foundations for radio frequency engineering

FOUNDATIONS FORRADIO FREQUENCY ENGINEERING

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N E W J E R S E Y L O N D O N S I N G A P O R E BE IJ ING S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

Wen GeyiNanjing University of Information Science & Technology, China

FOUNDATIONS FORRADIO FREQUENCY ENGINEERING

9040hc_9789814578707_tp.indd 2 17/9/13 10:08 AM

Published by

World Scientific Publishing Co. Pte. Ltd.

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Library of Congress Cataloging-in-Publication DataWen, Geyi. Foundations for radio frequency engineering / by Wen Geyi (Nanjing University of Information Science & Technology, China). pages cm Includes bibliographical references and index. ISBN 978-9814578707 (alk. paper) 1. Radio wave propagation--Mathematics. 2. Electromagnetic waves--Mathematical models. 3. Microwaves. I. Title. II. Title: Radio frequency engineering. QC665.T7W46 2015 621.384--dc23 2014045142

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Copyright 2015 by World Scientific Publishing Co. Pte. Ltd.

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To my parents

To Jun and Lan

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Preface

In an age of knowledge explosion, students have to meet the challengesof maintaining perspective amid a deluge of information and change intheir developments of expertise. The traditional university courses and theircontents must therefore be designed, planned or merged accordingly so thatthe students can master the core materials that are needed in their futurecareers, while having enough time to study the new courses to be frequentlyadded to the curriculum.

With the rapid development of wireless communication technologies,the demand on wireless spectrum has been growing dramatically. Thisresults in extensive and intensive research in radio frequency (RF) the-ory and techniques, and substantial advancements in the area of radioengineering, both in theory and practice, have emerged in recent years.RF engineering deals with various devices that are designed to operate inthe frequency range from 3kHz to 300GHz, and therefore covers all areaswhere electromagnetic elds must be transmitted or received as a carrierwave. For this reason, a good RF engineer must have in-depth knowledgein mathematics and physics, as well as specialized training in the areas ofapplied electromagnetics such as guided structures and microwave circuits,antenna and wave propagation, and electromagnetic compatibility (EMC)designs of electronic circuits.

RF engineering is closely linked to three IEEE (Institute of Electricaland Electronics Engineers) professional societies: Microwave Theory andTechniques (MTT), Antennas and Propagation (AP) and EMC. Tradition-ally, dierent courses have been created to meet the needs for dierentsocieties. For example, the students oriented to the MTT society musttake the courses such as Microwave Engineering, or Field Theory of

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Guided Waves. For the students specialized in AP and EMC societies, thecourses Antenna Theory and Design and Electromagnetic Compatibility arecompulsory. Nevertheless, these three professional societies are intimatelyrelated as they have numerous things in common in terms of theories andtechniques. Many professionals are often active in the three societies at thesame time. In fact, a typical RF department in a wireless company hasengineers working in dierent areas belonging to these societies, and mostof the time, they have to work together to solve an engineering problem asa team.

The above trends indicate that it is necessary and also possible to createa new course or a book that provides the fundamentals of microwaves,antennas and propagation, and EMC in a common framework for thestudents, engineers and applied physicists dedicated to the three IEEEsocieties. The topics in RF engineering are enormous. The contents ofthe book have been selected on the basis of their fundamentality andimportance to suit various needs arising in RF engineering. All areas ofRF engineering are established on the solutions of Maxwell equations,which can be solved either analytically or numerically. Before the inven-tion of computers, analytical methods were the dominant tools for theanalysis of electromagnetic phenomena, often involving the applications ofsophisticated mathematics and closed-form solutions. Nowadays, computertechnology plays a tremendous role in our daily life as well as in scienticresearch activities. By taking advantage of the capabilities of modern com-puter technologies and the state-of-the-art computer-aided design (CAD)tools, the numerical methods are capable of solving many complicatedproblems encountered in practice and they occupy a signicant piece ofcurrent academic research. In electromagnetic society, numerical methodshave been treated in many references. For this reason, this book essentiallyexamines analytical techniques while typical numerical methods and theirapplications will also be discussed.

One of the important research areas of RF engineering is the microwaveeld theory which may be applied to the analysis of guided waves,resonances, radiations and scattering. In many situations, a microwaveeld problem can be reduced to a network or circuit problem, allowingus to apply the circuit and network methods to solve the original eldproblem. The network formulation has eliminated unnecessary details in theeld theory while reserving useful global information, such as the terminalvoltages and currents. As a consequence, many RF engineers now largelyrely on CAD tools and circuit analysis with little or no eld analysis.

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This procedure, however, is not always successful. In fact, the initial RFcircuits resulting from CAD tools usually bear little resemblance to thenal design, and revisions are needed to achieve the required performances.One should always remember that the eld theory is the foundation ofthe circuit analysis, and its importance cannot be overemphasized in orderfor best innovation practices. For this reason, both microwave eld theoryand circuit design theory are discussed in this book, which features a widecoverage of the fundamental topics such as electromagnetic boundary valueproblems, waveguide theory, microwave resonators, microwave circuits,antennas and wave propagation, EMC techniques, and information theoryand typical application systems.

The book consists of 8 chapters. Chapter 1 reviews the fundamentalelectromagnetic theory. The basic properties and important theoremsderived from Maxwell equations are summarized. When applied properly,these properties and theorems may bring deep physical insight intothe practical problems and simplify them dramatically. Various solutionmethods for the boundary value problems related to Maxwell equations arediscussed, which includes the method of separation of variables, the methodof Greens functions, and the method of variations. Some important topicssuch as numerical techniques and potential theory are also addressed.

Chapter 2 deals with the waveguide theory. Waveguides are thecornerstone of microwave engineering and their counterparts are connectingwires in low frequency circuits. The waveguide theory can be formulated asan eigenvalue problem with the cut-o wavenumbers being the eigenvaluesand eigenfunctions being the corresponding vector modal functions. A vari-ational principle exists for the cut-o wavenumbers and can be expressed asa Rayleigh quotient. The vector modal functions are the extremal functionsthat make the Rayleigh quotient stationary and constitute a completeset. The typical waveguide eigenvalue problems are solved by the methodof separation of variables as well as by various numerical methods. Thewaveguide discontinuities or waveguide junctions are analyzed by the eldexpansions in terms of the vector modal functions as well as numericalmethods. Also presented in this chapter are inhomogeneous waveguidessuch as dielectric waveguides and microstrip lines, transient phenomena inwaveguides, and periodic structures.

A resonator is a device that oscillates at some frequencies (calledresonant frequencies) with greater amplitude than at others, and it isused to either generate waves of specic frequencies or to select specicfrequencies from a signal. Its counterpart is the LC resonant circuit at

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low frequency. A microwave resonator is an important building blockfor microwave circuits. A cavity resonator also constitutes an eigenvalueproblem and there exists a variational principle for the resonant frequencies.The theory of cavity resonators parallels that of waveguides. Chapter 3investigates the theory of microwave resonators. It includes the solutionsof vector modal functions for typical cavity resonators, coupling betweenthe waveguide and cavity resonator, dielectric resonators, microstrip patchresonators and open resonators.

A microwave circuit is composed of distributed elements with dimen-sions comparable to the wavelength. The amplitudes and phases of thevoltage and current may vary signicantly over the length of the circuit.The essential tools for the analysis and design of microwave circuits includethe theory of transmission lines, network analysis and synthesis, impedancematching, and lter theory. Many design problems in microwave integratedcircuit systems can be reduced to a circuit problem without too muchinvolvement of the eld theory. According to their functions, the RF circuitcomponents may be classied as frequency-related, impedance-related andpower-related. The fundamental aspects of the network theory and thedesign principles for various passive and active RF components, suchas phase shifters, attenuators, power combiners and dividers, directionalcouplers, lters, ampliers, oscillators, and mixers, are summarized andelucidated in Chapter 4.

An antenna is a device which converts a guided wave in a feeding lineinto a radio wave in free space, and vice versa. Antennas are essentialto all wireless systems and play a role in linking the components in thesystems through radio waves. In transmission mode, a radio transmittersupplies RF power to the antennas terminals through a transmission line,and the antenna radiates the energy into space as an electromagneticwave. In reception mode, the antenna intercepts the power from theincoming electromagnetic wave, inducing a weak voltage signal at itsterminal. The induced voltage signal is then amplied for further processing.Antennas can be designed to transmit or receive radio waves in a particulardirection (directional antennas) or in all directions equally (omnidirectionalantennas). Chapter 5 is devoted to the antenna theory as well as theradiation mechanisms of typical antenna systems, including wire antennas,slot and aperture antennas, broadband antennas, and array antennas. Itfeatures a wide coverage of advanced topics, such as the spherical vectorwavefunctions, Foster reactance theorem for ideal antennas, equivalentcircuits for transmitting and receiving antennas, the physical limitations

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of antennas, and the methods for the evaluation of antenna qualityfactor.

A radio propagation model is a mathematical formulation for the char-acterization of radio wave propagation as a function of frequency, distanceand other conditions. Many factors may change propagation propertiesof radio waves. The atmosphere, the ground, mountains, buildings, andweather conditions all have major inuences on wave propagations andcause variation in receiving signal strength. The propagation models canbe roughly divided into statistical and deterministic models. The statisticalmodels are derived from extensive eld measurements and statisticalanalysis, and are valid for similar environments where the measurementswere carried out. Sometimes site-specic deterministic propagation modelsare preferred for more accurate predictions of radio wave propagations.Chapter 6 is concerned with the propagation of radio waves in atmosphereand the ray tracing techniques, the statistical models for mobile channels,and the propagation models for deterministic multiple-input multiple-output (MIMO) systems.

