Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures Cam Nguyen Copyright  2000 John Wiley & Sons, Inc

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

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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 621.381031–dc21 99-086737

Printed in the United States of America.

10987654321 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 1.2 Analysis Methods for Planar Transmission Lines 7 1.3 Organization of the Book 9

2 Fundamentals of Electromagnetic Theory 12 2.1 Maxwell’s Equations 12 2.2 Constitutive Relations 14 2.3 Continuity Equation 15 2.4 Loss in Medium 15 2.5 Boundary Conditions 17 2.6 Skin Depth 18 2.7 Power Flow 19 2.8 Poisson’s and Laplace’s Equations 19 2.9 Wave Equations 20 2.10 Electric and Magnetic Potentials 21 2.11 Wave Types and Solutions 23 2.11.1 Wave Types 23 2.11.2 Wave Solutions 24 2.12 Orthogonality Relations 28 h 2.12.1 Orthogonality Relations Between mnx, y and e Between mnx, y 28 2.12.2 Orthogonality Relations Between Electric Fields and Between Magnetic Fields 31

vii 2.12.3 Orthogonality Relations Between Electric and Magnetic Fields 32 2.12.4 Power Orthogonality for Lossless Structures 35 References 37 Problems 37

3 Green’s Function 39 3.1 Descriptions of Green’s Function 39 3.1.1 Solution of Poisson’s Equation Using Green’s Function 39 3.1.2 Solution of the Wave Equation Using Green’s Function 41 3.2 Sturm–Liouville Equation 42 3.3 Solutions of Green’s Function 44 3.3.1 Closed-Form Green’s Function 44 3.3.2 Series-Form Green’s Function 49 3.3.3 Integral-Form Green’s Function 53 References 56 Problems 56 Appendix: Green’s Identities 62

4 Planar Transmission Lines 63 4.1 Transmission Line Parameters 64 4.1.1 Static Analysis 64 4.1.2 Dynamic Analysis 66 4.2 Microstrip Line 68 4.3 Coplanar Waveguide 71 4.4 Coplanar Strips 74 4.5 Strip Line 76 4.6 Slot Line 78 References 80 Problems 81

5 Conformal Mapping 85 5.1 Principles of Mappings 85 5.2 Fundamentals of Conformal Mapping 87 5.3 The Schwarz–Christoffel Transformation 95 5.4 Applications of the Schwarz–Christoffel Transformation in Transmisison Line Analysis 98 5.5 Conformal-Mapping Equations for Common Transmission Lines 106 References 112 Problems 113

6 Variational Methods 120 6.1 Fundamentals of Variational Methods 121 6.2 Variational Expressions for the Capacitance per Unit Length of Transmission Lines 123 6.2.1 Upper-Bound Variational Expression for C 124 6.2.2 Lower-Bound Variational Expression for C 125 6.2.3 Determination of C, Zo,andεeff 127 6.3 Formulation of Variational Methods in the Space Domain 128 6.3.1 Variational Formulation Using Upper-Bound Expression 128 6.3.2 Variational Formulation Using Lower-Bound Expression 130 6.4 Variational Methods in the Spectral Domain 135 6.4.1 Lower-Bound Variational Expression for C in the Spectral Domain 135 6.4.2 Determination of C, Zo,andεeff 137 6.4.3 Formulation 138 References 142 Problems 143 Appendix: Systems of Homogeneous Equations from the Lower-Bound Variational Formulation 148

7 Spectral-Domain Method 152 7.1 Formulation of the Quasi-static Spectral-Domain Analysis 152 7.2 Formulation of the Dynamic Spectral-Domain Analysis 162 References 176 Problems 177 Appendix A: Fourier Transform and Parseval’s Theorem 186 Appendix B: Galerkin’s Method 188

8 Mode-Matching Method 191 8.1 Mode-Matching Analysis of Planar Transmission Lines 191 8.1.1 Electric and Magnetic Field Expressions 193 8.1.2 Mode-Matching Equations 198 8.2 Mode-Matching Analysis of Planar Transmission Line Discontinuities 203 8.2.1 Electric and Magnetic Field Expressions 203 8.2.2 Single-Step Discontinuity 207 8.2.3 Double-Step Discontinuity 211 8.2.4 Multiple-Step Discontinuity 214 References 221 Problems 222 Appendix A: Coefﬁcients in Eqs. (8.62) 228 Appendix B: Inner Products in Eqs. (8.120)–(8.123) 233

Index 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 self- contained 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 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 Copyright  2000 John Wiley & Sons, Inc. Print ISBN 0-471-01750-7 Electronic ISBN 0-471-20067-0

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. BrandaoQ 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. Popovi´c(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 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

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 3

METAL SLOT

SUBSTRATE SLOT LINE

SUBSTRATE

METAL

INVERTED MICROSTRIP LINE

METAL

SUBSTRATE

COPLANAR WAVEGUIDE

METAL

SUBSTRATE

COPLANAR STRIPS

Figure 1.1 (Continued) 4

TABLE 1.1 Properties of Planar Transmission Lines Shown in Fig. 1.1 Operating Characteristic Solid-State Frequency Impedance Power Device Low-Cost Transmission Line (GHz) Range (Ohm) Dimension Loss Handling Mounting Production Microstrip line Ä110 GHz 10–100 Small High Low Fair Good Strip line Ä60 GHz 20–150 Moderate Low Low Moderate Good Suspended Ä220 GHz 20–150 Moderate Low Low Moderate Fair strip line Fin line Ä220 GHz 20–400 Moderate Moderate Low Easy Fair Slot line Ä110 GHz 60–200 Small High Low Easy Good Inverted microstrip line Ä220 GHz 25–130 Small Moderate Low Moderate Fair Coplanar waveguide Ä110 GHz 40–150 Small High Low Very easy Good Coplanar strips Ä110 GHz 30–250 Small High Low Easy Good 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 ﬂexi- bility, 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 9

Figure 1.7 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 ORGANIZATION OF THE BOOK

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. ORGANIZATION OF THE BOOK 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 spectral- domain 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. 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

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 W ð E D 2.1a ∂t ∂D W ð H D J C 2.1b ∂t W Ð D D 2.1c W Ð B D 0 2.1d

12 MAXWELL’S EQUATIONS 13

Integral Form d E Ð d D B Ð dS 2.2a dt d H Ð d D J Ð dS C D Ð dS 2.2b dt

B Ð dS D 0 2.2c D Ð dS D dv 2.2d

where Ex,y,z,t is electric field or electric field intensity in volts/meter (V/m) Hx,y,z,t is magnetic field or magnetic field intensity in amperes/meter (A/m) Dx,y,z,t is electric flux density in coulombs/meter2 (C/m2) Bx,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 dD J D 2.3 d dt is known as the displacement current density (in A/m2). The time (t) and location (e.g., x, y,andz) 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,weletd/dt equal zero and write the differential-form Maxwell equations as W ð EO D 0 2.4a W ð HO D JO 2.4b W Ð DO D 2.4c W Ð BO D 0 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 DjωB 2.5a W ð H D J C jωD 2.5b W ð D D 2.5c W Ð B D 0 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

Ex,y,z,tD 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 ε0εrωE 2.7

12 where ε0 D 8.854 ð 10 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 0rωH 2.8

7 where 0 D 4 ð 10 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 W Ð J D 2.10a dt

and W Ð J Djω 2.10b

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 W ð H D jωε 1 j E 2.11 ωε or W ð H D jωε1 j tan υE 2.12 16 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

where tan υ D 2.13 ωε

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

where 0 ε D ε D εoεr 2.14b

and ε00 D D ε tan υ2.14c ω

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

2 0 0 e 0.0012 e ′/ ′′/ e e

0.0008 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 Frequency (GHz) εr tan υ@25 C Styrofoam-103.7 3 1.03 0.0001 Rexolite-1422 3 2.54 0.0005 GaAs 10 12.9 0.006 Sapphire 10 9.4–11.5 0.0001 Alumina (96%) 10 8.9 0.0006 Alumina (99.5%) 10 9.8 0.0003 Quartz (fused) 10 3.78 0.0001 Teﬂon 10 2.1 0.0004 Silicon 10 11.9 0.004 RT/Duroid 5880 10 2.2 0.0009 RT Duroid 6010 10 10.2, 10.5, 10.8 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 lumped- element 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 0andJs D 0), between nonperfect dielectrics (Js D 0), and between a perfect dielectric and a perfect conductor. 18 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

Medium 1

e1, m1, s1 n

Medium 2

e2, m2, s2

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

n ð E D 0 2.16a

n ð H D Js 2.16b

n Ð D D s 2.16c n Ð B D 0 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 υ D 2.17 ω

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 D E ð H 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

1 Ł Sav D 2 ReE ð H 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 1 Pav D 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,

r2V 2.22 ε 20 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

Note that acts as a source producing the ﬁelds. When there is no charge, Poisson’s equation reduces to r2V D 0 2.23

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 r2A 2.25

where A is an arbitrary vector, we can then rewrite Eq. (2.24) as

r2E C k2E 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 W2H C k2H D 0 2.27

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 2.29 ∂x ∂y

for the rectangular coordinates, and

∂ Wz D az Dšaz 2.30 ∂z

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

2 2 2 r Drt C 2.31

2 where rt represents the transverse (to the z-axis) Laplacian operator. Substi- tuting Eq. (2.31) into (2.26) and (2.27) we obtain

2 2 rt Ex, y C kc Ex, y D 0 2.32a 2 2 rt Hx, y C kc Hx, y D 0 2.32b

where 2 2 2 2 kc D ωc ε D k C 2.33

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 DjωW ð yh 2.34 22 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

where yhx,y,zis a vector deﬁned as the magnetic vector potential. Substituting Eq. (2.34) into (2.5b) yields

W ð H D k2W ð yh 2.35

from which we can obtain H D k2yh C W˚2.36

where ˚ is an arbitrary scalar function. Now substituting Eq. (2.34) into the left-hand side of (2.5a), we get W ð E Djω WW Ð yh W2yh 2.37

The right-hand side of Eq. (2.5a) becomes, after replacing H by Eq. (2.36),

jωH Djωk2yh C W˚ 2.38

Equating Eqs. (2.37) and (2.38) gives

WW Ð yh r2yh D k2yh C W˚2.39

Choosing W Ð yh D ˚2.40

according to the Lorentz condition then yields

r2yh C k2yh 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 DjωW ð yh 2.42a H D k2yh C WW Ð yh2.42b

Following the same approach, we can also derive the following wave or Helmholtz equation for the electric vector potential yex, y, z as

r2ye C k2ye D 0 2.43

whose solution can be used to determine the magnetic and electric ﬁelds as

E D k2ye C WW Ð ye2.44a 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 ex, y, z and hx, 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 yhx, y, z and yex, y, z into the transverse and longitudinal components as

h h h y D yt C yz

h šhz h šhz D yt x, ye C az x, ye 2.45a e e e y D yt C yz

e šez e šez D yt x, ye C az x, ye 2.45b

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 h 2 h rt yt x, y C kc,hyt x, y D 0 2.46a 2 h 2 h rt x, y C kc,h x, y D 0 2.46b 2 e 2 e rt yt x, y C kc,eyt x, y D 0 2.46c 2 e 2 e rt x, y C kc,e x, y D 0 2.46d where

2 2 2 kc,h D k C h 2.47a 2 2 2 kc,e D k C e 2.47b

h e yt x, y and yt x, y are the transverse magnetic and electric vector potentials, respectively. hx, y and ex, 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 WAVE TYPES AND SOLUTIONS

2.11.1 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 FUNDAMENTALS OF ELECTROMAGNETIC THEORY different electromagnetic field distributions. It is these field distributions that dictate the nature of the waveguides. With reference to a particular wave- propagating 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 0andHz 6D 0. Transverse Magnetic (TM) Wave or Mode: This wave has only the magnetic field in the transverse plane. That is, Hz D 0andEz 6D 0. Hybrid Wave or Mode: This wave is characterized as having both Ez 6D 0and 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 yhx,y,zand yex,y,zhave only longitudinal z components such as

h h šhz y D az x, ye 2.48a

e e šez y D az x, ye 2.48b we can prove easily that the corresponding longitudinal electric and magnetic fields are equal to zero. This implies that the magnetic, hx, y, and electric, ex, 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 k2ye C WW Ð ye jωW ð yh 2.49a

‡ For TEM mode to exist exactly on a transmission line, all conductors must be perfect. WAVE TYPES AND SOLUTIONS 25

H D HTM C HTE D jωεW ð ye C k2yh C WW Ð yh2.49b

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 a k2 eešez C W C a š eešez jω W C a z t z ∂z e t z ∂z

h šhz ð e az 2.50

from which, we obtain the z and transverse components of the electric ﬁeld as

2 e šez Ez D kc,e e 2.51a

šez e šhz h Et Dšee Wt C jωe az ð Wt 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

∂ e ∂ h E Dš ešez jωešhz 2.52a x e ∂x ∂y ∂ e ∂ h E Dš ešez C jωešhz 2.52b y e ∂y ∂x

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

2 h šhz Hz D kc,h e 2.53a

šhz h šez e Ht Dšhe Wt jωεe az ð Wt 2.53b

Further expanding Eq. (2.53b) then yields

∂ h ∂ e H Dš ešhz C jωεešez 2.54a x h ∂x ∂y ∂ h ∂ e H Dš ešhz jωεešez 2.54b y h ∂y ∂x

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 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

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 hx, y. Therefore, to determine the TE ﬁelds, we simply let ex, y equal zero in the foregoing general equations. From Eqs. (2.51) and (2.53), we then have

Ez D 0 2.55a

šhz h Et D jωe az ð Wt 2.55b

2 h šhz Hz D kc,h e 2.55c

šhz h Ht Dšhe Wt 2.55d

Et and Ht are related by Et DšZhaz 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 Ex Ey Zh D D 2.58 Hy Hx

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

∂ h E Djωešhz 2.59a x ∂y ∂ h E D jωešhz 2.59b y ∂x ∂ h H Dš ešhz 2.59c x h ∂x ∂ h H Dš ešhz 2.59d y h ∂y

Note that h is obtained by solving Eq. (2.46b), which is repeated here for completeness: 2 h 2 h rt x, y C kc,h x, y D 0 2.60 WAVE TYPES AND SOLUTIONS 27

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

Hz D 0 2.61a

šez e Ht Djωεe az ð Wt 2.61b

2 e šez Ez D kc,e e 2.61c

šez e Et Dšee Wt 2.61d

Et and Ht are related by az ð Et Ht DÝ 2.62 Ze

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 E Dš ešez 2.63a x e ∂x ∂ e E Dš ešez 2.63b y e ∂y ∂ e H D jωεešez 2.63c x ∂y ∂ e H Djωεešez 2.63d y ∂x

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

2 e 2 e rt x, y C kc,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

2 h rt x, y D 0 2.65 28 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

which is basically Laplace’s equation in the transverse plane. The transverse ﬁelds can be obtained from Eqs. (2.55c) and (2.55d) as

šz h Et D jωe az ð Wt 2.66a šz h Ht Dše Wt 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 ex, y and hx, 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 mnx, y, mnx, y, and both of these potentials, respectively. Following the approach described in Collin [2], we h e can derive the orthogonality relations between mnx, y and mnx, y, between Emnx, y, between Hmnx, y, and between Emnx, y and Hmnx, y.

h e 2.12.1 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 The corresponding scalar potentials mnx, y or klx, y, with i being h or e,

n C 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,andaz form an orthogonal coordinate system. ORTHOGONALITY RELATIONS 29

satisfy the following wave equations:

2 i 2 i rt mn C kc,i,mn mn D 0 2.67a 2 i 2 i rt kl C kc,i,kl kl D 0 2.67b

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

i 2 i i 2 i 2 2 i i klrt mn mnrt kl D kc,i,kl kc,i,mn mn kl 2.68

Taking the surface integral and using Green’s second identity in two dimensions, ∂u ∂v ur2v vr2u dS D u v dl 2.69 t t ∂n ∂n S C

where u and v are arbitrary scalar functions, S is the surface, and C is the closed contour bounding that surface, we obtain ∂ i ∂ i k2 k2 i i dS D i mn i kl dl 2.70 c,i,kl c,i,mn kl mn kl ∂n mn ∂n S 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 Neumann’s Condition: j D 0 2.71a ∂n

e Dirichlet’s Condition: j D 0 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 e mn kl dS D 0,m6D k or n 6D l2.73a S h h mn kl dS D 0,m6D k or n 6D l2.73b S 30 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

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 i described as follows. Let «mn and «kl be the two new modes deﬁned by i i «mn D mn 2.74a i i i «kl D kl C mn 2.74b where C is a constant. These new functions are required to be mutually orthogonal; that is, i i «mn«kl D 0 2.75 S which then leads to i i mn kl dS S C D 2.76 i 2 mn dS S

i 2 provided that the integral mn dS exists and is nonzero. This process can be applied to more than two nonorthogonal modes to determine a new set of modes that are mutually orthogonal. i From Eq. (2.73) and the fact that the norm jj mnjj 6D 0 for any combination of 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 mnx, 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 2.78 i 2 mn dS S ORTHOGONALITY RELATIONS 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 e,mnše,klz e e Et,mn Ð Et,kl dS Dše,mne,kle Wt mn Ð Wt kl dS 2.79 S S Applying the two-dimensional Green’s ﬁrst identity, ∂v W u Ð W v C uW2v dS D u dl 2.80 t t t ∂n S C where u and v are arbitrary functions, and the Dirichlet condition (2.71b), we can rewrite Eq. (2.79) as

e e še,mnše,klz e 2 e Et,mn Ð Et,kl dS DÝe,mne,kle mnrt kl dS 2.81 S S

e Making use of the wave equation for kl,

2 e 2 e rt kl C kc,kl kl D 0 2.82

e e and the fact that mn and kl, with m 6D n and k 6D l, are mutually orthogonal, we finally obtain e e Et,mn Ð Et,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 e e Et,mn Ð Et,kl dS D 0 2.84a S e e Ht,mn Ð Ht,kl dS D 0 2.84b S 32 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

