Achieve optimal microwave system performance by mastering the principles and methods underlying today's powerful computational tools and commercial software in electromagnetics. This authoritative resource offers you clear and complete explanation of this essential electromagnetics knowledge, providing you with the analytical background you need to understand such key approaches as MoM (method of moments), FDTD (Finite Difference Time Domain) and FEM (Finite Element Method), and Green's functions. This comprehensive book includes all math necessary to master the material. Moreover, it features numerous solved problems that help ensure your understanding of key concepts throughout the book.
To keep you from being bogged down with complex mathematical details (vector calculus) and coding, this comprehensive volume places emphasis on the analysis of the scalar wave equation in Cartesian coordinates. The book also includes multiple choice questions, appropriate for self study or courses, that help clarify concepts without any mathematical burden. Packed with over 1,300 equations, most of the problems presented in the book can be solved using nothing more than calculator.
Software Included
CD-ROM-Included! Includes time-saving Matlab® source code for the problems presented in the book which can be easily modified to help you solve similar problems in the field.
Author(s): Ramesh Garg
Series: Artech House Electromagnetic Analysis
Edition: Har/Cdr
Publisher: Artech House Publishers
Year: 2008
Language: English
Pages: 551
Analytical and Computational Methods in Electromagnetics......Page 2
Contents......Page 8
Preface......Page 16
1.1 Maxwell’s Equations......Page 22
1.2 Constitutive Relations......Page 24
1.3 Electrical Properties of the Medium......Page 25
1.4 Interface and Boundary Conditions......Page 26
1.5 Skin Depth......Page 28
1.6 Poynting Vector and Power Flow......Page 29
1.7 Image Currents and Equivalence Principle......Page 30
1.9 Differential Equations in Electromagnetics......Page 33
1.10 Electric and Magnetic Vector Potentials......Page 35
1.11 Wave Types and Solutions......Page 36
1.12 Phase Velocity, Dispersion, and Group Velocity......Page 37
1.14 Charge and Current Singularities......Page 40
1.15 Classification of Methods of Analysis......Page 42
1.16 Mathematical Framework in Electromagnetics......Page 43
1.17 Overview of Analytical and Computational Methods......Page 44
1.18 Summary......Page 47
References......Page 48
2.1 Introduction......Page 50
2.2 Method of Separation of Variables......Page 52
2.3 Orthogonality Condition......Page 58
2.4 Sturm-Liouville Differential Equation......Page 62
2.4.1 Orthogonality of Eigenfunctions......Page 63
2.4.2 Boundary Conditions for Orthogonal Functions......Page 64
2.4.3 Examples of Sturm-Liouville Type of Differential Equations......Page 65
2.5 Eigenfunction Expansion Method......Page 68
2.6 Vector Space/Function Space......Page 72
2.6.1 Operators......Page 76
2.6.2 Matrix Representation of Operators......Page 80
2.7 Delta-Function and Source Representations......Page 83
2.8 Summary......Page 89
References......Page 90
CHAPTER 3: Green’s Function......Page 91
3.1 Introduction......Page 92
3.2 Direct Construction Approach for Green’s Function......Page 93
3.2.1 Green’s Function for the Sturm-Liouville Differential Equation......Page 96
3.2.2 Green’s Function for a Loaded Transmission Line......Page 97
3.3 Eigenfunction Expansion of Green’s Function......Page 101
3.4 Green’s Function in Two Dimensions......Page 102
3.4.1 Double Series Expansion Method......Page 103
3.4.2 Single Series Expansion Method......Page 105
3.5 Green’s Function for Probe Excitation of TE-Modes in Rectangular Waveguide......Page 108
3.6 Green’s Function for Unbounded Region......Page 114
Problems......Page 116
4.1 Introduction......Page 124
4.1.1 Analytic Function......Page 125
