Demystifying Maxwell’s Equations For Design, Verification and Quality Control

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Demystifying Maxwell's Equations: For Design, Veriﬁcation and Quality Control

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“Demystifying Maxwell’s Equations from Shiv Prasad Tripathy provides a holistic treatment to a complex subject, maintaining a narrative that sustains reader interest while providing requisite scholastic and applied knowledge. The book will be very useful to practicing engineers. Finally, one can pick a book on electromagnetics without getting intimidated.”

Vikram Kumar, Honorary Professor, Indian Institute of Technology Delhi & ex-Director, CSIR-National Physical Laboratory (India)

“Demystifying Maxwell’s Equations is a lucid account of electromagnetics that maintains a fine balance between theory and practice. It is the third book from Shiv Prasad Tripathy on RF/

Microwave technologies that is bound to ignite reader’s passion for the esoteric subject.”

V.D. Vankar, Adjunct Professor Netaji Subhas University of Technology New Delhi & ex-Professor Indian Institute of Technology Delhi

“Shiv Prasad Tripathy has written Demystifying Maxwell’s Equations for those who want to pursue physics knowledge for engineering applications. Written with the same fervor as his earlier book RF & μWave Measurements, it is yet another timeless desktop reference from the author.”

Vishal Gupta, Segment Manager and Solutions Architect, Keysight

Technologies

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For Design, Verification and Quality Control

Shiv Prasad Tripathy

Demystifying Maxwell’s Equations

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All rights reserved. No part of this book may be reproduced in any form or by an electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the copyright owner, except by a reviewer who may quote brief passages in a review.

ISBN: 9781656344595

Disclaimer: The author has made every effort to ensure the information in this book is correct. The author does not assume and hereby disclaims any liability to any party for any loss, damage, or disruption caused by errors or omissions, whether such errors or omissions result from negligence, accident, or any other cause.

Trademark Acknowledgement: All brand names and trademarks are acknowledged.

Cover Image: “Maxwell in Action”

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Foreword ... xv

Preface ... xvii

Chapter 1 Overview ... 1

Introduction ... 1

Background to Electromagnetics ... 1

Design Frameworks ... 1

Circuit Theory ...2

Field Theory ...3

Electromagnetic Applications ... 4

Electromagnetic Parameters ... 5

Mathematical Background ... 6

Fields and Coordinate Systems ... 6

Scalar & Vector Fields ...6

Coordinate Systems ...7

Vector Calculus Identities ... 8

The Del Operator ...9

Gradient ...9

Divergence ...9

Curl ... 10

Laplacian... 11

Vector Cross Product ... 12

Other Vector Identities ... 12

Theorems & Functions ... 12

Gradient Theorem ... 13

Stokes' Theorem... 13

Divergence Theorem ... 13

Helmholtz Decomposition Theorem ... 13

Green’s Theorem ... 14

Green’s Function ... 14

Bessel’s Function ... 14

Legendre’s Function ... 14

Helmholtz Equation ... 15

Fourier Transform ... 15

Boundary Conditions ... 15

Organization of the Book ... 16

About the Chapters ... 16

Guide to Symbols ... 18

Guide to Notation ... 19

Summary ... 19

Chapter 2 Maxwell’s Equations ... 21

Introduction ... 21

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The Maxwell Equations ... 21