EMC studies the unintentional generation, transmission and receptionof electromagnetic energy, and deals with the electromagnetic interferences(EMI) or disturbance that the unintentional electromagnetic energy (as anexternal source) may induce. Its aim is to ensure that the electronic equip-ment will not interfere with each others normal operation. Compliance withnational or international standards is usually required by laws passed byindividual nations before the electronic devices are brought to the market.Chapter 7 investigates the EMC in modern electronic circuits and systems.The relationship between the elds and circuits is discussed. The basic rulesfor emission reduction are expounded. The transmission line models for thestudy of susceptibility are introduced. The eective techniques to cope withthe EMC issues are investigated.

The pioneering work on information theory by Shannon (1948) andWiener (1949) has laid the foundation of modern communication theoryand information systems. The fundamental theorem of information theorystates that it is possible to transmit information through a noise channelat any rate less than channel capacity with an arbitrarily small probabilityof error. A signal chosen from a specied class is to be transmitted througha communication channel as an input and is received at the output of thechannel. During the transmission, the signal may be altered by noise anddistortion. For each permissible input, the output is determined statisticallyby a probability distribution assigned by the channel. At the output, a

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statistical decision must be made to identify the transmitted signal asreliably as possible. Chapter 8 gives a brief introduction to the informationtheory and typical information systems. It covers the probability theoryand random process, information theory, digital communication systems,and radar systems.

The book can be used as a graduate-level text or as a reference forresearchers and engineers in applied electromagnetics. The prerequisite forthe book is advanced calculus. The SI unit system is used throughout thebook. A ejt time variation is assumed for time-harmonic elds. A specialsymbol is used to indicate the end of an example or a remark. Thereferences are not meant to be complete but the author has tried to indicatethe origins of the important results included in the book.

In preparing this book, I have beneted from suggestions from manycolleagues and friends, and would like to take this occasion to extendmy sincere thanks to all of them. Particular thanks go to Prof. ThomasT. Y. Wong of Illinois Institute of Technology and Prof. Jun Xiang Ge ofNanjing University of Information Science and Technology. Last but notleast, I would like to express my deepest gratitude to my family membersfor their constant encouragement and support.

Wen GeyiNanjing, China

May 2014

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Contents

Preface vii

Chapter 1. Solutions of Electromagnetic Field Problems 11.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Maxwell Equations and Boundary Conditions . . . . . 41.1.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . 81.1.3 Wave Equations . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.5 Electromagnetic Field Theorems . . . . . . . . . . . . . 12

1.1.5.1 Superposition Theorem . . . . . . . . . . . . . 131.1.5.2 Conservation of Electromagnetic Energy . . . 131.1.5.3 Uniqueness Theorems . . . . . . . . . . . . . . 151.1.5.4 Equivalence Theorems . . . . . . . . . . . . . 161.1.5.5 Reciprocity . . . . . . . . . . . . . . . . . . . . 19

1.2 Method of Separation of Variables . . . . . . . . . . . . . . . . 201.2.1 Eigenvalue Problem of SturmLiouville Type . . . . . . 211.2.2 Rectangular Coordinate System . . . . . . . . . . . . . 231.2.3 Cylindrical Coordinate System . . . . . . . . . . . . . . 241.2.4 Spherical Coordinate System . . . . . . . . . . . . . . . 26

1.3 Method of Greens Functions . . . . . . . . . . . . . . . . . . . 281.3.1 Greens Functions for Helmholtz Equation . . . . . . . 291.3.2 Partial Dierential Equations and Integral

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.3 Dyadic Greens Functions . . . . . . . . . . . . . . . . . 321.3.4 Greens Functions and Spectral Representation . . . . . 35

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1.4 Variational Method and Numerical Techniques . . . . . . . . . 371.4.1 Functional Derivative . . . . . . . . . . . . . . . . . . . 371.4.2 Variational Expressions and RayleighRitz

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.4.3 Numerical Techniques: A General Approach . . . . . . 44

1.5 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 531.5.1 Vector Potential, Scalar Potential, and

Gauge Conditions . . . . . . . . . . . . . . . . . . . . . 531.5.2 Hertz Vectors and Debye Potentials . . . . . . . . . . . 551.5.3 Jump Relations . . . . . . . . . . . . . . . . . . . . . . 591.5.4 Multipole Expansion . . . . . . . . . . . . . . . . . . . 61

Chapter 2. Waveguides 652.1 Modal Theory for Metal Waveguides . . . . . . . . . . . . . . . 66

2.1.1 Eigenvalue Equation . . . . . . . . . . . . . . . . . . . . 672.1.2 Properties of Modal Functions . . . . . . . . . . . . . . 692.1.3 Mode Excitation . . . . . . . . . . . . . . . . . . . . . . 76

2.2 Vector Modal Functions . . . . . . . . . . . . . . . . . . . . . . 782.2.1 Rectangular Waveguide . . . . . . . . . . . . . . . . . . 782.2.2 Circular Waveguide . . . . . . . . . . . . . . . . . . . . 822.2.3 Coaxial Waveguide . . . . . . . . . . . . . . . . . . . . 842.2.4 Numerical Analysis for Metal Waveguides . . . . . . . . 86

2.2.4.1 Boundary Element Method . . . . . . . . . . . 862.2.4.2 Finite Dierence Method . . . . . . . . . . . . 902.2.4.3 Finite Element Method . . . . . . . . . . . . . 93

2.3 Inhomogeneous Metal Waveguides . . . . . . . . . . . . . . . . 952.3.1 General Field Relationships . . . . . . . . . . . . . . . . 952.3.2 Symmetric Formulation . . . . . . . . . . . . . . . . . . 972.3.3 Asymmetric Formulation . . . . . . . . . . . . . . . . . 982.3.4 Dielectric-Slab-Loaded Rectangular Waveguides . . . . 99

2.4 Waveguide Discontinuities . . . . . . . . . . . . . . . . . . . . . 1012.4.1 Network Representation of Waveguide

Discontinuities . . . . . . . . . . . . . . . . . . . . . . . 1012.4.2 Diaphragms in Waveguide-Variational Method . . . . . 1032.4.3 Conducting Posts in Waveguide Method

of Greens Function . . . . . . . . . . . . . . . . . . . . 1062.4.4 Waveguide Steps Mode Matching Method . . . . . . 1092.4.5 Coupling by Small Apertures . . . . . . . . . . . . . . . 1112.4.6 Numerical Analysis Finite Dierence Method . . . . 117

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2.5 Transient Fields in Waveguides . . . . . . . . . . . . . . . . . . 1202.5.1 Field Expansions . . . . . . . . . . . . . . . . . . . . . . 1212.5.2 Solution of Modied KleinGordon Equation . . . . . . 123

2.6 Dielectric Waveguides . . . . . . . . . . . . . . . . . . . . . . . 1252.6.1 Guidance Condition . . . . . . . . . . . . . . . . . . . . 1252.6.2 Circular Optical Fiber . . . . . . . . . . . . . . . . . . . 1282.6.3 Dielectric Slab Waveguide . . . . . . . . . . . . . . . . . 131

2.7 Microstrip Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 1322.7.1 Spectral-Domain Analysis . . . . . . . . . . . . . . . . . 1342.7.2 Closed Form Formulae for Microstrip Lines . . . . . . . 137

2.7.2.1 Analysis Formulae . . . . . . . . . . . . . . . . 1372.7.2.2 Synthesis Formulae . . . . . . . . . . . . . . . 138

2.7.3 Microstrip Discontinuities . . . . . . . . . . . . . . . . . 1382.7.3.1 Waveguide Models . . . . . . . . . . . . . . . . 1382.7.3.2 Method of Greens Function . . . . . . . . . . 140

2.8 Waveguide with Lossy Walls . . . . . . . . . . . . . . . . . . . 1432.9 Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . 146

2.9.1 Properties of Periodic Structures . . . . . . . . . . . . . 1472.9.2 Equivalent Circuit for Periodic Structures . . . . . . . . 1482.9.3 Diagram . . . . . . . . . . . . . . . . . . . . . . . . 151

Chapter 3. Microwave Resonators 155

3.1 Theory of Metal Cavity Resonators . . . . . . . . . . . . . . . 1563.1.1 Field Expansions for Cavity Resonators . . . . . . . . . 1573.1.2 Vector Modal Functions for Waveguide Cavity

Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 1653.1.2.1 Field Expansions in Waveguide Cavity

Resonator . . . . . . . . . . . . . . . . . . . . 1653.1.2.2 Rectangular Waveguide Cavity Resonator . . . 1683.1.2.3 Circular Waveguide Cavity Resonator . . . . . 1683.1.2.4 Coaxial Waveguide Cavity Resonator . . . . . 169

3.1.3 Integral Equation for Cavity Resonators . . . . . . . . . 1713.2 Coupling between Waveguide and Cavity Resonator . . . . . . 172

3.2.1 One-Port Microwave Network as a RLC Circuit . . . . 1723.2.2 Properties of RLC Resonant Circuit . . . . . . . . . . . 1743.2.3 Aperture Coupling to Cavity Resonator . . . . . . . . . 1753.2.4 Probe Coupling to Cavity Resonator . . . . . . . . . . . 178

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3.3 Dielectric Resonator . . . . . . . . . . . . . . . . . . . . . . . . 1833.3.1 Representation of the Fields in a Cylindrical

System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843.3.2 Circular Cylindrical Dielectric Resonator Mixed

Magnetic Wall Model . . . . . . . . . . . . . . . . . . . 1853.3.2.1 TE Modes . . . . . . . . . . . . . . . . . . . . 1853.3.2.2 TM Modes . . . . . . . . . . . . . . . . . . . . 187

3.3.3 Integral Equation for Dielectric Resonators . . . . . . . 1883.4 Microstrip Resonators . . . . . . . . . . . . . . . . . . . . . . . 1933.5 Open Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 194

3.5.1 Paraxial Approximations . . . . . . . . . . . . . . . . . 1943.5.2 Modes in Open Resonators . . . . . . . . . . . . . . . . 198

