Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures

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Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures Cam Nguyen Copyright  2000 John Wiley & Sons, Inc. Print ISBN 0-471-01750-7 Electronic ISBN 0-471-20067-0

Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures

Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures CAM NGUYEN Texas A&M University

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc., is aware of a claim, the product names appear in initial capital or ALL CAPITAL LETTERS. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Copyright  2001 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. ISBN 0-471-20067-0. This title is also available in print as ISBN 0-471-01750-7. For more information about Wiley products, visit our web site at www.Wiley.com. Library of Congress Cataloging-in-Publication Data: Nguyen, Cam Analysis methods for RF, microwave, and millimeter-wave planar transmission line structures/Cam Nguyen. p.cm. — (Wiley series in microwave and optical engineering) “Wiley-Interscience publication.” Includes index. ISBN 0-471-01750-7 (cloth : alk. paper) 1. Electric circuit analysis. 2. Microwave transmission lines. 3. Strip transmission lines. 4. Microwave integrated circuits. 5. Electric circuit analysis. I. Series. TK7876.N48 2000 621.3810 31–dc21 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

99-086737

To my wife, Ngo.c-Diê.p, and my children, Christine (Nhã-Uyên) and Andrew (An)

Contents Preface

xi

1 Introduction

1

1.1 Planar Transmission Lines and Microwave Integrated Circuits 1.2 Analysis Methods for Planar Transmission Lines 1.3 Organization of the Book 2 Fundamentals of Electromagnetic Theory

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Maxwell’s Equations Constitutive Relations Continuity Equation Loss in Medium Boundary Conditions Skin Depth Power Flow Poisson’s and Laplace’s Equations Wave Equations Electric and Magnetic Potentials Wave Types and Solutions 2.11.1 Wave Types 2.11.2 Wave Solutions

2.12 Orthogonality Relations h 2.12.1 Orthogonality Relations Between mn x, y and e Between mn x, y 2.12.2 Orthogonality Relations Between Electric Fields and Between Magnetic Fields

1 7 9 12

12 14 15 15 17 18 19 19 20 21 23 23 24 28 28 31

vii

2.12.3 Orthogonality Relations Between Electric and Magnetic Fields 2.12.4 Power Orthogonality for Lossless Structures

32 35

References Problems

37 37

3 Green’s Function

39

3.1 Descriptions of Green’s Function 3.1.1 Solution of Poisson’s Equation Using Green’s Function 3.1.2 Solution of the Wave Equation Using Green’s Function

39

3.2 Sturm–Liouville Equation 3.3 Solutions of Green’s Function 3.3.1 Closed-Form Green’s Function 3.3.2 Series-Form Green’s Function 3.3.3 Integral-Form Green’s Function

42 44 44 49 53

References Problems Appendix: Green’s Identities 4 Planar Transmission Lines

39 41

56 56 62 63

4.1 Transmission Line Parameters 4.1.1 Static Analysis 4.1.2 Dynamic Analysis

64 64 66

4.2 4.3 4.4 4.5 4.6

Microstrip Line Coplanar Waveguide Coplanar Strips Strip Line Slot Line

68 71 74 76 78

References Problems

80 81

5 Conformal Mapping

5.1 5.2 5.3 5.4

Principles of Mappings Fundamentals of Conformal Mapping The Schwarz–Christoffel Transformation Applications of the Schwarz–Christoffel Transformation in Transmisison Line Analysis 5.5 Conformal-Mapping Equations for Common Transmission Lines

85

85 87 95 98 106

References Problems 6 Variational Methods

112 113 120

6.1 Fundamentals of Variational Methods 6.2 Variational Expressions for the Capacitance per Unit Length of Transmission Lines 6.2.1 Upper-Bound Variational Expression for C 6.2.2 Lower-Bound Variational Expression for C 6.2.3 Determination of C, Zo , and εeff

121

6.3 Formulation of Variational Methods in the Space Domain 6.3.1 Variational Formulation Using Upper-Bound Expression 6.3.2 Variational Formulation Using Lower-Bound Expression

128

6.4 Variational Methods in the Spectral Domain 6.4.1 Lower-Bound Variational Expression for C in the Spectral Domain 6.4.2 Determination of C, Zo , and εeff 6.4.3 Formulation References Problems Appendix: Systems of Homogeneous Equations from the Lower-Bound Variational Formulation 7 Spectral-Domain Method

7.1 Formulation of the Quasi-static Spectral-Domain Analysis 7.2 Formulation of the Dynamic Spectral-Domain Analysis References Problems Appendix A: Fourier Transform and Parseval’s Theorem Appendix B: Galerkin’s Method 8 Mode-Matching Method

123 124 125 127

128 130 135 135 137 138 142 143 148 152

152 162 176 177 186 188 191

8.1 Mode-Matching Analysis of Planar Transmission Lines 8.1.1 Electric and Magnetic Field Expressions 8.1.2 Mode-Matching Equations

191 193 198

8.2 Mode-Matching Analysis of Planar Transmission Line Discontinuities 8.2.1 Electric and Magnetic Field Expressions 8.2.2 Single-Step Discontinuity

203 203 207

8.2.3 Double-Step Discontinuity 8.2.4 Multiple-Step Discontinuity References Problems Appendix A: Coefficients in Eqs. (8.62) Appendix B: Inner Products in Eqs. (8.120)–(8.123) Index

211 214 221 222 228 233 237

Preface RF integrated circuit (RFICs) and microwave integrated circuits (MICs), both hybrid and monolithic, have advanced rapidly in the last two decades. This progress has been achieved not only because of the advance of solid-state devices, but also due to the progression of planar transmission lines. Many milestones have been achieved: one of them being the development of various analysis methods for RF microwave and millimeter-wave passive structures, in general, and planar transmission lines, in particular. These methods have played an important role in providing accurate transmission line parameters for designing RFICs and MICs, as well as in investigating and developing new planar transmission lines. The primary objective of this book is to present the Green’s function, conformal-mapping, variational, spectral-domain, and mode-matching methods, which are some of the most useful and commonly used techniques for analyzing planar transmission lines. Information for these methods in the literature is at a level that is not very suitable for the majority of first-year graduate students and practicing RF and microwave engineers. The material in this book is selfcontained and presented in a way that allows readers with only fundamental knowledge in electromagnetic theory to easily understand and implement the techniques. The book also includes problems at the end of each chapter, allowing readers to reinforce their knowledge and to practice their understanding. Some of these problems are relatively long and difficult, and thus are more suitable for class projects. The book can therefore serve not only as a textbook for first-year graduate students, but also as a reference book for practicing RF and microwave engineers. Another purpose of the book is to use these methods as means to present the principles of applying electromagnetic theory to the analysis of microwave boundary-value problems. This knowledge is essential for microwave students and engineers, as it allows them to modify and improve these methods, as well as to develop new techniques. This book is based on the material of a graduate course on field theory for microwave passive structures offered at Texas A&M University. It is completely self-contained and requires readers to have only the fundamentals

of electromagnetic theory, which is normally fulfilled by the first undergraduate course in electromagnetics. I sincerely appreciate the patience of Professor Kai Chang, Editor of the Wiley Series in Microwave and Optical Engineering, and Mr. George Telecki, Executive Editor of Wiley-Interscience, during the writing of the manuscript for this book. I am also grateful to my former students who took the course and provided me with a purpose for writing this book. Finally, I wish to express my heartfelt thanks and deepest appreciation to my wife, Ngoc-Diep, for her constant encouragement and support, and my children, Christine and Andrew, for their understanding during the writing of this book. CAM NGUYEN College Station, Texas


Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures


WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANG, Editor Texas A&M University

FIBER-OPTIC COMMUNICATION SYSTEMS, Second Edition Govind P. Agrawal COHERENT OPTICAL COMMUNICATIONS SYSTEMS Silvello Betti, Giancarlo De Marchis and Eugenio Iannone HIGH-FREQUENCY ELECTROMAGNETIC TECHINQUES: RECENT ADVANCES AND APPLICATIONS Asoke K. Bhattacharyya COMPUTATIONAL METHODS FOR ELECTROMAGNETICS AND MICROWAVES Richard C. Booton, Jr. MICROWAVE RING CIRCUITS AND ANTENNAS Kai Chang MICROWAVE SOLID-STATE CIRCUITS AND APPLICATIONS Kai Chang RF AND MICROWAVE WIRELESS SYSTEMS Kai Chang DIODE LASERS AND PHOTONIC INTEGRATED CIRCUITS Larry Coldren and Scott Corzine RADIO FREQUENCY CIRCUIT DESIGN W. Alan Davis and Krishna Agarwal MULTICONDUCTOR TRANSMISSION-LINE STRUCTURES: MODAL ANALYSIS TECHNIQUES J. A. BrandaQ o Faria PHASED ARRAY-BASED SYSTEMS AND APPLICATIONS Nick Fourikis FUNDAMENTALS OF MICROWAVE TRANSMISSION LINES Jon C. Freeman OPTICAL SEMICONDUCTOR DEVICES Mitsuo Fukuda MICROSTRIP CIRCUITS Fred Gardiol HIGH-SPEED VLSI INTERCONNECTIONS: MODELING, ANALYSIS, AND SIMULATION A. K. Goel FUNDAMENTALS OF WAVELETS: THEORY, ALGORITHMS, AND APPLICATIONS Jaideva C. Goswami and Andrew K. Chan

ANALYSIS AND DESIGN OF INTERGRATED CIRCUIT ANTENNA MODULES K. C. Gupta and Peter S. Hall PHASED ARRAY ANTENNAS R. C. Hansen HIGH-FREQUENCY ANALOG INTEGRATED CIRCUIT DESIGN Ravender Goyal(ed.) MICROWAVE APPROACH TO HIGHLY IRREGULAR FIBER OPTICS Huang Hung-Chia NONLINEAR OPTICAL COMMUNICATION NETWORKS Eugenio Iannone, Franceso Matera, Antonio Mecozzi, and Marina Settembre FINITE ELEMENT SOFTWARE FOR MICROWAVE ENGINEERING Tatsuo Itoh, Giuseppe Pelosi and Peter P. Silvester (eds.) INFRARED TECHNOLOGY: APPLICATIONS TO ELECTROOPTICS, PHOTONIC DEVICES, AND SENSORS A. R. Jha SUPERCONDUCTOR TECHNOLOGY: APPLICATIONS TO MICROWAVE, ELECTROOPTICS, ELECTRICAL MACHINES, AND PROPULSION SYSTEMS A. R. Jha OPTICAL COMPUTING: AN INTRODUCTION M. A. Karim and A. S. S. Awwal INTRODUCTION TO ELECTROMAGNETIC AND MICROWAVE ENGINEERING Paul R. Karmel, Gabriel D. Colef, and Raymond L. Camisa MILLIMETER WAVE OPTICAL DIELECTRIC INTEGRATED GUIDES AND CIRCUITS Shiban K. Koul MICROWAVE DEVICES, CIRCUITS AND THEIR INTERACTION Charles A. Lee and G. Conrad Dalman ADVANCES IN MICROSTRIP AND PRINTED ANTENNAS Kai-Fong Lee and Wei Chen (eds.) OPTICAL FILTER DESIGN AND ANALYSIS: A SIGNAL PROCESSING APPROACH Christi K. Madsen and Jian H. Zhao THEORY AND PRACTICE OF INFRARED TECHNOLOGY FOR NONDESTRUCTIVE TESTING Xavier Maldague OPTOELECTRONIC PACKAGING A. R. Mickelson, N. R. Basavanhally, and Y. C. Lee (eds.) OPTICAL CHARACTER RECOGNITION Shunji Mori, Hirobumi Nishida, and Hiromitsu Yamada ANTENNAS FOR RADAR AND COMMUNICATIONS: A POLARIMETRIC APPROACH Harold Mott INTEGRATED ACTIVE ANTENNAS AND SPATIAL POWER COMBINING Julio A. Navarro and Kai Chang ANALYSIS METHODS FOR RF, MICROWAVE, AND MILLIMETER-WAVE PLANAR TRANSMISSION LINE STRUCTURES Cam Nguyen FREQUENCY CONTROL OF SEMICONDUCTOR LASERS Motoichi Ohtsu (ed.) SOLAR CELLS AND THEIR APPLICATIONS Larry D. Partain (ed.) ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES Clayton R. Paul

INTRODUCTION TO ELECTROMAGNETIC COMPATIBILITY Clayton R. Paul ELECTROMAGNETIC OPTIMIZATION BY GENETIC ALGORITHMS Yahya Rahmat-Samii and Eric Michielssen (eds.) INTRODUCTION TO HIGH-SPEED ELECTRONICS AND OPTOELECTRONICS Leonard M. Riaziat NEW FRONTIERS IN MEDICAL DEVICE TECHNOLOGY Arye Rosen and Harel Rosen (eds.) ELECTROMAGNETIC PROPAGATION IN MULTI-MODE RANDOM MEDIA Harrison E. Rowe ELECTROMAGNETIC PROPAGATION IN ONE-DIMENSIONAL RANDOM MEDIA Harrison E. Rowe NONLINEAR OPTICS E. G. Sauter COPLANAR WAVEGUIDE CIRCUITS, COMPONENTS, AND SYSTEMS Rainee N. Simons ELECTROMAGNETIC FIELDS IN UNCONVENTIONAL MATERIALS AND STRUCTURES Onkar N. Singh and Akhlesh Lakhtakia (eds.) FUNDAMENTALS OF GLOBAL POSITIONING SYSTEM RECEIVERS: A SOFTWARE APPROACH James Bao-yen Tsui InP-BASED MATERIALS AND DEVICES: PHYSICS AND TECHNOLOGY Osamu Wada and Hideki Hasegawa (eds.) DESIGN OF NONPLANAR MICROSTRIP ANTENNAS AND TRANSMISSION LINES Kin-Lu Wong FREQUENCY SELECTIVE SURFACE AND GRID ARRAY T. K. Wu (ed.) ACTIVE AND QUASI-OPTICAL ARRAYS FOR SOLID-STATE POWER COMBINING Robert A. York and Zoya B. Popović (eds.) OPTICAL SIGNAL PROCESSING, COMPUTING AND NEURAL NETWORKS Francis T. S. Yu and Suganda Jutamulia SiGe, GaAs, AND InP HETEROJUNCTION BIPOLAR TRANSISTORS Jiann Yuan ELECTRODYNAMICS OF SOLIDS AND MICROWAVE SUPERCONDUCTIVITY Shu-Ang Zhou


CHAPTER ONE

Introduction Microwave integrated circuits (MICs) were introduced in the 1950s. Since then, they have played perhaps the most important role in advancing the radiofrequency (RF) and microwave technologies. The most noticeable and important milestone was possibly the emergence of monolithic microwave integrated circuits (MMICs). This progress of MICs would not have been possible without the advances of solid-state devices and planar transmission lines. Planar transmission lines refer to transmission lines that consist of conducting strips printed on surfaces of the transmission lines’ substrates. These structures are the backbone of MICs, and represent an important and interesting research topic for many microwave engineers. Along with the advances of MICs and planar transmission lines, numerous analysis methods for microwave and millimeter-wave passive structures, in general, and planar transmission lines, in particular, have been developed in response to the need for accurate analysis and design of MICs. These analysis methods have in turn helped further investigation and development of new planar transmission lines. This book presents the Green’s function, conformal-mapping, variational, spectral-domain, and mode-matching methods. They are useful and commonly used techniques for analyzing microwave and millimeter-wave planar transmission lines, in particular, and passive structures, in general. Information for these methods in the literature is at a level that is not very suitable for the majority of first-year graduate students and practicing microwave engineers. This book attempts to present the materials in such a way as to allow students and engineers with basic knowledge in electromagnetic theory to understand and implement the techniques. The book also includes problems for each chapter so readers can reinforce and practice their knowledge. 1.1 PLANAR TRANSMISSION LINES AND MICROWAVE INTEGRATED CIRCUITS

Planar transmission lines are essential components of MICs. They have been used to realize many circuit functions, such as baluns, filters, hybrids, and couplers, as well as simply to carry signals. Figure 1.1 shows some commonly used planar 1

2

INTRODUCTION

METAL

SUBSTRATE METAL MICTROSTRIP LINE

SUBSTRATE METAL STRIP LINE METAL

SUBSTRATE SUSPENDED STRIP LINE DIELECTRIC

SLOT

METAL FIN LINE

Figure 1.1

Common planar transmission lines.

PLANAR TRANSMISSION LINES AND MICROWAVE INTEGRATED CIRCUITS

METAL SLOT

SUBSTRATE SLOT LINE

SUBSTRATE

METAL INVERTED MICROSTRIP LINE METAL

SUBSTRATE COPLANAR WAVEGUIDE METAL

SUBSTRATE COPLANAR STRIPS

Figure 1.1

(Continued )

3

4

GHz GHz GHz GHz GHz

Microstrip line Strip line Suspended

Ä220 Ä110 Ä220 Ä110 Ä110

Ä110 GHz Ä60 GHz Ä220 GHz

Transmission Line

Fin line Slot line Inverted microstrip line Coplanar waveguide Coplanar strips

Operating Frequency (GHz)

20–400 60–200 25–130 40–150 30–250

10–100 20–150 20–150

Characteristic Impedance Range (Ohm)

Moderate Small Small Small Small

Small Moderate Moderate

Dimension

TABLE 1.1 Properties of Planar Transmission Lines Shown in Fig. 1.1

Moderate High Moderate High High

High Low Low

Loss

Low Low Low Low Low

Low Low Low

Power Handling

Fair Moderate Moderate strip line Easy Easy Moderate Very easy Easy

Solid-State Device Mounting

Fair Good Fair Good Good

Good Good Fair

Low-Cost Production

PLANAR TRANSMISSION LINES AND MICROWAVE INTEGRATED CIRCUITS

5

transmission lines and Table 1.1 summarizes their properties. Each transmission line has its own unique advantages and disadvantages and, depending on circuit types, either an individual transmission line or a combination of them is needed to achieve desired circuit functions as well as optimum performances. The most viable planar transmission lines are perhaps the conventional microstrip line and coplanar waveguide (CPW), from which many other planar transmission lines have evolved. Multilayer planar transmission lines, such as that shown in Fig. 1.2, are especially attractive for MICs due to their flexibility and ability to realize complicated circuits, ultimately allowing very compact, high-density circuit integration. They also allow thin dielectric layers to be deposited on conductor-backed semiconducting substrates for achieving ultracompact MICs. Furthermore, multilayer transmission lines have significantly less cross talk and distortion via appropriate selection of dielectric layers. There are two classes of MICs: hybrid and monolithic circuits. Hybrid MIC refers to a planar circuit in which only parts of the circuit are formed on surfaces of the circuit’s substrates by some deposition schemes. A typical hybrid MIC has all the transmission lines deposited on the dielectric surfaces, except solid-state devices such as transistors and other passive components like capacitors. These solid-state devices and passive elements are discrete components and connected to the transmission lines by bonding, soldering, or conducting epoxy. The substrates of a hybrid MIC are generally low-loss insulators, used solely for supporting the circuit components and delivering the signals. Advantages of hybrid MICs include small size, light weight, easy fabrication, low cost, and high-volume production. In practice, hybrid MICs are normally referred to simply as MICs. Figures 1.3–1.7 show photographs of some hybrid MICs employing planar transmission lines. METAL

METAL SUBSTRATE

Figure 1.2

A multilayer planar transmission line.

6

INTRODUCTION

Figure 1.3 S-band (2–4 GHz) MIC push–pull field effect transistor (FET) amplifier using CPW and slot line.

Figure 1.4 W-band (75–110 GHz) MIC diode balanced mixer using fin line, CPW, and suspended strip line.

ANALYSIS METHODS FOR PLANAR TRANSMISSION LINES

7

(a)

(b)

Figure 1.5 Top (a) and bottom (b) sides of an S-band MIC bandpass filter using multilayer broadside-coupled CPW.

The monolithic MIC (MMIC) is a special class of MICs, in which all the circuit elements, including passive components and solid-state devices, are formed into the bulk or onto the surface of a semi-insulating semiconductor substrate by some deposition technique. In contrast to hybrid MICs, the substrates are used in MMICs not only as a signal-propagating medium and a supporting structure for passive components, but also as a material onto which semiconducting layers with good properties for realizing microwave solid-state devices are grown or deposited. Compared to hybrid MICs, the advantages of MMICs are lower-cost circuits through batch processing, improved reliability and reproducibility through minimization of wire bonds and discrete components, smaller size and weight, more circuit design flexibility, and multifunction performance on a single chip. MMICs are very important for microwave technology. Most microwave and millimeter-wave applications are expected eventually to employ all MMICs. Figure. 1.8 shows a photograph of a Ka-band (26.5–40 GHz) push–push MMIC oscillator. 1.2

ANALYSIS METHODS FOR PLANAR TRANSMISSION LINES

In using planar transmission lines in MICs, analysis methods are needed in order to determine the transmission lines’ characteristics such as characteristic

8

INTRODUCTION

Figure 1.6

A 5–20 GHz MIC balun using microstrip line.

impedance, effective dielectric constant, and loss. The design of MICs depends partly on accurate analysis of planar transmission lines. The microwave technology is changing rapidly and, in connecting with it, useful analysis methods for microwave and millimeter-wave planar transmission lines, either completely brand new or modifications of existing techniques, appear constantly. In fact, microwave engineers are now faced with many different techniques and a vast amount of information, making the techniques difficult to understand and hence to implement in the short time normally encountered in an industrial setting. Each method has its own unique advantages and disadvantages for particular problems and needs. However, they are all based on Maxwell’s

ORGANIZATION OF THE BOOK

Figure 1.7

9

Ka-band MIC bandstop filter using suspended strip line.

equations, in general, and wave equations and boundary conditions, in particular. These are the fundamentals of these methods and, while techniques can change steadily, the fundamentals always remain the same. They, in fact, provide a foundation for the derivation, modification, and implementation of all current and future analysis methods. In this book, we describe particularly the details of the Green’s function, conformal-mapping, variational, spectral-domain, and mode-matching methods. These methods not only represent some of the most useful and commonly used techniques for analyzing planar transmission lines, but also serve as means to present the fundamentals of applying electromagnetic theory to the analysis of microwave boundary-value problems. This knowledge would then allow readers to modify and improve these methods, or to develop new techniques.

1.3


The book is organized into eight chapters and is self-contained. Chapter 2 gives the fundamentals of electromagnetic theory, which are needed for the formulation of the methods addressed in this book. Chapter 3 covers Green’s functions used in various methods. Chapter 4 discusses the fundamentals of planar transmission lines and provides useful equations for commonly used transmission lines

10

INTRODUCTION

Figure 1.8

Ka-band MMIC push–push FET oscillator using microstrip line.


11

in MICs. Chapter 5 covers the principles of conformal mapping and demonstrates its use in analyzing planar transmission lines. Chapter 6 presents the variational methods in both the space and spectral domains and uses them to analyze planar transmission lines. Chapter 7 gives the foundation of the spectraldomain methods and then applies them in the analysis of planar transmission lines. Finally, Chapter 8 formulates the mode-matching method for both planar transmission lines and their discontinuities.


CHAPTER TWO

Fundamentals of Electromagnetic Theory Electromagnetic theory forms the foundation for electrical engineering. Not only can it be used to explain many phenomena of electronic, it can be employed to design and analyze accurately many electronic circuits operating across the electromagnetic spectrum. While circuit theory may fail to explain adequately an electrical phenomenon or to accurately analyze and design an electronic circuit, electromagnetic theory, in general, will not. In this chapter we will review sufficient fundamentals of electromagnetics to allow readers to understand the methods we will present in subsequent chapters.

2.1

MAXWELL’S EQUATIONS

Perhaps the most important equations in electromagnetic theory are Maxwell’s equations, which altogether create the foundation of electromagnetic theory. Maxwell’s equations can be written in a differential or integral form. For general time-varying electromagnetic fields, they are given as follows: Differential Form ∂B ∂t ∂D WðH DJC ∂t WÐD D WðE D

WÐB D0 12

2.1a 2.1b 2.1c 2.1d

MAXWELL’S EQUATIONS

Integral Form

d E Ð d D B Ð dS dt d H Ð d D J Ð dS C D Ð dS dt

13

B Ð dS D 0

2.2a 2.2b 2.2c

D Ð dS D

dv

2.2d

where E x, y, z, t is electric field or electric field intensity in volts/meter (V/m) H x, y, z, t is magnetic field or magnetic field intensity in amperes/meter (A/m) D x, y, z, t is electric flux density in coulombs/meter2 (C/m2 ) B x, y, z, t is magnetic flux density in webers/meter2 (Wb/m2 ) Jx, y, z, t is electric current density in amperes/meter2 (A/m2 ) x, y, z, t is electric charge density in coulombs/meter3 (C/m3 )

The integral Maxwell equations can be derived from their differential forms by using the Stokes and divergence theorems. The parameter defined by Jd D

dD dt

2.3

is known as the displacement current density (in A/m2 ). The time (t) and location (e.g., x, y, and z) dependence are assumed for all these fields. These Maxwell equations are general and hold for fields with arbitrary time dependence in any electronic structure and at any location in the structure. They become simpler for special cases such as static or quasi-static fields, sinusoidal time-varying (or time-harmonic) fields, and source-free media. Under the assumption of static or quasi-static, we let d/dt equal zero and write the differential-form Maxwell equations as O D0 WðE

2.4a

O D JO WðH

2.4b

O D WÐD

2.4c

O D0 WÐB

2.4d

Note that the static quantities are denoted with a hat ( O ). These field quantities are independent of time. It should be noted that these equations are only valid

14

FUNDAMENTALS OF ELECTROMAGNETIC THEORY

with direct current (dc). In most engineering practices, however, they can also be used when the operating frequencies are not high. For the case of time-harmonic fields, we can replace d/dt by jω and obtain Maxwell’s equations as W ð E D jωB

2.5a

W ð H D J C jωD

2.5b

WðDD

2.5c

WÐBD0

2.5d

where the fields now represent phasor fields, which are functions of location only. These Maxwell equations are commonly known as the time-harmonic Maxwell equations. The phasor representation such as E and its corresponding instantaneous field quantity E are related, with reference to cos ωt, by E x, y, z, t D Re[Ex, y, zejωt ]

2.6

The time-harmonic case is perhaps most commonly used in electrical engineering and will be considered in this book together with the static case. Maxwell’s equations under the source-free condition are obtained by letting D J D 0. These equations are applicable to passive microwave structures such as transmission lines.

2.2

CONSTITUTIVE RELATIONS

In order to solve for field quantities using Maxwell’s equations, three constitutive relations are needed. They basically describe the relations between the fields through the properties of the medium. Under the time-harmonic assumption, the (phasor) electric flux density D and electric field E in a simple medium are related by D D εωE D ε0 εr ωE 2.7 where ε0 D 8.854 ð 1012 F/m (farad/meter) and ε are the permittivity or dielectric constant of the vacuum and medium, respectively. Note that, in practice, free space is normally considered a vacuum. εr is called the relative permittivity or relative dielectric constant of the medium. The relation between the magnetic flux density B and magnetic field H in a simple medium is given as B D ωH D 0 r ωH

2.8

where 0 D 4 ð 107 H/m (henries/meter) and are the permeability of the vacuum and medium, respectively. r is the relative permeability of the medium.

LOSS IN MEDIUM

15

For time-harmonic fields and simple media, the current relates to the electric field by J D ωE 2.9 In general εr , r , and are a function of the location and direction in the medium as well as the power level applied to the medium. Most substrates used for electronic circuits, however, are homogeneous, isotropic, and linear, having constant εr , r , and . They are known as simple materials (media). Furthermore, most electronic substrates are nonmagnetic, having a relative permeability of 1. In this book, we will consider only simple and nonmagnetic substrates. There are also other materials classified as anisotropic (or nonisotropic), such as sapphire, and magnetic, such as ferrite. For these materials, the relative dielectric constant and permeability are described by the relative tensor dielectric constant and the permeability, respectively. In general, the conductivity and relative dielectric constant and permeability are also dependent on frequency. Good nonmagnetic substrates, however, have relative dielectric constants almost constant up to high frequencies. Good conductors have almost constant conductivity from dc up to the infrared frequencies. Their permittivity and permeability are approximately equal to those of a vacuum. 2.3

CONTINUITY EQUATION

The continuity equation is obtained from the conservation of charge as d dt

2.10a

W Ð J D jω

2.10b

WÐJ D and

for time-harmonic fields. 2.4

LOSS IN MEDIUM

Dielectrics used in electronic circuits are always nonperfect. Consequently, there is always loss present in any practical nonmagnetic dielectrics, known as dielectric loss, due to a nonzero conductivity of the medium. We can rewrite Maxwell’s Eq. (2.5b), making use of the constitutive relations (2.7) and (2.9), as E W ð H D jωε 1 j ωε

2.11

W ð H D jωε1 j tan υE

2.12

or

16


where tan υ D

ωε

2.13

is known as the loss tangent of the medium, which is normally used in practice to characterize the medium’s loss. Compared to the ideal case of a lossless medium, we can then define a complex dielectric constant of a lossy medium as εO D ε0 jε00

2.14a

ε0 D ε D εo εr

2.14b

where

and ε00 D

D ε tan υ ω

2.14c

Note that the real part ε0 of the complex dielectric constant is the dielectric property that contributes to the stored electric energy in the medium. The imaginary part ε00 contains the finite conductivity and results in loss in the medium. As for ε, ε00 is also dependent on frequency. Figure 2.1 shows ε0 and ε00 versus frequency for polystyrene [1]. It is apparent, from Eq. (2.14), that the loss tangent is equal to the ratio between the imaginary and real parts of the complex dielectric constant. The complex dielectric constant of a dielectric and, hence, its relative dielectric constant εr and loss tangent tan υ can be measured. The loss tangents and relative dielectric constants of substrates are supplied by manufacturers at

3 e′/e0

0.0012 0.0008

e′/e0

e′′/e0

2

1

0.0004 e′′/e0 0

0 10

102

103

104

105

106

107

108

109

1010

Frequency (hertz)

Figure 2.1 Real ε0 and imaginary ε00 parts of the complex dielectric constant of polystyrene versus frequency at 25° C.

BOUNDARY CONDITIONS

17

TABLE 2.1 Relative Dielectric Constant (er ) and Loss Tangent (tan d) of Typical Microwave Substrates

Material Styrofoam-103.7 Rexolite-1422 GaAs Sapphire Alumina (96%) Alumina (99.5%) Quartz (fused) Teflon Silicon RT/Duroid 5880 RT Duroid 6010

Frequency (GHz)

εr

tan υ@25° C

3 3 10 10 10 10 10 10 10 10 10

1.03 2.54 12.9 9.4–11.5 8.9 9.8 3.78 2.1 11.9 2.2 10.2, 10.5, 10.8

0.0001 0.0005 0.006 0.0001 0.0006 0.0003 0.0001 0.0004 0.004 0.0009 0.0028 max.

particular frequencies and used by RF and microwave engineers. Table 2.1 shows the parameters of substrates commonly used at microwave frequencies.

2.5

BOUNDARY CONDITIONS

Maxwell’s equations and constitutive relations may be used to obtain general solutions for electromagnetic fields existing in any microwave structures. To obtain unique solutions for the fields in a particular structure, such as coplanar waveguide, we must, however, enforce the structure’s boundary conditions. This is in fact similar to using Kirchhoff’s voltage and current laws in a lumpedelement circuit to obtain unique solutions for the voltages and currents in that circuit. For time-harmonic fields, the boundary conditions between two different media, shown in Fig. 2.2, are given as n ð E1 E2 D 0

2.15a

n ð H1 H2 D Js

2.15b

n Ð D1 D2 D s

2.15c

n Ð B1 B2 D 0

2.15d

where the subscripts 1 and 2 indicate media 1 and 2, respectively. n is the unit vector normal to the surface and pointing into medium 1. Js and s are the (linear) surface current density (in A/m) and surface charge density (in C/m2 ) existing at the boundary, respectively. General time-varying fields also follow these boundary conditions. These boundary conditions become simpler for special cases, such as between perfect dielectrics (s D 0 and Js D 0), between nonperfect dielectrics (Js D 0), and between a perfect dielectric and a perfect conductor.

18


Medium 1 e1, m1, s1 n

rs Medium 2 e2, m2, s2 Js

Figure 2.2

Boundary between two different media.

For instance, when media 1 and 2 are assumed to be a perfect dielectric and a perfect conductor, respectively, the boundary conditions become nðED0

2.16a

n ð H D Js

2.16b

n Ð D D s

2.16c

nÐBD0

2.16d

where the normal unit vector n points outward from the conductor surface. The tangential electric field along a perfect conductor is, therefore, always zero. For many practical problems, especially at low frequencies, good results can be obtained assuming good dielectrics and conductors are perfect. The boundary conditions (2.16b) and (2.16c) provide simple means for determining the current and charge induced on a conductor when fields are present. It should also be noted that the boundary conditions for the normal and tangential components of the fields between any two media are not independent of each other. 2.6

SKIN DEPTH

One of the most important parameters of a medium is its skin depth or depth of penetration. The skin depth is defined as the distance from the medium surface, over which the magnitudes of the fields of a wave traveling in the medium are reduced to 1/e, or approximately 37%, of those at the medium’s surface. The

POISSON’S AND LAPLACE’S EQUATIONS

19

skin depth υ of a good conductor is approximately given as

2 ω

υD

2.17

The skin depths of good conductors are very small, especially at high frequencies, causing currents to reside near the conductors’ surfaces. This subsequently results in a low conduction loss.

2.7

POWER FLOW

When a wave propagates in a medium, it carries the electric and magnetic fields and power. The power density at any location in the medium is given by the Poynting vector S DEðH 2.18 Note that S is instantaneous power with a unit of watt per square meter (W/m2 ). For time-harmonic fields, we define a phasor Poynting vector S D E ð HŁ

2.19

where HŁ is the complex conjugate of H. The time average of the instantaneous power density S or the average power density can be derived as Sav D

1 2

ReE ð HŁ

2.20

where ReÐ stands for the real part of a complex quantity. This power vector not only gives the magnitude of the power flow but also its direction. The direction of power flow or wave propagation is determined by the right-hand rule of the cross product and is always perpendicular to both E and H. The total average power crossing a surface S is then given as

Pav D

1 2

Re

E ð H Ð dS

2.21

S

2.8

POISSON’S AND LAPLACE’S EQUATIONS

Under the static assumption, the voltage Vx, y, z at any location of a structure having an electric charge density x, y, z is governed by Poisson’s equation, r2 V

ε

2.22

20


Note that acts as a source producing the fields. When there is no charge, Poisson’s equation reduces to 2.23 r2 V D 0 known as Laplace’s equation. Laplace’s equation is frequently employed to determine the static or quasi-static characteristic impedance and effective relative dielectric constant of a transmission line. These static parameters are easier to obtain than their dynamic counterparts but are only valid at dc. In practice, however, many RF and microwave engineers use the static parameters for the analysis and design of microwave circuits even when the operating frequencies are high, perhaps more than 18 GHz or so, and still obtain good results. 2.9

WAVE EQUATIONS

Electromagnetic fields may be determined by using Maxwell’s equations and constitutive relations directly. However, the most convenient way of obtaining these fields is solving a special class of equations known as the wave equations. We shall derive these equations as follows. We consider a medium that is source free ( D J D 0) and simple (homogeneous, isotropic, and linear) and assume that the fields are time harmonic. The medium is characterized by a dielectric constant ε and permeability . Taking the curl of Maxwell’s Eq. (2.5a) and making use of Maxwell’s Eq. (2.5b) and constitutive relations (2.7) and (2.8) yields W ð W ð E k2E D 0

2.24

p where k D ω ε is the wave number. Using the vector identity W ð W ð A D WW Ð A r2 A

2.25

where A is an arbitrary vector, we can then rewrite Eq. (2.24) as r2 E C k 2 E D 0

2.26

where r2 denotes the Laplacian operator. This equation is called the wave equation for the electric field. Similarly, the wave equation for the magnetic field can be derived as 2.27 W2 H C k 2 H D 0 Both of these wave equations are also known as Helmholtz equations. Other commonly used wave equations are those in the plane transverse to the direction of wave propagation. Let’s assume that the direction of propagation is z. We separate the operator r into the transverse, rt , and longitudinal, rz , components as W D Wt C Wz 2.28

ELECTRIC AND MAGNETIC POTENTIALS

21

where Wt D ax

∂ ∂ C ay ∂x ∂y

2.29

∂ D šaz ∂z

2.30

for the rectangular coordinates, and Wz D az

is the propagation constant with the š signs denoting the Ýz propagating directions, respectively. The Laplacian operator r2 can be written as r2 D r2t C 2

2.31

where r2t represents the transverse (to the z-axis) Laplacian operator. Substituting Eq. (2.31) into (2.26) and (2.27) we obtain r2t Ex, y C kc2 Ex, y D 0

2.32a

r2t Hx, y C kc2 Hx, y D 0

2.32b

kc2 D ωc2 ε D k 2 C 2

2.33

where

kc and ωc are referred to as the cutoff wave number and cutoff frequency, respectively, due to the fact that they reduce to the corresponding parameters at the cutoff ( D 0). Equation (2.32) is known as the wave or Helmholtz equation in the transverse plane, with z as the direction of propagation.

2.10

ELECTRIC AND MAGNETIC POTENTIALS

The fields in a microwave structure, in general, and in a transmission line, in particular, may be determined by directly solving the wave equations subject to appropriate boundary conditions. In practice, however, these fields are normally obtained via intermediate terms known as electric and magnetic vector or scalar potentials to simplify the mathematical analysis. These potentials are also solutions of the wave equations. In this section, we will derive these parameters for a source-free medium and their corresponding wave equations under the assumption of time-harmonic fields. From Maxwell’s Eq. (2.5c), we can describe the electric field as E D jωW ð yh

2.34

22


where yh x, y, z is a vector defined as the magnetic vector potential. Substituting Eq. (2.34) into (2.5b) yields W ð H D k 2 W ð yh

2.35

H D k 2 yh C W˚

2.36

from which we can obtain where ˚ is an arbitrary scalar function. Now substituting Eq. (2.34) into the left-hand side of (2.5a), we get W ð E D jω WW Ð yh W2 yh 2.37 The right-hand side of Eq. (2.5a) becomes, after replacing H by Eq. (2.36), jωH D jωk 2 yh C W˚

2.38

Equating Eqs. (2.37) and (2.38) gives WW Ð yh r2 yh D k 2 yh C W˚

2.39

W Ð yh D ˚

2.40

Choosing according to the Lorentz condition then yields r2 yh C k 2 yh D 0

2.41

which is also known as the wave or Helmholtz equation for the magnetic vector potential yh . Once solving for yh from Eq. (2.41), we can determine the electric and magnetic fields from Eqs. (2.34), (2.36), and (2.40) as E D jωW ð yh 2

h

2.42a h

H D k y C WW Ð y

2.42b

Following the same approach, we can also derive the following wave or Helmholtz equation for the electric vector potential ye x, y, z as r2 ye C k 2 ye D 0

2.43

whose solution can be used to determine the magnetic and electric fields as E D k 2 ye C WW Ð ye

2.44a

H D jωεW ð ye

2.44b

A remark needs to be made at this point. By setting the frequency to zero, we reduce the wave Eqs. (2.41) and (2.43) to the familiar Laplace’s Eq. (2.23) used

WAVE TYPES AND SOLUTIONS

23

under the static condition. Note that both e x, y, z and h x, y, z are now identical to Vx, y, z. We can also write the wave equations for the field vector potentials in the plane transverse to the propagating direction. For instance, let z be the direction of wave propagation, we separate the vector potentials yh x, y, z and ye x, y, z into the transverse and longitudinal components as yh D yht C yhz D yht x, yešh z C az e

y D

yet

C

h

x, yešh z

2.45a

x, yeše z

2.45b

yez

D yet x, yeše z C az

e

where h and e are the corresponding propagation constants, with the š signs indicating the Ýz-directions of propagation, respectively. Using these equations in (2.41) and (2.43), we obtain, after decomposing the Laplacian operator r2 into the transverse and longitudinal components, 2 yht x, y D 0 r2t yht x, y C kc,h

2.46a

r2t

x, y D 0

2.46b

2 r2t yet x, y C kc,e yet x, y D 0

2.46c

r2t

2.46d

h

2 x, y C kc,h

e

2 x, y C kc,e

h

e

x, y D 0

where 2 kc,h D k 2 C h2

2.47a

2 kc,e D k 2 C e2

2.47b

yht x, y and yet x, y are the transverse magnetic and electric vector potentials, respectively. h x, y and e x, y are the longitudinal components of the magnetic and electric vector potentials, respectively, and are referred to as the magnetic and electric scalar potentials. Note that, as for the case of the magnetic and electric fields in the transverse wave Eqs. (2.32), all these potentials are a function of x and y only. 2.11 2.11.1

WAVE TYPES AND SOLUTIONS Wave Types

The two most commonly used waveguides† are the rectangular waveguide and the transmission line. Waves propagating along different waveguides possess † We

define a waveguide as any structure that guides waves.

24


different electromagnetic field distributions. It is these field distributions that dictate the nature of the waveguides. With reference to a particular wavepropagating direction, we can classify different wave (or mode) types based on possible combinations of the electric and magnetic fields in that direction. These are, assuming z-direction of wave propagation: Transverse Electric (TE) Wave or Mode: This wave has the electric field only in the plane transverse to the direction of propagation. That is, the longitudinal components Ez D 0 and Hz 6D 0. Transverse Magnetic (TM) Wave or Mode: This wave has only the magnetic field in the transverse plane. That is, Hz D 0 and Ez 6D 0. Hybrid Wave or Mode: This wave is characterized as having both Ez 6D 0 and Hz 6D 0 and, therefore, is a combination of both TE and TM waves. Transverse Electromagnetic (TEM) Wave or Mode ‡ : The electric and magnetic fields of this wave have only transverse components. Both Ez and Hz are equal to zero. Note that, in general, different modes have different cutoff frequencies or cutoff wave numbers and hence different propagation constants. A rectangular waveguide is a special waveguide that has the same cutoff frequency for the corresponding TEmn and TMmn modes. When the modes have the same cutoff frequency, they are classified as degenerate modes. Otherwise, they are said to be nondegenerate modes. 2.11.2

Wave Solutions

Assuming yh x, y, z and ye x, y, z have only longitudinal z components such as yh D az e

y D az

h

x, yešh z

2.48a

e

še z

2.48b

x, ye

we can prove easily that the corresponding longitudinal electric and magnetic fields are equal to zero. This implies that the magnetic, h x, y, and electric, e x, y, scalar potentials may be used to determine the fields for TE and TM modes, respectively. This result will be used to derive fields for the four principal classes of modes discussed earlier. Using the principle of superposition, we can express the fields in any waveguide as a summation of those of TE and TM modes. These fields are given, making use of Eqs. (2.42) and (2.44), as E D ETM C ETE D k 2 ye C WW Ð ye jωW ð yh ‡ For

TEM mode to exist exactly on a transmission line, all conductors must be perfect.

