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NOTES ON COMPUTATIONAL ELECTROMAGNETICS Qing Huo Liu Duke University Spring 2008
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Contents 1 Basics of Electromagnetics Theory 1 1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Maxwell’s Equations in Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Maxwell’s Equations in Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Special Case I: Source-Free Media with a Finite Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Special Case II: Medium 1 is a Perfect Electric Conductor (PEC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Special Case III: Medium 1 is a Perfect Magnetic Conductor (PMC) . . . . . . . . . . 5 1.4 Power and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Time-Harmonic Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.1 Maxwell’s Equations for Phasor EM Fields . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.2 Power and Energy for Phasor EM Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5.3 Complex Permittivity and Complex Permeability . . . . . . . . . . . . . . . . . . . . . 8 1.6 Fields in a Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.2 Magnetic Vector Potential A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.3 Electric Vector Potential F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6.4 A Point Electric Dipole Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.5 Arbitrary Volume Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Volume Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Volume Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.9 Surface Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9.1 Love’s Equivalent—Love’s Equivalence Principle . . . . . . . . . . . . . . . . . . . . . 15 1.9.2 Equivalent PEC Scatterer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.9.3 Equivalent PMC Scatterer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.10 Surface Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.10.1 Surface Integral Equations for a Dielectric Object . . . . . . . . . . . . . . . . . . . . 16 1.10.2 Surface Integral Equations for a PEC Object . . . . . . . . . . . . . . . . . . . . . . . 17 1.10.3 Surface Integral Equations for a PEC Shell . . . . . . . . . . . . . . . . . . . . . . . . 17 1.11 Alternative Forms of Surface Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.11.1 Combined Field Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.11.2 Surface Integral Equations with Corners . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.12 Some Useful Formulas in Computational Electromagnetics . . . . . . . . . . . . . . . . . . . . 21 i
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ii CONTENTS 2 Computational Electromagnetics in 1D 25 2.1 Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 Finite-Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 The Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.5 Accuracy and Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.6 Sources and Their Time Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Method of Moments for Integral Euqations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 Simplifying 3-D VIE to 1-D VIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.2 1-D Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 Basis and testing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 1-D PDE in frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.2 Perfect Electric-Conductor (PEC) Boundaries . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.3 Perfect Magnetic-Conductor (PMC) Boundaries . . . . . . . . . . . . . . . . . . . . . 36 2.3.4 Radiation Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.5 Galerkin’s method with triangular functions . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.6 Elemental Matrices and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Computer Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 2-D Surface Integral Equation Methods 45 3.1 Weak-Form Surface Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 Objects with the Impedance Boundary Condition . . . . . . . . . . . . . . . . . . . . . 47 3.1.2 Perfect Electric Conductor Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 The General 2-D Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 The SIEs for TM Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 The SIEs for TE Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 2-D Scatterers with an Impedance Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 The SIEs for TM Incidence on an Impedance Surface . . . . . . . . . . . . . . . . . . . 58 3.3.2 The SIEs for TE Incidence on an Impedance Surface . . . . . . . . . . . . . . . . . . . 60 3.4 2-D PEC Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.1 TM Waves Scattered by 2-D PEC Scatterers . . . . . . . . . . . . . . . . . . . . . . . 63 3.4.2 TE Waves Scattered by 2-D PEC Scatterers . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Scalar Basis Functions and Application in 2D 69 4.1 1-D Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 1-D Linear Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.2 Higher-Order Lagrangian Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1.3
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