Chapter 4. Microwave Circuits 201

4.1 Circuit Theory of Transmission Lines . . . . . . . . . . . . . . 2034.1.1 Transmission Line Equations . . . . . . . . . . . . . . . 2034.1.2 Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . 209

4.2 Network Parameters . . . . . . . . . . . . . . . . . . . . . . . . 2114.2.1 One-Port Network . . . . . . . . . . . . . . . . . . . . . 2114.2.2 Multi-Port Network . . . . . . . . . . . . . . . . . . . . 2164.2.3 Foster Reactance Theorem . . . . . . . . . . . . . . . . 220

4.3 Impedance Matching Circuits . . . . . . . . . . . . . . . . . . . 2244.3.1 Basic Concept of Match . . . . . . . . . . . . . . . . . . 224

4.3.1.1 Impedance Matching for PureResistances . . . . . . . . . . . . . . . . . . . . 225

4.3.1.2 Impedance Matching for Complex Loads . . . 2274.3.2 Quarter-Wave Impedance Transformer . . . . . . . . . 2284.3.3 Tapered Line Transformer . . . . . . . . . . . . . . . . 229

4.4 Passive Components . . . . . . . . . . . . . . . . . . . . . . . . 2304.4.1 Electronically Controlled Phase Shifters . . . . . . . . . 2304.4.2 Attenuators . . . . . . . . . . . . . . . . . . . . . . . . 2324.4.3 Power Dividers and Combiners . . . . . . . . . . . . . . 2334.4.4 Directional Couplers . . . . . . . . . . . . . . . . . . . . 234

4.4.4.1 Hole Couplers . . . . . . . . . . . . . . . . . . 2354.4.4.2 Branch-Line Coupler . . . . . . . . . . . . . . 238

4.4.5 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.4.5.1 Insertion Loss . . . . . . . . . . . . . . . . . . 2394.4.5.2 Low-Pass Filter Prototypes . . . . . . . . . . . 243

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4.4.5.3 Frequency Transformations . . . . . . . . . . . 2514.4.5.4 Filter Implementation . . . . . . . . . . . . . . 255

4.5 Active Components . . . . . . . . . . . . . . . . . . . . . . . . 2604.5.1 Ampliers . . . . . . . . . . . . . . . . . . . . . . . . . 260

4.5.1.1 Power Gains for Two-Port Network . . . . . . 2604.5.1.2 Stability Criteria . . . . . . . . . . . . . . . . 2634.5.1.3 Noise Theory for Two-Port Network . . . . . . 2664.5.1.4 Amplier Design . . . . . . . . . . . . . . . . . 271

4.5.2 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 2794.5.2.1 Feedback Oscillators . . . . . . . . . . . . . . 2804.5.2.2 Negative Resistance Oscillators . . . . . . . . 2824.5.2.3 Dielectric Resonator Oscillators . . . . . . . . 285

4.5.3 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2854.5.3.1 Characteristics of Diode . . . . . . . . . . . . 2864.5.3.2 Mixer Designs . . . . . . . . . . . . . . . . . . 287

Chapter 5. Antennas 291

5.1 From Transmission Lines to Antennas . . . . . . . . . . . . . . 2935.1.1 Antennas Transformed from Two-Wire Lines . . . . . . 2945.1.2 Antennas Transformed from Coaxial Cables . . . . . . 2945.1.3 Antennas Transformed from Waveguides . . . . . . . . 295

5.2 Antenna Parameters . . . . . . . . . . . . . . . . . . . . . . . . 2955.2.1 Field Regions . . . . . . . . . . . . . . . . . . . . . . . . 2975.2.2 Radiation Patterns and Radiation Intensity . . . . . . . 2985.2.3 Radiation Eciency, Antenna Eciency and Matching

Network Eciency . . . . . . . . . . . . . . . . . . . . . 2995.2.4 Directivity and Gain . . . . . . . . . . . . . . . . . . . 3005.2.5 Input Impedance, Bandwidth and Antenna

Quality Factor . . . . . . . . . . . . . . . . . . . . . . . 3015.2.6 Vector Eective Length, Equivalent Area and

Antenna Factor . . . . . . . . . . . . . . . . . . . . . . 3025.2.7 Polarization and Coupling . . . . . . . . . . . . . . . . 3045.2.8 Specic Absorption Rate . . . . . . . . . . . . . . . . . 306

5.3 Spherical Vector Wavefunctions . . . . . . . . . . . . . . . . . 3075.4 Generic Properties of Antennas . . . . . . . . . . . . . . . . . . 310

5.4.1 Far Fields and Scattering Matrix . . . . . . . . . . . . . 3115.4.2 Poynting Theorem and Stored Energies . . . . . . . . . 3165.4.3 Equivalent Circuits for Antennas . . . . . . . . . . . . . 320

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5.4.3.1 Equivalent Circuit for TransmittingAntennas . . . . . . . . . . . . . . . . . . . . . 321

5.4.3.2 Equivalent Circuit for ReceivingAntennas . . . . . . . . . . . . . . . . . . . . . 323

5.4.4 Foster Reactance Theorem for Lossless Antennas . . . . 3305.4.5 Quality Factor and Bandwidth . . . . . . . . . . . . . . 339

5.4.5.1 Evaluation of Q from Input Impedance . . . . 3395.4.5.2 Evaluation of Q from Current

Distribution . . . . . . . . . . . . . . . . . . . 3405.4.5.3 Relationship between Q and Bandwidth . . . 3465.4.5.4 Minimum Possible Antenna Quality

Factor . . . . . . . . . . . . . . . . . . . . . . 3465.4.6 Maximum Possible Product of Gain and

Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 3495.4.6.1 Directive Antenna . . . . . . . . . . . . . . . . 3495.4.6.2 Omni-directional Antenna . . . . . . . . . . . 3515.4.6.3 Best Possible Antenna Performance . . . . . . 353

5.5 Wire Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3555.5.1 Asymptotic Solutions for Wire Antennas . . . . . . . . 3575.5.2 Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . 3615.5.3 Loop Antenna . . . . . . . . . . . . . . . . . . . . . . . 364

5.6 Slot Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3675.6.1 Babinets Principle . . . . . . . . . . . . . . . . . . . . 3675.6.2 Impedance of Slot Antennas . . . . . . . . . . . . . . . 369

5.7 Aperture Antennas . . . . . . . . . . . . . . . . . . . . . . . . 3715.8 Microstrip Patch Antennas . . . . . . . . . . . . . . . . . . . . 3775.9 Broadband Antennas . . . . . . . . . . . . . . . . . . . . . . . 380

5.9.1 Biconical Antenna . . . . . . . . . . . . . . . . . . . . . 3805.9.2 Helical Antenna . . . . . . . . . . . . . . . . . . . . . . 384

5.9.2.1 Normal Mode . . . . . . . . . . . . . . . . . . 3845.9.2.2 Axial Mode . . . . . . . . . . . . . . . . . . . 386

5.9.3 Frequency-Independent Antennas . . . . . . . . . . . . 3865.10 Coupling between Two Antennas . . . . . . . . . . . . . . . . . 388

5.10.1 A General Approach . . . . . . . . . . . . . . . . . . . . 3895.10.2 Coupling between Two Antennas with Large

Separation . . . . . . . . . . . . . . . . . . . . . . . . . 3935.10.3 Power Transmission between Two Antennas . . . . . . 395

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Contents xix

5.10.3.1 Power Transmission between TwoGeneral Antennas . . . . . . . . . . . . . . . . 395

5.10.3.2 Maximum Power Transmission between TwoPlanar Apertures . . . . . . . . . . . . . . . . 398

5.10.4 Antenna Gain Measurement . . . . . . . . . . . . . . . 4035.11 Array Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 405

5.11.1 A General Approach . . . . . . . . . . . . . . . . . . . . 4055.11.2 YagiUda Antenna . . . . . . . . . . . . . . . . . . . . 4085.11.3 Log Periodic Antennas . . . . . . . . . . . . . . . . . . 4095.11.4 Optimal Design of Multiple Antenna Systems . . . . . 410

5.11.4.1 Power Transmission between TwoAntenna Arrays . . . . . . . . . . . . . . . . . 413

5.11.4.2 Optimal Design of Antenna Arrays . . . . . . 415

Chapter 6. Propagation of Radio Waves 4216.1 Earths Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . 423

6.1.1 Structure of Atmosphere . . . . . . . . . . . . . . . . . 4236.1.2 Weather Phenomena . . . . . . . . . . . . . . . . . . . . 425

6.2 Wave Propagation in Atmosphere . . . . . . . . . . . . . . . . 4286.2.1 Propagation of Radio Waves over the Earth . . . . . . 429

6.2.1.1 A General Approach . . . . . . . . . . . . . . 4296.2.1.2 Vertical Current Element over the Earth . . . 4336.2.1.3 Two-Ray Propagation Model . . . . . . . . . . 438

6.2.2 Wave Propagation in Atmosphere: Ray-TracingMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

6.2.3 Ionospheric Wave Propagation . . . . . . . . . . . . . . 4516.2.4 Tropospheric-Scatter-Propagation . . . . . . . . . . . . 4546.2.5 Attenuation by Rain . . . . . . . . . . . . . . . . . . . . 456

6.3 Statistical Models for Mobile Radio Channels . . . . . . . . . . 4596.3.1 Near-Earth Large-Scale Models . . . . . . . . . . . . . . 460

6.3.1.1 Okumura Model . . . . . . . . . . . . . . . . . 4616.3.1.2 Hata Model . . . . . . . . . . . . . . . . . . . 4616.3.1.3 COST-231 Model . . . . . . . . . . . . . . . . 4626.3.1.4 Log-Distance Model . . . . . . . . . . . . . . . 463

6.3.2 Small-Scale Fading . . . . . . . . . . . . . . . . . . . . . 4636.4 Propagation Models for Deterministic MIMO System . . . . . 467

