# Theory and Computation of Electromagnetic Fields, 2nd Edition

ISBN: 978-1-119-10804-7
744 pages
September 2015, Wiley-IEEE Press

## Description

Reviews the fundamental concepts behind the theory and computation of electromagnetic fields

The book is divided in two parts. The first part covers both fundamental theories (such as vector analysis, Maxwell’s equations, boundary condition, and transmission line theory) and advanced topics (such as wave transformation, addition theorems, and fields in layered media) in order to benefit students at all levels. The second part of the book covers the major computational methods for numerical analysis of electromagnetic fields for engineering applications. These methods include the three fundamental approaches for numerical analysis of electromagnetic fields: the finite difference method (the finite difference time-domain method in particular), the finite element method, and the integral equation-based moment method. The second part also examines fast algorithms for solving integral equations and hybrid techniques that combine different numerical methods to seek more efficient solutions of complicated electromagnetic problems.

Theory and Computation of Electromagnetic Fields, Second Edition:

• Provides the foundation necessary for graduate students to learn and understand more advanced topics
• Discusses electromagnetic analysis in rectangular, cylindrical and spherical coordinates
• Covers computational electromagnetics in both frequency and time domains
• Includes new and updated homework problems and examples
Theory and Computation of Electromagnetic Fields, Second Edition is written for advanced undergraduate and graduate level electrical engineering students. This book can also be used as a reference for professional engineers interested in learning about analysis and computation skills.
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Preface xv

