Foundations of Applied Electromagnetics

615 Pages·2022·47.584 MB·English

by  Kamal Sarabandi

Foundations of Applied
Electromagnetics
KAMAL SARABANDI
FOUNDATIONS
OF APPLIED
ELECTROMAGNETICS
Kamal Sarabandi
The University of Michigan
Copyright ©2022KamalSarabandi
ThisbookispublishedbyMichiganPublishingunderanagreementwiththeauthor.Itismade
availablefreeofchargeinelectronicformtoanystudentorinstructorinterestedinthesubject
matter.
Published intheUnitedStatesofAmericaby
MichiganPublishing
Manufactured intheUnitedStatesofAmerica
ISBN 978-1-60785-819-5
The Free ECE Textbook Initiative is sponsored by the ECE
Department at the University of Michigan.
The image used in the chapter title pages was taken by theJames Webb Space Telescope,
courtesy ofNASA.
I dedicate this book to my family:
to my wife Shiva and our sons Arya and Sina
for their love and support,
and to the memory of my parents Abbasali and Jaleh,
who instilled in me a passion for science and engineering
iv
Preface
Field theory is one of the fundamental pillars of electrical engineering, with many threads
interwoven into numerous areas of science and technology. Classical electromagnetic theory
may be considered a mature field of science, but because of its importance in wireless
transmission ofdataandenergy, ithasremainedanareaofintenseresearchanddevelopment
for almost twocenturies. In recent years, the interest inthe field of applied electromagnetics
hasbeenfueledbythedemandforhighdata-ratewirelesscommunication, everywhereandat
any time. The implementation of such wireless systems relies on innovation in miniaturized
wideband and multiband antennas for handheld devices, vehicles, and infrastructures such
as base stations and wifi networks. In addition, knowledge of the characteristics of wave
propagation and wave interaction with terrain, vegetation, and manmade structures in urban
environmentsisanessentialandcriticaltoolfortheproperdesignofwirelesscommunication
networks. Moreover, over the past decade we have witnessed a boom in investment focused
ontherapiddevelopment ofautonomous vehicles. Sensorsenvisioned toenable autonomous
functionality include short-range and long-range sensors with different modalities. Some of
these sensors arebased onelectromagnetic waves,suchasmillimeter-wave radars toprovide
the range, direction, and velocity of objects present in traffic scenes. For these systems
to function with a high degree of reliability, we need to use applicable wave propagation
and scattering models together with highly sophisticated directional antennas, all of which
requires in-depth understanding of electromagnetic wave theory. In addition, traditional
applications of field theory in areas such as microwave remote sensing, military systems,
biomedical applications, and space exploration are active and ongoing. Graduate students
interested insuchexcitingfieldsofresearch needastrongfoundation infieldtheory, andthat
wasmymotivation forwriting this book onclassical electromagnetics but withan eye onits
modernapplications.
This book is the outgrowth of my class notes for an entry-level graduate course on
electromagnetictheoryattheUniversityofMichiganandwasinspiredbymyownresearchon
radar remote sensing, antenna theory, electromagnetic wave propagation, and more recently
onbioelectromagnetics.
Anytextbookbasedonafieldwithmorethan200yearsofhistorydrawsveryheavilyfrom
theworkofanenormousnumberofscientistsandengineers.Thisbookisnotanexceptionand
Ifounditimpossibletoprovideacomprehensivelistofreferencestotheoriginalcontributors
formosttopicsincluded inthisbook.
Publishing this book would not have been possible without the help and encouragement
ofseveralcolleagues andstudents. Ihavetofirstthankmydearcolleague andformeradvisor
Professor Fawwaz Ulaby, who has carefully reviewed and provided valuable comments and
suggestions for improving what is presented in this book. I am also indebted to Mr. Richard
Carnes, whospent asignificant amount of effort typing and formatting the book. I have also
to mention the contributions of my students Abdelhamid Nasr, Aditya Varma Muppala, and
Behzad Yektakhah for helping with some of the figures and computations presented in the
book.
