Theory and Computation of Electromagnetic Fields

THEORY AND COMPUTATION OF ELECTROMAGNETIC FIELDS......Page 4
CONTENTS......Page 6
PREFACE......Page 12
ACKNOWLEDGMENTS......Page 16
PART I: ELECTROMAGNETIC FIELD THEORY......Page 17
1.1.1 Vector Operations and Integral Theorems......Page 19
1.1.2 Symbolic Vector Method......Page 22
1.1.3 Helmholtz Decomposition Theorem......Page 24
1.2 MAXWELL ’ S EQUATIONS IN TERMS OF TOTAL CHARGES AND CURRENTS......Page 25
1.2.1 Maxwell ’ s Equations in Integral Form......Page 27
1.2.3 Current Continuity Equation......Page 30
1.3.1 Electric Polarization......Page 31
1.3.2 Magnetization......Page 33
1.3.4 Classi cation of Media......Page 35
1.4 MAXWELL ’ S EQUATIONS IN TERMS OF FREE CHARGES AND CURRENTS......Page 38
1.5 BOUNDARY CONDITIONS......Page 40
1.6 ENERGY, POWER, AND POYNTING ’ S THEOREM......Page 42
1.7.1 Time Harmonic Fields......Page 45
1.7.2 Fourier Transforms......Page 46
1.7.3 Complex Power......Page 48
1.7.4 Complex Permittivity and Permeability......Page 52
PROBLEMS......Page 53
2.1 SCALAR AND VECTOR POTENTIALS......Page 59
2.1.1 Static Fields......Page 60
2.1.2 Time Harmonic Fields and the Lorenz Gauge Condition......Page 61
2.2 SOLUTION OF VECTOR POTENTIALS IN FREE SPACE......Page 63
2.2.1 Delta Function and Green ’ s Function......Page 64
2.2.2 Green ’ s Function in Free Space......Page 65
2.2.3 Field – Source Relations in Free Space......Page 66
2.2.4 Why Use Auxiliary Potential Functions......Page 67
2.2.5 Free Space Dyadic Green ’ s Functions......Page 68
2.3.1 In nitesimal Electric Dipole......Page 71
2.3.2 Finite Electric Dipole......Page 73
2.3.3 Far Field Approximation and the Sommerfeld Radiation Condition......Page 75
2.3.4 Circular Current Loop and Magnetic Dipole......Page 77
2.4.1 Radiation by a Surface Current......Page 79
2.4.2 Radiation by a Phased Array......Page 81
PROBLEMS......Page 85
3.1 UNIQUENESS THEOREM......Page 89
3.2 IMAGE THEORY......Page 91
3.2.1 Basic Image Theory......Page 92
3.2.2 Half Space Field – Source Relations......Page 96
3.3.1 General Reciprocity Theorem......Page 98
3.3.3 Rayleigh – Carson Reciprocity Theorem......Page 99
3.4.1 Surface Equivalence Principle......Page 101
3.4.2 Application to Scattering by a Conducting Object......Page 103
3.4.3 Application to Scattering by a Dielectric Object......Page 106
3.4.4 Volume Equivalence Principle......Page 108
3.5 DUALITY PRINCIPLE......Page 110
3.6 APERTURE RADIATION AND SCATTERING......Page 111
3.6.1 Equivalent Problems......Page 112
3.6.2 Babinet ’ s Principle......Page 115
3.6.3 Complementary Antennas......Page 117
REFERENCES......Page 118
PROBLEMS......Page 119
4.1.1 Governing Differential Equations and General Solutions......Page 123
4.1.2 Ref ection and Transmission......Page 126
4.1.3 Green ’ s Function and Eigenfunction Expansion......Page 127
4.2 WAVE EQUATIONS AND GENERAL SOLUTIONS......Page 131
4.2.1 Wave Equations and Solution by Separation of Variables......Page 132
4.2.2 Characteristics of a Plane Wave......Page 133
4.2.3 Wave Velocities and Attenuation......Page 135
4.2.4 Linear, Circular, and Elliptical Polarizations......Page 137
4.2.5 Wave Propagation in Metamaterials......Page 140
4.3 PLANE WAVES GENERATED BY A CURRENT SHEET......Page 141
4.4.1 Ref ection and Transmission at Normal Incidence......Page 143
4.4.2 Ref ection and Transmission at Oblique Incidence......Page 145
4.4.3 Total Transmission and Total Ref ection......Page 148
4.4.4 Transmission into a Left Handed Medium......Page 151
4.5.1 Plane Waves in Uniaxial Media......Page 153
4.5.2 Plane Waves in Gyrotropic Media......Page 158
4.5.3 Plane Waves in Chiral Media......Page 161
PROBLEMS......Page 163
5.1.1 General Analysis......Page 168