TE Modes h h Et,mn Ð Et,kl dS D 0 2.85a S h h Ht,mn Ð Ht,kl dS D 0 2.85b S

Hybrid Modes e h Et,mn Ð Et,kl dS D 0 2.86a S e h Ht,mn Ð Ht,kl dS D 0 2.86b 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 e e h h Et D CmnEt,mn C CmnEt,mn 2.87a m n e e h h Ht D DmnHt,mn C DmnHt,mn 2.87b m n

e h e h where Cmn, Cmn, Dmn,andDmn 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

e e e Emnx, y, z D Et,mnx, y, z C Ez,mnx, y, z

e e,mnz e e,mnz D emnx, ye C ez,mnx, ye 2.88a e e e Hmnx, y, z D Ht,mnx, y, z C Hz,mnx, y, z

e e,mnz e e,mnz D hmnx, ye C hz,mnx, ye 2.88b ORTHOGONALITY RELATIONS 33

Note that we have introduced the notation e, h and ez, hz to signify the two- dimensional transverse and longitudinal components, respectively. The TM ﬁelds satisfy the Maxwell equations,

e e W ð Ej DjωHj 2.89a e e W ð Hj D jωεEj 2.89b where j D mn, kl. Making use of Eq. (2.89), we can write the following equation:

e e e e e e e e W Ð Emn ð Hkl Ekl ð Hmn D jω Hkl Ð Hmn Hkl Ð Hmn

e e e e C jωε Ekl Ð Emn Ekl Ð Emn D 0 2.90

Separating the W operator in Eq. (2.90) into the transverse and longitudinal parts gives

e e e e e e e e W Ð Emn ð Hkl Ekl ð Hmn D Wt Ð Emn ð Hkl Ekl ð Hmn ∂ C a Ð Ee ð He Ee ð He D 0 2.91 z ∂z mn kl kl mn

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

e e e e Wt Ð Emn ð Hkl Ekl ð Hmn dS D n ð Emn Ð Hkl n ð Ekl Ð Hmndl S C D 0 2.93 since e n ð Ej D 0; j D mn, kl 2.94 on perfectly conducting walls. Substituting Eq. (2.93) into (2.91), we have ∂ a Ð Ee ð He Ee ð He dS D 0 2.95 z ∂z mn kl kl mn S

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

e e e e e,mn C e,kl az Ð emnhkl ekl ð hmn dS D 0 2.96 S 34 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

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 e e to the z direction, then we can write Ekl and Hkl as

e e e,klz e e,klz Ekl D ekle ez,kle 2.97a

e e e,klz e e,klz Hkl Dhkle hz,kle 2.97b

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

e e e e e,mn e,kl az Ð emn ð hkl ekl ð hmn dS D 0 2.98 S

Adding and subtracting Eqs. (2.96) and (2.98) gives e e e e az Ð e,mnekl ð hmn dS D az Ð e,klemn ð hkl dS 2.99a S S e e e e az Ð e,klekl ð hmn dS D az Ð e,mnemn ð hkl dS 2.99b S S

Assuming nondegeneracy between these modes, that is, kl 6D mn,wecanthen obtain directly from Eq. (2.99) e e emn ð hkl Ð az dS D 0 2.100a S e e ekl ð hmn Ð az dS D 0 2.100b S

which is the orthogonality relationship between the transverse electric and magnetic ﬁelds. When the structure is lossless, we can prove that e eŁ emn ð hkl Ð az dS D 0 2.101a S e hŁ emn ð hkl Ð az dS D 0 2.101b S

eŁ e where hkl is the complex conjugate of hkl, assuming the modes are nondegenerate. Similarly, for two nondegenerate TEmn and TEkl modes, we can derive ORTHOGONALITY RELATIONS 35 h h emn ð hkl Ð az dS D 0 2.102a S h h ekl ð hmn Ð az dS D 0 2.102b S h hŁ emn ð hkl Ð az dS D 0 2.103a S h eŁ emn ð hkl 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

e h Emn D Emn C Emn

e jˇe,mnz e jˇe,mnz h jˇh,mnz h jˇh,mnz D emne C ez,mne C emne C ez,mne 2.104a e h Hmn D Hmn C Hmn

e jˇe,mnz e jˇe,mnz h jˇh,mnz h jˇh,mnz D hmne C hz,mne C hmne C hz,mne 2.104b

The total average power ﬂow along the waveguide is given by 1 Ł Pav D 2 Re E ð H Ð az dS S

1 Ł Ł D 2 Re Emn C Ekl ð Hmn C Hkl Ð az dS 2.105 S 36 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

Substituting Eq. (2.104) into (2.105) and expanding the cross products gives 1 e eŁ h hŁ e eŁ h hŁ Pav D 2 Re emn ð hmn C emn ð hmn C ekl ð hkl C ekl ð hkl Ð az dS S Ł Ł 1 e h jˇe,mnˇh,mnz e h jˇe,klˇh,klz C 2 Re emn ð hmne C ekl ð hkl e S Ł Ł e e jˇe,mnˇe,klz h h jˇh,mnˇh,klz C emn ð hkl e C emn ð hkl e Ł Ł e e jˇe,klˇe,mnz h h jˇh,klˇh,mnz C ekl ð hmne C ekl ð hmne Ł Ł C ee ð hh ejˇe,mnˇh,klz C eh ð he ejˇh,mnˇe,klz mn kl mn kl Ł Ł e h jˇe,klˇh,mnz h e jˇh,klˇe,mnz C ekl ð hmne C ekl ð hmne 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 1 e eŁ h hŁ e eŁ h hŁ Pav D 2 Re emn ð hmn C emn ð hmn C ekl ð hkl C ekl ð hkl Ð az dS S e h e h D Pmn C Pmn C Pkl C Pkl 2.107

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

0 Emn D Emn 2.109a 0 Hmn D Hmn 2.109b 0 Ekl D Ekl CEmn 2.109c 0 Hkl D Hkl CHmn 2.109d 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 showninFig.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 hx, y. 2.4 Derive Eqs. (2.51)–(2.54). 2.5 Show that TM modes can be characterized only by the electric scalar potential ex, 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:

∂ h TE Modes: D 0 (Neumann’s Condition) ∂n

TM Modes: e D 0 (Dirichlet’s Condition) 38 FUNDAMENTALS OF ELECTROMAGNETIC THEORY

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 PTM D 1 E ð HŁ Ð dS D 0 TE 2 TEmnn TMmnn S

and PTE D 1 E ð HŁ Ð dS D 0. TM 2 TMmnn TEmnn S 2.16 Prove that, for a lossless waveguide, the orthogonality relationship (2.101) holds for nondegenerate modes. 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

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 x0,y0,z0 . This charge distribution represents a source in the structure and

39 40 GREEN’S FUNCTION

(x′, y′, z′) (x, y, z) R

r′ r

e,m

Figure 3.1 A medium characterized by ε and .

therefore would produce a potential V at every location x, y, z ,whichis governed by Poisson’s equation,

x0,y0,z0 r2Vx, y, z D 3.1 ε

and the boundary conditions of the considered structure. Let Gx, y, z; x0,y0,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 x0,y0,z0 in the medium. This Green’s function must also satisfy Poisson’s equation and corresponding boundary conditions of the structure. That is,

υx, y, z; x0,y0,z0 r2Gx, y, z; x0,y0,z0 D 3.2 ε where υx, y, z; x0,y0,z0 is the Dirac delta function,deﬁnedas

υx, y, z; x0,y0,z0 D υx x0 υy y0 υz z0 1,xD x0; y D y0,z D z0 D 3.3 0, otherwise DESCRIPTIONS OF GREEN’S FUNCTION 41

The solution of Eq. (3.2) is 1 Gx, y, z; x0,y0,z0 D 3.4 4εR 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; x0,y0,z0 x0,y0,z0 dV0 3.5 V0 or, upon using Eq. (3.4), x0,y0,z0 Vx, y, z D dV0 3.6 4εR V0 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 x0,y0,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

r2Ax, y, z C k2Ax,y,z D Jx0,y0,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; x0,y0,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 (x0,y0,z0) in the medium. This Green’s function must also be the solution of the wave equation

2 0 0 0 2 0 0 0 0 0 0 r Gx,y,z; x ,y,z C k Gx,y,z; x ,y,z Dυ x, y, z; x ,y,z aJ 3.8

subject to the structure’s appropriate boundary conditions. aJ is the unit vector of the current at (x0,y0,z0). G can be derived as

ejkR Gx, y, z; x0,y0,z0 D a 3.9 4 R J 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; x0,y0,z0 J x0,y0,z0 dV0 3.10 4 V0 42 GREEN’S FUNCTION which becomes, after using Eq. (3.9), ejkR Ax, y, z D Jx0,y0,z0 dV0 3.11 4 R 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

1 Hx,y,z D W ð Ax, y, z 3.12 1 Ex, y, z D W ð Hx, y, z 3.13 jωε

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:

Ly D fx 3.14 where d d P L D P x C x 3.15 dx 1 dx 2 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 d2 d F D C x C C x C C x 3.17 1 dx2 2 dy 3

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 P1x gives

d2y 1 dP dy P fx C 1 C 2 y D . 2 3 18 dx P1 dx dx P1 P1

Dividing Eq. (3.16) by C1x produces

d2y C dy C Sx C 2 C 3 y D 3.19 2 dx C1 dx C1 C1

In order for Eqs. (3.18) and (3.19) to be equivalent, we set their coefﬁcients equal as

C x 1 dP x 2 D 1 3.20 C1x P1x dx C x P x 3 D 2 3.21 C1x P1x Sx fx D 3.22 C1x P1x which are then solved to obtain C2t P1x D exp dt 3.23 C1t

C3x P2x D P1x 3.24 C1x Sx fx D P1x 3.25 C1x

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: d2y dy x2 C x C x2 2 y D 0 3.26 dx2 dx Using Eqs. (3.23)–(3.25), we can derive t P x D exp dt D expln x D x3.27 1 t2 x2 2 P x D 3.28 2 x and fx D 0 3.29 which, after substituting into Eq. (3.14), gives the equivalent Sturm–Liouville form of the Bessel differential equation, d dy x2 2 x C y D 0 3.30 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 [L C P3x ]y D fx 3.31 where a Ä x Ä b; L is again the Sturm–Liouville operator; and is a constant. The Green’s function Gx; x0 is the solution of the Sturm–Liouville equation corresponding to a unit source and thus must satisfy the equation 0 0 [L C P3x ]Gx; x D υx; x 3.32 where υx; x0 is the one-dimensional Dirac delta function. The Green’s function has the following properties:

1. For any x 6D x0, Gx; x0 satisﬁes the equation

[L C P3x ]G D 0 3.33 SOLUTIONS OF GREEN’S FUNCTION 45

2. Gx; x0 satisﬁes appropriate boundary conditions at x D a and x D b. 3. Gx; x0 is symmetrical with respect to x and x0. 4. Gx; x0 is continuous at x D x0. 0 0 5. The derivative of Gx; x has a discontinuity of magnitude 1/P1x at x D x0.Thatis,

0C dGx 1 D 0 0C 0 0 D G x G x 0 3.34 dx xDx0 P1x Properties 1 and 2 are readily seen, while Properties 3, 4, and 5 can easily be proved. We assume that the function P1x is continuous and nonzero at any point within the interval [a, b]. The discontinuity of the derivative of Gx; x0 in Property 5 thus has a ﬁnite value. We also assume that P1x and P2x are continuous in the interval [a, b]. Once Gx; x0 is found, the solution to Eq. (3.31) can be obtained as b yx D Gx; x0 fx dx0 3.35 a We now ﬁnd the Green’s function by dividing the interval [a, b]intotwo separate regions: [a, x0]and[x0, b].

Region a ≤ x < x Let y1x be a nontrivial solution of the associated homogeneous differential equation of (3.31),

[L C P3x ]y D 0 3.36

which satisﬁes the boundary condition at x D a.Atx 6D x0, Gx; x0 must satisfy

[L C P3x ]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

0 Gx; x D ˛1y1x 3.38

where ˛1 is an unknown constant.

Region x < x ≤ b Similarly, we let y2x be a nontrivial solution of the homogeneous differential Eq. (3.36), which satisﬁes the boundary condition at x D b, and obtain 0 Gx; x D ˛2y2x 3.39

where ˛2 is an unknown constant. Using Property 4, we obtain from Eqs. (3.38) and (3.39)

0 0 ˛2y2x ˛1y1x 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

0 0 0 0 1 ˛2y2x ˛1y1x D 3.41 P1x

Solving Eqs. (3.40) and (3.41) gives

y x0 D 2 ˛1 0 0 3.42 P1x Wx y x0 D 1 ˛2 0 0 3.43 P1x Wx

where 0 0 0 0 0 0 0 Wx D y1x y2x y2x y1x 3.44

0 is called the Wronskian of y1 and y2 at x D x . 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   y x0  D 2 Ä Ä 0  y1x 0 0 ,a x x P1x Wx Gx; x0 D 3.45  y x0  D 1 0 Ä Ä y2x 0 0 ,x x b P1x Wx

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 2 0 0 rt C kc,i Ezix, y D jωiJzx ,y 3.46

and the boundary conditions

Ezi0,y D Ezia, y D 0,y2 [0,b] 3.47a

Ez1x, 0 D Ez0x, b D 0,x2 [0,a] 3.47b SOLUTIONS OF GREEN’S FUNCTION 47

0 Air d W

eO h

1 er

x 0 a

Figure 3.2 Cross section of shielded microstrip line.

2 2 p where kc,i D ki C ' is the cutoff wave number in region i, with ki D ω εii and ' being the corresponding wave number and propagation constant, respec- 2 2 2 2 2 tively. rt D ∂ /∂x C ∂ /∂y represents the transverse Laplacian operator. The Green’s function Gx, y; x0,y0 then satisﬁes

2 0 0 2 0 0 0 rt Gx, y; x ,y C kc,iGx, y; x ,y D υx; x υ y; y 3.48

and the boundary conditions

Gx D 0,y; x0,y0 D h D Ga, y; x0,h D 0 3.49a Gx, y D 0; x0,y0 D h D Gx, b; x0,h D 0 3.49b

We now express Gx, y; x0,y0 as a Fourier series of sine functions that satisfy the boundary conditions along the x axis (i.e., at x D 0, a):

1 mx Gx, y; x0,y0 D g y; x0,y0 sin 3.50 m a mD1

Substituting Eq. (3.50) into (3.48) yields

1 m 2 mx k2 g y; x0,h sin c,i a m a mD1 mx d2g y; x0,h C sin m D υx; x0 υy; y0 3.51 a dy2 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 , mx nx ,/2,mD n sin sin dx D 3.52 0 , , 0,m6D n

leads to d2g y; x0,h 2 mx0 m 2 g y; x0,h D sin υy; h 3.53 dy2 m,i m a a

where m 2 D k2 3.54 m,i a c,i

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 d2g y; x0,h m 2 g y; x0,h D 0 3.55 dy2 m,i m

This solution, which satisﬁes the boundary conditions at y D 0andb,isgiven as

1 1 0 gm D Cmx ,h sinhm,1y , 0 Ä y

0 1 0 0 The Wronskian of gm and gm at y D y D h becomes, after substituting y1 D gm 1 and y2 D gm in Eq. (3.44),

0 0 1 Wx ,h D Cm Cm Qm 3.58

where

Qm D m,0 sinhm,1h coshm,0 d

C m,1 sinhm,0 d coshm,1h 3.59

Comparing Eq. (3.53) with the Sturm–Liouville equation d du P y P y u C P y u D fy 3.60 dy 1 dy 2 3 SOLUTIONS OF GREEN’S FUNCTION 49

we obtain

P1y D P3y D 1

P2y D 0 2 Dm,i 3.61 The closed-form solution of Eq. (3.55) is   y y y h  1 2 Ä Ä  0 , 0 y h 0 P1h Wx ,h gmy; x ,h D 3.62  y h y y  1 2 Ä Ä 0 ,h y b P1h Wx ,h from which we can obtain the solution of Eq. (3.53) as  0  2 mx sinhm,1h sinh[m,0b y ]  sin , 0 Ä y Ä h a a Qm g y; x0,h D m  0  2 mx sinh[m,0b h ]sinhm,1y  sin ,hÄ y Ä b a a Qm 3.63

The Green’s function is then derived, from Eqs. (3.50) and (3.63), as   1 0  2 mx sinhm,1h sinh[m,0b y ]  sin ,  a a Q  mD1 m  0 0 Ä x Ä a, 0 Ä y Ä h Gy : x ,h D  3.64  2 1 mx0 sinh[ b h ]sinh y  m,0 m,1  sin ,  a a Qm  mD1 0 Ä x Ä a, h Ä y Ä b

which can be used to ﬁnd the electric ﬁeld component Ez as a 0 0 0 Ezi D jωi Jzx ,h Gx,y; x ,h dx 3.65 0 We can also derive another closed-form expression for the Green’s function by expanding it as a series of functions that satisfy the boundary conditions along the y axis (i.e., at y D 0andb).