Differential Form ... 22

Integral Form ... 23

Time Harmonic Form ... 26

Takeaways from Maxwell Equations ... 27

The Wave Equations – Field Form ... 28

The Field - Source Relationship ... 28

The Wave Equations – Potential Form ... 29

The Potential – Source Relationship ... 33

Static Case ... 33

Dynamic Case ... 35

Retarded Potentials ... 38

Applying Potential Functions ... 38

Practical Implication ... 39

Summary ... 39

Chapter 3 Static Fields ... 41

Introduction ... 41

Maxwell at Rest ... 41

Electrostatics ... 42

Field Properties ... 43

Potential Properties ... 43

Point Charge ... 44

Line Charge ... 44

Sheet of Charge ... 47

Short Dipole ... 50

Fundamental Quantities in Electrostatics ... 53

Magnetostatics... 54

Potential Properties ... 57

Current Element ... 57

Long Wire ... 58

Current Sheet ... 60

Small loop... 62

Fundamental Quantities in Magnetostatics ... 66

Summary ... 67

Chapter 4 Quasistatic Fields ... 69

Introduction ... 69

Quasistatic Laws ... 70

EQS Model ... 70

Mathematical Treatment ... 71

MQS Model ... 73

Mathematical Treatment ... 73

Which Model to use – EQS or MQS? ... 75

Summary ... 76

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Chapter 5 Properties of Materials ... 79

Introduction ... 79

Material Properties ... 79

Constitutive Relations ... 80

Conductivity ... 80

Permittivity ... 81

Polarization ... 81

Electric Displacement ... 82

Relation of D and E ... 84

Permeability ... 84

Magnetization ... 85

Magnetic Field H ... 86

Relation of H and B... 88

Maxwell’s Equation in Media ... 88

Scenario: Free-Charge, Free-Current ... 88

Scenario: No Free-Charge, No Free-Current ... 88

Scenario: No Free-Charge, Only Free-current ... 89

Conduction, Polarization & Magnetization Currents ... 89

Summary ... 91

Chapter 6 Unrestricted Dynamic Fields ... 93

Introduction ... 93

Propagation in Free-Space... 93

The Free-Space Equations ... 94

Field and Propagation... 94

Propagation in Material Media ... 96

Conductive Media ... 99

Low-loss Dielectric Media ... 99

Lossless Media ... 100

Penetration Depth ... 100

Radiation from Sources ... 101

Short Dipole ... 102

Field Characteristics ... 106

Small-Loop ... 108

Field Characteristics ... 114

Field – Distance Relationship ... 116

Unbounded Wave Properties ... 116

Propagation Constant ... 116

Phase Velocity ... 117

Group Velocity ... 117

Dispersion ... 117

Wave Impedance ... 117

Bandwidth ... 120

Wave Polarization ... 120

Energy Flow ... 120

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Summary ... 123

Chapter 7 Boundary Conditions ... 125

Introduction ... 125

Boundary Conditions ... 125

Electric Boundary ... 126

Scenario - Surface Charge, Surface Current ... 126

Scenario – No Surface Charge, No Surface Current ... 130

Scenario – PEC ... 131

Magnetic Boundary ... 132

Scenario - Surface Magnetic Charge, Surface Magnetic Current ... 132

Scenario – PMC ... 133

Impedance Boundary Conditions ... 134

Field Characteristics at Boundary ... 135

Reflected Power and Angle of Incidence ... 138

Brewster Angle ... 139

Transmitted Power ... 139

Critical Angle ... 140

Total Internal Reflection ... 140

Normal Incidence ... 141

Equivalence Principle ... 141

Uniqueness Theorem ... 142

Volume Equivalence ... 144

Surface Equivalence ... 145

Theorem of equivalent surface currents ... 146

Love’s Equivalence Principle ... 148

Schelkunoff’s Equivalence Principle ... 150

PEC Case ... 150

PMC Case ... 150

Source Reconstruction and Solution Steps ... 151

Reciprocity Theorem ... 152

Solving Boundary Value Problems ... 154

Electric/ Magnetic Wall ... 155

Applying Equivalence Principle... 155

Summary ... 156

Chapter 8 Confined Dynamic Fields - I ... 157

Transmission Line ... 157

General Solution ... 158

Coaxial Cables ... 159

Parallel Wires ... 161

Parallel Plate Lines ... 163

TE Scenario ... 165

TM Scenario ... 167

Transmission Line Properties ... 169

Multimode Propagation ... 169

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Characteristic Impedance ... 170

