# MATLAB-Based Electromagnetics, CourseSmart eTextbook

Published Date: Jun 19, 2013

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## Description

Can be used to either complement available electromagnetics text, or as an independent resource. Designed primarily for undergraduate electromagnetics, but can also be used in follow-up courses on antennas, propagation, microwaves, advanced electromagnetic theory, computational electromagnetics, electrical machines, signal integrity, etc.

MATLAB-Based Electromagentics provides engineering and physics students and other users with an operational knowledge and firm grasp of electromagnetic fundamentals aimed toward practical engineering applications, by teaching them “hands on” electromagnetics through a unique and comprehensive collection of MATLAB computer exercises and projects. Essentially, the book unifies two themes: it presents and explains electromagnetics using MATLAB on one side, and develops and discusses MATLAB for electromagnetics on the other.

MATLAB codes described (and listed) in TUTORIALS or proposed in other exercises provide prolonged benefits of learning. By running codes; generating results, figures, and diagrams; playing movies and animations; and solving a large variety of problems in MATLAB, in class, with peers in study groups, or individually, students gain a deep understanding of electromagnetics.

1 Electrostatic Field in Free Space 1
1.1 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electric Field Intensity Vector Due to Given Charge Distributions . . . . . . . . . 9
1.3 Electric Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Differential Relationship Between the Field and Potential in Electrostatics, Gradient 26
1.5 Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6 Gauss’ Law in Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7 Differential Form of Gauss’ Law, Divergence . . . . . . . . . . . . . . . . . . . . . . 31
1.8 Method of Moments for Numerical Analysis of Charged Metallic Bodies . . . . . . 33
2 Electrostatic Field in Dielectrics 41
2.1 Characterization of Dielectric Materials . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Dielectric—Dielectric Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Poisson’s and Laplace’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 Finite-Difference Method for Numerical Solution of Laplace’s Equation . . . . . . . 51
2.5 Evaluation of Capacitances of Capacitors and Transmission Lines . . . . . . . . . . 59
2.6 Capacitors with Inhomogeneous Dielectrics . . . . . . . . . . . . . . . . . . . . . . 69
2.7 Dielectric Breakdown in Electrostatic Systems . . . . . . . . . . . . . . . . . . . . . 70
3 Steady Electric Currents 73
3.1 Continuity Equation, Conductivity, and Ohm’s Law in Local Form . . . . . . . . . 73
3.2 Boundary Conditions for Steady Currents . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 Resistance and Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Magnetostatic Field in Free Space 86
4.1 Magnetic Force and Magnetic Flux Density Vector . . . . . . . . . . . . . . . . . . 86
4.2 Magnetic Field Computation Using Biot—Savart Law . . . . . . . . . . . . . . . . . 92
4.3 Ampere’s Law in Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Differential Form of Ampere’s Law, Curl . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6 Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Magnetostatic Field in Material Media 106
5.1 Permeability of Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Boundary Conditions for the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Magnetic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
vi Contents, Preface, and m Files on Instructor Resources
6 Time-Varying Electromagnetic Field 118
6.1 Faraday’s Law of Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Self-Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4 Displacement Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Maxwell’s Equations for the Time-Varying Electromagnetic Field . . . . . . . . . . 130
6.6 Boundary Conditions for the Time-Varying Electromagnetic Field . . . . . . . . . . 132
6.7 Time-Harmonic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.8 Complex Representatives of Time-Harmonic Field and Circuit Quantities . . . . . 137
6.9 Instantaneous and Complex Poynting Vector . . . . . . . . . . . . . . . . . . . . . 144
7 Uniform Plane Electromagnetic Waves 146
7.1 Time-Harmonic Uniform Plane Waves and Complex-Domain Analysis . . . . . . . 146
7.2 Arbitrarily Directed Uniform Plane Waves . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Theory of Time-Harmonic Waves in Lossy Media . . . . . . . . . . . . . . . . . . . 153
7.4 Wave Propagation in Good Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.5 Wave Propagation in Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . 158
7.6 Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.7 Wave Propagation in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.8 Polarization of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 163