Acknowledgments xxi

PART I Electromagnetic Field Theory 1

1 Basic Electromagnetic Theory 3

1.1 Review of Vector Analysis, 3

1.1.1 Vector Operations and Integral Theorems, 4

1.1.2 Symbolic Vector Method, 6

1.1.3 Helmholtz Decomposition Theorem, 9

1.1.4 Green’s Theorems, 9

1.2 Maxwell’s Equations in Terms of Total Charges and Currents, 11

1.2.1 Maxwell’s Equations in Integral Form, 12

1.2.2 Maxwell’s Equations in Differential Form, 17

1.2.3 Current Continuity Equation, 17

1.2.4 The Lorentz Force Law, 18

1.3 Constitutive Relations, 18

1.3.1 Electric Polarization, 19

1.3.2 Magnetization, 21

1.3.3 Electric Conduction, 22

1.3.4 Classification of Media, 23

1.4 Maxwell’s Equations in Terms of Free Charges and Currents, 25

1.5 Boundary Conditions, 27

1.6 Energy, Power, and Poynting’s Theorem, 31

1.7 Time-Harmonic Fields, 33

1.7.1 Time-Harmonic Fields, 33

1.7.2 Fourier Transforms, 35

1.7.3 Complex Power, 37

1.7.4 Complex Permittivity and Permeability, 42

References, 46

Problems, 46

2 Electromagnetic Radiation in Free Space 53

2.1 Scalar and Vector Potentials, 53

2.1.1 Static Fields, 54

2.1.2 Time-Harmonic Fields and the Lorenz Gauge Condition, 58

2.2 Solution of Vector Potentials in Free Space, 61

2.2.1 Delta Function and Green’s Function, 61

2.2.2 Green’s Function in Free Space, 62

2.2.3 Field–Source Relations in Free Space, 63

2.2.4 Why Use Auxiliary Potential Functions, 64

2.2.5 Free-Space Dyadic Green’s Functions, 66

2.3 Electromagnetic Radiation in Free Space, 69

2.3.1 Infinitesimal Electric Dipole, 69

2.3.2 Finite Electric Dipole, 72

2.3.3 Far-Field Approximation and the Sommerfeld Radiation Condition, 73

2.3.4 Circular Current Loop and Magnetic Dipole, 76

2.4 Radiation by Surface Currents and Phased Arrays, 78

2.4.1 Radiation by a Surface Current, 78

2.4.2 Radiation by a Phased Array, 81

References, 84

Problems, 85

3 Electromagnetic Theorems and Principles 89

3.1 Uniqueness Theorem, 90

3.2 Image Theory, 94

3.2.1 Basic Image Theory, 94

3.2.2 Half-Space Field–Source Relations, 99

3.3 Reciprocity Theorems, 101

3.3.1 General Reciprocity Theorem, 101

3.3.2 Lorentz Reciprocity Theorem, 102

3.3.3 Rayleigh–Carson Reciprocity Theorem, 103

3.4 Equivalence Principles, 107

3.4.1 Surface Equivalence Principle, 107

3.4.2 Application to Scattering by a Conducting Object, 109

3.4.3 Application to Scattering by a Dielectric Object, 114

3.4.4 Volume Equivalence Principle, 116

3.5 Duality Principle, 120

3.6 Aperture Radiation and Scattering, 121

3.6.1 Equivalent Problems, 121

3.6.2 Babinet’s Principle, 124

3.6.3 Complementary Antennas, 127

References, 128

Problems, 129

4 Transmission Lines and Plane Waves 135

4.1 Transmission Line Theory, 135

4.1.1 Governing Differential Equations and General Solutions, 135

4.1.2 Reflection and Transmission, 138

4.1.3 Green’s Function and Eigenfunction Expansion, 140

4.2 Wave Equations and General Solutions, 144

4.2.1 Wave Equations and Solution by Separation of Variables, 144

4.2.2 Characteristics of a Plane Wave, 146

4.2.3 Wave Velocities and Attenuation, 147

4.2.4 Linear, Circular, and Elliptical Polarizations, 151

4.2.5 Wave Propagation in Metamaterials, 154

4.3 Plane Waves Generated by a Current Sheet, 156

4.4 Reflection and Transmission, 159

4.4.1 Reflection and Transmission at Normal Incidence, 159

4.4.2 Reflection and Transmission at Oblique Incidence, 161

4.4.3 Total Transmission and Total Reflection, 164

4.4.4 Transmission into a Left-Handed Medium, 168

4.4.5 Plane Waves Versus Transmission Lines, 170

4.5 Plane Waves in Anisotropic and Bi-Isotropic Media, 174

4.5.1 Plane Waves in Uniaxial Media, 174

4.5.2 Plane Waves in Gyrotropic Media, 179

4.5.3 Plane Waves in Chiral Media, 183

References, 190

Problems, 191

5 Fields and Waves in Rectangular Coordinates 199

5.1 Uniform Waveguides, 199

5.1.1 General Analysis, 200

5.1.2 General Characteristics, 204

5.1.3 Uniform Rectangular Waveguide, 208

5.1.4 Losses in Waveguides and Attenuation Constant, 215

5.2 Uniform Cavities, 220

5.2.1 General Theory, 221

5.2.2 Rectangular Cavity, 223

5.2.3 Material and Geometry Perturbations, 226

5.3 Partially Filled Waveguides and Dielectric Slab Waveguides, 229

5.3.1 General Theory, 229

5.3.2 Partially Filled Rectangular Waveguide, 231