2.49a

25


H D HTM C HTE D jωεW ð ye C k 2 yh C WW Ð yh

2.49b

where the subscripts TE and TM indicate the TE and TM modes, respectively. Substituting Eq. (2.48) into (2.49a) and replacing the W operator by its transverse and longitudinal components yields E D az k ð

2

e še z

e

h šh z

e

∂

C Wt C az še ∂z

e še z

e

∂ jω Wt C az ∂z

az

2.50

from which, we obtain the z and transverse components of the electric field as 2 Ez D kc,e

e še z

e

2.51a

Et D še eše z Wt

e

C jωešh z az ð Wt

h

2.51b

respectively. Note that, for degenerate modes, e D h . Equation (2.51b) can then be used to derive individual x and y components of the electric field as Ex D še eše z

∂ e ∂ h jωešh z ∂x ∂y

2.52a

Ey D še eše z

∂ e ∂ h C jωešh z ∂y ∂x

2.52b

Similarly, by expanding Eq. (2.49b), we can express the longitudinal and transverse magnetic-field components as 2 Hz D kc,h

h šh z

e

Ht D šh ešh z Wt

2.53a h

jωεeše z az ð Wt

e

2.53b

Further expanding Eq. (2.53b) then yields Hx D šh ešh z

∂ h ∂ e C jωεeše z ∂x ∂y

2.54a

Hy D šh ešh z

∂ h ∂ e jωεeše z ∂y ∂x

2.54b

We can also determine the x and y components from the z components of the fields by using Maxwell’s equations directly. We can easily prove that both Ez and Hz cannot be even or odd with respect to x and y simultaneously. The results derived so far are very general and so are applicable to any possible modes

26


existing in any waveguide. We will now use them to obtain solutions for TE, TM, hybrid, and TEM modes. TE Modes The TE modes, as indicated earlier, correspond only to the magnetic scalar potential h x, y. Therefore, to determine the TE fields, we simply let e x, y equal zero in the foregoing general equations. From Eqs. (2.51) and (2.53), we then have

Ez D 0

2.55a

Et D jωešh z az ð Wt Hz D

2 kc,h

h šh z

e

h

2.55b 2.55c

Ht D šh ešh z Wt

h

2.55d

Et and Ht are related by Et D šZh az ð Ht

2.56

where Zh D jω/h is known as the characteristic wave impedance of the TE mode. Zh is real and inductive for propagating and evanescent TE modes, respectively. This impedance is defined as a ratio between the transverse electric and magnetic components, Ex Ey Zh D D 2.57 Hy Hx for the Cz direction, or Zh D

Ex Ey D Hy Hx

2.58

for the z direction. The individual transverse components of the TE fields may be obtained from Eqs. (2.52) and (2.54) or Eqs. (2.55b) and (2.55d), as Ex D jωešh z

∂ h ∂y

2.59a

Ey D jωešh z

∂ h ∂x

2.59b

Hx D šh ešh z

∂ h ∂x

2.59c

Hy D šh ešh z

∂ h ∂y

2.59d

Note that h is obtained by solving Eq. (2.46b), which is repeated here for completeness: 2 h r2t h x, y C kc,h x, y D 0 2.60


27

TM Modes The TM modes are determined solely from the electric scalar potential e x, y. Their longitudinal and transverse fields can thus be determined by letting h x, y equal zero in Eqs. (2.51) and (2.53). This gives

Hz D 0

2.61a

Ht D jωεe 2 Ez D kc,e

še z

az ð Wt

e

e še z

e

2.61b 2.61c

Et D še eše z Wt

e

2.61d

Et and Ht are related by Ht D Ý

az ð Et Ze

2.62

where Ze D jωε/e is called the characteristic wave impedance of the TM mode. This impedance is real and capacitive for propagating and evanescent TM modes, respectively. The transverse components of the TM fields can be expressed, from Eqs. (2.52) and (2.54) or Eqs. (2.61b) and (2.61d), as ∂ e ∂x ∂ e Ey D še eše z ∂y Ex D še eše z

Hx D jωεeše z

∂ e ∂y

Hy D jωεeše z e

∂ e ∂x

2.63a 2.63b 2.63c 2.63d

is the solution of Eq. (2.46d) and is given again below: r2t

e

2 x, y C kc,e

e

x, y D 0

2.64

Hybrid Modes A hybrid mode is a combination of both TE and TM modes. The general results Eqs. (2.51)–(2.54), derived earlier can therefore be used directly to determine the fields of the hybrid modes. TEM Modes Solution for the TEM mode can be viewed as a special solution of either the TE or TM mode when Hz or Ez is set to zero, respectively. For instance, we consider the TE mode and let Hz in Eq. (2.55c) equal zero. This leads to kc,h D 0 and, consequently, Eq. (2.46b) becomes

r2t

h

x, y D 0

2.65

28


which is basically Laplace’s equation in the transverse plane. The transverse fields can be obtained from Eqs. (2.55c) and (2.55d) as Et D jωešz az ð Wt Ht D še

šz

Wt

h

h

2.66a 2.66b

where is the TEM mode’s propagation constant. Proceeding with the TM mode also gives identical results. It should be noted that for the TEM case, both e x, y and h x, y are equal to the two-dimensional voltage or potential Vx, y.

2.12

ORTHOGONALITY RELATIONS

Consider a general waveguide as shown in Fig. 2.3 and assume that it has perfect conducting walls. In general, there exists an infinite number of modes in the structure. To signify different modes, we will use the subscripts m and n here, with their possible values from 0 to infinity. As indicated in Section 2.11.2, the h e TEmn , TMmn , and hybrid modes correspond to mn x, y, mn x, y, and both of these potentials, respectively. Following the approach described in Collin [2], we h e can derive the orthogonality relations between mn x, y and mn x, y, between Emn x, y, between Hmn x, y, and between Emn x, y and Hmn x, y. 2.12.1

h e Orthogonality Relations Between ymn .x , y / and Between ymn .x , y /

Let us consider two different TE or TM modes characterized by (m, n) and (k, l). i i x, y or kl x, y, with i being h or e, The corresponding scalar potentials mn

C

n

S

az

Figure 2.3 A waveguide of arbitrary shape. n is a unit vector perpendicular to the wall and pointing outward; t is a unit vector tangential to the wall; and az is a unit vector along the waveguide. n, t, and az form an orthogonal coordinate system.


29

satisfy the following wave equations: r2t

2 C kc,i,mn

i mn

r2t

i kl

2 C kc,i,kl

i mn

D0

2.67a

i kl

D0

2.67b

where kc,i,mn and kc,i,kl are the corresponding cutoff wave numbers. We multiply i i and mn , respectively, and subtract the resulting Eqs. (2.67a) and (2.67b) with kl equations to obtain i 2 kl rt

i mn

i 2 mn rt

i kl

2 2 D kc,i,kl kc,i,mn

i mn

i kl

2.68

Taking the surface integral and using Green’s second identity in two dimensions,

ur2t v

vr2t u

∂u ∂v v dS D u dl ∂n ∂n

S

2.69

C

where u and v are arbitrary scalar functions, S is the surface, and C is the closed contour bounding that surface, we obtain

2 2 kc,i,kl kc,i,mn

i kl

i mn

dS D

S

i i ∂ mn kl

∂n

i i ∂ kl dl mn

∂n

2.70

C

Note that ∂/∂n denotes the derivative with respect to the normal direction n. Along the perfectly conducting walls of the waveguide, the scalar potentials must satisfy the following Neumann’s and Dirichlet’s conditions: h j

∂

Neumann’s Condition:

∂n e j

Dirichlet’s Condition:

D0

2.71a

D0

2.71b

where j D mn or kl. Imposing these conditions on Eq. (2.70) gives

2 2 i i kc,i,kl kc,i,mn kl mn dS D 0

2.72

S

which implies that e mn

e kl

dS D 0,

m 6D k

or

n 6D l

2.73a

h mn

h kl

dS D 0,

m 6D k

or

n 6D l

2.73b

S

S

30


for two different TM and TE modes, respectively, provided that they have different cutoff wave numbers; that is, they are nondegenerate modes. The relationship described by Eq. (2.73) is known as the orthogonality condition between the scalar potentials. It states that the scalar electric or magnetic potentials of two nondegenerate modes are always orthogonal to each other. It should be noted that, due to Neumann’s and Dirichlet’s conditions, this orthogonality only holds for waveguides with perfectly conducting walls. When the modes are degenerate, Eq. (2.73) may not be satisfied, and so the scalar electric or magnetic potentials of two degenerate modes may not be orthogonal. Using a procedure analogous to the Gram–Schmidt process [3], however, we can construct a new set of mutually orthogonal modes, each of which is a linear combination of certain modes of the nonorthogonal degenerate modes. This process is i and *kli be the two new modes defined by described as follows. Let *mn i D *mn

i mn

*kli D

i kl

2.74a

C

i mn

2.74b

where C is a constant. These new functions are required to be mutually orthogonal; that is, i *mn *kli D 0 2.75 S

which then leads to

i mn S C D

i kl

dS

2 i mn

dS

2.76

S i 2 mn

dS exists and is nonzero. This process can be provided that the integral applied to more than two nonorthogonal modes to determine a new set of modes that are mutually orthogonal. i jj 6D 0 for any combination of From Eq. (2.73) and the fact that the norm jj mn i (m, n), the scalar potentials mn represent an orthogonal set and thus are linearly independent. Therefore, a field function at any location in a waveguide can be expressed as a summation of the scalar potentials of all possible modes as

i x, y D Cmn mn x, y 2.77 iDe,h m

n

where Cmn are called the orthogonal coefficients and may be computed by i mn dS S Cmn D

S

2 i mn

2.78 dS


31

Equation (2.77) implies that there is always a unique solution for the scalar potentials and, hence, electromagnetic fields for waveguides having perfect conductors. 2.12.2 Orthogonality Relations Between Electric Fields and Between Magnetic Fields

Let’s consider two different TMmn and TMkl modes. The surface integral of the dot product of the transverse electric fields is given, using Eq. (2.61d), as e e Eet,mn Ð Eet,kl dS D še,mn e,kl ee,mn še,kl z Wt mn Ð Wt kl dS 2.79 S

S

Applying the two-dimensional Green’s first identity,

∂v Wt u Ð Wt v C uW2t v dS D u dl ∂n S

2.80

C

where u and v are arbitrary functions, and the Dirichlet condition (2.71b), we can rewrite Eq. (2.79) as e 2 e Eet,mn Ð Eet,kl dS D Ýe,mn e,kl eše,mn še,kl z 2.81 mn rt kl dS S

S

Making use of the wave equation for r2t and the fact that we finally obtain

e mn

and

e kl ,

e kl

e kl ,

2 C kc,kl

e kl

D0

2.82

with m 6D n and k 6D l, are mutually orthogonal,

Eet,mn Ð Eet,kl dS D 0

2.83

S

This result indicates that the transverse electric fields of two different TM modes are always orthogonal to each other. Following the same approach, we can also derive the other orthogonality relationships for the transverse electromagnetic fields of TM, TE, and hybrid modes. All the orthogonality relationships are given as follows. TM Modes

Eet,mn Ð Eet,kl dS D 0

2.84a

Het,mn Ð Het,kl dS D 0

2.84b

S

S

32


TE Modes

Eht,mn Ð Eht,kl dS D 0

2.85a

Hht,mn Ð Hht,kl dS D 0

2.85b

Eet,mn Ð Eht,kl dS D 0

2.86a

Het,mn Ð Hht,kl dS D 0

2.86b

S

S

Hybrid Modes S

S

Note that Eq. (2.86) is valid even when m D k and n D l. Making use of the characteristics of an orthogonal system, we can then express the transverse fields at any location in a waveguide as a summation of the transverse fields of all possible modes as

Et D Cemn Eet,mn C Chmn Eht,mn 2.87a m

n

m

n

e h Ht D Dmn Het,mn C Dmn Hht,mn

2.87b

e h where Cemn , Chmn , Dmn , and Dmn are the orthogonal coefficients. As for the case of the scalar potentials, when the modes are degenerate and nonorthogonal, we can construct a new set of transverse fields, each of which is a linear combination of certain fields of the nonorthogonal degenerate modes, such that the new fields are mutually orthogonal.

2.12.3

Orthogonality Relations Between Electric and Magnetic Fields

Let’s consider again two different TMmn and TMkl modes. Assuming z is the direction of propagation, the fields of these modes can be expressed as a sum of the transverse and longitudinal fields. For instance, the fields of the TMmn mode are given as Eemn x, y, z D Eet,mn x, y, z C Eez,mn x, y, z D eemn x, yee,mn z C eez,mn x, yee,mn z Hemn x, y, z

D

Het,mn x, y, z

C

2.88a

Hez,mn x, y, z

D hemn x, yee,mn z C hez,mn x, yee,mn z

2.88b

33


Note that we have introduced the notation e, h and ez , hz to signify the twodimensional transverse and longitudinal components, respectively. The TM fields satisfy the Maxwell equations, W ð Eej D jωHej

2.89a

W ð Hej D jωεEej

2.89b

where j D mn, kl. Making use of Eq. (2.89), we can write the following equation:

W Ð Eemn ð Hekl Eekl ð Hemn D jω Hekl Ð Hemn Hekl Ð Hemn

C jωε Eekl Ð Eemn Eekl Ð Eemn D 0 2.90 Separating the W operator in Eq. (2.90) into the transverse and longitudinal parts gives

W Ð Eemn ð Hekl Eekl ð Hemn D Wt Ð Eemn ð Hekl Eekl ð Hemn C az

∂ e Ð Emn ð Hekl Eekl ð Hemn D 0 ∂z

2.91

Now taking the surface integral of Eq. (2.91) and applying the divergence theorem in two dimensions, Wt Ð A dS D n Ð A dl 2.92 S

C

where A is an arbitrary vector, we obtain

Wt Ð Eemn ð Hekl Eekl ð Hemn dS D n ð Emn Ð Hkl n ð Ekl Ð Hmn dl S

C

D0

2.93

since n ð Eej D 0;

j D mn, kl

2.94

on perfectly conducting walls. Substituting Eq. (2.93) into (2.91), we have

az Ð

∂ e Emn ð Hekl Eekl ð Hemn dS D 0 ∂z

2.95

S

Substituting Eq. (2.88) into (2.95) and taking the derivatives leads to

e,mn C e,kl az Ð eemn hekl eekl ð hemn dS D 0 S

2.96

34


Note that this equation is obtained assuming both TMmn and TMkl modes correspond to the Cz direction of propagation. If we now assume that the TMmn mode still corresponds to the Cz propagation direction, but the TMkl mode corresponds to the z direction, then we can write Eekl and Hekl as Eekl D eekl ee,kl z eez,kl ee,kl z

2.97a

Hekl

2.97b

D

hekl ee,kl z

hez,kl ee,kl z

which implies that we can derive the following relation by simply changing hkl to hkl in Eq. (2.96):

az Ð eemn ð hekl eekl ð hemn dS D 0

e,mn e,kl

2.98

S

Adding and subtracting Eqs. (2.96) and (2.98) gives

az Ð

e,mn eekl

ð

hemn dS

D

S

az Ð e,kl eemn ð hekl dS

2.99a

az Ð e,mn eemn ð hekl dS

2.99b

S

az Ð

e,kl eekl

ð

hemn dS

D

S

S

Assuming nondegeneracy between these modes, that is, kl 6D mn , we can then obtain directly from Eq. (2.99)

eemn ð hekl Ð az dS D 0

2.100a

eekl ð hemn Ð az dS D 0

2.100b

S

S

which is the orthogonality relationship between the transverse electric and magnetic fields. When the structure is lossless, we can prove that

Ł

eemn ð hekl Ð az dS D 0

2.101a

S

Ł

eemn ð hhkl Ð az dS D 0

2.101b

S Ł

where hekl is the complex conjugate of hekl , assuming the modes are nondegenerate. Similarly, for two nondegenerate TEmn and TEkl modes, we can derive


35

ehmn ð hhkl Ð az dS D 0

2.102a

ehkl ð hhmn Ð az dS D 0

2.102b

S

S

Ł

ehmn ð hhkl Ð az dS D 0

2.103a

S

Ł

ehmn ð hekl Ð az dS D 0

2.103b

S

for lossless waveguides. When the modes are degenerate, Eqs. (2.100)–(2.103) still hold when one mode is TE and the other is TM. In general, however, these equations may not be satisfied and so the fields are not mutually orthogonal. In that case, as for the case of the scalar potentials discussed earlier, we may define new fields, which are linear combinations of the old fields such that the orthogonality holds. 2.12.4

Power Orthogonality for Lossless Structures

Let’s assume that the waveguide is lossless and there are only two hybrid modes, characterized by (m, n) and (k, l), exist in the structure. Effectively, there will be four different modes propagating in the waveguide, namely, TEmn , TMmn , TEkl , and TMkl . The fields of these hybrid modes can therefore be expressed as a sum of those of the corresponding TE and TM modes. For instance, the fields of the hybrid (m, n) mode is given as Emn D Eemn C Ehmn D eemn ejˇe,mn z C eez,mn ejˇe,mn z C ehmn ejˇh,mn z C ehz,mn ejˇh,mn z 2.104a Hmn D D

Hemn

C

Hhmn

hemn ejˇe,mn z

C hez,mn ejˇe,mn z C hhmn ejˇh,mn z C hhz,mn ejˇh,mn z 2.104b

The total average power flow along the waveguide is given by Pav D 12 Re E ð HŁ Ð az dS S

D

1 2

Re S

Emn C Ekl ð HŁmn C HŁkl Ð az dS

2.105

36


Substituting Eq. (2.104) into (2.105) and expanding the cross products gives Ł Ł Ł Ł Pav D 12 Re eemn ð hemn C ehmn ð hhmn C eekl ð hekl C ehkl ð hhkl Ð az dS S

C

1 2

Ł Ł Re eemn ð hhmn ejˇe,mn ˇh,mn z C eekl ð hhkl ejˇe,kl ˇh,kl z S Ł

Ł

C eemn ð hekl ejˇe,mn ˇe,kl z C ehmn ð hhkl ejˇh,mn ˇh,kl z Ł

Ł

C eekl ð hemn ejˇe,kl ˇe,mn z C ehkl ð hhmn ejˇh,kl ˇh,mn z Ł

Ł

C eemn ð hhkl ejˇe,mn ˇh,kl z C ehmn ð hekl ejˇh,mn ˇe,kl z

Ł Ł C eekl ð hhmn ejˇe,kl ˇh,mn z C ehkl ð hemn ejˇh,kl ˇe,mn z Ð az dS

2.106

The second integral of the right-hand side represents the power resulting from the interaction between the TE and TM modes. This power term is zero by virtue of Eqs. (2.101) and (2.103). The total average power is therefore given as Ł Ł Ł Ł Pav D 12 Re eemn ð hemn C ehmn ð hhmn C eekl ð hekl C ehkl ð hhkl Ð az dS S

D

Pemn

C Phmn C Pekl C Phkl

2.107

e h e h , Pmn , Pkl , and Pkl are the powers carried by the TMmn , TEmn , TMkl , where Pmn and TEkl modes, respectively. Generalizing this result to multiple hybrid modes we obtain Pav D Pemn C Phmn 2.108 m

n

Equation (2.108) suggests that the total power flow in a lossless waveguide is equal to the summation of the powers carried by individual modes. This further implies that each mode carries power independent of the other modes. This condition is known as the power orthogonality. Equation (2.108) is only valid for hybrid, TE, and TM modes that are nondegenerate. For degenerate modes, it only holds when the modes are of the same kind (TE or TM). For degenerate modes not satisfying Eq. (2.108), we can, however, choose new modes related to these degenerate modes such that the new modes follow the relation (2.101). Possible choices for the fields of the new modes are E0mn D Emn

2.109a

H0mn

2.109b

D Hmn

E0kl D Ekl CEmn

2.109c

H0kl

2.109d

D Hkl CHmn

PROBLEMS

37

where the constant C is chosen to satisfy the power orthogonality. It should be noted that the power orthogonality is only approximately held for low-loss waveguides propagating nondegenerate modes. For degenerate modes, strong couplings between these modes occur. One remark needs to be made at this point. All the derived orthogonality relations are completely satisfied only if the modes were calculated exactly. Normally, the eigenmodes in planar transmission lines can only be determined approximately. Under this condition, it is easily proved that the orthogonality relations are not satisfied. The satisfaction of these orthogonality relations, such as Eq. (2.100) or (2.101), may therefore serve as a check of the accuracy of the computed modes. REFERENCES 1. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961, p. 24. 2. R. E. Collin, Field Theory of Guided Waves, IEEE Press, New York, 1991, pp. 329–337. 3. A. E. Taylor and W. R. Mann, Advanced Calculus, John Wiley & Sons, New York, 1983, pp. 277–279.

PROBLEMS

2.1 Derive the boundary conditions (2.15) between two different media as shown in Fig. 2.2. 2.2 Using the Poynting vector, prove that the average power density of a signal propagating in a waveguide is given by Eq. (2.20). 2.3 Show that TE modes can be characterized only by the magnetic scalar potential h x, y. 2.4 Derive Eqs. (2.51)–(2.54). 2.5 Show that TM modes can be characterized only by the electric scalar potential e x, y. 2.6 Prove that, in any waveguides, both Ez and Hz cannot be even or odd simultaneously. 2.7 Consider a general waveguide with perfectly conducting walls as shown in Fig. 2.3. Derive the following boundary conditions along the surface of the conductor for both TE and TM modes: TE Modes: TM Modes:

∂ h D0 ∂n e

D0

(Neumann’s Condition) (Dirichlet’s Condition)

38


2.8 Verify the power orthogonality relationship for nondegenerate modes in a lossless circular waveguide. 2.9 Verify the power orthogonality relationship for nondegenerate modes in a lossless rectangular waveguide. 2.10 The electric and magnetic fields of the eigenmodes existing in planar transmission lines are normally determined approximately. Show that, under this condition, the orthogonality relationship (2.100) does not hold. Show also that, if these fields were calculated with a good accuracy, the orthogonality relation is well satisfied. 2.11 Derive the constant C in Eq. (2.109) that describes the fields of a new set of modes in terms of the fields of nonorthogonal degenerate modes, such that the new modes satisfy the power orthogonality (2.108). 2.12 Derive the orthogonality relations (2.84b) and (2.85) for TM and TE modes, respectively. 2.13 Derive the orthogonality relations (2.86) for hybrid modes. 2.14 Derive the orthogonality relation (2.101) and (2.103) for a lossless waveguide. 2.15 In general, there exist many degenerate modes in a waveguide. Some are coupled together while others are not. Prove that the mode coupling does not take place between the two degenerate TEmn and TMmn modes in a rectangular waveguide: that is, prove that the power-interaction terms TM PTE D 12 ETEmnn ð HŁTMmnn Ð dS D 0 S

and

TE D PTM

1 2

ETMmnn ð HŁTEmnn Ð dS D 0. S

2.16 Prove that, for a lossless waveguide, the orthogonality relationship (2.101) holds for nondegenerate modes.


CHAPTER THREE

Green’s Function Green’s function is one of the most commonly used functions in solving microwave boundary-value problems. It represents a response (e.g., an electric field) due to a source of unit amplitude (e.g., a unit current). Green’s function has been used in finding solutions for many microwave problems such as scattering and transmission line analysis. Its particular use in analyzing transmission lines is described in Chapters 6 (Variational Methods) and 7 (Spectral-Domain Method). Green’s function is described in details in [1] and [2]. In this chapter, we will present essential information on Green’s function in the space domain. Its treatment in the spectral domain can be found in Chapters 6 and 7.

3.1

DESCRIPTIONS OF GREEN’S FUNCTION

Solution to a microwave boundary-value problem would involve finding the response due to a source in the microwave structure directly or indirectly. In essence, the main task of analyzing a microwave structure can thus revolve around finding the response caused by a source of unit amplitude (i.e., a Green’s function) in that particular structure. Once Green’s function is found, the total response can easily be determined by taking a summation or integral. To illustrate this principle, we will describe and use Green’s function in obtaining the solutions of the two basic but most commonly used equations in microwave boundary-value problems: Poisson’s equation and the wave equation. 3.1.1

Solution of Poisson’s Equation Using Green’s Function

We consider a medium characterized by a permittivity ε and permeability as shown in Fig. 3.1. We assume that the medium contains a charge density x 0 , y 0 , z0 . This charge distribution represents a source in the structure and 39

40

GREEN’S FUNCTION

z

(x ′, y ′, z ′) (x, y, z)

R

r′

r

y e,m

x

Figure 3.1

A medium characterized by ε and .

therefore would produce a potential V at every location x, y, z , which is governed by Poisson’s equation, r2 Vx, y, z D

x 0 , y 0 , z0 ε

3.1

and the boundary conditions of the considered structure. Let Gx, y, z; x 0 , y 0 , z0 represent the Green’s function of the structure, which is the potential at point x, y, z due to a unit charge located at point x 0 , y 0 , z0 in the medium. This Green’s function must also satisfy Poisson’s equation and corresponding boundary conditions of the structure. That is, r2 Gx, y, z; x 0 , y 0 , z0 D

υx, y, z; x 0 , y 0 , z0 ε

3.2

where υx, y, z; x 0 , y 0 , z0 is the Dirac delta function, defined as υx, y, z; x 0 , y 0 , z0 D υx x 0 υ y y 0 υ z z0 1, x D x 0 ; y D y 0 , z D z0 D 0, otherwise

3.3

DESCRIPTIONS OF GREEN’S FUNCTION

41

The solution of Eq. (3.2) is Gx, y, z; x 0 , y 0 , z0 D

1 4εR

3.4

The potential due to the total charge distributed over the entire medium contained in a volume V0 is obtained from the Green’s function as Vx, y, z D Gx, y, z; x 0 , y 0 , z0 x 0 , y 0 , z0 dV0 3.5 V0

or, upon using Eq. (3.4),

Vx, y, z D V0

x 0 , y 0 , z0 dV0 4εR

3.6

which is exactly the same as the solution obtained by solving Poisson’s equation directly. 3.1.2

Solution of the Wave Equation Using Green’s Function

We consider again the medium in Fig. 3.1 and assume that a (vector) current distribution J x 0 , y 0 , z0 exists in the region. This current distribution produces a magnetic vector potential A x, y, z , which can be obtained from the wave equation r2 Ax, y, z C k 2 Ax, y, z D Jx 0 , y 0 , z0

3.7

p

and the structure’s boundary conditions. k D ω ε is the wave number. As for the case of Poisson’s equation, we let Gx, y, z; x 0 , y 0 , z0 be the Green’s function of the structure, which now represents the magnetic vector potential at point x, y, z due to a unit current located at point (x 0 , y 0 , z0 ) in the medium. This Green’s function must also be the solution of the wave equation r2 Gx, y, z; x 0 , y 0 , z0 C k 2 Gx, y, z; x 0 , y 0 , z0 D υ x, y, z; x 0 , y 0 , z0 aJ 3.8 subject to the structure’s appropriate boundary conditions. aJ is the unit vector of the current at (x 0 , y 0 , z0 ). G can be derived as Gx, y, z; x 0 , y 0 , z0 D

ejkR aJ 4 R

3.9

The magnetic vector potential due to the total current distributed over the entire medium can be obtained from the Green’s function as Ax, y, z D Gx, y, z; x 0 , y 0 , z0 J x 0 , y 0 , z0 dV0 3.10 4 V0

42

GREEN’S FUNCTION

which becomes, after using Eq. (3.9), Ax, y, z D

4

Jx 0 , y 0 , z0

ejkR dV0 R

3.11

V0

This magnetic vector potential is identical to the direct solution of the wave equation. The magnetic and electric fields produced by the current can readily be obtained from the magnetic vector potential as Hx, y, z D

1 W ð Ax, y, z

3.12

Ex, y, z D

1 W ð Hx, y, z jωε

3.13

These simple analyses demonstrate that, instead of solving directly Poisson’s equation, wave equations, or other equations, resulting from a microwave boundary-value problem, for a desired field quantity, we can first determine the Green’s function of the considered problem and then use it to obtain the field quantity. As will be seen, the Green’s function satisfies the so-called Sturm–Liouville equation, which can uniquely be obtained for a microwave boundary-value problem.

3.2

STURM–LIOUVILLE EQUATION

The Sturm–Liouville equation is a differential equation of the following form:

where

LD

d dx

Ly D fx

3.14

d P P1 x C x dx 2

3.15

is called the Sturm–Liouville operator. The function fx represents a source. This function is given as a direct function of x but may also depend on another unknown function of x (e.g., yx ). Many microwave boundary-value problems produce differential equations that can be converted to the Sturm–Liouville equation. To show this, we consider the following one-dimensional, second-order differential equation: Fy D Sx

3.16

STURM–LIOUVILLE EQUATION

43

where Sx represents a source existing in the microwave structure and F is the operator defined as

FD

C1 x

d2 d C C x x C C 2 3 dx 2 dy

3.17

Equation (3.16) is a generalization of one-dimensional differential equations resulting from a microwave boundary-value problem (e.g., Poisson’s and wave equations). Equation (3.16) can easily be converted to the Sturm–Liouville Eq. (3.14) as follows: Expanding Eq. (3.14) and dividing it by P1 x gives P2 1 dP1 dy fx d2 y C yD C 2 dx P1 dx dx P1 P1

3.18

Dividing Eq. (3.16) by C1 x produces C2 dy Sx d2 y C3 C yD C 2 dx C1 dx C1 C1

3.19

In order for Eqs. (3.18) and (3.19) to be equivalent, we set their coefficients equal as C2 x 1 dP1 x D C1 x P1 x dx

3.20

P2 x C3 x D C1 x P1 x

3.21

fx Sx D C1 x P1 x

3.22

which are then solved to obtain

P1 x D exp

C2 t dt C1 t

3.23

P2 x D P1 x

C3 x C1 x

3.24

fx D P1 x

Sx C1 x

3.25

These equations facilitate the conversion of the one-dimensional, second-order differential Eq. (3.16) to the Sturm–Liouville Eq. (3.14).

44

GREEN’S FUNCTION

As an example, we consider the following Bessel differential equation, normally obtained in microwave problems involving cylindrical coordinates: x2

dy d2 y Cx C x 2 2 y D 0 2 dx dx

Using Eqs. (3.23)–(3.25), we can derive t dt D expln x D x P1 x D exp t2 P2 x D and

x 2 2 x

3.26

3.27 3.28

fx D 0

3.29

which, after substituting into Eq. (3.14), gives the equivalent Sturm–Liouville form of the Bessel differential equation, dy x 2 2 d yD0 3.30 x C dx dx x It is now apparent that the Green’s function for a microwave structure can be obtained as the solution of the Sturm–Liouville equation when the source has unit amplitude, subject to appropriate boundary conditions. 3.3

SOLUTIONS OF GREEN’S FUNCTION

In general, Green’s function can be described in three forms: closed form, series form, and integral form [2]. The formulation of these functions is given as follows. 3.3.1

Closed-Form Green’s Function

For the sake of generality, we now consider the more general Sturm–Liouville equation 3.31 [L C P3 x ]y D fx where a x b; L is again the Sturm–Liouville operator; and is a constant. The Green’s function Gx; x 0 is the solution of the Sturm–Liouville equation corresponding to a unit source and thus must satisfy the equation [L C P3 x ]Gx; x 0 D υx; x 0

3.32

where υx; x 0 is the one-dimensional Dirac delta function. The Green’s function has the following properties: 1. For any x 6D x 0 , Gx; x 0 satisfies the equation [L C P3 x ]G D 0

3.33


2. 3. 4. 5.

45

Gx; x 0 satisfies appropriate boundary conditions at x D a and x D b. Gx; x 0 is symmetrical with respect to x and x 0 . Gx; x 0 is continuous at x D x 0 . The derivative of Gx; x 0 has a discontinuity of magnitude 1/P1 x 0 at x D x 0 . That is, 0C dG x 1 D G0 x 0C G0 x 0 D dx xDx0 P1 x 0

3.34

Properties 1 and 2 are readily seen, while Properties 3, 4, and 5 can easily be proved. We assume that the function P1 x is continuous and nonzero at any point within the interval [a, b]. The discontinuity of the derivative of Gx; x 0 in Property 5 thus has a finite value. We also assume that P1 x and P2 x are continuous in the interval [a, b]. Once Gx; x 0 is found, the solution to Eq. (3.31) can be obtained as b Gx; x 0 fx dx 0 3.35 yx D a

We now find the Green’s function by dividing the interval [a, b] into two separate regions: [a, x 0 ] and [x 0 , b]. Region a ≤ x < x Let y1 x be a nontrivial solution of the associated homogeneous differential equation of (3.31),

[L C P3 x ]y D 0

3.36

which satisfies the boundary condition at x D a. At x 6D x 0 , Gx; x 0 must satisfy [L C P3 x ]G D 0

3.37

and the boundary condition at x D a, according to Properties 1 and 2 of the Green’s function, respectively. From Eqs. (3.36) and (3.37), we can then write Gx; x 0 D ˛1 y1 x

3.38

where ˛1 is an unknown constant. Region x < x ≤ b Similarly, we let y2 x be a nontrivial solution of the homogeneous differential Eq. (3.36), which satisfies the boundary condition at x D b, and obtain Gx; x 0 D ˛2 y2 x 3.39

where ˛2 is an unknown constant. Using Property 4, we obtain from Eqs. (3.38) and (3.39) ˛2 y2 x 0 ˛1 y1 x 0 D 0

3.40

46

GREEN’S FUNCTION

Applying Property 5, we take the derivatives of Eqs. (3.38) and (3.39) and substitute into Eq. (3.34) to obtain ˛2 y20 x 0 ˛1 y10 x 0 D

1 P1 x

3.41

Solving Eqs. (3.40) and (3.41) gives ˛1 D

y2 x 0 P1 x 0 Wx 0

3.42

˛2 D

y1 x 0 P1 x 0 Wx 0

3.43

where Wx 0 D y1 x 0 y20 x 0 y2 x 0 y10 x 0

3.44

is called the Wronskian of y1 and y2 at x D x 0 . The solutions ˛1 and ˛2 in Eqs. (3.42) and (3.43) exist and are unique when the Wronskian differs from zero, which is always valid unless y1 and y2 are linearly dependent. The closed-form solution of the Green’s function is now obtained from Eqs. (3.38), (3.39), (3.42), and (3.43) as

Gx; x 0 D

     y1 x D     y2 x D

y2 x 0 , a x x0 P1 x 0 Wx 0 y1 x 0 , x0 x b P1 x 0 Wx 0

3.45

It is apparent that this closed-form Green’s function is only useful if the solution to the homogeneous differential Eq. (3.36) can be found within the interval [a, b]. As a demonstration of the procedure for finding the Green’s function in closed form, we consider a shielded microstrip line as shown in Fig. 3.2. The shield and strip are assumed to be perfect conductors, and the strip thickness is negligible. We assume there exists a current source J on the strip. We wish to find the closed-form Green’s function and the z component of the electric field, Ez . The electric field component Ezi in region i (i D 0, 1) is produced by the current component Jz . It satisfies the two-dimensional Helmholtz wave equation 2 r2t C kc,i Ezi x, y D jωi Jz x 0 , y 0

3.46

and the boundary conditions Ezi 0, y D Ezi a, y D 0,

y 2 [0, b]

3.47a

Ez1 x, 0 D Ez0 x, b D 0,

x 2 [0, a]

3.47b

47


y

b

0

d

Air

W eO

h 1

er

a

0

Figure 3.2

x

Cross section of shielded microstrip line.

p where kc,i D ki2 C ' 2 is the cutoff wave number in region i, with ki D ω εi i and ' being the corresponding wave number and propagation constant, respectively. r2t D ∂2 /∂x 2 C ∂2 /∂y 2 represents the transverse Laplacian operator. The Green’s function Gx, y; x 0 , y 0 then satisfies 2 Gx, y; x 0 , y 0 D υx; x υ y; y 0 r2t Gx, y; x 0 , y 0 C kc,i

3.48

and the boundary conditions Gx D 0, y; x 0 , y 0 D h D Ga, y; x 0 , h D 0

3.49a

Gx, y D 0; x 0 , y 0 D h D Gx, b; x 0 , h D 0

3.49b

We now express Gx, y; x 0 , y 0 as a Fourier series of sine functions that satisfy the boundary conditions along the x axis (i.e., at x D 0, a): Gx, y; x 0 , y 0 D

1

gm y; x 0 , y 0 sin

mx

a

mD1

3.50

Substituting Eq. (3.50) into (3.48) yields 1

2 kc,i

mD1

C sin

m 2

a

gm y; x 0 , h sin

mx d2 g y; x 0 , h m a dy 2

mx

a

D υx; x 0 υ y; y 0

3.51

48

GREEN’S FUNCTION

Multiplying both sides by sinmx/a , integrating from 0 to a with respect to x, and applying the orthogonality relation, we find

,

sin 0

mx

,

sin

nx

,

dx D

,/2, 0,

mDn m 6D n

3.52

leads to 2 d2 gm y; x 0 , h 2m,i gm y; x 0 , h D sin 2 dy a where 2m,i D

m

mx 0 a

υ y; h

2 kc,i

a

3.53

3.54

Equation (3.53) is a one-dimensional differential equation, which can be solved using the procedure discussed earlier as follows. We break the interval [0, b] along the y axis into two separate regions, [0, h] and [h, b], and find the solution of the corresponding homogeneous differential equation d2 gm y; x 0 , h 2m,i gm y; x 0 , h D 0 3.55 dy 2 This solution, which satisfies the boundary conditions at y D 0 and b, is given as g1m D C1m x 0 , h sinhm,1 y ,

0y p εr

2W 4a t 1 2a C t AD1C C ln 2a t ( 2a t t 2a 1 4(W 0.414t BD1C C ln 0.5 C 0.5W C 0.7t W 2( t

4.96

4.97 4.98

The dielectric attenuation constant is given by ˛d D

4.6

p 27.3 εr tan υ 0

dB/m

4.99

SLOT LINE

The slot line, as shown in Fig. 4.6, is also useful for RF and microwave ICs. Its balanced nature is especially attractive for circuits requiring a balanced topology. In contrast with conventional transmission lines, the slot line does not support

SLOT LINE

79

y

W t

er

h

x

Figure 4.6

Cross section of a slot line.

a TEM propagation mode. Modes propagating on the slot the line are quasi-TE modes, which resemble the TE type. The dominant mode is quasi-TE10 , similar to the TE10 of the rectangular waveguide. The quasi-TE10 mode of the slot line, however, has no cutoff frequency. In order to be used as a good transmission line, the slot line must be fabricated using a substrate with a high dielectric constant to minimize radiation. On the other hand, an antenna using the slot line should be implemented using a low dielectric constant substrate. Closed-form expressions for the characteristic impedance, Zo , based on voltage and power and the wavelength, g , for the slot line on high dielectric constant substrates 9.7 εr 20 were obtained [15] by curve fitting to Cohn’s [16] numerical results. They are given as follows. For 0.02 W/h 0.2, g / 0 D 0.923 0.195 ln εr C 0.2W/h 0.126W/h C 0.02 lnh/ 0 ð 102 Zo D 72.62 15.283 ln εr C 50

4.100

W/h 0.02W/h 0.1 W/h

C lnW/h ð 102 19.23 3.693 ln εr [0.139 ln εr 0.11 C W/h0.465 ln εr C 1.44] ð 11.4 2.636 ln εr h/ 0 ð 102 2

4.101

For 0.2 W/h 1.0, g / 0 D 0.987 0.21 ln εr C W/h0.111 0.0022εr 0.053 C 0.041W/h 0.0014εr lnh/ 0 ð 102

4.102

80

PLANAR TRANSMISSION LINES

Zo D 113.19 23.257 ln εr C 1.25W/h114.59 22.531 ln εr C 20W/h 0.21 W/h [0.15 C 0.1 ln εr C W/h 0.79 C 0.899 ln εr ] ð [10.25 2.171 ln εr C W/h2.1 0.617 ln εr h/ 0 ð 102 ]2 4.103 These equations were obtained assuming infinitesimally thin conductors and are accurate to within 2% for the following ranges: 9.7 εr 20 h h 0.01 0 0 c

4.104 4.105

where h/ 0 c is the cutoff value for the TE10 surface-wave mode on the slot line and is given as h D 0.25 εr 1 4.106 0 c Closed-form expressions for the characteristic impedance and wavelength for low dielectric constant substrates were also derived by curve fitting to results of the spectral-domain method [17].

REFERENCES 1. D. D. Grieg and H. F. Englemann, “Microstrip — A New Transmission Technique for the Kilomegacycle Range,” Proc. IRE, Vol. 40, pp. 1644–1650, Dec. 1952. 2. E. O. Hammerstad and O. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” 1980 IEEE MTT-S Digest, pp. 407–409. 3. E. O. Hammerstad, “Equation for Microstrip Circuit Design,” Proc. Eur. Microwave Conf., pp. 268–272, 1975. 4. R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Losses in Microstrip,” IEEE Trans. Microwave Theory Tech., Vol. MTT-16, pp. 342–350, June 1968. 5. R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Correction to Losses in Microstrip,” IEEE Trans. Microwave Theory Tech., Vol. MTT-16, p. 1064, Dec. 1968. 6. J. D. Welch and H. J. Pratt, “Losses in Microstrip Transmission Systems for Integrated Microwave Circuits,” NEREM Rec., Vol. 8, pp. 100–101, 1966. 7. C. P. Wen, “Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonreciprocal Gyromagnetic Device Application,” IEEE Trans. Microwave Theory Tech., Vol. MTT-17, pp. 1087–1090, Dec. 1969. 8. G. Ghione and C. Naldi, “Analytical Formulas for Coplanar Lines in Hybrid and Monolithic MICs,” Electron. Lett., Vol. 20, No. 4, pp. 179–181, Feb. 1984. 9. G. Ghione and C. Naldi, “Parameters of Coplanar Waveguides with Lower Ground Plane,” Electron. Letters, Vol. 19, No. 18, pp. 734–735, Sept. 1983.

