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Contents
Preface xv
CHAPTER 1
Basic Principles of Electromagnetic Theory 1
1.1 Maxwell’s Equations 1
1.2 Constitutive Relations 3
1.3 Electrical Properties of the Medium 4
1.4 Interface and Boundary Conditions 5
1.5 Skin Depth 8
1.6 Poynting Vector and Power Flow 8
1.7 Image Currents and Equivalence Principle 9
1.8 Reciprocity Theorem 12
1.9 Differential Equations in Electromagnetics 12
1.10 Electric and Magnetic Vector Potentials 14
1.11 Wave Types and Solutions 15
1.12 Phase Velocity, Dispersion, and Group Velocity 16
1.13 Characteristics of Transmission Lines 19
1.14 Charge and Current Singularities 19
1.15 Classification of Methods of Analysis 21
1.16 Mathematical Framework in Electromagnetics 22
1.17 Overview of Analytical and Computational Methods 23
1.18 Summary 26
References 27
CHAPTER 2
Analytical Methods and Orthogonal Functions 29
2.1 Introduction 29
2.2 Method of Separation of Variables 31
2.3 Orthogonality Condition 37
2.4 Sturm-Liouville Differential Equation 42
2.4.1 Orthogonality of Eigenfunctions 42
2.4.2 Boundary Conditions for Orthogonal Functions 43
2.4.3 Examples of Sturm-Liouville Type of Differential Equations 44
2.5 Eigenfunction Expansion Method 47
2.6 Vector Space/Function Space 51
2.6.1 Operators 55
2.6.2 Matrix Representation of Operators 59
2.6.3 Generic Solution of Sturm-Liouville Type Differential
Equations 62
2.7 Delta-Function and Source Representations 62
2.8 Summary 68
References 69
Problems 70
CHAPTER 3
Green’s Function 71
3.1 Introduction 71
3.2 Direct Construction Approach for Green’s Function 72
3.2.1 Green’s Function for the Sturm-Liouville Differential Equation 75
3.2.2 Green’s Function for a Loaded Transmission Line 76
3.3 Eigenfunction Expansion of Green’s Function 80
3.4 Green’s Function in Two Dimensions 81
3.4.1 Double Series Expansion Method 82
3.4.2 Single Series Expansion Method 84
3.4.3 Green’s Function in Spectral Domain 87
3.5 Green’s Function for Probe Excitation of TE-Modes in Rectangular
Waveguide 87
3.6 Green’s Function for Unbounded Region 93
3.7 Summary 95
References 95
Problems 95
CHAPTER 4
Contour Integration and Conformal Mapping 103
4.1 Introduction 103
4.1.1 Analytic Function 104
4.1.2 Analytic Continuation 105
4.2 Calculus of Residues 106
4.2.1 Poles and Branch-Point Singularities 106
4.2.2 Cauchy Integral Theorem 106
4.2.3 Residue Theorem 109
4.3 Evaluation of Definite Improper Integrals 110
4.3.1 Improper Integral Along the Real Axis 111
4.3.2 Fourier Transform Improper Integrals 115
4.3.3 Some Other Methods Useful for Solving Improper Integrals 120
4.4 Conformal Mapping of Complex Functions 121
4.4.1 Mapping 121
4.4.2 Properties of Conformal Mapping 122
4.4.3 Applications of Conformal Mapping 125
4.5 Schwarz-Christoffel Transformation 125
4.5.1 Elliptic Sine Function 129
4.5.2 Application to Coplanar Strips 131
4.6 Quasi-Static Analysis of Planar Transmission Lines 134
4.6.1 Strip Line 135
4.6.2 Microstrip Line with a Cover Shield 141
4.7 Some Useful Mappings for Planar Transmission Lines 144
4.7.1 Transformation of Finite Dielectric Thickness to Infinite
Thickness 145
4.7.2 Transformations for Finite Width Lateral Ground Planes and
Finite Dielectric Thickness 146
4.7.3 Transformation from Asymmetric to Symmetric Metallization 148
4.8 Summary 149
References 150
Problems 150
CHAPTER 5
Fourier Transform Method 153
5.1 Introduction 153
5.2 Reduction of PDE to Ordinary Differential Equation/Algebraic
