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Foundations of Applied Electrodynamics P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come FOUNDATIONS OF APPLIED ELECTRODYNAMICS P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come FOUNDATIONS OF APPLIED ELECTRODYNAMICS Wen Geyi Waterloo, Canada A John Wiley and Sons, Ltd., Publication P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come This edition first published 2010 C 2010 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Wen, Geyi. Foundations of applied electrodynamics / Geyi Wen. p. cm. Includes bibliographical references and index. ISBN 978-0-470-68862-5 (cloth) 1. Electrodynamics–Mathematics. I. Title. QC631.3.W46 2010 537.60151–dc22 A catalogue record for this book is available from the British Library. ISBN 978-0-470-68862-5 (H/B) Typeset in 10/12pt Times by Aptara Inc., New Delhi, India Printed and Bound in P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come To my parents To Jun and Lan P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come Contents Preface xv 1 Maxwell Equations 1 1.1 Experimental Laws 2 1.1.1 Coulomb’s Law 2 1.1.2 Ampere’s` Law 5 1.1.3 Faraday’s Law 9 1.1.4 Law of Conservation of Charge 9 1.2 Maxwell Equations, Constitutive Relation, and Dispersion 10 1.2.1 Maxwell Equations and Boundary Conditions 11 1.2.2 Constitutive Relations 15 1.2.3 Wave Equations 18 1.2.4 Dispersion 20 1.3 Theorems for Electromagnetic Fields 22 1.3.1 Superposition Theorem 22 1.3.2 Compensation Theorem 23 1.3.3 Conservation of Electromagnetic Energy 23 1.3.4 Conservation of Electromagnetic Momentum 25 1.3.5 Conservation of Electromagnetic Angular Momentum 27 1.3.6 Uniqueness Theorems 27 1.3.7 Equivalence Theorems 32 1.3.8 Reciprocity 36 1.4 Wavepackets 39 1.4.1 Spatial Wavepacket and Temporal Wavepacket 40 1.4.2 Signal Velocity and Group Velocity 42 1.4.3 Energy Density for Wavepackets 42 1.4.4 Energy Velocity and Group Velocity 45 1.4.5 Narrow-band Stationary Stochastic Vector Field 47 2 Solutions of Maxwell Equations 49 2.1 Linear Space and Linear Operator 50 2.1.1 Linear Space, Normed Space and Inner Product Space 50 2.1.2 Linear and Multilinear Maps 52 P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come viii Contents 2.2 Classification of Partial Differential Equations 54 2.2.1 Canonical Form of Elliptical Equations 56 2.2.2 Canonical Form of Hyperbolic Equations 57 2.2.3 Canonical Form of Parabolic Equations 57 2.3 Modern Theory of Partial Differential Equations 58 2.3.1 Limitation of Classical Solutions 58 2.3.2 Theory of Generalized Functions 60 2.3.3 Sobolev Spaces 66 2.3.4 Generalized Solutions of Partial Differential Equations 67 2.4 Method of Separation of Variables 69 2.4.1 Rectangular Coordinate System 69 2.4.2 Cylindrical Coordinate System 70 2.4.3 Spherical Coordinate System 71 2.5 Method of Green’s Function 73 2.5.1 Fundamental Solutions of Partial Differential Equations 73 2.5.2 Integral Representations of Arbitrary Fields 74 2.5.3 Integral Representations of Electromagnetic Fields 78 2.6 Potential Theory 83 2.6.1 Vector Potential, Scalar Potential, and Gauge Conditions 83 2.6.2 Hertz Vectors and Debye Potentials 87 2.6.3 Jump Relations in Potential Theory 89 2.7 Variational Principles 93 2.7.1 Generalized Calculus of Variation 93 2.7.2 Lagrangian Formulation 95 2.7.3 Hamiltonian Formulation 100 3 Eigenvalue Problems 105 3.1 Introduction to Linear Operator Theory 106 3.1.1 Compact Operators and Embeddings 106 3.1.2 Closed Operators 109 3.1.3 Spectrum and Resolvent of Linear Operators 110 3.1.4 Adjoint Operators and Symmetric Operators 112 3.1.5 Energy Space 114 3.1.6 Energy Extension, Friedrichs Extension and Generalized Solution 116 3.2 Eigenvalue Problems for Symmetric Operators 120 3.2.1 Positive-Bounded-Below Symmetric Operators 120 3.2.2 Compact