PROBLEMS

81

10. W. Hilberg, “From Approximation to Exact Relations for Characteristic Impedances,” IEEE Trans. Microwave Theory Tech., Vol. MTT-13, pp. 29–38, Jan. 1965. 11. K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines, Artech House, Dedham, MA, 1979, pp. 277–280. 12. K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design of Microwave Circuits, Artech House, Dedham, MA, 1981, p. 72. 13. S. B. Cohn, “Problems in Strip Transmission Lines,” IRE Trans. Microwave Theory Tech., Vol. MTT-2, pp. 52–55, July 1954. 14. H. A. Wheeler, “Transmission Line Properties of a Stripline Between Parallel Planes,” IEEE Trans. Microwave Theory Tech., Vol. MTT-26, pp. 866–876, Nov. 1978. 15. R. Garg and K. C. Gupta, “Expressions for Wavelength and Impedance of Slotline,” IEEE Trans. Microwave Theory Tech., Vol. MTT-24, p. 532, Aug. 1976. 16. S. B. Cohn, “Slot Line on a Dielectric Substrate,” IEEE Trans. Microwave Theory Tech., Vol. MTT-17, pp. 768–778, Oct. 1969. 17. R. Janaswamy and D. H. Schaubert, “Characteristic Impedance of a Wide Slotline on Low-Permittivity Substrates,” IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 900–902, Aug. 1986.

PROBLEMS

4.1 The quasi-TEM characteristic impedance of a lossless transmission line may be determined from 1 Zo D p c CCa where C and Ca are the capacitances per unit length with and without the dielectrics, and c is the speed of light in air. Use this equation to derive Zo of a lossless coaxial line having inner and outer radii of a and b, respectively. The coaxial line is filled with a dielectric having relative dielectric constant εr . 4.2 Prove that the three commonly used definitions of the dynamic characteristic impedance given in Eqs. (4.8)–(4.10) are identical if Pavg D 1 ReVIŁ . 2 4.3 Show that the three definitions of characteristic impedance in Eqs. (4.8)– (4.10) are derived from the formula for the characteristic impedance of a TEM transmission line. 4.4 Derive the following characteristic impedances of a rectangular waveguide, having dimensions a and b along the x and y axes, respectively, for TEm0 modes: Vo m(b D (a) ZVI Zh ; m D 1, 3, 5, . . . o D Io 2a 2Pavg m2 ( 2 b D D (b) ZPI Zh ; m D 1, 3, 5, . . . o jIo j2 8a

82


(c) ZPV o D

jVo j2 2b Zh D 2Pavg a

where Zh D jω/ is the wave impedance, and D is the propagation constant.

m(/a2 2(/ g 2

4.5 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a microstrip line, with zero strip thickness, as a function of the normalized strip width, W/h. Compute and plot Zo and εeff versus W/h from 0.05 to 5 for relative dielectric constant, εr , of 2.2 and 10.5. 4.6 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a microstrip line with zero strip thickness as a function of frequency, f. Compute and plot Zo and εeff versus f from 0 to 40 GHz for normalized strip width, W/h, of 0.1, 0.5, 1, 2, and 5, and two different substrates: εr D 10.5, h D 0.635 mm (RT/Duroid 6010) and εr D 2.2, h D 0.635 mm (RT/Duroid 5880). Based on these results, compare the dispersions between high- and low-dielectricconstant substrates, and between thin and thick substrates. 4.7 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a microstrip line with finite strip thickness as a function of the normalized strip width, W/h. Compute and plot Zo and εeff versus W/h from 0.05 to 5 for a relative dielectric constant, εr , of 2.2 and 10.5, and normalized strip thickness, t/h, of 0, 0.01, 0.02, 0.05, and 0.1. 4.8 Write a computer program to calculate the conductor, ˛c , and dielectric, ˛d , attenuation constants of a microstrip line as a function of the normalized strip width, W/h. Compute and plot ˛c and ˛d versus W/h from 0.05 to 5 for RT/Duroid 5880 substrate with relative dielectric constant, εr , of 2.2 and loss tangent of 0.0009, normalized strip thickness, t/h, of 0, 0.01, 0.02, 0.05, and 0.1, and a strip conductivity, 0, of 5.8 ð 107 S/m (copper). 4.9 Write a computer program to calculate the normalized strip width, W/h, of a microstrip line with zero strip thickness as a function of the characteristic impedance, Zo . Compute and plot W/h versus Zo from 30 to 120 ohms for a relative dielectric constant, εr , of 2.2 and 10.5. 4.10 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a conventional CPW with zero strip thickness as a function of the dimension ratio, a/b. Compute and plot Zo and εeff versus a/b from 0.1 to 0.9 for a relative dielectric constant, εr , of 2.2 and 10.5 and normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10. 4.11 Repeat Problem 4.10 for a conductor-backed CPW with zero strip thickness.

PROBLEMS

83

4.12 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a conventional CPW with finite strip thickness as a function of the dimension ratio, a/b. Compute and plot Zo and εeff versus a/b from 0.1 to 0.9 for a relative dielectric constant, εr , of 10.5, normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10, and normalized strip thickness, t/a, of 0, 0.01, 0.02, 0.05, and 0.1. 4.13 Repeat Problem 4.12 for a conductor-backed CPW. 4.14 Write a computer program to calculate the conductor, ˛c , and dielectric, ˛d , attenuation constants of a conventional CPW as a function of the dimension ratio, a/b. Compute and plot ˛c and ˛d versus a/b from 0.1 to 0.9 for RT/Duroid 6010 substrate with a relative dielectric constant, εr , of 10.5 and loss tangent of 0.0028, normalized strip thickness, t/a, of 0, 0.01, 0.02, 0.05, and 0.1, and a strip conductivity, 0, of 5.8 ð 107 S/m (copper). 4.15 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a CPS with zero strip thickness as a function of the dimension ratio, a/b. Compute and plot Zo and εeff versus a/b from 0.1 to 0.9, for a relative dielectric constant, εr , of 2.2 and 10.5, and normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10. 4.16 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a CPS as a function of the dimension ratio, a/b. Compute and plot Zo and εeff versus a/b from 0.1 to 0.9 for a relative dielectric constant, εr , of 10.5, normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10, and normalized strip thickness, t/2a, of 0, 0.01, 0.02, 0.05, and 0.1. 4.17 Write a computer program to calculate the conductor, ˛c , and dielectric, ˛d , attenuation constants of a CPS as a function of the dimension ratio a/b. Compute and plot ˛c and ˛d versus a/b from 0.1 to 0.9 for RT/Duroid 6010 substrate with a relative dielectric constant, εr , of 10.5 and loss tangent of 0.0028, normalized strip thickness, t/2a, of 0, 0.01, 0.02, 0.05, and 0.1, and a strip conductivity, 0, of 5.8 ð 107 S/m (copper). 4.18 Write a computer program to calculate the characteristic impedance, Zo , of a strip line with zero strip thickness as a function of the normalized strip width, W/a. Compute and plot Zo versus W/a from 0.1 to 5 for a relative dielectric constant, εr , of 2.2 and 10.5. 4.19 Write a computer program to calculate the characteristic impedance, Zo , of a strip line with finite strip thickness as a function of the normalized strip width, W/a. Compute and plot Zo versus W/a from 0.1 to 5 for a relative dielectric constant, εr , of 2.2 and 10.5, and normalized strip thickness, t/a, of 0, 0.01, 0.02, 0.050, 0.1, and 0.2. 4.20 Write a computer program to calculate the normalized strip width, W/a, of a strip line with finite strip thickness as a function of the characteristic impedance, Zo . Compute and plot W/a versus Zo from 10 to 100 ohms

84


for a relative dielectric constant, εr , of 2.2 and 10.5, and normalized strip thickness, t/a, of 0, 0.01, 0.02, 0.05, 0.1, and 0.2. 4.21 Write a computer program to calculate the conductor, ˛c , and dielectric, ˛d , attenuation constants of a strip line as a function of the normalized strip width, W/a. Compute and plot ˛c and ˛d versus W/a from 0.1 to 5 for RT/Duroid 5880 substrate with a relative dielectric constant, εr , of 2.2 and loss tangent of 0.0009, normalized strip thickness, t/a, of 0, 0.01, 0.02, 0.050, 0.1, and, 0.2, and a strip conductivity, 0, of 5.8 ð 107 S/m (copper). 4.22 Write a computer program to calculate the characteristic impedance, Zo , and effective dielectric constant, εeff , of a slot line as a function of the normalized slot width, W/h. Compute and plot Zo and εeff versus W/h from 0.1 to 2 for a relative dielectric constant, εr , of 10.5, and frequency of 1, 10, and 20 GHz.


CHAPTER FIVE

Conformal Mapping Conformal mapping is a useful mathematical analysis for solving many boundary problems in various engineering disciplines. Perhaps the most elegant features of conformal mapping are its simplicity and ability to produce closed-form formulas. Conformal mapping can be employed to obtain solutions for many electromagnetic problems for both static and dynamic situations. In particular, this analytical method has been used to analyze various transmission lines used in RF and microwave integrated circuits (e.g., [1]). In this chapter, we will present the fundamentals of this method [2] and apply it to determine the static or quasi-static parameters, such as characteristic impedances and effective dielectric constants, of planar transmission lines.

5.1

PRINCIPLES OF MAPPINGS

Mapping by itself is a very elementary geometric technique. In many mathematical problems, we have encountered a function f describing the relation between two complex variables z D x C jy and w D u C jv as w D fz . Geometrically, these complex variables are represented by points z D x, y and w D u, v , respectively. Consequently, u and v are effectively a function of both x and y. In polar coordinates, z and w are given as z D rej and w D ej , where r, and

, are the corresponding magnitudes or moduli and the angles or arguments of the complex numbers. In this mathematical description, the function f performs a transformation or mapping between the two corresponding points z and w in general or, specifically, transforms or maps point z into the corresponding point w according to a rule described by the function f. With this concept, f is thus referred to as a transformation or mapping. Geometrically, the function f then maps or transforms a curve or a region containing points (x, y) into another curve or region consisting of points (u, v). For simplicity and convenience of mathematical analysis, separate planes are normally used for the coordinates of points z and w. We will refer to the planes containing points z and w as the 85

86

CONFORMAL MAPPING

(complex) z and w planes, respectively. Figure 5.1 illustrates the principle of the mapping of z into w, in which C and C0 contain the corresponding points z and w, respectively. As an example, we consider point z in the z plane described in polar form as z D rej . We assume that point z is related to point w(u, v) in the w plane by the function or transformation w D fz D ln z 5.1 Assume that point z lies in a circle of radius a. We wish to map this circle into a curve in the w plane. From Eq. (5.1), we have u D ln r and v D , which indicate that points z on the circle with r D a and 0 2 in the z plane correspond to points w having u D ln a and 0 v 2 in the w plane. The circle of radius a in the z plane is thus mapped onto the line segment u D a, 0 v 2 of the w plane. This mapping of a circle onto a line segment is useful in analyses of some microwave structures. Figure 5.2 shows this mapping. y

v

C C′

x

z-plane

Figure 5.1

w-plane

u

General mapping from z D x, y into w D u, v D fz .

y

v

2p

r

θ

a

z = re jθ

Figure 5.2

x

ln a

w = ln z = u + jv

Mapping between a circle and a line segment.

u

FUNDAMENTALS OF CONFORMAL MAPPING

87

This example demonstrates that we can map a rather complex geometry into a simpler one. This leads to the possibility that we may be able to transform mathematically a geometrically complicated transmission line into a much simpler structure whose solution is known or can easily be found. This is the principle behind applications of conformal mapping in microwave problems. Figure 5.3 shows some useful transformations of regions along with the mapping functions [2]. 5.2


We consider a transformation w D fz that maps point zx, y into point wu, v . If the function f is analytic at point z and f0 z D df/dz 6D 0, then the mapping is said to be conformal. Conformal mapping, or conformal transformation, is therefore a mapping that employs an analytic function whose derivative is always nonzero on the z plane where the function is defined. As will be seen, this kind of mapping is especially useful for analyzing planar transmission lines. A function f is analytic at a point z if its derivative exists and is unique at z. The function is then said to be analytic in a particular region if it is analytic at every point in that region. This requirement is met if f satisfies both the necessary and sufficient conditions for analyticity. A necessary, but not sufficient, condition for f to be analytic is that the first-order partial derivatives of its real part u and imaginary part v with respect to x and y must satisfy the Cauchy–Riemann equations: ∂v ∂u D ∂x ∂y

5.2a

∂v ∂u D ∂y ∂x

5.2b

Any two functions that satisfy the Cauchy–Riemann equations are called harmonic conjugate functions or simply conjugate functions. A sufficient condition for the existence of analyticity is that the first-order partial derivatives of u and v must exist in the neighborhood of z, where f is defined, and are continuous and satisfy the Cauchy–Riemann equations at z. Taking the partial derivatives of Eqs. (5.2a) and (5.2b) with respect to x and y, respectively, and adding the resulting equations, we obtain ∂2 u ∂2 u C 2 D0 ∂x 2 ∂y

5.3

Similarly, differentiating Eqs. (5.2a) and (5.2b) with respect to y and x, respectively, and subtracting the results gives ∂2 v ∂2 v D D0 ∂x 2 ∂y 2

5.4

88

CONFORMAL MAPPING

v

y D

jp

E

F

1

A

B

C

F′

x

E′ D ′ C′ B ′

A′

u

(a) w = e z

y v A

B

B′

j

A′

C′

C

1 E′

x

u

D D′ E (b) w =

z−1 z +1

y

v D′

C′

D′

E ′ A′

jp

B′

1

A

B

C

D

E

x

(c) w = ln

Figure 5.3

z −1 z+1

Some useful mappings.

B′ u

89


y

v

C E′

B

D E

A

1

D′

x

C

B′ 2

A′ u

l (d) w = z + z

Figure 5.3

(Continued )

The real and imaginary parts of an analytic function of complex variables thus satisfy the two-dimensional Laplace’s equation and, hence, either one may represent the potential. This fact implies that we can solve for the potential and the corresponding static electric and magnetic fields in a particular microwave structure by choosing an analytical function wz , determining its real part u and imaginary part v, and taking either u or v as the potential subject to the structure’s boundary conditions. For transmission line problems, the obtained potential and electric field can be used to determine the capacitance per unit length of the transmission line and hence the transmission line’s characteristic impedance and effective dielectric constant. We now consider an analytical function w D fz . Under the transformation f, any infinitesimal linear element dz in the z plane is transformed into a small linear element dw in the w plane, according to the chain rule, as dw D

dw dz dz

5.5

provided that dw/dz is nonzero. In general, dw/dz is complex and hence may be expressed in terms of its magnitude a and phase as dw D aej dz

5.6

dw D aej dz

5.7

from which, we can write

This result indicates that the transformed element dw is obtained from the original element dz by multiplying its length by a and rotating it through an angle . Figure 5.4 illustrates this transformation. It hence follows that any infinitesimal area in a neighborhood of the point z would be transformed into a similar

90

CONFORMAL MAPPING

v

y

C1′

C1

f

C2

x

Figure 5.4

f

C2′

u

A mapping from the curve C in the z plane into the curve C0 in the w plane.

infinitesimal area, that is, having approximately the same shape, in a neighborhood of the point w. This kind of mapping preserves the form of the original element and, hence, the name conformal mapping. It should be noted that this mapping is only achieved by means of an analytical function defined on a domain whose derivative is never zero. The transformed area is obtained by scaling each linear dimension of the original area by a factor of a and rotating the original element by an angle . It should be noted that since the scale factor a and the rotational angle generally change from point to point, a large region in the z plane may be transformed into another region in the w plane that has no resemblance to the original one. It is also recognized that if two curves in the z plane intersect at a particular angle, then their transformed curves in the w plane will also intersect at the same angle, due to the fact that both the transformed curves will be turned through the same angle . Any conformal mapping would therefore transform orthogonal or parallel curves in the z plane into orthogonal or parallel curves in the w plane, respectively. Figure 5.5 illustrates the transformation of orthogonal curves between the two planes. In mapping region to region, however, parallel curves of the original region may not be transformed into parallel curves, or parallel curves in the transformed region may not correspond to parallel curves in the original region due to the physical constraints of the original region. However, if the curves in the transformed region are orthogonal, then their corresponding curves in the original region must also be orthogonal and vice versa. As an example of conformal mapping, we consider again the logarithmic function given in Eq. (5.1). By expressing z in the polar form as z D rej , we can obtain u D ln x 2 C y 2 D ln r 5.8a y v D tan1 D 5.8b x


y

v2

B

91

v

u1

A′

v1 A

u1

D′

A

C u2

D

B′

v1

v2

C′ u

x u2 z plane

Figure 5.5

w plane

A mapping of perpendicular curves between the z and w planes.

y

v q0

C D

00

r = r1 q0 A

C′

D′

r = r2 A′ B

x

0

u1 = ln r1

B′ u 2 = ln r2

u

w plane

z plane

Figure 5.6

The mapping between a circular and a rectangular region.

Equation (5.8) indicates that a circular region in the z plane would transform onto a rectangular region in the w plane as shown in Fig. 5.6. This mapping performs several transformations. It maps the circular region ABCD onto the rectangular region A0 B0 C0 D0 , the arcs AD and BC into the u-constant segments A0 D0 and B0 C0 , and the straight-line segments DC and AB into the v-constant segments D0 C0 and A0 B0 , respectively. The mapping transforms orthogonal curves, such as AB and BC, into orthogonal curves; such as A0 B0 and B0 C0 , respectively. It also maps the parallel segments AD and BC into the parallel segments A0 D0 and B0 C0 , respectively. It should be noted that, although the segments DC and AB intersect at an angle 0 6D , the transformed curves D0 C0 and A0 B0 are in parallel. This is due to the constraints imposed by the AB and CD or the angle 0 .

92

CONFORMAL MAPPING

Had the segments DC and AB been transformed as two independent curves, the transformed curves D0 C0 and A0 B0 would have intersected at the same angle 0 . We can also see that the straight lines u D constant and v D constant in the w plane correspond to the circle of radius eu centered at the origin (0,0) and the straight line passing through the origin and intersecting the circle at a right angle. In electromagnetic problems, in general, and transmission line problems, in particular, the boundary conditions are normally given by the function itself or by the normal derivative of the function along the boundary. These boundaryvalue problems are known as Dirichlet and Neumann problems, respectively. Under conformal mapping these conditions remain unchanged in the transformed region. Let’s consider an analytic function w D fz D ux, y C jvx, y that conformally maps a curve C in the z plane into a curve C0 in the w plane. Let a function hu, v be defined that is differentiable on C0 . If hu, v satisfies either of the boundary conditions hu, v D d 5.9a where d is a constant, or dhx, y D0 dn

5.9b

along C0 , then the function Hx, y D h[ux, y , vx, y ] also satisfies the corresponding boundary condition along C; namely,

or

Hx, y D d

5.10a

dHx, y D0 dn

5.10b

This property of conformal mapping implies that, instead of solving the boundaryvalue problem directly in the z plane, we can transform this problem into a simpler one in the w plane, determine its solution, and transfer back to the z plane to obtain the solution of the original problem. As a demonstration of the above principle, we consider a general twoconductor transmission line, whose transverse plane is in the z plane, embedded in a linear, homogeneous, and isotropic medium of permittivity ε. Under conformal mapping, this two-conductor transmission line is transformed into another twoconductor structure in the w plane. The energy per unit length stored in the electrostatic field of the transmission line in the z plane is given as We D 12 εjEj2 dS 5.11 S

where E is the electrostatic field. Replacing E by Wt V, with V being the potential distribution between the two conductors, expanding the Wt operator, and


utilizing the Cauchy–Riemann Eq. (5.2a), we can write 2 2 1 ∂V ∂V We D ε C dx dy 2 x y ∂x ∂y 2 2 2 2 1 ∂V ∂u ∂V ∂v D ε C dx dy 2 x y ∂u ∂x ∂v ∂y 2 2 1 ∂V ∂V ∂u ∂v D ε C dx dy 2 x y ∂u ∂v ∂x ∂y 2 2 ∂V ∂V 1 C du dv D ε 2 u v ∂u ∂v

93

5.12

which represents the electrostatic energy per unit length for the transformed structure in the w plane. This energy is also given as We D 12 CV2o

5.13

where C is the transmission line’s capacitance per unit length and Vo stands for the potential difference between the two conductors. The per-unit-length capacitances of the corresponding transmission lines in the z and w planes are thus equal. This result shows that we can determine the capacitance per unit length of the transformed structure in the w plane and use it for that of the original transmission line in the z plane. Note that a transformation back to the z plane is not needed in this case. The solution for the transformed structure in the w plane would be easier to solve if the original structure is mapped in such a way that either the curves u D constant or the curves v D constant will coincide with the boundary of the transformed structure. This desired mapping wz D u C jv may be determined by expressing x and y of the original problem in parametric forms. Let fx, y D 0 be the equation describing a curve C in the z plane. Assume that f can also be expressed in parametric forms as x D f1 t and y D f2 t , where f1 and f2 are real functions, and t is the real independent variable whose domain corresponds to the entire curve C. Now consider the following transformation: z D f1 aw C jf2 aw

5.14

where a is a real constant. Letting v D 0 in Eq. (5.14), we obtain x D f1 au

5.15a

y D f2 au

5.15b

Note that both f1 au and f2 au are real numbers. These equations indicate that the mapping (5.14) would transform the curve fx, y D 0 in the z plane into

94

CONFORMAL MAPPING

the curve v D 0 in the w plane. This procedure may enable one to determine an appropriate conformal mapping that can produce a transformed structure that has its boundary coinciding with the curves u D constant or the curves v D constant. As an example, we consider a circle in the z plane described by fx, y D x 2 C y 2 D r 2

5.16

x D f1 t D r cos t

5.17a

y D f2 t D r sin t

5.17b

or

in parametric form. The transformation is given from Eq. (5.14) as z D r cosaw C jr sinaw

5.18

which can be rewritten in polar form as

and hence

z D rejaw

5.19

j z w D ln a r

5.20

This mapping function would transform a circle of radius r in the z plane into a segment corresponding to 0 u 2/a and v D 0, which coincides with the curve v D 0. In practice, conformal-mapping problems usually require several transformations in sequence. That is, the original structure is mapped successively to yield a final structure whose solution is known or can easily be found. To illustrate this multiple transformation, we consider the upper plane y > 0 and wish to transform it onto a region in the w plane. We perform this transformation in two successive steps. First, we use the mapping defined by w1 D u1 C jv1 D to obtain

z1 zC1

5.21

u1 D

x2 C y 2 1 x C 1 2 C y 2

5.22a

v1 D

2y x C 1 2 C y 2

5.22b

This mapping transforms the upper plane y > 0 onto the upper plane v1 > 0 in the w1 plane. Next, we employ the mapping w D u C jv D ln w1

5.23

95

THE SCHWARZ–CHRISTOFFEL TRANSFORMATION

v1

y

v D ′′

A −∞

B C −1

D 1

E x ∞

C′ −1

u1

D ′′ −∞

w1 plane

z plane

Figure 5.7

D ′ B ′ E ′ A′ 1

C ′′

jp

E ′′ A′′

B ′′

B ′′ u ∞

w plane

Successive mappings from the half-plane y > 0 onto the strip 0 < v < .

which yields, upon expressing w1 in the polar form of w1 D rej , u D ln r

5.24a

v D j

5.24b

Equation (5.24) shows that the upper plane v1 > 0 is mapped onto the strip 0 < v < in the w plane. This composite mapping is shown in Fig. 5.7. Available transformations such as those listed in Fig. 5.3 are useful for conformal-mapping problems.

5.3


Perhaps the most commonly used conformal mapping in transmission line analysis is the Schwarz–Christoffel transformation. This transformation maps the x axis and the upper half of the z plane onto a closed polygon and its interior in the w plane, respectively. We consider a conformal mapping w D fz and assume that its derivative with respect to z is given as f0 z D

dw D Az x1 k1 z x2 k2 Ð Ð Ð z xN kN dz

DA

N

z xn kn

5.25

nD1

where A is a complex constant, kn n D 1, 2, . . . , N) are real constants, and xn are real numbers representing points on the x axis and satisfying x1 < x2 < Ð Ð Ð < xN1 < xN

5.26

96

CONFORMAL MAPPING

v

y

k4 p

w4 f4

N

kN p x1

x2

x3

xn

xn +1 xN

f3

f

wN

f1 f1

x w1`

z plane (a)

Figure 5.8

k3p

k1p

w3 k2p w2 u

w plane (b)

The Schwarz–Christoffel transformation.

as shown in Fig. 5.8(a). It should be noted that one or two points of xn might be at infinity. The argument or angle of f0 z can be written as arg f0 z D arg A k1 argz x1 k2 argz x2 Ð Ð Ð kN argz xN 5.27 Now we consider a point z lying on the x axis. z xn is thus positive or negative when z > xn or z < xn , respectively, leading to

argz xn D

0, ,

z > xn z < xn

5.28

Assuming the point z traverses to the right through different points xn and letting

n D arg f0 z corresponding to xn < z < xnC1 , we then have from Eqs. (5.27) and (5.28)

n D arg A knC1 C knC2 C Ð Ð Ð C kn

5.29a

nC1 D arg A knC2 C knC3 C Ð Ð Ð C kn

5.29b

nC1 n D knC1

5.30

and hence,

The argument of f0 z may also be written as arg

dw du C j dv D arg dz dx dv D tan1 du

5.31


97

which is essentially the angle between the element dw and the real u axis in the w plane. It is now apparent that as the point z traverses the x axis of the z plane through the points xn , the corresponding point w, transformed by the mapping (5.25), traverses a polygon in the w plane through the vertices wn D fxn as seen in Fig. 5.8(b). The angle ( nC1 n ) of Eq. (5.30) measures the polygon’s exterior angle at the vertex wn . Note that the exterior angles are limited within š, and so 1 < kn < 1. It is also seen from Eq. (5.30) that as the point z passes through the point xnC1 , the direction of the path through which the transformed point w traverses changes by an angle of knC1 as shown in Fig. 5.8(b). We assume that the sides of the polygon do not cross one another and, hence, the polygon is a closed contour. The positive sense of the polygon is assumed to be counterclockwise. For the closed polygon, its interior angle at the vertex wnC1 is defined as the angle nC1 between the two adjacent sides and can thus be obtained from the corresponding exterior angle as nC1 D nC1 n D 1 knC1

5.32

by making use of Eq. (5.30). Integrating Eq. (5.25) with respect to z and using Eq. (5.32) gives w D fz D A

z N

z xn n /1 dz C B

5.33

nD1

where B is an arbitrary constant that determines the position of the polygon. The magnitude and angle of the constant A control the size and orientation of the polygon in the w plane. This mapping is known as the Schwarz–Christoffel transformation. This transformation maps the interior points of the upper half of the z plane into the points lying to the left sides of the polygon in the counterclockwise direction. This implies that the upper half of the z plane is mapped onto the interior region of the polygon in the w plane. In practice, a polygon is normally defined in the w plane for a given problem, and the Schwarz–Christoffel transformation is determined such that the x axis of the z plane is mapped onto the polygon. This process is accomplished by determining the polygon’s interior angles n and the points xn and evaluating the integral in Eq. (5.33). All these interior angles are readily obtained from the polygon. For the points xn , however, only a maximum of three points or three conditions of xn can be chosen arbitrarily. Hence, for polygons with more than three sides (N > 3), some of the points xn must be determined so that the x axis is transformed into the polygon. This is often difficult. The integration encountered in the transformation may also be quite complicated, further imposing another difficulty in obtaining the solution. For transmission line problems, the x axis of the z plane corresponding to the polygon that represents the transmission line in the w plane always has parts of different potentials. This is due to the fact

98

CONFORMAL MAPPING

that a transmission line with two or more conductors has two or more different potentials, respectively; one of these potentials is zero. In these cases, successive mappings would be needed.

5.4 APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION IN TRANSMISISON LINE ANALYSIS

To illustrate the concept of the Schwarz–Christoffel transformation for transmission line analysis, we now use it to analyze a microstrip transmission line with an upper conducting cover as shown in Fig. 5.9. We assume that the strip and ground planes are perfect conductors, the strip is infinitesimally thin, and the ground planes are infinitely large. We also suppose that the interface between the dielectric and air, except where the strip resides, represents a perfect magnetic surface. It should be noted that this assumption is exact only when the upper and lower regions are identical and thus maintaining a perfect symmetry. For the considered microstrip line, this symmetry does not exist and thus error in calculations is expected. Nevertheless, as will be seen, good results for practical microstrip lines are obtained by this conformal mapping. We can now decompose the total capacitance per unit length of the microstrip line into two independent per-unit-length capacitances: Co corresponding to the upper (air) and Cr corresponding to the lower (dielectric) planes. Each of these capacitances is determined as though the other capacitance was not present. To determine Cr , we thus only need to consider the lower part of the microstrip line. Because the structure is symmetrical with respect to the central vertical plane and, hence, imposes a perfect magnetic wall there, we can finally consider only a half of the lower part as shown in Fig. 5.10(a). We visualize this lower area as a polygon in the w plane whose vertices are w1 , w2 , w3 , and w4 and wish to transform it onto the upper half of the z plane as shown in Fig. 5.10(b). It is observed that while the different potentials of the strip and ground plane are on

ho

e0 2a

er

Figure 5.9

h

Cross section of microstrip line with an upper conducting cover.

99

APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION

v

w-plane

w3

(− ∞ ←) w4

h MW

er −a (− ∞ ←) w1

0

w2

wa

MW

u

(a)

y z-plane

er −1 MW 0

MW (− ∞ ←) x1

−xa

x2

x3

x

x4 (→∞)

(b)

Figure 5.10 The Schwarz–Christoffel mapping of a half of the microstrip’s lower part in the w plane (a) into the z plane (b). MW stands for magnetic wall.

different surfaces in the w plane, they appear on the same surface in the z plane. Using the Schwarz–Christoffel transformation (5.33), we can write the mapping from the z plane into the w plane as

wDA

z

z x1 1 /1 z x2 2 /1 z x3 3 /1 z x4 4 /1 dz C B 5.34

Since the considered polygon is rectangular, the interior angles are /2. To cover the entire x axis, we let x1 and x4 approach 1 and C1. We also choose x2 D 1 and x3 D 0, and rewrite the mapping as 1/2 z z z A z x2 z x3 1 1 dz C B wD p x1 x4 xi x4 z dz p CB 5.35 D A1 zz C 1

100

CONFORMAL MAPPING

p where A1 D A/ x1 x4 . Multiplying the numerator and denominator of the integrand by [z1/2 C z C 1 1/2 ][z1/2 C z C 1 1/2 ], we obtain

1/2 z C z C 1 1/2 z1/2 C z C 1 1/2 w D A1 dz C B

2 z1/2 C z C 1 1/2 2 z 1/2 d z C z C 1 1/2 D A1

2 C B z1/2 C z C 1 1/2

D 2A1 ln z1/2 C z C 1 1/2 C B

z

5.36

Note that under this mapping we have transformed the points xn in the z plane into the points wn in the w plane as follows: x1 ! 1 $ w1 ! 1

5.37a

x2 D 1 $ w2 D 0

5.37b

x3 D 0 $ w3 D jh

5.37c

x4 ! 1 $ w4 ! 1 C jh

5.37d

Note also that x1 < x2 < x3 < x4 as required by Eq. (5.26). Using the boundary conditions (5.37b) and (5.37c), we obtain 2A1 ln

p 1 D B

5.38a

B D jh

5.38b

Replacing 1 in Eq. (5.38a) by ej and solving for A1 gives h

5.39

2h 1/2 ln z C z C 1 1/2 C jh

5.40

A1 D The mapping (5.36) is now given as wD

Using this mapping, we can then write the relation between the two corresponding points z D xa , where xa > 0, and w D wa D a as jxa1/2 C jxa 1 1/2 D ea/2h ej/2

5.41

from which we can write xa 1 1/2 D ea/2h xa1/2

5.42


101

Taking the square of both sides of Eq. (5.41), we can derive xa1/2 D D

ea/h C 1 2ea/2h ea/2h C ea/2h 2

and hence xa D cosh2

5.43

a

5.44

2h

To obtain a structure whose capacitance can be determined easily, we further map the upper half-plane in the z plane of Fig. 5.10(b) onto a polygon in the w0 D u0 C jv0 plane using the Schwarz–Christoffel transformation again. For an easy calculation of the capacitance Cr , we choose a rectangular polygon. This mapping is illustrated in Fig. 5.11. It should be noted that we have transferred

y z-plane

εr MW

−∞←

−xa

x1

−1 MW 0

x2

x 4 (→∞)

x3

x

(a)

v′

w a′

va′ MW

w 3′

er

MW

0

w 1′

w 2′

u′

(b)

Figure 5.11 Conformal mapping from the upper half in the z plane (a) into the w plane (b) MW denotes magnetic wall.

102

CONFORMAL MAPPING

the different potentials along the x axis to different surfaces in the w0 plane. Since the polygon is rectangular we again have each of the interior angles equal to /2. For this mapping, the points along the x axis of the z plane are already known from the previous mapping. Note, however, that point x1 is now chosen to be xa , while points x2 , x3 , and x4 are still at 1, 0, and 1, respectively, as in Fig. 5.10. We choose the four points in the w0 plane corresponding to the points x1 , x2 , x3 , and x4 in the z plane as follows: x1 D xa D cosh2

a

2h

$ w10 D 0

x2 D 1 $ w20 D 1 x3 D 0 $

w30

D1C

5.45a 5.45b

jv0a

5.45c

x4 ! 1 $ w40 D j0 v0a

5.45d

We can write the Schwarz–Christoffel transformation for this mapping as

z

w 0 D A0

p

dz C B0 zz C 1 z C xa

5.46

where A0 and B0 are constants. Since B0 is as yet arbitrary, we may choose a lower limit for the integration. Setting this limit to zero and letting z D t2 , A01 D 2jA0 xa1/2 , we have 0

w D

A01

p

z

0

dt 1

t2 1

t2 /xa

C B0

5.47

Making use of the inverse elliptic function

sn1 s, k D 0

s

dt 1 t2 1 k 2 t2

,

0k1

5.48

we can rewrite Eq. (5.47) as

w0 D A01 sn1 z 1/2 , xa1/2 C B0

5.49

Substituting the boundary conditions (5.45a)–(5.45c) into Eq. (5.49), we get A01 sn1 xa1/2 , xa1/2 C B0 D 0

5.50a

A01 sn1 1, xa1/2 C B0 D 1 0

B D1C

5.50b jv0a

5.50c

103


which can then be solved to obtain, upon replacing k D xa1/2 , v0a D

j sn1 1, k sn1 1, k

5.51

sn1 k 1 , k

sn1 1, k is equal to the complete elliptic integral of the first kind, Kk , as 1

sn 1, k D Kk D 0

1

dt 1

t2 1

5.52

k 2 t2

Using Wallis formula

/2

sinn t dt D 0

1 ð 3 ð 5 ð Ð Ð Ð ð n 1 , 2 2 ð 4 ð 6 ð ÐÐÐ ð n

n D even integer

we can derive 2 1 1ð3 2 4 1ð3ð5 2 6 2 Kk D 1C k C k C k CÐÐÐ 2 2 2ð4 2ð4ð6

5.53

5.54

which can be used to compute Kk forpvarious values of k. To simplify Eq. (5.51), we let k 0 D 1 k 2 and obtain the complete integral of the first kind for the modulus k 0 as 1 dt Kk 0 D 2 1 t 1 t2 C k 2 t2 0 k 1 dt 5.55 D j 1 t2 1 k 2 t2 1 The inverse elliptic function sn1 k 1 , k is given as

k 1

sn1 k 1 , k D

0

D 0

1

dt 1 t2 1 k 2 t2 dt

1 t2 1 k 2 t2

D Kk jKk 0

k 1

C 1

dt 1 t2 1 k 2 t2 5.56

upon using Eqs. (5.52) and (5.55). Substituting Eqs. (5.52) and (5.56) into Eq. (5.51) then gives Kk 5.57 v0a D Kk 0 The upper and lower conducting plates of the polygon in Fig. 5.11(b) have different potentials and constitute a pure parallel-plate capacitor due to the

104

CONFORMAL MAPPING

existence of the magnetic walls along the u0 D 0 and 1 surfaces. As indicated earlier from the result of Eq. (5.13), the capacitance of the transformed structure is equal to that of the original structure. Therefore, the capacitance per unit length of the parallel-plate capacitor is equal to a half of the per-unit-length capcitance Cr corresponding to the microstrip’s lower half. Hence, we have 2ε0 εr Kk 0 D 2ε ε 0 r v0a Kk

Cr D

5.58

where k D sech

k0 D

a

5.59a

2h

1 k 2 D tanh

a

2h

5.59b

The per-unit-length capacitance Co corresponding to the upper half of the microstrip line with air as the dielectric can be derived from Eq. (5.58) as Co D 2ε0 where

ko D sech ko0

D

1

a 2ho ko2

Kko0 Kko

5.60

5.61a

D tanh

a 2ho

5.61b

The total capacitance per unit length of the microstrip line can now be obtained as a summation of the capacitances Cr and Co as C D 2ε0 εr

Kk 0 Kko0 C 2ε0 Kk Kko

5.62

The effective dielectric constant is given as εeff D

C Ca

5.63

where Ca is the capacitance per unit length with the dielectric replaced by air. Using Eq. (5.62) we obtain Kk 0 Kko0 C Kk Kko D Kk 0 Kko0 C Kk Kko εr

εeff

5.64


105

When h D ho , the effective dielectric constant becomes εeff D

εr C 1 2

5.65

which is independent of the strip width. The characteristic impedance is given as 1 Zo D p c CCa

5.66

Multiplying the numerator and denominator by Ca and making use of c D ε0 ,0 1/2 , Eq. (5.62), and Eq. (5.63), we get Zo D

p

εeff

60 Kk 0 Kko0 C Kk Kko

5.67

The ratio Kk /Kk 0 can be determined using Eq. (5.54). It can also be conveniently evaluated using the following accurate approximation [3]:  p  1C k 1   p , 0.5 k 2 1 ln 2    1 k   Kk D 5.68 Kk 0  0 k 2 0.5 p ,   0  1C k     ln 2 1 pk 0 For the conventional open microstrip line with no conducting cover, ho tends to infinity and hence ko becomes 1. This results in εeff equal to εr , which is incorrect for the microstrip line. Equations (5.64) and (5.67) cannot therefore be used for the open microstrip line by simply letting ho approach infinity. This limitation is due to the fact that the assumption of a perfect magnetic surface at the upper dielectric surface is no longer valid as ho becomes very large. To determine the valid range for these conformal-mapping equations, we compute the effective dielectric constant and characteristic impedance of the microstrip line versus 2a/h for different ho /h and commonly used relative dielectric constants of 2.2, 6.15, and 10.5. We also compare these results with those obtained more accurately by the spectral-domain method described in Chapter 7. We found that Eqs. (5.64) and (5.67) are valid within 6% for ho /h up to 5. Note that the results when ho /h D 5 can also be used for an open microstrip line. We then have the following conditions for ko and ko0 :  ho a   , 0 0, y > 0

0

5.10 Verify the transformation between the figures in the z and w planes in Fig. 5.3(a) using the given mapping function. 5.11 Verify the transformation between the figures in the z and w planes in Fig. 5.3(b) using the given mapping function. 5.12 Use the Schwarz–Christoffel transformation to derive the mapping function given in Fig. 5.3(c). 5.13 Derive the capacitance per unit length, effective dielectric constant, and characteristic impedance of the microstrip line, shown in Fig. 5.9, through successive Schwarz–Christoffel transformations. The first mapping between the z and w planes is described in Fig. P5.2. 5.14 Consider an asymmetric strip line shown in Fig. P5.3. We assume that the strip is negligibly thin and a perfect conductor and the ground planes are infinitely large and perfect conductors. Use the Schwarz–Christoffel transformation to derive a closed-form equation for the characteristic impedance of this strip line. 5.15 Calculate the characteristic impedance of the strip line in Problem 5.14 for W/h2 from 0.1 to 10, εr of 2.2 and 10.5, and h1 /h2 of 1 and 2. 5.16 Consider a CPW with an upper conducting cover as shown in Fig. 5.14. Assume that the central and ground strips have negligible thickness and are perfect conductors, the ground strips are infinitely large, and the upper cover is infinitely large and a perfect conductor. We also suppose that the slots behave as perfect magnetic walls. Use the Schwarz–Christoffel transformation to derive closed-form formulas for the capacitance per unit length, effective dielectric constant, and characteristic impedance of this CPW. 5.17 Calculate the effective dielectric constant and characteristic impedance of the CPW in Problem 5.16 for a/b from 0.1 to 0.8, εr of 2.2 and 10.5, h of 0.635 and 1.27 mm, and h1 /h of 1, 2, and 4. 5.18 Consider a conductor-backed coplanar waveguide (CPW) with an upper conducting cover as shown in Fig. 5.16. Assume that the central and ground strips have negligible thickness and are perfect conductors, the

116

CONFORMAL MAPPING

v w-plane

w1 (→∞)

w2 h er

a

0

w3

wa

w4

(→∞)

u

(a)

y z- plane

er

0 (− ∞ ←)

x1

1

x2

x3

xa

x 4 (→∞)

(b)

Figure P5.2

h1 W er

h2

Figure P5.3

Cross section of an asymmetric strip line.

x

PROBLEMS

117

ground strips are infinitely large, and the upper and lower conducting plates are infinitely large and perfect conductors. We also suppose that the slots behave as perfect magnetic walls. Use the Schwarz–Christoffel transformation to derive closed-form formulas for the capacitance per unit length, effective dielectric constant, and characteristic impedance of this CPW. 5.19 Calculate the effective dielectric constant and characteristic impedance of the CPW in Problem 5.18 for a/b from 0.1 to 0.8, εr of 2.2 and 10.5, h of 0.635 and 1.27 mm, and h1 /h of 1, 2, and 4. 5.20 Consider a CPW with finite ground planes as shown in Fig. 5.17. Assume that the central and ground strips have negligible thickness and are perfect conductors, and the ground strips are infinitely large. We also suppose that the slots behave as perfect magnetic walls. Use the Schwarz–Christoffel transformation to derive closed-form formulas for the capacitance per unit length, effective dielectric constant, and characteristic impedance of this CPW. 5.21 Calculate the effective dielectric constant and characteristic impedance of the CPW in Problem 5.20 for a/b from 0.1 to 0.8, εr of 2.2 and 10.5, h of 0.635 and 1.27 mm, and b/c of 0.2, 0.4, and 0.8. 5.22 Consider an asymmetric CPW as shown in Fig. 5.18. Assume that the central and ground strips have negligible thickness and are perfect conductors, and the ground strip is infinitely large. We also suppose that the upper air–dielectric interface, except where the central and ground strips reside, behaves as a perfect magnetic wall. Use the Schwarz–Christoffel transformation to derive closed-form formulas for the capacitance per unit length, effective dielectric constant, and characteristic impedance of this CPW. 5.23 Calculate the effective dielectric constant and characteristic impedance of the CPW in Problem 5.22 for W/S from 0.1 to 1.5, εr of 2.2 and 10.5, and h of 0.635 and 1.27 mm. 5.24 Consider a coplanar strip (CPS) as shown in Fig. 5.19. The strips are assumed to be infinitely thin and perfect conductors. We also suppose that the upper air–dielectric interface, except where the strips reside, appears as a perfect magnetic surface. Use the Schwarz–Christoffel transformation to derive closed-form equations for the effective dielectric constant and characteristic impedance of this CPS. 5.25 Calculate the effective dielectric constant and characteristic impedance of the CPS in Problem 5.24 for a/b from 0.1 to 0.8, εr of 2.2 and 10.5, and h of 0.635 and 1.27 mm. 5.26 Consider an asymmetrical CPS as shown in Fig. 5.20. The strips are assumed to be infinitely thin and perfect conductors. We also suppose that the upper air–dielectric interface, except where the strips reside, appears as a perfect magnetic surface. Use the Schwarz–Christoffel transformation

118

CONFORMAL MAPPING

to derive closed-form equations for the effective dielectric constant and characteristic impedance of this CPS. 5.27 Calculate the effective dielectric constant and characteristic impedance of the CPS in Problem 5.26 for S/S C W1 C W2 from 0.1 to 0.8, εr of 2.2 and 10.5, and h of 0.635 and 1.27 mm. 5.28 Consider two parallel-coupled strip lines as shown in Fig. 5.21. We assume that the strips are negligibly thin and perfect conductors and the ground planes are infinitely large and perfect conductors. Use the Schwarz–Christoffel transformation to derive closed-form equations for the even- and odd-mode capacitances per unit length and characteristic impedances of these parallel-coupled strip lines. 5.29 Calculate the even- and odd-mode characteristic impedances of the parallel-coupled strip lines in Problem 5.28 for W/b from 0.1 to 2, εr of 2.2 and 10.5, and W C S /b of 0.4, 0.8, and 1.6. 5.30 Consider the microstrip line as shown in Fig. 5.9. Its effective dielectric constant, εeff , and characteristic impedance, Zo , can be determined using the conformal-mapping Eqs. (5.64) and (5.69), respectively. Verify Eq. (5.69) by calculating εeff and Zo versus 2a/h for different ho /h and εr D 2.2, 6.15, and 10.5, and compare them to more accurate results using techniques such as the spectral-domain method presented in Chapter 7. 5.31 Consider a coaxial transmission line, whose cross section is shown in Fig. P5.4. The transmission line is filled with two dielectric materials

er 2

er 1

a b

Figure P5.4

Cross section of a coaxial transmission line having two different dielectrics.