5.1.2 General Characteristics......Page 172
5.1.3 Uniform Rectangular Waveguide......Page 176
5.1.4 Losses in Waveguides and Attenuation Constant......Page 183
5.2 UNIFORM CAVITIES......Page 186
5.2.1 General Theory......Page 187
5.2.2 Rectangular Cavity......Page 189
5.2.3 Material and Geometry Perturbations......Page 191
5.3.1 General Theory......Page 194
5.3.2 Partially Filled Rectangular Waveguide......Page 196
5.3.3 Dielectric Slab Waveguide on a Ground Plane......Page 199
5.4.1 Excitation by Planar Surface Currents......Page 203
5.4.2 Excitation by General Volumetric Currents......Page 205
5.5.1 Spectral Green ’ s Function and Sommerfeld Identity......Page 207
5.5.2 Vertical Electric Dipole above a Layered Medium......Page 208
5.5.3 Horizontal Electric Dipole above a Layered Medium......Page 210
5.5.4 Dipoles on a Grounded Dielectric Slab......Page 212
NOTE......Page 213
PROBLEMS......Page 214
6.1.1 Solution by Separation of Variables......Page 216
6.1.2 Cylindrical Wave Functions......Page 219
6.2 CIRCULAR AND COAXIAL WAVEGUIDES AND CAVITIES......Page 220
6.2.1 Circular Waveguide......Page 221
6.2.2 Coaxial Waveguide......Page 224
6.2.3 Cylindrical Cavity......Page 228
6.3 CIRCULAR DIELECTRIC WAVEGUIDE......Page 229
6.3.1 Analysis of Hybrid Modes......Page 230
6.3.2 Characteristics of Hybrid Modes......Page 233
6.4 WAVE TRANSFORMATION AND SCATTERING ANALYSIS......Page 238
6.4.1 Wave Transformation......Page 239
6.4.2 Scattering by a Circular Conducting Cylinder......Page 240
6.4.3 Scattering by a Circular Dielectric Cylinder......Page 243
6.4.4 Scattering by a Circular Multilayer Dielectric Cylinder......Page 246
6.5.1 Line Current Radiation in Free Space......Page 250
6.5.2 Radiation by a Cylindrical Surface Current......Page 252
6.5.3 Radiation in the Presence of a Circular Conducting Cylinder......Page 254
6.5.4 Radiation in the Presence of a Conducting Wedge......Page 257
6.5.5 Radiation by a Finite Current......Page 259
PROBLEMS......Page 261
7.1.1 Solution by Separation of Variables......Page 264
7.1.2 Spherical Wave Functions......Page 267
Modes......Page 269
7.2 SPHERICAL CAVITY......Page 271
7.3.1 In nitely Long Model......Page 275
7.3.2 Finite Biconical Antenna......Page 278
7.4.1 Wave Transformation......Page 280
7.4.2 Expansion of a Plane Wave......Page 282
7.4.3 Scattering by a Conducting Sphere......Page 285
7.4.4 Scattering by a Dielectric Sphere......Page 290
7.4.5 Scattering by a Multilayer Dielectric Sphere......Page 292
7.5.1 Addition Theorem for Spherical Wave Functions......Page 297
7.5.2 Radiation of a Spherical Surface Current......Page 299
7.5.3 Radiation in the Presence of a Sphere......Page 302
7.5.4 Radiation in the Presence of a Conducting Cone......Page 304
REFERENCES......Page 307
PROBLEMS......Page 308
PART II: ELECTROMAGNETIC FIELD COMPUTATION......Page 311
THE FINITE DIFFERENCE METHOD......Page 313
8.1 FINITE DIFFERENCING FORMULAS......Page 314
8.2.1 Solution of the Diffusion Equation......Page 316
8.2.3 Stability Analysis......Page 318
8.2.4 Numerical Dispersion Analysis......Page 321
8.3.1 Analysis in the Time Domain......Page 322
8.3.2 Analysis in the Frequency Domain......Page 324
8.4.1 Two Dimensional Analysis......Page 325
8.4.2 Three Dimensional Analysis......Page 327
8.5.1 One Dimensional ABC......Page 330
8.5.2 Two Dimensional ABCs......Page 332
8.5.3 Perfectly Matched Layers......Page 334
8.6.1 Recursive Convolution Approach......Page 345
8.6.2 Auxiliary Differential Equation Approach......Page 347
8.7 WAVE EXCITATION AND FAR FIELD CALCULATION......Page 349
8.7.1 Modeling of Wave Excitation......Page 350
8.7.2 Near to Far Field Transformation......Page 353
8.8 SUMMARY......Page 354
REFERENCES......Page 355
PROBLEMS......Page 356