5.3.3 Dielectric Slab Waveguide on a Ground Plane, 236

5.4 Field Excitation in Waveguides, 241

5.4.1 Excitation by Planar Surface Currents, 242

5.4.2 Excitation by General Volumetric Currents, 243

5.5 Fields in Planar Layered Media, 245

5.5.1 Spectral Green’s Function and Sommerfeld Identity, 245

5.5.2 Vertical Electric Dipole above a Layered Medium, 247

5.5.3 Horizontal Electric Dipole above a Layered Medium, 249

5.5.4 Dipoles on a Grounded Dielectric Slab, 251

References, 257

Problems, 257

6 Fields and Waves in Cylindrical Coordinates 261

6.1 Solution of Wave Equation, 261

6.1.1 Solution by Separation of Variables, 262

6.1.2 Cylindrical Wave Functions, 263

6.2 Circular and Coaxial Waveguides and Cavities, 266

6.2.1 Circular Waveguide, 267

6.2.2 Coaxial Waveguide, 273

6.2.3 Cylindrical Cavity, 276

6.3 Circular Dielectric Waveguide, 279

6.3.1 Analysis of Hybrid Modes, 279

6.3.2 Characteristics of Hybrid Modes, 283

6.4 Wave Transformation and Scattering Analysis, 287

6.4.1 Wave Transformation, 288

6.4.2 Scattering by a Circular Conducting Cylinder, 289

6.4.3 Scattering by a Circular Dielectric Cylinder, 293

6.4.4 Scattering by a Circular Multilayer Dielectric Cylinder, 296

6.5 Radiation by Infinitely Long Currents, 300

6.5.1 Line Current Radiation in Free Space, 300

6.5.2 Radiation by a Cylindrical Surface Current, 304

6.5.3 Radiation in the Presence of a Circular Conducting Cylinder, 306

6.5.4 Radiation in the Presence of a Conducting Wedge, 309

6.5.5 Radiation by a Finite Current, 312

References, 319

Problems, 320

7 Fields and Waves in Spherical Coordinates 325

7.1 Solution of Wave Equation, 325

7.1.1 Solution by Separation of Variables, 325

7.1.2 Spherical Wave Functions, 328

7.1.3 TEr and TMr Modes, 329

7.2 Spherical Cavity, 331

7.3 Biconical Antenna, 335

7.3.1 Infinitely Long Model, 335

7.3.2 Finite Biconical Antenna, 339

7.4 Wave Transformation and Scattering Analysis, 341

7.4.1 Wave Transformation, 342

7.4.2 Expansion of a Plane Wave, 344

7.4.3 Scattering by a Conducting Sphere, 347

7.4.4 Scattering by a Dielectric Sphere, 352

7.4.5 Scattering by a Multilayer Dielectric Sphere, 357

7.5.1 Addition Theorem for Spherical Wave Functions, 360

7.5.2 Radiation of a Spherical Surface Current, 362

7.5.3 Radiation in the Presence of a Sphere, 368

7.5.4 Radiation in the Presence of a Conducting Cone, 370

References, 377

Problems, 377

PARTII Electromagnetic Field Computation 383

8 The Finite Difference Method 385

8.1 Finite Differencing Formulas, 385

8.2 One-Dimensional Analysis, 387

8.2.1 Solution of the Diffusion Equation, 387

8.2.2 Solution of the Wave Equation, 389

8.2.3 Stability Analysis, 390

8.2.4 Numerical Dispersion Analysis, 392

8.3 Two-Dimensional Analysis, 393

8.3.1 Analysis in the Time Domain, 393

8.3.2 Analysis in the Frequency Domain, 395

8.4 Yee’s FDTD Scheme, 397

8.4.1 Two-Dimensional Analysis, 397

8.4.2 Three-Dimensional Analysis, 399

8.5 Absorbing Boundary Conditions, 402

8.5.1 One-Dimensional ABC, 403

8.5.2 Two-Dimensional ABCs, 404

8.5.3 Perfectly Matched Layers, 406

8.6 Modeling of Dispersive Media, 417

8.6.1 Recursive Convolution Approach, 417

8.6.2 Auxiliary Differential Equation Approach, 420

8.7 Wave Excitation and Far-Field Calculation, 422

8.7.1 Modeling of Wave Excitation, 422

8.7.2 Near-to-Far-Field Transformation, 426

8.8 Summary, 427

References, 428

Problems, 429

9 The Finite Element Method 433

9.1 Introduction to the Finite Element Method, 434

9.1.1 The General Principle, 434

9.1.2 One-Dimensional Example, 435

9.2 Finite Element Analysis of Scalar Fields, 439

9.2.1 The Boundary-Value Problem, 439

9.2.2 Finite Element Formulation, 440

9.2.3 Application Examples, 446

9.3 Finite Element Analysis of Vector Fields, 450

9.3.1 The Boundary-Value Problem, 450

9.3.2 Finite Element Formulation, 452

9.3.3 Application Examples, 456

9.4 Finite Element Analysis in the Time Domain, 465

9.4.1 The Boundary-Value Problem, 465

9.4.2 Finite Element Formulation, 466

9.4.3 Application Examples, 470

9.5 Discontinuous Galerkin Time-Domain Method, 472

9.5.1 Basic Idea, 473

9.5.2 Central-Flux DGTD Method, 475

9.5.3 Upwind-Flux DGTD Method, 478

9.5.4 Application Example, 481

9.6 Absorbing Boundary Conditions, 483

9.6.1 Two-Dimensional ABCs, 483

9.6.2 Three-Dimensional ABCs, 486