PROBLEMS

119

having relative dielectric constants of εr1 and εr2 . The inner and outer radii are a and b, respectively. The conductors are assumed to be perfect, and the dielectrics are lossless. Use conformal mapping to derive an expression for the characteristic impedance Zo . Verify the result with those obtained in Problem 5.2 and by the well-known equation 60 b Zo D p ln εr a when εr1 D εr2 .


CHAPTER SIX

Variational Methods In electromagnetic problems, in general, and microwave problems, in particular, solutions are normally obtained directly by solving appropriate differential or integral equations. For example, the resonant frequency of a resonator can be determined by solving the wave equation of the electric or magnetic field. On the other hand, variational methods operate by indirectly looking for a solution. Generally, a variational method seeks a functional that gives a maximum or minimum of a desired quantity. For example, it searches for the charge distribution (functional) on a transmission line that produces a maximum of the capacitance per unit length of the transmission line. So a variational method is essentially a maximization or minimization technique. The main advantage of the variational method is that it produces stationary formulas, which yield results insensitive to the first-order errors in the unknown function. There are, in general, three kinds of variational methods, depending on the technique used to obtain approximate solutions of problems expressed in a variational form: the direct method based on the classical Rayleigh–Ritz or simply Ritz procedure, the indirect method such as Galerkin and least squares, and the semidirect method based on separation of variables. Variational methods can be formulated in both the space and Fourier-transform or spectral domain. The use of the spectral domain simplifies the derivation of the Green’s function needed in the analysis, as will be seen later in this chapter and also in Chapter 7 on the spectral-domain method. Applications of variational methods include analysis of transmission lines to obtain characteristic impedances, effective dielectric constants, and losses, analysis of discontinuities, determination of resonant frequencies of resonators, and determination of impedances of antennas and obstacles in waveguides. In this chapter, we will describe variational methods for analyzing transmission lines in both the space and spectral domain. Coaxial and microstrip transmission lines will be used to illustrate the formulation process. The Ritz procedure will be implemented to obtain numerical results for the characteristic impedances and effective dielectric constants. 120

FUNDAMENTALS OF VARIATIONAL METHODS

6.1

121

FUNDAMENTALS OF VARIATIONAL METHODS

As indicated earlier, a variational method generally seeks a functional that results in a maximum or minimum of a desired quantity. Let’s consider a functional y, which depends on a function ux as y D f[ux], and assume that the variational method will look for ux that gives a maximum or minimum of y. To this end, we assume a function for ux and call it a trial function. Let x D x0 to be the exact or true solution that produces the absolute maximum or minimum of y. Now applying Taylor’s series around this true x D x0 gives df 1 d2 f fx D fx0 C x x0 C dx xDx0 2! dx 2 xDx0 1 dn f ð x x0 2 C Ð Ð Ð C x x0 n C RnC1 6.1 n! dx n xDx0 from which we obtain df 1 d2 f x x0 C x x0 2 C Ð Ð Ð ffx fx0 D dx xDx0 2! dx 2 xDx0

6.2

or f D υfx C υ2 fx C Ð Ð Ð

6.3

where υf is called the first variation of f or the first-order error, and υ2 f represents the second variation of f or the second-order error. The functional y D f[ux] is called the stationary or variational expression if υf D 0 for x 6D x0 , and nonstationary otherwise. υf D 0 leads to the derivative of fx at x D x0 to be equal to zero, implying that yx is a minimum or maximum at x D x0 . The value of x corresponding to the minimum or maximum of yx is referred to as the upper- or lower-bound solution, respectively. Figure 6.1 illustrates stationary and non stationary solutions for y D fx. Note that the stationary formula, corresponding to υf D 0, gives a smaller error in y than does the non stationary formula, which has υf 6D 0, for the same approximate solution x1 with a small deviation from the true solution x0 . An important criterion in applying a variational method is that y D f[ux] must be stationary. In order to prove that, we can let x D x0 C υx and y D y0 C υy, where x0 denotes the true solution for x and y0 is the corresponding value for y, and then we prove that υy D 0. In this process, we neglect the higher-order terms, which are second-order error. For problems involving multiple dimensions, we let y be a multidimensional functional as y D f[u1 x1 , x2 , . . . , xN , u2 x1 , x2 , . . . , xN , . . . , uN x1 , x2 , . . . , xN ]. The variational method then assumes trial functions for ui i D 1, 2, . . . , N. We can obtain stationary and nonstationary formulas as well as the upper- and lower-bound solutions in the same fashion as for the one-dimensional variational formula discussed earlier.

122

VARIATIONAL METHODS

y = f(x)

y = f(x)

f(x1) •

•

x0

•

f(x1) •

•

•

•

x1

•

x0

x

x1

(a)

x (b)

y = f(x) f(x1) •

•

•

•

x0

•

x1

x (c)

Figure 6.1 Illustration of (a, b) stationary and (c) nonstationary solutions. (a) Upper-bound solution; (b) lower-bound solution. For the same value of x1 , a stationary formula gives a smaller error in y than does the nonstationary formula.

Various techniques can be employed to determine approximate solutions of problems expressed in variational forms. In order to implement these techniques, we must derive the trial function ux such that the variational expression y has a minimum or a maximum. ux may be selected from a physical consideration; that is, choose ux to meet or closely approximate the actual behavior of the parameter it represents. The closers ux is to the true value, the better the result for y and the less computation time. The so-called Rayleigh–Ritz or simply Ritz method is perhaps the most commonly used procedure. It is applied by expanding the charge distribution as a sum of known functions whose coefficients are the variational parameters and by solving for the coefficient values by maximizing or minimizing y with respect to them. For the one-dimensional variational problem, the Ritz procedure essentially involves two steps. It first approximates the trial function ux by a linear combination of a finite number of functions as

ux D

N iD1

ai ui x

6.4

VARIATIONAL EXPRESSIONS FOR THE CAPACITANCE

123

where ai , i D 1, 2, . . . , N, are the (independent) unknown constants, referred to as the variational parameters. The ai need to be determined so that the functional y has a minimum or maximum. The ui x are referred to as the basis functions and should be so selected that Eq. (6.4) satisfies the boundary conditions of the considered physical structure for any choice of ai . In practical transmission lines, the general behavior of the potential or charge distribution of a transmission line is often known. This can be exploited to choose suitable basis functions ui x so that a linear combination of them, representing the trial function ux, may approximate closely the potential or charge’s actual behavior. In general, the ui x should be selected to form a complete set of orthogonal functions or polynomials. Once the ui x are chosen, we can write y as a function of ai as y D fai . Finally, the Ritz method continues by finding the variational parameters ai so that y is maximized or minimized. This is satisfied by taking the derivatives of y with respect to ai for i D 1, 2, . . . , N and letting them be zero as ∂y D 0, ∂ai

i D 1, 2, . . . , N

6.5

For multidimensional problems, a similar approach is also employed for the Ritz method. We can see that a variational method produces only approximate solutions for a problem. The main characteristic of this method is that the formula for the desired quantity is stationary about the true or exact solution, and hence the name stationary or variational formula. This means that a variational expression is relatively insensitive to small changes in the approximation around the true solution, which implies that a first-order error in the approximation (i.e., a trial function) produces only a second-order error in the desired solution. A variational formula may produce an upper-bound (maximum) or a lower-bound (minimum) solution of the desired quantity.

6.2 VARIATIONAL EXPRESSIONS FOR THE CAPACITANCE PER UNIT LENGTH OF TRANSMISSION LINES

We learn in Chapter 4 that the static or quasi-static parameters of a transmission line, including the characteristic impedance Zo and effective dielectric constant εeff , can be obtained from the transmission line’s capacitance per unit length C. Therefore, to determine these parameters using a variational method, variational formulas for C are needed. As for any variational expression, a variational expression for C may produce an upper- or a lower-bound solution for it. In this section, we will derive two kinds of expression for C: the upper bound based on the potential and the lower bound based on the charge density. It will be seen later that the upper and lower bounds to C correspond to the lower and upper bounds for the corresponding characteristic impedance, respectively.

124

VARIATIONAL METHODS

6.2.1

Upper-Bound Variational Expression for C

We begin by noting that the electrostatic energy stored per unit length of a general two-conductor transmission line, shown in Fig. 6.2, is given as We D 12 ε0 εr E Ð EŁ dx dy

D 12 ε0 εr

Wt Ð Wt dx dy

D

1 ε ε 2 0 r

∂ ∂x

2

C

∂ ∂y

2

dx dy

6.6

where E is the electric field between the two conductors S1 and S2 . In practice, one conductor, for example, S1 , is normally held at zero potential representing the ground and the other is kept at a potential Vo . ε0 denotes the permittivity of free space; εr represents the relative dielectric constant of the medium surrounding the two conductors; and x, y is the potential at any location in the plane transverse to the direction of propagation. The double integral is carried over the transmission line’s cross section. This energy may also be obtained as We D 12 CV2 where

6.7

S2

Wt Ð d

V D 2 1 D

6.8

S1

is the difference in potentials between the two conductors. d represents a differential-length vector along any path from conductor S1 to conductor S2 .

y

f2

f1

S2

S1

x

Figure 6.2 General two-conductor transmission line. 1 and 2 are the potentials on conductors S1 and S2 , respectively.


125

Equating Eq. (6.6) to (6.7) and making use of Eq. (6.8) gives

Wt 2 dx dy

C D ε0 εr

S2

6.9

Wt Ð d S1

It can be proved that Eq. (6.9) is stationary. That is, a first-order error in the potential causes a second-order error in the true or exact value Co of C. The true solution Co would result from Eq. (6.9) if we use the true value o for . Note that o satisfies Laplace’s equation exactly. An approximate value for C obtained from Eq. (6.9) corresponding to an approximate is always greater than Co . This equation therefore represents an upper-bound expression for C. Co is the stationary value, which is an absolute minimum, for C. Figure 6.3 shows a possible curve for the capacitance C as a function of , calculated using the upper-bound variational expression (6.9). The fact that Eq. (6.9) gives upperbound values to C implies that we should use a trial function for that results in calculated values of C that are as small as possible. This is an important criterion in selecting a proper trial function for . 6.2.2

Lower-Bound Variational Expression for C

We consider again the two-conductor transmission line shown in Fig. 6.2, in which conductors S1 and S2 are held at zero and Vo potentials, respectively. Now let x, y be the potential at any point in the transverse plane due to the (perunit-length) charge distribution x 0 , y 0 located at an arbitrary location x 0 , y 0 on conductor S2 . For a unit charge at x 0 , y 0 , the resulting potential is basically the Green’s function Gx, y; x 0 , y 0 associated with the transmission line, as discussed

C (f )

Co

0

fo

f

Figure 6.3 Possible values of C calculated using the upper-bound variational expression. The true value Co represents an absolute minimum.

126

VARIATIONAL METHODS

in Chapter 3. Note that this Green’s function is the (potential) response due to a unit (charge) source located at x 0 , y 0 on conductor S2 and can be determined from the physical parameters of a transmission line. The potential and the Green’s function G must satisfy the corresponding Poisson equations subject to appropriate boundary conditions. Using the principle of superposition, it can be shown that the potential at any location x, y due to a charge distribution x 0 , y 0 at a location x 0 , y 0 on conductor S2 is given as

Gx, y; x 0 , y 0 s x 0 , y 0 d0

x, y D

6.10

S2

where the integral is performed over the surface of conductor S2 in the transverse plane where the charge is distributed. This potential is equal to Vo for locations on S2 . That is,

Gx, y; x 0 , y 0 s x 0 , y 0 d0

Vo D

6.11

S2

where x, y indicates a location on S2 only. The capacitance per unit length of the transmission line is obtained as Q Q2 D Vo QVo

CD

6.12

where

s x, y d

QD

6.13

S2

denotes the total charge on S2 . Substituting Eqs. (6.10) and (6.13) into Eq. (6.12), we have

1 D C

S2

S2

Gx, y; x 0 , y 0 s x, ys x 0 , y 0 d d0

2 s x, y d

6.14

S2

which is stationary. Note that the true value Co for C can be obtained from Eq. (6.14) if we use the true value o for . It can be proved easily that an approximate value for C calculated using Eq. (6.14) corresponding to an approximate is always smaller than the true Co . This suggests that this equation is a lower-bound expression for C. Co is the stationary value, which is an absolute maximum, for C in this case. Figure 6.4 illustrates possible values of the capacitance C as a function of the charge distribution , calculated using the lower-bound variational expression (6.14). The lower-bound characteristic of Eq. (6.14) presents a useful criterion in selecting an appropriate trial function for , in which we should choose that trial function leading to calculated results of C as large as possible.


127

C (r)

Co

ro

0

r

Figure 6.4 Possible values of C calculated using the upper-bound variational expression. The true value Co represents an absolute maximum.

6.2.3

Determination of C , Zo , and eeff

The capacitance per unit length C can be determined using either the upper- or lower-bound expression. The choice of a particular formula depends on individual problems. In general, if the potential is easily approximated, then one should use the upper-bound formula. If, however, can be determined easily, then using the lower-bound equation is more appropriate. For complex problems such as multilayer, multiconductor transmission lines, both upper- and lower-bound expressions may be used together for the best accuracy. As discussed in Section 6.1, in order to implement a variational method using the upper- or lower-bound expression, we must derive a trial function for or , respectively. Let ux be the (unknown) trial function representing either or , and let y D f[ux] represent either the variational expression (6.9) for C or 6.14 for 1/C, respectively. We must determine ux so that y has a minimum or a maximum. ux should be selected to meet or closely approximate the actual behavior of or . Often, we determine ux using Eq. (6.4) based on the Ritz procedure as N ux D ai ui x 6.15 iD1

where ui x, i D 1, 2, . . . , N, are basis functions. The ai are the unknown variational parameters. Once the ui x are chosen, we can write y as a function of ai as y D fai . We take the derivatives of y with respect to ai for i D 1, 2, . . . , N and let them be zero to maximize or minimize y as ∂y D 0, ∂ai

i D 1, 2, . . . , N

6.16

This produces a system of N homogeneous linear equations, from which we can solve for C as discussed in the Appendix at the end of this chapter. It should

128

VARIATIONAL METHODS

be noted that a more accurate result for C is obtained by using more variational parameters ai but at the expense of more computation time. Once C is determined, we can then calculate the characteristic impedance and effective dielectric constant as 1 Zo D p c CCa and εeff D

C Ca

6.17

6.18

respectively, where c is the free-space velocity and Ca is the capacitance per unit length with the dielectrics removed. From Eq. (6.17) we can see that the upper and lower bounds to C correspond to the lower and upper bounds to Zo . It should be noted again that an exact value for C and, hence, Zo and εeff , can be obtained only if the potential or charge distribution is given exactly. In practice, only approximate values for C, Zo , and εeff may be determined because values for and are obtained approximately through trial functions. 6.3 FORMULATION OF VARIATIONAL METHODS IN THE SPACE DOMAIN

The formulation process is best illustrated via practical transmission lines. We will first implement the upper-bound variational expression to determine the perunit-length capacitance and characteristic impedance of a coaxial transmission line. We will then analyze a three-layer microstrip line based on the lower-bound variational expression. 6.3.1

Variational Formulation Using Upper-Bound Expression

Figure 6.5 shows the considered coaxial transmission line. The conductors are assumed to be perfect conductors, and the substrate is lossless. Without lost of generality we assume that the potentials on the outer conductor S1 and inner conductor S2 are zero and Vo , respectively. Before beginning the analysis, we should note that, for this problem, the true or exact values for the potential at any location corresponding to a radius r in the transverse plane, o , and the per-unit-length capacitance Co can be derived as o D

r Vo ln lna/b b

6.19

where a and b are the radii of the inner and outer conductors, respectively, and Co D

2%ε0 εr lnb/a

6.20

FORMULATION OF VARIATIONAL METHODS IN THE SPACE DOMAIN

129

er

r

a

b

Figure 6.5 Coaxial transmission line. εr is the substrate’s relative dielectric constant; a and b are radii of the inner and outer conductors, respectively.

where εr represents the substrate’s relative dielectric constant. These equations may be used to verify the accuracy of results computed here. Now using the upper-bound variational expression (6.9) and following Collin [1], we express C as Wt 2 dx dy 6.21 C D ε0 εr 2 S2

Wt Ð d S1

Using the Ritz method described in Section 6.1, we first assume a trial function for as r D a0 C a1 r C a2 r 2 6.22 where a1 and a2 are the (unknown) variational parameters. Next, taking the partial derivatives of Eq. (6.22) in the cylindrical-coordinate system and using them in the transverse operator Wt , we can write Wt D a1 C 2a2 r ar

6.23

Substituting Eq. (6.23) into (6.21) gives

2%

b

a1 C 2a2 r2 r dr d& C D ε0 ε r

0

a b

2

a1 C 2a2 r dr

a

b3 a3 b2 a2 2 a1 C 4 a1 a2 C b4 a4 a22 2 3 D 2 2 2 2 b a a1 C 2b a b a a1 a2 C b2 a2 2 a22

2%ε0 εr

6.24

130

VARIATIONAL METHODS

The Ritz procedure continues by letting the partial derivatives of C with respect to a1 and a2 be zero. This will produce the minimum value for C and lead to a system of homogeneous equations, 8 2C 2C 2 b a2 2b2 a2 a1 C b a2 b a b3 a3 a2 D 0 %ε %ε 3 8 3 2C 2 2C 2 2 3 2 4 4 b a b a b a C b a 4b a a2 D 0 %ε 3 %ε 6.25 as outlined in Section 6.2.3. Note that these equations contain (unknown) C in the coefficient matrix, so the (unknown) variational parameters a1 and a2 cannot be solved prior to the determination of C. To solve for C, we require that the determinant of the coefficient matrix equal zero, which is necessary in order to have nontrivial solutions for these equations. Enforcing this condition yields a characteristic equation of C, from which we can solve for C directly. Assuming b is equal to 2a, the calculated value for C is 2.889%ε0 εr , which is only 0.1% different from the exact value of 2.886%ε0 εr for C calculated from Eq. (6.20). This small error is achieved even though the assumed trial function for does not follow closely the true o given in Eq. (6.19). This is the principal advantage of variational methods. We can, of course, obtain more accurate results for C by employing more terms in the trial function of but with an increase in the computation time. Once C is obtained, we can then calculate the characteristic impedance, using Eq. (6.17): 1 Zo D p 6.26 c CCa We can also solve Eq. (6.25) for a1 and a2 using the calculated value of C. Applying the boundary condition for the inner conductor we obtain Vo D a0 C a1 b C a2 b2

6.27

which can be solved for a0 by making use of the results obtained for a1 and a2 . The potential at any location corresponding to a radius r in the transverse plane can now be determined completely from Eq. (6.22). 6.3.2

Variational Formulation Using Lower-Bound Expression

The formulation of the variational method based on the lower-bound expression is illustrated using a three-layer shielded microstrip line shown in Fig. 6.6. The enclosure and conducting strip are assumed to be perfect conductors. The strip thickness is assumed negligible, and the dielectric substrates are assumed lossless. This structure includes the open and shielded single-layer microstrip line, suspended strip line, strip line, and two-layer microstrip line. For example, the (open single-layer) microstrip line is obtained by letting h1 and a be very large,


131

y

er 1

S

h1 W

er 2

h2

er 3

h3

b

x a

0

Figure 6.6

Cross section of a shielded three-layer microstrip line.

h3 D 0, and εr1 D 1. This structure, which is embedded in inhomogeneous media, supports a quasi-TEM mode. For this problem, the upper-bound variational expression would require a twodimensional trial function for the potential . On the other hand, only a onedimensional trial function for the charge distribution would be needed, due to the fact that the strip thickness is assumed infinitely thin. A two-dimensional function is more difficult to derive and consumes more computation time than that of one dimension. The upper-bound expression based on the charge distribution is therefore preferred over the lower-bound formula for the considered structure. Using the lower-bound variational expression (6.14), we now express the inverse of the per-unit-length capacitance of the transmission line 1/C as

1 D C

S

Gx; x 0 s xs x 0 dx dx 0

2 s x dx

6.28

s

where S represents the surface of the conducting strip along the x direction, and the integral is taken over the strip’s surface in the transverse plane. s x is the (per-unit-length) surface charge density at the upper interface where the strip resides; it exists only on the strip and solely depends on x for this problem. Gx; x 0 is the Green’s function, which is symmetric in x and x 0 due to reciprocity. Equation (6.28) is valid for any transmission line whose strips are infinitesimally thin.

132

VARIATIONAL METHODS

To calculate C and, hence, Zo and εeff , we need to determine the Green’s function G and a suitable trial function for the charge density . The Green’s function is given as [2] 0

Gx; x D

1

gn gxgx 0

6.29

nD1

where 2 n% Y gx D sin˛n x

6.30

gn D

0

6.31

0

6.32 gx D sin˛n x n% ˛n D , n D 1, 2, . . . 6.33 a εr1 coth˛n h1 C εr2 tanh˛n h2 Y D ε0 εr2 C ε0 εr3 coth˛n h3 6.34 εr2 C εr1 coth˛n h1 tanh˛n h2 εri and hi , i D 1, 2, 3, are the relative dielectric constant and thickness of the ih dielectric layer. a denotes the transmission line’s enclosure width. This Green’s function can also be derived using the procedure described in Chapter 3 (see Problem 3.19). Now using the Ritz method, we let the charge density on the strip be s x D

N

ai si x

6.35

iD1

where ai , i D 1, 2, . . . , N, are the unknown variational parameters, and si x are the basis functions, chosen to approximate the charge density as  xS   cos i 1 %    W   , si x D 2x S W 2  1   W     0, otherwise

Sx SCW 6.36

Sketches of these charge basis functions are shown in Fig. 6.7. Note that the basis function s1 , corresponding to i D 1, is minimum at the center of the strip and increases rapidly near the strip edges and, thus, approximates closely the actual charge distribution on the strip. We can therefore expect that good results for C may be obtained even if only one basis function N D 1 is used. Proceeding with the Ritz procedure, we take the partial derivatives of 1/C in


133

y

i =1 i =3

1

x S

−1

S +W S+

W 2 i=2

Figure 6.7

Sketches of the charge basis functions.

Eq. (6.24) with respect to a1 and a2 and let them be zero. This will produce the maximum value for 1/C and lead to a system of homogeneous equations as outlined in Section 6.2.3. In the Appendix at the end of this chapter, we derive two different systems of homogeneous equations with and without 1/C in the coefficient matrices. The equations that contain 1/C are given in Eq. (A6.14) as 1 N iD1

1 gn SQni SQnj Qi Qj ai D 0; C nD1

j D 1, 2, . . . , N

6.37

where gn is given in Eqs. (6.30) and (6.34), and the total charge on the strip, Qi , is given from Eq. (A6.6) as

SCW

si x dx

Qi D

6.38

S

Upon substitution of the charge basis function si x from Eq. (6.32), Qi becomes %W J0 Qi D 2

i 1 % 2

cos

i 1 % 2

6.39

134

VARIATIONAL METHODS

where J0 stands for the zero-order Bessel function of the first kind. The other parameter, SQi , is obtained from Eq. (A6.7) as

SCW

SQni D

gxsi x dx

6.40

S

Substituting Eqs. (6.31) and (6.36) into Eq. (6.40), we obtain, after several manipulations, %W ˛n W C i 1 % ˛n W C i 1 % J0 sin ˛n S C SQni D 4 2 2 ˛n W i 1 % ˛n W i 1 % C J0 sin ˛n S C 6.41 2 2

Now using Eqs. (6.39) and (6.41) in Eq. (6.37), we can obtain the final N linear equations in N unknowns ai , i D 1, 2, . . . , N. These equations contain the (unknown) C in the coefficient matrix, so the (unknown) variational parameters ai cannot be solved prior to the determination of C. However, in order for the nontrivial solutions of ai to exist, the determinant of the coefficient matrix must be zero. Doing so results in a characteristic equation of C, from which we can solve for C directly. Once C is determined, we can then calculate the characteristic impedance and effective dielectric constant using Eqs. (6.17) and (6.18), respectively, as 1 Z0 D p c CCa

6.42

C Ca

6.43

εeff D

Using the calculated value of C, we can also solve the system of equations for ai and then determine the charge density x. The system of equations that does not contain 1/C is given in Eq. (A6.16) as N 1

[gn SQnj Ð Qi gn SQni Ð Qj ] ai D 0;

j D 1, 2, . . . , N

6.44

iD1 nD1

Substituting Eqs. (6.39) and (6.41) into Eq. (6.44), we obtain the final system of N homogeneous equations in N unknowns ai , i D 1, 2, . . . , N, from which we can solve for ai . Using the results for ai we can calculate C obtained in

135

VARIATIONAL METHODS IN THE SPECTRAL DOMAIN

Eq. (A6.5) as 1

1 D C

nD1

gn

N

2

ai SQni

iD1

N

2

6.45

ai Qi

iD1

Once C is determined, we can of course compute the characteristic impedance and effective dielectric constant. If a simple expression such as

s x D

x2, 0,

Sx SCW otherwise

6.46

is used for x, we can derive a closed-form expression for C. It should be noted again that the closer x is to the true value o the better the result for C. Also, a more accurate result can be obtained by using more variational parameters ai , which is, of course, achieved at the expense of increased computation time. Perhaps the most difficult step in implementing a variational method based on the lower-bound expression is the derivation of the Green’s function Gx, y; x 0 , y 0 . This Green’s function, however, can be found easily if we perform the variational analysis in the Fourier-transform or spectral domain as described in the following section.

6.4


As for the case of variational methods in the space domain, in order to determine the static or quasi-static parameters of a transmission line using variational methods in the spectral domain, we need to have variational formulas for the per-unit-length capacitance C in the spectral domain. Here, we will derive the lower-bound expression for C in the spectral domain and then formulate the variational process. 6.4.1

Lower-Bound Variational Expression for C in the Spectral Domain

We begin by considering a general two-conductor transmission line as shown in Fig. 6.2. The potentials on conductors S1 and S2 are assumed to be zero and Vo , respectively. The lower-bound variational expression for C in the space domain given in Eq. (6.14) can be rewritten as

136

VARIATIONAL METHODS

1 D C

s x, yx, y d S2

Q2

6.47

where x, y is the potential at any point in the transverse plane given in Eq. (6.10) and Qx, y is the total charge on conductor S2 given in Eq. (6.13). We will now perform the analysis in the spectral domain using the following Fourier transform definition 1 Q f˛, y D fx, yej˛x dx 6.48 1

where ˛ is the Fourier transform variable and the tilde ¾ indicates the Fouriertransformed quantity. This Fourier transform along with Parseval’s theorem are discussed in Appendix A at the end of Chapter 7. Applying Parseval’s theorem 1 1 1 Ł Q fx, yg x, y dx D 6.49 f˛, yQgŁ ˛, y d˛ 2% 1 1 to the right-hand side of Eq. (6.47) leads to 1 1 1 Q D Q s ˛, y˛, y d˛ C 2%Q2 1

6.50

Note that Eqs. (6.49) and (6.50) are general and specifically apply to (open) transmission lines. For a transmission line enclosed in a conducting channel, the Fourier transform is finite and corresponds to the discrete Fourier transform variable ˛n , which must be determined based on the transmission line’s boundary conditions. For instance, for the three-layer shielded microstrip line shown in Fig. 6.6, the Fourier transform can be defined as a Q ˛n , y D 2 f fx, y sin ˛n x dx 6.51 a 0 where ˛n D n %/a, n D 1, 2, . . .. This Fourier transform, given in Appendix A of Chapter 7, is also used in the spectral-domain formulation presented in Chapter 7. The corresponding Parseval’s relation is

aQ f ˛n , y gQ Ł ˛n , y 2 nD1 1

a

fx, ygŁ x, y dx D 0

6.52

and Eq. (6.50) becomes 1 a 1 Q s ˛n , yQ ˛n , y D C 2Q2 nD1

6.53

Similar to the case of the (space-domain) Green’s function described in SecQ tion 6.2.2, the Green’s function in the spectral domain, G˛, is defined as the


137

resultant potential due to a unit charge located on conductor S2 and so may be written as Q s ˛, y Q Q G˛ D ˛, y 6.54 ε0 Equation (6.50) can now be rewritten, upon making use of Eq. (6.54), as 1 1 1 Q Q 2 ˛, yG˛, y d˛ 6.55 D C 2%ε0 Q2 1 s For the case of the three-layer shielded microstrip line shown in Fig. 6.6, using Eq. (6.54) in (6.53) gives 1 1 a 2 Q n , y Q ˛n , yG˛ D C 2ε0 Q2 nD1 s

6.56

Both Eqs. (6.55) and (6.56) are stationary. Because these equations are derived directly from the lower-bound variational formula, they also represent lowerbound expressions for C in the spectral domain. This lower-bound characteristic is useful in selecting an appropriate trial function for the charge distribution x, y, in which we should choose a trial function that results in as large a calculated value of C as possible. 6.4.2

Determination of C , Zo , and eeff

To obtain numerical results for C using a spectral-domain variational expression such as Eq. (6.55) and, hence, Zo and εeff , we must determine a trial function for the charge density x, y. This trial function must be so chosen that the resultant capacitance per unit length C is maximized. Often, we use the Ritz method and express the charge density as a truncated summation of known basis functions in the space domain as N ai si x 6.57 s x ¾ D iD1

where i x, i D 1, 2, . . . , N, are the basis functions chosen to describe the charge distribution on the strip and are nonzero only on the strip; ai are the unknown coefficients known as variational parameters; and N denotes the number of basis functions used to approximate the strip’s charge density. In the spectral domain, the charge density given in Eq. (6.57) becomes Q s ˛ ¾ D

N

ai Q si ˛

6.58

iD1

Criteria for choosing the basis functions were already described earlier. In addition to these standards, the basis functions used in the spectral-domain variational

138

VARIATIONAL METHODS

method should have closed-form Fourier transforms to reduce the computation time. Using Eq. (6.58) in (6.55) then gives us an expression for 1/C whose only unknowns are ai . Continuing with the Ritz method, we set the derivatives of 1/C with respect to ai for i D 1, 2, . . . , N to zero. This results in a system of N homogeneous linear equations, from which we can solve for C. The derivation of these equations is given in the Appendix at the end of this chapter. Once C is determined, we can compute the characteristic impedance and effective dielectric constant of the transmission line using Eqs. (6.17) and (6.18), respectively. Equation (6.17) produces a value that is an upper bound to the true characteristic impedance. The obtained characteristic impedance and effective dielectric constant are also approximate due to the fact that the charge density on the strip is chosen approximately.

6.4.3

Formulation

The formulation process of the variational method based on the lower-bound expression in the spectral domain is illustrated by using the same three-layer shielded microstrip line shown in Fig. 6.6. We will present details in determining the per-unit-length capacitance, characteristic impedance, and effective dielectric constant of the transmission line. The process is very similar to the one we present for the quasi-static spectral-domain method in Chapter 7. We begin with the two-dimensional Laplace equation for the electric potential in a plane transverse to the direction of wave propagation, ∂2 i x, y ∂2 i x, y C D 0, ∂x 2 ∂y 2

i D 1, 2, 3

6.59

where i x, y is the unknown potential in the ith region. The boundary conditions for the considered structure are i 0, y D i a, y D 0

6.60

1 x, b D 3 x, 0 D 0

6.61 U

1 x, h2 C h3 D 2 x, h2 C h3 D V x

εr2

2 x, h3 D 3 x, h3 x ∂2 ∂1 εr1 D ∂y yDh2 Ch3 ∂y yDh2 Ch3 ε0 ∂3 ∂2 εr2 D0 εr3 ∂y yDh3 ∂y yDh3

6.62 6.63 6.64 6.65

where ε0 is the free-space permittivity and εri , i D 1, 2, 3, is the relative dielectric constant of the ith layer; VU x denotes the potential at the upper interface, where


139

the strip resides, and can be expressed as VU x D Vo x C Vx

6.66

Vo x D Vo on the strip and zero elsewhere; we choose the value of Vo . Vx is the unknown potential at the upper interface and is zero on the strip. x represents the unknown charge density on the strip and is nonzero only on the strip. To perform the variational method in the spectral domain for the considered Q n , y of fx, y as problem, we define the Fourier transform f˛ Q n , y D 2 f˛ a

a

fx, y sin ˛n x dx

6.67

0

where ˛n D n %/a, n D 1, 2, 3, . . . , denoting the spectral order or term. This Fourier transform is the sine transform used for odd functions; it is chosen here because the functions i x, y are required to be zero at x D 0 and a. As can be verified, the choice for the Fourier transform variable ˛n will cause the boundary conditions of Eq. (6.60) on i x, y to be met automatically. Laplace’s equation in the Fourier-transform or spectral domain can now be obtained, upon applying Eq. (6.67) to (6.59), as ∂2 Q i ˛2n Q i D 0 ∂y 2

6.68

This equation is derived in Section 7.2. The boundary conditions of the considered structure in the spectral domain are obtained by Fourier transforming Eqs. (6.61)–(6.65) as Q 1 ˛n , b D Q 3 ˛n , 0 D 0 Q U ˛n Q 1 ˛n , h2 C h3 D Q 2 ˛n , h2 C h3 D V

εr2

Q 2 ˛n , h3 D Q 3 ˛n , h3 Q s ˛n ∂Q 2 ∂Q 1 εr1 D ∂y ∂y ε0 yDh2 Ch3 yDh2 Ch3 ∂Q 3 ∂Q 2 εr2 D0 εr3 ∂y ∂y yDh3

6.69 6.70 6.71 6.72

6.73

yDh3

Note that the boundary condition (6.60) is not transformed since it is already satisfied by the choice of ˛n . The potential along the upper interface is given in the spectral domain as Q U ˛n D V Q o ˛n C V˛ Q n V

6.74

140

VARIATIONAL METHODS

The solution of Eq. (6.68), which satisfies the boundary conditions (6.69)–(6.73), is derived in Chapter 7 as Q n Q s ˛n D V Q U ˛n G˛

6.75

where

εr3 tanh ˛n h1 tanh ˛n h2 C tanh ˛n h3 ε Q n D r2 G˛ εr1 ε0 ˛n tanh ˛n h2 εr3 C tanh ˛n h1 coth ˛n h2 εr2 C tanh ˛n h3 εr1 coth ˛n h2 C εr2 tanh ˛n h1

6.76

Q U ˛n when the charge ˛ Q n is set to This function is equal to the potential V Q U ˛n is one and, thus, represents the Green’s function in the spectral domain. V given in Eq. (6.74) as Q U ˛n D V Q 0 ˛n C V˛ Q n V

6.77

Q 0 ˛n and V˛ Q n stand for the Fourier transforms of Vo x and Vx, respecV tively. Vo x is chosen and is thus known. Vx is given as Q n D V˛

2 a

a

Vx sin ˛n x dx

6.78

0

with ˛n D n %/a, n D 1, 2, 3 . . . . It should be noted here that the process of deriving Eq. (6.75), and hence Eq. (6.76), would be simpler if we assume h1 and h2 are always nonzero. Furthermore, the spectral-domain Green’s function is independent of the dimensions along the x axis. It is basically the total potential, corresponding to a unit charge, along the interface where the conductor resides. It is recognized that the Green’s function in the spectral domain is easier to derive than that in the space domain. Up to now, the formulation is exact and the same as that for the spectraldomain method described in Section 7.2. The reciprocal of the per-unit-length capacitance, 1/C, is derived in Eq. (6.56) as 1 1 a 2 Q ˛n , y Q ˛n , y G D C 2ε0 Q2 nD1 s

6.79

Q n is given in Eq. (6.36) and where G˛

s x dx

QD S

6.80


141

To solve for C, we therefore need to determine the strip’s charge density x. Using the Ritz method, we express the charge density as a truncated summation of basis functions in the space domain as x ¾ D

N

ai i x

6.81

iD1

where i x, i D 1, 2, . . . , are the basis functions describing the charge distribution on the strip and are nonzero only on the strip. These basis functions strongly influence the numerical efficiency of the solution process and the accuracy of the solutions and thus should be chosen carefully following the criteria discussed previously. The ai are the unknown coefficients referred to as the variational parameters; and N denotes the number of basis functions used for the strip’s charge density. In the spectral domain, ˛ Q n ¾ D

N

ai Q i ˛n

6.82

iD1

The basis functions used here are  xS   cos i 1 %    W   , i x D 2x S W 2  1   W     0, otherwise

Sx SCW 6.83

which are the same as those used for the space-domain variational method of Section 6.3.2. The Fourier transforms of these basis functions are obtained using Eq. (6.67) as %W [sink1 C k2 J0 k1 C k3 C sink2 k1 J0 k1 k3 ] 6.84 2 a % k1 D j 1 6.85 2 W 6.86 k2 D ˛n S C 2

Q si ˛n D

k3 D ˛n

W 2

6.87

where J0 stands for the zero-order Bessel function of the first kind. Substitute Eq. (6.82) into (6.56), take the derivatives with respect to ai , and let them be zero according to the Ritz method. This can result in two systems of linear homogeneous equations — one containing 1/C and one not. The system

142

VARIATIONAL METHODS

of equations that does not contain 1/C is given by Eq. (A6.21), in the Appendix at the end of this chapter, as N iD1

ai

1

Q n Q sj ˛n Q si ˛n G˛

D0

6.88

nD1

Now using Eqs. (6.84) and (6.36) in Eq. (6.88), we can obtain the final N linear equations in N unknowns ai , i D 1, 2, . . . , N, from which we can solve for ai . Using the results for ai we can calculate C obtained in Eq. (A6.19) of the Appendix at the end of this chapter as 1 N 1 1 a Q n Q sj ˛n a a Q si ˛n G˛ D i j C 2ε0 Q2 iD1 jD1 nD1

6.89

Once C is determined, it can then be used to calculate the characteristic impedance and effective dielectric constant as 1 Z0 D p c CCa and εeff D

C Ca

6.90

6.91

respectively, where c is the free-space velocity and Ca is the capacitace per unit length with the dielectrics removed. Because of the use of an approximate charge density for the strip, C evaluated using Eq. (6.79) only gives an approximate result. However, as a characteristic of the variational method, this expression is stationary; that is, a first-order error in the choice of the charge density basis functions i will only produce a second-order error. Therefore, Eq. (6.79) should give a value for C that is very close to the exact result, provided that good choices for the basis functions are used. It can be proved that C from Eq. (6.79) is always smaller than the exact result and, hence, this equation represents the lower-bound expression for C. It should be noted that both the number of basis functions and the number of spectral terms, used in Eq. (6.79) for determining C, affect the accuracy of the numerical results. The larger these numbers the more accurate the results, but at the expense of increased computation time. For most engineering purposes, three basis functions and 200 spectral terms are sufficient. REFERENCES 1. R. E. Collin, Field Theory of Guided Waves, 2nd edition, IEEE Press, New York, 1991, Chap. 4.