6.4.1 Channel Matrix . . . . . . . . . . . . . . . . . . . . . . 4676.4.2 Computation of Channel Matrix Elements . . . . . . . 470

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xx Contents

Chapter 7. Electromagnetic Compatibility 477

7.1 Fields and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 4787.1.1 Impressed Field and Scattered Field . . . . . . . . . . . 4807.1.2 Kirchhos Laws . . . . . . . . . . . . . . . . . . . . . . 4817.1.3 Low-frequency Approximations and Lumped

Circuit Parameters . . . . . . . . . . . . . . . . . . . . 4827.1.3.1 RLC Circuits . . . . . . . . . . . . . . . . . . 4827.1.3.2 Lumped Circuit Elements . . . . . . . . . . . 486

7.1.4 Mutual Coupling between Low-Frequency Circuits . . . 4897.1.4.1 Inductive Coupling . . . . . . . . . . . . . . . 4897.1.4.2 Capacitive Coupling . . . . . . . . . . . . . . . 490

7.2 Electromagnetic Emissions and Susceptibility . . . . . . . . . . 4917.2.1 Rules for Emission Reductions . . . . . . . . . . . . . . 4917.2.2 Fields of Electric Dipoles . . . . . . . . . . . . . . . . . 493

7.2.2.1 Innitesimal Electric Dipole . . . . . . . . . . 4947.2.2.2 Electrically Short Dipole Antennas . . . . . . 497

7.2.3 Fields of Magnetic Dipoles . . . . . . . . . . . . . . . . 4997.2.4 Emissions from Common Mode Current and

Dierential Mode Current . . . . . . . . . . . . . . . . 5017.2.5 Multi-Conductor Transmission Line Models for

Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 5027.3 Electromagnetic Coupling through Apertures . . . . . . . . . . 510

7.3.1 Coupling through Arbitrary Apertures . . . . . . . . . 5117.3.2 Coupling through Small Apertures . . . . . . . . . . . . 513

7.4 EMC Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 5167.4.1 Shielding Method . . . . . . . . . . . . . . . . . . . . . 517

7.4.1.1 Shielding Eectiveness: Far-Field Sources . . . 5177.4.1.2 Shielding Eectiveness: Near-Field

Sources . . . . . . . . . . . . . . . . . . . . . . 5207.4.1.3 Electrostatic Shielding . . . . . . . . . . . . . 522

7.4.2 Filtering Method . . . . . . . . . . . . . . . . . . . . . . 5267.4.2.1 Line Impedance Stabilization Network . . . . 5267.4.2.2 Common-Mode and Dierential-Mode . . . . . 5287.4.2.3 Power Supply Filters . . . . . . . . . . . . . . 529

7.4.3 Grounding Method . . . . . . . . . . . . . . . . . . . . 5307.4.3.1 Safety Ground . . . . . . . . . . . . . . . . . . 5317.4.3.2 Signal Ground . . . . . . . . . . . . . . . . . . 531

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Contents xxi

7.5 Lightning Protection . . . . . . . . . . . . . . . . . . . . . . . . 5327.5.1 Lightning Discharge and Lighting Terminology . . . . . 5327.5.2 Lightning Protection . . . . . . . . . . . . . . . . . . . 536

Chapter 8. Information Theory and Systems 539

8.1 Probability Theory and Random Process . . . . . . . . . . . . 5408.1.1 Probability Space . . . . . . . . . . . . . . . . . . . . . 5408.1.2 Probability Distribution Function . . . . . . . . . . . . 5428.1.3 Mathematical Expectations and Moments . . . . . . . . 5458.1.4 Stochastic Process . . . . . . . . . . . . . . . . . . . . . 546

8.1.4.1 Time-Average and Ensemble-Average . . . . . 5478.1.4.2 Power Spectral Density . . . . . . . . . . . . . 548

8.1.5 Gaussian Process . . . . . . . . . . . . . . . . . . . . . 5508.1.6 Complex Gaussian Density Function . . . . . . . . . . . 5518.1.7 Analytic Representation . . . . . . . . . . . . . . . . . . 5528.1.8 Narrow-Band Stationary Stochastic Process . . . . . . 554

8.2 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . 5578.2.1 System with One Random Variable . . . . . . . . . . . 5578.2.2 System with Two Random Variables . . . . . . . . . . . 5598.2.3 System with More Than Two Random Variables . . . . 5608.2.4 Channel Capacity of Deterministic MIMO System . . . 562

8.3 Digital Communication Systems . . . . . . . . . . . . . . . . . 5698.3.1 Digital Modulation Techniques . . . . . . . . . . . . . . 571

8.3.1.1 Baseband Transmission . . . . . . . . . . . . . 5718.3.1.2 Modulation and Demodulation . . . . . . . . . 577

8.3.2 Probability of Error . . . . . . . . . . . . . . . . . . . . 5868.3.3 Link Budget Analysis . . . . . . . . . . . . . . . . . . . 591

8.3.3.1 Link Margin, Noise Figure and NoiseTemperature . . . . . . . . . . . . . . . . . . . 592

8.3.3.2 Link Budget Analysis for Mobile Systems . . . 5958.3.4 Mobile Antennas and Environments . . . . . . . . . . . 596

8.3.4.1 Incident Signal . . . . . . . . . . . . . . . . . . 5978.3.4.2 Received Signal by Mobile Antenna . . . . . . 6008.3.4.3 Mean Eective Gain . . . . . . . . . . . . . . . 602

8.4 Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 6028.4.1 Radar Signals . . . . . . . . . . . . . . . . . . . . . . . 6028.4.2 Radar Cross Section . . . . . . . . . . . . . . . . . . . . 605

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xxii Contents

8.4.2.1 Scattering by Conducting Targets . . . . . . . 6068.4.2.2 Scattering by Rain . . . . . . . . . . . . . . . 6108.4.2.3 Eect of Polarization . . . . . . . . . . . . . . 612

8.4.3 Radar Range Equation . . . . . . . . . . . . . . . . . . 615

Bibliography 617

Index 639

January 28, 2015 11:5 Foundations for Radio Frequency Engineering - 9in x 6in b1914-ch01 page 1

Chapter 1

Solutions of ElectromagneticField Problems

One scientific epoch ended and another began with James Clerk

Maxwell.

Albert Einstein

The electromagnetic theory is the foundation of radio frequency (RF)engineering. In 1873, J.C. Maxwell (18311879) summarized the theory onelectricity and magnetism discovered by many great physicists includingH.C. Oersted (17771851), A.M. Ampe`re (17751836), and M. Faraday(17911861), and formulated a set of equations since known as Maxwellequations, representing one of the great achievements in physics. TheMaxwell equations describe the behavior of electric and magnetic elds,as well as their interactions with matter, and they are the starting pointfor the investigation of all macroscopic electromagnetic phenomena.

Radio frequency (RF) refers to the frequency range from 3KHz to300GHz. RF engineering deals with various wireless systems, and is animportant subject in electrical engineering. RF technologies are widely usedin xed and mobile communication, broadcasting, radar and navigationsystems, satellite communication, computer networks and innumerableother applications. Dierent frequencies of radio waves have dierentpropagation characteristics in the Earths atmosphere. Table 1.1 showsvarious frequency bands and their major applications.

Microwave frequency often refers to the frequency range from 1GHzto 300GHz. Table 1.2 gives the old and new names for typical microwavefrequency bands. At the low end of the microwave spectrum, the traditionallumped circuit theory starts to become ineective, and the eld theory thusenters the picture. Microwave eld theory is one of the important researchareas of RF engineering, which may be applied to solve various boundary

1


2 Foundations for Radio Frequency Engineering

Table 1.1 RF spectrum

Frequency/wavelength Designation Applications

3Hz30Hz/105 km104 km

ELF (Extremely lowfrequency)

Submarines

30Hz300Hz/104 km103 km

SLF (Super low frequency) Power grids, submarines

300Hz3 kHz/103 km102 km

ULF (Ultra low frequency) Earthquake studies

3 kHz30 kHz/100 km10 km

VLF (Very low frequency) Submarines near the surface

30 kHz300 kHz/10 km1 km

LF (Low frequency) Submarines, aircraft beacons,AM broadcast, navigation

300 kHz3MHz/1 km100m

MF (Medium frequency) AM broadcast, navigation

3MHz30MHz/100m10m

HF (High frequency) Shortwave broadcast, over thehorizon radar

30MHz300MHz/10m1m

VHF (Very high frequency) FM, TV

300MHz3GHz/1m10 cm

UHF (Ultra high frequency) TV, LAN, cellular, GPS

3GHz30 GHz/10 cm1 cm

SHF (Super high frequency) Radar, GSO satellites, datacommunications

30GHz300GHz/1 cm1mm

EHF (Extremely highfrequency)

Radar, automotive, datacommunications

300GHz3THz/1mm0.1mm

THF (Tremendously highfrequency)

Sensing and imaging, securityscreening, high-altitudecommunications

Table 1.2 Microwave frequency bands

Frequency (GHz) Old names New names

12 L D24 S E, F48 C G, H812 X I, J

1218 Ku J1826 K J2640 Ka K

value problems such as the analysis of guided waves, resonances, radiationsand scattering.

In mathematics, a boundary value problem consists of a dierentialequation together with a set of additional constraints on the boundary


Solutions of Electromagnetic Field Problems 3

Table 1.3 Some trinities for dierential equations

Trinity Description

Three types of dierentialequations:

Elliptical, hyperbolic and parabolic.

Three types of problems: Boundary value problems, initial valueproblems, and eigenvalue problems.

Three types of boundaryconditions:

Dirichlet boundary condition, named after theGerman mathematician Johann Peter GustavLejeune Dirichlet (18051859); Neumannboundary condition, named after the Germanmathematician Carl Gottfried Neumann(18321925); and Robin boundary condition,named after the French mathematician VictorGustave Robin (18551897).

Three importantmathematical tools:

Divergence theorem, inequalities andconvergence theorems.

