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What is an Open Textbook? Open textbooks are complete textbooks that have been funded, published, and licensed to be freely used, adapted, and distributed. As a particular type of Open Educational Resource (OER), this open textbook is intended to provide authoritative, accurate, and comprehensive subject content at no cost, to anyone including those who utilize technology to read and those who cannot afford traditional textbooks. This book is licensed with a Creative Commons Share-Alike 4.0 license (see p. iv.), which allows it to be adapted, remixed, and shared under the same license with attribution. Instructors and other readers may be interested in localizing, rearranging, or adapting content, or in transforming the content into other formats with the goal of better addressing student learning needs, and/or making use of various teaching methods. Open textbooks in a variety of disciplines are available via the Open Textbook Library: https://open.umn.edu/opentextbooks. Feedback Requested The editor and author of this book seek content-related suggestions from faculty, students, and others using the book. Methods for providing feedback are presented in the User Feedback Guide at: http://bit.ly/userfeedbackguide • Submit suggestions (anonymous) http://bit.ly/electromagnetics-suggestion • Annotate using Hypothes.is http://web.hypothes.is (instructions are in the feedback guide) • Send suggestions via email: publishing@vt.edu Additional Resources These following resources for Electromagnetics Vol 1 are available at: http://hdl.handle.net/10919/84164 • Downloadable PDF of the text • Source files (LaTeX tarball) • Errata for Volume 1 • Problem sets & solution manual • Information for book’s community portal and listserv If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form: http://bit-ly/vtpublishing-updates v	Electromagnetics_Vol1_Page_6_Chunk1301
Contents 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 . . . . . . . . . . . . . . . . . . . . . . .	Electromagnetics_Vol1_Page_7_Chunk1302
. . . . . . . 27 vi	Electromagnetics_Vol1_Page_7_Chunk1303
CONTENTS vii 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 . . . . . . . . .	Electromagnetics_Vol1_Page_8_Chunk1304
. . . . . . . . . . . . . . . 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	Electromagnetics_Vol1_Page_8_Chunk1305
viii CONTENTS 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 . . . . . . . . . . . . . . . . . . . . . . .	Electromagnetics_Vol1_Page_9_Chunk1306
. . . . . . . 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	Electromagnetics_Vol1_Page_9_Chunk1307
CONTENTS ix 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 . . . . . .	Electromagnetics_Vol1_Page_10_Chunk1308
. . . . . . . . . . . . . . . 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	Electromagnetics_Vol1_Page_10_Chunk1309
x CONTENTS 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 . . . . . . . . . . . . . . . . . . . . . . .	Electromagnetics_Vol1_Page_11_Chunk1310
. . . . . . . . . . . . . . . . . . . . 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	Electromagnetics_Vol1_Page_11_Chunk1311
CONTENTS xi 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	Electromagnetics_Vol1_Page_12_Chunk1312
Preface About This Book [m0146] 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. What’s new. This version of the book is the second public release of this book. The ﬁrst release, known as “Volume 1 (Beta),” was released in January 2018. Improvements from the beta version include the following: • Correction of errors identiﬁed in the beta version errata and many minor improvements. • Addition of an index. 