PROBLEMS

143

2. B. Bhat and S. K. Koul, “Unified Approach to Solve a Class of Strip and Microstrip-like Transmission Lines,” IEEE Trans. Microwave Theory Tech., Vol. MTT-30, pp. 679–686, May 1982. 3. E. Yamashita and R. Mittra, “Variational Method for the Analysis of Microstrip Lines,” IEEE Trans. Microwave Theory Tech., Vol. MTT-16, pp. 251–256, Apr. 1968.

PROBLEMS

6.1 Prove that Eq. (6.9) is stationary. 6.2 Prove that C, given by Eq. (6.9), produces an upper bound to the true value Co . 6.3 Prove that Eq. (6.14) is stationary. 6.4 Prove that C, given by Eq. (6.14), produces a lower bound to the true value Co . 6.5 Derive Eq. (6.41). 6.6 Derive a system of linear equations using Eqs. (6.37), (6.39), and (6.41), whose unknowns are ai , i D 1, 2, . . . , N, and the corresponding characteristic equation for C. 6.7 Consider a three-layer shielded microstrip as shown in Fig. 6.6. The enclosure and conducting strip are assumed to be perfect conductors. The strip thickness is assumed negligible, and the dielectric substrates are assumed lossless. The charge basis functions are given in Eq. (6.36). (a) Formulate the lower-bound variational analysis for determining the characteristic impedance, Zo , and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff . Calculate and plot Zo and εeff versus W/h2 from 0.1 to 5, εr2 D 2.2, h1 D h3 D 0.66 mm, h2 D 0.254 mm, a D 2.54 mm, and S D 0.25a W and 0.5a W. Compare results using one to five basis functions. 6.8 Use the program developed in Problem 6.7 to calculate and plot Zo and εeff versus W/h from 0.05 to 5 for the microstrip line, as shown in Fig. 4.2, having εr D 2.2 and 10.5 and h D 0.635 and 1.27 mm. The strip thickness is negligible. Compare results with those calculated using the program developed in Problem 6.15. Provide an assessment of the basis functions used. 6.9 Consider the strip line as shown in Fig. P3.2 of Problem 3.18. It is assumed that the strip is a perfect conductor and has zero thickness. Also, the ground planes are assumed to be infinitely wide and perfect conductors. The dielectric is assumed to be perfect. (a) Derive the Green’s function in closed form (see Problem 3.18). (b) Derive the upper-bound expression for the characteristic impedance Zo , assuming the charge distribution is

144

VARIATIONAL METHODS

x 2 , x 2 strip 0, otherwise

s x D

(c) Calculate and plot Zo versus the normalized strip width, W/a, from 0.1 to 5 for εr D 2.2 and 10.5 and a D 0.635, and 1.27 mm. Compare results to those obtained in Problem 4.18. 6.10 Consider a shielded microstrip line as shown in Fig. 3.2. The enclosure and conducting strip are assumed to be perfect conductors. The strip thickness is assumed negligible, and the dielectric substrates are assumed lossless. (a) Derive expressions for the capacitance per unit length, C, characteristic impedance, Zo , and effective dielectric constant, εeff , using a variational method, assuming the following charge distribution:

s x D

  A, 

0,

aCW aW x 2 2 otherwise

where A is a constant. (b) Calculate and plot Zo and εeff versus W/h from 0.05 to 5 for εr D 2.2 and 10.5 and h D 0.635 and 1.27 mm. Compare results to those obtained using the program developed in Problem 6.7 and provide an evaluation of the basis functions used. (c) Determine the potential at x D 5a/6 and y D h. 6.11 Repeat Problem 6.10, parts (a) and (b), using     

s x D

   

1

1 jx a/2j W/2

2 ,

0,

aCW aW x 2 2

otherwise

Compare the calculated results with those obtained in parts (a) and (b) of Problem 6.10 and provide an assessment of the basis functions used. 6.12 Determine Eq. (A6.14) for the microstrip line shown in Fig. 6.6. The charge basis functions are given in Eq. (6.36). The resultant equations contain the zero-order Bessel function of the first kind, 1 J0 ˛ D %

1

1

ej˛x p dx 1 x2

6.13 Prove that both Eqs. (6.55) and (6.56) are stationary.

PROBLEMS

145

y

W

h2 er 2

er 1

h1

x Figure P6.1

Cross section of a two-layer microstrip line.

6.14 Consider a three-layer shielded microstrip as shown in Fig. 6.6. The enclosure is assumed to be a perfect conductor. Determine the Fourier transform variable ˛n in Eq. (6.67). 6.15 Consider a microstrip line as shown in Fig. P6.1. It is assumed that the strip is a perfect conductor and has zero thickness. Also, the ground plane is assumed to be infinitely wide and a perfect conductor. The dielectrics are assumed to be perfect. (a) Derive the Green’s function in the spectral domain. (b) Formulate the lower-bound variational analysis in the spectral domain for determining the capacitance per unit length, characteristic impedance, Zo , and effective dielectric constant, εeff , using the following charge distribution:     

s x D

   

1 1

0,

jxj W/2

2 ,

x

W 2

otherwise

(c) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus W/h2 from 0.1 to 5 for εr1 D 10.5, h1 D 1.27 mm and εr1 D 2.2, h2 D 0.508 mm.

146

VARIATIONAL METHODS

6.16 Consider a three-layer shielded microstrip as shown in Fig. 6.6. The enclosure and conducting strip are assumed to be perfect conductors. The strip thickness is assumed negligible, and the dielectric substrates are assumed lossless. Assume the following distribution for the charge:

s x D

x, 0,

Sx SCW otherwise

(a) Derive the lower-bound expression for 1/C in the spectral domain, and then Zo and εeff . (b) Calculate and plot Zo and εeff versus W/h2 from 0.1 to 5, for εr2 D 2.2, h1 D h3 D 0.026 in., h2 D 0.254 mm, a D 2.54 mm, and S D 0.25 a W and 0.5a W. Compare results to those obtained in Problem 6.7 and provide an assessment of the basis functions used. 6.17 Consider a shielded CPW as shown in Fig. 7.1. The enclosure and ground and central strips are assumed to be perfect conductor. The metallization thickness of the ground and central strips is negligible, and the dielectric substrates are considered lossless. Assume that the basis functions for the charge distributions on the central strip, s1i , and ground strips, s2i and s3i , are given by Eqs. (7.47)–(7.49). (a) Formulate the variational analysis in the spectral domain for determining the characteristic impedance, Zo and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus W/h2 from 0.1 to 5, for εr1 D 1, εr2 D 2.2, εr3 D 10.5, h1 D h3 D 0.66 mm, h2 D 0.254 mm, a D 2.54 mm, S1 D S2 D 0.254 mm, and G1 D G2 . Compare results using one to five basis functions. 6.18 Use the program developed in Problem 6.17 to calculate and plot Zo and εeff for the conventional CPW shown in Fig. 4.3(a) versus the dimension ratio a/b from 0.1 to 0.9 for a relative dielectric constant, εr , of 2.2 and 10.5 and a normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10. Assume the ground and central strips have zero thickness. Compare the results to those obtained in Problem 4.10. 6.19 Repeat Problem 6.18 for the conductor-backed CPW with zero strip thickness shown in Fig. 4.3(b). Compare the results to those obtained in Problem 4.11. 6.20 Consider a CPS as shown in Fig. 4.4. The strips are assumed to be perfect conductor with negligible thickness, and the dielectric substrate is assumed lossless. Assume that the basis functions for the charge distributions on the strips are given by

PROBLEMS

 xCb   cos i 1 %   ba   ,  2 s1i x D 2x C b b a  1   ba     0,  xa   cos i 1 %   ba   ,  2 s2i x D 2x a b a  1   ba     0,

147

b x a

otherwise

axb

otherwise

(a) Formulate the variational analysis in the spectral domain for determining the characteristic impedance, Zo , and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus a/b from 0.1 to 0.9, for a relative dielectric constant, εr , of 2.2 and 10.5, and a normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10. Compare results using one to five basis functions, and to those obtained in Problem 4.15. 6.21 Consider the strip line of Problem 6.9. Assume that the basis functions for the charge distribution on the strip are given as  xCW   cos i 1 %   2W   ,  2 si x D x C W W  1   W     0,

W x W

otherwise

(a) Formulate the variational analysis in the spectral domain for determining the characteristic impedance, Zo . (b) Write a computer program to calculate Zo . Compute and plot Zo versus the normalized strip width, W/d, from 0.1 to 5 for εr D 2.2 and 10.5 and d D 0.635 and 1.27 mm. Compare results using one to five basis functions, and to those obtained in Problems 4.18 and 6.9. 6.22 Consider a shielded CPW as shown in Fig. 7.1. The enclosure and ground and central strips are assumed to be perfect conductors. The metallization thickness of the ground and central strips is negligible, and the dielectric

148

VARIATIONAL METHODS

substrates are considered lossless. Assume that the charge distributions on the central strip, s1 , and ground strips, s2 and s3 , are defined only on the strips as follows:

s x D %

W 2

2

1 , W 2 x G1 S1 2 G1 C S1 < x < G1 C S1 C W

s2 x D 2G1

s3 x D 2G2

x

0 < x < G1

,

G21 x 2 ax

,

G22 a x2

a G2 < x < a

(a) Formulate the variational analysis in the space domain for determining the characteristic impedance, Zo , and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus W/h2 from 0.1 to 5, for εr1 D 1, εr2 D 2.2, εr3 D 10.5, h1 D h3 D 0.66 mm, h2 D 0.254 mm, a D 2.54 mm, S1 D S2 D 0.254 mm, and G1 D G2 . Compare results to those in Problem 6.17. 6.23 Repeat Problem 6.22 in the spectral domain. APPENDIX: SYSTEMS OF HOMOGENEOUS EQUATIONS FROM THE LOWER-BOUND VARIATIONAL FORMULATION

The lower-bound variational expression for the capacitance per unit length of a transmission line from Eq. (6.28) is repeated here as

Gx; x 0 s xs x 0 dx dx 0 1 S A6.1 D

2 C s x dx S

Here we assume the transmission line has an infinitely thin strip. The Green’s function Gx; x 0 may be assumed to be of the form Gx; x 0

1 nD1

gn gxgx 0

A6.2

APPENDIX: SYSTEMS OF HOMOGENEOUS EQUATIONS

149

where gx and gx 0 are identical in form, and gn depends on the transmission line parameters. Gx; x 0 is given in Eqs. (6.29)–(6.34) for the three-layer microstrip line considered in Section 6.3.1. The charge distribution x on the strip can be approximated using the Ritz method as x D

N

ai i x

A6.3

iD1

Now substituting Eq. (A6.2) into (A6.1), we obtain

1 gn gxgx 0 s xs x 0 dx dx 0 S 1 nD1 D

2 C s x dx 1

D

nD1

S

gn

2

gxs x dx

S

2

A6.4

s x dx S

Using Eq. (A6.3) in (A6.4) then yields 1

1 D C

nD1

gn

N

2

ai SQni

iD1

N

2

A6.5

ai Qi

iD1

where

Qi D

si x dx,

i D 1, 2, . . . , N

gxsi x dx,

i D 1, 2, . . . , N and n D 1, 2, . . . , 1 A6.7

A6.6

S

SQni D S

Note that gn , Qi , and SQni can all be computed for a particular transmission line. The capacitance given by Eq. (A6.5) is now a function only of the unknowns ai , i D 1, 2, . . . , N. Let 1/C in Eq. (A6.5) be 1 Ha1 , a2 , . . . , aN D C Ia1 , a2 , . . . , aN

A6.8

150

VARIATIONAL METHODS

where HD

1

gn

nD1

and

2

ai SQni

A6.9

iD1

ID

N

N

2

ai Qi

A6.10

iD1

Taking the partial derivative of Eq. (A6.8) with respect to aj and dividing the denominator and numerator by I then yields ∂ 1 ∂H 1 ∂I 1 D A6.11 ∂aj C I ∂aj C ∂aj where

and

N 1 ∂H D2 ai gn SQni SQnj ∂aj iD1 nD1

A6.12

∂I D2 ai Qi Qj ∂aj iD1

A6.13

N

To maximize the value of C in accordance with the lower-bound formula, we let the partial derivatives of 1/C with respect to aj in Eq. (A6.11) be zero. This leads to 1 N 1 gn S Qni S Qnj Qi Qj ai D 0, j D 1, 2, . . . , N A6.14 C iD1 nD1 which is a system of N linear homogeneous equations in N unknowns ai , i D 1, 2, . . . , N. In order to have nontrivial solutions, the coefficient matrix must be singular. To satisfy this requirement, we can set the determinant of the matrix to zero, from which we can solve directly 1/C. Another way to solve for 1/C is described as follows. We take the derivative of Eq. (A6.8) and let it be zero to obtain I

∂H ∂I H D0 ∂aj ∂aj

A6.15

Substituting Eqs. (A6.9), (A6.10), (A6.12), and (A6.13) into Eq. (A6.15), we obtain N 1 iD1 nD1

gn SQnj Ð Qi gn SQni Ð Qj ai D 0,

j D 1, 2, . . . , N

A6.16

APPENDIX: SYSTEMS OF HOMOGENEOUS EQUATIONS

151

which represents a system of N linear homogeneous equations in N unknowns ai , i D 1, 2, . . . , N. Note that these equations do not contain 1/C as does Eq. (A6.14). We can now solve Eq. (A6.16) for ai and substitute the results into Eq. (A6.5) along with values of the parameters Qi and SQni from Eqs. (A6.6) and (A6.7), respectively, to determine a value for C. A system of equations similar to Eqs. (A6.14) and (A6.16) can also be obtained in the spectral domain. For instance, the system of equations that does not contain 1/C, corresponding to the shielded three-layer microstrip line considered in Section 6.4.2, is derived as follows. We rewrite the expression for 1/C in Eq. (6.56) as 1 1 a Q n , yQ s ˛n , y Q s ˛n , yG˛ D C 2ε0 Q2 nD1

A6.17

Q n is given in Eq. (6.76), and where the spectral-domain Green’s function G˛ the charge density Q s ˛n is expanded in Eq. (6.58) as Q s ˛n ¾ D

N

ai Q si ˛n

A6.18

iD1

Substituting Eq. (A6.18) into (A6.17), we obtain, with reordering of summations, 1 N 1 1 a Q n Q sj ˛n a a Q si ˛n G˛ D i j C 2ε0 Q2 iD1 jD1 nD1

A6.19

Now we apply the Ritz method to obtain the unknown coefficients ai by setting ∂1/C D 0, ∂am

m D 1, 2, . . . , N

A6.20

This yields a set of linear homogeneous equations, N iD1

ai

1

Q n Q sj ˛n Q si ˛n G˛

D0

A6.21

nD1

It is worthwhile to note that the coefficient matrix in Eq. (A6.21) is independent of the total charge on the strip. Also, the matrix is symmetric as expected by reciprocity. We can now solve Eq. (A6.21) for ai and substitute the results back into Eq. (A.19) along with the Green’s function from Eq. (6.76) to determine the capacitance C.


CHAPTER SEVEN

Spectral-Domain Method Many numerical methods exist to date for analyzing microwave and millimeterwave passive structures. Among them, the spectral-domain analysis (SDA) is one of the most popular ones. It was developed in 1974 [1]. A SDA version for quasi-static analysis was also presented [2]. SDA is basically a Fouriertransformed version of the integral equation method. However, as compared to the conventional space-domain integral equation method, the SDA has several advantages. Its formulation results in a system of coupled algebraic equations instead of coupled integral equations. Closed-form expressions can easily be obtained for the Green’s functions. In addition, incorporation of physical conditions of analyzed structures via the so-called basis functions is achieved, and the obtained solutions are stationary. These features make the SDA numerically simpler and more efficient than the conventional integral equation method. SDA has been used extensively in analyzing planar transmission lines (e.g., [3]). In this chapter, we will present a detailed formulation of the SDA for planar transmission lines in both quasi-static and dynamic domains. Other applications of SDA, to resonators and antenna and scattering problems, can be found in Itoh [4], Zhang and Itoh [5], and Scott [6], respectively.

7.1 FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS

Figure 7.1 shows a cross section of the three-layer coplanar waveguide (CPW) to be used in illustrating the quasi-static SDA formulation. The central and ground strips are assumed to be perfect electric conductors of zero thickness, uniform and infinite in the z direction. The dielectric substrates are assumed to be lossless. The enclosure or channel is assumed to be a perfect electric conductor and is used to simplify some of the analysis and computations, but the resulting analysis can be used for some transmission line structures discussed in Chapter 4 by choosing appropriate parameters. For instance, the conductor-backed CPW is obtained by 152

FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS

153

y

er 1

h1 G1

S1

W

S2

G2

er 2

h2

er 3

h3

0

Figure 7.1 SDA.

a

b

x

Cross section of the asymmetric CPW used for illustrating the quasi-static

letting h3 D 0, S1 D S2 , h1 and a be large, and εr1 D 1; the microstrip line is obtained by letting h3 D 0, G1 D G2 D 0, h1 and a be large, and εr1 D 1; the strip line is realized when we let h3 D 0, G1 D G2 D 0, h1 D h2 , a be large, and εr1 D εr2 . Due to the difference of multiple dielectrics surrounding the metallic strips, the dominant propagating mode is quasi-TEM. A quasi-static analysis solves the two-dimensional Laplace equation for the electric potential in a plane transverse to the direction of wave propagation subject to appropriate boundary conditions in the space domain, ∂2 i x, y ∂2 i x, y C D 0, ∂x 2 ∂y 2

i D 1, 2, 3

7.1

where i x, y is the unknown potential in the ith region. The quasi-static SDA, on the other hand, solves Laplace’s equation by applying a moment method, Galerkin’s technique, discussed in Appendix B at the end of this chapter, in the Fourier-transform or spectral domain. The analysis obtains the charge density on the central strip, and from this the per-unit-length (PUL) capacitance is obtained. The PUL capacitance can then be used to determine the effective dielectric constant and characteristic impedance of the transmission line. The boundary conditions are derived from the fact that x, y is continuous everywhere and n Ð D2 D1 D s at the upper interface (between the first and second layers.) Di , i D 1, 2, is the electric flux density in region i, n is the unit vector normal to the interface, and s stands for the surface charge density. These boundary

154

SPECTRAL-DOMAIN METHOD

conditions are i 0, y D i a, y D 0

(7.2)

1 x, b D 3 x, 0 D 0

(7.3) U

1 x, h2 C h3 D 2 x, h2 C h3 D V x

εr2

2 x, h3 D 3 x, h3 s x ∂ 2 ∂ 1 εr1 D ∂y yDh2 Ch3 ∂y yDh2 Ch3 ε0 ∂ 3 ∂ 2 εr2 D0 εr3 ∂y yDh3 ∂y yDh3

(7.4) (7.5) (7.6) (7.7)

where ε0 is the free-space permittivity and εri , i D 1, 2, 3, is the relative dielectric constant of the ith layer. VU x denotes the potential at the upper interface and can be expressed as VU x D Vo x C Vx 7.8 Vo x D Vo on the central strip and zero elsewhere; we choose the value of the Vo . The ground strips are assumed to be at zero potential. Vx is the unknown potential on the two slots and is zero on the central and ground strips. The charge density at the upper interface, s , can be described as s x D s1 x C s2 x C s3 x

7.9

where s1 x, s2 x, and s3 x are unknown charge densities on the central, left, and right ground strips, respectively, and are nonzero only on the corresponding strips. To perform the SDA, a Fourier transform is needed. For the considered Q n , y of fx, y as follows: problem, we define the Fourier transform f˛ Q n , y D 2 f˛ a

a

fx, y sin ˛n x dx

7.10

0

where ˛n D n/a, n D 1, 2, 3, . . . , denoting the spectral order or term. This choice for the Fourier-transform variable ˛n will cause the boundary condition of Eq. (7.2) on i x, y to be met automatically. This Fourier transform along with the corresponding Parseval’s theorem is given in Appendix A at the end of this chapter. Multiply both sides of Eq. (7.1) and taking the integral with respect to x from 0 to a gives

a 0

∂2 i x, y sin ˛n x dx C ∂x 2

0

a

∂2 i x, y sin ˛n x dx D 0 ∂y 2

7.11


155

Using the following derivative property of the Fourier transform,

dk fx j˛x e dx D j˛k dx k

1

1

1

fxej˛x dx

7.12

1

we obtain

a

0

dk fx sin ˛x dx D ˛k dx k

a

fx cos ˛x dx,

k odd

0

a

k

fx sin ˛x dx,

D ˛

k even

7.13

0

assuming fx is real. The first term on the left-hand side of Eq. (7.11) is the Fourier transform of ∂2 i x, y/∂x 2 . Using Eq. (7.13) in this term gives 0

a

∂2 i x, y sin ˛n x dx D ˛2n ∂x 2

a

i x, y sin ˛n x dx 0

D ˛2n Q i ˛n , y

7.14

The second term is obtained, upon using Eq. (7.10), as 0

a

∂2 Q i ˛n , y ∂2 i x, y sin ˛ x dx D n ∂y 2 ∂y 2

7.15

Substituting Eqs. (7.14) and (7.15) into Eq. (7.11) then yields ∂2 Q i ˛2n Q i D 0 ∂y 2

7.16

which represents Laplace’s equation in the Fourier-transform or spectral domain. Fourier-transforming Eqs. (7.3)–(7.9), we obtain Q 1 ˛n , b D Q 3 ˛n , 0 D 0

(7.17)

Q U ˛n Q 1 ˛n , h2 C h3 D Q 2 ˛n , h2 C h3 D V

(7.18)

Q 2 ˛n , h3 D Q 3 ˛n , h3 Q s ˛n ∂ Q 1 εr1 D ∂y ε0 yDh2 Ch3 yDh2 Ch3 ∂ Q 3 ∂ Q 2 εr3 εr2 D0 ∂y ∂y

(7.19)

εr2

∂ Q 2 ∂y

yDh3

yDh3

(7.20)

(7.21)

156


which are the boundary conditions of the considered CPW structure in the spectral domain, and Q o ˛n C V˛ Q n Q U ˛n D V V

7.22

Q s ˛n D Q s1 ˛n C Q s2 ˛n C Q s3 ˛n

7.23

The boundary condition (7.2) is not transformed since it is already satisfied by the choice of ˛n . The solution of Eq. (7.16) is well known. A judicious choice yields the following forms: Q 1 ˛n , y D A sinh ˛n b y C E cosh ˛n b y

7.24

Q 2 ˛n , y D B sinh ˛n y h3 C C cosh ˛n y h3

7.25

Q 3 ˛n , y D D sinh ˛n y C F cosh ˛n y

7.26

where A, B, C, D, E, and F are unknown constants. In order to satisfy the first boundary condition (7.17), E and F must be equal to zero. This fact will be used implicitly in subsequent applications of the boundary conditions. Now substituting Eqs. (7.25) and (7.26) into Eq. (7.19) yields D sinh ˛n h3 D C

7.27

Applying Eq. (7.21) to Eqs. (7.25) and (7.26), we obtain ˛n εr2 B ˛n εr3 D cosh ˛n h3 D 0

7.28

Using Eq. (7.18) in Eqs. (7.24) and (7.25) gives A sinh ˛n h1 D B sinh ˛n h2 C C cosh ˛n h2

7.29

Substituting Eqs. (7.24) and (7.25) into Eq. (7.20), we have ˛n A cosh ˛n h1 C ˛n εr2 B cosh ˛n h2 C C sinh ˛n h2 D

Q s ˛n ε0

7.30

Now manipulating Eqs. (7.27)–(7.30) and making use of Eq. (7.18), we can solve Q U ˛n in terms of Q s ˛n as for V Q n Q s ˛n D V Q U ˛n G˛

7.31

Q n is called the Green’s function in the spectral domain and is given as where G˛ εr3 tanh ˛n h1 tanh ˛n h2 C tanh ˛n h3 ε Q n D r2 G˛ 7.32 ε0 ˛n tanh ˛n h2 εr3 εr1 /εr2 C tanh ˛n h1 coth ˛n h2 C tanh ˛n h3 εr1 coth ˛n h2 C εr2 tanh ˛n h1 ]


157

Q U ˛n is given in Eq. (7.22) as V Q o ˛n C V˛ Q n Q U ˛n D V V Q o ˛n and V˛ Q n stand for the Fourier transforms of Vo x and Vx, respecV tively. Vo x is chosen and is thus known. Vx is given as Q n D V˛

2 a

a

Vx sin ˛n x dx

7.33

0

with ˛n D n/a, n D 1, 2, 3, . . . . It should be noted here that the process of deriving Eq. (7.31) and hence (7.32) would be simpler if we assume h1 and h2 are always nonzero. Furthermore, the spectral-domain Green’s function is independent of the dimensions along the x axis. It is basically the total potential, corresponding to a unit charge, along the interface where the conductors reside. As seen in Chapter 6 on variational methods, the Green’s function in the spectral domain is easier to derive than that in the space domain. Up to now, the formulation is exact. There are two unknowns in Q n . We now apply Galerkin’s technique in the Eq. (7.31) — Q s ˛n and V˛ Q so that Eq. (7.31) can spectral domain to eliminate the unknown voltage V be solved. We begin by expressing each strip’s charge density as a truncated summation of basis functions in the space domain as s x ¾ D

Nj 3

dji sji x

7.34

jD1 iD1

where sji x describes the charge distribution on the jth strip and is nonzero only on that strip; dji is the unknown coefficient; and Nj denotes the number of basis functions used for the jth strip’s charge density. In the spectral domain, Q s ˛n ¾ D

Nj 3

dji Q sji ˛n

7.35

jD1 iD1

Substitute Eq. (7.35) into (7.31) and take the inner product of the resultant equation with respect to Q sji ˛n for j D 1, 2, 3 and i D 1, 2, . . . , Nj . That is, multiply the resultant equation by Q sji ˛n , j D 1, 2, 3 and i D 1, 2, . . . , Nj , and sum over ˛n , letting the spectral order n go from 1 to infinity. This results in a system of coupled linear algebraic equations, referred to as the Galerkin or Rayleigh–Ritz equations: N1 iD1

ij

P11 d1i C

N2 iD1

ij

P12 d2i C

N3 iD1

ij

j

P13 d3i D Q1 ,

j D 1, 2, . . . , N1

7.36

158

SPECTRAL-DOMAIN METHOD N1

ij

P21 d1i C

N2

iD1

ij

P22 d2i C

N3

iD1

N1

ij

P31 d1i C

N2

iD1

ij

j

j D 1, 2, . . . , N2

7.37

ij

j

j D 1, 2, . . . , N3

7.38

P23 d3i D Q2 ,

iD1 ij

P32 d2i C

N3

iD1

P33 d3i D Q3 ,

iD1

where ij Q Q sli i D Pkl D hQ skj , G

1

Q Q sli , Q skj G

k D 1, 2, 3 and l D 1, 2, 3

nD1 j

Q o C Vi Q D hQ skj , V Q o i C hQ skj , Vi Q Qk D hQ skj , V D

1

Qo C Q skj V

nD1

1

Q Q skj V,

k D 1, 2, 3

nD1

The notation hÐi indicates an inner product discussed in Appendix B at the end of this chapter. Applying Parseval’s theorem

a

aQ fx, yg x, y dx D f˛n , yQgŁ ˛n , y 2 nD1 1

Ł

0

7.39

to the right-hand sides of Eqs. (7.36)–(7.38) produces N1

ij

P11 d1i C

iD1

D N1

2 Vo a

iD1

ij

P12 d2i C

iD1

ij

P13 d3i

iD1

G1 CS1 CW

j D 1, 2, . . . , N1

7.40

ij

j D 1, 2, . . . , N2

7.41

ij

j D 1, 2, . . . , N3

7.42

s1j x dx,

ij

P21 d1i C

N2

ij

P22 d2i C

iD1 ij

N3

G1 CS1

iD1 N1

N2

P31 d1i C

N2 iD1

N3

P23 d3i D 0,

iD1 ij

P32 d2i C

N3

P33 d3i D 0,

iD1

The infinity upper limit of the spectral order n in the summations represents the number of spectral terms that need to be carried out. In practice, this number is truncated to a finite value N to reduce the computation time. Parseval’s theorem eliminates the summations involving the unknown voltage Q in Eqs. (7.36)–(7.38), due to the fact that the charge densities and voltages V are nonzero in complementary regions along the plane of the strips in the space domain. Therefore, the use of Galerkin’s technique allows us to eliminate


159

Q Furthermore, two summations involving known voltage V Q o in the unknown V. Eqs. (7.36)–(7.38) are eliminated. Equations (7.40)–(7.42) can now be solved for the unknown coefficients dji of the charge density basis functions sij , where i D 1, 2, . . . , Ni and j D 1, 2, 3. The transmission line’s PUL capacitance is given as a s x dx Qo D 0 CD Vo Vo N1 G1 CS1 CW d1i s1i x dx iD1

D

G1 CS1

Vo

7.43

It should be observed that the integral in the numerator of Eq. (7.43) is the same as the integral on the right-hand side of Eq. (7.40). So if we let the right-hand side of Eq. (7.40), which is already calculated, to be Rj , then we can rewrite Eq. (7.43) as N1 a CD d1i Ri 7.44 2V2o iD1 j

Note that Rj is equal to Q1 in Eq. (7.36). C evaluated using Eq. (7.44) only gives an approximate result. However, analogous with variational methods in Chapter 6, Eq. (7.44) is a stationary expression; that is, a first-order error in the choice of the charge density basis functions sij will produce only a second-order error. Therefore, Eq. (7.44) should give a value for C that is very close to the exact result provided that good choices for the basis functions are used. It can also be proved that C from Eq. (7.44) is always smaller than the exact result and, hence, Eq. (7.44) represents the lower-bound expression for C in much the same way as the lower-bound variational expression discussed in Chapter 6. C can then be used to calculate the characteristic impedance and effective dielectric constant as 1 Zo D p 7.45 c CCa and εeff D

C Ca

7.46

respectively, where c is the free-space velocity and Ca is the capacitace per unit length with the dielectrics removed. To obtain numerical results for C and, hence, Zo and εeff , we need to choose basis functions for the charge densities, sji x, with j D 1, 2, 3 for the central strip and left and right ground strips, respectively. These basis functions influence strongly the numerical efficiency of the solution process and the accuracy of the solutions. Computation time can be reduced significantly if the chosen

160


basis functions closely describe actual behaviors of the charge distributions and have closed-form Fourier transforms. In addition, the basis functions should form complete sets, so that solution accuracy can be enhanced by increasing the number of basis functions. This is an important criterion and should be implemented strictly when choosing the basis functions in the SDA solution. Furthermore, they should be twice continuously differentiable to avoid spurious solutions. Moreover, as C obtained from Eq. (7.44) represents a lower bound to the true capacitance value similar to that from the lower-bound variational expression in Chapter 6, the charge basis functions should be selected so that the calculated value for C will be as large as possible. The same criteria are also used for the choice of basis functions needed for the variational methods in Chapter 6. The basis functions used here for the considered problem have the form x S1 G1 cos i 1 W s1i x D 2x S1 G1 W 2 1 W 1 x G1 cos i 2 G1 s2i x D 1 x/G1 2 1 x a C G2 cos i 2 G2 s3i x D 2 ax 1 G2

7.47

7.48

7.49

These basis functions are defined only on the strips. Sketches of the charge basis functions (7.47) for the central strip are shown in Fig. 7.2. Note that the basis function s11 corresponding to i D 1 is very large near the central strip’s edges and minimum at the strip’s center, and s21 and s31 are also very large near the edges of the left and right ground strips. These functions thus approximate closely the actual charge distributions on the central and ground strips. We then expect that good results for C may be obtained even if only one basis function is used for each of the charges on these strips. The Fourier transforms of these basis functions are obtained using Eq. (7.10) as Q s1i ˛n D

i 1 C ˛n W J0 2

i 1 C ˛n S1 C G1 C ð sin 2

i 1 ˛n S1 C G1 C ð sin 2

W 2a

i 1 ˛n W J0 2 W 7.50 2

W 2


161

y

i =1 i =3

1

x −1

x1

x2

x3

x4

x5

i =2

Figure 7.2 Sketches of the charge distributions’ basis functions. x1 D G1 , x2 D x1 C S1 , x3 D x2 C W, x4 D x3 C S2 , and x5 D x4 C G2 .

1 G1 J0 i C ˛n G1 2a 2 1 J0 i ˛n G1 2 1 iCnC1 G2 Q s3i ˛n D 1 J0 i C ˛n G2 2a 2 1 J0 i ˛n G2 2

Q s2i ˛n D 1i

7.51

7.52

where J0 stands for the zero-order Bessel function of the first kind. The parameter Ri , representing the right-hand side of Eq. (7.40), that appears in Eq. (7.44) can be obtained upon using Eq. (7.47) as WVo i1 i1 Ri D 7.53 J0 cos a 2 2 A remark needs to be made at this point that both the numbers of basis functions Nj and spectral terms N used in Eq. (7.40)–(7.42) affect the accuracy of the

162


64

2.7

62

2.6

60

2.5

58

2.4

56

2.3

54 0

5

10

15

20 S2 (mils)

25

30

35

Effective Dielectric Constant, eeff

Characteristic Impedance, Z o (ohms)

66

2.2 40

Figure 7.3 Calculated characteristic impedance and effective dielectric constant of the CPW using the quasi-static SDA. a D 2.54 mm, 2W D G D 0.508 mm, h1 D h3 D 4h2 D 0.508 mm, εr1 D 1, εr2 D 2.2, and εr3 D 10.5.

numerical results. The larger these numbers the more accurate the results, but at the expense of increased computation time. For most engineering purposes, three basis functions Nj D 3 and 200 spectral terms N D 200 are sufficient. As a demonstration of the quasi-static SDA, we show in Fig. 7.3 the calculated values of the characteristic impedance and effective dielectric constant for the considered CPW versus the right gap. 7.2

FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS

A three-layer CPW is also used to illustrate the dynamic SDA formulation. However, to simplify the analysis without loss of generality, we consider only a symmetrical CPW shown in Fig. 7.4. Analysis of the asymmetrical CPW shown in Fig. 7.1 using the dynamic SDA can be found in Rahman and Nguyen [7] and Nguyen [8]. The dynamic SDA allows us to examine all the eigenmodes existing in the transmission line structure. For the considered CPW, these modes consist of both TE and TM fields. Although SDA can produce results for all of the real and complex eigenmodes for this structure [9], in this section we are restricted to the analysis of the real eigenmodes, including the dominant (CPW) mode, for the purpose of illustrating the SDA. The dominant CPW mode is the quasiTEM mode discussed in Section 7.1, which is the principal propagating mode in integrated circuits and is the one in which we are most interested in this book. A dynamic analysis solves the wave equations or Helmholtz equations for the electric and magnetic fields or the electric and magnetic potentials in the space


163

y

er 1

h1 G

2W

S

G

S

er 2

h2

er 3

h3

b

x

−a

a

0

Figure 7.4

Cross section of the CPW used in the dynamic SDA.

domain subject to proper boundary conditions. The dynamic SDA, instead, solves these equations in the spectral domain using Galerkin’s technique, discussed in Appendix B at the end of this chapter. The analysis can obtain the propagation constants, effective dielectric constants, and characteristic impedances of the transmission line for all of the eigenmodes. In essence, its formulation process is similar to that for the quasi-static case. Let ie x, y and ih x, y represent the scalar electric and magnetic potentials associated with the TM and TE modes, respectively, in the ith region i D 1, 2, 3 in the space domain. The wave equations of these potentials are obtained from Eqs. (2.46b) and (2.46d) as r2t

p i x, y

ˇ2 ki2

where r2t D

p i x, y

D 0;

∂2 ∂2 C 2 2 ∂x ∂y

p D e, h

7.54

7.55

p ˇ is the propagation constant and ki D ω εi 5i denotes the wave number in region i. ω is the angular frequency, and εi and 5i represent the permittivity and permeability, respectively, of medium i. The boundary conditions for the considered CPW are as follows. For a x a, y D h3 : Ex3 x, h3 D Ex2 x, h3

7.56

Ez3 x, h3 D Ez2 x, h3

7.57

164


Hx3 x, h3 D Hx2 x, h3

7.58

Hz3 x, h3 D Hz2 x, h3

7.59

For a x a, y D h2 C h3 : Ex2 x, h2 C h3 D Ex1 x, h2 C h3 D Ex x

7.60

Ez2 x, h2 C h3 D Ez1 x, h2 C h3 D Ez x

7.61

Hx2 x, h2 C h3 Hx1 x, h2 C h3 D Jz x

7.62

Hz2 x, h2 C h3 Hz1 x, h2 C h3 D Jx x

7.63

For a x a, y D 0 and b: Ex3 x, y D Ex1 x, y D Ez3 x, y D Ez1 x, y D 0

7.64

For x D ša and 0 y b: Ez3 x, y D Ez1 x, y D 0

7.65

where Ei and Hi , i D 1, 2, 3, stand for the electric and magnetic fields respectively, in region i. Ex x and Ez x are the x and z components of the unknown electric field on the two slots; they are nonzero on the slots and zero elsewhere. Jx x and Jz x denote the total respective unknown x- and z-directed current densities on the central and ground strips, and they are nonzero only on these strips. The last boundary condition (7.65) is only used to determine the variable ˛n for the Fourier transform defined later. In order to carry out the SDA, we now need to define an appropriate Fourier transform. For the considered structure, we define the Fourier transform of fx, y as a Q f˛n , y D fx, yej˛n x dx 7.66 a

or

a

Q n , y D 2 f˛

fx, y sin ˛n x dx

7.67

fx, y cos ˛n x dx

7.68

0

and

Q n , y D 2 f˛

a

0

for even and odd functions with respect to x, respectively. ˛n is chosen to satisfy the boundary conditions at the central plane x D 0 and the two side walls


x D ša and thus is given by  n    a , ˛n D 1   ,  n 2 a

165

odd function 7.69 even function

In actual computations, however, the choice of ˛n does not make any difference because a large number of the spectral terms is normally used. So either expression for ˛n may be used. Discussion of these Fourier transforms and their associated Parseval’s theorem are given in Appendix A at the end of this chapter. There are two kinds of modes existing in CPW: even and odd modes. The even mode is the dominant mode, normally referred to as the CPW mode. This mode corresponds to Ez x and Hz x, which are even and odd functions with respect to x, respectively. The odd mode is normally referred to as the slot line mode and corresponds to odd Ez x and even Hz x. The z-directed magnetic and electric fields at the plane of symmetry x D 0 are zero for the even and odd modes, respectively. Now following the same approach in Section 7.1 for the quasi-static SDA formulation, we take the Fourier transform of Eq. (7.54) and use the differentiation property to obtain ∂2 Q p p ˛n , y 7i2 Q i ˛n , y D 0; ∂y 2 i

p D e, h and i D 1, 2, 3

7.70

where 7i2 D ˇ2 C ˛2n ki2

7.71

Q ie ˛n , y and Q ih ˛n , y are the scalar electric and magnetic potentials, respectively, in the spectral domain. Equation (7.70) is the wave equation or Helmholtz equation of these potentials in the Fourier-transform or spectral domain. Taking the Fourier transform of Eqs. (7.56)–(7.64) gives the boundary conditions of the considered CPW in the spectral domain as Q x2 ˛n , h3 Q x3 ˛n , h3 D E E

7.72

Q z3 ˛n , h3 D E Q z2 ˛n , h3 E

7.73

Q x3 ˛n , h3 D H Q x2 ˛n , h3 H

7.74

Q z3 ˛n , h3 D H Q z2 ˛n , h3 H

7.75

Q x2 ˛n , h2 C h3 D E Q x1 ˛n , h2 C h3 D EQ x ˛n E

7.76

Q z2 ˛n , h2 C h3 D E Q z1 ˛n , h2 C h3 D E Q z ˛n E

7.77

Q x2 ˛n , h2 C h3 H Q x1 ˛n , h2 C h3 D JQ z ˛n H

7.78

Q z1 ˛n , h2 C h3 D JQ x ˛n Q z2 ˛n , h2 C h3 H H

7.79

166


Q z1 ˛n , 0 D 0 Q x3 ˛n , 0 D EQ x1 ˛n , 0 D EQ z3 ˛n , 0 D E E

7.80

Q x3 ˛n , b D EQ x1 ˛n , b D EQ z3 ˛n , b D E Q z1 ˛n , b D 0 E

7.81

The fields in each region i may be given in terms of the potential functions as Ezi x, y, z D j

ki2 ˇ2 ˇ

e jˇz i x, ye

7.82

ki2 ˇ2 h jˇz i x, ye ˇ ω5i Eti x, y, z D Wt ie x, y az ð Wt ih x, y ejˇz ˇ ωεi Hti x, y, z D Wt ih x, y C az ð Wt ie x, y ejˇz ˇ Hzi x, y, z D j

7.83 7.84 7.85

where the subscript t indicates the transverse (x or y) component, and Wt D

∂ ∂ ax C ay ∂x ∂y

7.86

In the spectral domain, they are Q xi ˛n , y D j˛n Q ie ˛n , y C E

ω5i ∂ Q ih ˛n , y ˇ ∂y

7.87

EQ yi ˛n , y D j

˛n ω5i Q h ∂ Q ie ˛n , y ˛ , y C n i ˇ ∂y

7.88

EQ zi ˛n , y D j

ki2 ˇ2 Q e i ˛n , y ˇ

7.89

ωεi ∂ Q ie ˛n , y ˇ ∂y

7.90

˛n ωεi Q e ∂ Q ih ˛n , y i ˛n , y C ˇ ∂y

7.91

Q xi ˛n , y D j˛n Q ih ˛n , y C H Q yi ˛n , y D j H Q zi ˛n , y D j H

ki2 ˇ2 Q h i ˛n , y ˇ

7.92

General solutions of Eq. (7.70) are Q 1e ˛n , y D Ae sinh 71 b y C Ee cosh 71 b y

7.93

Q 1h ˛n , y D Ah cosh 71 b y C Eh sinh 71 b y

7.94


167

Q 2e ˛n , y D Be sinh 72 y h3 C Ce cosh 72 y h3

7.95

Q 2h ˛n , y D Bh sinh 72 y h3 C Ch cosh 72 y h3

7.96

Q 3e ˛n , y D De sinh 73 y C Fe cosh 73 y

7.97

Q 3h ˛n , y

7.98

D Dh cosh 73 y C Fh sinh 73 y

where Ae,h , Be,h , Ce,h , De,h , Ee,h , and Fe,h are unknown constants. In order to satisfy the last two boundary conditions, Eqs. (7.80) and (7.81), Ee,h , and Fe,h must all be equal to zero. This fact will be used implicitly in later applications of the remaining boundary conditions. The fields in the three regions in the spectral domain can now be derived by substituting Eqs. (7.93)–(7.98) into (7.87)–(7.92) as Q x1 ˛n , y D j˛n Ae sinh 71 b y E