Bandwidth ... 170

Waveguides ... 171

Superposition of Plane Waves ... 171

General Solution ... 173

Solution Types ... 179

Rectangular Waveguide ... 179

TE Case ... 179

TM Case ... 184

Circular Waveguide ... 188

TE Case ... 189

TM Case ... 195

Waveguide Properties ... 197

Multimode Propagation ... 197

Waveguide Impedance ... 198

Bandwidth ... 199

Transmission Line vs Waveguide ... 199

Waveguide Selection ... 199

Rectangular vs Circular Waveguides ... 200

Summary ... 200

Chapter 9 Confined Dynamic Fields - II ... 203

Cavity Resonators ... 203

Rectangular Cavity... 203

TE Mode ... 204

TM Mode ... 206

Circular Cavity ... 208

TE Mode ... 209

TM Mode ... 209

Spherical Cavity ... 210

TE Mode ... 215

TM Mode ... 215

Resonator Properties ... 216

Degeneracy ... 216

Operating Frequency ... 216

Bandwidth ... 217

Q-Factor ... 217

Circuit Resonator vs Cavity Resonator ... 217

Rectangular, Circular and Spherical Resonator ... 218

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Summary ... 218

Chapter 10 Maxwell in Action ... 219

At Length ... 219

Radiation Resistance ... 219

Short Dipole ... 220

Small Loop ... 222

Practical Antennas ... 223

Shielding Effectiveness ... 223

Method of Images ... 227

Point Charge Potential ... 227

Electric Dipoles ... 230

Characteristic Impedance ... 230

Waveguide Operating Bandwidth ... 235

Rectangular Waveguide ... 235

Circular Waveguide ... 236

Attenuation in Waveguide ... 237

Wall Attenuation... 237

Dielectric Attenuation ... 239

Cavity Resonator Quality Factor ... 240

Unloaded Q ... 242

Loaded Q ... 245

Dielectric Waveguide ... 246

TE Mode ... 248

TM Mode ... 251

Guided vs Radiation Mode ... 254

Cut-off Frequency... 254

Antenna Reciprocity ... 254

In Brief ... 256

Microstrip Line ... 256

Dielectric Resonator ... 258

Isolator Principle ... 259

Waveguide Transition ... 260

Open-ended Waveguide Radiation ... 261

Conduction/ Displacement Current Type Antennas ... 262

Summary ... 263

Chapter 11 Electromagnetic Measurements ... 265

The Design Process... 265

Numerical Methods ... 266

Numerical Techniques ... 267

Applying Numerical Methods ... 268

Electromagnetic Parameters ... 268

Wave Properties ... 269

Material Properties ... 271

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Circuit Properties ... 271

Device Properties ... 272

Measuring Instruments ... 273

EMF Meter ... 273

Bolometer ... 274

Wavemeter ... 275

Power Meter ... 275

Spectrum Analyzer ... 275

EMI Receiver ... 275

Network Analyzer ... 276

Oscilloscopes/ TDR ... 276

RF Impedance Analyzer ... 276

Signal Generator ... 277

GTEM cell ... 277

Antenna Positioner ... 277

Anechoic Chamber ... 278

Usage Scenarios ... 278

EMF Measurement ... 278

Specific Absorption Rate ... 279

Radiation Pattern ... 279

Device Modeling ... 280

Device Verification ... 281

Electromagnetic Compatibility ... 282

Material Characterization ... 283

Radome Testing ... 285

Radar Cross Section ... 286

Measurement Challenges ... 286

Summary ... 287

Appendix – The Wave Equation ... 289

Acronyms & Keywords ... 297

Index ... 301

About the Author ... 307

Feedback ... 309

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Foreword

Shiv Prasad Tripathy’s book is a monumental achievement of conciseness as he flies us through the amazing universe of Maxwell's electromagnetic theory. I do recommend the interested reader should come to this journey with a full quiver of vector calculus familiarity, those who do so will find many wonderful friends presented in the pages within briefly and concisely, ready to be assimilated, or re-assimilated into their working knowledge, yielding solutions for their most difficult electromagnetic problems.

