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Electromagnetics Volume 2

Instrumentation, Electronics & Control Sensing Devices

Electromagnetics, Volume 2
Steven W. Ellingson
Virginia Tech
230 Pages

Open: Electromagnetics, Volume 2
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Preface:

This book is intended to serve as a primary textbook for the second semester of a two-semester course in undergraduate engineering electromagnetics. The presumed textbook for the first semester is Electromagnetics Vol. 1,1 which addresses 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. The book you are now reading – Electromagnetics Vol. 2 – addresses the following topics:

TOC

1 Preliminary Concepts 1
1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Electromagnetic Field Theory: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Magnetostatics Redux 11
2.1 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Magnetic Force on a Current-Carrying Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Torque Induced by a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 The Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Force, Energy, and Potential Difference in a Magnetic Field . . . . . . . . . . . . . . . . . . . 20

3 Wave Propagation in General Media 25
3.1 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Wave Equations for Lossy Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Loss Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Plane Waves in Lossy Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Wave Power in a Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Decibel Scale for Power Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.9 Attenuation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.10 Poor Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.11 Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.12 Skin Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Current Flow in Imperfect Conductors 48
4.1 AC Current Flow in a Good Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Impedance of a Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Wave Reflection and Transmission 56
5.1 Plane Waves at Normal Incidence on a Planar Boundary . . . . . . . . . . . . . . . . . . . . . 56
5.2 Plane Waves at Normal Incidence on a Material Slab . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Total Transmission Through a Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Propagation of a Uniform Plane Wave in an Arbitrary Direction . . . . . . . . . . . . . . . . . 67
5.5 Decomposition of a Wave into TE and TM Components . . . . . . . . . . . . . . . . . . . . . 70
5.6 Plane Waves at Oblique Incidence on a Planar Boundary: TE Case . . . . . . . . . . . . . . . 72
5.7 Plane Waves at Oblique Incidence on a Planar Boundary: TM Case . . . . . . . . . . . . . . . 76
5.8 Angles of Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.9 TE Reflection in Non-magnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.10 TM Reflection in Non-magnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.11 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.12 Evanescent Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Waveguides 95
6.1 Phase and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Parallel Plate Waveguide: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Parallel Plate Waveguide: TE Case, Electric Field . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Parallel Plate Waveguide: TE Case, Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Parallel Plate Waveguide: TM Case, Electric Field . . . . . . . . . . . . . . . . . . . . . . . . 104
6.6 Parallel Plate Waveguide: The TM0 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.7 General Relationships for Unidirectional Waves . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.8 Rectangular Waveguide: TM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.9 Rectangular Waveguide: TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.10 Rectangular Waveguide: Propagation Characteristics . . . . . . . . . . . . . . . . . . . . . . 117

7 Transmission Lines Redux 121
7.1 Parallel Wire Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Microstrip Line Redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Attenuation in Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Power Handling Capability of Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.5 Why 50 Ohms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Optical Fiber 138
8.1 Optical Fiber: Method of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Acceptance Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Dispersion in Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9 Radiation 145
9.1 Radiation from a Current Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Solution of the Wave Equation for Magnetic Vector Potential . . . . . . . . . . . . . . . . . . 150
9.4 Radiation from a Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.5 Radiation from an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.6 Far-Field Radiation from a Thin Straight Filament of Current . . . . . . . . . . . . . . . . . . 159
9.7 Far-Field Radiation from a Half-Wave Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.8 Radiation from Surface and Volume Distributions of Current . . . . . . . . . . . . . . . . . . 162

10 Antennas 166
10.1 How Antennas Radiate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.2 Power Radiated by an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.3 Power Dissipated by an Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . 169
10.4 Reactance of the Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.5 Equivalent Circuit Model for Transmission; Radiation Efficiency . . . . . . . . . . . . . . . . 173
10.6 Impedance of the Electrically-Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.7 Directivity and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.8 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.9 Equivalent Circuit Model for Reception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.10 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.11 Potential Induced in a Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.12 Equivalent Circuit Model for Reception, Redux . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.13 Effective Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
10.14 Friis Transmission Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

A Constitutive Parameters of Some Common Materials 205
A.1 Permittivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A.2 Permeability of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A.3 Conductivity of Some Common Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B Mathematical Formulas 209
B.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.3 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C Physical Constants 212
Index 213

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