Three analytical solutionmethods:

Separation of variables, Greens functionmethod, and variational method.

Three numerical solution

methods:

Finite element method, nite dierence method,

moment method.

of the domain of the equation (called the boundary conditions). Variousmethods for the solution of dierential equations have been proposed.Linear dierential equations are generally solved by means of variationalmethod, the method of separation of variables, and the method of Greensfunction, named after the British mathematician George Green (17931841). Some usual trinities for dierential equations are summarized inTable 1.3 (Gustafson, 1987).

1.1 Maxwell Equations

Maxwell equations are a set of partial dierential equations that formthe foundation of electrical and optical engineering. Maxwell equationsdescribe how electric and magnetic elds are generated by charges andcurrents and altered by each other. Maxwell equations have been provedto be very successful in explaining and predicting a variety of macroscopicphenomena. However, in some special situations such as extremely strongelds and extremely short distances, they may fail and can be noticeablyinaccurate. Moreover, Maxwell equations must be replaced by quantumelectrodynamics in order for dealing with microscopic phenomena.



1.1.1 Maxwell Equations and Boundary Conditions

Maxwell equations in the time domain can be expressed as follows

H(r, t) = D(r, t)t

+ J(r, t),

E(r, t) = B(r, t)t

,

D(r, t) = (r, t), B(r, t) = 0.

(1.1)

In (1.1), r is the observation point of the elds in meter and t is the timein second; H is the magnetic eld intensity measured in amperes per meter(A/m); B is the magnetic induction intensity measured in tesla (N/Am); Eis electric eld intensity measured in volts per meter (V/m); D is the electricinduction intensity measured in coulombs per square meter (C/m2); J iselectric current density measured in amperes per square meter (A/m2); isthe electric charge density measured in coulombs per cubic meter (C/m3).The rst equation is Ampe`res law, and it describes how the electric eldchanges according to the current density and magnetic eld. The secondequation is Faradays law, and it characterizes how the magnetic eld variesaccording to the electric eld. The minus sign is required by Lenzs law, i.e.,when an electromotive force is generated by a change of magnetic ux, thepolarity of the induced electromotive force is such that it produces a currentwhose magnetic eld opposes the change, which produces it. The thirdequation is Coulombs law, and it says that the electric eld depends onthe charge distribution and obeys the inverse square law. The last equationshows that there are no free magnetic monopoles and that the magneticeld also obeys the inverse square law. It should be understood that noneof the experiments had anything to do with waves at the time when Maxwellderived his equations. Maxwell equations imply more than the experimentalfacts. The continuity equation can be derived from (1.1) as

J(r, t) = (r, t)t

. (1.2)

The charge density and the current density J in Maxwell equations arefree charge density and currents and they exclude charges and currentsforming part of the structure of atoms and molecules. The bound chargesand currents are regarded as material, which are not included in andJ. The current density normally consists of two parts: J = Jcon + Jimp.Here Jimp is referred to as external or impressed current source, which



is independent of the elds and delivers energy to electric charges in asystem. The impressed current source can be of electric and magnetic typeas well as of non-electric or non-magnetic origin. Jcon = E, where isthe conductivity of the medium in mhos per meter, denotes the conductioncurrent induced by the impressed source Jimp. Sometimes it is convenientto introduce an external or impressed electric eld Eimp dened by Jimp =Eimp. In more general situation, one may write J = Jind(E,B) + Jimp,where Jind(E,B) is the induced current by the impressed current Jimp.

Sometimes it is convenient to introduce magnetic current Jm andmagnetic charges m, which are related by

Jm(r, t) = m(r, t)t

(1.3)

and Maxwell equations must be modied as

H(r, t) = D(r, t)t

+ J(r, t),

E(r, t) = B(r, t)t

Jm(r, t),

D(r, t) = (r, t), B(r, t) = m(r, t).

(1.4)

The inclusions of Jm and m make Maxwell equations more symmetricalthough there has been no evidence that the magnetic current and chargeare physically present. The validity of introducing such concepts in Maxwellequations is justied by the equivalence principle, i.e., they are introducedas a mathematical equivalent to electromagnetic elds.

If all the sources are of magnetic type, Equations (1.4) reduce to

H(r, t) = D(r, t)t

,

E(r, t) = B(r, t)t

Jm(r, t),

D(r, t) = 0, B(r, t) = m(r, t).

(1.5)

Mathematically (1.1) and (1.5) are similar. One can obtain one of themby simply interchanging symbols between the left and right columns inTable 1.4, where and denote the permeability and permittivity of themedium respectively. This property is called duality. The importance of



Table 1.4 Duality

Electric source Magnetic source

E HH EJ Jm m

the duality is that one can obtain the solution of magnetic type from thesolution of electric type by interchanging symbols and vice versa.

For the time-harmonic (sinusoidal) elds, Equations (1.1) and (1.2) canbe expressed as

H(r) = jD(r) + J(r),E(r) = jB(r), D(r) = (r), B(r) = 0, J(r) = j(r),

(1.6)

where the eld quantities denote the complex amplitudes (phasors)dened by

E(r, t) = Re[E(r)ejt], etc.

We use the same notations for both time-domain and frequency-domainquantities.

The force acting on a point charge q, moving with a velocity v withrespect to an observer, by the electromagnetic eld is given by

F(r, t) = q[E(r, t) + v(r, t) B(r, t)] (1.7)where E and B are the total elds, including the eld generated by themoving charge q. Equation (1.7) is referred to as Lorentz force equation,named after Dutch physicist Hendrik Antoon Lorentz (18531928). It isknown that there are two dierent formalisms in classical physics. One ismechanics that deals with particles, and the other is electromagnetic eldtheory that deals with radiated waves. The particles and waves are coupledthrough Lorentz force equation, which usually appears as an assumptionseparate from Maxwell equations. The Lorentz force is the only means



to detect electromagnetic elds. For a continuous charge distribution, theLorentz force equation becomes

f(r, t) = E(r, t) + J(r, t) B(r, t) (1.8)where f is the force density acting on the charge distribution , i.e., theforce acting on the charge distribution per unit volume. Maxwell equations,Lorentz force equation and continuity equation constitute the fundamentalequations in electrodynamics.

The boundary conditions on the surface between two dierent mediacan be easily obtained as follows

un (H1 H2) = Js,un (E1 E2) = 0,un (D1 D2) = s,un (B1 B2) = 0,

(1.9)

where un is the unit normal of the boundary directed from medium 2 tomedium 1; Js and s are the surface current density and surface chargedensity respectively.

Remark 1.1:To derive the boundary conditions (1.9), we may draw a smallcylinder of height h and base area S so that the boundary S betweenmedium 1 and medium 2 intersects the middle section of the cylinder asillustrated in Figure 1.1. If the base area is suciently small the elds maybe assumed to be a constant value over each end of the cylinder. Takingthe integration of the rst equation of (1.4) over the surface of the cylinder,we obtain

un H1S un H2S +Kh Dt

Sh = JSh, (1.10)

Medium 1

Medium 2

nu

S

h Boundary S

Figure 1.1 Derivation of boundary conditions.



where Kh denotes the integral of unH over the side walls of the cylinder.In the limit as h 0, the ends of the cylinder lie just on either side ofthe boundary S and the integral over the side walls becomes vanishinglysmall. Thus

limh0

(Kh D

tSh

)= 0, lim

h0Jh = Js.

Here Js stands for the surface current density. Equation (1.10) can bewritten as

un (H1 H2) = Js. (1.11)The rest of the equations in (1.9) can be derived in a similar way.

1.1.2 Constitutive Relations

Maxwell equations are a set of 7 equations involving 16 unknowns (i.e.,ve vector functions E,H,B,D,J and one scalar function and the lastequation of (1.1) is not independent). To determine the elds, nine moreequations are needed, and they are given by the generalized constitutiverelations:

D = f1(E,H), B = f2(E,H)

together with the generalized Ohms law:

J = f3(E,H)

if the medium is conducting. The constitutive relations establish theconnections between eld quantities and reect the properties of themedium, and they are totally independent of the Maxwell equations. Ananisotropic medium is dened by

Di(r, t) =

j=x,y,z

[aji (r)Ej(r, t) + (Gji Ej)(r, t)],

Bi(r, t) =

j=x,y,z

[dji (r)Hj(r, t) + (Fji Hj)(r, t)],

where i = x, y, z; denotes the convolution with respect to time; aji , djiare independent of time; and Gji , F

ji are functions of (r, t). A biisotropic

medium is dened by

D(r, t) = a(r)E(r, t) + b(r)H(r, t) + (G E)(r, t) + (K H)(r, t)B(r, t) = c(r)E(r, t) + d(r)H(r, t) + (L E)(r, t) + (F H)(r, t)



where a, b, c, d are independent of time and G,K,L, F are functions of (r, t).An isotropic medium is dened by

D(r, t) = a(r)E(r, t) + (G E)(r, t),B(r, t) = d(r)H(r, t) + (F H)(r, t).

For monochromatic elds, the constitutive relations for an anisotropicmedium are usually expressed by

D = E, B = H,

where and

are dyads. For an introduction of dyadic analysis, please

refer to Bladel (2007).The constitutive relations are often written as

D(r, t) = 0E(r, t) +P(r, t) + ,B(r, t) = 0[H(r, t) +M(r, t) + ],

(1.12)

where 0 and 0 are permeability and permittivity in vacuum respectively;M is the magnetization vector and P is the polarization vector. Equa-tions (1.12) may contain higher order terms, which have been omittedsince in most cases only the magnetization and polarization vectors aresignicant. The vector M and P reect the eects of the Lorentz forceon elemental particles in the medium and therefore they depend on bothE and B in general. Since the elemental particles in the medium have nitemasses and are mutually interacting, M and P are also functions of timederivatives of E and B as well as their magnitudes. The same applies for thecurrent density Jind. In most cases, M is only dependent on the magneticeld B and its time derivatives while P and J are only dependent on theelectric eld E and its time derivatives. If these dependences are linear,the medium is said to be linear. These linear dependences are usuallyexpressed as

D = E+ 1Et

+ 22Et2

+ ,

B = H+ 1Ht

+ 22Ht2

+ ,

Jind = E+ 1Et

+ 22Et2

+ ,

(1.13)

where all the scalar coecients are constants. For the monochromatic elds,the rst two expressions of (1.13) reduce to

D = E, B = H



where

= j, = j, = 22 + , = 22 + , (1.14) = 1 + 33 , = 1 + 33 .

