Analytical and Computational Methods in Electromagnetics

Name: Analytical and Computational Methods in Electromagnetics
Brand: Artech House
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Ramesh Garg(Author)

Artech House (Publisher)

1st Edition

Published on 1. January 2008

470 pages

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978-1-59693-386-6 (ISBN)

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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
11.2.1 Integral Equation
11.2.2 Static Charge Distribution on a Wire
11.2.3 Analysis of Strip Line
11.2.4 Analysis of Wire Dipole Antenna
11.2.5 Scattering from a Conducting Cylinder of Infinite Length
11.3 Fast Multipole Solution Methods for MoM
11.4 Comparison of FDM, FDTD, FEM, and MoM
11.5 Hybrid Computational Methods
11.6 Summary
References
Problems
APPENDIX A: Solution Methods for the Set ofSimultaneous Equations
A.1 Processor Time Considerations [1]
A.2 Matrix Solution Techniques
A.2.1 Gauss Elimination [3, 4]
A.2.2 L-U Factorization [3, 4]
A.3 Sparse Matrix Techniques
A.3.1 Reordering of Equations
A.3.2 Preconditioned Conjugate Gradient Method
References
APPENDIX B: Evaluation of Singular Integrals
References
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Analytical and Computational Methods in Electromagnetics

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