear models, 221–26 HBT, 222–23 HEMT, 223–26 PHEMT, 236 Nonlinear resistance high-Q oscillators, 34 N-shaped, 17 Nonlinear stability, 169–72 global, 169 local, 169–72 See also Stability Normalized determinant function (NDF), 167 Norm-reducing Newton method, 346, 348 Norton current source, 134 Norton generator, 232 Norton theorem, 112

N-shaped characteristic, 22 Numerical method, 178–82 defined, 178 noise analysis, 179–82 steady state, 179 Numerical simulations, 235–39 negative conductance, 237 phase-noise, 237–38, 241, 242 relationship between steady-state and phase-noise results, 238–39, 242, 243 small signal (AC) open-loop, 235–37 steady-state, 237, 240, 241 Nyquist analysis periodicity property and, 271 for time-periodic steady states, 270–74 Nyquist contours, 28, 29 Nyquist criterion, 24, 52–53 applying, 271 for generic periodic steady state, 272–73 modifying with high Q, 53 Nyquist diagram, 163 Nyquist locus, 172 Nyquist loop, 28, 38, 53 Nyquist plots, 171 as bounded closed curve, 272 for broadband VCO, 297 double-crossing, 281 for large circuits, 273 for simple oscillator, 26–27 for stable dc state of DRO, 289 for stable periodic state (point S) of DRO, 293–94 for unstable dc state of DRO, 291 for unstable period state (point U) of DRO, 292–93 One-autonomous solution path, 366–68 D-type bifurcations of, 369–70 Hopf bifurcations of, 366–68 See also Solution paths One-port oscillators, 183 Open-loop approach, 159–60 circuit simulation, 237 representation, 159 simulated circuit, 238–39

Index simulation, 235–37 simulation results, 164 Open-loop gain, 159, 164 complex, defined, 235 defined, 159 Operating point stability, 23–29 Optimization broadband, 310 of circuit energy, 190–93 continuously tunable oscillators, 308–10 direct-Newton, 344–49 discrete tuning oscillators, 310–12 electromagnetics-based, 343–51 free-running oscillators, 303–8 general methods, 303–12 of load capacitance, 185–86 Newton-iteration based, 307, 312 of power at oscillation frequency, 186–87 with substitution methods, 312–15 Oscillation buildup, 41, 42 after bias turn-on, 318 condition, 318 optimization, 318–19 Oscillations linear theory of, 50 locked, 48 N-shaped characteristic, 22 relaxation, 21, 23 spurious, 294–301 steady-state, 29–31, 41 theory, 36 Oscillations (in presence of external signal), 43–48 circuit equations, 43–45 introduction, 43 resonance characteristics, 45–48 Oscillator benchmark simulations, 235–39 negative conductance, 237 phase noise, 237–38, 241, 242 relation between steady-state and phase-noise results, 238–39, 242, 243 small signal (AC) open-loop, 235–37 steady-state, 237, 240, 241

393

Oscillator design, 15–50 for asynchronous stability, 337–43 CAD-oriented, 301–43 CAD tools and, 15–16 complexity, 302 high-Q oscillators, 29–43 introduction, 15–16 nonlinear dynamics, 16–23 operating point stability, 23–29 oscillations in presence of external signal, 43–48 spurious-free, 340–43 summary, 48–50 Oscillator equation, 16–19 Oscillator/harmonic mixer multifunction, 208–10 block diagram, 210 defined, 208–10 phase noise, 211 photograph, 211 Oscillators breadboard, 195–200 cavity, 30 circuit design tools, 156–82 Clapp, 80 Colpitts, 79 continuously tunable, 308–10 dielectric resonator (DRO), 30, 208, 287–94 with discrete tuning, 310–12 doubling, 197 free-running, 160–65, 182–95 function of, 1 high-Q, 22, 29–43, 89 high-Q sapphire resonator, 199–200 ideal, 64–68 LC, 20, 79–84 low-Q, 89 microwave, 1–13 MMIC-based, 200–212 multiple arrangements, 8 nonlinearity, 1 one-port, 183 operating regimes, 63 output waveform, 175–76 phase equation, 4–5 quarter-wave lumped feedback, 195