4.1.2 Analytic Continuation......Page 126
4.2.2 Cauchy Integral Theorem......Page 127
4.2.3 Residue Theorem......Page 130
4.3.1 Improper Integral Along the Real Axis......Page 131
4.3.2 Fourier Transform Improper Integrals......Page 135
4.3.3 Some Other Methods Useful for Solving Improper Integrals......Page 141
4.4.1 Mapping......Page 142
4.4.2 Properties of Conformal Mapping......Page 143
4.4.3 Applications of Conformal Mapping......Page 145
4.5 Schwarz-Christoffel Transformation......Page 146
4.5.1 Elliptic Sine Function......Page 150
4.5.2 Application to Coplanar Strips......Page 152
4.6 Quasi-Static Analysis of Planar Transmission Lines......Page 155
4.6.1 Strip Line......Page 156
4.6.2 Microstrip Line with a Cover Shield......Page 162
4.7 Some Useful Mappings for Planar Transmission Lines......Page 165
4.7.1 Transformation of Finite Dielectric Thickness to Infinite Thickness......Page 166
4.7.2 Transformations for Finite Width Lateral Ground Planes and FiniteDielectric Thickness......Page 167
4.7.3 Transformation from Asymmetric to Symmetric Metallization......Page 169
4.8 Summary......Page 170
Problems......Page 171
5.1 Introduction......Page 174
5.2 Reduction of PDE to Ordinary Differential Equation/Algebraic Equation Using Fourier Transform......Page 177
5.3.1 Free-Space Green’s Function in One Dimension......Page 178
5.3.2 Fourier Sine Transform and Half-Space Green’s Function......Page 181
5.3.3 Free-Space Green’s Function in Two Dimensions......Page 183
5.3.4 Electric Line Source Above a Perfectly Conducting Ground Plane......Page 194
5.3.5 Free-Space Green’s Function in Three Dimensions......Page 196
5.4 Radiation from Two-Dimensional Apertures......Page 197
5.5 Stationary Phase Method......Page 199
5.5.1 Radiation Pattern......Page 201
5.5.2 Asymptotic Value of Bessel Functions......Page 207
5.6 Green’s Function for the Quasi-Static Analysis of Microstrip Line......Page 209
5.7 Summary......Page 211
Appendix 5A: Evaluation of the Integral in (5.120)......Page 212
Problems......Page 213
6.1 Elements of Computational Methods......Page 220
6.2.1 Subdomain Basis Functions......Page 223
6.2.2 Entire Domain Basis Functions......Page 227
6.3 Convergence and Discretization Error......Page 233
6.3.2 Order of Convergence......Page 235
6.3.3 Disctretization Error and Extrapolation......Page 236
6.3.4 Discretization of Operators......Page 238
6.3.5 Discretization Error in FDM, FDTD, and FEM......Page 240
6.4.1 Stability of FDTD Solution......Page 244
6.4.2 Stability of Matrix Solution......Page 246
6.5.3 Round-Off Error......Page 248
6.6 Spurious Solutions......Page 249
6.8 Summary......Page 250
References......Page 251
Problems......Page 252
7.1.1 Difference Form of the First Derivative......Page 254
7.1.2 Difference Form of the Second Derivative......Page 256
7.1.3 Difference Form of Laplace and Poisson Equations......Page 257
7.2.1 Nodes on the Interface......Page 264
7.2.2 Dielectric Inhomogeneity in One Quadrant About a Node......Page 266
7.2.3 Neumann Boundary Condition and the Nodes on the Edge......Page 267
7.2.4 Node at a Corner......Page 269
7.2.6 Treatment of Curved Boundaries......Page 270
7.2.7 Finite Difference Analysis of an Inhomogeneously Filled Parallel PlateCapacitor......Page 273
7.3.1 Analysis of Enclosed Microstrip Line......Page 275
7.3.2 Analysis of Geometries with Open Boundaries......Page 282
7.3.3 Wave Propagation and Numerical Dispersion......Page 283
7.3.4 Analysis of Ridge Waveguide......Page 285
7.4 Summary......Page 289
References......Page 291
Problems......Page 292
8.1 Pulse Propagation in a Transmission Line......Page 302
8.2 FDTD Analysis in One Dimension......Page 305
8.2.1 Spatial Step Dx and Numerical Dispersion......Page 309
8.2.2 Time Step Dt and Stability of the Solution......Page 313
8.2.3 Source or Excitation of the Grid......Page 316