3.3.2 Series-Form Green’s Function We consider again the Sturm–Liouville Eq. (3.31) over the interval [a, b] along with the following (general) mixed boundary conditions: 50 GREEN’S FUNCTION

dyx dya ˛ yx j C ˛ j D ˛ ya C ˛ D 0 3.66 1 xDa 2 dx xDa 1 2 dx dyx dyb ˇ yx j C ˇ j D ˇ yb C ˇ D 0 3.67 1 xDb 2 dx xDb 1 2 dx

where ˛i and ˇi (i D 1, 2) are constants. Accordingly, the Green’s function Gx; x0 must satisfy Eq. (3.32) and the boundary conditions dGa; x0 ˛ Ga; x0 C ˛ D 0 3.68 1 2 dx dGb; x0 ˇ Gb; x0 C ˇ D 0 3.69 1 2 dx

Let ˚nx be the eigenfunctions with the eigenvalues n. ˚nx satisfy the homogeneous differential equation

[L C nP3x ]˚nx D 0 3.70 and the same boundary conditions for Gx; x0 in Eqs. (3.68) and (3.69). Note that ˚nx form a complete set of orthonormal eigenfunctions, and so, for a Ä x Ä b, these functions follow the orthogonality condition b ˚mx ˚nx P3x dx D υmn 3.71 a where 1,mD n υ D 3.72 mn 0,m6D n is the Kronecker delta function. We can therefore let 1 0 0 Gx; x D anx ˚nx 3.73 nD1

where an is the amplitude coefﬁcient. Multiplying both sides of Eq. (3.73) by ˚nx P3x , integrating from a to b, and applying the orthogonality condition (3.71), we get b 0 0 anx D Gx; x ˚nx P3x dx 3.74 a 0 Now multiplying Eq. (3.32) by ˚nx and Eq. (3.70) by Gx; x , subtracting from each other, and taking the integral from a to b gives b 0 0 [˚nx LG x; x Gx; x L˚nx ] dx a b b 0 0 D n Gx; x ˚nx P3x dx C υx; x ˚nx dx 3.75 a a SOLUTIONS OF GREEN’S FUNCTION 51

whose left-hand side vanishes upon using Eqs. (3.73) and (3.70). Equation (3.75) hence becomes b b 0 0 n Gx; x ˚nx P3x dx C υx; x ˚nx dx D 0 3.76 a a

Substituting Eq. (3.74) into (3.76) and solving for an produces

0 ˚nx anx D 3.77 n

where b 0 0 ˚nx D υx; x ˚nx dx 3.78 a

The series-form Green’s function of Eq. (3.73) now becomes, upon substitution of Eq. (3.77), 1 ˚ x0 Gx; x0 D n ˚ x 3.79 n nD1 n

The Green’s function in series form is useful for bounded microwave structures, in which the orthogonal functions ˚nx used to describe the Green’s function represent standing waves, and when the eigenvalue spectrum formed by n is discrete. As an illustration for the formulation of the series-form Green’s function, we consider a rectangular-waveguide resonator whose dimensions along the x, y,and z axes are a, b,andd, respectively. We assume that a current source, represented by the current density Jx, y, z , exists in the resonator and TE modes are excited. The wave equation governing the magnetic ﬁeld component Hz produced by the current can be derived as

∂J x, y, z ∂J x, y, z r2H x, y, z C ω2εH x, y, x D x y 3.80 z z ∂y ∂x

Hz satisﬁes the following boundary conditions:

Hzx, y, 0 D Hzx, y, d D 0 3.81a ∂H ∂H z D z D 0 3.81b ∂x xD0,a ∂y yD0,b

The Green’s function Gx, y, z; x0,y0,z0 is the solution of

r2Gx, y, z; x0,y0,z0 C k2Gx, y, z; x0,y0,z0 D υx; x0 υy; y0 υz; z0 3.82 52 GREEN’S FUNCTION

Let ˚mnpx, y, z represent the orthogonal eigenfunctions, which satisfy the homogeneous differential equation

2 2 r ˚mnpx, y, z C k ˚mnpx, y, z D 0 3.83

and the same boundary conditions as in Eq. (3.81). Using the method of separation of variables, we can write

˚mnpx,y,z D Cmnp˚mx ˚ny ˚py 3.84

where ˚mx , ˚ny ,and˚pz are functions of only x, y,andz, respectively, and Cmnp is a constant. Substituting Eq. (3.84) into (3.83) and solving the resulting equation, subject to the boundary conditions, yields mx ny pz ˚ x,y,z D C cos cos sin 3.85 mnp mnp a b d The eigenvalues are derived as m 2 n 2 p 2 D C C 3.86 mnp a b d

Multiplying Eq. (3.85) by ˚qrs, taking the integral from 0 to a, b,andd,and applying the orthogonality relation a b d ˚mnpx, y, z ˚qrsx, y, z dz dy dx 0 0 0 1; m D q, n D r, p D s D 3.87 0; m 6D q, n 6D r, p 6D s

leads to p 2 2 Cmnp D p 3.88 abd

Substituting Eq. (3.88) into (3.85) gives p 2 2 mx ny pz ˚mnpx, y, z D p cos cos sin 3.89 abd a b d

The Green’s function is now determined from Eq. (3.79), noting that for TE modes, m D 0, 1, 2,...; n D 0, 1, 2,...; p D 1, 2 ...;andm and n cannot be zero simultaneously, as

1 1 1 ˚ x0,y0,z0 ˚ x, y, z Gx, y; x0,y0 D mnp mnp 3.90 k2 2 mD0 nD0 pD1 mnp SOLUTIONS OF GREEN’S FUNCTION 53

or, upon using Eqs. (3.89) and (3.86), p 2 2 1 1 1 Gx, y; x0,y0 D p abd mD0 nD0 pD1 mx0 ny0 pz0 mx ny pz cos cos sin cos cos sin a b d a b d ð m 2 n 2 p 2 ω2ε C C a b d 3.91 0 0 The magnetic ﬁeld component Hz can now be found, using Gx, y; x ,y in Eq. (3.91), as p mx ny pz 1 1 1 cos cos sin 2 2 a b d Hzx, y, z D p abd m 2 n 2 p 2 mD0 nD0 pD1 ω2ε C C a b d a b d ∂J ∂J mx0 ð x y cos 0 0 0 ∂y ∂x a ny0 pz0 ð cos sin dz0 dy0 dx0 3.92 b d

Note that Hz has a singularity at 1 m 2 n 2 p 2 1/2 f D p C C 3.93 r 2 ε a b d

which is known as the resonant frequency of the rectangular-waveguide resonator.

3.3.3 Integral-Form Green’s Function We now discuss another form of the Green’s function, the integral form. This form is useful when the eigenvalue spectrum is continuous. Under this condition, the summation in Eq. (3.79) becomes an integral. The integral-form Green’s function is especially desirable when at least one of the boundary conditions is at inﬁnity. For instance, a source (e.g., a biased transistor) integrated with an antenna is radiating in an unbounded medium. The orthogonal functions ˚nx can also be used to describe the integral-form Green’s function, but here they represent traveling waves. We consider the one-dimensional Helmholtz equation

d2 x C ˇ2 x D fx 3.94 dx 0 54 GREEN’S FUNCTION that describes the field x produced by the source represented by fx .Here, x is assumed to vanish at infinity; that is, the source is radiating in an 0 unbounded medium. ˇ0 is a constant. The Green’s function Gx; x then satisfies

d2Gx; x0 C ˇ2Gx; x0 D υx; x0 3.95 dx2 0 subject to the same boundary conditions of x . Making use of the Fourier transform, we can write 1 1 Gx; x0 D p GˇQ ; x0 ejˇx dˇ 3.96 2 1 and 1 1 GˇQ ; x0 D p Gx; x0 ejˇx dx 3.97 2 1

Gx; x0 represented by Eq. (3.96) can be interpreted as consisting of a continuous spectrum of waves having amplitudes GˇQ ; x0 andtravelinginalossless medium in the Cx direction, which is in essence similar to the series form of Eq. (3.79). Similarly, the function υx; x0 and its Fourier transform, υˇQ ; x0 ,are related by 1 1 υˇQ ; x0 D p υx; x0 ejˇx dx 2 1

1 0 D p ejˇx 3.98 2 and 1 1 υx; x0 D p υˇQ ; x0 ejˇx dˇ 2 1 1 1 1 0 D p p ejˇx ejˇx dˇ 3.99 2 1 2

Substituting Eqs. (3.96) and (3.99) into Eq. (3.95) yields 1 1 2 2 Q 0 1 jˇx0 jˇx p ˇ0 ˇ Gˇ; x p e e dˇ D 0 3.100 2 1 2 from which we obtain 0 1 ejˇx GˇQ ; x0 D p 3.101 2 2 2 ˇ0 ˇ SOLUTIONS OF GREEN’S FUNCTION 55

The Green’s function then becomes, upon substituting Eq. (3.101) into (3.96),

0 1 1 ejˇxx Gx; x0 D p dˇ 3.102 2 2 2 1 ˇ0 ˇ

To illustrate the derivation of this Green’s function, we consider an open, linear, isotropic and homogeneous charge-free medium characterized by ε and . The medium contains a current source Jx0,y0,z0; t . The time-domain wave equation for the electric ﬁeld is derived as

∂2Er; t ∂Jr0; t r2Er; t ε D 3.103 ∂t2 ∂t

where r D x, y, z and r0 D x0,y0,z0 . We now deﬁne the Green’s functions in the time domain, gr r0; t , and frequency domain, GrQ r0; ω , which are related by 1 1 gr r0; t D p GQ r r0; ω ejωt dω 3.104 2 1 1 1 GQ r r0; ω D p gr r0; t ejωt dt 3.105 2 1

gr r0; t satisﬁes ∂2 r2 ε gr r0; t Dυr r0 υt; t0 3.106 ∂t2

where υt; t0 is the unit impulse function, which is derived in the frequency domain as 1 1 υω Q D p υ t; t0 ejωt dt 2 1

1 0 D p ejωt 3.107 2

υt; t0 is then given by, upon using Eq. (3.107), 1 1 υt; t0 D p υω eQ jωt dω 2 1 1 1 0 D ejωtt dω 3.108 2 1

The Green’s function is due to a point source, and so we can use the spherical coordinates and assume spherical symmetry. Equation (3.106) then becomes, 56 GREEN’S FUNCTION

letting R Djr r0j and making use of Eqs. (3.104) and (3.108), 1 1 1 ∂ ∂ ∂2 p R2 ε GQ r r0; ω ejωt dω 2 2 2 1 R ∂R ∂R ∂t 1 1 0 D υ r r0 ejωtt dω 3.109 2 1

This leads to 1 ∂ ∂ υr r0 R2 GQ r r0; ω C ω2εGQ r r0; ω D p 3.110 R2 ∂R ∂R 2

We have chosen t0 D 0. The solution to Eq. (3.110) is

1 ejkR GQ r r0; ω D p 3.111 2 R

Using Eq. (3.111) in (3.104) gives 1 1 1 p gr r0; t D p ejωtR ε dω 3.112 R 2 1

which becomes, after applying Eq. (3.108),

1 p gr r0; t D υt R ε 3.113 R

REFERENCES

1. R. E. Collin, Field Theory of Guided Waves, 2nd edition, IEEE Press, New York, 1991, Chap. 2. 2. C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989, Chap. 14.

PROBLEMS

3.1 Show that the two-dimensional Laplace equation,

∂2V ∂2V C D 0 P3.1 ∂x2 ∂y2 can be converted to two one-dimensional Sturm–Liouville equations. PROBLEMS 57

3.2 In solving microwave boundary-value problems involving spherical coordinates, one often encounters the following partial differential equation: 1 d dR r2 nn C 1 C k2r2 D 0 P3.2 R dr dr where R D f(r), n is an integer, and k is a constant. Convert the above equation to the Sturm–Liouville equation. 3.3 Prove the following properties of the Green’s function G(x; x0): (a) G(x; x0) is symmetrical with respect to x and x0. (b) G(x; x0) is continuous at x D x0. 0 0 0 (c) The derivative of G(x; x ) has a discontinuity of 1/P1(x ) at x D x :

0C dGx 1 D 0 0C 0 0 D G x G x 0 P3.3 dx xDx0 P1x 3.4 Show that the Green’s function associated with the Bessel differential equation d dy n2 x y D fx P3.4 dx dx x and the boundary conditions y0 D y1 D 0isgivenas   1 x n  1 x0 2n ,xx0 2n x when n 6D 0. 3.5 Consider the Sturm–Liouville equation

Ly D fx P3.6

whose domain is [a, b], and the boundary conditions

dyx dya ˛ yx j C ˛ j D ˛ ya C ˛ D 0 P3.7a 1 xDa 2 dx xDa 1 2 dx dyx dyb ˇ yx j C ˇ j D ˇ yb C ˇ D 0 P3.7b 1 xDb 2 dx xDb 1 2 dx

where ˛i and ˇi (i D 1, 2) are constants. The solution to Eq. (P3.6) subject to the boundary conditions (P3.7) is given as 58 GREEN’S FUNCTION b yx D Gx; x0 fx0 dx0 P3.8 a where G(x; x0) is the Green’s function. Derive this equation by performing the following: (a) Show that b b Gx; x0 Lyx dx D Gx; x0 fx dx P3.9 a a

(b) Show that the left-hand side of Eq. (P3.9), upon division into two separate regions [a, x0]and[x0,b], can be written as b x0 b 0 0 0 Gx; x Lyx dx D G1x; x Lyx dxC G2x; x Lyx dx 0 a a x 0 0 0 0 0 0 ∂G1x; x 0 DP x G x ; x y x j 0 yx 1 1 ∂x xDx 0 0 0 0 0 0 ∂G2x; x 0 0 C P x G x ; x y x j 0 yx D yx 1 2 ∂x xDx P3.10

0 where G1 and G2 are the Green’s functions in the intervals [a, x ]and [x0,b], respectively. (c) Deduce Eq. (P3.8) by appropriate changes of variables. 3.6 Consider a rectangular waveguide whose dimensions along the x and y directions are a and b, respectively. Assume that the waveguide is infinitely long and its dielectric is air. Let the source in the waveguide be represented by a current density J(x, y, z). Find the series-form expressions for the Green’s function and the z component of the electric field for TM modes. 3.7 Consider an infinitely long and thin conducting wire enclosed by an infinitely long rectangular conducting pipe (i.e., a rectangular waveguide) as shown in Fig. P3.1. The medium inside the pipe has a permittivity ε. The pipe is held at zero potential. The charge distribution (x, y) along the wire is assumed to be uniform. Find the Green’s function in closed form and the corresponding potential distribution in the medium. The Green’s function may be expressed as a Fourier series of sine functions that satisfy the boundary conditions at x D 0anda. 3.8 Repeat Problem 3.7 using the series-form Green’s function. 3.9 Consider the shielded microstrip line of Fig. 3.2 having a current distribution J on the strip. Derive the closed-form Green’s function and the corresponding electric field component Ez by expressing the Green’s function PROBLEMS 59

V = 0 V = 0 b V = 0

y′ e x 0 V = 0 x ′ a

Figure P3.1 A conducting wire inside a rectangular waveguide.

as a Fourier series of sine functions that satisfy the boundary conditions along the y direction. 3.10 Repeat Problem 3.9 using the Green’s function in series form. 3.11 Consider again the shielded microstrip line of Fig. 3.2 having a current distribution J on the strip. Derive the Green’s function and the magnetic field component Hz in closed form. The Green’s function may be expressed as a Fourier series of cosine functions that satisfy the boundary conditions along the x axis. 3.12 Repeat Problem 3.11 using the Green’s function described as a Fourier series of cosine functions that satisfy the boundary conditions along the y axis. 3.13 Derive Eq. (A3.1) and (A3.2) in the Appendix describing Green’s first identity. 3.14 Derive Eq. (A3.3) and (A3.4) of Green’s second identity. 3.15 Consider a rectangular-waveguide resonator with dimensions a, b,and d along the x, y,andz directions, respectively. Let J(x, y, z)represent the current source in the resonator and assume that only TM modes are excited. Derive the Green’s function in series form and the associated electric field component Ez. 60 GREEN’S FUNCTION

3.16 Consider a circular-waveguide resonator whose radius is a and length along the z direction is d. The resonator is excited by a current source J(r, :,z). For TE modes, derive the Green’s function and the magnetic field component Hz in series form. 3.17 Repeat problem 3.16 for TM modes and electric field. 3.18 A strip line is shown in Fig. P3.2. 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. Derive the Green’s function and the potential distribution in closed form. 3.19 A shielded asymmetrical three-layer microstrip line is shown in Fig. P3.3. The shield and strip are perfect conductors and the strip thickness is infinitesimally thin. The dielectrics are perfect. Derive the closed-form expressions for the Green’s function and the potential distribution.

a er W

Figure P3.2 Cross section of strip line.

h e 1 r1 W

h2 er2

h3 er3

Figure P3.3 Cross section of shielded asymmetrical three-layer microstrip line. PROBLEMS 61

3.20 Consider the following differential equation, which appears in problems involving transmission lines, waveguides, and so on:

d2ϕx C ˇ2ϕx D fx P3.11 dx2 0

2 2 where ˇ0 D ω ε. The boundary conditions are ϕ0 D ϕL D 0. (a) Derive the Green’s function in series form. (b) Derive the Green’s function in closed form at dc. 3.21 A shielded stripline is shown in Fig. P3.4. It is assumed that the shield and strip are perfect conductors, the strip has negligible thickness, and the dielectric is perfect. The charge distribution (x, y) along the strip is uniform. Derive closed-form expressions for the Green’s function and the potential distribution. 3.22 Consider an inﬁnitely long coaxial transmission line, whose inner and outer radii are a and b, respectively. The dielectric between these conductors has a permittivity of ε. The outer conductor is grounded and the inner conductor is held at potential V0. The charge distribution along the inner conductor is assumed to be uniform. Find the Green’s function and potential distribution in closed form. 3.23 Repeat Problem 3.22 using series form. 3.24 Consider a microstrip line shown in Fig. P3.5 having a current distribution J on the strip. It is assumed that the ground plane and strip are perfect conductors, the strip’s thickness is inﬁnitely thin, and the substrate is

a m

W a 2

x 0 b b 2

Figure P3.4 Cross section of shielded strip line. 62 GREEN’S FUNCTION

er h

Figure P3.5 Cross section of microstrip line.

perfect. Derive the integral-form Green’s function and the corresponding electric ﬁeld.