394


Oscillators (continued) ring, 84–88 RLC, 61 SAW, 30 simple, 16–23, 26 sinusoidal, 156–67 symmetrical, 9, 10, 165–66 symmetrical negative resistance, 83 transistor, 5, 12, 167–69 two-port microwave, 284–87 VCO, 82, 83, 203–8, 212–13, 295–99, 325–36 XTO, 30 YIG, 30 Output power, 5 Output spectrum broadband VCO, 336 double-ring mixer, 373, 374 observable, 7–8 voltage, 78 Parameter switching, 370 Parseval’s theorem, 75 Paths solution, 261–66 subcritical, 282 supercritical, 282 Period-doubling bifurcations, 278 approximately finding, 368 of periodic solution path, 368–69 See also Bifurcations Periodicity error, 278 Periodic solution path, 274–77 for DRO, 292 period-doubling bifurcations of, 368–69 turning points, 301 two-port microwave oscillator, 285 two-port microwave oscillator (expanded scale), 286 two-port oscillator parameterized by 1/T, 323 See also Solution paths Phase changing with time, 6 difference, 11 equation, 4–5

Phase-locking condition, 2, 10 Phase modulation (PM) noise modulation, 176 at offset frequency, 177 simulation methods, 181 spectra processing, 178–82 See also Noise Phase noise, 59–88 38.25 GHz VCO, 206, 209 calculation, 175–78 characterization, 172–82 circuit noise evolution into, 75 close-in, 78–79 detailed considerations, 64–68 domination, 60–61 five-stage, 85 ideal oscillator, 64–68 linearity and, 68–79 low-frequency dynamic influence on, 242 numerical method description, 178–82 in one-port circuits, 184–86 oscillator/harmonic mixer multifunction, 211 output signal spectrum, 173–75 predications with LTV model, 81 ring oscillators, 84 simulation, 237–38 simulation illustration, 242 spectrum, 67 time variation and, 68–79 VCOs, 83 See also Noise Phase-plane analysis, 19–23 generalizations of, 22–23 power of, 20 Phase space, 261 Phase-to-voltage conversion, 74 Piecewise HB method, 245 Pierce, J.R. (1943), 3–4 Pi FET model, 105 with all parasitics, 106 nonlinear equivalent circuits, 105 Pitchfork bifurcation, 276, 280–81 Power consumption, 62 Preconditioner, 354

Index Pseudomorphic HEMT (PHEMT), 105 current source parameters, 225 extrinsic elements, 226 GaAs-based, 149 gate-drain capacitance, 225 gate-source capacitance, 226 low-frequency noise limitations, 204 See also High-electron mobility transistors (HEMTs) Pulsed S-parameter measurements, 117–18 Pulse I-V measurement principle, 116 pulse duration/duty cycle compromise, 116–17 setup, 115–17 Push-pull circuits, 8 Quarter-wave lumped feedback oscillators, 195 Quasi-Newton method, 255 Quasi-periodic steady states, 267–70, 298 Reference harmonics, 250 Regular turning points, 264, 283 Relaxation oscillations, 21, 23 Resonance characteristics oscillations (in presence of external signal), 45–48 for oscillator with cross-coupled BJTs, 47 for van der Pol equation, 46 Resonators admittance, 33 bandwidth, 67 dielectric, 197 high-Q, 23, 29, 38 RLC, 61 tapped, 80 Ring oscillators, 84–88 circuit examples, 84–88 differential, 87–88 jitter, 87–88 phase-noise performance, 84 single-ended CMOS, 87 RLC oscillators, 61 RLC resonator, 61 Rucker, C.T. (1969), 8–10

395

Saddle-node bifurcation, 276, 280 SAW oscillators, 30 Self-oscillating mixer drain voltage spectrum, 258 schematic topology, 256 Shot noise defined, 148 HBT MMIC-based oscillator, 211 ISFs for, 81, 82 See also Noise Signal spectrum, 247 Simple oscillators illustrated, 17 linearization of, 26 nonlinear dynamics, 16–23 Nyquist plots for, 26–27 Sinusoidal oscillators, 156–67 linear operating principles, 156 negative-conductance concept, 157–59 open-loop approach, 159–60 start conditions, 160–65 start in multitransistor circuits, 167 start in symmetrical circuits, 165–66 Slotted section, 4 Smith chart, 4 Solution paths, 261–66 dc, 279–82 defined, 263 forced, 368 mixed-mode Newton iteration and, 264 numerical construction of, 262–66 one-autonomous, 366–68, 369–70 periodic, 274–77 quasi-periodic, 298 state-space representation, 261–62 Space-mapping, 343 S-parameters, 25, 26 large-signal, 36–38 measured and computed, 121 measured/modeled, comparison, 129 Nyquist plots for, 26–27 pulsed measurements, 117–18 Spurious-free design, 340–43 Spurious oscillations, 294–301 Stability, 23–29 asynchronous, 337–43