9.6.3 Perfectly Matched Layers, 488

9.7 Some Numerical Aspects, 494

9.7.1 Mesh Generation, 494

9.7.2 Matrix Solvers, 494

9.7.3 Higher-Order Elements, 495

9.7.4 Curvilinear Elements, 496

9.7.5 Adaptive Finite Element Analysis, 496

9.8 Summary, 497

References, 497

Problems, 499

10 The Method of Moments 505

10.1 Introduction to the Method of Moments, 506

10.2 Two-Dimensional Analysis, 510

10.2.1 Formulation of Integral Equations, 510

10.2.2 Scattering by a Conducting Cylinder, 514

10.2.3 Scattering by a Conducting Strip, 518

10.2.4 Scattering by a Homogeneous Dielectric Cylinder, 522

10.3 Three-Dimensional Analysis, 523

10.3.1 Formulation of Integral Equations, 523

10.3.2 Scattering and Radiation by a Conducting Wire, 528

10.3.3 Scattering by a Conducting Body, 532

10.3.4 Scattering by a Homogeneous Dielectric Body, 538

10.3.5 Scattering by an Inhomogeneous Dielectric Body, 542

10.4 Analysis of Periodic Structures, 544

10.4.1 Scattering by a Planar Periodic Conducting Patch Array, 544

10.4.2 Scattering by a Discrete Body-of-Revolution Object, 549

10.5 Analysis of Microstrip Antennas and Circuits, 551

10.5.1 Formulation of Integral Equations, 552

10.5.2 The Moment-Method Solution, 556

10.5.3 Evaluation of Green’s Functions, 556

10.5.4 Far-Field Calculation and Application Examples, 561

10.6 The Moment Method in the Time Domain, 561

10.6.1 Time-Domain Integral Equations, 563

10.6.2 Marching-On-in-Time Solution, 564

10.7 Summary, 568

References, 568

Problems, 571

11 Fast Algorithms and Hybrid Techniques 575

11.1 Introduction to Fast Algorithms, 576

11.2.1 Scattering by a Conducting Strip or Wire, 578

11.2.2 Scattering by a Conducting Plate, 579

11.2.3 Scattering by a Dielectric Object, 585

11.3.1 Planar Structures, 592

11.3.2 Three-Dimensional Objects, 596

11.4 Fast Multipole Method, 602

11.4.1 Two-Dimensional Analysis, 602

11.4.2 Three-Dimensional Analysis, 606

11.4.3 Multilevel Fast Multipole Algorithm, 610

11.5.1 Low-Rank Matrix, 615

11.5.3 Application to the Moment-Method Solution, 619

11.6 Introduction to Hybrid Techniques, 623

11.7 Hybrid Finite Difference–Finite Element Method, 624

11.7.1 Relation Between FETD and FDTD, 625

11.7.2 Hybridization of FETD and FDTD, 627

11.7.3 Application Example, 629

11.8 Hybrid Finite Element–Boundary Integral Method, 630

11.8.2 Symmetric Formulation, 635

11.8.3 Numerical Examples, 638

11.9 Summary, 642

References, 643

Problems, 649

12 Concluding Remarks on Computational Electromagnetics 651

12.1 Overview of Computational Electromagnetics, 651

12.1.1 Frequency-Versus Time-Domain Analysis, 651

12.1.2 High-Frequency Asymptotic Techniques, 653

12.1.3 First-Principle Numerical Methods, 654

12.1.4 Time-Domain Simulation Methods, 656

12.1.5 Hybrid Techniques, 658

12.2 Applications of Computational Electromagnetics, 659

12.3 Challenges in Computational Electromagnetics, 670

References, 671

Appendix A Vector Identities, Integral Theorems, and Coordinate Transformation 681

A.1 Vector Identities, 681

A.2 Integral Theorems, 682

A.3 Coordinate Transformation, 682

Appendix B Bessel Functions 683

B.1 Definition, 683

B.2 Series Expressions, 683

B.3 Integral Representation, 685

B.4 Asymptotic Expressions, 685

B.5 Recurrence and Derivative Relations, 685

B.6 Symmetry Relations, 686

B.7 Wronskian Relation, 686

B.8 Useful Integrals, 686

Appendix C Modified Bessel Functions 687

C.1 Definition, 687

C.2 Series Expressions, 687

C.3 Integral Representations, 688

C.4 Asymptotic Expressions, 688

C.5 Recurrence and Derivative Relations, 689

C.6 Symmetry Relations, 690

C.7 Wronskian Relation, 690

C.8 Useful Integrals, 690

Appendix D Spherical Bessel Functions 691

D.1 Definition, 691

D.2 Series Expressions, 692

D.3 Asymptotic Expressions, 693

D.4 Recurrence and Derivative Relations, 693

D.5 Symmetry Relations, 694

D.6 Wronskian Relation, 695

D.7 Riccati–Bessel Functions, 695

D.8 Modified Spherical Bessel Functions, 695

Appendix E Associated Legendre Polynomials 697

E.1 Definition, 697

E.2 Series Expression, 698

E.3 Special Values, 700

E.4 Symmetry Relations, 701

E.5 Recurrence and Derivative Relations, 701

E.6 Orthogonal Relations, 702

E.7 Fourier–Legendre Series, 702

Index 703

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