APPENDIX: GREEN’S IDENTITIES

Green’s First Identity urv Ð dS D ur2v Cru Ðrv dV A3.1

S V or ∂v u dS D ur2v Cru Ðrv dV A3.2 ∂n S V where V is a volume and S is the surface enclosing that volume. u and v are two scalar functions. ∂v/∂n is the directional derivative of v along the direction of the normal vector n.

Green’s Second Identity urv vru Ð dS D ur2v vr2u dV A3.3

S V or ∂v ∂u u v dS D ur2v D vr2u dV A3.4 ∂n ∂n S V

Green’s second identity is also called Green’s theorem. 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

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 rejÂ and w D ej,wherer, and Â, 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,inwhichC 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 rejÂ. We assume that point z is related to point w(u, v)inthew plane by the function or transformation w D fz D ln z5.1

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 Â, which indicate that points z on the circle with r D a and 0 Ä Â Ä 2 in the z plane correspond to points w having u D ln a and 0 Ä v Ä 2 in the w plane. The circle of radius a in the z plane is thus mapped onto the line segment u D a,0Ä v Ä 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′

x u z-plane w-plane

Figure 5.1 General mapping from z D x, y into w D u, v D fz .

y v

r 2p

θ a x ln a u

θ z = re j w = ln z = u + jv

Figure 5.2 Mapping between a circle and a line segment. 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 math- ematically 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 FUNDAMENTALS OF CONFORMAL MAPPING

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 f0z D df/dz 6D 0, then the mapping is said to be conformal. Conformal mapping,orconformal 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: ∂u ∂v D 5.2a ∂x ∂y ∂u ∂v D 5.2b ∂y ∂x

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,wheref 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

∂2u ∂2u C D 0 5.3 ∂x2 ∂y2

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

∂2v ∂2v D D 0 5.4 ∂x2 ∂y2 88 CONFORMAL MAPPING

y v

D E jp F

C B A x F′ E′ D′ C′ B′ A′ u

(a) w = e z

y v A

B′ B j

C C ′ A′ x 1 u E′ D D′

z−1 (b) w = z+1

y v

D ′ C′ jp B ′

A B C D E x D ′ E′ A′ B ′ u

z−1 (c) w = ln z+1

Figure 5.3 Some useful mappings. FUNDAMENTALS OF CONFORMAL MAPPING 89

y v

′ ′ ′ ′ D B E D C B A E A 1 x 2 u

l w = z + (d) 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 dw D dz 5.5 dz

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 5.6 dz

from which, we can write dw D aej dz 5.7

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 inﬁnites- imal area in a neighborhood of the point z would be transformed into a similar 90 CONFORMAL MAPPING

y v

C1′ C1

f f C2 C2′

x u

Figure 5.4 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 resem- blance 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 rejÂ ,we can obtain u D ln x2 C y2 D ln r5.8a y v D tan1 D Â5.8b x FUNDAMENTALS OF CONFORMAL MAPPING 91

y v 2 B v u1 A′ B′

v1 A u1 D′ C A

D u2 v1 v 2 C′ x u u2

z plane w plane

Figure 5.5 A mapping of perpendicular curves between the z and w planes.

y v

D′ C′ C q0

D r = r2 r = r 1 ′ ′ q0 A B A B 00 x 0 u1 = ln r1 u2 = 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 A0B0C0D0, the arcs AD and BC into the u-constant segments A0D0 and B0C0, and the straight-line segments DC and AB into the v-constant segments D0C0 and A0B0, respectively. The mapping transforms orthogonal curves, such as AB and BC, into orthogonal curves; such as A0B0 and B0C0, respectively. It also maps the parallel segments AD and BC into the parallel segments A0D0 and B0C0, respectively. It should be noted that, although the segments DC and 0 0 0 0 AB intersect at an angle Â0 6D , the transformed curves D C and A B are in parallel. This is due to the constraints imposed by the AB and CD or the angle Â0. 92 CONFORMAL MAPPING

Had the segments DC and AB been transformed as two independent curves, the 0 0 0 0 transformed curves D C and A B would have intersected at the same angle Â0. 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 boundary- value 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 confor- mally 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.Ifhu, v satisﬁes either of the boundary conditions hu, v D d5.9a

where d is a constant, or dhx, y D 0 5.9b dn along C0, then the function Hx, y D h[ux, y , vx, y ] also satisﬁes the corresponding boundary condition along C;namely,

Hx, y D d5.10a

or dHx, y D 0 5.10b dn This property of conformal mapping implies that, instead of solving the boundary- value 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 two- conductor 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 two- conductor 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 1 2 We D 2 εjEj dS 5.11 S

where E is the electrostatic ﬁeld. Replacing E by WtV, with V being the potential distribution between the two conductors, expanding the Wt operator, and FUNDAMENTALS OF CONFORMAL MAPPING 93

utilizing the Cauchy–Riemann Eq. (5.2a), we can write 1 ∂V 2 ∂V 2 We D ε C dx dy 2 x y ∂x ∂y 1 ∂V 2 ∂u 2 ∂V 2 ∂v 2 D ε C dx dy 2 x y ∂u ∂x ∂v ∂y 1 ∂V 2 ∂V 2 ∂u ∂v D ε C dx dy 2 x y ∂u ∂v ∂x ∂y 1 ∂V 2 ∂V 2 D ε C du dv 5.12 2 u v ∂u ∂v

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

1 2 We D 2 CVo 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.Letfx, y D 0 be the equation describing a curve C in the z plane. Assume that f canalsobe expressed in parametric forms as x D f1t and y D f2t ,wheref1 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 f1aw C jf2aw 5.14

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

x D f1au 5.15a

y D f2au 5.15b

Note that both f1au and f2au are real numbers. These equations indicate that the mapping (5.14) would transform the curve fx, y D 0inthez plane into 94 CONFORMAL MAPPING

the curve v D 0inthew 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 x2 C y2 D r2 5.16 or

x D f1t D r cos t5.17a

y D f2t D r sin t5.17b 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 z D rejaw 5.19

and hence j z w D ln 5.20 a r This mapping function would transform a circle of radius r in the z plane into a segment corresponding to 0 Ä u Ä 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 z 1 w D u C jv D 5.21 1 1 1 z C 1 to obtain x2 C y2 1 u D 5.22a 1 x C 1 2 C y2 2y v D 5.22b 1 x C 1 2 C y2

This mapping transforms the upper plane y>0 onto the upper plane v1 > 0in the w1 plane. Next, we employ the mapping

w D u C jv D ln w1 5.23 THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 95

y v1 v

D′′ C′′ j p B′′

ABC D Ex C′ D′ B′ E′ A′ u1 D′′ E′′ A′′ B′′ u −∞ −11∞ −11−∞ ∞

z plane w1 plane w plane

Figure 5.7 Successive mappings from the half-plane y>0 onto the strip 0 < v <.

jÂ which yields, upon expressing w1 in the polar form of w1 D re ,

u D ln r5.24a v D jÂ 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 THE SCHWARZ–CHRISTOFFEL TRANSFORMATION

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

dw f0z D D Az x k1 z x k2 ÐÐÐz x kN dz 1 2 N N kn D A z xn 5.25 nD1 where A is a complex constant, knn D 1, 2,...,N) are real constants, and xn are real numbers representing points on the x axis and satisfying

v y k4p w4 f 4 k3p

w3 f3 w f k2p N N f 1 w2 kN p f1 x x x x u 1 2 3 n xn+1 xN x k1p w1`

z plane w plane (a) (b)

Figure 5.8 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 f0z can be written as

0 arg f z D arg A k1 argz x1 k2 argz x2 ÐÐÐkN 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 zxn argz xn D 5.28 , z < xn

Assuming the point z traverses to the right through different points xn and letting 0 Ân D arg f z corresponding to xn

Ân D arg A knC1 C knC2 CÐÐÐCkn 5.29a

ÂnC1 D arg A knC2 C knC3 CÐÐÐCkn 5.29b

and hence, ÂnC1 Ân D knC1 5.30

The argument of f0z may also be written as

dw du C jdv arg D arg dz dx dv D tan1 5.31 du THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 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 (ÂnC1 Ân) of Eq. (5.30) measures the polygon’s exterior angle at the vertex wn. Note that the exterior angles are limited within š,andso1

nC1 D ÂnC1 Ân

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 z N n/1 w D fz D A z xn dz C B5.33 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 counter- clockwise 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,andw4 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

h er

Figure 5.9 Cross section of microstrip line with an upper conducting cover. APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 99

w-plane w3 −∞←) h ( w4

er MW − a 0 −∞←) ( w1 MW wa w2 u

(a)

y z-plane

MW −1 MW 0 −∞←) − →∞ x ( x1 xa x2 x3 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 z 1/1 2/1 3/1 4/1 w D A z x1 z x2 z x3 z x4 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 D1andx3 D 0, and rewrite the mapping as A z z z 1/2 w D p z x2 z x3 1 1 dz C B x1x4 xi x4 z dz D A p C B5.35 1 zz C 1 100 CONFORMAL MAPPING p where A1 D A/ x1x4. Multiplying the numerator and denominator of the inte- grand by [z1/2 C z C 1 1/2][z1/2 C z C 1 1/2], we obtain z z1/2 C z C 1 1/2 z1/2 C z C 1 1/2 w D A1 dz C B z1/2 C z C 1 1/2 2 z d z1/2 C z C 1 1/2 2 D A1 C B z1/2 C z C 1 1/2 2 1/2 1/2 D 2A1 ln z C z C 1 C B5.36

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 D1 $ w2 D 0 5.37b

x3 D 0 $ w3 D jh 5.37c

x4 !1$w4 !1Cjh 5.37d

Note also that x1

j Replacing 1 in Eq. (5.38a) by e and solving for A1 gives

h A D 5.39 1

The mapping (5.36) is now given as

2h w D ln z1/2 C z C 1 1/2 C jh 5.40

Using this mapping, we can then write the relation between the two corresponding points z Dxa,wherexa > 0, and w D wa Da as

1/2 1/2 a/2h j/2 jxa C jxa 1 D e e 5.41

from which we can write

1/2 a/2h 1/2 xa 1 D e xa 5.42 APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 101

Taking the square of both sides of Eq. (5.41), we can derive

ea/h C 1 x1/2 D a 2ea/2h ea/2h C ea/2h D 5.43 2

and hence a x D cosh2 5.44 a 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

ε r

− − MW xa 1 MW 0 −∞← (→∞) x x1 x2 x3 x4

(a)

v ′

′ ′ w a w3 ′ va

MW er MW

0 ′ ′ w1 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,andx4 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,andx4 in the z plane as follows: a x Dx Dcosh2 $ w0 D 0 5.45a 1 a 2h 1 0 x2 D1 $ w2 D 1 5.45b 0 0 x3 D 0 $ w3 D 1 C jva 5.45c 0 0 0 x4 !1$w4 D j va 5.45d

We can write the Schwarz–Christoffel transformation for this mapping as z dz w0 D A0 p C B0 5.46 zz C 1 z C xa

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 Dt2, 0 0 1/2 A1 D 2jA xa ,wehave

p z dt w0 D A0 C B0 5.47 1 2 2 0 1 t 1 t /xa

Making use of the inverse elliptic function s dt sn1s, k D , 0 Ä k Ä 1 5.48 0 1 t2 1 k2t2

we can rewrite Eq. (5.47) as 0 0 1 1/2 1/2 0 w D A1 sn z ,xa C B 5.49

Substituting the boundary conditions (5.45a)–(5.45c) into Eq. (5.49), we get

0 1 1/2 1/2 0 A1 sn xa ,xa C B D 0 5.50a 0 1 1/2 0 A1 sn 1,xa C B D 1 5.50b 0 0 B D 1 C jva 5.50c APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 103

1/2 which can then be solved to obtain, upon replacing k D xa ,

jsn11,k v0 D 5.51 a sn1k1,k sn11,k sn11,k is equal to the complete elliptic integral of the ﬁrst kind, Kk ,as 1 dt sn11,k D Kk D 5.52 0 1 t2 1 k2t2

Using Wallis formula /2 1 ð 3 ð 5 ðÐÐÐðn 1 sinn tdt D ,nD even integer 5.53 0 2 2 ð 4 ð 6 ðÐÐÐðn we can derive 1 2 1 ð 3 2 1 ð 3 ð 5 2 Kk D 1 C k2 C k4 C k6 CÐÐÐ 5.54 2 2 2 ð 4 2 ð 4 ð 6 which can be used to compute Kk forp various values of k. To simplify Eq. (5.51), we let k0 D 1 k2 and obtain the complete integral of the ﬁrst kind for the modulus k0 as 1 dt Kk0 D 0 1 t2 1 t2 C k2t2 k1 dt Dj 5.55 1 1 t2 1 k2t2

The inverse elliptic function sn1k1,k is given as k1 dt sn1k1,k D 0 1 t2 1 k2t2 1 dt k1 dt D C 0 1 t2 1 k2t2 1 1 t2 1 k2t2 D Kk jKk0 5.56 upon using Eqs. (5.52) and (5.55). Substituting Eqs. (5.52) and (5.56) into Eq. (5.51) then gives Kk v0 D 5.57 a Kk0 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 0and1surfaces.Asindicated 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

2ε ε Kk0 D 0 r D Cr 0 2ε0εr 5.58 va Kk where a k D sech 5.59a 2h a k0 D 1 k2 D tanh 5.59b 2h

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

0 Kko Co D 2ε0 5.60 Kko where a ko D sech 5.61a 2ho a 0 2 ko D 1 ko D tanh 5.61b 2ho The total capacitance per unit length of the microstrip line can now be obtained as a summation of the capacitances Cr and Co as

0 0 Kk Kko C D 2ε0εr C 2ε0 5.62 Kk Kko The effective dielectric constant is given as

C εeff D 5.63 Ca

where Ca is the capacitance per unit length with the dielectric replaced by air. Using Eq. (5.62) we obtain

Kk0 Kk0 ε C o r Kk Kk ε D o 5.64 eff Kk0 Kk0 C o Kk Kko APPLICATIONS OF THE SCHWARZ–CHRISTOFFEL TRANSFORMATION 105

When h D ho, the effective dielectric constant becomes ε C 1 ε D r 5.65 eff 2 which is independent of the strip width. The characteristic impedance is given as

1 Zo D p 5.66 c CCa

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

60 Zo D 0 0 5.67 p Kk Kko εeff C Kk Kko

The ratio Kk /Kk0 can be determined using Eq. (5.54). It can also be conve- niently evaluated using the following accurate approximation [3]:  p  1 1 C k  2  ln 2 p , 0.5 Ä k Ä 1  1 k Kk D 5.68 Kk0  p , 0 Ä k2 Ä 0.5  0  1 C k  ln 2 p 1 k0

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,whichis 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 0 following conditions for ko and ko:   a h  sech , 0 Ä o < 5 2ho h ko D 5.69a  a h  sech , 5 Ä o 10h h 106 CONFORMAL MAPPING   a ho  tanh , 0 Ä < 5 0 2 2ho h ko D 1 ko D 5.69b  a ho tanh , 5 Ä 10h h

When εr D 1andho D h, the computed results are exact as expected.

5.5 CONFORMAL-MAPPING EQUATIONS FOR COMMON TRANSMISSION LINES

Equations for various commonly used transmission lines, obtained using the conformal-mapping technique are given as follows.

Strip Line (Fig. 5.12) [4] 30 Kk0 Zo D p 5.70a εr Kk W k D tanh 5.70b 4a

Coplanar Waveguide (CPW) (Fig. 5.13) [5]

0 εr 1 Kk Kk1 εeff D 1 C 0 5.71a 2 Kk Kk1 30 Kk0 Zo D p 5.71b εeff Kk a k D 5.71c b sinha/2h k D 5.71d 1 sinhb/2h

Figure 5.12 Cross section of a strip line. CONFORMAL-MAPPING EQUATIONS FOR COMMON TRANSMISSION LINES 107

e r h

Figure 5.13 Cross section of a CPW.

2b e0 ho

er h

Figure 5.14 Cross section of a CPW with an upper conducting cover.

CPW with Upper Conducting Cover (Fig. 5.14) [6]

Kk1 Kk0 ε D 1 C ε 1 1 5.72a eff r Kk Kk C 2 0 0 Kk Kk2 60 1 Zo D p 5.72b ε Kk Kk2 eff C 0 0 Kk Kk2 k D a/b 5.72c sinha/2h k D 5.72d 1 sinhb/2h

tanha/2ho k2 D 5.72e tanh b/2ho 108 CONFORMAL MAPPING

er h

Figure 5.15 Cross section of conductor-backed CPW.