Equation Using Fourier Transform 156
5.3 Solution of Differential Equations with Unbounded Regions 157
5.3.1 Free-Space Green’s Function in One Dimension 157
5.3.2 Fourier Sine Transform and Half-Space Green’s Function 160
5.3.3 Free-Space Green’s Function in Two Dimensions 162
5.3.4 Electric Line Source Above a Perfectly Conducting Ground
Plane 173
5.3.5 Free-Space Green’s Function in Three Dimensions 175
5.4 Radiation from Two-Dimensional Apertures 176
5.5 Stationary Phase Method 178
5.5.1 Radiation Pattern 180
5.5.2 Asymptotic Value of Bessel Functions 186
5.6 Green’s Function for the Quasi-Static Analysis of Microstrip Line 189
5.7 Summary 190
References 191
Appendix 5A: Evaluation of the Integral in (5.120) 191
Problems 192
CHAPTER 6
Introduction to Computational Methods 199
6.1 Elements of Computational Methods 199
6.2 Basis Functions 202
6.2.1 Subdomain Basis Functions 202
6.2.2 Entire Domain Basis Functions 206
6.3 Convergence and Discretization Error 212
6.3.1 Convergence Test 214
6.3.2 Order of Convergence 214
6.3.3 Disctretization Error and Extrapolation 215
6.3.4 Discretization of Operators 217
6.3.5 Discretization Error in FDM, FDTD, and FEM 219
6.3.6 Vector and Matrix Norms 223
6.4 Stability of Numerical Solutions 223
6.4.1 Stability of FDTD Solution 224
6.4.2 Stability of Matrix Solution 225
6.5 Accuracy of Numerical Solutions 227
6.5.1 Modeling Errors 227
6.5.2 Truncation Error 227
6.5.3 Round-Off Error 227
6.5.4 Validation 228
6.6 Spurious Solutions 229
6.7 Formulations for the Computational Methods 229
6.8 Summary 229
References 230
Problems 231
CHAPTER 7
Method of Finite Differences 233
7.1 Finite Difference Approximations 233
7.1.1 Difference Form of the First Derivative 233
7.1.2 Difference Form of the Second Derivative 235
7.1.3 Difference Form of Laplace and Poisson Equations 236
7.2 Treatment of Interface and Boundary Conditions 243
7.2.1 Nodes on the Interface 243
7.2.2 Dielectric Inhomogeneity in One Quadrant About a Node 245
7.2.3 Neumann Boundary Condition and the Nodes on the Edge 246
7.2.4 Node at a Corner 248
7.2.5 Node at an Edge with Dielectric Inhomogeneity About the
Node 249
7.2.6 Treatment of Curved Boundaries 249
7.2.7 Finite Difference Analysis of an Inhomogeneously Filled
Parallel Plate Capacitor 252
7.3 Finite Difference Analysis of Guiding Structures 254
7.3.1 Analysis of Enclosed Microstrip Line 254
7.3.2 Analysis of Geometries with Open Boundaries 261
7.3.3 Wave Propagation and Numerical Dispersion 262
7.3.4 Analysis of Ridge Waveguide 264
7.4 Summary 268
References 270
Problems 271
CHAPTER 8
Finite-Difference Time-Domain Analysis 281
8.1 Pulse Propagation in a Transmission Line 281
8.2 FDTD Analysis in One Dimension 284
8.2.1 Spatial Step x and Numerical Dispersion 288
8.2.2 Time Step t and Stability of the Solution 292
8.2.3 Source or Excitation of the Grid 295
8.2.4 Absorbing Boundary Conditions for One-Dimensional
Propagation 305
8.3 Applications of One-Dimensional FDTD Analysis 309
8.3.1 Reflection at an Interface 309
8.3.2 Determination of Propagation Constant 312
8.3.3 Design of Material Absorber 313
8.3.4 Exponential Time-Stepping Algorithm in the Lossy Region 316
8.3.5 Extraction of Frequency Domain Information from the Time
Domain Data 316
8.3.6 Simulation of Lossy, Dispersive Materials 317
8.4 FDTD Analysis in Two Dimensions 323
8.4.1 Unit Cell in Two Dimensions 325
8.4.2 Numerical Dispersion in Two Dimensions 327