Symmetric Operators 126 3.3 Interior Electromagnetic Problems 130 3.3.1 Mode Theory for Waveguides 130 3.3.2 Mode Theory for Cavity Resonators 140 3.4 Exterior Electromagnetic Problems 145 3.4.1 Mode Theory for Spherical Waveguides 145 3.4.2 Singular Functions and Singular Values 149 3.5 Eigenfunctions of Curl Operator 150 P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come Contents ix 4 Antenna Theory 153 4.1 Antenna Parameters 154 4.1.1 Radiation Patterns and Radiation Intensity 155 4.1.2 Radiation Efficiency, Antenna Efficiency and Matching Network Efficiency 156 4.1.3 Directivity and Gain 157 4.1.4 Input Impedance, Bandwidth and Antenna Quality Factor 157 4.1.5 Vector Effective Length, Equivalent Area and Antenna Factor 158 4.1.6 Polarization and Coupling 161 4.2 Properties of Far Fields 162 4.3 Spherical Vector Wavefunctions 165 4.3.1 Field Expansions in Terms of Spherical Vector Wavefunctions 165 4.3.2 Completeness of Spherical Vector Wavefunctions 171 4.4 Foster Theorems and Relationship Between Quality Factor and Bandwidth 172 4.4.1 Poynting Theorem and the Evaluation of Antenna Quality Factor 173 4.4.2 Equivalent Circuit for Transmitting Antenna 176 4.4.3 Foster Theorems for Ideal Antenna and Antenna Quality Factor 178 4.4.4 Relationship Between Antenna Quality Factor and Bandwidth 182 4.5 Minimum Possible Antenna Quality Factor 183 4.5.1 Spherical Wavefunction Expansion for Antenna Quality Factor 183 4.5.2 Minimum Possible Antenna Quality Factor 185 4.6 Maximum Possible Product of Gain and Bandwidth 186 4.6.1 Directive Antenna 186 4.6.2 Omni-Directional Antenna 189 4.6.3 Best Possible Antenna Performance 192 4.7 Evaluation of Antenna Quality Factor 193 4.7.1 Quality Factor for Arbitrary Antenna 193 4.7.2 Quality Factor for Small Antenna 194 4.7.3 Some Remarks on Electromagnetic Stored Energy 200 5 Integral Equation Formulations 203 5.1 Integral Equations 204 5.2 TEM Transmission Lines 205 5.3 Waveguide Eigenvalue Problems 207 5.3.1 Spurious Solutions and their Discrimination 208 5.3.2 Integral Equations Without Spurious Solutions 210 5.4 Metal Cavity Resonators 211 5.5 Scattering Problems 213 5.5.1 Three-Dimensional Scatterers 213 5.5.2 Two-Dimensional Scatterers 224 5.5.3 Scattering Cross-Section 230 5.5.4 Low Frequency Solutions of Integral Equations 231 5.6 Multiple Metal Antenna System 233 5.7 Numerical Methods 238 5.7.1 Projection Method 238 P1: OTE/OTE/SPH P2: OTE FM BLBK281-Wen March 13, 2010 16:38 Printer Name: Yet to Come x Contents 5.7.2 Moment Method 239 5.7.3 Construction of Approximating Subspaces 240 6 Network Formulations 245 6.1 Transmission Line Theory 245 6.1.1 Transmission Line Equations 246 6.1.2 Signal Propagations in Transmission Lines 249 6.2 Scattering Parameters for General Circuits 250 6.2.1 One-Port Network 250 6.2.2 Multi-Port Network 252 6.3 Waveguide Junctions 254 6.4 Multiple Antenna System 258 6.4.1 Impedance Matrix 258 6.4.2 Scattering Matrix 262 6.4.3 Antenna System with Large Separations 263 6.5 Power Transmission Between Antennas 267 6.5.1 Universal Power Transmission Formula 267 6.5.2 Power Transmission Between Two Planar Apertures 270 6.5.3 Power Transmission Between Two Antenna Arrays 273 6.6 Network Parameters in a Scattering Environment 275 6.6.1 Compensation Theorem for Time-Harmonic Fields 275 6.6.2 Scattering Parameters in a Scattering Environment 276 6.6.3 Antenna Input Impedance in a Scattering Environment 279 6.7 RLC Equivalent Circuits 280 6.7.1 RLC Equivalent Circuit for a One-Port Microwave Network 280 6.7.2 RLC Equivalent Circuits for Current Sources 282 7 Fields in Inhomogeneous Media 287 7.1 Foundations of Spectral Analysis 288 7.1.1 The Spectrum 288 7.1.2 Spectral Theorem 289 7.1.3 Generalized Eigenfunctions of Self-Adjoint Operators 290 7.1.4 Bilinear
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