1Are you an instructor who is not a fan of the “transmission lines ﬁrst” approach? Then see “What are those little numbers in square brackets?” later in this section. • Accessibility features: Figures now include “alt text” suitable for screen reading software. • Addition of a separate manual of examples and solutions (see the web site). • Addition of source ﬁles for the book (see the web site). Target audience. This book is intended for electrical engineering students in the third year of a bachelor of science degree program. It is assumed that readers are familiar with the fundamentals of electric circuits and linear systems, which are normally taught in the second year of the degree program. It is also assumed that readers have received training in basic engineering mathematics, including complex numbers, trigonometry, vectors, partial differential equations, and multivariate calculus. Review of the relevant principles is provided at various points in the text. In a few cases (e.g., phasors, vectors) this review consists of a separate stand-alone section. Notation, examples, and highlights. Section 1.7 summarizes the mathematical notation used in this book. Examples are set apart from the main text as follows: Example 0.1. This is an example. “Highlight boxes” are used to identify key ideas as follows: This is a key idea. What are those little numbers in square brackets? This book is a product of the Open Electromagnetics Project. This project provides a large number of sections (“modules”) which are Electromagnetics Vol 1. c⃝2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1	Electromagnetics_Vol1_Page_13_Chunk1313
xiii assembled (“remixed”) to create new and different versions of the book. The text “[m0146]” that you see at the beginning of this section uniquely identiﬁes the module within the larger set of modules provided by the project. This identiﬁcation is provided because different remixes of this book may exist, each consisting of a different subset and arrangement of these modules. Prospective authors can use this identiﬁcation as an aid in creating their own remixes. Why do some sections of this book seem to repeat material presented in previous sections? In some remixes of this book, authors might choose to eliminate or reorder modules. For this reason, the modules are written to “stand alone” as much as possible. As a result, there may be some redundancy between sections that would not be present in a traditional (non-remixable) textbook. While this may seem awkward to some at ﬁrst, there are clear beneﬁts: In particular, it never hurts to review relevant past material before tackling a new concept. And, since the electronic version of this book is being offered at no cost, there is not much gained by eliminating this useful redundancy. Why cite Wikipedia pages as additional reading? Many modules cite Wikipedia entries as sources of additional information. Wikipedia represents both the best and worst that the Internet has to offer. Most authors of traditional textbooks would agree that citing Wikipedia pages as primary sources is a bad idea, since quality is variable and content is subject to change over time. On the other hand, many Wikipedia pages are excellent, and serve as useful sources of relevant information that is not strictly within the scope of the curriculum. Furthermore, students beneﬁt from seeing the same material presented differently, in a broader context, and with additional references cited by Wikipedia pages. We trust instructors and students to realize the potential pitfalls of this type of resource and to be alert for problems. Acknowledgments. Here’s a list of talented and helpful people who contributed to this book: The staff of VT Publishing, University Libraries, Virginia Tech: Editor: Anita Walz Advisors: Peter Potter, Corinne Guimont Cover: Robert Browder, Anita Walz Other VT contributors: Assessment: Tiffany Shoop, Anita Walz Accessibility: Christa Miller Virginia Tech students: Alt text writer: Stephanie Edwards Figure designer: Michaela Goldammer Figure designer: Kruthika Kikkeri Figure designer: Youmin Qin Copyediting: Melissa Ashman, Kwantlen Polytechnic University External reviewers: Samir El-Ghazaly, University of Arkansas Stephen Gedney, University of Colorado Denver Randy Haupt, Colorado School of Mines Karl Warnick, Brigham Young University Also, thanks are due to the students of the Spring 2018 and Summer 2018 sections of ECE3105 at Virginia Tech who used the beta version of this book and provided useful feedback. Thanks also to Justin Yonker, instructor of the Summer 2018 section.	Electromagnetics_Vol1_Page_14_Chunk1314