Conductor-Backed CPW (Fig. 5.15) [7]

0 Kk Kk1 1 C εr 0 Kk Kk1 εeff D 0 5.73a Kk Kk1 1 C 0 Kk Kk1 60 1 Zo D p 5.73b ε Kk Kk1 eff C 0 0 Kk Kk1 a k D 5.73c b tanha/2h k D 5.73d 1 tanhb/2h

Conductor-Backed CPW with Upper Conducting Cover (Fig. 5.16) [6] Kk Kk0 ε D 1 C ε 1 5.74a eff r Kk Kk C 1 0 0 Kk Kk1 60 1 Zo D p 5.74b ε Kk Kk1 eff C 0 0 Kk Kk1 tanha/2h k D 5.74c tanhb/2h

tanha/2ho k1 D 5.74d tanhb/2ho CONFORMAL-MAPPING EQUATIONS FOR COMMON TRANSMISSION LINES 109

2b ho

e r h

Figure 5.16 Cross section of conductor-backed CPW with an upper conducting cover.

2b 2a

er h

Figure 5.17 Cross section of a CPW with ﬁnite ground planes.

CPW with Finite Ground Planes (Fig. 5.17) [6]

ε 1 Kk Kk0 D C r 1 εeff 1 0 5.75a 2 Kk Kk1

0 30 Kk1 Zo D p 5.75b εeff Kk1 sinha/2h 1 sinh2b/2h / sinh2c/2h k D 5.75c sinhb/2h 1 sinh2a/2h / sinh2c/2h a 1 b/c 2 k D 5.75d 1 b 1 a/c 2 110 CONFORMAL MAPPING

W S

h er

Figure 5.18 Cross section of an asymmetric CPW.

Asymmetrical CPW (Fig. 5.18) [8] ε 1 Kk Kk D C r 1 εeff 1 0 0 5.76a 2 Kk Kk1 60 Kk D Zo p 0 5.76b εeff Kk S k D 5.76c W C S t1 t2 1 t2 k1 D 5.76d t1 C t2 1 C t2

edi 1 tii D 1, 2 D 5.76e edi C 1 2W S d D C 5.76f 1 2 h h S d D 5.76g 2 2h

Coplanar Strip (CPS) (Fig. 5.19) [5]

0 εr 1 Kk Kk1 εeff D 1 C 0 5.77a 2 Kk Kk1 120 Kk D Zo p 0 5.77b εeff Kk a k D 5.77c b sinha/2h k D 5.77d 1 sinhb/2h CONFORMAL-MAPPING EQUATIONS FOR COMMON TRANSMISSION LINES 111

er h

Figure 5.19 Cross section of a CPS.

S W W1 2

e r h

Figure 5.20 Cross section of an asymmetric CPS.

Asymmetrical CPS (Fig. 5.20) [8] ε 1 Kk Kk D C r 1 εeff 1 0 0 5.78a 2 Kk Kk1 60 Kk D Zo p 0 5.78b εeff Kk 1 k D 1 5.78c 1 C S/W1 1 C S/W2 t1 t2 t3 t2 k1 D 5.78d t1 C t2 t3 C t2

edi 1 tii D 1, 2, 3 D 5.78e edi C 1 2W S d D 2 C 5.78f 1 2 h h S d D 5.78g 2 2h 2W S d D 1 C 5.78h 3 2 h h 112 CONFORMAL MAPPING

S W W1 2 h er

Figure 5.21 Cross section of parallel-coupled strip lines.

Parallel-Coupled Strip Lines (Fig. 5.21) [9] Even-mode characteristic impedance:

0 30 Kke Zoe D p 5.79a εr Kke

Odd-mode characteristic impedance:

0 30 Kko Zoo D p 5.79b εr Kko W W C S k D tanh tanh 5.79c e 2 b 2 b W W C S k D tanh coth 5.79d o 2 b 2 b

In all these equations, the complementary modulus k0 is obtained from the modulus k by k0 D 1 k2 5.80

REFERENCES

1. R. E. Collin, Field Theory of Guided Waves, 2nd edition, IEEE Press, New York, 1991, pp. 259–273. 2. R. V. Churchill, J. W. Brown, and R. F. Verhey, Complex Variables and Applications, 3rd edition, McGraw-Hill, New York, 1976. 3. W. Hilberg, “From Approximations to Exact Relations for Characteristic Impedance,” IEEE Trans. Microwave Theory Tech., Vol. MTT-17, pp. 259–265, May 1969. 4. S. B. Cohn, “Characteristic Impedance of Shielded Strip Transmission Line,” IRE Trans. Microwave Theory Tech., Vol. MTT-2, pp. 52–55, July 1954. PROBLEMS 113

5. 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. 6. G. Ghione and C. Naldi, “Coplanar Waveguides for MMIC Applications: Effect of Upper Shielding, Conductor Backing, Finite-Extent Ground Planes, and Line-to-Line Coupling,” IEEE Trans. Microwave Theory Tech., Vol. MTT-35, No. 3, pp. 260–267, Mar. 1987. 7. G. Ghione and C. Naldi, “Parameters of Coplanar Waveguides with Lower Ground Plane,” Electron. Lett., Vol. 19, No. 18, pp. 734–735, Sept. 1983. 8. R. K. Hoffmann, Handbook of Microwave Integrated Circuits, Artech House, Dedham, MA, 1987. 9. S. B. Cohn, “Shielded Coupled-Strip Transmission Line,” IRE Trans. Microwave Theory Tech., Vol. MTT-3, pp. 29–38, Oct. 1955.

PROBLEMS

5.1 Derive the Cauchy–Riemann equations, (5.2a) and (5.2b), where u and v are the respective real and imaginary parts of an analytical function w D fz with z given as z D x C jy. 5.2 Consider a coaxial transmission line with the respective inner and outer conductors’ radii of a and b. The relative dielectric constant of the medium between the two conductors is εr. The conductors are assumed to be perfect, and the medium is lossless. The potentials on the inner and outer conductors are Vo and zero, respectively. Using the conformal mapping deﬁned by the logarithmic function

w D A ln z C B

and choosing its real part u as the potential function, derive the potential distribution, electric and magnetic ﬁelds, capacitance per unit length, and characteristic impedance of the transmission line. Compare the derived characteristic impedance to the well-known equation:

60 b Zo D p ln εr a

5.3 Repeat Problem 5.2 with the function’s imaginary part v chosen as the potential function. 5.4 Use the mapping function w D ln z to transform the region in the z plane shown in Fig. P5.1 onto another region in the w plane. 5.5 Prove that, under a conformal mapping, the two families of curves in the z plane corresponding to the curves u D constant and the curves v D constant in the w plane form an orthogonal system. 114 CONFORMAL MAPPING

r r1 2 x

Figure P5.1

5.6 Find a transformation that maps a curve fx, y D 0inthez plane into the curve u D 0inthew plane. Use this mapping to transform a circle of radius r in the z plane into the w plane. 5.7 Consider an ellipse described by

x2 y2 C D r2 a2 b2

where a, b,andr are positive real constants. Determine the transform that maps this ellipse into the curve v D 0inthew plane. 5.8 Verify the mappings that transform the upper plane y>0 onto the strip 0 < v <, including the correspondence between the marked points and boundaries in Fig. 5.7. 5.9 Consider the following Schwarz–Christoffel transformation: z dz w D j p 0 zz C 1 z 1 PROBLEMS 115

Show that this transformation maps the x axis onto the square with vertices w1 D ja, w2 D 0, w3 D a,andw4 D a C ja,wherea is a positive constant given as 1 1 1 a D 2 B 4 , 2

with Bx, y being the beta function of two real variables x and y given by 1 Bx, y D tx11 t y1 dt, x > 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

w2 w1 ( →∞) h

0 a →∞) w3 wa w4 ( u

(a)

y z-plane

0 1 −∞←) →∞) x ( x1 x2 x3 xa x4 (

(b)

Figure P5.2

W er

h 2

Figure P5.3 Cross section of an asymmetric strip line. 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 assumedtobeinfinitely 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 assumedtobeinfinitely 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 1 er 2

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. 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

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 121

6.1 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.Letx 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 d2f fx D fx C x x C 0 0 2 dx xDx 2! dx xDx 0 0 n 2 1 d f n ð x x0 CÐÐÐC x x0 C RnC1 6.1 n dxn ! xDx0

from which we obtain 2 df 1 d f 2 f fx fx0 D x x0 C x x0 CÐÐÐ 6.2 dx dx2 xDx0 2! xDx0 or f D υfx C υ2fx CÐÐÐ 6.3 where υf is called the ﬁrst variation of f or the ﬁrst-order error, and υ2f 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 0for x 6D x0,andnonstationary 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,wherex0 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[u1x1,x2,...,xN, u2x1,x2,...,xN,...,uNx1,x2, ...,xN]. The variational method then assumes trial functions for uii 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) • • f(x1)

• • • • x x x x 0 1 x 0 1 x (a) (b)

y = f(x)

• • f(x1)

•

• • x x 0 1 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 smallererroriny 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

N ux D aiuix 6.4 iD1 VARIATIONAL EXPRESSIONS FOR THE CAPACITANCE 123

where ai,iD 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 uix 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 uix 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 uix should be selected to form a complete set of orthogonal functions or polynomials. Once the uix 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,iD 1, 2,...,N 6.5 ∂ai

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 1 Ł We D 2 ε0εr E Ð E dx dy 1 D 2 ε0εr Wt Ð Wtdxdy ∂ 2 ∂ 2 D 1 ε ε C dx dy 6.6 2 0 r ∂x ∂y 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

1 2 We D 2 CV 6.7

where S2 V D 2 1 D Wt Ð d6.8 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.

f 2 S2

f 1 S1

Figure 6.2 General two-conductor transmission line. 1 and 2 are the potentials on conductors S1 and S2, respectively. VARIATIONAL EXPRESSIONS FOR THE CAPACITANCE 125

Equating Eq. (6.6) to (6.7) and making use of Eq. (6.8) gives 2 Wt dx dy C D ε0εr 6.9 S2 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 upper- bound 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 (per- unit-length) charge distribution x0,y0 located at an arbitrary location x0,y0 on 0 0 conductor S2. For a unit charge at x ,y, the resulting potential is basically the Green’s function Gx, y; x0,y0 associated with the transmission line, as discussed

C(f )

f 0 o f

Figure 6.3 Possible values of C calculated using the upper-bound variational expression. ThetruevalueCo represents an absolute minimum. 126 VARIATIONAL METHODS

in Chapter 3. Note that this Green’s function is the (potential) response due to 0 0 a unit (charge) source located at x ,y 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 x0,y0 0 0 at a location x ,y on conductor S2 is given as 0 0 0 0 0 x, y D Gx, y; x ,ysx ,yd 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.Thatis, 0 0 0 0 0 Vo D Gx, y; x ,ysx ,yd 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 C D D 6.12 Vo QVo

where

Q D sx, y d 6.13 S2

denotes the total charge on S2. Substituting Eqs. (6.10) and (6.13) into Eq. (6.12), we have Gx, y; x0,y0 x, y x0,y0dd0 1 s s D S2 S2 6.14 C 2 sx, y d 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. VARIATIONAL EXPRESSIONS FOR THE CAPACITANCE 127

C(r)

0 ro r

Figure 6.4 Possible values of C calculated using the upper-bound variational expression. ThetruevalueCo represents an absolute maximum.

6.2.3 Determination of C, Zo,andeeff 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 ,andlety 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 aiuix 6.15 iD1

where uix, i D 1, 2,...,N, are basis functions. The ai are the unknown variational parameters. Once the uix 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,iD 1, 2,...,N 6.16 ∂ai 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 6.17 c CCa

and C εeff D 6.18 Ca

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,andεeff 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 per- unit-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 V r D o ln 6.19 o lna/b b

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

2%ε ε C D 0 r 6.20 o lnb/a FORMULATION OF VARIATIONAL METHODS IN THE SPACE DOMAIN 129

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 2 Wt dx dy

C D ε0εr 2 6.21 S2 Wt Ð d S1

Using the Ritz method described in Section 6.1, we ﬁrst assume a trial function for as 2 r D a0 C a1r C a2r 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 2a2r ar 6.23

Substituting Eq. (6.23) into (6.21) gives 2% b 2 a1 C 2a2r rdrdÂ 0 a C D ε0εr b 2 a1 C 2a2r dr a b2 a2 b3 a3 2%ε ε a2 C 4 a a C b4 a4a2 0 r 2 1 3 1 2 2 D 6.24 2 2 2 2 2 2 2 2 b a a1 C 2b a b a a1a2 C b a a2 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 andleadtoa system of homogeneous equations, 2C 2C 8 b a2 2b2 a2 a C b2 a2b a b3 a3 a D 0 %ε 1 %ε 3 2 2C 8 2C b2 a2b a b3 a3 C b2 a2 4b4 a4 a D 0 %ε 3 %ε 2 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%ε0εr, which is only 0.1% different from the exact value of 2.886%ε0εr 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

2 Vo D a0 C a1b C a2b 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, FORMULATION OF VARIATIONAL METHODS IN THE SPACE DOMAIN 131

er1 h1 S W

er 2 h2 b

er 3 h3

x 0 a

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 two- dimensional trial function for the potential . On the other hand, only a one- dimensional 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 Gx; x0 x x0dxdx0 1 s s D S 6.28 C 2 sx dx 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. sx 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; x0 is the Green’s function, which is symmetric in x and x0 due to reciprocity. Equation (6.28) is valid for any transmission line whose strips are inﬁnitesi- mally 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]

1 0 0 Gx; x D gngxgx 6.29 nD1

where 2 gn D 6.30 n% Y

gx D sin˛nx 6.31 0 0 gx D sin˛nx 6.32 n% ˛ D ,nD 1, 2,... 6.33 n a

εr1 coth˛nh1 C εr2 tanh˛nh2 Y D ε0εr2 C ε0εr3 coth˛nh36.34 εr2 C εr1 coth˛nh1 tanh˛nh2

εri and hi,iD 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

N sx D aisix 6.35 iD1

where ai,iD 1, 2,...,N, are the unknown variational parameters, and six are the basis functions, chosen to approximate the charge density as   x S  cos i 1%  W  ,SÄ x Ä S C W 2 six D  2x S W 6.36  1  W  0, otherwise

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 FORMULATION OF VARIATIONAL METHODS IN THE SPACE DOMAIN 133

i =1 i =3

x SS+W −1 W S + 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 N 1 1 g SQ SQ Q Q a D 0; j D 1, 2,...,N 6.37 n ni nj C i j i iD1 nD1 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 Qi D six dx 6.38 S

Upon substitution of the charge basis function six from Eq. (6.32), Qi becomes %W i 1% i 1% Q D J cos 6.39 i 2 0 2 2 134 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 gxsix dx 6.40 S

Substituting Eqs. (6.31) and (6.36) into Eq. (6.40), we obtain, after several manip- ulations, %W ˛ W C i 1% ˛ W C i 1% SQ D J n sin ˛ S C n ni 4 0 2 n 2 ˛ W i 1% ˛ W i 1% C J n sin ˛ S C n 6.41 0 2 n 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,iD 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 6.42 c CCa C εeff D 6.43 Ca

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,iD 1, 2,...,N, from which we can solve for ai. Using the results for ai we can calculate C obtained in VARIATIONAL METHODS IN THE SPECTRAL DOMAIN 135

Eq. (A6.5) as 1 N 2 gn aiSQni 1 D nD1 iD1 6.45 C N 2 aiQi iD1

Once C is determined, we can of course compute the characteristic impedance and effective dielectric constant. If a simple expression such as 2 Ä Ä C x D x ,S x S W 6.46 s 0, otherwise

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; x0,y0. 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 VARIATIONAL METHODS IN THE SPECTRAL DOMAIN

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 areassumedtobezeroandVo, respectively. The lower-bound variational expression for C in the space domain given in Eq. (6.14) can be rewritten as 136 VARIATIONAL METHODS x, yx, y d 1 s D S2 6.47 C Q2 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 f˛,Q y D fx, yej˛x dx 6.48 1 where ˛ is the Fourier transform variable and the tilde ¾ indicates the Fourier- transformed 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 fx, ygŁx, y dx D f˛,Q ygQŁ˛, y d˛ 6.49 1 2% 1 to the right-hand side of Eq. (6.47) leads to 1 1 1 D Q ˛, y˛,Q y d˛ 6.50 2 s C 2%Q 1 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 2 a fQ ˛n,y D fx, y sin ˛nxdx 6.51 a 0

where ˛n D n%/a,nD 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 a a 1 fx, ygŁx, y dx D fQ ˛ ,y gQŁ ˛ ,y 6.52 2 n n 0 nD1 and Eq. (6.50) becomes

1 a 1 D Q ˛ ,yQ ˛ ,y 6.53 C 2Q2 s n n nD1 Similar to the case of the (space-domain) Green’s function described in Sec- tion 6.2.2, the Green’s function in the spectral domain, G˛Q , is deﬁned as the VARIATIONAL METHODS IN THE SPECTRAL DOMAIN 137

resultant potential due to a unit charge located on conductor S2 and so may be written as Q ˛, y G˛Q s D ˛,Q y 6.54 ε0 Equation (6.50) can now be rewritten, upon making use of Eq. (6.54), as 1 1 1 D Q 2˛, yG˛,Q y d˛ 6.55 2 s C 2%ε0Q 1

For the case of the three-layer shielded microstrip line shown in Fig. 6.6, using Eq. (6.54) in (6.53) gives