The parameters and are real and are called capacitivity anddielectric loss factor respectively. The parameters and are realand are called inductivity and magnetic loss factor respectively.

1.1.3 Wave Equations

The electromagnetic wave equations are second order partial dierentialequations that describe the propagation of electromagnetic waves througha medium. If the medium is homogeneous and isotropic and non-dispersive,we have B = H and D = E, where and are constants. On eliminationof E or H in the generalized Maxwell equations, we obtain

E+ 2Et2

= Jm Jt

,

H+ 2Ht2

= J Jmt

.

(1.15)

These are known as the wave equations. For the time-harmonic elds,Equations (1.15) reduce to

E k2E = Jm jJ,H k2H = J jJm,

(1.16)

where k = is the wavenumber. It can be seen that the source terms

on the right-hand side of (1.15) and (1.16) are very complicated. To simplifythe analysis, the electromagnetic potential functions may be introduced (seeSection 1.5). The wave equations may be used to solve the following threedierent eld problems:

(1) Electromagnetic elds in source-free region: Wave propagations in spaceand waveguides, wave oscillation in cavity resonators, etc.

(2) Electromagnetic elds generated by known source distributions:Antenna radiations, excitations in waveguides and cavity resonators,etc.

(3) Interaction of elds and sources: Wave propagation in plasma, couplingbetween electron beams and propagation mechanism, etc.



If the medium is inhomogeneous and anisotropic so that D = E and

B = H, the wave equations for the time-harmonic elds are

1 E(r) 2 E(r) = jJ(r)1 Jm,

1 H(r) 2 H(r) = jJm(r) +1 J.

(1.17)

1.1.4 Dispersion

If the speed of the wave propagation and the wave attenuation in a mediumdepend on the frequency, the medium is said to be dispersive. Dispersionarises from the fact that the polarization and magnetization and the currentdensity cannot follow the rapid changes of the electromagnetic elds, whichimplies that the electromagnetic energy can be absorbed by the medium.Thus, the dissipation or absorption always occurs whenever the mediumshows the dispersive eects. In reality, all media show some dispersiveeects. The medium can be divided into normal dispersive and anomalousdispersive. A normal dispersive medium refers to the situation wherethe refractive index increases as the frequency increases. Most naturallyoccurring transparent media exhibit normal dispersion in the visible rangeof electromagnetic spectrum. In an anomalous dispersive medium, therefractive index decreases as frequency increases. The dispersive eects areusually recognized by the existence of elementary solutions (plane wavesolution) of Maxwell equations in source-free region

A(k)ej(tkr), (1.18)

where A(k) is the amplitude, k is wave vector and is the frequency.When the elementary solutions are introduced into Maxwell equations, arelationship between k and may be found as follows

f(,k) = 0. (1.19)

This is called dispersion relation. For a single linear dierential equationwith constant coecients, there is a oneone correspondence between theequation and the dispersion relation. We only need to consider the followingcorrespondences:

t j, jk,

which yield a polynomial dispersion relation. To nd the dispersion relationof the medium, the plane wave solutions may be assumed for Maxwell



equations as follows

E(r, t) = Re[E(r)ejtjkr], etc. (1.20)

Similar expressions hold for other quantities. In the following, it is assumedthat the wave vector k is allowed to be a complex vector and there isno impressed source inside the medium. Introducing (1.20) into Maxwellequations (1.6) with Jimp = 0 and using the calculation ejkr =jkejkr, we obtain

jkH(r) +H(r) = jD(r) + Jcon(r),jkE(r) +E(r) = jB(r).

In most situations, the complex amplitudes of the elds are slowlyvarying functions of space coordinates. The above equations may thusreduce to

kH(r) = D(r) + jJcon(r),kE(r) = B(r).

(1.21)

If the medium is isotropic, dispersive and lossy, we may write

Jcon = E, D = ( j)E, B = ( j)H.

Substituting these into (1.21) yields

k k = 2( j)[ j( + /)].

Assuming k = uk( j)(uk is a unit vector), then we have

j =

( j)[ j( + /)]

from which we may nd that

=2

(A2 + B2)1/2 + A, =

2

(A2 + B2)1/2 A

where A = ( + /), B = + ( + /).

1.1.5 Electromagnetic Field Theorems

A number of theorems can be derived from Maxwell equations, andthey usually bring deep physical insight into the electromagnetic eld



problems. When applied properly, these theorems can simplify the problemsdramatically.

1.1.5.1 Superposition Theorem

Superposition theorem applies to all linear systems. Suppose that theimpressed current source Jimp can be expressed as a linear combinationof independent impressed current sources Jkimp(k = 1, 2, . . . , n)

Jimp =n

k=1

akJkimp,

where ak(k = 1, 2, . . . , n) are arbitrary constants. If Ek and Hk areelds produced by the source Jkimp, the superposition theorem forelectromagnetic elds asserts that the elds E =

nk=1 akE

k and H =nk=1 akH

k are a solution of Maxwell equations produced by the sourceJimp.

1.1.5.2 Conservation of Electromagnetic Energy

The law of conservation of electromagnetic energy is known asthe Poynting theorem, named after the English physicist John HenryPoynting (18521914). It can be found from (1.1) that

Jimp E Jind E = S+E Dt

+H Bt

. (1.22)

In a region V bounded by S, the integral form of (1.22) is

V

Jimp E dV =V

Jind E dV +S

S un dS

+V

(E D

t+H B

t

)dV, (1.23)

where un is the unit outward normal of S, and S = EH is the Poyntingvector representing the electromagnetic power-ow density measured inwatts per square meter (W/m2). It is assumed that this explanation holdsfor all media. Thus, the left-hand side of the above equation stands for thepower supplied by the impressed current source. The rst term on the right-hand side is the work done per second by the electric eld to maintain thecurrent in the conducting part of the system. The second term denotes theelectromagnetic power owing out of S. The last term can be interpreted



as the work done per second by the impressed source to establish the elds.The energy density w required to establish the electromagnetic elds maybe dened as follows

dw =(E D

t+H B

t

)dt. (1.24)

Assuming all the sources and elds are zero at t = , we havew = we + wm, (1.25)

where we and wm are the electric eld energy density and magneticeld energy density respectively

we =12E D+

t

12

(E D

tD E

t

)dt,

wm =12H B +

t

12

(H B

tB H

t

)dt.

Equation (1.23) can be written as

V

Jimp E dV =V

Jind E dV +S

S un dS + t

V

(we + wm)dV .

(1.26)

In general, the energy densityw does not represent the stored energy densityin the elds: the energy temporarily located in the elds and completelyrecoverable when the elds are reduced to zero. The energy density w givenby (1.25) can be considered as the stored energy density only if the mediumis lossless (i.e., S = 0). If the medium is isotropic and time-invariant,we have

we =12E D, wm = 12H B.

If the elds are time-harmonic, the Poynting theorem takes the followingform

12

V

E Jimp dV = 12V

E Jind dV +S

12(E H) un dS

+ j2V

(14B H 1

4E D

)dV , (1.27)

where the bar denotes complex conjugate. The time averages of Poyntingvector, energy densities over one period of the sinusoidal wave ejt, denoted



T , are

1T

T0

EHdt = 12Re(E H),

1T

T0

12E Ddt = 1

4Re(E D),

1T

T0

12H Bdt = 1

4Re(H B).

It should be noted that the Poynting theorem (1.23) in time domain andthe Poynting theorem (1.27) in frequency domain are independent. Thisproperty can be used to nd the stored energies around a small antenna(see Chapter 5).

1.1.5.3 Uniqueness Theorems

It is important to know the conditions under which the solution of Maxwellequations is unique. Let us consider a multiple-connected region V boundedby S =

Ni=0 Si, as shown in Figure 1.2. Assume that the medium inside V

is linear, isotropic and time invariant, and it may contain some impressedsource Jimp. So we have D = E,B = H, and Jind = E. The uniquenesstheorem for time-domain elds can be expressed as follows:

Uniqueness theorem for time-domain elds: Suppose that theelectromagnetic sources are turned on at t = 0. The electromagnetic eldsin a region are uniquely determined by the sources within the region, theinitial electric eld and the initial magnetic eld at t = 0 inside the region,

nu

NS1S

2S 0S

impJ

Figure 1.2 Multiple-connected region.



together with the tangential electric eld (or the tangential magnetic eld)on the boundary for t > 0, or together with the tangential electric eld onpart of the boundary and the tangential magnetic eld on the rest of theboundary for t > 0.

The uniqueness theorem for time-harmonic elds may be stated asfollows:

Uniqueness theorem for time-harmonic elds: For a region thatcontains the dissipation loss or radiation loss, the electromagnetic elds areuniquely determined by the sources within the region, together with thetangential electric eld (or the tangential magnetic eld) on the boundary,or together with the tangential electric eld on part of the boundary andthe tangential magnetic eld on the rest of the boundary.

The uniqueness for time-harmonic elds is guaranteed if the systemhas radiation loss, regardless of the medium is lossy or not. This propertyhas been widely validated by the study of antenna radiation problems, inwhich the surrounding medium is often assumed to be lossless. Note thatthe uniqueness for time-harmonic elds fails for a system that contains nodissipation loss and radiation loss. The uniqueness in a lossless medium isusually obtained by considering the elds in a lossless medium to be thelimit of the corresponding elds in a lossy medium as the loss goes to zero,which is based on an assumption that the limit of a unique solution is alsounique. However, this limiting process may lead to physically unacceptablesolutions. Also notice that there is no need to introduce losses for a uniquesolution in the time-domain analysis (Geyi, 2010).