xiv PREFACE About the Open Electromagnetics Project [m0148] The Open Electromagnetics Project was established at Virginia Tech in 2017 with the goal of creating no-cost openly-licensed textbooks for courses in undergraduate engineering electromagnetics. While a number of very ﬁne traditional textbooks are available on this topic, we feel that it has become unreasonable to insist that students pay hundreds of dollars per book when effective alternatives can be provided using modern media at little or no cost to the student. This project is equally motivated by the desire for the freedom to adopt, modify, and improve educational resources. This work is distributed under a Creative Commons BY SA license which allows – and we hope encourages – others to adopt, modify, improve, and expand the scope of our work. About the Author [m0153] Steven W. Ellingson (ellingson@vt.edu) is an Associate Professor at Virginia Tech in Blacksburg, Virginia in the United States. He received PhD and MS degrees in Electrical Engineering from the Ohio State University and a BS in Electrical & Computer Engineering from Clarkson University. He was employed by the US Army, Booz-Allen & Hamilton, Raytheon, and the Ohio State University ElectroScience Laboratory before joining the faculty of Virginia Tech, where he teaches courses in electromagnetics, radio frequency electronics, wireless communications, and signal processing. His research includes topics in wireless communications, radio science, and radio frequency instrumentation. Professor Ellingson serves as a consultant to industry and government and is the author of Radio Systems Engineering (Cambridge University Press, 2016).	Electromagnetics_Vol1_Page_15_Chunk1315
Chapter 1 Preliminary Concepts 1.1 What is Electromagnetics? [m0037] The topic of this book is applied engineering electromagnetics. This topic is often described as “the theory of electromagnetic ﬁelds and waves,” which is both true and misleading. The truth is that electric ﬁelds, magnetic ﬁelds, their sources, waves, and the behavior these waves are all topics covered by this book. The misleading part is that our principal aim shall be to close the gap between basic electrical circuit theory and the more general theory that is required to address certain topics that are of broad and common interest in the ﬁeld of electrical engineering. (For a preview of topics where these techniques are required, see the list at the end of this section.) In basic electrical circuit theory, the behavior of devices and systems is abstracted in such a way that the underlying electromagnetic principles do not need to be considered. Every student of electrical engineering encounters this, and is grateful since this greatly simpliﬁes analysis and design. For example, a resistor is commonly deﬁned as a device which exhibits a particular voltage V = IR in response to a current I, and the resistor is therefore completely described by the value R. This is an example of a “lumped element” abstraction of an electrical device. Much can be accomplished knowing nothing else about resistors; no particular knowledge of the physical concepts of electrical potential, conduction current, or resistance is required. However, this simpliﬁcation makes it impossible to answer some frequently-encountered questions. Here are just a few: • What determines R? How does one go about designing a resistor to have a particular resistance? • Practical resistors are rated for power-handling capability; e.g., discrete resistors are frequently identiﬁed as “1/8-W,” “1/4-W,” and so on. How does one determine this, and how can this be adjusted in the design? • Practical resistors exhibit signiﬁcant reactance as well as resistance. Why? How is this determined? What can be done to mitigate this? • Most things which are not resistors also exhibit signiﬁcant resistance and reactance – for example, electrical pins and interconnects. Why? How is this determined? What can be done to mitigate this? The answers to the these questions must involve properties of materials and the geometry in which those materials are arranged. These are precisely the things that disappear in lumped element device models, so it is not surprising that such models leave us in the dark on these issues. It should also be apparent that what is true for the resistor is also going to be true for other devices of practical interest, including capacitors (and devices unintentionally exhibiting capacitance), inductors (and devices unintentionally exhibiting inductance), transformers (and devices unintentionally exhibiting mutual impedance), and so on. From this perspective, electromagnetics may be viewed as a generalization of electrical circuit theory that addresses these considerations. Conversely basic electric circuit theory may be viewed a special case of Electromagnetics Vol 1. c⃝2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1	Electromagnetics_Vol1_Page_16_Chunk1316