1 a 1 D Q 2˛ ,yG˛Q ,y 6.56 C 2ε Q2 s n n 0 nD1

Both Eqs. (6.55) and (6.56) are stationary. Because these equations are derived directly from the lower-bound variational formula, they also represent lower- bound 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,andeeff 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 ¾ sx D aisix 6.57 iD1

where ix, 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

N ¾ Q s˛ D aiQ 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 x, y ∂2 x, y i C i D 0,iD 1, 2, 3 6.59 ∂x2 ∂y2 where ix, y is the unknown potential in the ith region. The boundary conditions for the considered structure are

i0,yD ia, y D 0 6.60

1x, b D 3x, 0 D 0 6.61 U 1x, h2 C h3 D 2x, h2 C h3 D V x 6.62 x, h D x, h 6.63 2 3 3 3 ∂2 ∂1 x εr2 εr1 D 6.64 ∂y yDh Ch ∂y yDh Ch ε0 2 3 2 3 ∂3 ∂2 εr3 εr2 D 0 6.65 ∂y ∂y yDh3 yDh3

where ε0 is the free-space permittivity and εri,iD 1, 2, 3, is the relative dielectric constant of the ith layer; VUx denotes the potential at the upper interface, where VARIATIONAL METHODS IN THE SPECTRAL DOMAIN 139

the strip resides, and can be expressed as

U V x D Vox C Vx 6.66

Vox 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 problem, we deﬁne the Fourier transform f˛Q n,y of fx, y as 2 a f˛Q n,yD fx, y sin ˛nxdx 6.67 a 0

where ˛n D n%/a,nD 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 ix, y are required to be zero at x D 0anda.Ascanbe veriﬁed, the choice for the Fourier transform variable ˛n will cause the boundary conditions of Eq. (6.60) on ix, 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

∂2Q i ˛2 Q D 0 6.68 ∂y2 n i

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,bD Q 3˛n, 0 D 0 6.69 U Q 1˛n,h2 C h3 D Q 2˛n,h2 C h3 D VQ ˛n6.70 Q ˛ ,h D Q ˛ ,h 6.71 2 n 3 3 n 3 ∂Q ∂Q Q ˛ 2 1 s n εr2 εr1 D 6.72 ∂y ∂y ε0 yDh Ch yDh Ch 2 3 2 3 ∂Q ∂Q ε 3 ε 2 D 0 6.73 r3 ∂y r2 ∂y yDh3 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

U VQ ˛n D VQo˛n C V˛Q n6.74 140 VARIATIONAL METHODS

The solution of Eq. (6.68), which satisﬁes the boundary conditions (6.69)–(6.73), is derived in Chapter 7 as

U G˛Q nQ s˛n D VQ ˛n6.75

where εr3 tanh ˛nh1 tanh ˛nh2 C tanh ˛nh3 εr2 G˛Q n D εr1 ε0˛n tanh ˛nh2 εr3 C tanh ˛nh1 coth ˛nh2 εr2 6.76

C tanh ˛nh3εr1 coth ˛nh2 C εr2 tanh ˛nh1

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

U VQ ˛n D VQ 0˛n C V˛Q n6.77

VQ0˛n and V˛Q n stand for the Fourier transforms of Vox and Vx,respectively. Vox is chosen and is thus known. Vx is given as 2 a V˛Q n D Vx sin ˛nxdx 6.78 a 0

with ˛n D n%/a,nD 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 spectral- domain method described in Section 7.2. The reciprocal of the per-unit-length capacitance, 1/C, is derived in Eq. (6.56) as

1 a 1 D Q 2˛ ,yG˛Q ,y 6.79 C 2ε Q2 s n n 0 nD1

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

Q D sx dx 6.80 S VARIATIONAL METHODS IN THE SPECTRAL DOMAIN 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

N ¾ x D aiix 6.81 iD1

where ix, 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,

N ¾ ˛Q n D aiQ i˛n6.82 iD1

The basis functions used here are   x S  cos i 1%  W  ,SÄ x Ä S C W 2 ix D  2x S W 6.83  1  W  0, otherwise

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 Q ˛ D [sink C k J k C k C sink k J k k ] 6.84 si n 2 a 1 2 0 1 3 2 1 0 1 3 % k D j 1 6.85 1 2 W k D ˛ S C 6.86 2 n 2 W k D ˛ 6.87 3 n 2

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 1 ai Q si˛nG˛Q nQ sj˛n D 0 6.88 iD1 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,iD 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 a N 1 1 D a a Q ˛ G˛Q Q ˛ 6.89 C 2ε Q2 i j si n n sj n 0 iD1 jD1 nD1

Once C is determined, it can then be used to calculate the characteristic impedance and effective dielectric constant as

1 Z0 D p 6.90 c CCa

and C εeff D 6.91 Ca

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,iD 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 2 2 x D x ,x strip s 0, otherwise

(c) Calculate and plot Zo versus the normalized strip width, W/a, from 0.1to5forεr 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,usinga variational method, assuming the following charge distribution:   a W a C W A, Ä x Ä sx D  2 2 0, 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/6andy D h. 6.11 Repeat Problem 6.10, parts (a) and (b), using  1 a W a C W   , Ä x Ä jx a/2j 2 2 2 sx D 1  W/2  0, 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 1 ej˛x J0˛ D p dx % 1 1 x2

6.13 Prove that both Eqs. (6.55) and (6.56) are stationary. PROBLEMS 145

h2 er 2

e r1 h1

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:   1 W  ,xÄ jxj 2 2 sx D 1   W/2 0, 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: x, S Ä x Ä S C W x D s 0, 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 Prob- lem 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,of2.2and 10.5 and a normalized substrate thickness, h/b,of0.1,0.5,4,and10. 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 Pro- blem 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 147   x C b  cos i 1%   b a 2 , b Ä x Äa s1ix D  2x C b b a  1  b a  0, otherwise   x a  cos i 1%   b a 2 ,aÄ x Ä b s2ix D  2x a b a  1  b a  0, 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,of0.1, 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   x C W  cos i 1%   2W 2 , W Ä x Ä W six D  x C W W  1  W  0, 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:

1 sx D , W 2 W 2 % x G S 2 1 1 2

G1 C S1

(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; x0 x x0dxdx0 1 s s D S A6.1 C 2 sx dx S

Here we assume the transmission line has an inﬁnitely thin strip. The Green’s function Gx; x0 maybeassumedtobeoftheform

1 0 0 Gx; x gngxgx A6.2 nD1 APPENDIX: SYSTEMS OF HOMOGENEOUS EQUATIONS 149

0 where gx and gx are identical in form, and gn depends on the transmission line parameters. Gx; x0 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

N x D aiix A6.3 iD1

Now substituting Eq. (A6.2) into (A6.1), we obtain 1 0 0 0 gngxgx sxsx dxdx 1 S D nD1 C 2 sx dx S 1 2 gn gxsx dx S nD1 D 2 A6.4 sx dx S

Using Eq. (A6.3) in (A6.4) then yields 1 N 2 gn ai SQni 1 D nD1 iD1 A6.5 C N 2 ai Qi iD1

where

Qi D six dx, i D 1, 2,...,N A6.6 S

SQni D gxsix dx, i D 1, 2,...,N and n D 1, 2,...,1 A6.7 S

Note that gn,Qi,andSQni 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,iD 1, 2,...,N. Let 1/C in Eq. (A6.5) be

1 Ha ,a ,...,a D 1 2 N A6.8 C Ia1,a2,...,aN 150 VARIATIONAL METHODS

where 1 N 2 H D gn ai SQni A6.9 nD1 iD1

and N 2 I D 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 1 ∂H 1 ∂I D A6.11 ∂aj C I ∂aj C ∂aj

where ∂H N 1 D 2 a g SQ SQ A6.12 ∂a i n ni nj j iD1 nD1 and ∂I N D 2 a Q Q A6.13 ∂a i i j j iD1 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 N 1 1 g SQ SQ Q Q a D 0,jD 1, 2,...,N A6.14 n ni nj C i j i 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

∂H ∂I I H D 0 A6.15 ∂aj ∂aj

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

N 1 gnSQnj Ð Qi gnSQni Ð Qjai D 0,jD 1, 2,...,N A6.16 iD1 nD1 APPENDIX: SYSTEMS OF HOMOGENEOUS EQUATIONS 151

which represents a system of N linear homogeneous equations in N unknowns ai,iD 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 a 1 D Q ˛ ,yG˛Q ,yQ ˛ ,y A6.17 C 2ε Q2 s n n s n 0 nD1 where the spectral-domain Green’s function G˛Q n is given in Eq. (6.76), and the charge density Q s˛n is expanded in Eq. (6.58) as

N ¾ Q s˛n D aiQ si˛nA6.18 iD1

Substituting Eq. (A6.18) into (A6.17), we obtain, with reordering of summations,

1 a N 1 1 D a a Q ˛ G˛Q Q ˛ A6.19 C 2ε Q2 i j si n n sj n 0 iD1 jD1 nD1

Now we apply the Ritz method to obtain the unknown coefﬁcients ai by setting

∂1/C D 0,mD 1, 2,...,N A6.20 ∂am

This yields a set of linear homogeneous equations, N 1 ai Q si˛nG˛Q nQ sj˛n D 0 A6.21 iD1 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. 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

CHAPTER SEVEN

Spectral-Domain Method

Many numerical methods exist to date for analyzing microwave and millimeter- wave 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 Fourier- transformed 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

er 1 h1

G1 S1 W S2 G2

er 2 h 2 b

h er 3 3

x 0 a

Figure 7.1 Cross section of the asymmetric CPW used for illustrating the quasi-static SDA.

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 x, y ∂2 x, y i C i D 0,iD 1, 2, 3 7.1 ∂x2 ∂y2 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,yD 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 (7.4) x, h D x, h (7.5) 2 3 3 3 ∂ 2 ∂ 1 s x εr2 εr1 D (7.6) ∂y yDh Ch ∂y yDh Ch ε0 2 3 2 3 ∂ 3 ∂ 2 εr3 εr2 D 0 (7.7) ∂y ∂y yDh3 yDh3 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 U V x D Vo x C V x 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. V x 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,ands3 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 problem, we deﬁne the Fourier transform f ˛Q n,y of f x, y as follows: 2 a f ˛Q n,yD f x, y sin ˛nxdx 7.10 a 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 0toa gives a ∂2 x, y a ∂2 x, y i sin ˛ xdxC i sin ˛ xdxD 0 7.11 2 n 2 n 0 ∂x 0 ∂y FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS 155

Using the following derivative property of the Fourier transform, 1 dkf x 1 ej˛x dx D j˛k f xej˛x dx . k 7 12 1 dx 1 we obtain a dkf x a sin ˛x dx D ˛k f x cos ˛x dx, k odd dxk 0 0 a D˛k f x sin ˛x dx, k even 7.13 0 assuming f x is real. The ﬁrst term on the left-hand side of Eq. (7.11) is the Fourier transform of 2 2 ∂ i x, y/∂x . Using Eq. (7.13) in this term gives a ∂2 x, y a i sin ˛ xdxD˛2 x, y sin ˛ xdx 2 n n i n 0 ∂x 0 2 Q D˛n i ˛n,y 7.14

The second term is obtained, upon using Eq. (7.10), as a ∂2 x, y ∂2 Q ˛ ,y i ˛ xdxD i n . 2 sin n 2 7 15 0 ∂y ∂y

Substituting Eqs. (7.14) and (7.15) into Eq. (7.11) then yields

∂2 Q i ˛2 Q D 0 7.16 ∂y2 n i which represents Laplace’s equation in the Fourier-transform or spectral domain. Fourier-transforming Eqs. (7.3)–(7.9), we obtain

Q 1 ˛n,bD Q 3 ˛n, 0 D 0 (7.17) U Q 1 ˛n,h2 C h3 D Q 2 ˛n,h2 C h3 D VQ ˛n (7.18) Q ˛ ,h D Q ˛ ,h (7.19) 2 n 3 3 n 3 ∂ Q ∂ Q Q ˛ 2 1 s n εr2 εr1 D (7.20) ∂y ∂y ε0 yDh Ch yDh Ch 2 3 2 3 ∂ Q ∂ Q ε 3 ε 2 D 0 (7.21) r3 ∂y r2 ∂y yDh3 yDh3 156 SPECTRAL-DOMAIN METHOD

which are the boundary conditions of the considered CPW structure in the spectral domain, and

U VQ ˛n D VQ o ˛n C V ˛Q n 7.22

Q s ˛n DQs1 ˛n CQs2 ˛n CQs3 ˛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,yD A sinh ˛n b y C E cosh ˛n b y 7.24

Q 2 ˛n,yD B sinh ˛n y h3 C C cosh ˛n y h3 7.25

Q 3 ˛n,yD D sinh ˛ny C F cosh ˛ny 7.26

where A, B, C, D, E,andF 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 ˛nh3 D C 7.27

Applying Eq. (7.21) to Eqs. (7.25) and (7.26), we obtain

˛nεr2B ˛nεr3D cosh ˛nh3 D 0 7.28

Using Eq. (7.18) in Eqs. (7.24) and (7.25) gives

A sinh ˛nh1 D B sinh ˛nh2 C C cosh ˛nh2 7.29

Substituting Eqs. (7.24) and (7.25) into Eq. (7.20), we have

Q s ˛n ˛nA cosh ˛nh1 C ˛nεr2 B cosh ˛nh2 C C sinh ˛nh2 D 7.30 ε0 Now manipulating Eqs. (7.27)–(7.30) and making use of Eq. (7.18), we can solve U for VQ ˛n in terms of Q s ˛n as

U G ˛Q nQ s ˛n D VQ ˛n 7.31

where G ˛Q n is called the Green’s function in the spectral domain and is given as εr3 tanh ˛nh1 tanh ˛nh2 C tanh ˛nh3 εr2 G ˛Q n D 7.32 ε0˛n tanh ˛nh2 εr3 εr1/εr2 C tanh ˛nh1 coth ˛nh2 C tanh ˛nh3 εr1 coth ˛nh2 C εr2 tanh ˛nh1] FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS 157

U VQ ˛n is given in Eq. (7.22) as

U VQ ˛n D VQ o ˛n C V ˛Q n

VQ o ˛n and V ˛Q n stand for the Fourier transforms of Vo x and V x,respectively. Vo x is chosen and is thus known. V x is given as 2 a V ˛Q n D V x sin ˛nxdx 7.33 a 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 Eq. (7.31) — Q s ˛n and V ˛Q n. We now apply Galerkin’s technique in the spectral domain to eliminate the unknown voltage VQ so that Eq. (7.31) can be solved. We begin by expressing each strip’s charge density as a truncated summation of basis functions in the space domain as

3 Nj ¾ s x D djisji 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,

3 Nj ¾ Q s ˛n D djiQ 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, 3andi D 1, 2,...,Nj.Thatis, multiply the resultant equation by Q sji ˛n, j D 1, 2, 3andi 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 N2 N3 ij ij ij j P11d1i C P12d2i C P13d3i D Q1,jD 1, 2,...,N1 7.36 iD1 iD1 iD1 158 SPECTRAL-DOMAIN METHOD

N1 N2 N3 ij ij ij j P21d1i C P22d2i C P23d3i D Q2,jD 1, 2,...,N2 7.37 iD1 iD1 iD1

N1 N2 N3 ij ij ij j P31d1i C P32d2i C P33d3i D Q3,jD 1, 2,...,N3 7.38 iD1 iD1 iD1

where

1 ij Q Q Pkl DhQskj, GQ sliiD Q skjGQ sli,kD 1, 2, 3andl D 1, 2, 3 nD1 j Q Q Q Q Qk DhQskj, Vo C ViDhQskj, VoiChQskj, Vi 1 1 D Q skjVQ o C Q skjV,Q k D 1, 2, 3 nD1 nD1

The notation hÐi indicates an inner product discussed in Appendix B at the end of this chapter. Applying Parseval’s theorem a a 1 f x, ygŁ x, y dx D f ˛Q ,ygQŁ ˛ ,y 7.39 2 n n 0 nD1

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

N1 N2 N3 ij ij ij P11d1i C P12d2i C P13d3i iD1 iD1 iD1 2 G1CS1CW D Vo s1j x dx, j D 1, 2,...,N1 7.40 a G1CS1

N1 N2 N3 ij ij ij P21d1i C P22d2i C P23d3i D 0,jD 1, 2,...,N2 7.41 iD1 iD1 iD1

N1 N2 N3 ij ij ij P31d1i C P32d2i C P33d3i D 0,jD 1, 2,...,N3 7.42 iD1 iD1 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 VQ in Eqs. (7.36)–(7.38), due to the fact that the charge densities and voltages 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 FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS 159

the unknown VQ . Furthermore, two summations involving known voltage VQ o in 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,wherei D 1, 2,...,Ni and j D 1, 2, 3. The transmission line’s PUL capacitance is given as a x dx Q s C D o D 0 Vo Vo N1 G1CS1CW d1i s1i x dx G1CS1 D iD1 7.43 Vo

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 a N1 C D d R 7.44 2V2 1i i o 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 C εeff D 7.46 Ca

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 SPECTRAL-DOMAIN METHOD

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 S G cos i 1 1 1 W s1i x D 7.47 2 x S G W 2 1 1 1 W 1 x G cos i 1 2 G1 s2i x D 7.48 2 1 x/G1 1 x a C G cos i 2 2 G2 s3i x D 7.49 a x 2 1 G2

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 W i 1 C ˛nW Q s1i ˛n D J0 2a 2 i 1 W i 1 ˛nW ð sin C ˛n S1 C G1 C J0 2 2 2 i 1 W ð sin ˛ S C G C 7.50 2 n 1 1 2 FORMULATION OF THE QUASI-STATIC SPECTRAL-DOMAIN ANALYSIS 161

i =1 i =3

x x x x x x −1 1 2 3 4 5

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,andx5 D x4 C G2.