ω51 71 Ah sinh 71 b y ˇ

Q x2 ˛n , y D j˛n [Be sinh 72 y h3 C Ce cosh 72 y h3 ] E ω52 72 [Bh cosh 72 y h3 C Ch sinh 72 y h3 ] C ˇ ω53 73 Q x3 ˛n , y D j˛n De sinh 73 y C Dh sinh 73 y E ˇ Q z1 ˛n , y D jAe E Q z2 ˛n , y D j E

k12 ˇ2 sinh 71 b y ˇ

k32 ˇ2 sinh 73 y ˇ

Q x1 ˛n , y D j˛n Ah cosh 71 b y C H

Q z2 ˛n , y D j H

ωε1 71 Ae cosh 71 b y ˇ

k12 ˇ2 cosh 71 b y ˇ

k22 ˇ2 [Bh sinh 72 y h3 C Ch cosh 72 y h3 ] ˇ

Q z3 ˛n , y D jDh H

k32 ˇ2 cosh 73 y ˇ

7.101

7.103 7.104

Q x2 ˛n , y D j˛n [Bh sinh 72 y h3 C Ch cosh 72 y h3 ] H ωε2 72 [Be cosh 72 y h3 C Ce sinh 72 y h3 ] ˇ ωε 7 Q x3 ˛n , y D j˛n Dh cosh 73 y 3 3 De cosh 73 y H ˇ Q z1 ˛n , y D jAh H

7.100

7.102

k22 ˇ2 [Be sinh 72 y h3 C Ce cosh 72 y h3 ] ˇ

Q z3 ˛n , y D jDe E

7.99

7.105

7.106 7.107 7.108 7.109 7.110

168


Now we apply the boundary conditions (7.72)–(7.79). Applying Eq. (7.72) and using Eqs. (7.100) and (7.101) yields j˛n De sinh ˛n h3 C

ω53 73 ω52 72 Dh sinh ˛n h3 D j˛n Ce C Bh ˇ ˇ

Using Eq. (7.73) in Eqs. (7.103) and (7.104), we obtain 2 k3 ˇ2 De sinh 73 h3 D k22 ˇ2 Ce

7.111a

7.111b

Applying Eq. (7.74) to Eqs. (7.106) and (7.107), we get j˛n Dh cosh ˛n h3 C

ωε3 73 ωε2 72 De cosh ˛n h3 D j˛n Ch Be ˇ ˇ

Substituting Eqs. (7.109) and (7.110) into Eq. (7.75) gives 2 k3 ˇ2 Dh cosh 73 h3 D k22 ˇ2 Ch

7.112

7.113

Substituting Eqs. (7.99) and (7.100) into Eq. (7.76), we obtain j˛n Be sinh 72 h2 C Ce cosh 72 h2 C D j˛n Ae sinh 71 h1

ω52 72 Bh cosh 72 h2 C Ch sinh 72 h2 ˇ

ω51 71 Ah sinh 71 h1 D EQ x ˛n ˇ

7.114

Substituting Eqs. (7.102) and (7.103) into Eq. (7.77) yields 2 k2 ˇ2 Be sinh 72 h2 C Ce cosh 72 h2 D k12 ˇ2 Ae sinh 71 h1 D jˇEQ z ˛n 7.115 Substituting Eqs. (7.105) and (7.106) into Eq. (7.78) leads to j˛n Bh sinh 72 h2 C Ch cosh 72 h2 C j˛n Ah cosh 71 h1

ωε2 72 Be cosh 72 h2 C Ce sinh 72 h2 ˇ

ωε1 71 Ae cosh 71 h1 D JQ z ˛n ˇ

7.116

Finally, applying Eq. (7.79) in Eqs. (7.108) and (7.109), we have 2 k2 ˇ2 Bh sinh 72 h2 C Ch cosh 72 h2 k12 ˇ2 Ah cosh 71 h1 D jˇJQ x ˛n 7.117 From Eqs. (7.111a)–(7.117) we can solve for JQ x ˛n and JQ z ˛n in terms of Q z ˛n as EQ x ˛n and E Q 11 ˛n , ˇEQ x ˛n C G Q 12 ˛n , ˇEQ z ˛n D JQ x ˛n G

7.118

Q 21 ˛n , ˇEQ x ˛n C G Q 22 ˛n , ˇEQ z ˛n D JQ z ˛n G

7.119


169

Q ij ˛n , ˇ, i, j D 1, 2, are the Green’s functions in the spectral domain where G and are given as 1 2 Q 2Q ˛ ˇ G G e h ˛2n C ˇ2 n Qe CG Q 21 D G Q 12 D j˛n ˇ G Qh G ˛2n C ˇ2 1 2Q 2 Q Q 22 D G G G ˇ ˛ e h n ˛2n C ˇ2   2 3 2  k0  εr2 ˛3 n tanh ˛n h2 tanh ˛n h3 ˛n εr3  D 2  ˛2 ε tanh ˛2 h  ˛n  3 n r3 n 2 ˛n tanh ˛3 n h3 C εr2 k0 εr1 C 1 ˛n tanh ˛1 n h1   2 tanh ˛3 n h3 2 1 ˛n tanh ˛n h2  3 1  ˛1   n n 2 ˛ D    k tanh ˛1 k0  tanh ˛n h2 tanh ˛3 n h3 n h1 0 C ˛2 ˛3 n n k12 ˇ2 ˛2n ˛1 n D ˛2 D k22 ˇ2 ˛2n n ˛3 D k32 ˇ2 ˛2n n

Q 11 D G

Qe G

Qh G

7.120 7.121 7.122

7.123

7.124

7.125 7.126 7.127

with k0 being the free-space wave number. Equations (7.118) and (7.119) are the Fourier transforms of the conventional coupled integral equations in the space domain. As for the case of quasi-static SDA, the spectral-domain Green’s functions obtained here are also independent of the dimensions along the x axis. These functions are easier to obtain than those involved in the coupled integral equations encountered in the space domain. There are four unknowns in Eqs. (7.118) and (7.119); they are EQ x , EQ z , JQ x , and JQ z along the plane where the conductors are located. The analysis, up to this stage, is exact. We now apply Galerkin’s technique in the spectral domain to cancel the two unknowns JQ x and JQ z and solve Eqs. (7.118) and (7.119) approximately. To this end, we express the slots’ electric fields as truncated summations of basis functions in the space domain as Ex x ¾ D

M mD0

cm Exm x

7.128

170

SPECTRAL-DOMAIN METHOD K

Ez x ¾ D

dk Ezk x

7.129

kD1

where cm and dk are the unknown coefficients. The basis functions Exm x and Ezk x describe the x- and z-electric field distributions on the slots and are nonzero only on these slots. In the spectral domain, the unknown electric fields are Q x ˛n ¾ E D

M

Q xm ˛n cm E

7.130

dk EQ zk ˛n

7.131

mD0

Q z ˛n ¾ E D

K kD1

Substituting Eqs. (7.130) and (7.131) into Eqs. (7.118) and (7.119) gives M

Q 11 ˛n , ˇEQ xm ˛n cm C G

mD0 M

K

Q 12 ˛n , ˇEQ zk ˛n dk D JQ x ˛n 7.132 G

kD1

Q 21 ˛n , ˇEQ xm ˛n cm C G

mD0

K

Q 22 ˛n , ˇEQ zk ˛n dk D JQ z ˛n 7.133 G

kD1

Q xi ˛n , We now take the inner product of Eqs. (7.132) and (7.133) with E Q i D 0, 1, . . . , M, and Ezj ˛n , j D 0, 1, . . . , K, respectively. That is, multiply Eqs. (7.132) and (7.133) with EQ xi ˛n , i D 0, 1, . . . , M, and EQ zj ˛n , j D 0, 1, . . . , K, respectively, and sum over ˛n . This process results in the following system of coupled linear algebraic equations, referred to as the Galerkin or Rayleigh–Ritz equations: M

im P11 ˇcm C

mD0 M

ik P12 ˇdk D 0;

i D 0, 1, 2, . . . , M

7.134

j D 0, 1, 2, . . . , K

7.135

kD1 jm

P21 ˇcm C

mD0

where

K

K

jk

P22 ˇdk D 0;

kD1

im Q 11 ˛n , ˇEQ xm ˛n D EQ xi ˛n , G P11

D

1

Q 11 ˛n , ˇEQ xm ˛n EQ xi ˛n G

7.136

nD1

ik Q 12 ˛n , ˇEQ zk ˛n D EQ xi ˛n G P12

D

1 nD1

Q 12 ˛n , ˇEQ zk ˛n EQ xi ˛n G

7.137


171

jm Q 21 ˛n , ˇEQ xm ˛n P21 D EQ zj ˛n G

D

1

Q 21 ˛n , ˇEQ xm ˛n EQ zj ˛n G

7.138

nD1

jk Q 22 ˛n , ˇEQ zk ˛n P22 D EQ zj ˛n G

D

1

Q 22 ˛n , ˇEQ zk ˛n EQ zj ˛n G

7.139

nD1

The notation hÐi again indicates an inner product. The use of Galerkin’s technique enables us to eliminate the unknown current densities JQ x,z ˛n by applying Parseval’s theorem to the right-hand sides of Eqs. (7.134) and (7.135). Specifically, Parseval’s theorem eliminates the summations occurring on the right-hand sides, due to the fact that the currents and electric fields are nonzero in complementary regions in the space domain. Equations (7.134) and (7.135) form a system of homogeneous equations and, hence, their nontrivial solutions exist only if the determinant of the coefficient matrix vanishes. Setting this determinant to zero, we obtain a characteristic equation of the propagation constant, ˇ, which can then be solved for ˇ. The roots for ˇ are values of the propagation constants associated with (discrete) eigenmodes of the considered CPW. The effective dielectric constant of the transmission line for a particular mode is obtained as εeff D ˇ/k0 2 , where p k0 D ω ε0 50 . Upon using the computed value for ˇ, we can solve Eqs. (7.134) and (7.135) for cm and dk , from which Ex and Ez can be determined. We can then calculate all the other field components and the (dynamic) characteristic impedance as a function of frequency. As discussed in Chapter 4, because of the non-TEM nature of the considered transmission line, its characteristic impedance is not unique; that is, there exists various definitions for it. Using the definition based on power and voltage, we can express the characteristic impedance as jVo j2 Zo D 7.140 2Pavg where

WCS

Ex x, h2 C h3 dx

Vo D

7.141

W

is the voltage across the slot, and Pavg D

1 Re 2

a

a

0

b

Ex HyŁ Ey HxŁ dy dx

7.142

represents the transmitted power across the transmission line’s cross section.

172


To obtain numerical results for ˇ and consequently for εeff and Z0c , suitable basis functions for the electric field components, Exm and Ezk with m D 0, 1, . . . , M and k D 1, 2, . . . , K, are needed. These functions affect substantially the numerical efficiency of the solution process and the accuracy of the solutions. As for the choice of the charge basis functions discussed in Section 7.1 for the quasi-static SDA, the selected basis functions should resemble the expected physical behaviors of the electric fields over the slots. In order to be able to increase the solution accuracy by increasing the number of basis functions, these functions must also belong to complete sets. Moreover, the basis functions should have closed-form Fourier transforms to reduce computing time. Lastly, they should be twice continuous differentiable to eliminate spurious solutions. In choosing the basis function to satisfy the actual field behavior, particular attention should be paid to the singularities of the fields in the vicinity of sharp edges. Incorporation of these singularities in the solution process leads to some favorable features, including enhancement of the numerical solution accuracy, the possibility of avoiding an undesirable effect called the relative convergence (i.e., converging to nonphysical solutions) [10,11], and the achievement of high computing efficiency. The principal concern of satisfying the edge requirement is to safeguard against the convergence toward a nonphysical solution. However, other benefits including enhancement of the rate of convergence and improvement of the matrix condition, if applicable, usually follow. It is shown in Mirshekar-Syahkal [3] that for CPW with strips of infinitely thin perfect conductors, the transverse electric and magnetic field components are unbounded at the strip edges and approach infinity with r 1/2 . The longitudinal field components are bounded and behave as r 1/2 near the edges. With regard to these criteria, we employ the following basis functions according to Uwano and Itoh [12]:  x C d x d   cos cos m m   S S   ; m D 0, 2, . . .    1 [2x C d/S]2 1 [2x d/S]2 Exm x D  x C d x d    sin m sin m   S S   C ; m D 1, 3, . . .  1 [2x C d/S]2 1 [2x d/S]2 7.143  x C d x d   cos cos k k   S S     2 C 2 ;    1 2x C d/S 1 2x d/S Ezk x D  x C d x d    sin k sin k   S S     2 ; 2  1 2x C d/S 1 2x d/S

k D 1, 3, . . .

k D 2, 4, . . . 7.144


173

8 6 4

Exm

2

m=2

m=0

0 −2

m=3

m=1

−4 −6

Slot

Slot −8 −0.5

−0.4

−0.3

−0.2

−0.1

0.0 x

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

(a) 1.5

k =2 k =1

1.0

Ezk

0.5 0.0 −0.5 −1.0

k=3

Slot −1.5 −0.5

−0.4

−0.3

−0.2

−0.1

0.0 x

Slot 0.1

0.2

(b)

Figure 7.5

Plots of basis functions for the (a) x and (b) z electric fields.

174


which are defined only over the slots, where d D W C S/2. These functions satisfy all the constraints discussed earlier. Figure 7.5 illustrates the shapes of the first few basis functions. Using Eq. (7.66), their Fourier transforms are found to be

 ˛n S C m ˛n S m S  ;  sin ˛ j d J C J  n 0 0  2 2 2      m D 0, 2, . . . Q xm ˛n D 7.145

E ˛n S C m ˛n S m S   ;  cos ˛ j d J J n 0 0   2 2 2     m D 1, 3, . . .

 ˛n S C k ˛n S k S  ;  d J C J cos ˛ n 0 0   2 2 2     k D 1, 3, . . . Q zk ˛n D 7.146 E

   S sin ˛n d J0 ˛n S C k J0 ˛n S k ;    2 2   2 k D 2, 4, . . . where J0 denotes the zero-order Bessel function of the first kind. An example of computed results from the dynamic SDA is shown in Fig. 7.6, in which the propagation constants of the first four modes for a CPW are plotted 3.5 4th mode

Propagation Constant (rad /mm)

3.0

2.5

2.0 3rd mode

1.5

2nd mode

1.0

0.5 1st mode 0.0 10

15

20

25

30

35

Frequency (GHz)

Figure 7.6 Propagation constants of the first four modes for the CPW calculated using the dynamic SDA. a D 1.524 mm, h1 D 0.762 mm, h2 D 0.254 mm, h3 D 0.508 mm, 2W D 0.5 mm, S D 6 mm, εr1 D 1, εr2 D 9.6, and εr3 D 13.

Ex (arbitrary unit)


7 6 5 4 3 2 1 Ground plane 0 −1

175

Slot Ground plane

Strip

Slot

−2 −3 −4 −5 −6 −7 0.0

0.5

1.0

1.5 x−a (mm)

2.0

2.5

3.0

Ey (arbitrary unit)

(a) 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5 −4.0 −4.5 −5.0 −5.5 −6.0

Strip

Slot

Slot

Ground plane

0.0

0.5

Ground plane

1.0

1.5 x−a (mm)

2.0

2.5

3.0

(b)

Figure 7.7 (a) x component and (b) y component of the dominant mode’s electric field at the interface y D h2 C h3 . a D 1.422 mm, h1 D 4.55 mm, h2 D 0.127 mm, h3 D 1.016 mm, S D 0.889 mm, 2W D 0.102 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

against frequency. The first mode, which is the dominant mode, is found to be propagating up to about 26.5 GHz. As a check for the validity of the numerically calculated eigenmodes, we calculate the electric fields at the interface y D h2 C h3 for the dominant and first higher-order evanescent modes. They are shown in Figs. 7.7 and 7.8, respectively. The expected asymmetry of the x-directed fields

176

SPECTRAL-DOMAIN METHOD 0.4 0.3

Ex (arbitrary unit)

0.2

Ground plane

0.1

Strip

Slot

0.0 Slot −0.1

Ground plane

−0.2 −0.3 −0.4 0.0

0.5

1.0

1.5 x−a (mm)

2.0

2.5

3.0

(a) 0.075

Strip

0.050

Ground plane

Ey (arbitrary unit)

0.025

Ground plane

0.000 Slot

Slot

−0.025 −0.050 −0.075 −0.100 −0.125 −0.150 −0.175 −0.200 0.0

0.5

1.0

1.5 x −a (mm)

2.0

2.5

3.0

(b)

Figure 7.8 (a) x component and (b) y component of the first higher-order mode’s electric field at the interface y D h2 C h3 . Dimensions are the same as those in Fig. 7.7.

and the edge conditions are clearly visible in these plots. In addition, the tangential fields on the conducting strips are very close to zero. REFERENCES 1. T. Itoh and R. Mittra, “Spectral-Domain Approach for Calculating the Dispersion Characteristic of Microstrip Line,” IEEE Trans. Microwave Theory Tech., Vol. MTT-2, pp. 498–499, July 1973.

PROBLEMS

177

2. T. Itoh and A. S. Hebert, “A Generalized Spectral Domain Analysis for Coupled Suspended Microstrip Lines with Tuning Septums,” IEEE Trans. Microwave Theory Tech., Vol. MTT-26, pp. 820–826, Oct. 1978. 3. D. Mirshekar-Syahkal, Spectral Domain Method for Microwave Integrated Circuits, Research Studies Press Ltd., Somerset, England, 1990. 4. T. Itoh, “Analysis of Microstrip Resonators,” IEEE Trans. Microwave Theory Tech., Vol. MTT-22, pp. 946–952, Nov. 1974. 5. Q. Zhang and T. Itoh, “Spectral-Domain Analysis of Scattering from E-Plane Circuit Elements,” IEEE Trans. Microwave Theory Tech., Vol. MTT-35, pp. 138–150, Feb. 1987. 6. C. Scott, The Spectral Domain Method in Electromagnetics, Artech House, Norwood, MA, 1989. 7. K. M. Rahman and C. Nguyen, “Frequency Dependent Analysis of Shielded Asymmetric Coplanar Waveguide Step Discontinuity,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 1013–1015, 1994. 8. C. Nguyen, Spectral-Domain Analysis, in Wiley Encyclopedia of Electrical and Electronics Engineering, Vol. 20, J. G. Webster, Ed., John Wiley & Sons, New York, 1998, pp. 105–111. 9. K. M. Rahman and C. Nguyen, “On the Computation of Complex Modes in Lossless Shielded Asymmetric Coplanar Waveguides,” IEEE Trans. Microwave Theory Tech., Vol. MTT-43, pp. 2713–2716, Dec. 1995. 10. R. Mittra, “Relative Convergence of the Solution of a Doubly Infinite Set of Equations,” J. Res. NBS, Vol. 670, pp. 245–254, 1963. 11. S. W. Lee, W. R. Jones, and J. J. Campbell, “Convergence of Numerical Solutions of Iris-Type Discontinuity Problems,” IEEE Trans. Microwave Theory Tech., Vol. MTT-19, pp. 528–536, 1971. 12. T. Uwano and T. Itoh, Spectral-Domain Approach, in Numerical Techniques for Microwave and Millimeter-Wave Structures, T. Itoh, Ed., John Wiley & Sons, New York, 1989, pp. 334–380. 13. R. F. Harrington, Field Computation by Moment Methods, Robert E. Krieger Publishing, Melbourne, FL, 1983, Chap. 1.

PROBLEMS

7.1 Derive Eqs. (7.50) and (7.51). 7.2 Derive Eq. (7.53). 7.3 Derive Eq. (7.69). 7.4 Derive Eq. (7.70). 7.5 Derive Eqs. (7.145) and (7.146). 7.6 Consider a microstrip line as shown in Fig. 4.2. It is assumed that the strip is a perfect conductor and has zero thickness. Also, the ground plane is assumed to be infinitely wide and a perfect conductor. The dielectric is assumed to be perfect. Assume the charge distribution on the strip is

178


given as

s x D

1, 0,

jxj W/2 otherwise

(a) Formulate the quasi-static SDA for determining the capacitance per unit length, characteristic impedance, Zo , and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff using the formulation in part (a). Compute and plot Zo and εeff versus W/h from 0.05 to 5 for εr D 2.2 and 10.5 and h D 0.635 and 1.27 mm. 7.7 Repeat Problem 7.6 using the following charge distribution on the strip:  W 1   jxj 2 1 [jxj/W/2]2 s x D   0, otherwise Compare the numerical results of Zo and εeff to those obtained in Problem 7.6, and provide an assessment of the charge distributions used. 7.8 Repeat Problem 7.6 using the following basis functions for charge distribution on the strip: x C W/2 cos i 1 W si x D 2x C W/2 W 2 1 W Compare the numerical results of Zo and εeff to those obtained in Problems 7.6 and 7.7, and provide an assessment of the charge distributions used. Also, compare results using one to five basis functions and 50 to 400 spectral terms. 7.9 Consider the CPW shown in Fig. 7.1. Use the quasi-static SDA formulation for this transmission line, presented in Section 7.1, to write a computer program to calculate its characteristic impedance, Zo , and effective dielectric constant, εeff . Compute and plot Zo and εeff versus S1 D S2 from 0.0254 to 0.762 mm, for a D 2.54 mm, h1 D h3 D 0 and 508 mm, h2 D 0.127 mm, εr1 D 1, εr2 D 2.2, εr3 D 2.2, W D 0.0.508 mm, and G1 D G2 . 7.10 Consider a CPW as shown in Fig. 4.3(a). The ground and strip are assumed to be perfect conductors and to have zero thickness, and the dielectric substrate is assumed lossless. (a) Formulate the quasi-static SDA for determining the capacitance per unit length, characteristic impedance, Zo , and effective dielectric constant, εeff . Choose appropriate charge distributions for the strip and grounds.

PROBLEMS

179

(b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus the dimension ratio a/b from 0.1 to 0.9 for εr D 2.2 and 10.5 and h/b D 0.1, 0.5, 4, and 10. Compare results to those obtained in Problems 4.10 and 6.18. 7.11 Use the program developed in Problem 7.9 to calculate and plot Zo and εeff of the CPW shown in Fig. 4.3(a) versus the dimension ratio a/b from 0.1 to 0.9 for εr D 2.2 and 10.5 and h/b D 0.1, 0.5, 4, and 10. Compare results to those obtained in Problems 4.10, 6.18, and 7.10. 7.12 Consider a CPS as shown in Fig. 4.4. The strips are assumed to be perfect conductors with negligible thickness, and the dielectric substrate is assumed lossless. Assume that the basis functions for the charge distributions on the strips are given by  x C b   cos i 1    ba   , b x a s1i x D 2x C b b a 2  1   ba     0, otherwise  x a   cos i 1    ba   , axb s2i x D 2x a b a 2  1   ba     0, otherwise (a) Formulate the quasi-static SDA for determining the characteristic impedance, Zo , and effective dielectric constant, εeff . (b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus a/b from 0.1 to 0.9, for a relative dielectric constant, εr , of 2.2 and 10.5, and a normalized substrate thickness, h/b, of 0.1, 0.5, 4, and 10. Compare results to those obtained in Problem 6.20. 7.13 Consider a broadside-coupled CPW as shown in Fig. P7.1. The enclosure is assumed to be a perfect conductor, and all the ground and central strips are assumed to be perfectly conducting with zero thickness. The dielectric substrates are assumed to be lossless. This structure supports two dominant propagating modes: the even and odd modes corresponding to an open circuit or magnetic wall (MW) and a short-circuit or electric wall (EW) at the center of the central dielectric substrate (y D 0), respectively. These modes are balanced modes corresponding to a MW at the center of the strip (x D 0). Note that the structure corresponding to the odd mode is the same as the shielded conductor-backed CPW. Assume the charge distributions U U U L L on the central strips, s1i and s1i , and ground strips, s2i , s2i and s3i ,

180


y

er 1

h

G 2b

S

2W

S

G 2d

er 2

er 1

x

h

2a

Figure P7.1

Cross section of a broadside-coupled CPW.

L s3i , are given as

x C W cos i 1 2W U L s1i x D s1i x D 2x C W 2W 2 1 2W 1 x C W C S cos i 2 G U L s2i x D s2i x D xCWCSCG 2 1 G 1 x W S cos i 2 G U L s3i x D s3i x D xWSG 2 1 G

where the superscripts U and L refer to the upper and lower strips, respectively. The subscripts 1, 2, and 3 denote the central strip and left and right ground strips, respectively. (a) Formulate the quasi-static SDA analysis for determining the characteristic impedance, Zoe , and effective dielectric constant, εeff,e , of the even mode.

PROBLEMS

181

(b) Formulate the quasi-static SDA analysis for determining the characteristic impedance, Zoo , and effective dielectric constant, εeff,o , of the odd mode. (c) Write a computer program to calculate the Zoe , Zoo , and εeff,e , εeff,o . Compute and plot these parameters versus 2W from 0.0254 to 1.016 mm for 2a D 2.54 mm, h D 0.508 mm, 2d D 0.127 mm, εr1 D 1, εr2 D 10.5, εr3 D 2.2, and S D 0.508 mm. 7.14 Consider the broadside-coupled CPW in Problem 7.13. (a) Formulate an analysis of the effective dielectric constant, εeff,e , of the dominant even mode using the dynamic SDA. (b) Formulate an analysis of the effective dielectric constant, εeff,o , of the dominant odd mode using the dynamic SDA. (c) Write a computer program to calculate εeff,e and εeff,o as a function of frequency, using the electric field’s basis functions given in Eqs. (7.143) and (7.144). Compute and plot these parameters versus frequency from 1 to 50 GHz for 2a D 2.54 mm, h D 0.508 mm, 2d D 0.127 mm, εr1 D 1, εr2 D 10.5, 2W D 0.508 mm, and S D 0.0254, 0.254, and 0.762 mm. Compare results using one to five basis functions and 50 to 400 spectral terms. 7.15 Consider again the broadside-coupled CPW in Problem 7.13, and define its characteristic impedance as Zo D

V2o 2Pavg

where Vo is the voltage at the slot, and Pavg denotes the average power transported across the transmission line’s cross section. (a) Derive Pavg in terms of the field components in the Fourier-transform domain. (b) Formulate an analysis of the characteristic impedance for the dominant even mode, Zoe , using the dynamic SDA. (c) Formulate an analysis of the characteristic impedance for the dominant odd mode, Zoo , using the dynamic SDA. (d) Write a computer program to calculate Zoe and Zoo as a function of frequency. The electric field’s basis functions are given in Eqs. (7.143) and (7.144). Compute and plot these characteristic impedances versus frequency from 1 to 50 GHz for 2a D 3.556 mm, h D 2.54 mm, 2d D 0.635 mm, εr1 D 1, εr2 D 10, S D 0.635, 0.254, and 0.787 mm. Compare results using one to five basis functions and 50 to 400 spectral terms. 7.16 Consider again the broadside-coupled CPW in Problem 7.13. At high frequencies, due the hybrid nature of the propagation, this structure can

182


generate higher-order modes, classified as the even and odd eigenmodes corresponding to a MW and an EW at y D 0, respectively. (a) Formulate an analysis of the even eigenmodes using the dynamic SDA. (b) Formulate an analysis of the odd eigenmodes using the dynamic SDA. (c) Write a computer program to calculate the effective dielectric constants of the even and odd eigenmodes as a function of frequency. The electric field’s basis functions are given in Eqs. (7.143) and (7.144). Compute and plot the effective dielectric constants of the dominant and first higher-order even and odd modes versus frequency from 1 to 50 GHz for 2a D 3.556 mm, h D 2.54 mm, 2d D 0.635 mm, 2W D 0.787 mm, S D 0.635 mm, εr1 D 1, and εr2 D 2.2 and 10. These results can be used to determine the operating frequency range of the broadside-coupled CPW (i.e., the range of dominant-mode operation). 7.17 Use the formulation derived in Problem 7.16 to write a computer program to calculate the characteristic impedances of the even and odd eigenmodes as a function of frequency. The electric field’s basis functions are still given in Eqs. (7.143) and (7.144). Compute and plot the characteristic impedances of the dominant and first higher-order even and odd modes versus frequency from 1 to 50 GHz for 2a D 3.556 mm, h D 2.54 mm, 2d D 0.635 mm, 2W D 0.787 mm, and S D 0.635 mm, εr1 D 1, and εr2 D 2.2 and 10. 7.18 Consider the CPW shown in Fig. 7.4. Use the dynamic SDA formulation for this transmission line, presented in Section 7.2, to write a computer program to calculate its effective dielectric constants, εeff , for the eigenmodes. Compute and plot εeff for the first four modes versus frequency from 10 to 35 GHz for 2a D 1.524 mm, h1 D 0762 mm, h2 D 0.254 mm, h3 D 0.508 mm, 2W D 0.5 mm, S D 0.6 mm, εr1 D 1, εr2 D 9.6, and εr3 D 13. 7.19 Consider the CPW shown in Fig. 7.4. Use the dynamic SDA formulation for this transmission line, presented in Section 7.2, to write a computer program to calculate its characteristic impedance, Zo , for the dominant (CPW) mode. Compute and plot Zo versus frequency from 1 to 25 GHz for 2a D 1.524 mm, h1 D 0762 mm, h2 D 0.254 mm, h3 D 0.508 mm, 2W D 0.5 mm, S D 0.6 mm, εr1 D 1, εr2 D 9.6, and εr3 D 13. 7.20 Consider a slot line as shown in Fig. 4.6. The ground planes are assumed to be perfect conductors and to have zero thickness. The dielectric substrate is assumed to be lossless. The x- and z-directed electric fields are approximated by M cm Exm x Ex x ¾ D mD1

Ez x ¾ D

K kD1

dk Ezk x

183

PROBLEMS

where cm and dk are the unknown coefficients. The basis functions Exm x and Ezk x are assumed as  W[W/22 x 2 ]m1   , 2 W/22 x 2 Exm x D   0,

jxj

W 2

otherwise

  2x W 2 W 2   j 2 x x2 W 2 2 Ezk x D    0,

k1

, jxj

W 2

otherwise

The Fourier transform of a function fx, y is defined as

1

fx, yej˛x

Q f˛, y D 1

(a) Plot the first four basis functions for Exm and Ezk versus x. (b) Formulate an analysis for the effective dielectric constant, εeff , of the dominant mode using the dynamic SDA. The Fourier transforms of Exm and Ezk would involve the Bessel functions of order (m 1) and (k C 1), respectively. (c) Write a computer program to calculate εeff . Compute and plot εeff as a function of the normalized slot width, W/h, from 0.1 to 2 for εr of 10.5 and frequency of 1, 10, and 20 GHz. Compare results to those obtained in Problem 4.22. 7.21 Repeat Problem 7.20, parts (b) and (c), for the characteristic impedance, Zo , of the dominant mode, defined as Zo D

V2o 2Pavg

where Vo is the voltage at the slot, and Pavg denotes the average power transported across the slot line’s cross section. 7.22 Consider a three-layer shielded microstrip line as shown in Fig. 6.6. The enclosure and conducting strip are assumed to be perfect conductors. The strip thickness is assumed negligible, and the dielectric substrates are assumed to be lossless. The basis functions for the currents on the strip

184


are assumed as x S W/2 sin 2n W Jxn x D 2x S W/2 2 1 W x S W/2 cos 2 n 1 W Jzn x D 2x S W/2 2 1 W

(a) Plot the first four basis functions for Jxn and Jzn versus x. (b) Formulate the dynamic SDA for determining the effective dielectric constant, εeff , of the dominant mode. (c) Write a computer program to calculate εeff . Calculate and plot εeff versus W/h2 from 0.1 to 5, for εr2 D 2.2, h1 D h3 D 0.66 mm, h2 D 0.254 mm, a D 2.54 mm, S D 0.25a W and 0.5(a W), and a frequency of 1, 10, and 100 GHz. Compare results to the quasi-static results obtained in Problem 6.7. 7.23 Repeat Problem 7.22, parts (b) and (c), for the characteristic impedance Zo of the dominant mode. Zo is defined as Zo D

Pavg I2o

where Io is the current flow on the strip along the z direction, and Pavg represents the time-average power transmitted across the microstrip line’s cross section. 7.24 Consider a multi layer slot line as shown in Fig. P7.2. The ground planes are assumed to be perfect conductors and to have zero thickness. The dielectric substrates are assumed to be lossless. The electric fields’ basis functions are the same as those given in Problem 7.20. (a) Formulate an analysis for the effective dielectric constant, εeff , of the dominant mode using the dynamic SDA. The Fourier transforms of Exm and Ezk would involve the Bessel functions of order m 1 and k C 1, respectively. (b) Write a computer program to calculate εeff . Compute and plot εeff as a function of the normalized slot width, W/h, from 0.1 to 2 for εr1 D εr2 D εr4 D 1, εr3 D 10.5, and a frequency of 1, 10, and 20 GHz. Compare results to those obtained in Problems 4.22 and 7.20.

PROBLEMS

185

y

er 1

er 2

h1 W

h

er 3

x er 4

Figure P7.2

Cross section of a multilayer slot line.

(c) Compute and plot εeff as a function of frequency from 1 to 30 GHz for W/h D 0.05, 0.1, 0.8, 1, and 1.5 and εr1 D εr4 D 1, εr2 D 2.2, and εr3 D 10.5. 7.25 Consider the slot line in Problem 7.24, and define its characteristic impedance as V2o Zo D 2Pavg where Vo is the voltage at the slot, and Pavg denotes the average power transported across the slot line’s cross section. (a) Formulate an analysis for the characteristic impedance, Zo , of the dominant mode using the dynamic SDA. (b) Write a computer program to calculate Zo . Compute and plot Zo as a function of the normalized slot width, W/h, from 0.1 to 2 for εr1 D εr2 D εr4 D 1, εr3 D 10.5, and a frequency of 1, 10, and 20 GHz. Compare results to those obtained in Problems 4.22 and 7.20. (c) Compute and plot Zo as a function of frequency from 1 to 30 GHz for W/h D 0.05, 0.1, 0.8, 1, and 1.5 and εr1 D εr4 D 1, εr2 D 2.2, and εr3 D 10.5. 7.26 Consider a microstrip resonator as shown in Fig. P7.3. Define the following two-dimensional Fourier transform of fx, y, z: 1 a Q fx, y, zej˛n xCˇz dx dz f˛n , y, ˇ D 1

a

186


y

2W

d

e0 2l

2W

h

x

er 1

x z

2a (a) End View

Figure P7.3

(b) Top View

(a) End view and (b) top view of a microstrip resonator.

where ˛n and ˇ are the Fourier transform variables. Formulate the process to determine the resonant frequency using the SDA.

APPENDIX A: FOURIER TRANSFORM AND PARSEVAL’S THEOREM Open Transmission Line Structures

For open structures, such as the open microstrip line shown in Fig. 4.2, the Fourier transform of a function fx, y with respect to x is a continuous transform and can be defined as 1 Q f˛, y D fx, yej˛x dx A7.1 1

where ˛ is the Fourier transform variable, signifying the number of cycles per unit of x. The inverse Fourier transform or Fourier integral is 1 fx, y D 2

1

Q f˛, yej˛x d˛

A7.2

1

This Fourier transform is the same as the sine transform 1 Q fx, y sin ˛x dx f˛, y D 2

A7.3

0

when fx, y is odd with respect to x, that is, fx, y D fx, y, and the cosine transform 1 Q fx, y cos ˛x dx A7.4 f˛, y D 2 0

APPENDIX A: FOURIER TRANSFORM AND PARSEVAL’S THEOREM

187

if fx, y is even with respect to x, that is, fx, y D fx, y. Note that fx, y Q ˛, y is defined is defined only over a certain part of the domain of x, whereas f over the entire domain of ˛ . Other definitions for the Fourier transform may also be used. For example, the cosine transform may be written as

1

Q f˛, y D

fx, y cos ˛x dx

A7.5

0

Parseval’s theorem associated with the continuous Fourier transform has the form 1 1 1 Q f x, y gŁ x, y dx D A7.6 f˛, yQgŁ ˛, y d˛ 2 1 1 where the superscript Ł denotes the complex conjugate. This Parseval’s formula indicates that if the two functions fx, y and gx, y are nonzero in complementary regions in the space domain, the integral of their Fourier transforms’ product is zero with respect to the Fourier transform variable ˛. This result is useful for spectral-domain analysis. Closed Transmission Line Structures

For closed structures, such as the shielded microstrip line shown in Fig. 6.6, the Fourier transform of a function fx, y with respect to x is a discrete or finite transform and can be defined as a Q n , y D 1 f˛ fx, yej˛n x dx A7.7 a a where ˛n is the Fourier transform variable and has discrete values. This Fourier transform is equivalent to the sine transform, Q n , y D 2 f˛ a

a

fx, y sin ˛n x dx

A7.8

0

when fx, y is odd with respect to x, that is, fx, y D fx, y, and the cosine transform a Q n , y D 2 f˛ fx, y cos ˛n x dx A7.9 a 0 when fx, y is even with respect to x; that is, fx, y D fx, y. The inverse Fourier transforms of Eqs. (A7.8) and (A7.9) are just the Fourier series of fx, y: fx, y D

1 nD1

Q n , y cos ˛n x f˛

A7.10

188


fx, y D

1

Q n , y sin ˛n x f˛

A7.11

nD1

respectively. Other versions of the discrete Fourier transform can also be used. Parseval’s theorem associated with these transforms is written as

a

aQ fx, yg x, y dx D f˛n , yQgŁ ˛n , y 2 nD1 1

Ł

0

A7.12

which implies that if the two functions fx, y and gx, y are nonzero in complementary regions in the space domain, then the sum of their Fourier transforms’ product, over the Fourier transform variable ˛n , is zero. This theorem is useful in implementing spectral-domain analysis.

APPENDIX B: GALERKIN’S METHOD

In this appendix, we will present Galerkin’s method [13]. Galerkin’s method is basically a method of moment, which can be used for solving a linear system of equations. To this end, we consider the following inhomogeneous equation: Lf D g

B7.1

where L is a linear operator, f is an unknown function, and g is a known function. This kind of equation is often encountered in the analysis of microwave structures, in which L represents the system, f represents the response, and g denotes the source. Note that Eq. (B7.1) has a unique solution only if the boundary conditions on f are specified. For example, in Poisson’s equation r2 Vx, y, z D

x, y, z ε

B7.2

L D r2 , which is the Laplacian operator; f D V, representing the unknown voltage; and g D is the known charge distribution. Solution of Eq. (B7.1) using Galerkin’s method is now described as follows. We begin by expanding the unknown function f in a series of known basis functions, fn , which exist in the domain of L as

fD

N

an fn

B7.3

nD1

where an are the unknown coefficients. Note that if N approaches infinity and ffn g forms a complete set, for example, a Fourier series, then Eq. (B7.3)

189

APPENDIX B: GALERKIN’S METHOD

represents an exact solution for f. Substituting Eq. (B7.3) into (B7.1) gives N

an L fn D g

B7.4

nD1

Now we define a suitable inner product between two functions f and g, hf, gi, which satisfies the following properties: hf, gi D hg, fi

B7.5

haf C bg, hi D a hf, hi C b hg, hi ! Ł " f , f > 0, if f > 0 ! Ł " f , f D 0, if f D 0

B7.6 B7.7 B7.8

where h is another function, a and b are scalars, and the superscriptŁ indicates complex conjugate. For example, we can define

hf, gi D

wxfxgx dx

B7.9

where wx is referred to as the weighting or testing function. For the considered problem, we use N different weighting functions, wm m D 1, 2, . . . , N, in the range of L. Taking the inner product of Eq. (B7.4) with each of these functions, we obtain the following system of linear equations: N

an hwm , L fn i D hwm , gi

B7.10

nD1

which can be rewritten as

[lmn ] [an ] D gm

B7.11

where  hw

1 , Lf1 i

 hw2 , Lf1 i [lmn ] D  ..  . hwN , Lf1 i

[an ] D [a1 a2 gn D g1 g2 D hw1 , gi

ÐÐÐ ÐÐÐ

hw1 , Lf2 i hw2 , Lf2 i .. .

ÐÐÐ ÐÐÐ .. .

hw1 , LfN i hw2 , LfN i .. .

   

B7.12

hwN , Lf2 i Ð Ð Ð hwN , LfN i aN ]T T gN

hw2 , gi

ÐÐÐ

B7.13

hwN , gi

T

B7.14

190


with T indicating a transpose matrix. Equation (B7.11) can be solved to determine the unknown coefficients an . If [lmn ] is nonsingular, then we can obtain an directly as [an ] D [lmn ]1 gm B7.15 Substituting Eq. (B7.15) into (B7.3) yields the final solution for the unknown function f as f D fn [lmn ]1 gm B7.16 where [fn ] D [f1 f2 Ð Ð Ð fN ]. Various choices for fn and wn can be used, and they are critical for the numerical efficiency of the solution process and the accuracy of the solution. fn should be linearly independent and form a complete set so the solution accuracy can be enhanced by increasing the number of fn . ) They should be chosen such that N nD1 an fn satisfies the boundary conditions of f; individual fn , however, do not have to satisfy these boundary conditions closely. In practice, fn are normally chosen to closely describe f. Moreover, fn should be twice continuously differentiable to avoid spurious solutions. wn should also be linearly independent and form a complete set. Very often in microwave problems, wn and fn are chosen to be equal, and the method is referred to as Galerkin’s method. Thus, the spectral-domain method is basically a method of moment implemented using Galerkin’s technique in the Fourier-transform domain.