2 CHAPTER 1. PRELIMINARY CONCEPTS electromagnetic theory that applies when these considerations are not important. Many instances of this “electromagnetics as generalization” vs. “lumped-element theory as special case” dichotomy appear in the study of electromagnetics. There is more to the topic, however. There are many devices and applications in which electromagnetic ﬁelds and waves are primary engineering considerations that must be dealt with directly. Examples include electrical generators and motors; antennas; printed circuit board stackup and layout; persistent storage of data (e.g., hard drives); ﬁber optics; and systems for radio, radar, remote sensing, and medical imaging. Considerations such as signal integrity and electromagnetic compatibility (EMC) similarly require explicit consideration of electromagnetic principles. Although electromagnetic considerations pertain to all frequencies, these considerations become increasingly difﬁcult to avoid with increasing frequency. This is because the wavelength of an electromagnetic ﬁeld decreases with increasing frequency.1 When wavelength is large compared to the size of the region of interest (e.g., a circuit), then analysis and design is not much different from zero-frequency (“DC”) analysis and design. For example, the free space wavelength at 3 MHz is about 100 m, so a planar circuit having dimensions 10 cm × 10 cm is just 0.1% of a wavelength across at this frequency. Although an electromagnetic wave may be present, it has about the same value over the region of space occupied by the circuit. In contrast, the free space wavelength at 3 GHz is about 10 cm, so the same circuit is one full wavelength across at this frequency. In this case, different parts of this circuit observe the same signal with very different magnitude and phase. Some of the behaviors associated with non-negligible dimensions are undesirable, especially if not taken into account in the design process. However, these behaviors can also be exploited to do some amazing and useful things – for example, to launch an electromagnetic wave (i.e., an antenna) or to create 1Most readers have encountered the concepts of frequency and wavelength previously, but can refer to Section 1.3, if needed, for a quick primer. ﬁlters and impedance matching devices consisting only of metallic shapes, free of discrete capacitors or inductors. Electromagnetic considerations become not only unavoidable but central to analysis and design above a few hundred MHz, and especially in the millimeter-wave, infrared (IR), optical, and ultraviolet (UV) bands.2 The discipline of electrical engineering encompasses applications in these frequency ranges even though – ironically – such applications may not operate according to principles that can be considered “electrical”! Nevertheless, electromagnetic theory applies. Another common way to answer the question “What is electromagnetics?” is to identify the topics that are commonly addressed within this discipline. Here’s a list of topics – some of which have already been mentioned – in which explicit consideration of electromagnetic principles is either important or essential:3 • Antennas • Coaxial cable • Design and characterization of common discrete passive components including resistors, capacitors, inductors, and diodes • Distributed (e.g., microstrip) ﬁlters • Electromagnetic compatibility (EMC) • Fiber optics • Generators • Magnetic resonance imaging (MRI) • Magnetic storage (of data) • Microstrip transmission lines • Modeling of non-ideal behaviors of discrete components • Motors 2See Section 1.2 for a quick primer on the electromagnetic spec- trum and this terminology. 3Presented in alphabetical order so as to avoid the appearance of any bias on the part of the author!	Electromagnetics_Vol1_Page_17_Chunk1317