8 Reflection and Transmission of Plane Waves 173
8.1 Normal Incidence on a Perfectly Conducting Plane . . . . . . . . . . . . . . . . . . 173
8.2 Normal Incidence on a Penetrable Planar Interface . . . . . . . . . . . . . . . . . . 180
8.3 Oblique Incidence on a Perfect Conductor . . . . . . . . . . . . . . . . . . . . . . . 189
8.4 Oblique Incidence on a Dielectric Boundary . . . . . . . . . . . . . . . . . . . . . . 192
8.5 Wave Propagation in Multilayer Media . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 Field Analysis of Transmission Lines 204
9.1 Field Analysis of Lossless Transmission Lines . . . . . . . . . . . . . . . . . . . . . 204
9.2 Transmission Lines with Small Losses . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.3 Evaluation of Primary and Secondary Circuit Parameters of Transmission Lines . . 212
9.4 Transmission Lines with Inhomogeneous Dielectrics . . . . . . . . . . . . . . . . . . 213
9.5 Multilayer Printed Circuit Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10 Circuit Analysis of Transmission Lines 222
10.1 Telegrapher’s Equations and Their Solution . . . . . . . . . . . . . . . . . . . . . . 222
10.2 Reflection Coefficient for Transmission Lines . . . . . . . . . . . . . . . . . . . . . . 225
10.3 Transmission-Line Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.4 Complete Solution for Line Voltage and Current . . . . . . . . . . . . . . . . . . . 233
10.5 Short-Circuited, Open-Circuited, and Matched Transmission Lines . . . . . . . . . 235
10.6 Impedance-Matching Using Short- and Open-Circuited Stubs . . . . . . . . . . . . 237
10.7 The Smith Chart — Construction and Basic Properties . . . . . . . . . . . . . . . . 241
10.8 Circuit Analysis of Transmission Lines Using the Smith Chart . . . . . . . . . . . . 247
10.9 Transient Analysis of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 263
10.10 Step Response of Transmission Lines with Purely Resistive Terminations . . . . . 267
10.11 Analysis of Transmission Lines with Pulse Excitations . . . . . . . . . . . . . . . . 272
10.12 Bounce Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
10.13 Transient Response for Reactive Terminations . . . . . . . . . . . . . . . . . . . . 286
11 Waveguides and Cavity Resonators 291
11.1 Analysis of Rectangular Waveguides Based on Multiple Reflections of Plane Waves 291
11.2 Arbitrary TE and TM Modes in a Rectangular Waveguide . . . . . . . . . . . . . . 299
11.3 Wave Impedances of TE and TM Waves . . . . . . . . . . . . . . . . . . . . . . . . 306
11.4 Power Flow Along a Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
11.5 Waveguides With Small Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
11.6 Waveguide Dispersion and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . 312
11.7 Rectangular Cavity Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
11.8 Electromagnetic Energy Stored in a Cavity Resonator . . . . . . . . . . . . . . . . 316
11.9 Quality Factor of Rectangular Cavities with Small Losses . . . . . . . . . . . . . . 319
12 Antennas and Wireless Communication Systems 321
12.1 Electromagnetic Field due to a Hertzian Dipole . . . . . . . . . . . . . . . . . . . . 321
12.2 Far Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
12.3 Steps in Far Field Evaluation of an Arbitrary Antenna . . . . . . . . . . . . . . . . 326
12.4 Radiation and Ohmic Resistances of an Antenna . . . . . . . . . . . . . . . . . . . 328
12.5 Antenna Radiation Patterns, Directivity, and Gain . . . . . . . . . . . . . . . . . . 331
12.6 Wire Dipole Antennas of Arbitrary Lengths . . . . . . . . . . . . . . . . . . . . . . 337
12.7 Theory of Receiving Antennas. Wireless Links with Nonaligned Wire Antennas . . 344
12.8 Friis Transmission Formula for a Wireless Link . . . . . . . . . . . . . . . . . . . . 350
12.9 Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

Appendix 1: Quantities, Symbols, Units, and Constants A-1
Appendix 2: Mathematical Facts and Identities A-4
A2.1 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4
A2.2 Exponential, Logarithmic, and Hyperbolic Identities . . . . . . . . . . . . . . . . . A-4
A2.3 Solution of Quadratic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5
A2.4 Approximations for Small Quantities . . . . . . . . . . . . . . . . . . . . . . . . . A-5
A2.5 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5
A2.6 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5
A2.7 Vector Algebraic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5
A2.8 Vector Calculus Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6
A2.9 Gradient, Divergence, Curl, and Laplacian in Orthogonal Coordinate Systems . . A-6
A2.10 Vector Algebra and Calculus Index . . . . . . . . . . . . . . . . . . . . . . . . . . A-7
Appendix 3: List of MATLAB Exercises A-8
Bibliography A-22
Index A-25

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MATLAB-Based Electromagnetics, CourseSmart eTextbook
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