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Electromagnetics Volume 1

Instrumentation, Electronics & Control Sensing Devices

Electromagnetics, Volume 1
Steven W. Ellingson
Virginia Tech
240 Pages

Open: Electromagnetics, Volume 1
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Preface:

Goals for this book. This book is intended to serve as a primary textbook for a one-semester introductory course in undergraduate engineering electromagnetics, including the following topics: electric and magnetic fields; electromagnetic properties of materials; electromagnetic waves; and devices that operate according to associated electromagnetic principles including resistors, capacitors, inductors, transformers, generators, and transmission lines.

This book employs the “transmission lines first” approach, in which transmission lines are introduced using a lumped-element equivalent circuit model for a differential length of transmission line, leading to one-dimensional wave equations for voltage and current.1 This is sufficient to address transmission line concepts, including characteristic impedance, input impedance of terminated transmission lines, and impedance matching techniques. Attention then turns to electrostatics, magnetostatics, time-varying fields, and waves, in that order.

TOC

Preface xii
1 Preliminary Concepts 1
1.1 What is Electromagnetics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Fundamentals of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Guided and Unguided Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Electric and Magnetic Fields 17
2.1 What is a Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Electric Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Electric Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Magnetic Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Magnetic Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Electromagnetic Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Transmission Lines 30
3.1 Introduction to Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Types of Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Transmission Lines as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Lumped-Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Telegrapher’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Wave Equation for a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.8 Wave Propagation on a TEM Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.9 Lossless and Low-Loss Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.10 Coaxial Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.11 Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.12 Voltage Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.13 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.14 Standing Wave Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Input Impedance of a Terminated Lossless Transmission Line . . . . . . . . . . . . . . . . . . . 52
3.16 Input Impedance for Open- and Short-Circuit Terminations . . . . . . . . . . . . . . . . . . . . 54
3.17 Applications of Open- and Short-Circuited Transmission Line Stubs . . . . . . . . . . . . . . . 55
3.18 Measurement of Transmission Line Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 56
3.19 Quarter-Wavelength Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.20 Power Flow on Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.21 Impedance Matching: General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.22 Single-Reactance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.23 Single-Stub Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Vector Analysis 70
4.1 Vector Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.8 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10 The Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Electrostatics 93
5.1 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Electric Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Electric Field Due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . . . . . . 96
5.5 Gauss’ Law: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Electric Field Due to an Infinite Line Charge using Gauss’ Law . . . . . . . . . . . . . . . . . . 101
5.7 Gauss’ Law: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 Force, Energy, and Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.9 Independence of Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 Kirchoff’s Voltage Law for Electrostatics: Integral Form . . . . . . . . . . . . . . . . . . . . . 108
5.11 Kirchoff’s Voltage Law for Electrostatics: Differential Form . . . . . . . . . . . . . . . . . . . 109
5.12 Electric Potential Field Due to Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.13 Electric Potential Field due to a Continuous Distribution of Charge . . . . . . . . . . . . . . . . 112
5.14 Electric Field as the Gradient of Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.15 Poisson’s and Laplace’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.16 Potential Field Within a Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.17 Boundary Conditions on the Electric Field Intensity (E) . . . . . . . . . . . . . . . . . . . . . . 118
5.18 Boundary Conditions on the Electric Flux Density (D) . . . . . . . . . . . . . . . . . . . . . . 120
5.19 Charge and Electric Field for a Perfectly Conducting Region . . . . . . . . . . . . . . . . . . . 122
5.20 Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.21 Dielectric Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.22 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.23 The Thin Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.24 Capacitance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.25 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Steady Current and Conductivity 134
6.1 Convection and Conduction Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Current Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6 Power Dissipation in Conducting Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7 Magnetostatics 146
7.1 Comparison of Electrostatics and Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 Gauss’ Law for Magnetic Fields: Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.3 Gauss’ Law for Magnetism: Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4 Ampere’s Circuital Law (Magnetostatics): Integral Form . . . . . . . . . . . . . . . . . . . . . 149
7.5 Magnetic Field of an Infinitely-Long Straight Current-Bearing Wire . . . . . . . . . . . . . . . 150
7.6 Magnetic Field Inside a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.7 Magnetic Field of a Toroidal Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.8 Magnetic Field of an Infinite Current Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.9 Ampere’s Law (Magnetostatics): Differential Form . . . . . . . . . . . . . . . . . . . . . . . . 159
7.10 Boundary Conditions on the Magnetic Flux Density (B) . . . . . . . . . . . . . . . . . . . . . . 160
7.11 Boundary Conditions on the Magnetic Field Intensity (H) . . . . . . . . . . . . . . . . . . . . . 161
7.12 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.13 Inductance of a Straight Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.14 Inductance of a Coaxial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.15 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.16 Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8 Time-Varying Fields 174
8.1 Comparison of Static and Time-Varying Electromagnetics . . . . . . . . . . . . . . . . . . . . 174
8.2 Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.4 Induction in a Motionless Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5 Transformers: Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.6 Transformers as Two-Port Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.7 The Electric Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.8 The Maxwell-Faraday Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.9 Displacement Current and Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9 Plane Waves in Lossless Media 194
9.1 Maxwell’s Equations in Differential Phasor Form . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.2 Wave Equations for Source-Free and Lossless Regions . . . . . . . . . . . . . . . . . . . . . . 196
9.3 Types of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.4 Uniform Plane Waves: Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.5 Uniform Plane Waves: Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.6 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.7 Wave Power in a Lossless Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A Constitutive Parameters of Some Common Materials 213
A.1 Permittivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2 Permeability of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B Mathematical Formulas 217
B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.3 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
C Physical Constants 220
Index 221

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