CHAPTER EIGHT

Mode-Matching Method The mode-matching method [1] is a useful technique for formulation of boundary-value problems, especially for structures consisting of two or more separate regions. It is based on matching the fields at the boundaries of different regions and hence lends itself naturally to the analysis of microwave boundaryvalue problems. The mode-matching method has widely been used for scattering and transmission problems, as well as transmission line analysis. Scattering problems include discontinuities in waveguides and transmission lines, such as the microstrip line and the coplanar waveguide (CPW), and obstacles in a medium. Transmission problems include analysis of filters, such as fin line bandpass filters, impedance transformers, and power dividers and transitions such as waveguide-to-microstrip line. Transmission line analyses include determination of the transmission line’s propagation constant and characteristic impedance, such as those of a microstrip line, a CPW, and coplanar strips (CPSs). In this chapter, we will describe the mode-matching method for analyzing planar transmission lines. Coplanar strips with finite strip metallization thickness will be used to illustrate the formulation process [2]. We will also describe the mode-matching method for the analysis of planar transmission line discontinuities. A CPW will be used to serve this purpose [3].

8.1

MODE-MATCHING ANALYSIS OF PLANAR TRANSMISSION LINES

Figure 8.1 shows a cross section of the coplanar strips (CPSs) used to illustrate the formulation process. The structure is assumed to be uniform and infinitely long. The conducting strips are assumed to be perfect, and the dielectric substrate is considered lossless. This structure supports, in general, hybrid modes consisting of both TE and TM. The dominant mode however, is, quasi-TEM. Modes in the structure are odd modes corresponding to an electric wall (EW) at the plane of symmetry of the two strips. 191

192

MODE-MATCHING METHOD

2a

2b

t

h

Figure 8.1

y

Cross section of a CPS.

W a b

2

1

t

Air 3

x er

5

EW

h

4

Air

MW

Figure 8.2 Half-structure of the CPS used for analysis. EW and MW stand for electric wall and magnetic wall, respectively.

In view of the symmetry, only one-half of the structure (x ½ 0) shown in Fig. 8.2 needs to be considered. To facilitate the solution process, a longitudinal magnetic wall (MW) is inserted perpendicular to the structure, sufficiently apart from the strips, as seen in Fig. 8.2. It should be noted here that a longitudinal electric wall (EW) can also be used in place of the MW and will yield the same results. Of course, appropriate boundary conditions must be used corresponding to the EW. These walls are needed for Fourier series expansions of the fields. The fields are normally confined between the two strips; hence the effect of this hypothetical MW can be neglected. This step is avoided for closed transmission line structures such as a shielded microstrip line. The considered half structure is now divided into five subregions. In mode-matching analysis, hybrid fields in all the subregions are determined first. These fields are then


193

matched along the transverse plane at the boundaries of different regions to yield a set of equations, from which parameters such as the propagation constants and characteristic impedances are determined. 8.1.1

Electric and Magnetic Field Expressions

The hybrid fields in each of the subregions i, i D 1, 2, . . . , 5, can be expressed in terms of the scalar electric, e , and magnetic, h , potentials using Eqs. (2.51a), (2.51b), (2.53a), and (2.53c) given in Chapter 2: Ez D kc2

e z

e

Et D e z Wt Hz D

kc2

8.1 e

C jωe z az ð Wt

h

h z

e

Ht D e z Wt

8.2

8.3

h

jωεe z az ð Wt

e

8.4

where kc is the cutoff wave number and is the propagation constant. We assume that the wave propagates in the Cz direction. The subscript t denotes the transverse (x and y) components. Note that e and h equal zero for TE and TM modes, respectively. e and h are solutions of the wave equations (2.46d) and (2.46b), r2t

e

x, y C kc2

e

x, y D 0

8.5a

r2t

h

kc2

h

x, y D 0

8.5b

x, y C

and can be expanded in a series of eigenfunctions with respect to the y direction as follows: e 1 x, y

D

N

An sin ˇn x exp[˛1n y t]

8.6

Bn cos ˇn x exp[˛1n y t]

8.7

sin ˇpn x[C1n sin ˛2n y C C2n cos ˛2n y]

8.8

cos ˇpn x[D1n sin ˛2n y C D2n cos ˛2n y]

8.9

nD1 h 1 x, y

D

N nD1

e 2 x, y

D

P nD1

h 2 x, y

D

P nD0

e 3 x, y

D

Q

sin[ˇqn x a][P1n sin ˛3n y C P2n cos ˛3n y]

8.10

cos[ˇqn x a][Q1n sin ˛3n y C Q2n cos ˛3n y]

8.11

nD1 h 3 x, y

D

Q nD1

194


e 4 x, y

D

N

sin ˇn x[R1n sin ˛4n y C R2n cos ˛4n y]

8.12

cos ˇn x[S1n sin ˛4n y C S2n cos ˛4n y]

8.13

Kn sin ˇn x exp[˛1n y C h]

8.14

Ln sin ˇn x exp[˛1n y C h]

8.15

nD1 h 4 x, y

D

N nD1

e 5 x, y

D

N nD1

h 5 x, y

D

N nD1

where n 12 ' ˇn D , W n' ˇpn D , b n 12 ' , ˇqn D Wa

n D 1, 2, 3, . . .

8.16

n D 0, 1, 2, . . .

8.17

n D 1, 2, 3, . . .

8.18

˛21n D ˇn2 2 ω2 ε0 0

8.19

˛22n

2 ˇpn

8.20

2 ˛23n D ω2 ε0 0 C 2 ˇqn

8.21

˛24n D ω2 ε0 εr 0 C 2 ˇn2

8.22

2

2

D ω ε0 0 C

An , Bn , . . . , Ln are the unknown Fourier coefficients; P, Q, and N represent the numbers of Fourier terms in regions 2, 3, and all the other regions, respectively. The scalar potentials, expressed in Eqs. (8.6)–(8.15), already satisfy the boundary conditions at the magnetic wall at x D W and the electric walls at x D 0, b, and a. Substituting Eqs. (8.6)–(8.15) into (8.1)–(8.4) then yields the components of the electric and magnetic fields in all the regions as Ex1 D

N nD1

Ex2 D

P

An ˛1n ˇn Bn jωε0

P ˛2n ˇpn nD1

cos ˇn x exp[˛1n y t]

8.23

[D1n sin ˛2n y C D2n cos ˛2n y]

nD0

C

jωε0

[C1n cos ˛2n y C2n sin ˛2n y] cos ˇpn x 8.24


Ex3 D

Q

195

[Q1n sin ˛3n y C Q2n cos ˛3n y]

nD1

Ex4

˛3n ˇqn [P1n cos ˛3n y P2n sin ˛3n y] cos ˇqn x a 8.25 C jωε0 N D [S1n sin ˛4n y C S2n cos ˛4n y] nD1

Ex5

˛4n ˇn [R1n cos ˛4n y R2n sin ˛4n y] cos Pn x C jωε0 εr N Kn ˛1n ˇn D Ln C cos ˇn x exp[˛1n y C h] jωε0 nD1

Ey1 D

N

nD1

Ey2 D

P

˛2 jω0 C 1n jωε0

nD1

Ey3 D

nD1

˛2 jω0 2n jωε0

An sin ˇn x exp[˛1n y t]

Ey4 D

nD1

[C1n sin ˛2n y

˛2 jω0 3n jωε0

Ey5 D

nD1

Ez1 D

N

[P1n sin ˛3n y

˛24n jω0 jωε0 εr

nD1

˛1n An jωε0

8.30

[R1n sin ˛4n y

˛21n jω0 Kn C Kn sin ˇn x exp[˛1n y C h] jωε0

Bn ˇn C

8.29

C R2n cos ˛4n y] sin ˇn x N

8.28

C P2n cos ˛3n y] sin ˇqn x N

8.27

C C2n cos ˛2n y] sin ˇpn x Q

8.26

8.31

8.32

sin ˇn x exp[˛1n y t]

8.33

196


Ez2 D

P

ˇpn [D1n sin ˛2n y C D2n cos ˛2n y]

nD1

˛2n [C1n cos ˛2n y C2n sin ˛2n y] sin ˇpn x jωε0 Q Ez3 D ˇpn [Q1n sin ˛3n y C Q2n cos ˛3n y]

8.34

nD1

˛3n [P1n cos ˛3n y C P2n sin ˛3n y] sin[ˇqn x a] jωε0 N Ez4 D ˇn [S1n sin ˛4n y C S2n cos ˛4n y]

8.35

nD1

˛4n [R1n cos ˛4n y R2n sin ˛4n y] sin Pn x jωε0 εr N ˛1n Kn Ez5 D L n ˇn sin ˇn x exp[˛1n y C h] jωε0 nD1

Hx1 D

N nD1

Hx2 D

P

ˇn Bn ˛1n An C jω0

8.36

8.37

sin ˇn x exp[˛1n y t]

8.38

[C1n sin ˛2n y C C2n cos ˛2n y]

nD1

˛2n ˇpn [D1n cos ˛2n y D2n sin ˛2n y] sin ˇpn x jω0 Q [P1n sin ˛3n y C P2n cos ˛3n y] Hx3 D

8.39

nD1

˛3n ˇqn [Q1n cos ˛3n y Q2n sin ˛3n y] sin[ˇqn x a] 8.40 jω0 N Hx4 D [R1n sin ˛4n y C R2n cos ˛4n y]

nD1

ˇn ˛4n [S1n cos ˛4n y S2n sin ˛4n y] sin ˇn x jω0 N ˇn Ln ˛1n Hx5 D Kn sin ˇn x exp[˛1n y C h] jω0 nD1

8.41

8.42


˛21n Bn Hy1 D jωε0 Bn C cos ˇn x exp[˛1n y t] jω0 nD0 P ˛22n Hy2 D jωε0 [D1n sin ˛2n y jωε0 nD0 N

C D2n cos ˛2n y] cos ˇpn x

Hy3 D

Q

˛2 jωε0 3n jωε0

nD0

8.43

8.44

[Q1n sin ˛3n y

C Q2n cos ˛3n y] cos[ˇqn x a] ˛24n Hy4 D jωε0 εr [S1n sin ˛4n y C S2n cos ˛4n y] jω0 nD0 N ˛21n Kn jωε0 Ln C cos ˇn x exp[˛1n y C h] Hy5 D jω0 nD0 N

197

8.45

N Bn ˛1n ˇn An C cos ˇn x exp[˛1n y t] Hz1 D jω0 nD1 P Hz2 D ˇpn [C1n sin ˛2n y C C2n cos ˛2n y]

8.46

8.47

8.48

nD1

P ˛2n [D1n cos ˛2n y D2n sin ˛2n y] cos ˇpn x jω0 nD0

Hz3 D

8.49

Q ˇqn [P1n sin ˛3n y C P2n cos ˛3n y] nD1

˛3n [Q1n cos ˛3n y Q2n cos ˛3n y] cos[ˇqn x a] 8.50 jω0 N Hz4 D ˇn [R1n sin ˛4n y C R2n cos ˛4n y]

nD1

˛4n [S1n cos ˛4n y S2n sin ˛4n y] cos ˇn x jω0 N ˛1n Ln Hz5 D Kn ˇn cos ˇn x exp[˛1n y C h] jω0 nD1

8.51

8.52

198


8.1.2

Mode-Matching Equations

We now impose the following remaining boundary conditions along each of the interfaces at y D t, 0, and h: At y D t, Ex1 x, t D Ex2 x, t,

x 2 [0, b]

8.53

Ez1 x, t D Ez2 x, t,

x 2 [0, b]

8.54

Ex1 x, t D Ex3 x, t,

x 2 [a, W]

8.55

Ex4 x, 0 D Ex2 x, t,

x 2 [0, b]

8.56

Ez4 x, 0 D Ez2 x, t,

x 2 [0, b]

8.57

Ex4 x, 0 D Ex3 x, t,

x 2 [a, W]

8.58

Ez4 x, 0 D Ez3 x, t,

x 2 [a, W]

8.59

At y D 0,

At y D h, Ex4 x, h D Ex5 x, h,

x 2 [0, W]

8.60

Ez4 x, h D Ez5 x, h,

x 2 [0, W]

8.61

we utilize the orthogonality properties of the eigenfunctions to obtain the following system of homogeneous equations: P

H1mn C1n C H2mn C2n C

nD1

P

H3mn D1n C H4mn D2n

nD0

C

Q

H5mn P1n C H6mn P2n C

nD1

Q

H7mn Q1n C H8mn Q2n D 0,

nD1

m D 1, 2, . . . , P P

I1mn C1n C I2mn C2n C

nD1

8.62a P

I3mn D1n C I4mn D2n

nD0

C

Q

I5mn P1n C I6mn P2n C

nD1

m D 0, 2, . . . , P

Q

I7mn Q1n C I8mn Q2n D 0,

nD1

8.62b

199

MODE-MATCHING ANALYSIS OF PLANAR TRANSMISSION LINES P

J1mn C1n C J2mn C2n C

P

nD1

J3mn D1n C J4mn D2n

nD0

C

Q

Q

J5mn P1n C J6mn P2n C

nD1

J7mn Q1n C J8mn Q2n D 0,

nD1

m D 1, 2, . . . , Q P

8.62c

K1mn C1n C K2mn C2n C

nD1

P

K3mn D1n C K4mn D2n

nD0

C

Q

Q

K5mn P1n C K6mn P2n C

nD1

K7mn Q1n C K8mn Q2n D 0,

nD1

m D 1, 2, . . . , Q P

L1mn C1n C L2mn C2n C

nD1

8.62d P

L3mn D1n C L4mn D2n

nD0

C

Q

Q

L5mn P1n C L6mn P2n C

nD1

L7mn Q1n C L8mn Q2n D 0,

nD1

m D 1, 2, . . . , P P

8.62e

M1mn C1n C M2mn C2n C

nD1

P

M3mn D1n C M4mn D2n

nD0

C

Q

M5mn P1n C M6mn P2n C

nD1

Q

M7mn Q1n C M8mn Q2n D 0,

nD1

m D 0, 1, 2, . . . , P P

N1mn C1n C N2mn C2n C

nD1

8.62f P

N3mn D1n C N4mn D2n

nD0

C

Q

N5mn P1n C N6mn P2n C

nD1

Q

N7mn Q1n C N8mn Q2n D 0,

nD1

m D 1, 2, . . . , Q P

T1mn C1n C T2mn C2n C

nD1

C

8.62g P

T3mn D1n C T4mn D2n

nD0 Q

T5mn P1n C T6mn P2n C

nD1

m D 1, 2, . . . , Q

Q

T7mn Q1n C T8mn Q2n D 0,

nD1

8.62h

200


where the coefficients H, I, J, K, L, M, N, and T are given in Appendix A at the end of this chapter. These equations are referred to as the mode-matching equations. For nontrivial solutions, the determinant of the coefficient matrix of Eq. (8.62) must vanish, from which the propagation constant can be solved. The effective dielectric constant is then obtained as εeff D

ˇ2 0 0

ω2 ε

8.63

where ˇ is the phase constant. The characteristic impedance of a CPS, due to its hybrid nature of propagation, cannot be uniquely defined as discussed in Chapter 4. Here we use the following definition: jVo j2 Z0 D

8.64 2Pavg where Vo represents the voltage between the strips and Pavg is the average power flowing through the cross section of the structure. Vo is given by

b

Vo D 2

Ex x, 0 dx

8.65

E ð HŁ Ð dx dy az

8.66

0

Pavg is obtained as

1

Pavg D 2 1

W

0

and equals the summation of individual powers contributed by all five regions. Figure 8.3 shows an example of the numerical results obtained from the modematching analysis of a CPS. The effective dielectric constant and characteristic impedance are plotted as a function of frequency with the strip thickness as a parameter. An illustration of the numerical convergence of the mode-matching analysis is given in Fig. 8.4, which shows the effective dielectric constant and characteristic impedance of the CPS as a function of the numbers of eigenmodes in regions 2, P, and 3, Q. The abscissa is the total number of modes in the two regions. For each curve, P is kept constant while Q is varied. It is readily seen that the convergence is approached as the number of modes is increased in both regions. All numerical results in Fig. 8.3 were computed by using P D Q D 10, which ensures the convergence of the solutions for the dimensions used.


201

Effective Dielectric Constant eeff

2.95

2.90

f =0 m 2.85

f =10 m

f =5 m 2.80

f =20 m 2.75 5.0

7.5

10.0

12.5

15.0

17.5

20.0

15.0

17.5

20.0

Frequency f (GHz) (a)

Characteristic Impedance Zo (ohms)

125.0 122.5

f =0 m 120.0 117.5 115.0

f =5 m

112.5

f =10 m

110.0

f =20 m

107.5 105.0 5.0

7.5

10.0

12.5 Frequency f (GHz) (b)

Figure 8.3 (a) Effective dielectric constant and (b) characteristic impedance of a CPS versus frequency for different values of the metallization thickness t. a D 0.5 mm, b D 0.2 mm, h D 0.127 mm, and εr D 9.6.

202


3.0

P =1 2

Effective Dielectric Constant eeff

2.9 2.8 2.7 2.6 2.5 13 3 4 5 6 7 8 9 10 1112 14 15

2.4 2.3 2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

20

22

24

26

28

30

Total Modes (P +Q ) (a)

Characteristic Impedance Zo (ohms)

115

110 5 4 105

3

100

6 7 8 9 10 11 13 14 15 12

2

P =1

95

90

85 2

4

6

8

10

12

14

16

18

Total Modes (P +Q ) (b)

Figure 8.4 Relative convergence of (a) the effective dielectric constant and (b) characteristic impedance as a function of the total number of modes. P D number of modes in region 2; Q D number of modes in region 3. a D 0.5 mm, b D 0.2 mm, h D 0.127 mm, t D 10 µm, εr D 9.6, and f D 10 GHz.

203

MODE-MATCHING ANALYSIS OF PLANAR TRANSMISSION LINE DISCONTINUITIES

8.2 MODE-MATCHING ANALYSIS OF PLANAR TRANSMISSION LINE DISCONTINUITIES

Figures 8.5 and 8.6 show a cross section of the shielded three-layer CPW and step discontinuities used to demonstrate the mode-matching analysis. The enclosure, ground planes, and central strip are assumed to be perfect conductors, and the dielectric substrates are assumed to be lossless. Also, the thickness of the ground planes and strip is neglected. The discontinuities are classified as singlestep [Fig. 8.6(a)], double-step [Fig. 8.6(b)], and multiple-step [Fig. 8.6(c)]. They are general in that the widths of the strip and ground planes can be changed separately or simultaneously. Both symmetrical and asymmetrical changes can also be accommodated. In order to implement the mode matching method, all the hybrid modes, including the propagating and evanescent modes, of the shielded CPW must first be determined. 8.2.1

Electric and Magnetic Field Expressions

The electric and magnetic fields in all regions i, i D 1, 2, 3, of the CPW shown in Fig. 8.5 can be expressed in Fourier series, which satisfy the wave equations and the enclosure’s boundary conditions at y D 0, 2a and x D 0, 2b as follows. E 1 x D

1

1 A 1 n cos ˛n x 2b sin ˛n y

8.67

nD1

x

2a

1

er 1

W

S

t2

W

er 2

2

er 3

3

h

2b

t1 y

Figure 8.5

Cross section of a shielded three-layer CPW.

204


a

Wa 2a

Wb

Sa

Sb b

z=0 (a)

Wa 2a

a

Wb Sc

Sa b

Sb

c

Wc

z=L

z=0 (b)

d

a

c • • • •

2a b

(c)

Figure 8.6

E 1 y D

E 1 z D

(a) Single-step, (b) double-step, and (c) multiple-step CPW discontinuities.

1

1 1 1 An ˛n ˛n jˇk0 Bn 1 2 Cˇ 1 ð sin ˛n x 2b cos ˛n y

˛2 nD1 n

1

˛2 nD1 n

8.68

1

1

1 jˇ˛ 1 sin ˛ 1 n An C k0 ˛n Bn n x 2b sin ˛n y 8.69 2 Cˇ


Hx 1 D

1

Bn 1 sin ˛ 1 n x 2b cos ˛n y

205

8.70

nD1

Hy 1 D

Hz 1 D

E 2 x D

1

1

1

1 jˇk0 εr1 A 1 n ˛n ˛n Bn 2 Cˇ 1 ð cos ˛n x 2b sin ˛n y

˛2 nD1 n

1

1 1 k0 εr1 ˛n A 1 n jˇ˛n Bn 2 C ˇ2 ˛ nD1 n ð cos ˛ 1 n x 2b cos ˛n y 1

2 A 2 n sin[˛n x t1 ] sin ˛n y C

nD1

E 2 y D

E 2 z D

Hx 2 D

E 3 x D

8.72

Pn 2

t1 ] sin ˛n y

8.73

1

2

2 ˛n ˛ 2 cos ˛ 2 n An jˇk0 Bn n x t1 cos ˛n y 2 C ˇ2 ˛ nD1 n

2

2 C ˛n ˛ 2 sin ˛ 2

8.74 n Pn jˇk0 Qn n x t1 cos ˛n y 1

1

2

2 jˇ˛ 2 cos ˛ 2 n An C ˛n k0 Bn n x t1 sin ˛n y 2 Cˇ

2

2 C jˇ˛ 2 sin ˛ 2

8.75 n Pn C ˛n k0 Qn n x t1 sin ˛n y

˛2 nD1 n

1

1 Bn 2 cos ˛ 2

x t cos ˛ y C Qn 2 1 n n

ð sin ˛ 2 n x t1 cos ˛n y

Hz 2 D

1 nD1

ð cos[˛ 2 n x 1

nD1

Hy 2 D

8.71

1

nD1

8.76

1

2 1

2

2

2 jˇk ε A C ˛ ˛ B sin ˛

x t sin ˛n y 0 r2 n 1 n n n n ˛2 C ˇ2 nD1 n

2 C jˇk0 εr2 Pn 2 ˛ 2 cos ˛ 2

8.77 n ˛n Qn n x t1 cos ˛n y 1

1

2 2 k0 εr2 ˛n A 2 sin ˛ 2 n C jˇ˛n Bn n x t1 cos ˛n y 2 Cˇ

2 C k0 εr2 ˛n Pn 2 C jˇ˛ 2 cos ˛ 2

8.78 n Qn n x t1 cos ˛n y

˛2 nD1 n

1

3 A 3 n cos ˛n x sin ˛n y

8.79

nD1

E 3 y D

1

˛2 nD1 n

3 1

3 ˛n ˛n A 3 sin ˛ 3 n jˇk0 Bn n x cos ˛n y 2 Cˇ

8.80

206


E 3 z D Hx 3 D

1

˛2 nD1 n 1

1

3

3 jˇ˛ 3 sin ˛ 3 n An C ˛n k0 Bn n x sin ˛n y 2 Cˇ

Bn 3 sin ˛ 3 n x cos ˛n y

8.81

8.82

nD1

Hy 3 D Hz 3 D

1

˛2 nD1 n 1

˛2 nD1 n

1

3

3 jˇk0 εr3 A 3 cos ˛ 3 n ˛n ˛n Bn n x sin ˛n y 2 Cˇ

8.83

1

3 3 k0 εr3 ˛n A 3 cos ˛ 3 n jˇ˛n Bn n x cos ˛n y 2 Cˇ

8.84

where ˛ 1 n D ˛ 2 n D ˛ 3 n D ˛n D

εr1 k02 ˇ2 ˛2n

8.85

εr2 k02 ˇ2 ˛2n

8.86

εr3 k02 ˇ2 ˛2n

8.87

n' 2a

8.88

k0 is the free-space wave number, and ˇ represents the propagation constants of all the eigenmodes. The superscript (1, 2, or 3) indicates the corresponding region. The expansion coefficients, the A’s, B’s, P’s and Q’s, can be determined by matching the tangential electric and magnetic fields across the interface at x D t1 and the tangential electric fields at x D t1 C h as A 2 n D

8.89

Q 2 ˛n EQ 2 zn ˛n C jˇEyn ˛n

8.90 2 

3 sin ˛ h ˛ n n  k0 a cos ˛ 2 n h C

2 ˛n tan ˛ 3 n t1  2 

2 ε sin ˛ h ˛ ˛ 2 n r3 n cos ˛ 2 n  A 2 D

8.91 n h C 3 n

1

1 ˛n sin ˛n t2 ˛n εr2 tan ˛ 3 n t1

Bn 2 D

A 1 n

Q 2 jˇEQ 2 zn ˛n C ˛n Eyn ˛n 2 

2 ε sin ˛ h ˛

2 n r3 n  a˛ 2 n cos ˛n h C

3 ˛n εr2 tan ˛ 3 n t1 





Bn 1

2  2 sin ˛ h ˛ 3 cos ˛n h C n n  Bn 2 D

1

2 sin ˛n t2 ˛n tan ˛ 3 n t1

1

8.92


207

A 3 n D

˛ 2 n A 2 n

3 ˛ 3 sin ˛ t n n 1

8.93

Bn 3 D

B 2 n sin ˛ 3 n t1

8.94

Pn 2 D

Qn 2 D

˛ 2 r3 n ε A 2 n

3

3 ˛n εr2 tan ˛n t1

˛ 3 n Bn 2

2

3 ˛n tan ˛n t1

8.95

8.96

Q 2 where EQ 2 yn and Ezn are the Fourier transforms of the unknown y and z components of the slot field at x D t1 C h, obtained using the spectral-domain technique described in Chapter 7. The electric and magnetic fields given in Eqs. (8.67)–(8.84) will be used in the following analysis of the single-, double-, and multiple-step discontinuities using the mode-matching method. 8.2.2

Single-Step Discontinuity

The single-step discontinuity, as shown in Fig. 8.6(a), is divided into two regions, a and b. Region b is assumed to extend to infinity. The hybrid modes on the left-hand side (region a) and the right-hand side (region b) of the plane of the discontinuity z D 0 are first determined. For the mode-matching analysis, it is sufficient to impose only the continuity condition of the normal transverse fields at the discontinuity. The boundary conditions for the longitudinal fields will automatically be satisfied through Maxwell’s equations. The transverse electric and magnetic fields for a mode i in region a can be expressed for each of the three regions, 1, 2, and 3, shown in Fig. 8.5 as follows: Region 1: E 1 ai D

1

1 A 1 cos ˛

x 2b sin ˛n yax nai nai

nD1

C

1

1 M 1 nai sin ˛nai x 2b cos ˛n yay

8.97

nD1

H 1 ai D

1

1 Bnai sin ˛ 1 nai x 2b cos ˛n yax

nD1

C

1 nD1

1 Snai cos ˛ 1

x 2b sin ˛n yay nai

8.98

208


where 1

1

1 A 1 nai ˛nai ˛n jˇi k0 Bnai 2 C ˇi 1

1

1 D 2 jˇi k0 εr1 A 1 nai ˛nai ˛n Bnai 2 ˛n C ˇi

M 1 nai D

1 Snai

˛2n

8.99

8.100

Region 2: E 2 ai D

1

2

2

2 A 2 sin ˛n yax nai sin ˛nai x t1 C Pnai cos ˛nai x t1

nD1

C

1

2

2

2 M 2 sin ˛

x t C R cos ˛

x t cos ˛n yay 1 1 nai nai nai nai

nD1

8.101 H 2 ai D

1

2

2

2 Bnai cos ˛ 2

x t C Q sin ˛

x t cos ˛n yax 1 1 nai nai nai

nD1

C

1

2

2

2 Snai cos ˛ 2 sin ˛n yay nai x t1 C Tnai sin ˛nai x t1

nD1

8.102 where 1

2 2

2 ˛ ˛ A jˇ k B n i 0 nai nai nai ˛2n C ˇi2 1

2 2

2 D 2 ˛ ˛ P jˇ k Q n i 0 nai nai nai ˛n C ˇi2 1

2

2

2 D 2 jˇ k ε P ˛ ˛ Q i 0 r2 nai nai n nai ˛n C ˇi2 1

2

2 D 2 jˇi k0 εr2 A 2 nai ˛nai ˛n Bnai 2 ˛n C ˇi

2 Rnai D

M 2 nai

2 Snai

T 2 nai

8.103

8.104

8.105

8.106

Region 3: E 3 ai D

1

3 A 3 nai cos ˛nai x sin ˛n yax

nD1

C

1 nD1

3 M 3 sin ˛ x cos ˛n yay nai nai

8.107


H 3 ai D

1

209

3 Bnai sin ˛ 3 x cos ˛n yax nai

nD1

C

1

3 Snai cos ˛ 3 nai x sin ˛n yay

8.108

nD1

where 1

3 3

3 A ˛ ˛ jˇ k B n i 0 nai nai nai ˛2n C ˇi2 1

3

3

3 D 2 jˇ k ε A ˛ ˛ B i 0 r1 n nai nai nai ˛n C ˇi2

M 3 nai D

3 Snai

8.109

8.110

2

3

1

2

1

2

3 3 ˛ 1 nai , ˛nai , ˛nai , ˛n , Anai , Anai , Anai , Bnai , Bnai , and Bnai are given in Eqs. (8.85)–(8.94). Note that the subscripts a and i have been added to signify region a and mode i, respectively. The fields in region b can also be obtained similarly. The transverse electric, ei x, y, z, and magnetic, hi x, y, z, fields for mode i are written as:

ei x, y, z D ai ei x, yeš i z

8.111

š i z

8.112

hi x, y, z D ai hi x, ye

where ei x, y and hi x, y are given in Eqs. (8.97)–(8.110) for region a. ai is the amplitude for mode i. i denotes the propagation constant of mode i and is imaginary and real for the propagating and evanescent modes, respectively. We now assume that a wave, corresponding to i D 1, with a unit amplitude

a1 D 1 is incident upon the junction at z D 0 from region a. To satisfy the boundary conditions, this wave will excite all the other modes, including propagating and evanescent modes, at the junction. Although the evanescent modes die down with distance away from the junction, their effects would be significant near the junction and thus should be considered. The propagating and evanescent modes, generated at the discontinuity, are then reflected into region a and transmitted into region b. The total transverse electric and magnetic fields in region a, just to the left of the plane of the discontinuity, can be expressed in terms of the hybrid modes in this region as Ea D 1 C 5ea1 C

1

ai eai

8.113

ai hai

8.114

iD2

Ha D 1 5ha1

1 iD2

210


where eai and hai are the normalized electric and magnetic fields of mode i

i D 1, 2, 3, . . . in region a, respectively. 5 is the reflection coefficient of the incident mode characterized by ea1 and ha1 . ai is the complex amplitude of mode i scattered into region a from the incident mode at the discontinuity. Expanding the total transverse fields, just to the right of the discontinuity, in region b in terms of the transmitted modes from region a, we get Eb D

1

bj ebj

8.115

bj hbj

8.116

jD1

Hb D

1 jD1

where ebj and hbj are the normalized electric and magnetic fields of mode j

j D 1, 2, 3, . . . in region b, respectively. bj , j D 1, 2, 3, . . . , is the complex amplitude of mode j transmitted into region b, which is generated by the incident mode at the discontinuity. We assume that region b is continuous in the z direction, so there are no back-scattered modes in this region. Imposing the continuity condition of the transverse electric and magnetic fields at the discontinuity yields the following mode-matching equations:

1 C 5ea1 C

1

ai eai D

iD2

1

5ha1

1 iD2

1

bj ebj

8.117

bj hbj

8.118

jD1

ai hai

D

1 jD1

In order to solve Eqs. (8.117)–(8.118), we introduce an inner product defined by

6 6 Iij D ei x, y, hj x, y D ei x, y ð h6j x, y Ð dS

8.119 S

where D a or b, 6 D a or b, and S is the cross section of the considered CPW. Here, the inner product is performed in the spectral domain to keep the computation consistent with the mode solution of the CPW obtained using the spectral-domain approach. The inner product in the spectral domain is 1

1 2b Q6 6 6 eQ i , hj D eQ ix x, ˛n hQ jy

x, ˛n eQ iy x, ˛n hQ jx

x, ˛n dx 4a nD1 0

8.120 Using the terminology of Wexler [1], waveguide step discontinuities can be classified into two types: boundary reduction and boundary enlargement. A boundary reduction occurs when the waveguide cross section for the incident wave is larger than that for the transmitted wave. Otherwise, it is referred to as the boundary enlargement. The difference in the formulation for the


211

two problems is in the definition of the inner product, which optimizes the convergence of the solution as a function of the number of modes used. For many waveguide discontinuities, such as those in circular and rectangular waveguides, the boundary enlargement and boundary reduction are well defined. This is, however, not the case for the CPW, as the enclosure has the same dimension on both sides of the discontinuity. Therefore, the definition of boundary enlargement or reduction must be studied carefully for CPW step discontinuities to obtain an optimally convergent solution with a relatively small number of modes. When boundary enlargement or reduction is clearly defined, the most rapid convergence is obtained when the inner product is defined such as when the electric and magnetic fields are taken from the smaller and larger waveguides, respectively, because of the effective enforcement of the discontinuity’s boundary conditions through the inner products. This enables the boundary condition at the discontinuity to be satisfied fairly well with a minimum number of modes. For a CPW step discontinuity having Wa < Wb , which can be referred to as the boundary enlargement or step-up problem, we take the inner products of Eqs. (8.117) and (8.118) with hbm and ean , respectively. By truncating the infinite series to Na and Nb , which signify the numbers of eigenmodes in regions a and b, respectively, we obtain the following system of Na C Nb linear equations:

1 C 5Iab 1m C

Na

Nb

ai Iab im

iD2

1 5Iaa n1

Na

bj Ibb jm D 0,

m D 1, . . . , Nb

bj Iab nj D 0,

n D 1, . . . , Na

8.121b

8.121a

jD1

ai Iaa ni

iD2

Nb jD1

where the inner product I’s are given in Appendix B at the end of this chapter. For the step-down junction with Wb > Wa (boundary reduction problem), we have

1 C 5Iaa 1m C

Na

ai Iaa im

iD2

1 5Iba n1

Na

Nb

bj Iba jm D 0,

m D 1, . . . , Na

8.122

bj Ibb nj D 0,

n D 1, . . . , Nb

8.123

jD1

ai Iba ni

iD2

Nb jD1

The above equations can immediately be solved to obtain the solutions for the

Na C Nb unknowns, 5, ai i D 2, . . . , Na , and bj j D 1, 2, . . . , Nb , which characterize the scattering S parameters of the discontinuity. 8.2.3

Double-Step Discontinuity

The double-step discontinuity is shown in Fig. 8.6(b). It contains three different regions: a z 0, b 0 z L, and c z ½ L. Regions a and c are assumed

212


to be semi-infinitely long. The hybrid modes in regions a, b, and c are first determined, similar to the case of a single-step discontinuity. For instance, the transverse fields in region a are given by Eqs. (8.97)–(8.112). We assume that a wave of unit amplitude a1 D 1 is incident upon the first junction z D 0 from region a. This incident wave excites all the propagating and evanescent modes at this junction. These generated modes are then reflected and transmitted into regions a and b, respectively. The total transverse electric and magnetic fields, just to the left of the plane of the discontinuity at z D 0, in region a can thus be described in terms of the hybrid modes in this region as given in Eqs. (8.113) and (8.114), respectively. The modes in region b consist of transmitted modes from region a (through the first junction at z D 0 and the reflected modes from the second junction at z D L (due to the modes incident upon this junction from region b). The transverse electric field of mode j j D 1, 2, 3, . . . transmitted into region b from region a is given as bj ebj , where ebj is the normalized transverse electric field of mode j in region b, and bj is the complex amplitude of mode j transmitted into region b, which is generated by the incident mode at the first junction. Each transmitted mode j (propagating or evanescent), upon reaching the second junction, excites all the other modes at that junction, which are then reflected and transmitted into regions b and c, respectively. The transverse electric field of a mode k reflected from the second junction is contributed from an infinite number of modes j incident at that junction and can thus be written as 1 b rk ebk D bj ebk ejˇj L

8.124 jD1

where rk is the overall reflection amplitude of mode k at the second junction. The total transverse electric and magnetic fields in region b can now be given as Eb D

1

b bj ebj ejˇ1 z

C

jD1

Hb D

1 jD1

1

b

rk ebk ejˇk Lz

8.125

kD1 b

bj hbj ejˇ1 z

1

b

rk hbk ejˇk Lz

8.126

kD1

where ebj and hbj are the normalized transverse electric and magnetic fields of mode j j D 1, 2, 3, . . . in region b, respectively. bj , j D 1, 2, 3, . . . , is the complex amplitude of mode j at the first junction, and rk is the complex amplitude of mode k reflected from the second junction. In region c, there are only transmitted modes from region b since region c is assumed to be semi-infinitely long. The total transverse fields in region c can therefore be expressed as


Ec D

1

c

cr ecr ejˇr zL

213

8.127

rD1

Hc D

1

c

cr hcr ejˇr zL

8.128

rD1

where ecr and hcr are the normalized electric and magnetic fields of mode r r D 1, 2, 3, . . . in region c, respectively. cr is the complex amplitude of mode r transmitted into region c, which is generated by an incident mode at the second junction. We now impose the boundary conditions at the two junctions. By applying the continuity of the transverse electric and magnetic fields at the z D 0 plane, we obtain 1

1 C 5ea1 C

ai eai D

1

iD2

1 5ha1

1

1

bj ebj C

jD1

ai hai D

8.129

kD1

1

iD2

b

rk ebk ejˇk L

bj hbj

jD1

1

b

rk hbk ejˇk L

8.130

cr ecr

8.131

cr hcr

8.132

kD1

Similarly, we have at the z D L plane 1

b

bj ebj ejˇ1 L C

jD1 1

1

rk ebk D

kD1 b bj hbj ejˇ1 L

jD1

1

1 rD1

rk hbk

D

kD1

1 rD1

Taking the inner products of Eqs. (8.129)–(8.132) with ham , ebn , hcp , and ebq , respectively, in accordance with the inner product defined in Eq. (8.119), with and 6 now denoting a, b, or c, yields the following system of Na C 2Nb C Nc linear equations:

1 C 5Iaa 1m C

Na

ai Iaa im

iD2

Nb

Nb

bj Iba jm

jD1

b rk Iba km exp jˇk L D 0,

m D 1, . . . , Na

8.133

n D 1, . . . , Nb

8.134

kD1

1 5Iba n1

Na iD2

C

Nb kD1

ai Iba ni

Nb

bj Ibb nj

jD1

b rk Ibb nk exp jˇk L D 0,

214

MODE-MATCHING METHOD Nb

b bj Ibc jp exp jˇj L C

jD1

Nb

rk Ibc kp

kD1

Nc

cp Icc rp D 0,

rD1

p D 1, . . . , Nc Nb

8.135

b bj Ibb qj exp jˇj L

jD1

Nb kD1

rk Ibb qk

Nc

cr Ibc qr D 0,

rD1

q D 1, . . . , Nb

8.136

where the inner product I’s are given in Appendix B at the end of this chapter. The infinite series are truncated to Na , Nb , and Nc that approximate the number of eigenmodes in regions a, b, and c, respectively. These equations can now be solved to determine the unknowns, 5, ai i D 2, 3, . . . , Na , bj and rk j, k D 1, 2, . . . , Nb , and cr r D 1, 2, . . . , Nc .

8.2.4

Multiple-Step Discontinuity

Figure 8.6(c) shows the multiple-step discontinuity. Its analysis is very similar to that of the double-step discontinuity. In the multistep problem, the composite generalized S matrix is obtained, taking into account the interactions between the junctions via the fundamental as well as higher-order modes. As examples of numerical results obtained from the mode-matching analysis of step discontinuities of a shielded three-layer CPW, we show in Figs. 8.7–8.9 the calculated S parameters of single-, double-, and triple-step discontinuities versus frequency, respectively. A remark needs to be made at this point that the eigenmodes of the considered CPW should satisfy the following modal orthogonality condition:

ISS ij D

S

eSi ð hSj Ð dS D υij

8.137

where υij is the Kronecker delta function and S D a, b, c, or d. However, as the CPW eigenmode calculations are only approximate due to the necessary numerical truncations, deviation from the orthogonality criterion may occur. To illustrate this phenomenon, we compare the S parameters of single-step CPW discontinuities, calculated with and without the assumption of the orthogonality condition, in Figs. 8.10 and 8.11. In Fig. 8.10, the mode calculations are very accurate, and hence the results agree very well. However, there is a small variation between the results in Fig. 8.11 due to slightly inaccurate calculations of the eigenmodes.


215

1.0

S21

Magnitudes of S11 and S21

0.9 0.8 0.7 0.6 0.5 0.4

S11

0.3 0.2 0.1 0.0 10.0

12.5

15.0

17.5

20.0

22.5

25.0

20.0

22.5

25.0

Frequency (GHz) (a) 100

Phases of S11 and S21 (degree)

90

S21

80 70 60 50 40 30 20 10

S11

0 −10 10.0

12.5

15.0

17.5 Frequency (GHz) (b)

Figure 8.7 Calculated (a) magnitudes and (b) phases of S11 and S21 of a single-step discontinuity. a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, Wa D 0.3 mm, Sa D 0.527 mm, Wb D 0.5 mm, Sb D 0.127 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

In order to assess the accuracy of the calculated S parameters, the validity of the power conservation jS11 j2 C jS21 j2 D 1, the boundary condition at the x D t1 C h plane, the orthogonality condition, and the convergence of the S parameters should be verified numerically. In our calculations, the power conservation condition is satisfied to within 0.9% when the orthogonality is not assumed, but to

216


1.0

S21


0.9 0.8 0.7 0.6 0.5 0.4 0.3

S11

0.2 0.1 0.0 10.0

12.5

15.0

17.5

20.0

22.5

25.0

20.0

22.5

25.0

Frequency (GHz) (a)


225 200

S11

175 150 125 100 75

S21

50 25 10.0

12.5

15.0


Figure 8.8 Calculated (a) magnitudes and (b) phases of S11 and S21 of a double-step discontinuity. a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, L D 1 mm, Wa D 0.9 mm, Sa D 0.2 mm, Wb D 0.6 mm, Sb D 0.6 mm, Wc D 0.8 mm, Sc D 0.4 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

217

MODE-MATCHING ANALYSIS OF PLANAR TRANSMISSION LINE DISCONTINUITIES L2

2a

Sc

Wb

Wa

Wd Sd

Sa Sb

Wc

L1

1.0


0.9

S21

0.8 0.7 0.6 0.5

S11

0.4 0.3 0.2 0.1 0.0 10.0

12.5

15.0

17.5

20.0

225

25.0

20.0

225

25.0


S21 Phases of S11 and S21 (degree)

260 240 220

S11 200 180 160 140 120 10.0

12.5

15.0


Figure 8.9 Calculated (a) magnitudes and (b) phases of S11 and S21 of a triple-step discontinuity. a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, L1 D 2 mm, L2 D 4.5 mm, Wa D 0.9 mm, Sa D 0.2 mm, Wb D 0.6 mm, Sb D 0.6 mm, Wc D 0.8 mm, Sc D 0.4 mm, Wd D 0.6 mm, Sd D 0.6 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

218


1.0

S21


0.9 0.8 0.7

Orthogonality not enforced

0.6

Orthogonally enforced

0.5 0.4 0.3

S11

0.2 0.1 0.0 10.0

12.5

15.0

17.5

20.0

22.5

25.0



90

S21

80 70 60


50

Orthogonally enforced

40 30 20 10

S11

0 −10 10.0

12.5

15.0

17.5 Frequency (GHz)

20.0

22.5

25.0

(b)

Figure 8.10 Comparison of the calculated (a) magnitudes and (b) phases of S11 and S21 of a single-step discontinuity with and without the orthogonality enforced. a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, Wa D 0.3 mm, Sa D 0.6 mm, Wb D 0.9 mm, Sb D 0.2 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

within only 0.005% when it is assumed. Furthermore, a better convergence of the numerically computed S parameters is achieved if the orthogonality was assumed. Consequently, we enforced the orthogonality condition in all the calculations presented in Figs. 8.7–8.9. However, the actual orthogonality conditions for these results had been checked and verified to assure their accuracy.