KAMAL SARABANDI
ANN ARBOR, AUGUST 2022
CONTENTS v
Contents
Preface iv
1 Electromagnetic Fields 1
1-1 TheFieldEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1-2 Faraday’sLawforaMovingSurfaceinaTime-VaryingMagneticField . . . 10
1-3 Ampère’sLawforaMovingSurfaceinaTime-VaryingElectricField . . . . 13
1-4 ConstitutiveRelations: Macroscopic PropertiesofMatter . . . . . . . . . . . 14
1-5 Kramers-KrönigRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1-6 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1-7 DriftCurrentinMetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1-8 HallEffectinConducting Media . . . . . . . . . . . . . . . . . . . . . . . . 35
1-9 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Electromagnetic Concepts,Tools,andTheorems 62
2-1 EquivalentMagneticChargeandCurrentDensities . . . . . . . . . . . . . . 63
2-2 ImageTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2-3 MethodofImagesforOtherProblems . . . . . . . . . . . . . . . . . . . . . 71
2-4 Polarization Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2-5 StoredElectromagnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . 83
2-6 FlowofEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2-7 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2-8 Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2-9 EquivalencePrincipleforElectromagnetic Sources . . . . . . . . . . . . . . 92
3 TheElectromagnetic PotentialsandRadiation 107
3-1 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3-2 Time-HarmonicElectromagnetic Waves . . . . . . . . . . . . . . . . . . . . 122
3-3 Time-HarmonicRetardedPotential . . . . . . . . . . . . . . . . . . . . . . . 131
3-4 Far-FieldDistanceCriterion . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3-5 SmallLoopofCurrent:AHertzianMagnetic Dipole . . . . . . . . . . . . . 139
3-6 WireAntennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3-7 EquivalentCircuitforReceivingAntennas . . . . . . . . . . . . . . . . . . . 147
vi CONTENTS
4 FormalSolutionstoMaxwell’sEquationsandTheirApplications 161
4-1 FormalSolutionoftheHelmholtzEquation . . . . . . . . . . . . . . . . . . 162
4-2 SolutionoftheHelmholtzEquationforaComplexMedium . . . . . . . . . . 168
4-3 IntegralEquationsforElectromagnetic Fields . . . . . . . . . . . . . . . . . 176
4-4 IntegralEquationFormulationBasedonEquivalentSources . . . . . . . . . 178
4-5 IntegralEquationFormulationforDielectric Scatterers . . . . . . . . . . . . 181
4-6 Reciprocity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4-7 Applications oftheReciprocity Theorem . . . . . . . . . . . . . . . . . . . . 189
4-8 Babinet’sPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5 Electromagnetic PlaneWaves 227
5-1 Plane-WavePropagation inHomogeneous Media . . . . . . . . . . . . . . . 228
5-2 Polarization ofPlaneWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5-3 kDBCoordinate forPlaneWavesinBianisotropic Media . . . . . . . . . . . 247
5-4 Transverse Electric (TE) and Transverse Magnetic (TM) Field Solutions of
theHelmholtzEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5-5 Plane-Wave Reflection at the Interface between a Dielectric Medium and a
Good-Conducting Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5-6 WavePropagationinanInhomogeneousMedium:Geometric-OpticsApprox-
imation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
5-7 Plane-WaveReflectionandTransmissionfromaHalf-SpaceUniaxialMedium 275
5-8 PlaneWavesinLayeredMedia . . . . . . . . . . . . . . . . . . . . . . . . . 280
5-9 Plane-WavePropagation inaNegative-Index Medium . . . . . . . . . . . . . 290
5-10 NegativeRefractive-Index Lens . . . . . . . . . . . . . . . . . . . . . . . . 292
6 CartesianWaveFunctions:GuidingStructuresandResonators 313
6-1 TheDielectricPlateWaveguide . . . . . . . . . . . . . . . . . . . . . . . . . 315
6-2 GuidedWavesonImpedance Surfaces . . . . . . . . . . . . . . . . . . . . . 321
6-3 PracticalRealization ofReactiveImpedanceSurfaces . . . . . . . . . . . . . 324
6-4 Isotropic ReactiveImpedanceSurfaces . . . . . . . . . . . . . . . . . . . . . 330
6-5 TheRectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6-6 Transmission-Line CircuitModelforWaveguides . . . . . . . . . . . . . . . 357
6-7 OtherModalSolutions forRectangular Waveguides . . . . . . . . . . . . . . 363
6-8 ModalExpansionofFieldQuantities . . . . . . . . . . . . . . . . . . . . . . 367
6-9 Calculus of Variations for Estimation of Resonant Frequencies in General
Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
7 CylindricalWaveFunctionsandTheirApplications 404
7-1 WaveFunctionsintheCylindricalCoordinate System . . . . . . . . . . . . . 406
7-2 TheCircularDielectric Waveguide . . . . . . . . . . . . . . . . . . . . . . . 421
7-3 Green’sFunctionsSolutions forSomeCanonical Problems . . . . . . . . . . 436
7-4 ScatteringfromaMetallicCircularCylinder . . . . . . . . . . . . . . . . . . 452
7-5 IntegralRepresentation ofBesselFunctions . . . . . . . . . . . . . . . . . . 460
7-6 2-DGreen’s Function forHomogeneous Media inthe Presence ofaMetallic
Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
CONTENTS vii
