Analytical and Computational Methods in Electromagnetics (Artech House Electromagnetic Analysis)

551 Pages·2008·6.1 MB·English

by  Ramesh Garg

Analytical and Computational Methods
in Electromagnetics
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Analytical and Computational Methods
in Electromagnetics
Ramesh Garg
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ISBN-13: 978-1-59693-385-9
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To my family Madhu, Geetika, and Prashant, and in the memory of my teacher and
friend, the late Professor K. C. Gupta

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
vii
viii Contents
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
Contents ix
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