1.1.5.4 Equivalence Theorems

It is known that there is no answer to the question of whether eld or sourceis primary. The equivalence principles just indicate that the distinctionbetween the eld and source is kind of blurred. Let V be an arbitrary regionbounded by S, as shown in Figure 1.3. Two sources that produce the sameelds inside a region are said to be equivalent within that region. Similarly,two electromagnetic elds {E1,D1,H1,B1} and {E2,D2,H2,B2} are saidto be equivalent inside a region if they both satisfy the Maxwell equationsand are equal in that region.

The main application of the equivalence theorem is to nd equivalentsources to replace the inuences of substance (the medium is homogenized),so that the formulae for retarding potentials can be used. The equivalentsources may be located inside S (equivalent volume sources) or on S



Snu

V3R V

Figure 1.3 Equivalence theorem.

(equivalent surface sources). The most general form of the equivalentprinciples is stated as follows.

General equivalence principle: Let us consider two electromagnetic eldproblems in two dierent media:

Problem 1:

H1(r, t) = D1(r, t)/t + J1(r, t),E1(r, t) = B1(r, t)/t Jm1(r, t), D1(r, t) = 1(r, t), B1(r, t) = m1(r, t),D1(r, t) = 1(r)E1(r, t),B1(r, t) = 1(r)H1(r, t)

Problem 2:

H2(r, t) = D2(r, t)/t + J2(r, t),E2(r, t) = B2(r, t)/t Jm2(r, t), D2(r, t) = 2(r, t), B2(r, t) = m2(r, t),D2(r, t) = 2(r)E2(r, t),B2(r, t) = 2(r)H2(r, t).

If a new set of electromagnetic elds {E,D,H,B} satisfying

H(r, t) = D(r, t)/t + J(r, t),E(r, t) = B(r, t)/t Jm(r, t), D(r, t) = (r, t), B(r, t) = m(r, t),D(r, t) = (r)E(r, t),B(r, t) = (r)H(r, t),

(1.28)

is constructed in such a way that the sources of the elds {E,D,H,B} andthe parameters of the medium satisfy

J = J1,Jm = Jm1 = 1, m = m1, r V ; = 1, = 2

J = J2,Jm = Jm2 = 2, m = m2, r R3 V = 2, = 2



and

J = un (H2+ H1)Jm = un (E2+ E1) = un (D2+ D1)m = un (B2+ B1)

, r S

where un is the unit outward normal to S, and the subscripts + and signify the values obtained as S is approached from outside S and inside Srespectively, then we have

{E,D,H,B} = {E1,D1,H1,B1} , r V{E,D,H,B} = {E2,D2,H2,B2} , r R3 V

.

By the equivalence principle, the magnetic current Jm and magnetic chargem, introduced in the generalized Maxwell equations, are justied in thesense of equivalence. If E1 = D1 = H1 = B1 = J1 = Jm1 = 0 in the generalequivalence theorem, we may choose = 2, = 2 in (1.28) inside S. Ifall the sources for Problem 2 are contained inside S, the following sources{

Js = un H2+,Jms = un E2+s = un D2+, ms = un B2+

, r S

produce the electromagnetic elds {E,D,H,B} in (1.28). In other words,the above sources generate the elds {E2,D2,H2,B2} in R3V and a zeroeld in V . Thus we have:

SchelkunoLove equivalence (named after the American mathemati-cian Sergei Alexander Schelkuno, 18971992; and the English mathemati-cian Augustus Edward Hough Love, 18631940): Let {E,D,H,B} be theelectromagnetic elds with source conned in S. The following surfacesources {

Js = un H, Jms = un Es = un D, ms = un B

, r S (1.29)

produce the same elds {E,D,H,B} outside S and a zero eld inside S.Since the sources in (1.29) produce a zero eld inside S, the interior

of S may be lled with a perfect electric conductor. By use of the Lorentzreciprocity theorem [see (1.32)], it can be shown that the surface electriccurrent pressed tightly on the perfect conductor does not produce elds. Asa result, only the surface magnetic current is needed in (1.29). Similarly,



the interior of S may be lled with a perfect magnetic conductor, and inthis case the surface magnetic current does not produce elds and onlythe surface electric current is needed in (1.29). In both cases, one cannotdirectly apply the vector potential formula even if the medium outside S ishomogeneous.

1.1.5.5 Reciprocity

A linear system is said to be reciprocal if the response of the system witha particular load and a source is the same as the response when the sourceand the load are interchanged. Consider two sets of time-harmonic sources,J1,Jm1 and J2,Jm2, of the same frequency in the same linear medium. Theelds produced by the two sources are respectively denoted by E1,H1 andE2,H2, and they satisfy the Maxwell equations{

Hi(r) = jEi(r) + Ji(r)Ei(r) = jHi(r) Jmi(r)

, (i = 1, 2).

The reciprocity can be stated asV

(E2 J1 H2 Jm1)dV =V

(E1 J2 H1 Jm2)dV

+S

(E1 H2 E2 H1) un dS, (1.30)

where V is a nite region bounded by S. If both sources are outside S,the surface integral in (1.30) is zero. If both sources are inside S, it can beshown that the surface integral is also zero by using the radiation condition.Therefore, we obtain the Lorentz form of reciprocity

S

(E1 H2 E2 H1) un dS = 0 (1.31)

and the RayleighCarson form of reciprocityV

(E2 J1 H2 Jm1)dV =V

(H1 Jm2 +E1 J2)dV . (1.32)

If the surface S only contains the sources J1(r) and Jm1(r), (1.30) becomesV

(E2 J1 H2 Jm1)dV =S

(E2 un H1 H2 E1 un)dS.



This is the familiar form of Huygens principle. The electromagneticreciprocity theorem can also be generalized to an anisotropic medium(Kong, 1990; Tai, 1961; Harrington, 1958). Let us consider a special casewhere the region V does not contain any sources. We denote the elds insidethe region V by (E1,H1) or (E2,H2) when it is endowed with medium

parameters ( , ) or with transposed medium parameters (

t,

t). It

follows from the Maxwell equations in source-free region that

(E1 H2) = H2 E1 E1 H2= jH2 H1 jE1

t E2, (E2 H1) = H1 E2 E2 H1

= jH1 t H2 jE2 E1.

This gives

(E1 H2 E2 H1) = 0.

After integration over V , we obtain (1.31).

1.2 Method of Separation of Variables

Let us consider a dierential equation

Lu = f, (1.33)

where L is a dierential operator, f is a known source function and u isthe unknown function. One method of solving (1.33) is to nd the spectralrepresentation of L by studying the solution of the following eigenvalueequation

Lu = u,

where is called eigenvalue and u is the corresponding eigenfunction.The method of eigenfunction expansion is also called the method ofseparation of variables if L is a partial dierential operator. The basicidea of separation of variables is to seek a solution in the form of a product offunctions, each of which depends on one variable only, so that the solution oforiginal partial dierential equations may reduce to the solution of ordinarydierential equations. We will use the Helmholtz equation to illustrate theprocedure in this section. The Helmholtz equation, named after the



German physicist Hermann Ludwig Ferdinand von Helmholtz (18211894),is the time-independent form of wave equation, and is dened by

(2 + k2)u = 0, (1.34)

where k is a constant. When k is zero, the Helmholtz equation reduces tothe Laplace equation, named after the French mathematician Pierre-Simonmarquis de Laplace (17491827). The Helmholtz equation is separable in11 orthogonal coordinate systems (Eisenhart, 1934).

1.2.1 Eigenvalue Problem of SturmLiouville Type

First let us consider the most common eigenvalue problem for the ordinarydierential equation known as the SturmLiouville equation [namedafter the French mathematicians Jacques Charles Francois Sturm (18031855) and Joseph Liouville (18091882)][

ddx

p(x)d

dx+ q(x)

]vn(x) = nw(x)vn(x), a < x < b (1.35)

subject to the homogeneous boundary conditions of impedance type:

p(x)dvn(x)

dx+ (x)vn(x) = 0, x = a, b. (1.36)

In the above, n is the eigenvalue and vn is the corresponding eigenfunction.The functions p, q and the weight function w are assumed to be realfunctions of x in [a, b] and furthermore w > 0. Multiplying (1.35) by vn,integrating over x between a and b and using integration by parts, we obtain

n =

ba

p(dvndx

)2dx +

ba

qv2n dx + (b)v2n(b) (a)v2n(a)

ba

wv2n dx

. (1.37)

This indicates that n is real. We now multiply (1.35) by the eigenfunctionvm and integrate over the x domain to obtain

ba

vmd

dx

(pdvndx

)dx

ba

qvmvn dx + n

ba

wvmvn dx = 0. (1.38)



Interchanging m and n gives another equation

ba

vnd

dx

(pdvmdx

)dx

ba

qvnvm dx + m

ba

wvnvm dx = 0. (1.39)

Subtracting (1.38) from (1.39) yields

(m n)b

a

wvnvm dx =[p

(vn

dvmdx

vm dvndx

)]ba

.

In view of the boundary conditions (1.36), we obtain the followingorthogonal relationship

ba

wvnvm dx = 0, m = n. (1.40)

The eigenfunctions may be normalized as follows

ba

w(x)v2n(x)dx = 1. (1.41)

The set of eigenfunctions {vn} is said to be orthonormal if both (1.40)and (1.41) are satised. If we assume vn(a) = vn(b) = 0, then all theeigenvalues are positive from (1.37). Suppose that the eigenfunctions arecomplete and therefore every square integrable function f(x) in [a, b] canbe represented by

f(x) =n

fnvn, (1.42)

where the sum is over all eigenfunctions, and

fn =

ba

w(x)f(x)vn(x)dx.

