Analytical and Computational Methods in Electromagnetics -- Contents -- Preface -- CHAPTER 1: Basic Principles of Electromagnetic Theory -- 1.1 Maxwell's Equations -- 1.2 Constitutive Relations -- 1.3 Electrical Properties of the Medium -- 1.4 Interface and Boundary Conditions -- 1.5 Skin Depth -- 1.6 Poynting Vector and Power Flow -- 1.7 Image Currents and Equivalence Principle -- 1.8 Reciprocity Theorem -- 1.9 Differential Equations in Electromagnetics -- 1.10 Electric and Magnetic Vector Potentials -- 1.11 Wave Types and Solutions -- 1.12 Phase Velocity, Dispersion, and Group Velocity -- 1.13 Characteristics of Transmission Lines -- 1.14 Charge and Current Singularities -- 1.15 Classification of Methods of Analysis -- 1.16 Mathematical Framework in Electromagnetics -- 1.17 Overview of Analytical and Computational Methods -- 1.18 Summary -- References -- CHAPTER 2: Analytical Methods and Orthogonal Functions -- 2.1 Introduction -- 2.2 Method of Separation of Variables -- 2.3 Orthogonality Condition -- 2.4 Sturm-Liouville Differential Equation -- 2.4.1 Orthogonality of Eigenfunctions -- 2.4.2 Boundary Conditions for Orthogonal Functions -- 2.4.3 Examples of Sturm-Liouville Type of Differential Equations -- 2.5 Eigenfunction Expansion Method -- 2.6 Vector Space/Function Space -- 2.6.1 Operators -- 2.6.2 Matrix Representation of Operators -- 2.6.3 Generic Solution of Sturm-Liouville Type Differential Equations -- 2.7 Delta-Function and Source Representations -- 2.8 Summary -- References -- Problems -- CHAPTER 3: Green's Function -- 3.1 Introduction -- 3.2 Direct Construction Approach for Green's Function -- 3.2.1 Green's Function for the Sturm-Liouville Differential Equation -- 3.2.2 Green's Function for a Loaded Transmission Line -- 3.3 Eigenfunction Expansion of Green's Function -- 3.4 Green's Function in Two Dimensions.

3.4.1 Double Series Expansion Method -- 3.4.2 Single Series Expansion Method -- 3.4.3 Green's Function in Spectral Domain -- 3.5 Green's Function for Probe Excitation of TE-Modes in Rectangular Waveguide -- 3.6 Green's Function for Unbounded Region -- 3.7 Summary -- References -- Problems -- CHAPTER 4: Contour Integration and Conformal Mapping -- 4.1 Introduction -- 4.1.1 Analytic Function -- 4.1.2 Analytic Continuation -- 4.2 Calculus of Residues -- 4.2.1 Poles and Branch-Point Singularities -- 4.2.2 Cauchy Integral Theorem -- 4.2.3 Residue Theorem -- 4.3 Evaluation of Definite Improper Integrals -- 4.3.1 Improper Integral Along the Real Axis -- 4.3.2 Fourier Transform Improper Integrals -- 4.3.3 Some Other Methods Useful for Solving Improper Integrals -- 4.4 Conformal Mapping of Complex Functions -- 4.4.1 Mapping -- 4.4.2 Properties of Conformal Mapping -- 4.4.3 Applications of Conformal Mapping -- 4.5 Schwarz-Christoffel Transformation -- 4.5.1 Elliptic Sine Function -- 4.5.2 Application to Coplanar Strips -- 4.6 Quasi-Static Analysis of Planar Transmission Lines -- 4.6.1 Strip Line -- 4.6.2 Microstrip Line with a Cover Shield -- 4.7 Some Useful Mappings for Planar Transmission Lines -- 4.7.1 Transformation of Finite Dielectric Thickness to Infinite Thickness -- 4.7.2 Transformations for Finite Width Lateral Ground Planes and FiniteDielectric Thickness -- 4.7.3 Transformation from Asymmetric to Symmetric Metallization -- 4.8 Summary -- References -- Problems -- CHAPTER 5: Fourier Transform Method -- 5.1 Introduction -- 5.2 Reduction of PDE to Ordinary Differential Equation/Algebraic Equation Using Fourier Transform -- 5.3 Solution of Differential Equations with Unbounded Regions -- 5.3.1 Free-Space Green's Function in One Dimension -- 5.3.2 Fourier Sine Transform and Half-Space Green's Function.

5.3.3 Free-Space Green's Function in Two Dimensions -- 5.3.4 Electric Line Source Above a Perfectly Conducting Ground Plane -- 5.3.5 Free-Space Green's Function in Three Dimensions -- 5.4 Radiation from Two-Dimensional Apertures -- 5.5 Stationary Phase Method -- 5.5.1 Radiation Pattern -- 5.5.2 Asymptotic Value of Bessel Functions -- 5.6 Green's Function for the Quasi-Static Analysis of Microstrip Line -- 5.7 Summary -- References -- Appendix 5A: Evaluation of the Integral in (5.120) -- Problems -- CHAPTER 6: Introduction to Computational Methods -- 6.1 Elements of Computational Methods -- 6.2 Basis Functions -- 6.2.1 Subdomain Basis Functions -- 6.2.2 Entire Domain Basis Functions -- 6.3 Convergence and Discretization Error -- 6.3.1 Convergence Test -- 6.3.2 Order of Convergence -- 6.3.3 Disctretization Error and Extrapolation -- 6.3.4 Discretization of Operators -- 6.3.5 Discretization Error in FDM, FDTD, and FEM -- 6.3.6 Vector and Matrix Norms -- 6.4 Stability of Numerical Solutions -- 6.4.1 Stability of FDTD Solution -- 6.4.2 Stability of Matrix Solution -- 6.5 Accuracy of Numerical Solutions -- 6.5.1 Modeling Errors -- 6.5.2 Truncation Error -- 6.5.3 Round-Off Error -- 6.5.4 Validation -- 6.6 Spurious Solutions -- 6.7 Formulations for the Computational Methods -- 6.8 Summary -- References -- Problems -- CHAPTER 7: Method of Finite Differences -- 7.1 Finite Difference Approximations -- 7.1.1 Difference Form of the First Derivative -- 7.1.2 Difference Form of the Second Derivative -- 7.1.3 Difference Form of Laplace and Poisson Equations -- 7.2 Treatment of Interface and Boundary Conditions -- 7.2.1 Nodes on the Interface -- 7.2.2 Dielectric Inhomogeneity in One Quadrant About a Node -- 7.2.3 Neumann Boundary Condition and the Nodes on the Edge -- 7.2.4 Node at a Corner -- 7.2.5 Node at an Edge with Dielectric Inhomogeneity About the Node.