i G1 1 Q s2i ˛n D 1 J0 i C ˛nG1 2a 2 1 J0 i ˛nG1 7.51 2 iCnC1 G2 1 Q s3i ˛n D 1 J0 i C ˛nG2 2a 2 1 J0 i ˛nG2 7.52 2

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 WV i 1 i 1 R D o J cos 7.53 i a 0 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 SPECTRAL-DOMAIN METHOD

64 2.7 eff e (ohms) o Z 62 2.6

60 2.5

58 2.4

56 2.3 Dielectric Constant, Effective Characteristic Impedance, Characteristic Impedance,

54 2.2 0 5 10 15 20 25 30 35 40

S2 (mils)

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 quasi- TEM 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 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 163 y

er1 h1 G S 2W S G

er 2 h2 b

h er 3 3

x − a 0 a 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. e h Let i x, y and i 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

2 p 2 2 p rt i x, y ˇ ki i x, y D 0; p D e, h 7.54 where ∂2 ∂2 r2 D C 7.55 t ∂x2 ∂y2 p ˇ is the propagation constant and ki D ω εi5i 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 Ä x Ä a, y D h3:

Ex3 x, h3 D Ex2 x, h3 7.56

Ez3 x, h3 D Ez2 x, h3 7.57 164 SPECTRAL-DOMAIN METHOD

Hx3 x, h3 D Hx2 x, h3 7.58

Hz3 x, h3 D Hz2 x, h3 7.59

For a Ä x Ä 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 DJx x 7.63

For a Ä x Ä a, y D 0andb:

Ex3 x, y D Ex1 x, y D Ez3 x, y D Ez1 x, y D 0 7.64

For x Dša and 0 Ä y Ä 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-andz-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 f x, y as a j˛nx f ˛Q n,yD f x, ye dx 7.66 a

or a f ˛Q n,yD 2 f x, y sin ˛nxdx 7.67 0

and a f ˛Q n,yD 2 f x, y cos ˛nxdx 7.68 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 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 165

x Dša and thus is given by   n  , odd function a ˛n D  1 7.69  n , even function 2 a

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.Thez-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 differentia- tion property to obtain

∂2 Q p ˛ ,y 72 Q p ˛ ,yD 0; p D e, h and i D 1, 2, 3 7.70 ∂y2 i n i i n where 2 2 2 2 7i D ˇ C ˛n ki 7.71

Q e Q h i ˛n,y and i ˛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

EQ x3 ˛n,h3 D EQ x2 ˛n,h3 7.72

EQ z3 ˛n,h3 D EQ z2 ˛n,h3 7.73

HQ x3 ˛n,h3 D HQ x2 ˛n,h3 7.74

HQ z3 ˛n,h3 D HQ z2 ˛n,h3 7.75

EQ x2 ˛n,h2 C h3 D EQ x1 ˛n,h2 C h3 D EQ x ˛n 7.76

EQ z2 ˛n,h2 C h3 D EQ z1 ˛n,h2 C h3 D EQ z ˛n 7.77

HQ x2 ˛n,h2 C h3 HQ x1 ˛n,h2 C h3 D JQ z ˛n 7.78

HQ z2 ˛n,h2 C h3 HQ z1 ˛n,h2 C h3 DJQ x ˛n 7.79 166 SPECTRAL-DOMAIN METHOD

EQ x3 ˛n, 0 D EQ x1 ˛n, 0 D EQ z3 ˛n, 0 D EQ z1 ˛n, 0 D 0 7.80

EQ x3 ˛n,bD EQ x1 ˛n,bD EQ z3 ˛n,bD EQ z1 ˛n,bD 0 7.81

The ﬁelds in each region i may be given in terms of the potential functions as

k2 ˇ2 E x, y, z D j i e x, yejˇz 7.82 zi ˇ i k2 ˇ2 H x, y, z D j i h x, yejˇz 7.83 zi ˇ i ω5 E x, y, z D W e x, y i a ð W h x, y ejˇz 7.84 ti t i ˇ z t i ωε H x, y, z D W h x, y C i a ð W e x, y ejˇz 7.85 ti t i ˇ z t i where the subscript t indicates the transverse (x or y) component, and

∂ ∂ W D a C a 7.86 t ∂x x ∂y y

In the spectral domain, they are

ω5 ∂ Q h ˛ ,y EQ ˛ ,yDj˛ Q e ˛ ,yC i i n 7.87 xi n n i n ˇ ∂y ˛ ω5 ∂ Q e ˛ ,y EQ ˛ ,yD j n i Q h ˛ ,yC i n 7.88 yi n ˇ i n ∂y k2 ˇ2 EQ ˛ ,yD j i Q e ˛ ,y 7.89 zi n ˇ i n ωε ∂ Q e ˛ ,y HQ ˛ ,yDj˛ Q h ˛ ,yC i i n 7.90 xi n n i n ˇ ∂y ˛ ωε ∂ Q h ˛ ,y HQ ˛ ,yDj n i Q e ˛ ,yC i n 7.91 yi n ˇ i n ∂y k2 ˇ2 HQ ˛ ,yD j i Q h ˛ ,y 7.92 zi n ˇ i n

General solutions of Eq. (7.70) are

Q e 1 ˛n,yD Ae sinh 71 b y C Ee cosh 71 b y 7.93 Q h 1 ˛n,yD Ah cosh 71 b y C Eh sinh 71 b y 7.94 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 167

Q e 2 ˛n,yD Be sinh 72 y h3 C Ce cosh 72 y h3 7.95 Q h 2 ˛n,yD Bh sinh 72 y h3 C Ch cosh 72 y h3 7.96 Q e 3 ˛n,yD De sinh 73y C Fe cosh 73y 7.97 Q h 3 ˛n,yD Dh cosh 73y C Fh sinh 73y 7.98

where Ae,h, Be,h, Ce,h, De,h, Ee,h,andFe,h are unknown constants. In order to satisfy the last two boundary conditions, Eqs. (7.80) and (7.81), Ee,h,andFe,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 ω5 7 EQ ˛ ,yDj˛ A sinh 7 b y 1 1 A sinh 7 b y 7.99 x1 n n e 1 ˇ h 1

EQ x2 ˛n,yDj˛n[Be sinh 72 y h3 C Ce cosh 72 y h3] ω5 7 C 2 2 [B cosh 7 y h C C sinh 7 y h ] 7.100 ˇ h 2 3 h 2 3 ω5 7 EQ ˛ ,yDj˛ D sinh 7 y C 3 3 D sinh 7 y 7.101 x3 n n e 3 ˇ h 3 k2 ˇ2 EQ ˛ ,yD jA 1 sinh 7 b y 7.102 z1 n e ˇ 1 k2 ˇ2 EQ ˛ ,yD j 2 [B sinh 7 y h C C cosh 7 y h ] 7.103 z2 n ˇ e 2 3 e 2 3 k2 ˇ2 EQ ˛ ,yD jD 3 sinh 7 y 7.104 z3 n e ˇ 3 ωε 7 HQ ˛ ,yDj˛ A cosh 7 b y C 1 1 A cosh 7 b y 7.105 x1 n n h 1 ˇ e 1

HQ x2 ˛n,yDj˛n[Bh sinh 72 y h3 C Ch cosh 72 y h3] ωε 7 2 2 [B cosh 7 y h C C sinh 7 y h ] 7.106 ˇ e 2 3 e 2 3 ωε 7 HQ ˛ ,yDj˛ D cosh 7 y 3 3 D cosh 7 y 7.107 x3 n n h 3 ˇ e 3 k2 ˇ2 HQ ˛ ,yD jA 1 cosh 7 b y 7.108 z1 n h ˇ 1 k2 ˇ2 HQ ˛ ,yD j 2 [B sinh 7 y h C C cosh 7 y h ] 7.109 z2 n ˇ h 2 3 h 2 3 k2 ˇ2 HQ ˛ ,yD jD 3 cosh 7 y 7.110 z3 n h ˇ 3 168 SPECTRAL-DOMAIN METHOD

Now we apply the boundary conditions (7.72)–(7.79). Applying Eq. (7.72) and using Eqs. (7.100) and (7.101) yields ω5 7 ω5 7 j˛ D sinh ˛ h C 3 3 D sinh ˛ h Dj˛ C C 2 2 B 7.111a n e n 3 ˇ h n 3 n e ˇ h Using Eq. (7.73) in Eqs. (7.103) and (7.104), we obtain 2 2 2 2 k3 ˇ De sinh 73h3 D k2 ˇ Ce 7.111b Applying Eq. (7.74) to Eqs. (7.106) and (7.107), we get ωε 7 ωε 7 j˛ D cosh ˛ h C 3 3 D cosh ˛ h Dj˛ C 2 2 B 7.112 n h n 3 ˇ e n 3 n h ˇ e Substituting Eqs. (7.109) and (7.110) into Eq. (7.75) gives 2 2 2 2 k3 ˇ Dh cosh 73h3 D k2 ˇ Ch 7.113 Substituting Eqs. (7.99) and (7.100) into Eq. (7.76), we obtain ω5 7 j˛ B sinh 7 h C C cosh 7 h C 2 2 B cosh 7 h C C sinh 7 h n e 2 2 e 2 2 ˇ h 2 2 h 2 2 ω5 7 Dj˛ A sinh 7 h 1 1 A sinh 7 h D EQ ˛ 7.114 n e 1 1 ˇ h 1 1 x n Substituting Eqs. (7.102) and (7.103) into Eq. (7.77) yields 2 2 2 2 Q k2 ˇ Be sinh 72h2 C Ce cosh 72h2 D k1 ˇ Ae sinh 71h1 DjˇEz ˛n 7.115

Substituting Eqs. (7.105) and (7.106) into Eq. (7.78) leads to ωε 7 j˛ B sinh 7 h C C cosh 7 h 2 2 B cosh 7 h C C sinh 7 h n h 2 2 h 2 2 ˇ e 2 2 e 2 2 ωε 7 C j˛ A cosh 7 h 1 1 A cosh 7 h D JQ ˛ 7.116 n h 1 1 ˇ e 1 1 z n Finally, applying Eq. (7.79) in Eqs. (7.108) and (7.109), we have 2 2 2 2 Q k2 ˇ Bh sinh 72h2 C Ch cosh 72h2 k1 ˇ Ah cosh 71h1 DjˇJx ˛n 7.117

From Eqs. (7.111a)–(7.117) we can solve for JQ x ˛n and JQ z ˛n in terms of EQ x ˛n and EQ z ˛n as

GQ 11 ˛n,ˇEQ x ˛n C GQ 12 ˛n,ˇEQ z ˛n D JQ x ˛n 7.118

GQ 21 ˛n,ˇEQ x ˛n C GQ 22 ˛n,ˇEQ z ˛n D JQ z ˛n 7.119 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 169

where GQ ij ˛n,ˇ, i, j D 1, 2, are the Green’s functions in the spectral domain and are given as 1 GQ D ˛2 GQ ˇ2GQ 7.120 11 2 2 n e h ˛n C ˇ j˛ ˇ GQ DGQ D n GQ C GQ 7.121 21 12 2 2 e h ˛n C ˇ 1 GQ D ˇ2GQ ˛2 GQ 7.122 22 ˛2 C ˇ2 e n h  n   3 2 3 2  Q k0 εr2˛n tanh ˛n h2 tanh ˛n h3 ˛n εr3  Ge D 2  2 2  ˛n 3 3 ˛n εr3 tanh ˛n h2 ˛n tanh ˛n h3 C εr2 k ε C 0 r1 7.123 1 1 ˛n tanh ˛n h1   3 tanh ˛ h3 1 ˛ 2 tanh ˛ 2h n   n n 2 3  1 1 ˛n ˛n GQ h D   7.124 k  tanh ˛ 2h tanh ˛ 3h  1 0 n 2 C n 3 k0 tanh ˛n h1 ˛ 2 ˛ 3 n n 1 2 2 2 ˛n D k1 ˇ ˛n 7.125 2 2 2 2 ˛n D k2 ˇ ˛n 7.126 3 2 2 2 ˛n D k3 ˇ ˛n 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

M ¾ Ex x D cmExm x 7.128 mD0 170 SPECTRAL-DOMAIN METHOD

K ¾ Ez x D dkEzk x 7.129 kD1

where cm and dk are the unknown coefficients. The basis functions Exm x and Ezk x describe the x-andz-electric field distributions on the slots and are nonzero only on these slots. In the spectral domain, the unknown electric fields are M ¾ EQ x ˛n D cmEQ xm ˛n 7.130 mD0 K ¾ EQ z ˛n D dkEQ zk ˛n 7.131 kD1 Substituting Eqs. (7.130) and (7.131) into Eqs. (7.118) and (7.119) gives M K GQ 11 ˛n,ˇEQ xm ˛ncm C GQ 12 ˛n,ˇEQ zk ˛ndk D JQ x ˛n 7.132 mD0 kD1 M K GQ 21 ˛n,ˇEQ xm ˛ncm C GQ 22 ˛n,ˇEQ zk ˛ndk D JQ z ˛n 7.133 mD0 kD1

We now take the inner product of Eqs. (7.132) and (7.133) with EQ xi ˛n, i D 0, 1,...,M,andEQ zj ˛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,andEQ 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 K im ik P11 ˇcm C P12 ˇdk D 0; i D 0, 1, 2,...,M 7.134 mD0 kD1 M K jm jk P21 ˇcm C P22 ˇdk D 0; j D 0, 1, 2,...,K 7.135 mD0 kD1 where im Q Q Q P11 D Exi ˛n, G11 ˛n,ˇExm ˛n 1 D EQ xi ˛nGQ 11 ˛n,ˇEQ xm ˛n 7.136 nD1 ik Q Q Q P12 D Exi ˛nG12 ˛n,ˇEzk ˛n 1 D EQ xi ˛nGQ 12 ˛n,ˇEQ zk ˛n 7.137 nD1 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 171 jm Q Q Q P21 D Ezj ˛nG21 ˛n,ˇExm ˛n 1 D EQ zj ˛nGQ 21 ˛n,ˇEQ xm ˛n 7.138 nD1 jk Q Q Q P22 D Ezj ˛nG22 ˛n,ˇEzk ˛n 1 D EQ zj ˛nGQ 22 ˛n,ˇEQ zk ˛n 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). Specifi- cally, 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 ε D ˇ/k 2 transmissionp line for a particular mode is obtained as eff 0 ,where k0 D ω ε050. 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 2 jVoj Zo D 7.140 2Pavg

where WCS Vo D Ex x, h2 C h3dx 7.141 W

is the voltage across the slot, and a b 1 Ł Ł Pavg D Re ExHy EyHx dy dx 7.142 2 a 0

represents the transmitted power across the transmission line’s cross section. 172 SPECTRAL-DOMAIN METHOD

0 To obtain numerical results for ˇ and consequently for εeff and Zc, suitable basis functions for the electric field components, Exm and Ezk with m D 0, 1,...,Mand 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 r1/2. The longitudinal field components are bounded and behave as r1/2 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 m cos m  S S  ; m D 0, 2,...  1 [2 x C d/S]2 1 [2 x d/S]2 Exm x D   x C d x d  sin m sin m  S S  C ; m D 1, 3,... 1 [2 x C d/S]2 1 [2 x d/S]2 7.143   x C d x d  cos k cos k  S S  C ; k D 1, 3,...  2 2  1 2 x C d/S 1 2 x d/S Ezk x D   x C d x d  sin k sin k  S S  D  ; k 2, 4,... 1 2 x C d/S 2 1 2 x d/S 2 7.144 FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 173

m=0 2 m=2

xm 0 E

−2 m=1 m=3 −4

−6 Slot Slot −8 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 x (a)

1.5 k =2

1.0 k=1

0.5

zk 0.0 E

−0.5

−1.0

Slotk=3 Slot −1.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 x (b)

Figure 7.5 Plots of basis functions for the (a) x and (b) z electric ﬁelds. 174 SPECTRAL-DOMAIN METHOD

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   S ˛nS C m ˛nS m  j sin ˛nd J0 C J0 ;  2 2 2   m D 0, 2,... Q Exm ˛n D  7.145  S ˛nS C m ˛nS m  j cos ˛nd J0 J0 ;  2 2 2  m D 1, 3,...   S ˛nS C k ˛nS k  cos ˛nd J0 C J0 ;  2 2 2  k D 1, 3,... EQ zk ˛n D 7.146  S ˛ S C k ˛ S k  n n  sin ˛nd J0 J0 ;  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 3.0

2.5

2.0

3rd mode 1.5

1.0 2nd mode Propagation Constant (rad/mm) 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. FORMULATION OF THE DYNAMIC SPECTRAL-DOMAIN ANALYSIS 175

7 6 5 4 3 2 Slot 1 Ground plane 0 − 1 Slot Strip Ground plane (arbitrary unit) x −2 E −3 −4 −5 −6 −7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x−a (mm) (a)

2.0 1.5 1.0 Strip 0.5 Slot Slot 0.0 − 0.5 Ground plane Ground plane −1.0 −1.5 −2.0 −2.5

(arbitrary unit) −3.0 y E −3.5 −4.0 −4.5 −5.0 −5.5 −6.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x−a (mm) (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

0.2 Ground plane 0.1 Slot Strip 0.0 Slot (arbitrary unit) −0. x 1 Ground E plane −0.2

−0.3

−0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x−a (mm) (a)

0.075 Strip 0.050 Ground Ground 0.025 plane plane 0.000 Slot Slot −0.025 −0.050 −0.075 (arbitrary unit) y −0.100 E −0.125 −0.150 −0.175 −0.200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x−a (mm) (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 Asym- metric 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 Elec- tronics 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 SPECTRAL-DOMAIN METHOD

given as 1, jxjÄW/2 x D s 0, 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 to5forεr 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:   1 W  jxjÄ x D 1 [jxj/ W/2]2 2 s  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 2 x C W/2 W 2 1 W

Compare the numerical results of Zo and εeff to those obtained in Prob- lems 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 0.1to0.9forεr 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  b a  , b Ä x Äa 2 s1i x D  2 x C b b a  1  b a  0, otherwise   x a  cos i 1  b a  ,aÄ x Ä b 2 s2i x D  2 x a b a  1  b a  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 L U L U on the central strips, s1i and s1i, and ground strips, s2i, s2i and s3i, 180 SPECTRAL-DOMAIN METHOD

e r 1 h G S 2W S G

2d 2b er 2 x

er 1 h

Figure P7.1 Cross section of a broadside-coupled CPW.