1.2. ELECTROMAGNETIC SPECTRUM 3 • Non-contact sensors • Photonics • Printed circuit board stackup and layout • Radar • Radio wave propagation • Radio frequency electronics • Signal integrity • Transformers • Waveguides In summary: Applied engineering electromagnetics is the study of those aspects of electrical engineering in situations in which the electromagnetic proper- ties of materials and the geometry in which those materials are arranged is important. This re- quires an understanding of electromagnetic ﬁelds and waves, which are of primary interest in some applications. Finally, here are two broadly-deﬁned learning objectives that should now be apparent: (1) Learn the techniques of engineering analysis and design that apply when electromagnetic principles are important, and (2) Better understand the physics underlying the operation of electrical devices and systems, so that when issues associated with these physical principles emerge one is prepared to recognize and grapple with them. 1.2 Electromagnetic Spectrum [m0075] Electromagnetic ﬁelds exist at frequencies from DC (0 Hz) to at least 1020 Hz – that’s at least 20 orders of magnitude! At DC, electromagnetics consists of two distinct disciplines: electrostatics, concerned with electric ﬁelds; and magnetostatics, concerned with magnetic ﬁelds. At higher frequencies, electric and magnetic ﬁelds interact to form propagating waves. Waves having frequencies within certain ranges are given names based on how they manifest as physical phenomena. These names are (in order of increasing frequency): radio, infrared (IR), optical (also known as “light”), ultraviolet (UV), X-rays, and gamma rays (γ-rays). See Table 1.1 and Figure 1.1 for frequency ranges and associated wavelengths. The term electromagnetic spectrum refers to the various forms of electromagnetic phenomena that exist over the continuum of frequencies. The speed (properly known as “phase velocity”) at which electromagnetic ﬁelds propagate in free space is given the symbol c, and has the value ∼= 3.00 × 108 m/s. This value is often referred to as the “speed of light.” While it is certainly the speed of light in free space, it is also speed of any electromagnetic wave in free space. Given frequency f, wavelength is given by the expression λ = c f in free space Table 1.1 shows the free space wavelengths associated with each of the regions of the electromagnetic spectrum. This book presents a version of electromagnetic theory that is based on classical physics. This approach works well for most practical problems. However, at very high frequencies, wavelengths become small enough that quantum mechanical effects may be important. This is usually the case in the X-ray band and above. In some applications, these	Electromagnetics_Vol1_Page_18_Chunk1318
4 CHAPTER 1. PRELIMINARY CONCEPTS Regime Frequency Range Wavelength Range γ-Ray > 3 × 1019 Hz < 0.01 nm X-Ray 3 × 1016 Hz – 3 × 1019 Hz 10–0.01 nm Ultraviolet (UV) 2.5 × 1015 – 3 × 1016 Hz 120–10 nm Optical 4.3 × 1014 – 2.5 × 1015 Hz 700–120 nm Infrared (IR) 300 GHz – 4.3 × 1014 Hz 1 mm – 700 nm Radio 3 kHz – 300 GHz 100 km – 1 mm Table 1.1: The electromagnetic spectrum. Note that the indicated ranges are arbitrary but consistent with common usage. effects become important at frequencies as low as the optical, IR, or radio bands. (A prime example is the photoelectric effect; see “Additional References” below.) Thus, caution is required when applying the classical version of electromagnetic theory presented here, especially at these higher frequencies. Theory presented in this book is applicable to DC, radio, IR, and optical waves, and to a lesser extent to UV waves, X-rays, and γ-rays. Cer- tain phenomena in these frequency ranges – in particular quantum mechanical effects – are not addressed in this book. The radio portion of the electromagnetic spectrum alone spans 12 orders of magnitude in frequency (and wavelength), and so, not surprisingly, exhibits a broad range of phenomena. This is shown in Figure 1.1. For this reason, the radio spectrum is further subdivided into bands as shown in Table 1.2. Also shown in Table 1.2 are commonly-used band identiﬁcation acronyms and some typical applications. Similarly, the optical band is partitioned into the familiar “rainbow” of red through violet, as shown in Figure 1.1 and Table 1.3. Other portions of the spectrum are sometimes similarly subdivided in certain applications. Additional Reading: • “Electromagnetic spectrum” on Wikipedia. • “Photoelectric effect” on Wikipedia. 400 nm 500 nm 600 nm 700 nm 1000 m 100 m 10 m 1 m 10 cm 1 cm 1 mm 1000 µm 100 µm 10 µm 1 µm 1000 nm 100 nm 10 nm 0.1 Å 0.1 nm 1Å 1 nm Wavelength 1017 1016 1015 1014 1013 1012 1011 1010 109 108 106 Frequency (Hz) 1018 1019 UHF VHF 7-13 FM VHF 2-6 1000 MHz 500 MHz 100 MHz 50 MHz Gamma-rays X-rays Ultraviolet Visible Near IR Infra-red Thermal IR Far IR Microwaves Radar Radio, TV AM Long-waves 107 c⃝V. Blacus CC BY-SA 3.0 Figure 1.1: Electromagnetic Spectrum.	Electromagnetics_Vol1_Page_19_Chunk1319