THE FINITE ELEMENT METHOD......Page 358
9.1.1 The General Principle......Page 359
9.1.2 One Dimensional Example......Page 360
9.2.1 The Boundary Value Problem......Page 363
9.2.2 Finite Element Formulation......Page 364
9.2.3 Application Examples......Page 370
9.3.1 The Boundary Value Problem......Page 375
9.3.2 Finite Element Formulation......Page 376
9.3.3 Application Examples......Page 380
9.4 FINITE ELEMENT ANALYSIS IN THE TIME DOMAIN......Page 388
9.4.1 The Boundary Value Problem......Page 389
9.4.2 Finite Element Formulation......Page 390
9.4.3 Application Examples......Page 394
9.5 ABSORBING BOUNDARY CONDITIONS......Page 395
9.5.1 Two Dimensional ABC s......Page 396
9.5.2 Three Dimensional ABCs......Page 399
9.5.3 Perfectly Matched Layers......Page 402
9.6 SOME NUMERICAL ASPECTS......Page 406
9.6.2 Matrix Solvers......Page 407
9.6.3 Higher Order Elements......Page 408
9.7 SUMMARY......Page 409
REFERENCES......Page 410
PROBLEMS......Page 411
10.1 INTRODUCTION TO THE METHOD OF MOMENTS......Page 415
10.2.1 Formulation of Integral Equations......Page 420
10.2.2 Scattering by a Conducting Cylinder......Page 423
10.2.3 Scattering by a Conducting Strip......Page 427
10.2.4 Scattering by a Homogeneous Dielectric Cylinder......Page 431
10.3.1 Formulation of Integral Equations......Page 432
10.3.2 Scattering and Radiation by a Conducting Wire......Page 437
10.3.3 Scattering by a Conducting Body......Page 441
10.3.4 Scattering by a Homogeneous Dielectric Body......Page 446
10.3.5 Scattering by an Inhomogeneous Dielectric Body......Page 449
10.4.1 Scattering by a Planar Periodic Conducting Patch Array......Page 452
10.4.2 Scattering by a Discrete Body of Revolution Object......Page 457
10.5.1 Formulation of Integral Equations......Page 459
10.5.3 Evaluation of Green ’ s Functions......Page 462
10.5.4 Far Field Calculation and Application Examples......Page 467
10.6 THE MOMENT METHOD IN THE TIME DOMAIN......Page 468
10.6.1 Time Domain Integral Equations......Page 469
10.6.2 Marching On in Time Solution......Page 470
REFERENCES......Page 474
PROBLEMS......Page 476
11.1 INTRODUCTION TO FAST ALGORITHMS......Page 479
11.2 CONJUGATE GRADIENT – FFT METHOD......Page 481
11.2.2 Scattering by a Conducting Plate......Page 482
11.2.3 Scattering by a Dielectric Object......Page 488
11.3.1 Planar Structures......Page 494
11.3.2 Three Dimensional Objects......Page 498
11.4.1 Two Dimensional Analysis......Page 503
11.4.2 Three Dimensional Analysis......Page 508
11.4.3 Multilevel Fast Multipole Algorithm......Page 511
11.5.1 Low Rank Matrix......Page 516
11.5.2 Adaptive Cross Approximation......Page 518
11.5.3 Application to the Moment Method Solution......Page 520
11.6 INTRODUCTION TO HYBRID TECHNIQUES......Page 524
11.7 HYBRID FINITE DIFFERENCE – FINITE ELEMENT METHOD......Page 525
11.7.1 Relation between FETD and FDTD......Page 526
11.7.2 Hybridization of FETD and FDTD......Page 528
11.7.3 Application Example......Page 530
11.8 HYBRID FINITE ELEMENT – BOUNDARY INTEGRAL METHOD......Page 531
11.8.1 Traditional Formulation......Page 533
11.8.2 Symmetric Formulation......Page 536
11.8.3 Numerical Examples......Page 539
11.9 SUMMARY......Page 542
REFERENCES......Page 543
PROBLEMS......Page 547
Frequency versus Time Domain Analysis......Page 549
12.1.2 High Frequency Asymptotic Techniques......Page 550
12.1.3 First Principle Numerical Methods......Page 552
12.1.4 Time Domain Simulation Methods......Page 554
12.2 APPLICATIONS OF COMPUTATIONAL ELECTROMAGNETICS......Page 556
12.3 CHALLENGES IN COMPUTATIONAL ELECTROMAGNETICS......Page 567
REFERENCES......Page 568
INTEGRAL THEOREMS......Page 575
COORDINATE TRANSFORMATION......Page 576
INDEX......Page 577
Color Plates......Page 589