7-7 AsymptoticEvaluationofaFieldDiffractedbyaMetallicWedge . . . . . . . 474
8 SphericalWaveFunctionsandTheirApplications 502
8-1 WaveFunctionsintheSphericalCoordinateSystem . . . . . . . . . . . . . . 504
8-2 WaveTransformation toSpherical WaveFunctions . . . . . . . . . . . . . . 533
8-3 MultipoleRepresentation ofSphericalWaves . . . . . . . . . . . . . . . . . 538
8-4 Plane-WaveScattering fromSpheres . . . . . . . . . . . . . . . . . . . . . . 542
8-5 WavePropagation inaConicalWaveguide . . . . . . . . . . . . . . . . . . . 553
8-6 BiconicalStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
8-7 OtherSphericalWaveguides . . . . . . . . . . . . . . . . . . . . . . . . . . 568
A PropertiesofComplexFunctions 586
A-1 Cauchy–RiemannConditions . . . . . . . . . . . . . . . . . . . . . . . . . . 587
A-2 ConformalMapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
A-3 BranchCutandBranchPoint . . . . . . . . . . . . . . . . . . . . . . . . . . 588
A-4 Cauchy’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
A-5 CauchyFormulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
A-6 PolesandResidues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
A-7 Jordan’sLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
B MethodofSteepestDescent 592
B-1 SaddlePoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
B-2 Integration alongtheSteepestDescentPath . . . . . . . . . . . . . . . . . . 594
C UsefulVectorIdentities,Operators,andCoordinateTransformations 596
Index 601
1
Chapter
Electromagnetic Fields
Chapter Contents Objectives
Overview, 2 Upon learning the material presented in this chapter, you
1-1 The Field Equations, 2 should be able to:
1-2 Faraday’s Law for a Moving Surface
in a Time-Varying Magnetic Field, 10 1. Explain the basic physics behind Maxwell’s
1-3 Ampère’s Law for a Moving Surface equations.
in a Time-Varying Electric Field, 13
1-4 Constitutive Relations: Macroscopic 2. Understand the relations between the field
Properties of Matter, 14 intensities and the flux densities in a material
1-5 Kramers-Krönig Relations, 28 medium (known as the constitutive relations)
1-6 Boundary Conditions, 30 and their behavior as a function of time and
1-7 Drift Current in Metals, 34 frequency.
1-8 Hall Effect in Conducting Media, 35
1-9 Generalized Coordinates, 36 3. Incorporate complexities associated with field
Chapter Summary, 44 discontinuities across abrupt boundaries between
Problems, 46 dissimilar media.
4. Express the expanded form of Maxwell’s equa-
tions and relevant differential operators in both
standard and arbitrary coordinate systems.
1
2 Chapter1 Electromagnetic Fields
Overview
All material media—from an isolated atom to an entire galaxy of stars and planets—radiate
electromagnetic waves, all the time! We are constantly getting bombarded by a spectrum
of EM waves, including light waves, microwaves, and radio waves. Electromagnetics, the
bidirectionalinteractionbetweentheelectricandmagneticfields,isatthecoreofwhatmakes
electronic circuits function, communication systems transfer data, and computer systems
process information. This book is about how electromagnetic fields interact with material
media, including reflection by and refraction across boundaries between electromagnetically
dissimilar media, transmission across boundaries, absorption by lossy media, and scattering
withininhomogeneous media.
Chapter 1 starts with an examination of the four famous equations known as Maxwell’s
equations, in both differential and integral forms. Maxwell’s equations are then formulated
for the condition when the electric and magnetic fields are in the presence of a time-varying
(moving)surface,andalsousedtodefinetheconstitutivepropertiesofmaterialmedia,namely
the electrical permittivity and magnetic permeability, in both homogeneous and anisotropic
materials.Thesefoundationalformulationswillservetofacilitatethetreatmentsofthevarious
EM-relatedtopicscoveredinforthcoming chapters.
1-1 The Field Equations
Electromagnetism, asthestructure ofthewordimplies, encompasses certain lawsofphysics
that define the interrelationships between the electric and magnetic fields. Electromagnetic
phenomena are observed only when the two field quantities are time-varying. The nature of
the interaction between the time-varying electric and magnetic fields was first discovered by
Michael Faraday (1791–1867) and later formulated into mathematical expressions by James
Clerk Maxwell (1831–1879). Faraday’s significant discovery was based on experimental
observations and Maxwell’s formulation was based on mathematical deduction. Faraday’s
extensive experimental work was motivated by the belief that every cause and effect has its
converse. That is, if electricity can produce a magnetic field, a phenomenon discovered by
Oersted,then,conversely, amagneticfieldshouldbeabletoproduceanelectricfield.
The fundamental laws of electricity and magnetism are encapsulated by Maxwell’s
equations:*
∂D
∇ H=J+ (modifiedAmpère’slaw), (1.1a)
× ∂t
∂B
∇ E= (Faraday’s law), (1.1b)
× −∂t
∇ D=ρ (Gauss’lawforelectricity), (1.1c)
·
∇ B=0 (Gauss’lawformagnetism). (1.1d)
·
*JamesClerkMaxwell,ATreatiseonElectricityandMagnetism,ConstableandCo.,London,1873.