8.4.3 Time Step t for Two-Dimensional Propagation 329
8.4.4 Absorbing Boundary Conditions for Propagation in Two
Dimensions 329
8.4.5 Perfectly Matched Layer ABC 333
8.5 FDTD Analysis in Three Dimensions 339
8.5.1 Yee Cell 339
8.5.2 Numerical Dispersion in Three Dimensions 343
8.5.3 Time Step t for Three-Dimensional Propagation 343
8.5.4 Absorbing Boundary Conditions and PML for Three
Dimensions 344
8.6 Implementation of Boundary Conditions in FDTD 345
8.6.1 Perfect Electric and Magnetic Wall Boundary Conditions 345
8.6.2 Interface Conditions 346
8.7 Advances in FDTD 347
8.8 Summary 347
References 348
Problems 349
CHAPTER 9
Variational Methods 355
9.1 Calculus of Variations 355
9.1.1 Stationarity 355
9.1.2 Extremum 357
9.1.3 Functional 358
9.1.4 Variation or Increment of a Function, (x) 359
9.1.5 Variation and Stationarity of Functionals 360
9.2 Stationary Functionals and Euler Equations 363
9.3 The Ritz Variational Method 366
9.4 Applications of Ritz Approach 367
9.4.1 Variational Solution of Laplace Equation 368
9.4.2 Cutoff Frequencies for Waveguide Modes 373
9.4.3 Resonant Frequency for Cavity Modes 374
9.4.4 Variational Formulation in Spectral Domain for
the Microstrip Line 378
9.5 Construction of Functionals from the PDEs 381
9.6 Method of Weighted Residuals 383
9.6.1 Galerkin’s Method 384
9.6.2 Point Matching Method 385
9.7 Summary 387
References 388
Problems 388
CHAPTER 10
Finite Element Method 393
10.1 Basic Steps in Finite Element Analysis 393
10.1.1 Segmentation or Meshing of the Geometry 394
10.1.2 Derivation of the Element Matrix 395
10.1.3 Assembly of Element Matrices 397
10.1.4 Solution of System Matrix 397
10.1.5 Postprocessing 398
10.2 FEM Analysis in One Dimension 398
10.2.1 Treatment of Boundary and Interface Conditions 402
10.2.2 Accuracy and Numerical Dispersion 406
10.3 FEM Analysis in Two Dimensions 409
10.3.1 Solution of Two-Dimensional Wave Equation 410
10.3.2 Element Matrix for Rectangular Elements 411
10.3.3 Element Matrix for Triangular Elements 415
10.3.4 Assembly of Element Matrices and System Equations 418
10.3.5 Capacitance of a Parallel Plate Capacitor 422
10.3.6 Cutoff Frequency Waveguide Modes 429
10.3.7 FEM Analysis of Open Boundary Problems 436
10.4 Mesh Generation and Node Location Table 436
10.5 Weighted Residual Formulation for FEM 440
10.6 Summary 441
References 442
Problems 442
CHAPTER 11
Method of Moments 445
11.1 Introduction 445
11.1.1 MoM Procedure 446
11.1.2 Point Matching and Galerkin’s Methods 448
11.1.3 Eigenvalue Analysis Using MoM 449
11.2 Solution of Integral Equations Using MoM 452
11.2.1 Integral Equation 452
11.2.2 Static Charge Distribution on a Wire 455
11.2.3 Analysis of Strip Line 462
11.2.4 Analysis of Wire Dipole Antenna 469
11.2.5 Scattering from a Conducting Cylinder of Infinite Length 476
11.3 Fast Multipole Solution Methods for MoM 485
11.4 Comparison of FDM, FDTD, FEM, and MoM 486
11.5 Hybrid Computational Methods 487
11.6 Summary 487
References 487
Problems 488
APPENDIX A
Solution Methods for the Set of Simultaneous Equations 493
A.1 Processor Time Considerations 493
A.2 Matrix Solution Techniques 494
A.2.1 Gauss Elimination 495
A.2.2 L-U Factorization 498
A.3 Sparse Matrix Techniques 500
A.3.1 Reordering of Equations 500
A.3.2 Preconditioned Conjugate Gradient Method 502
References 502
APPENDIX B
Evaluation of Singular Integrals 505
References 507
About the Author 509
Index 511
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