219

1.0

S21

0.9


0.8 Orthogonality enforced 0.7 0.6


0.5 0.4

S11

0.3 0.2 0.1 0.0 10.0

12.5

15.0

17.5

20.0

Frequency (GHz)

Figure 8.11 Comparison of the calculated magnitudes of S11 and S21 of another single-step discontinuity with and without the orthogonality enforced. a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, Wa D 0.3 mm, Sa D 0.6 mm, Wb D 0.9 mm, Sb D 0.2 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

The numerical convergence for the S parameters of the discontinuities calculated using the mode-matching method is affected by the number of eigenmodes used. Figure 8.12 shows this convergence behavior of the magnitude and phase of S11 for a single-step CPW discontinuity. For each of the curves in Fig. 8.12, the number of modes, P, to the left of the discontinuity (region a) is kept constant, while the number of modes, Q, to the right of the discontinuity (region b) is varied. The convergence error is very small. For instance, a good convergence is obtained for P and Q greater than 9. The data presented for single steps in Figs. 8.7, 8.10, and 8.11 were all calculated with six eigenmodes in each region, which, as can be seen from the conversion analysis presented in Fig. 8.12, gives fairly accurate results. Similar behavior was also observed for S21 . Figure 8.13 shows the numerical convergence for the magnitude and phase of S11 of a doublestep CPW discontinuity as a function of the number of eigenmodes. For each curve, the numbers of modes, P and Q, in the outer regions, a and c, respectively, are kept equal and constant, whereas the number of modes in the middle region, R, is varied. It is readily seen that the numbers of modes in regions a and c have little effect on the convergence of the S parameters. On the other hand, the number of modes in region b significantly affects the convergence, which is expected, because there are more interactions between modes in the midregion than in the outer regions. The interactions are especially strong for structures with small midregion lengths since evanescent modes created at one junction are able

220

MODE-MATCHING METHOD 0.2606

P =1

0.2604

2

Magnitude of S11

3 0.2602

4

0.2600

5 6

0.2598

7 9

8

0.2596

11 12

10 0.2594 2

4

6

8

10

12

14

16

18

20

22

24

Number of modes (P +Q ) (a) 0.5

Phase of S11

0.0 −0.5 10 11

−1.0

12 −1.5

8

9

6 7 −2.0

4

2

P =1

−2.5 2

4

6

8

10

12

5

3 14

16

18

20

22

24

Number of modes (P +Q ) (b)

Figure 8.12 Relative convergence of the (a) S11 of a single-step discontinuity. P D number Q D number of modes in region b. f D 20 GHz, t2 D 4.5466 mm, h D 0.127 mm, Wa D 0.3 mm, Sb D 0.127 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

magnitude and of modes in a D 1.4224 mm, Sa D 0.527 mm,

(b) phase of region a and t1 D 1.016 mm, Wb D 0.5 mm,

to reach other junctions before decaying completely. Consequently, more modes are generally needed in the midregions for double-step as well as multiple-step discontinuities in order for the convergence to occur. Our double- and triple-step S-parameter data in Figs. 8.8 and 8.9 were obtained with six modes in the outer regions and twelve modes in the midregions.

REFERENCES

221

0.234 10

11

12

Magnitude of S11

0.233

8

0.232 6 0.231 2

4

9

7

5

3

P =Q =1

0.230

0.229 3

6

9

12

15 18 21 24 Number of modes (P +Q +R ) (a)

27

30

33

10

11

36

203.8 12

Phase of S11 (degree)

203.6 203.4 203.2 6

7 8

203.0 2

4 3

9

5

202.8 P =Q =1 202.6 3

6

9

12

15 18 21 24 Number of modes (P +Q +R ) ( b)

27

30

33

36

Figure 8.13 Relative convergence of the (a) magnitude and (b) phase of S11 of a double-step discontinuity. P D number of modes in region a; Q D number of modes in region c; and R D number of modes in region b. f D 10 GHz, a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, L D 2 mm, Wa D 0.9 mm, Sa D 0.2 mm, Wb D 0.6 mm, Sb D 0.6 mm, Wc D 0.8 mm, Sc D 0.4 mm, εr1 D 3.7, εr2 D 10, and εr3 D 2.2.

REFERENCES 1. A. Wexler, “Solution of Waveguide Discontinuities by Modal Analysis,” IEEE Trans. Microwave Theory Tech., Vol. MTT-15, pp. 508–517, Sept. 1967.

222


2. K. M. Rahman and C. Nguyen, “Full-Wave Analysis of Coplanar Strips Considering the Finite Metallization Thickness,” IEEE Trans. Microwave Theory Tech., Vol. MTT42, pp. 2177–2179, Nov. 1994. 3 K. M. Rahman and C. Nguyen, “On the Analysis of Single and Multiple Step Discontinuities for a Shielded Three-Layer Coplanar Waveguide,” IEEE Trans. Microwave Theory Tech., Vol. MTT-41, pp. 1484–1488, Sept. 1993.

PROBLEMS

8.1 Consider the CPS as shown in Fig. 8.1. Use the formulation presented in Section 8.1 to write a computer program for calculating the effective dielectric constant, εeff , and characteristic impedance, Zo , for the quasiTEM mode. Compute and plot εeff and Zo versus h/6 from 0.01 to 1 for 2b/h D 1, a b/h D 1.5, εr D 9, and t D 0, 5, 10, and 20 m. 6 is the free-space wavelength. 8.2 Consider a microstrip line as shown in Fig. P8.1. The enclosure and strip are assumed to be perfect conductors. The dielectric substrates are assumed to be lossless. By placing a magnetic wall at the center of the strip and dividing the structure into subregions as shown, formulate the mode-matching process for determining the effective dielectric constant, εeff , and characteristic impedance, Zo , for the eigenmodes. Zo is defined as in Eq. (4.9). 8.3 Use the formulation derived in Problem 8.2 to write a computer program for calculating εeff and Zo for the quasi-TEM mode. Compute and plot these parameters versus frequency from 1 to 40 GHz for εr1 D 10.5, y

1

εr 3

h3

2

2W t

3

εr 2

5

εr 1

−a

0

Figure P8.1

4

h2

6

h1

a

Cross section of a shielded three-layer microstrip line.

x

PROBLEMS

223

εr2 D 2.2, εr3 D 1, a D 1.27 mm, h1 D h2 D 0.254 mm, h3 D 0.635 mm, W D 0.254 mm, and t/ h1 C h2 of 0, 0.01, 0.02, 0.05, and 0.1. 8.4 Repeat Problem 8.3 for εeff of the first three higher-order modes. The frequency is from 1 to 60 GHz. 8.5 Use the computer program developed in Problem 8.3 to compute and plot Zo and εeff for the quasi-TEM mode versus W/h2 from 0.05 to 5 for εr2 of 2.2 and 10.5, t/h2 of 0, 0.01, 0.02, 0.05, and 0.1, h1 D 0, h3 and a very large, and a frequency of 0.1, 10, and 20 GHz. This structure is basically an open microstrip line considered in Problem 4.7. Compare results at 0.1 GHz to the quasi-static results obtained in Problem 4.7. 8.6 Consider a two-layer conductor-backed CPW as shown in Fig. P8. 2. The central and ground strips are assumed to be perfect conductors, and the dielectrics are assumed lossless. (a) Formulate the mode-matching method for determining the characteristic impedance, Zo , and effective dielectric constant, εeff , for the quasi-TEM mode. Zo is defined as in Eq. (4.10). (b) Write a computer program to calculate Zo and εeff . Compute and plot Zo and εeff versus a/b from 0.1 to 0.9 for εr1 D 1, εr2 D 10.5, h2 /b D 0.1, 0.5, 4, and 10, t/a D 0, 0.01, 0.02, 0.05, and 0.1, h1 approaching infinity, and f D 0.1, 10, and 20 GHz. This structure is the conventional CPW as shown in Fig. 4.3(a). Compare results at 0.1 GHz to the quasi-static results obtained in Problem 4.12. (c) Compute and plot Zo and εeff versus a/b from 0.1 to 0.9 for εr1 D 12.9, εr2 D 3.9, h2 /b D 0.1, 0.5, 4, and 10, h1 /h2 D 1, 5, 10, and 15, t/a D 0, 0.01, 0.02, 0.05, and 0.1, and f D 0.1, 10, and 20 GHz. 8.7 Derive Eqs. (8.23)–(8.52). 2b 2a

t

er 2

h2

er 3

h1

Figure P8.2

Cross section of a two-layer CPW.

224


8.8 Derive Eq. (8.62) and the coefficients given in Appendix A at the end of this chapter. 8.9 Derive Eqs. (8.67)–(8.88). 8.10 Derive Eqs. (8.89)–(8.96). 8.11 Derive Eqs. (8.97)–(8.110). 8.12 Derive Eq. (8.120). 8.13 Derive Eqs. (8.121)–(8.123) and the inner products given in Appendix B at the end of this chapter. 8.14 Derive Eqs. (8.133)–(8.136). 8.15 Consider the CPW shown in Fig. 8.5. Use the formulation presented in Section 8.2.2 to write a computer program to calculate S11 and S21 of a single-step discontinuity. Compute and plot S11 and S21 versus frequency from 20 to 40 GHz for a D 1.78 mm, t1 D t2 D 3.432 mm, h D 0.254 mm, Wa D 0.1 mm, Sa D 0.8 mm, Wb D 0.6 mm, εr1 D εr3 D 1, and εr2 D 9.6. The orthogonality condition is enforced in the calculations. Also, verify the power conservation described by jS11 j2 C jS21 j2 D 1. 8.16 Repeat Problem 8.15 without enforcement of the orthogonality condition and compare the results to those obtained in Problem 8.15. 8.17 Consider the CPW shown in Fig. 8.5. Use the formulation presented in Section 8.2.3 to write a computer program to calculate S11 and S21 of a double-step discontinuity. Compute and plot S11 and S21 versus the length of the central section for a D 1.4224 mm, t1 D 1.016 mm, t2 D 4.5466 mm, h D 0.127 mm, Wa D Wc D 0.5 mm, Sa D Sc D 0.127 mm, Wb D 0.3 mm, Sb D 0.527 mm, εr1 D 3.7, εr2 D 10, εr3 D 2.2, and f D 10 GHz. The orthogonality condition is enforced. Also, verify the power conservation. 8.18 Reproduce the results in Fig. 8.12. 8.19 Consider again the CPW shown in Fig. 8.5. Use the formulation presented in Sections 8.2.3 and 8.2.4 to write a computer program to calculate S11 and S21 of a triple-step discontinuity. Reproduce the results in Fig. 8.9. Also, verify the power conservation. 8.20 Consider a shielded asymmetrical CPW as shown in Fig. 7.1. The enclosure and ground and central strips are assumed to be perfect conductors. The dielectric substrates are assumed to be lossless. Also, the strip metallization thickness is neglected. Due to the asymmetry in the structure, the CPW ', slot-line (c), and parallel-plate modes will all be excited. Here, however, we assume an equal potential for the two ground planes, so only ' and parallel-plate modes exist in the structure. (a) Formulate an analysis of the eigenmodes using the dynamic SDA. Note that different unknown coefficients are needed for the expansions of

PROBLEMS

225

the electric fields on different slots. The following basis functions can be used:  m' x Gi xi   cos    Si    2 , m D 0, 2, . . .   2 x G x  i i   1   Si E i

x D xm  m' x Gi xi    sin   Si    , m D 1, 3, . . .    2 x Gi xi 2    1 Si  m' x Gi xi   cos    Si    2 , m D 1, 3, . . .   2 x G x  i i  1    Si

i Ezm x D  m' x Gi xi    sin S i   Si    , m D 2, 4, . . .    2 x Gi xi 2    1 Si where y1 D S1 /2 and y2 D W C S1 C S2 /2, and i D 1, 2 denoting the right and left slots, respectively. (b) Formulate the mode-matching method for analyzing the single-step discontinuity shown in Fig. P8.3.

Gb1

Ga1 Wa1

a

b

Wb1 2a

Sa

Sb

Wa2

Wb2

Ga2

Gb2

z=0

Figure P8.3

Single-step discontinuity in the CPW of Fig. 7.1.

226


(c) Write a computer program to calculate S11 and S21 . Compute and plot S11 and S21 versus frequency from 5 to 40 GHz for a D 3.556 mm, h1 D h3 D 3.429 mm, h2 D 0.254 mm, Sa1 D 0.254 mm, Sa2 D 0.381 mm, Wa D 0.889 mm, Ga1 D 1.143 mm, Sb1 D 0.762 mm, Sb2 D 0.635 mm, Wb D 0.381 mm, Gb1 D 0.8255 mm, εr1 D εr3 D 1, and εr2 D 2.2. Compare results with and without the assumption of modal orthogonality. Also, verify the power conservation with and without enforcing the orthogonality. 8.21 Repeat parts (b) and (c) of Problem 8.20 for the double-step discontinuity shown in Fig. P8.4. a D 3.556 mm, h1 D h3 D 3.429 mm, h2 D 0.254 mm, Sa1 D 0.508 mm, Sa2 D 0.762 mm, Wa D 0.508 mm, Ga1 D 1.016 mm, Sb1 D 0.254 mm, Sb2 D 0.381 mm, Wb D 0.889 mm, Gb1 D 1.143 mm, Sc1 D 0.762 mm, Sc2 D 0.635 mm, Wc D 0.381 mm, Gc1 D 0.8255 mm, L D 1.5 mm, εr1 D εr3 D 1, and εr2 D 2.2. 8.22 Repeat parts (b) and (c) of Problem 8.20 for the triple-step discontinuity shown in Fig. P8.5. a D 3.556 mm, h1 D h3 D 3.429 mm, h2 D 0.254 mm, Sa1 D 0.508 mm, Sa2 D 0.762 mm, Wa D 0.508 mm, Ga1 D 1.016 mm, Sb1 D Sd1 D 0.254 mm, Sb2 D Sd2 D 0.381 mm, Wb D Wd D 0.889 mm, Gb1 D Gd1 D 1.143 mm, Sc1 D 0.762 mm, Sc2 D 0.635 mm, Wc D 0.381 mm, Gc1 D 0.8255 mm, L1 D 1.5 mm, L2 D 2 mm, εr1 D εr3 D 1, and εr2 D 2.2. 8.23 Use the computer program developed in Problem 8.21 to calculate S11 and S21 of the double-step discontinuity, as shown in Fig. P8.4, versus the length L of the middle section. a D 3.556 mm, h1 D h3 D 3.429 mm, h2 D 0.254 mm, Sa1 D Sc1 D 0.508 mm, Sa2 D Sc2 D 0.762 mm, Wa D Wc D 0.508 mm, Ga1 D Gc1 D 1.016 mm, Sb1 D 0.254 mm, Sb2 D 0.381 mm, Wb D 0.889 mm, Gb1 D 1.143 mm, εr1 D Gc1

Gb1 Ga1 W b1 Wa1

a

W c1

Sb Sc

Sa 2a

Wa2

Wb2

b

c

Gb2

Ga2

G c2

z=0

Figure P8.4

Wc2

z =L

Double-step discontinuity in the CPW of Fig. 7.1.

PROBLEMS

227

Gc1

G b1

Gd1

Ga1

Sb

Wa1

Wb1

Sc

c

Wc1

Wd1 Sd

Sa

2a

Wc2 Wa2

a

Wb2

d

b

Wd 2 Gd 2

Ga2 Gc 2

Gb2 z=0

Figure P8.5

z = L1

z = L1 + z =L1

Triple-step discontinuity in the CPW of Fig. 7.1.

εr3 D 1, and εr2 D 2.2. Numerical results would indicate that S11 and S21 approach zero and unity, respectively, as L is reduced. Why? 8.24 Consider a suspended strip line as shown in Fig. P8.6. The enclosure and strip are assumed to be perfect conductors. The dielectric substrate is assumed to be lossless. Also, the strip metallization thickness is infinitely thin. (a) Formulate an analysis of the eigenmodes using the dynamic SDA. The following basis functions are used for the current distributions: sin[2m' x a/W] Jxm x D ! 1 [2 x a/W]2 Jzm x D

cos[2' m 1 x a/W] ! 1 [2 x a/W]2

(b) Formulate the mode-matching method for analyzing the single-step discontinuity shown in Fig. P8.7. (c) Write a computer program to calculate S11 and S21 . Compute and plot S11 and S21 versus frequency from 10 to 30 GHz for 2a D h1 D 3.556 mm, h3 D 3.302 mm, h2 D 0.254 mm, Sa D 1.7372 mm, Sb D 1.143 mm, and εr D 2.2. Compare results with and without the assumption of modal orthogonality. Also, verify the power conservation with and without enforcing the orthogonality.

228


y

h3

e0

W

er

h2

e0

h1 x 2a

0

Figure P8.6

Cross section of a suspended strip line.

Wb

Wa

2a

z

Figure P8.7

Suspended strip line’s single-step discontinuity.

APPENDIX A: COEFFICIENTS IN EQS. (8.62)

The coefficients H, I, J, K, L, M, N, and T in Eqs. (8.62) are derived as follows: H1mn D h1n As12 sm cos ˛2n t υmn Wb/4 sin ˛2m t H2mn D h1n As12 sm sin ˛2n t υmn Wb/4 cos ˛2m t H3mn D h2n As12 sm sin ˛2n t C υmn Wb/4˛2m ˇpm 1/jω0 cos ˛2m t


229

H4mn D h2n As12 sm cos ˛2n t υmn Wb/4˛2m ˇpm 1/jω0 sin ˛2m t H5mn D h3n As12 sm cos ˛3n t H6mn D h3n As12 sm sin ˛3n t H7mn D h4n As12 sm sin ˛3n t H8mn D h4n As12 sm cos ˛3n t I1mn D f1n Ac12 sm cos ˛2n t υmn Wb/4ˇpm sin ˛2m t I2mn D f1n Ac12 sm sin ˛2n t υmn Wb/4ˇpm cos ˛2m t I3mn D f2n Ac12 sm sin ˛2n t C υmn Wb/4

˛2m 1 C υm0 cos ˛2m t jω0

I4mn D f2n Ac12 sm cos ˛2n t υmn Wb/4

˛2m 1 C υm0 sin ˛2m t jω0

I5mn D f3n Ac12 sm cos ˛3n t I6mn D f3n Ac12 sm sin ˛3n t I7mn D f4n Ac12 sm sin ˛3n t I8mn D f4n Ac12 sm cos ˛3n t J1mn D h1n As13 sm cos ˛2n t J2mn D h1n As13 sm sin ˛2n t J3mn D h2n As13 sm sin ˛2n t J4mn D h2n As13 sm cos ˛2n t J5mn D h3n As13 sm cos ˛3n t υmn [ W aW/4] sin ˛3m t J6mn D h3n As13 sm sin ˛3n t υmn [ W aW/4] cos ˛2m t J7mn D h4n As13 sm sin ˛3n t C υmn [ W aW/4]˛3m ˇqm 1/jω0 cos ˛3m t J8mn D h4n As13 sm cos ˛3n t υmn [ W aW/4]˛3m ˇqm 1/jω0 sin ˛3m t K1mn D f1n Ac13 sm cos ˛2n t K2mn D f1n Ac13 sm sin ˛2n t K3mn D f2n Ac13 sm sin ˛2n t K4mn D f2n Ac13 sm cos ˛2n t K5mn D f3n Ac13 sm cos ˛3n t υmn [ W aW/4]ˇqm sin ˛3m t

230


K6mn D f3n Ac13 sm sin ˛3n t υmn [ W aW/4]ˇqm cos ˛3m t ˛3m K7mn D f4n Ac13 cos ˛3m t sm sin ˛3n t C υmn [ W aW/4] jω0 ˛3m K8mn D f4n Ac13 sin ˛3m t sm cos ˛3n t υmn [ W aW/4] jω0 N εr R2s 1 ˇs ˛4s S1s 2 L1mn D Tsn Tsn As12 sm ˛2n 2 2 ˇ ˛ R jωε S 4s 1s 0 2s s sD1 Wb 4 Wb ˛2m ˇpm D υmn 4 jω0 N jωε0 εr 2 R2s 3 1 S1s 4 s12 D T ˇ ˛ T s 4s sn sn Asm 2 2 ˇ ˛ R S 4s 1s 2s sD1 s

L2mn D υmn L3mn L4mn

L5mn D

N sD1

2 ˇs 2

εr R2s 5 ˇs ˛4s S1s 6 Tsn Tsn As12 sm ˛3n ˛4s R1s jωε0 S2s

1 2 ˇs 2

jωε0 εr 2 R2s 7 S1s 8 T ˇs ˛4s T As12 sm ˛4s R1s sn S2s sn

2 ˇs 2

εr ˇs R2s 1 2 ˛4s S1s 2 T T Ac12 sm ˛2n ˛4s R1s sn jωε0 S2s sn

L6mn D 0 L7mn D 0 L8mn D

N sD1

M1mn D

N sD1

Wb ˇpm 4 Wb ˛2m D υmn

1 C υm0 4 jω0 N jωε0 εr 1 R2s 3 S1s 4 c12 D ˇ T ˛ T s 4s sn sn Asm 2 2 ˇ ˛ R S 4s 1s 2s sD1 s

M2mn D υmn M3mn M4mn

M5mn D

N sD1

M6mn D 0 M7mn D 0

1 2 ˇs 2

εr R2s 5 2 ˛4s S1s 6 ˇs Tsn Tsn Ac12 sm ˛3n ˛4s R1s jωε0 S2s


M8mn D

N sD1

N1mn D

N sD1

1 2 ˇs 2

jωε0 εr R2s 7 S1s 8 ˇs Tsn ˛4s Tsn Ac12 sm ˛4s R1s S2s

2 ˇs 2

εr R2s 1 ˇs ˛4s S1s 2 T T As13 sm ˛2n ˛4s R1s sn jωε0 S2s sn

1 2 ˇs 2

jωε0 εr 2 R2s 3 S1s 4 T ˇs ˛4s T As13 sm ˛4s R1s sn S2s sn

ˇs2 2

εr R2s 5 ˇs ˛4s S1s 6 Tsn Tsn As13 sm ˛3n ˛4s R1s jωε0 S2s

N2mn D 0 N3mn D 0 N4mn D

N sD1

N5mn D

N sD1

W W a 4 W W a ˛3m ˇqm D υmn 4 jω0 N jωε0 εr 2 R2s 7 1 S1s 8 s13 D T ˇ ˛ T s 4s sn sn Asm 2 2 ˇ ˛ R S 4s 1s 2s sD1 s

N6mn D υmn N7mn N8mn

T1mn D

N sD1

2 ˇs 2

εr ˇs R2s 1 2 ˛4s S1s 2 T T Ac13 sm ˛2n ˛4s R1s sn jωε0 S2s sn

1 ˇs2 2

jωε0 εr R2s 3 S1s 4 ˇs T ˛4s T Ac13 sm ˛4s R1s sn S2s sn

ˇs2 2

εr R2s 5 2 ˛4s S1s 6 ˇs Tsn Tsn Ac13 sm ˛3n ˛4s R1s jωε0 S2s

T2mn D 0 T3mn D 0 T4mn D

N sD1

T5mn D

N sD1

W W a ˇqm 4 W W a ˛3m D υmn

1 C υm0 4 jω0 N 1 R2s 7 S1s 8 jωε0 εr D ˇs Tsn ˛4s Tsn Ac13 sm 2 2 ˇ ˛ R S 4s 1s 2s s sD1

T6mn D υmn T7mn T8mn

231

232


where h1n D

N sD1

h2n D

N sD1

h3n D

N sD1

h4n D

N sD1

f1n D

N sD1

f2n D

N sD1

f3n D

N sD1

f4n D

N sD1

T1sn T2sn

1 2 ˇs 2 1 ˇs2 2 1 2 ˇs 2 1 ˇs2 2 1 2 ˇs 2 1 2 ˇs 2 1 ˇs2 2 1 2 ˇs 2

˛1s ˇs 2 1 1 T C T ˛2n ˛1s sn jωε0 sn

j 2 ωε0 3 Tsn C ˛1s ˇs T2sn ˛1s

˛1s ˇs 6 1 5 T C T ˛3n ˛1s sn jωε0 sn

j 2 ωε0 7 Tsn C ˛1s ˇs T8sn ˛1s

˛1s 2 2 ˇs 1 T C T ˛2n ˛1s sn jωε0 sn

jωε0 ˇs 3 Tsn C ˛1s T4sn ˛1s

˛1s 2 6 ˇs 5 T C T ˛3n ˛1s sn jωε0 sn

jωε0 ˇs 7 Tsn C ˛1s T8sn ˛1s

c12 D 2 As12 sn ˇpn ˇs Asn 1 s12 D

ˇpn Ac12 sn ˇs Asn jω0

s12 T3sn D ˇs Ac12 sn ˇpn ˇs Asn 1 2 c12 T4sn D

ˇs ˇpn As12 sn Asn jω0 c13 T5sn D 2 As13 sn ˇqn ˇs Asn 1 s13 T6sn D

ˇqn Ac13 sn ˇs Asn jω0 s13 T7sn D ˇs Ac13 sn ˇqn Asn 1 2 c13 T8sn D

ˇs ˇqn As13 sn Asn jω0

b As12 D sin ˇs x sin ˇpn x dx sn 0

233

APPENDIX B: INNER PRODUCTS IN EQS. (8.120)–(8.123)

Ac12 sn

b

cos ˇs x cos ˇpn x dx

D 0

Ac13 sn D

W

cos ˇs x cos ˇqn x dx a

Ac13 sn D

W

cos ˇs x cos ˇqn x dx a

υmn is the Kronecker delta function. R2s /R1s and S1s /S2s are obtained from the boundary condition at y D h.


The inner products Iab ij used in Eqs. (8.120)–(8.123) are given as follows:

ab Iij D eai ð hbj Ð dx dy az s

C Iab 2 C Iab 3 ij ij 

1

1 1 sin ˛ ˛ t2 nai nbj a

1

1 1  A 1 Iab 1 D ij na Snb Bnb Mna

1

1 2 nD1 ˛nai ˛nbj D

Iab 1 ij

1

1 1 C A 1 na Snb C Mna Bnb

sin

B8.1



1 ˛ 1 nai C ˛nbj t2 

1 ˛ 1 C ˛ nai nbj

B8.2

 

2

2 1 cos ˛ C ˛ nai nbj h a

2

2 2   A 2 Iab 2 D ij na Snb C Mna Bnb

2

2 2 nD1 ˛nai C ˛nbj

cos C

2 ˛ 2 nai ˛nbj h

2 ˛ 2 nai C˛nbj



ð

sin



2 ˛ 2 ˛ nai nbj h

2 ˛ 2 nai ˛nbj

˛ 2 nai

2˛ 2 nai 2

˛ 2 nbj



2 2

2 2 2  C Ana Tnb Mna Qnb

sin



2 2

2 2 C Pna Snb Rna Bnb ð 

sin



2 ˛ 2 C ˛ nai nbj h 

2 ˛ 2 C ˛ nai nbj

2 ˛ 2 nai C˛nbj h

2 ˛ 2 nai C˛nbj

sin C



2 ˛ 2 nai ˛nbj h 

2 ˛ 2 nai ˛nbj

234




2 2

2 2 C Pna Tnb ð Rna Qnb ð 

cos

˛ 2 nai

˛ 2 nai

D Iab 3 ij

˛ 2 nbj

˛ 2 nbj

1 a

2



cos

2 ˛ 2 nai C ˛nbj h

2 ˛ 2 nai C ˛nbj



h

C

˛ 2 nai

2˛ 2 nbj 2

3

3 3  A 3 na Snb Bnb Mna

˛ 2 nbj

sin

3

3 3 C A 3 na Snb C Mna Bnb

sin

B8.3

3 ˛ 3 ˛ nai nbj t1

3 ˛ 3 nai ˛nbj

nD1

 2 



3 ˛ 3 nai C ˛nbj t1 

3 ˛ 3 nai C ˛nbj

B8.4

Iaa ii

Iaa 1 ii

D s

eai ð hai Ð dx dy az

D Iaa 1 C Iaa 2 C Iaa 3 ij ij ij  1 a  1 1

1 D Ana Sna M 1 na Bna t2 2 nD1 

1 sin 2˛ t 2 nai

1

1 1  C A 1 na Sna C Mna Bna

1 2˛nai

Iaa 2 ii

B8.5

B8.6

 1 a  2 2

2 2 D Ana Sna Pna Tna 2 nD1   cos 2˛ 2 h 2 2 nai 1

2 2 C Mna Bna C Rna Qna  2 

2 2˛nai 2˛nai 

2 sin 2˛ h nai

2

2 2   C A 2 h na Tna Mna Qna

2 2˛nai 

  sin 2˛ 2 h 2 2 nai

2 2  C Pna Sna Rna Bna h C 2˛ 2 nai

B8.7


Iaa 3 ii

235

 1 a  3 3

3 D Ana Sna M 3 na Bna t1 2 nD1 

3 sin 2˛ t 1 nai

3

3 3  C A 3 na Sna C Mna Bna

3 2˛nai

B8.8

where M 1 nai D

1 Snai D

2 Rnai D

M 2 nai D

2 Snai D

T 2 nai D M 3 nai D

3 Snai D

1 1 1

1 A ˛ ˛ C jˇ k B n i 0 nai nai nai ˛2n C ˇi2 1

1

1

1 jˇ k ε A C ˛ ˛ B i 0 r1 n nai nai nai ˛2n C ˇi2 1

2 2

2 ˛ ˛ A jˇ k B n i 0 nai nai nai ˛2n C ˇi2 1

2 2

2 ˛ ˛ P C jˇ k Q n i 0 nai nai nai 2 ˛2n C ˇi 1

2

2

2 jˇ k ε P C ˛ ˛ Q i 0 r2 n nai nai nai 2 ˛2n C ˇi 1

2

2

2 jˇ k ε A C ˛ ˛ B i 0 r2 nai nai n nai ˛2n C ˇi2 1

3 3

3 A ˛ ˛ C jˇ k B i 0 nai nai nai n ˛2n C ˇi2 1

3

3

3 jˇ k ε A C ˛ ˛ B i 0 r3 n nai nai nai ˛2n C ˇi2

B8.9

B8.10

B8.11

B8.12

B8.13

B8.14

B8.15

B8.16

The subscripts a and b refer to the respective regions, while the subscripts i and j denote the eigenmodes i and j. Note that these expressions are derived in the spectral domain to keep the computation consistent with the field solutions of the considered CPW. The same expressions for the I’s can also be used for the double- and multiple-step CPW discontinuities.


Index Analysis methods, 1, 7–9 Analytic function, 87, 89 Anisotropic, 15 Attenuation constant, 66 conductor attenuation constant, 70, 78 coplanar strips, 76 coplanar waveguide, 74 strip line, 77 dielectric attenuation constant, 70, 71 coplanar strips, 76 coplanar waveguide, 74 strip line, 77 Average power, 35, 36, 67, 200 density, 19, 37 Basis functions, 123, 127, 132, 137, 141, 142, 160, 172, 188 Bessel differential equation, 44, 57 Boundary enlargement, 210, 211 reduction, 210, 211 Boundary conditions, 17, 18, 37, 46, 47, 49–51, 57, 138, 153, 154, 163, 164 in the spectral domain, 139, 155, 165 Capacitance per unit length, 65 Cauchy–Riemann equations, 87 Characteristic impedance, 4, 7–8, 20, 64–66, 68, 70, 128, 134 asymmetrical coplanar strips, 111 asymmetrical coplanar waveguide, 110 conductor-backed coplanar waveguide, 73, 108 with upper conducting cover, 108 coplanar strips, 74, 75, 110, 200

coplanar waveguide, 67, 71, 106 with finite ground plane, 109 with upper conducting cover, 107 microstrip line, 67 parallel-coupled strip lines, 112 slot line, 67, 79, 80 strip line, 77, 106 transverse (TE) mode, 26 transverse (TM) mode, 27 Circular resonator, 60 Circular waveguide, 38 Complete integral of the first kind, 72, 73, 103 Conformal mapping, 1, 9, 10, 85, 87, 90 Conformal mapping equations asymmetrical coplanar strips, 111 asymmetrical coplanar waveguide, 110 conductor-backed coplanar waveguide, 108 with upper conducting cover, 108 coplanar strips, 110 coplanar waveguide, 106 with finite ground planes, 109 with upper conducting cover, 107 fundamentals of, 87 parallel-coupled strip lines, 112 strip line, 106 Conjugate functions, 87 Constitutive relations, 14 Continuity equation, 15 Coplanar strips (CPS), 3, 4, 74, 75, 110, 111 asymmetrical coplanar strips, 111 effective dielectric constant of, 74, 75, 110 Coplanar waveguide (CPW), 3, 4, 71, 106–107, 109, 110 conductor-backed, 71, 109, 108 step discontinuities, 203, 204

238

INDEX

CPS, see Coplanar strips CPW, see Coplanar waveguides Cross talk, 5 Cutoff, 21 frequency, 21, 24, 64 wave number, 21, 24, 29, 30, 47 Degenerate modes, 24, 25, 30, 36–38 Depth of penetration, see Skin depth Dielectric, 15 Dielectric constant, 14 complex, 16 Dirac delta function, 40, 44 Dirichlet conditions, 29, 30, 37 Dirichlet problem, 92 Discontinuities, 11 Distortion, 5 Divergence theorem, 13, 33 Dominant mode, 64 Dynamic analysis, 66 approach, 64 parameters, 20 spectral-domain analysis, 162 Dynamic characteristic impedance, 66, 67 Effective dielectric constant, 8, 20, 64–66, 68, 70, 128, 134, 200 asymmetrical coplanar waveguide, 110 conductor-backed coplanar waveguide, 73, 108 with upper conducting cover, 108 conventional coplanar waveguide, 71, 106 coplanar waveguide with finite ground plane, 109 with upper conducting cover, 107 Eigenfunctions, 50, 52 Eigenmodes, 37, 38, 162, 175, 211 Eigenvalues, 50, 52 Electric potential, 21 scalar, 21, 23, 37, 163, 193 vector, 21 Electric wall, 192 Electromagnetic, 12 fields, 17, 20, 23, 31 theory, 1, 9, 12 Electrostatic energy per unit length, 92, 93 Evanescent modes of the shielded coplanar waveguide, 203 Exact solution, 121 Fin line, 2, 4 Finite metallization thickness, 71. See also Finite strip thickness

Finite strip thickness, 69, 73, 77 Fourier series, 187 Fourier transform, 54, 136, 139–141, 154, 155, 164, 186 variable, 139, 186, 187 Fourier-transformed domain, see Spectral domain Full-wave approach, 64 Galerkin equation, 157, 170 Galerkin technique, 153, 163, 171, 188, 190 Gram–Schmidt process, 30 Green’s first identity, 31, 62 Green’s function, 1, 9, 39–41, 44–47, 49, 50–52, 54, 55, 57, 58, 132, 148 closed-form, 44, 46, 49, 58 in the spectral domain, 140, 156, 157 integral-form, 44, 53 series-form, 44, 49, 51, 58 solutions of, 44 Green’s second identity, 29, 62 Green’s theorem, 62 Harmonic conjugate functions, 87 Helmholtz equation, 20, 21, 46, 53 for the electric vector potential, 22 for the magnetic vector potential, 22 Higher order modes, 64 Homogeneous medium, 64 Hybrid fields, 193 Hybrid modes, 24, 26–28, 32, 35, 36, 64 Hybrid wave, 24 Inductance per unit length, 65 Inner product, 157, 158, 170, 189, 210, 211, 213 Inverse elliptic function, 102, 103 Kronecker delta function, 50, 214 Laplace’s equation, 19, 20, 56, 138, 153 in the spectral domain, 139, 155 Laplacian operator, 20, 21, 23 Longitudinal operator, 20 Lorentz condition, 22 Loss, 4, 8, 16, 70 conductor, 71 dielectric, 15, 71 tangent, 16, 17, 71 Magnetic potential, 21 scalar, 21, 23, 26, 30, 37, 163, 193 vector, 21, 22, 41, 42 Magnetic wall, 192

INDEX

Mapping, 85 Maxwell’s equations, 8, 9, 12–15, 17, 20, 21, 33 differential form, 12 integral form, 12 time-harmonic, 14 Method of separation of variables, 52 Microstrip line, 2, 4, 5, 46, 60, 65, 68, 98, 131 inverted, 3, 4 Microwave and millimeter-wave passive structures, 1 Microwave boundary-value problems, 9, 39, 42, 43 Microwave integrated circuits (MIC), 1, 5, 7, 8 Modal orthogonality condition, 214 Mode coupling, 38 analysis of planar transmission lines, 191 analysis of planar transmission line discontinuities, 203 equations, 198, 210 method, 1, 9, 11, 191, 192 types, 24 Mode types, 24 Moment method, 153, 188, 190 Monolithic circuits, 5 Monolithic microwave integrated circuits (MMIC), 1, 7 Multilayer planar transmission lines, 5, 7 Neumann conditions, 29, 30, 37 Neumann problem, 92 Nondegenerate modes, 24, 30, 34, 36–38 Nonisotropic, see Anisotropic Nonmagnetic, 15 Nonorthogonal modes, 30, 32, 38 Nonstationary, 121 Numerical convergence of the mode-matching analysis, 200, 219 Operating frequency, 4, 20 Orthogonal coefficients, 30, 32 Orthogonality conditions, 218 between the scalar potentials, 30 Orthogonality relations, 28, 31, 32, 34, 37, 52. See also Orthogonality conditions Orthogonal modes, 30 Parallel-coupled strip lines, 112 Parallel-plate mode, 76 Parseval’s theorem, 136, 154, 158, 171, 186–188 Passive components, 7 structures, 1

239

Permeability, 14, 15 relative, 14, 15 Permittivity, 14, 15 relative, 14 Phase constant, 66 velocity, 65 Phasor fields, 14 Poynting vector, 19 Planar transmission lines, 1, 4, 5, 7–9, 37, 38, 63 Poisson’s equation, 19, 20, 39–41, 188 Power conservation, 215 flow, 19 handling, 4 orthogonality, 35–38 Power-interaction terms, 38 Poynting vector, 19, 37 Principle of superposition, 24 Propagating mode of the shielded coplanar waveguide, 203 Propagation constant, 21, 24, 28, 47, 66, 206 Quasi-static, 13, 20 approach, 64 spectral-domain method, 137, 138 Quasi-TE modes, 79 Quasi-TE10 , 79 Quasi-TEM mode, 64 Rayleigh–Ritz equation, 157, 170 Rayleigh–Ritz method, 122 Rectangular waveguide, 23, 24, 38, 58 resonator, 51, 53 Relative dielectric constant, 14–17 tensor, 15 Resonant frequency, 53, 120 RF, 1, 17, 20 Ritz method, 122, 127, 132, 137, 141. See also Rayleigh–Ritz method Schwarz–Christoffel transformation, 95, 98, 102 Skin depth, 18, 19 Slot line, 3, 4, 78 Solid-state device, 4, 5, 7 Space domain, 11, 39 Spectral domain, 11, 39, 137 Spectral-domain method, 1, 9, 11, 39, 152 Spectral order, 139, 142, 154, 165 Spectral term, see Spectral order Static, 13, 20, 23 analysis, 64

240

INDEX

Static (Continued) approach, 64 parameters, 20 Stationary, 121, 123 formula, 123 Stoke theorem, 13 Strip line, 2, 4, 61, 76, 106 strip width for, 77 Sturm–Liouville equation, 42–44, 48, 49, 56, 57 operator, 42, 44 Substrates, 1, 5, 7, 15–17 Surface resistivity, 71 Suspended strip line, 2, 4 Taylor’s series, 121 TE, see Transverse electric TE10 , 79 TEM, see Transverse electromagnetic Time harmonic, 14, 20 fields, 14, 15, 17, 21 Time-varying electromagnetic fields, 12, 17 TM, see Transverse magnetic Transformation, 85 Transmission lines, 1, 4, 5, 9, 20, 21, 23, 39 Transverse electric (TE) fields, 26 mode, 24, 26, 28, 32, 35–38, 51, 52, 64 wave, 24 Transverse electric vector potential, 23 Transverse electromagnetic (TEM) mode, 24, 27, 64 wave, 24 Transverse magnetic (TM) fields, 27, 33 mode, 24, 27, 28, 35–38, 64, 67 wave, 24

Transverse magnetic vector potential, 23 Transverse operator, 20 Laplacian, 21, 47 Trial function, 121 True solution, 121 Variational expressions for the capacitance per unit length of transmission lines, 123 upper-bound, 124, 125, 127, 129 lower-bound, 125–127, 131, 148 Variational formula, 123. See also Variational expressions Variational methods, 1, 9, 10, 39, 120 formulation in the space domain, 128 formulation in the spectral domain, 135, 138 formulation using lower-bound expression, 130 formulation using upper-bound expression, 18 fundamentals of, 121 Variational parameters, 121, 123, 127 Wallis formula, 103 Wave equations, 9, 20, 21, 23, 29, 41, 51, 163 for the electric vector potential, 22 for the magnetic vector potential, 22 in the spectral domain, 165 number, 20, 41, 47 solution, 24 types, 23 Waveguides, 23, 24, 28–30, 36, 37 Wavelength of slot line, 66, 79, 80 Wronskian, 46, 48 Zero-order Bessel function of the first kind, 161, 174

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