1.3. FUNDAMENTALS OF WAVES 5 Band Frequencies Wavelengths Typical Applications EHF 30-300 GHz 10–1 mm 60 GHz WLAN, Point-to-point data links SHF 3–30 GHz 10–1 cm Terrestrial & Satellite data links, Radar UHF 300–3000 MHz 1–0.1 m TV broadcasting, Cellular, WLAN VHF 30–300 MHz 10–1 m FM & TV broadcasting, LMR HF 3–30 MHz 100–10 m Global terrestrial comm., CB Radio MF 300–3000 kHz 1000–100 m AM broadcasting LF 30–300 kHz 10–1 km Navigation, RFID VLF 3–30 kHz 100–10 km Navigation Table 1.2: The radio portion of the electromagnetic spectrum, according to a common scheme for naming ranges of radio frequencies. WLAN: Wireless local area network, LMR: Land mobile radio, RFID: Radio frequency identiﬁcation. Band Frequencies Wavelengths Violet 668–789 THz 450–380 nm Blue 606–668 THz 495–450 nm Green 526–606 THz 570–495 nm Yellow 508–526 THz 590–570 nm Orange 484–508 THz 620–590 nm Red 400–484 THz 750–620 nm Table 1.3: The optical portion of the electromagnetic spectrum. 1.3 Fundamentals of Waves [m0074] In this section, we formally introduce the concept of a wave and explain some basic characteristics. To begin, let us consider not electromagnetic waves, but rather sound waves. To be clear, sound waves and electromagnetic waves are completely distinct phenomena. Sound waves are variations in pressure, whereas electromagnetic waves are variations in electric and magnetic ﬁelds. However, the mathematics that govern sound waves and electromagnetic waves are very similar, so the analogy provides useful insight. Furthermore, sound waves are intuitive for most people because they are readily observed. So, here we go: Imagine standing in an open ﬁeld and that it is completely quiet. In this case, the air pressure everywhere is about 101 kPa (101,000 N/m2) at sea level, and we refer to this as the quiescent air pressure. Sound can be described as the differential air pressure p(x, y, z, t), which we deﬁne as the absolute air pressure at the spatial coordinates (x, y, z) minus the quiescent air pressure. So, when there is no sound, p(x, y, z, t) = 0. The function p as an example of a scalar ﬁeld.4 Let’s also say you are standing at x = y = z = 0 and you have brought along a friend who is standing at x = d; i.e., a distance d from you along the x axis. Also, for simplicity, let us consider only what is happening along the x axis; i.e., p(x, t). At t = 0, you clap your hands once. This forces the air between your hands to press outward, creating a region of increased pressure (i.e., p > 0) that travels outward. As the region of increased pressure moves outward, it leaves behind a region of low pressure where p < 0. Air molecules immediately move toward this region of lower pressure, and so the air pressure quickly returns to the quiescent value, p = 0. The traveling disturbance in p(x, t) is the sound of the clap. The disturbance continues to travel outward until it reaches your friend, who then hears the clap. At each point in time, you can make a plot of p(x, t) versus x for the current value of t. This is shown in Figure 1.2. At times t < 0, we have simply p(x, t) = 0. A short time after t = 0, the peak pressure is located at slightly to the right of x = 0. The pressure is not a simple impulse because interactions between air molecules constrain the 4Although it’s not important in this section, you can read about the concept of a “ﬁeld” in Section 2.1.	Electromagnetics_Vol1_Page_20_Chunk1320