8.2.4 Absorbing Boundary Conditions for One-Dimensional Propagation......Page 326
8.3.1 Reflection at an Interface......Page 330
8.3.2 Determination of Propagation Constant......Page 333
8.3.3 Design of Material Absorber......Page 334
8.3.4 Exponential Time-Stepping Algorithm in the Lossy Region......Page 336
8.3.5 Extraction of Frequency Domain Information from the Time Domain Data......Page 337
8.3.6 Simulation of Lossy, Dispersive Materials......Page 338
8.4 FDTD Analysis in Two Dimensions......Page 344
8.4.1 Unit Cell in Two Dimensions......Page 346
8.4.2 Numerical Dispersion in Two Dimensions......Page 348
8.4.4 Absorbing Boundary Conditions for Propagation in Two Dimensions......Page 350
8.4.5 Perfectly Matched Layer ABC......Page 354
8.5.1 Yee Cell......Page 360
8.5.4 Absorbing Boundary Conditions and PML for Three Dimensions......Page 364
8.6.1 Perfect Electric and Magnetic Wall Boundary Conditions......Page 366
8.6.2 Interface Conditions......Page 367
8.8 Summary......Page 368
References......Page 369
Problems......Page 370
CHAPTER 9: Variational Methods......Page 375
9.1.1 Stationarity......Page 376
9.1.2 Extremum......Page 378
9.1.3 Functional......Page 379
9.1.4 Variation or Increment of a Function......Page 380
9.1.5 Variation and Stationarity of Functionals......Page 381
9.2 Stationary Functionals and Euler Equations......Page 384
9.3 The Ritz Variational Method......Page 387
9.4.1 Variational Solution of Laplace Equation......Page 388
9.4.2 Cutoff Frequency for Waveguide Modes......Page 394
9.4.3 Resonant Frequency for Cavity Modes......Page 395
9.4.4 Variational Formulation in Spectral Domain for the Microstrip Line......Page 399
9.5 Construction of Functionals from the PDEs......Page 402
9.6 Method of Weighted Residuals......Page 403
9.6.1 Galerkin’s Method......Page 405
9.6.2 Point Matching Method......Page 406
9.7 Summary......Page 408
Problems......Page 409
10.1.1 Segmentation or Meshing of the Geometry......Page 414
10.1.2 Derivation of the Element Matrix......Page 416
10.1.5 Postprocessing......Page 418
10.2 FEM Analysis in One Dimension......Page 419
10.2.1 Treatment of Boundary and Interface Conditions......Page 423
10.2.2 Accuracy and Numerical Dispersion......Page 427
10.3 FEM Analysis in Two Dimensions......Page 430
10.3.1 Solution of Two-Dimensional Wave Equation......Page 431
10.3.2 Element Matrix for Rectangular Elements......Page 432
10.3.3 Element Matrix for Triangular Elements......Page 436
10.3.4 Assembly of Elements and System Equations......Page 439
10.3.5 Capacitance of a Parallel Plate Capacitor......Page 443
10.3.6 Cutoff Frequency of Waveguide Modes......Page 450
10.4 Mesh Generation and Node Location Table......Page 457
10.5 Weighted Residual Formulation for FEM......Page 461
10.6 Summary......Page 462
Problems......Page 463
11.1 Introduction......Page 466
11.1.1 MoM Procedure......Page 467
11.1.2 Point Matching and Galerkin’s Methods......Page 469
11.1.3 Eigenvalue Analysis Using MoM......Page 470
11.2.1 Integral Equation......Page 473
11.2.2 Static Charge Distribution on a Wire......Page 476
11.2.3 Analysis of Strip Line......Page 483
11.2.4 Analysis of Wire Dipole Antenna......Page 490
11.2.5 Scattering from a Conducting Cylinder of Infinite Length......Page 497
11.3 Fast Multipole Solution Methods for MoM......Page 506
11.4 Comparison of FDM, FDTD, FEM, and MoM......Page 507
References......Page 508
Problems......Page 509
APPENDIX A: Solution Methods for the Set ofSimultaneous Equations......Page 513
A.1 Processor Time Considerations [1]......Page 514
A.2 Matrix Solution Techniques......Page 515
A.2.1 Gauss Elimination [3, 4]......Page 516
A.2.2 L-U Factorization [3, 4]......Page 519
A.3.1 Reordering of Equations......Page 521
References......Page 523
APPENDIX B: Evaluation of Singular Integrals......Page 526
References......Page 528
About the Author......Page 530
Index......Page 532