396


Stability (continued) circuit linearization, 24–25 conclusions, 29 constant amplitude, 57 counterexample, 25–28 exchange at bifurcations, 282–84 global, 169 high-Q oscillators, 41–43 importance, 302 introduction, 23–24 intuitive criterion, 28–29 local, 169–72 nonlinear, 169–72 Nyquist criterion, 24, 52–53 portrait, 299, 300–301 steady-state synchronous, 315–17 Stability analysis, 260–301 applications, 284–94 global, 274–84, 366–74 quasi-periodic steady states, 267–70 solution paths, 261–66 spurious oscillations and, 294–301 time-periodic steady states, 270–74 State space autonomous, 262 harmonic, 261 representation, 261–62 Stationary solution, 254 Steady-state analysis, 167–69 simulation conditions, 168–69 simulation results, 168 simulation techniques, 167–68 Steady-state oscillations amplitude, 41 periodic, 29–31 Steady states asynchronously unstable, 268 autonomous, 273, 274 generic periodic, 272–73 output power spectrum of, 300–301, 302 quasi-periodic, 267–70, 298 stable, 268 synchronously unstable, 268 time-periodic, 270–74 Steady-state simulation, 237

Steady-state synchronous stability, 315–17 ensuring, 316 margin, increasing, 317 Subcritical paths, 282 Substitution algorithm, 312–15 advantages, 314 defined, 312 Substrate trap model, 111 Supercritical paths, 282 Symmetrical negative resistance oscillator, 83 Symmetrical oscillators, 9, 10 illustrated, 165 oscillation start in, 165–66 Synchronous instability, 268, 315 Tank capacitance, 63 conductance, 64 inductance, 63 LC, 70 RLC, 65 Tapped resonators, 80 Taylor series, 24 Thermal circuit extraction, 126–27 Thermal model extraction, 122–23 thermal resistance determination, 122 thermal time constant determination, 123 Thermal noise amplitude and, 65 phase and, 65 white, 64 Time domain integration (TDI), 167 Time domain Monte Carlo (TDMC), 174–75 defined, 175 drawback, 175 Time invariance, 69 Time-periodic steady states, 270–74 Time variation, 68–79 Total correlation matrix, 180 Trajectory tracking, 19 Transcapacitances HBT base-collector, 223 HBT base-emitter, 222–23 nonlinear, 120–22

Index Transcritical bifurcation, 276, 280–81 Transistor bias, 144 Transistor CAD-oriented models, 95–156 black-box, 95 circuit-CAD-oriented noise sources, 150–56 defined, 95 extraction procedure, 118–30 FET operating/modeling, 102–15 homojunction/heterojunction bipolar transistor, 96–102 introduction, 95–96 I-V and S-parameter measurement system, 115–18 low-frequency noise characterization, 134–50 noise sources in semiconductor devices, 130–34 physically based, 95 See also HF circuit design Transistor load-line, 187–88 Transistor low-frequency noise characterization, 134–50 experimental procedure, 148–50 experimental setup, 137–38 FET noise measurements, 138–48 introduction, 134–35 in linear two-port devices, 135–37 Transistor oscillators circuit parameters, 5 circuits, 186–87 functional diagram, 157 Hartley and Colpitts, 12 negative-conductance concept, 157–59 steady-state analysis, 167–69 Transistors input admittance, 192 intrinsic, 227, 228 low-frequency noise characterization, 226–35 maximum added power of, 188–90 Tuning characteristic, 309 conditions of DRO, 326–27 discrete, 310–12 parameters, 249 physically different, 311

397

Turning point, 265 on periodic solution paths, 301 regular, 264, 283 Two-port microwave oscillator, 284–87 condition number of Jacobian matrix, 286 periodic-solution path, 285 periodic-solution path (expanded scale), 286 schematic circuit layout, 345 schematic topology, 284 See also Microwave oscillators Vacuum Tube Oscillators, 4 Van der Pol (1927), 2–3 nonlinear resonance characteristics, 46 results validity, 17 VCOs 38.25 GHz, 205–8 behavior, 340 broadband, 295–99, 325–36 design problem, 340 for FMCW radar, 203 multifunction on MMIC, 205–8 multifunction using external resonator, 203–5 phase noise, 83 pulsed radar, 203 region of convergence of Newton iteration for, 259 schematic, 82 schematic topology, 259 transistor current source load line, 204 tuning characteristics, 332, 333 tuning range, 340 X-band HBT-based, 212–13 Voltage bias, 304 breakdown, 98 current transformation, 56–57 mean-square noise, 65, 66 noise, 64 output spectrum, 78 Voltage-controlled oscillators. See VCOs White noise sources, 131, 132–33 sources (HBT), 152

398


White noise (continued) thermal, 64 See also Noise X-band HBT-based VCO, 212–13 measured frequency tuning characteristic, 213

measured phase noise, 213 simulated phase noise, 212 See also VCOs XTO oscillators, 30 YIG oscillators, 30 Zero-crossing timings, 70

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