The completeness and orthonormality of the set {vn} can be expressedconcisely in a symbolic manner by choosing f(x) = (x x) in (1.42)

(x x)w(x)

=n

vn(x)vn(x), a < x, x < b. (1.43)



1.2.2 Rectangular Coordinate System

In rectangular coordinate system (x, y, z), Helmholtz equation (1.34)becomes

2u

x2+

2u

y2+

2u

z2+ k2u = 0. (1.44)

We seek a solution in the form of product of three functions of onecoordinate each

u = X(x)Y (y)Z(z). (1.45)

Substituting (1.45) into (1.44) gives

1X

d2X

dx2+

1Y

d2Y

dy2+

1Z

d2Z

dz2+ k2 = 0. (1.46)

Since k is a constant and each term depends on one variable only andcan change independently, the left-hand side of (1.46) can sum to zero forall coordinate values only if each term is a constant. Thus we have

d2X

dx2+ k2xX = 0,

d2Y

dy2+ k2yY = 0,

d2Z

dz2+ k2zZ = 0,

(1.47)

where kx, ky and kz are separation constants and satisfy

k2x + k2y + k

2z = k

2. (1.48)

The solutions of (1.47) are harmonic functions, denoted by X(kxx),Y (kyy) and Z(kzz), and they are any linear combination of the followingindependent harmonic functions:

eik, eik, cos k, sin k ( = x, y, z). (1.49)

Consequently, the solution (1.45) may be expressed as

u = X(kxx)Y (kyy)Z(kzz). (1.50)

The separation constants kx, ky and kz are also called eigenvalues, and theyare determined by the boundary conditions. The corresponding solutions(1.50) are called eigenfunctions or elementary wavefunctions. The general



solution of (1.44) can be expressed as a linear combination of the elementarywavefunctions.

1.2.3 Cylindrical Coordinate System

In cylindrical coordinate system (, , z), (1.34) can be written as

1

(u

)+

12

2u

2+

2u

z2+ k2u = 0. (1.51)

By the method of separation of variables, the solutions may be assumedto be

u = R()()Z(z). (1.52)

Introducing (1.52) into (1.51) yields

d2R

d2+

1

dR

d+(2 p

2

2

)R = 0,

d2d2

+ p2 = 0,

d2Z

dz2+ 2Z = 0,

(1.53)

where , p and are separation constants and satisfy

2 + 2 = k2. (1.54)

The rst equation of (1.53) is Bessel equation, named after the Germanmathematician Friedrich Wilhelm Bessel (17841846), whose solutions areBessel functions:

Jp(), Np(), H(1)p (), H(2)p (),

where Jp() and Np() are the Bessel functions of the rst and secondkind, H(1)p () and H

(2)p () are the Bessel functions of the third and fourth

kind, also called Hankel functions of rst and second kind respectively,named after German mathematician Hermann Hankel (18391873). TheBessel function of the rst kind is dened by

Jp(z) =

m=0

(1)m(m + 1)(p + m + 1)

(z2

)p+2m, (1.55)



where () is the gamma function dened by

() =

0

x1exdx, > 0.

If p is not an integer, a second independent solution is Jp(z). If p = n isan integer, Jn(z) is related to Jn(z) by

Jn(z) = (1)nJn(z).

The Bessel function of the second kind (also known as Neumannfunction) dened by

Np(z) =cos pJp(z) Jp(z)

sin p, (1.56)

and the Bessel functions of the third (Hankel function of the rst kind) andfourth kind (Hankel function of the second kind) are dened by

H(1)p (z) = Jp(z) + jNp(z),

H(2)p (z) = Jp(z) jNp(z).

(1.57)

The solutions of second and third equation of (1.53) are harmonicfunctions. Note that only Jp() is nite at = 0. The separation constants and p are determined by the boundary conditions. For example, if the eldu is nite and satises homogeneous Dirichlet boundary condition u = 0at = a, the separation constant is determined by Jp() = 0. If thecylindrical region contains all from 0 to 2, the separation constant p isusually determined by the requirement that the eld is single-valued, i.e.,(0) = (2). In this case, p must be integers. If the cylindrical region onlycontains a circular sector, p will be fractional numbers.

Let Rp(z) = AJp(z) + BNp(z), where A and B are constant. Wehave the recurrence relations

2pz

Rp(z) = Rp1(z) + Rp+1(z),

1

d

dzRp(z) =

12[Rp1(z)Rp+1(z)] ,

zd

dzRp(z) = pRp(z) zRp+1(z),



d

dz[zpRp(z)] = zpRp1(z),

d

dz[zpRp(z)] = zpRp+1(z).

1.2.4 Spherical Coordinate System

In spherical coordinate system (r, , ), (1.34) can be expressed as

1r2

r

(r2

u

r

)+

1r2 sin

(sin

u

)+

1r2 sin

2u

2+ k2u = 0.

(1.58)

By means of the separation of variables, we may let

u = R(r)()(). (1.59)

Substitution of (1.59) into (1.58) leads to

1R

d

dr

(r2

dR

dr

)+ k2r2 = 2,

1 sin

d

d

(sin

dd

) m

2

sin2 = 2,

d2d2

+ m2 = 0.

(1.60)

Let x = cos and P (x) = (), the second equation of (1.60) becomes

(1 x2)d2P

dx2 2xdP

dx+(2 m

2

1 x2)P = 0. (1.61)

This is called Legendre equation, named after the French mathematicianAdrien-Marie Legendre (17521833). The points x = 1 are singular.Equation (1.61) has two linearly independent solutions and can be expressedas a power series at x = 0. In general, the series solution diverges at x = 1.But if we let 2 = n(n + 1), n = 0, 1, 2, . . . , the series will be nite atx = 1 and have nite terms. Thus the separation constant is determined



naturally and (1.60) can be written as

d

dr

(r2

dR

dr

)+[k2r2 n(n + 1)]R = 0,

(1 x2)d2P

dx2 2xdP

dx+[n(n + 1) m

2

1 x2]P = 0,

d2d2

+ m2 = 0.

(1.62)

The solutions of the rst equation of (1.62) are spherical Besselfunctions

jn(kr) =

2krJn+1/2(kr), nn(kr) =

2krNn+1/2(kr),

h(1)n (kr) =

2krH

(1)n+1/2(kr), h

(2)n (kr) =

2krH

(2)n+1/2(kr).

(1.63)

Let zn(kr) = Ajn(kr) + Bnn(kr), where A and B are constants. We havethe recurrence relations:

2n + 1kr

zn(kr) = zn1(kr) + zn+1(kr),

2n + 1k

d

drzn(kr) = nzn1(kr) (n + 1)zn+1(kr),

d

dr[rn+1zn(kr)] = krn+1zn1(kr),

d

dr[rnzn(kr)] = krnzn+1(kr).

The solutions of the second equation of (1.62) are associated Legendrefunctions of rst and second kind dened by

Pmn (x) =(1 x2)m/2

2nn!dm+n

dxm+n(x2 1)n, (1.64)

and

Qmn (x) = (1 x2)m2

dm

dxmQn(x), m n, (1.65)



respectively, with

Qn(x) =12P 0n(x) ln

1 + x1 x

nr=1

1rP 0r1(x)P

0nr(x)

being the Legendre function of the second kind.The following integrations are useful

11

Pmn (x)Pkn (x)

1 x2 dx =1m

(n + m)!(nm)!mk,

11

Pmk (x)Pmn (x)dx =

22k + 1

(k + m)!(k m)!kn,

0

[dPmn (cos )

d

dPmk (cos )d

+m2

sin2 Pmn (cos )P

mk (cos )

]sin d

=2

2n+ 1(n + m)!(nm)!n(n + 1)nk.

The solutions of the third equation of (1.62) are harmonic functions. Notethat the separation coecients are not related in spherical coordinatesystem.

1.3 Method of Greens Functions

Physically, the Greens function represents the eld produced by a pointsource, and provides a general method to solve dierential equations.Through the use of the Greens function, the solution of a dierentialequation can be represented by an integral dened over the source regionor on a closed surface enclosing the source. Mathematically, the solution ofa partial dierential equation in the source region V

Lu(r) = f(r), r V (1.66)can be expressed as

u(r) = L1f(r),

where L1 stands for the inverse of L and is often represented by an integraloperator whose kernel is the Greens function. Let us assume that thereexists a function G such that

L1f(r) = V

G(r, r)f(r)dV (r), (1.67)



Applying L to both sides of the above equation yields

LL1f(r) = f(r) = V

LG(r, r)f(r)dV (r).

This equation implies that the function G satises

LG(r, r)f(r) = (r r) (1.68)

where denotes the delta function. The function G is called the funda-mental solution or Greens function of the Equation (1.66).

1.3.1 Greens Functions for Helmholtz Equation

Let = (x, y), r = (x, y, z) and v be a constant. The fundamental solutionsof wave equations are summarized in Table 1.5, where H(x) is the unitstep function. It can be seen that the Greens functions are symmetricG(r, r) = G(r, r).

Example 1.1: The Greens function for one-dimensional Helmholtz equa-tion satises

d2G(z, z)dz2

+ k2G(z, z) = (z z),

limz

(dG

dz jkG

)= 0.

(1.69)

The second equation denotes the radiation condition at innity. Let

G(z, z) =

{G1(z, z), z < z

G2(z, z), z > z.

Then we may write

G1(z, z) = a1ejk(zz) + b1ejk(zz

),

G2(z, z) = a2ejk(zz) + b2ejk(zz

),

where a1, b1, a2, b2 are constants to be determined. Taking the radiationcondition into account, we have a1 = b2 = 0. Thus

G1(z, z) = b1ejk(zz),

G2(z, z) = a2ejk(zz).

(1.70)


30 Foundations for

foundations for radio frequency engineering

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