Demystifying Maxwell’s Equations is especially useful for the experienced EM worker who might have forgotten the details of one or another of the many nooks and crannies in Maxwell's theory. With this book at hand, your questions will be answered in short order.

December 2021 James C. Rautio

Founder & Board Chair, Sonnet Software Syracuse

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Preface

Demystifying Maxwell’s Equations is a continuation of my works to simplify and exemplify RF/ Microwave topics for the benefit of engineers and scientists engaged in the field of design, verification and quality control of electromagnetic products. It is for physics and engineering students who want to grasp the subject beyond the traditional theoretical construct. It is aimed to motivate the readers to develop interest beyond physics into engineering applications. It will be of interest to anyone who wants to decode electromagnetics. Comfort with higher mathematics is definitely an advantage.

Demystifying Maxwell’s Equations covers concepts and principles with direct relevance to engineering applications, without losing sight of associated mathematical complexities. It includes design scenarios that illustrate and expound electromagnetics under realistic conditions and bring out the significance of Maxwell’s Equations from engineering perspective.

All mathematical sciences must be expressed and analyzed in numbers, and a measureable assessment is critical to product engineering. Understanding of electromagnetic parameters and their measurement techniques is integral to applied-electromagnetics and the book covers these along with instrumentation knowledge.

Demystifying Maxwell’s Equations presents electromagnetics with a holistic overview as it navigates through the electromagnetic space comprising spatial, temporal and material effects. The application-oriented approach ensures discussions are not limited to abstract models and related mathematics, but provide engineering linkage that assist solution design and execute truthful measurements. The book offers a balanced coverage on the breadth and depth of core topics while maintaining a portable size.

Demystifying Maxwell’s Equations maintains focus on Maxwell’s Equations as topics of increased complexity get introduced across the chapters. The arrangement of themes ensures seamless flow of knowledge as the mystery of the equations gets unraveled in space, time and medium. Maxwell’s Equations are analyzed in static, quasistatic and dynamic scenarios and in unbound and bounded conditions. Material electromagnetic properties, electromagnetic sources, near-field and far-field radiation are investigated. Field behavior is examined in guided and confined structures.

Demystifying Maxwell’s Equations illustrates diverse scenarios that illuminate the reader on the practical side of electromagnetics. Exhaustive list of electromagnetic measurement parameters is presented and instrumentation systems are introduced. The appendix acts as a refresher on wave equation. Additional readings are provided for the inquisitive reader who may want to explore further.

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Demystifying Maxwell’s Equations completes my book trilogy on RF/ Microwave. I am confident the knowledge source shall encourage the reader to transcend the electromagnetics theory towards engineering applications.

December 2021 Shiv Prasad Tripathy

New Delhi

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About the Author

Shiv Prasad Tripathy (SPT) is engaged in professional services in the field of electronic test & measurement, including design and development of products and solutions on new technologies. His books simplify and present core topics with a holistic approach and are targeted at people with passion for RF/ Microwave technologies.

He has earlier authored two books “RF & μWave Measurements: For Design, Verification and Quality Control” and “The 100-Page Book on Antenna Design Parameters” and this book completes his trilogy on RF/ Microwave topics.

SPT has extensive experience in architecting, design, engineering, management of hardware/ software based communication systems, network management systems, cloud platform based solutions and multi-vendor system integration projects. He has spent many years in R&D Labs on analog circuit design, digital design, high frequency design, embedded programming and design of real-time state-machine based systems. His industrial work experience includes companies like HCL Ltd., Agilent Technologies (now Keysight), Hewlett Packard and others.

SPT is a bachelor of technology in electrical engineering from Indian Institute of Technology Delhi and a master of business administration (International Business) from Indian Institute of Foreign Trade, New Delhi. He is a Senior Member of IEEE.

His hobbies include amateur radio, drawing oil paintings and jazz drumming.

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