L , are given as s3i x C W cos i 1 U L 2W s1i x D s1i x D 2 x C W 2W 2 1 2W 1 x C W C S cos i U L 2 G s2i x D s2i x D x C W C S C G 2 1 G 1 x W S cos i U L 2 G s3i x D s3i x D x W S G 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,ofthe even mode. PROBLEMS 181

(b) Formulate the quasi-static SDA analysis for determining the characteristic impedance, Zoo, and effective dielectric constant, εeff,o,ofthe odd mode.

(c) Write a computer program to calculate the Zoe, Zoo,andεeff,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,ofthe dominant even mode using the dynamic SDA.

(b) Formulate an analysis of the effective dielectric constant, εeff,o,ofthe 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

2 Vo Zo D 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 SPECTRAL-DOMAIN METHOD

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,forthe 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-andz-directed electric fields are approximated by M ¾ Ex x D cmExm x mD1 K ¾ Ez x D dkEzk x kD1 PROBLEMS 183

where cm and dk are the unknown coefﬁcients. The basis functions Exm x and Ezk x are assumed as   W[ W/22 x2]m1 W  , jxjÄ E x D 2 W/22 x2 2 xm  0, otherwise

  2 2 k1  2x W W W j x2 x2 , jxjÄ W 2 2 2 Ezk x D   0, otherwise

The Fourier transform of a function f x, y is deﬁned as 1 f ˛,Q y D f x, yej˛x 1

(a) Plot the ﬁrst four basis functions for Exm and Ezk versus x.

(b) Formulate an analysis for the effective dielectric constant, εeff,ofthe 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

2 Vo 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.

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 SPECTRAL-DOMAIN METHOD

are assumed as x S W/2 sin 2n W Jxn x D 2 x S W/2 2 1 W x S W/2 cos 2 n 1 W Jzn x D 2 x 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 0.25 a 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

P Z D avg o 2 Io

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,ofthe 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

er1

er 2 h 1 W

er 3 h

er4

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 2 Vo 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,ofthe 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 f x, y, z: 1 a j ˛nxCˇz f ˛Q n,y,ˇD f x, y, ze dx dz 1 a 186 SPECTRAL-DOMAIN METHOD

d e0 2W 2l x

h er 1 x z 2a

(a) (b) End View Top View

Figure P7.3 (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 f x, y with respect to x is a continuous transform and can be deﬁned as 1 f ˛,Q y D f x, 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 1 f x, y D f ˛,Q yej˛x d˛ A7.2 2 1

This Fourier transform is the same as the sine transform 1 f ˛,Q y D 2 f x, y sin ˛x dx A7.3 0 when f x, y is odd with respect to x,thatis,f x, y Df x, y,andthe cosine transform 1 f ˛,Q y D 2 f x, y cos ˛x dx A7.4 0 APPENDIX A: FOURIER TRANSFORM AND PARSEVAL’S THEOREM 187

if f x, y is even with respect to x,thatis,f x, y D f x, y. Note that f x, y is defined only over a certain part of the domain of x, whereas fQ ˛, y is defined 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 f ˛,Q y D f x, y cos ˛x dx A7.5 0

Parseval’s theorem associated with the continuous Fourier transform has the form 1 1 1 f x, y gŁ x, y dx D f ˛,Q ygQŁ ˛, y d˛ A7.6 1 2 1 where the superscript Ł denotes the complex conjugate. This Parseval’s formula indicates that if the two functions f x, y and g x, 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 f x, y with respect to x is a discrete or ﬁnite transform and can be deﬁned as 1 a j˛nx f ˛Q n,yD f x, ye 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, 2 a f ˛Q n,yD f x, y sin ˛nxdx A7.8 a 0

when f x, y is odd with respect to x,thatis,f x, y Df x, y,andthe cosine transform 2 a f ˛Q n,yD f x, y cos ˛nxdx A7.9 a 0 when f x, y is even with respect to x;thatis,f x, y D f x, y. The inverse Fourier transforms of Eqs. (A7.8) and (A7.9) are just the Fourier series of f x, y: 1 f x, y D f ˛Q n,ycos ˛nx A7.10 nD1 188 SPECTRAL-DOMAIN METHOD

1 f x, y D f ˛Q n,ysin ˛nx 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 a 1 f x, ygŁ x, y dx D f ˛Q ,ygQŁ ˛ ,y A7.12 2 n n 0 nD1

which implies that if the two functions f x, y and g x, 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:

L f 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

x, y, z r2V x, y, z D B7.2 ε

L Dr2, 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

N f D anfn B7.3 nD1

where an are the unknown coefﬁcients. Note that if N approaches inﬁnity and ffng forms a complete set, for example, a Fourier series, then Eq. (B7.3) APPENDIX B: GALERKIN’S METHOD 189

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

N anL 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 B7.6 ! " fŁ,f > 0, if f>0 B7.7 ! " fŁ,f D 0, if f D 0 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 w xf xg x dx B7.9

where w x 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 fni D hwm,gi B7.10 nD1

which can be rewritten as [lmn][an] D gm B7.11

where   hw1,L f1ihw1,L f2i ÐÐÐ hw1,L fNi    hw2,L f1ihw2,L f2i ÐÐÐ hw2,L fNi  [lmn] D  . . . .  B7.12 . . . . hwN,L f1ihwN,L f2i ÐÐÐ hwN,L fNi T [an] D [a1 a2 ÐÐÐ aN] B7.13 T gn D g1 g2 ÐÐÐ gN T D hw1,gihw2,gi ÐÐÐ hwN,gi B7.14 190 SPECTRAL-DOMAIN METHOD

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 1 [an] D [lmn] gm B7.15

Substituting Eq. (B7.15) into (B7.3) yields the ﬁnal solution for the unknown function f as 1 f D fn [lmn] gm B7.16

where [fn] D [f1 f2 ÐÐÐ 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. N They should be chosen such that nD1 anfn 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. 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

Index

Analysis methods, 1, 7–9 coplanar waveguide, 67, 71, 106 Analytic function, 87, 89 with finite ground plane, 109 Anisotropic, 15 with upper conducting cover, 107 Attenuation constant, 66 microstrip line, 67 conductor attenuation constant, 70, 78 parallel-coupled strip lines, 112 coplanar strips, 76 slot line, 67, 79, 80 coplanar waveguide, 74 strip line, 77, 106 strip line, 77 transverse (TE) mode, 26 dielectric attenuation constant, 70, 71 transverse (TM) mode, 27 coplanar strips, 76 Circular resonator, 60 coplanar waveguide, 74 Circular waveguide, 38 strip line, 77 Complete integral of the first kind, 72, 73, 103 Average power, 35, 36, 67, 200 Conformal mapping, 1, 9, 10, 85, 87, 90 density, 19, 37 Conformal mapping equations asymmetrical coplanar strips, 111 Basis functions, 123, 127, 132, 137, 141, 142, asymmetrical coplanar waveguide, 110 160, 172, 188 conductor-backed coplanar waveguide, 108 Bessel differential equation, 44, 57 with upper conducting cover, 108 Boundary coplanar strips, 110 enlargement, 210, 211 coplanar waveguide, 106 reduction, 210, 211 with finite ground planes, 109 Boundary conditions, 17, 18, 37, 46, 47, with upper conducting cover, 107 49–51, 57, 138, 153, 154, 163, 164 fundamentals of, 87 in the spectral domain, 139, 155, 165 parallel-coupled strip lines, 112 strip line, 106 Capacitance per unit length, 65 Conjugate functions, 87 Cauchy–Riemann equations, 87 Constitutive relations, 14 Characteristic impedance, 4, 7–8, 20, 64–66, Continuity equation, 15 68, 70, 128, 134 Coplanar strips (CPS), 3, 4, 74, 75, 110, 111 asymmetrical coplanar strips, 111 asymmetrical coplanar strips, 111 asymmetrical coplanar waveguide, 110 effective dielectric constant of, 74, 75, 110 conductor-backed coplanar waveguide, 73, Coplanar waveguide (CPW), 3, 4, 71, 108 106–107, 109, 110 with upper conducting cover, 108 conductor-backed, 71, 109, 108 coplanar strips, 74, 75, 110, 200 step discontinuities, 203, 204 238 INDEX

CPS, see Coplanar strips Finite strip thickness, 69, 73, 77 CPW, see Coplanar waveguides Fourier series, 187 Cross talk, 5 Fourier transform, 54, 136, 139–141, 154, 155, Cutoff, 21 164, 186 frequency, 21, 24, 64 variable, 139, 186, 187 wave number, 21, 24, 29, 30, 47 Fourier-transformed domain, see Spectral domain Degenerate modes, 24, 25, 30, 36–38 Full-wave approach, 64 Depth of penetration, see Skin depth Dielectric, 15 Galerkin equation, 157, 170 Dielectric constant, 14 Galerkin technique, 153, 163, 171, 188, 190 complex, 16 Gram–Schmidt process, 30 Dirac delta function, 40, 44 Green’s first identity, 31, 62 Dirichlet conditions, 29, 30, 37 Green’s function, 1, 9, 39–41, 44–47, 49, Dirichlet problem, 92 50–52, 54, 55, 57, 58, 132, 148 Discontinuities, 11 closed-form, 44, 46, 49, 58 Distortion, 5 in the spectral domain, 140, 156, 157 Divergence theorem, 13, 33 integral-form, 44, 53 Dominant mode, 64 series-form, 44, 49, 51, 58 Dynamic solutions of, 44 analysis, 66 Green’s second identity, 29, 62 approach, 64 Green’s theorem, 62 parameters, 20 spectral-domain analysis, 162 Harmonic conjugate functions, 87 Dynamic characteristic impedance, 66, 67 Helmholtz equation, 20, 21, 46, 53 for the electric vector potential, 22 Effective dielectric constant, 8, 20, 64–66, 68, for the magnetic vector potential, 22 70, 128, 134, 200 Higher order modes, 64 asymmetrical coplanar waveguide, 110 Homogeneous medium, 64 conductor-backed coplanar waveguide, 73, Hybrid fields, 193 108 Hybrid modes, 24, 26–28, 32, 35, 36, 64 with upper conducting cover, 108 Hybrid wave, 24 conventional coplanar waveguide, 71, 106 coplanar waveguide Inductance per unit length, 65 with finite ground plane, 109 Inner product, 157, 158, 170, 189, 210, 211, with upper conducting cover, 107 213 Eigenfunctions, 50, 52 Inverse elliptic function, 102, 103 Eigenmodes, 37, 38, 162, 175, 211 Eigenvalues, 50, 52 Kronecker delta function, 50, 214 Electric potential, 21 scalar, 21, 23, 37, 163, 193 Laplace’s equation, 19, 20, 56, 138, 153 vector, 21 in the spectral domain, 139, 155 Electric wall, 192 Laplacian operator, 20, 21, 23 Electromagnetic, 12 Longitudinal operator, 20 fields, 17, 20, 23, 31 Lorentz condition, 22 theory,1,9,12 Loss, 4, 8, 16, 70 Electrostatic energy per unit length, 92, 93 conductor, 71 Evanescent modes of the shielded coplanar dielectric, 15, 71 waveguide, 203 tangent, 16, 17, 71 Exact solution, 121 Magnetic potential, 21 Fin line, 2, 4 scalar, 21, 23, 26, 30, 37, 163, 193 Finite metallization thickness, 71. See also vector, 21, 22, 41, 42 Finite strip thickness Magnetic wall, 192 INDEX 239

Mapping, 85 Permeability, 14, 15 Maxwell’s equations, 8, 9, 12–15, 17, 20, 21, relative, 14, 15 33 Permittivity, 14, 15 differential form, 12 relative, 14 integral form, 12 Phase time-harmonic, 14 constant, 66 Method of separation of variables, 52 velocity, 65 Microstrip line, 2, 4, 5, 46, 60, 65, 68, 98, 131 Phasor inverted, 3, 4 fields, 14 Microwave and millimeter-wave passive Poynting vector, 19 structures, 1 Planar transmission lines, 1, 4, 5, 7–9, 37, 38, Microwave boundary-value problems, 9, 39, 42, 63 43 Poisson’s equation, 19, 20, 39–41, 188 Microwave integrated circuits (MIC), 1, 5, 7, 8 Power Modal orthogonality condition, 214 conservation, 215 Mode coupling, 38 flow, 19 analysis of planar transmission lines, 191 handling, 4 analysis of planar transmission line orthogonality, 35–38 discontinuities, 203 Power-interaction terms, 38 equations, 198, 210 Poynting vector, 19, 37 method, 1, 9, 11, 191, 192 Principle of superposition, 24 types, 24 Propagating mode of the shielded coplanar Mode types, 24 waveguide, 203 Moment method, 153, 188, 190 Propagation constant, 21, 24, 28, 47, 66, 206 Monolithic circuits, 5 Monolithic microwave integrated circuits Quasi-static, 13, 20 (MMIC), 1, 7 approach, 64 Multilayer planar transmission lines, 5, 7 spectral-domain method, 137, 138 Quasi-TE modes, 79 Neumann conditions, 29, 30, 37 Quasi-TE10,79 Neumann problem, 92 Quasi-TEM mode, 64 Nondegenerate modes, 24, 30, 34, 36–38 Rayleigh–Ritz equation, 157, 170 Nonisotropic, see Anisotropic Rayleigh–Ritz method, 122 Nonmagnetic, 15 Rectangular waveguide, 23, 24, 38, 58 Nonorthogonal modes, 30, 32, 38 resonator, 51, 53 Nonstationary, 121 Relative dielectric constant, 14–17 Numerical convergence of the mode-matching tensor, 15 analysis, 200, 219 Resonant frequency, 53, 120 RF, 1, 17, 20 Operating frequency, 4, 20 Ritz method, 122, 127, 132, 137, 141. See also Orthogonal coefficients, 30, 32 Rayleigh–Ritz method Orthogonality conditions, 218 between the scalar potentials, 30 Schwarz–Christoffel transformation, 95, 98, Orthogonality relations, 28, 31, 32, 34, 37, 52. 102 See also Orthogonality conditions Skin depth, 18, 19 Orthogonal modes, 30 Slot line, 3, 4, 78 Solid-state device, 4, 5, 7 Parallel-coupled strip lines, 112 Space domain, 11, 39 Parallel-plate mode, 76 Spectral domain, 11, 39, 137 Parseval’s theorem, 136, 154, 158, 171, Spectral-domain method, 1, 9, 11, 39, 152 186–188 Spectral order, 139, 142, 154, 165 Passive Spectral term, see Spectral order components, 7 Static, 13, 20, 23 structures, 1 analysis, 64 240 INDEX

Static (Continued) Transverse magnetic vector potential, 23 approach, 64 Transverse operator, 20 parameters, 20 Laplacian, 21, 47 Stationary, 121, 123 Trial function, 121 formula, 123 True solution, 121 Stoke theorem, 13 Strip line, 2, 4, 61, 76, 106 Variational expressions for the capacitance per strip width for, 77 unit length of transmission lines, 123 Sturm–Liouville upper-bound, 124, 125, 127, 129 equation, 42–44, 48, 49, 56, 57 lower-bound, 125–127, 131, 148 operator, 42, 44 Variational formula, 123. See also Variational Substrates, 1, 5, 7, 15–17 expressions Surface resistivity, 71 Variational methods, 1, 9, 10, 39, 120 Suspended strip line, 2, 4 formulation in the space domain, 128 formulation in the spectral domain, 135, 138 Taylor’s series, 121 formulation using lower-bound expression, TE, see Transverse electric 130 TE10,79 formulation using upper-bound expression, TEM, see Transverse electromagnetic 18 Time harmonic, 14, 20 fundamentals of, 121 fields, 14, 15, 17, 21 Variational parameters, 121, 123, 127 Time-varying electromagnetic fields, 12, 17 TM, see Transverse magnetic Wallis formula, 103 Transformation, 85 Wave Transmission lines, 1, 4, 5, 9, 20, 21, 23, 39 equations, 9, 20, 21, 23, 29, 41, 51, 163 Transverse electric (TE) for the electric vector potential, 22 fields, 26 for the magnetic vector potential, 22 mode, 24, 26, 28, 32, 35–38, 51, 52, 64 in the spectral domain, 165 wave, 24 number, 20, 41, 47 Transverse electric vector potential, 23 solution, 24 Transverse electromagnetic (TEM) types, 23 mode, 24, 27, 64 Waveguides, 23, 24, 28–30, 36, 37 wave, 24 Wavelength of slot line, 66, 79, 80 Transverse magnetic (TM) Wronskian, 46, 48 fields, 27, 33 mode, 24, 27, 28, 35–38, 64, 67 Zero-order Bessel function of the first kind, wave, 24 161, 174