7.2.6 Treatment of Curved Boundaries -- 7.2.7 Finite Difference Analysis of an Inhomogeneously Filled Parallel PlateCapacitor -- 7.3 Finite Difference Analysis of Guiding Structures -- 7.3.1 Analysis of Enclosed Microstrip Line -- 7.3.2 Analysis of Geometries with Open Boundaries -- 7.3.3 Wave Propagation and Numerical Dispersion -- 7.3.4 Analysis of Ridge Waveguide -- 7.4 Summary -- References -- Problems -- CHAPTER 8: Finite-Difference Time-Domain Analysis -- 8.1 Pulse Propagation in a Transmission Line -- 8.2 FDTD Analysis in One Dimension -- 8.2.1 Spatial Step Dx and Numerical Dispersion -- 8.2.2 Time Step Dt and Stability of the Solution -- 8.2.3 Source or Excitation of the Grid -- 8.2.4 Absorbing Boundary Conditions for One-Dimensional Propagation -- 8.3 Applications of One-Dimensional FDTD Analysis -- 8.3.1 Reflection at an Interface -- 8.3.2 Determination of Propagation Constant -- 8.3.3 Design of Material Absorber -- 8.3.4 Exponential Time-Stepping Algorithm in the Lossy Region -- 8.3.5 Extraction of Frequency Domain Information from the Time Domain Data -- 8.3.6 Simulation of Lossy, Dispersive Materials -- 8.4 FDTD Analysis in Two Dimensions -- 8.4.1 Unit Cell in Two Dimensions -- 8.4.2 Numerical Dispersion in Two Dimensions -- 8.4.3 Time Step Dt for Two-Dimensional Propagation -- 8.4.4 Absorbing Boundary Conditions for Propagation in Two Dimensions -- 8.4.5 Perfectly Matched Layer ABC -- 8.5 FDTD Analysis in Three Dimensions -- 8.5.1 Yee Cell -- 8.5.2 Numerical Dispersion in Three Dimensions -- 8.5.3 Time Step Dt for Three-Dimensional Propagation -- 8.5.4 Absorbing Boundary Conditions and PML for Three Dimensions -- 8.6 Implementation of Boundary Conditions in FDTD -- 8.6.1 Perfect Electric and Magnetic Wall Boundary Conditions -- 8.6.2 Interface Conditions -- 8.7 Advances in FDTD -- 8.8 Summary -- References -- Problems.

CHAPTER 9: Variational Methods -- 9.1 Calculus of Variations -- 9.1.1 Stationarity -- 9.1.2 Extremum -- 9.1.3 Functional -- 9.1.4 Variation or Increment of a Function -- 9.1.5 Variation and Stationarity of Functionals -- 9.2 Stationary Functionals and Euler Equations -- 9.3 The Ritz Variational Method -- 9.4 Applications of Ritz Approach -- 9.4.1 Variational Solution of Laplace Equation -- 9.4.2 Cutoff Frequency for Waveguide Modes -- 9.4.3 Resonant Frequency for Cavity Modes -- 9.4.4 Variational Formulation in Spectral Domain for the Microstrip Line -- 9.5 Construction of Functionals from the PDEs -- 9.6 Method of Weighted Residuals -- 9.6.1 Galerkin's Method -- 9.6.2 Point Matching Method -- 9.7 Summary -- References -- Problems -- CHAPTER 10: Finite Element Method -- 10.1 Basic Steps in Finite Element Analysis -- 10.1.1 Segmentation or Meshing of the Geometry -- 10.1.2 Derivation of the Element Matrix -- 10.1.3 Assembly of Element Matrices -- 10.1.4 Solution of System Matrix -- 10.1.5 Postprocessing -- 10.2 FEM Analysis in One Dimension -- 10.2.1 Treatment of Boundary and Interface Conditions -- 10.2.2 Accuracy and Numerical Dispersion -- 10.3 FEM Analysis in Two Dimensions -- 10.3.1 Solution of Two-Dimensional Wave Equation -- 10.3.2 Element Matrix for Rectangular Elements -- 10.3.3 Element Matrix for Triangular Elements -- 10.3.4 Assembly of Elements and System Equations -- 10.3.5 Capacitance of a Parallel Plate Capacitor -- 10.3.6 Cutoff Frequency of Waveguide Modes -- 10.3.7 FEM Analysis of Open Boundary Problems -- 10.4 Mesh Generation and Node Location Table -- 10.5 Weighted Residual Formulation for FEM -- 10.6 Summary -- References -- Problems -- CHAPTER 11: Method of Moments -- 11.1 Introduction -- 11.1.1 MoM Procedure -- 11.1.2 Point Matching and Galerkin's Methods -- 11.1.3 Eigenvalue Analysis Using MoM.

11.2 Solution of Integral Equations Using MoM.