6 CHAPTER 1. PRELIMINARY CONCEPTS p(x,t) x p(x,t) x c⃝Y. Qin CC BY 4.0 Figure 1.2: The differential pressure p(x, t) (top) a short time after the clap and (bottom) a slightly longer time after the clap. pressure to be continuous over space. So instead, we see a rounded pulse representing the rapid build-up and similarly rapid decline in air pressure. A short time later p(x, t) looks very similar, except the pulse is now further away. Now: What precisely is p(x, t)? Completely skipping over the derivation, the answer is that p(x, t) is the solution to the acoustic wave equation (see “Additional References” at the end of this section): ∂2p ∂x2 −1 c2s ∂2p ∂t2 = 0 (1.1) where cs is the speed of sound, which is about 340 m/s at sea level. Just to emphasize the quality of the analogy between sound waves and electromagnetic waves, know that the acoustic wave equation is mathematically identical to equations that that govern electromagnetic waves. Although “transient” phenomena – analogous to a clap – are of interest in electromagnetics, an even more common case of interest is the wave resulting from a sinusoidally-varying source. We can demonstrate this kind of wave in the context of sound as well. Here we go: In the previous scenario, you pick up a trumpet and blow a perfect A note. The A note is 440 Hz, meaning that the air pressure emerging from your trumpet is varying sinusoidally at a frequency of 440 Hz. Let’s say you can continue to blow this note long enough for the entire ﬁeld to be ﬁlled with the sound of your trumpet. Now what does the pressure-versus-distance curve look like? Two simple observations will settle that question: • p(x, t) at any constant position x is a sinusoid as a function of x. This is because the acoustic wave equation is linear and time invariant, so a sinusoidal excitation (i.e., your trumpet) results in a sinusoidal response at the same frequency (i.e., the sound heard by your friend). • p(x, t) at any constant time t is also a sinusoid as a function of x. This is because the sound is propagating away from the trumpet and toward your friend, and anyone in between will also hear the A note, but with a phase shift determined by the difference in distances. This is enough information to know that the solution must have the form: p(x, t) = Am cos (ωt −βx + ψ) (1.2) where ω = 2πf, f = 440 Hz, and Am, β, and ψ remain to be determined. You can readily verify that Equation 1.2 satisﬁes the acoustic wave equation when β = ω cs (1.3) In this problem, we ﬁnd β ∼= 8.13 rad/m. This means that at any given time, the difference in phase measured between any two points separated by a distance of 1 m is 8.13 rad. The parameter β goes by at least three names: phase propagation constant, wavenumber, and spatial frequency. The term “spatial frequency” is particularly apt, since β plays precisely the same role for distance (x) as ω plays for time (t) – This is apparent from Equation 1.2. However, “wavenumber” is probably the more commonly-used term. The wavenumber β (rad/m) is the rate at which the phase of a sinusoidal wave progresses with distance.	Electromagnetics_Vol1_Page_21_Chunk1321
1.3. FUNDAMENTALS OF WAVES 7 Note that Am and ψ are not determined by the wave equation, but instead are properties of the source. Speciﬁcally, Am is determined by how hard we blow, and ψ is determined by the time at which we began to blow and the location of the trumpet. For simplicity, let us assume that we begin to blow at time t ≪0; i.e., in the distant past so that the sound pressure ﬁeld has achieved steady state by t = 0. Also, let us set ψ = 0 and set Am = 1 in whatever units we choose to express p(x, t). We then have: p(x, t) = cos (ωt −βx) (1.4) Now we have everything we need to make plots of p(x) at various times. Figure 1.3(a) shows p(x, t = 0). As expected, p(x, t = 0) is periodic in x. The associated period is referred to as the wavelength λ. Since λ is the distance required for the phase of the wave to increase by 2π rad, and because phase is increasing at a rate of β rad/m, we ﬁnd: λ = 2π β (1.5) In the present example, we ﬁnd λ ∼= 77.3 cm. Wavelength λ = 2π/β is the distance required for the phase of a sinusoidal wave to increase by one complete cycle (i.e., 2π rad) at any given time. Now let us consider the situation at t = +1/4f, which is t = 568 µs and ωt = π/2. We see in Figure 1.3(b) that the waveform has shifted a distance λ/4 to the right. It is in this sense that we say the wave is propagating in the +x direction. Furthermore, we can now compute a phase velocity vp: We see that a point of constant phase has shifted a distance λ/4 in time 1/4f, so vp = λf (1.6) In the present example, we ﬁnd vp ∼= 340 m/s; i.e., we have found that the phase velocity is equal to the speed of sound cs. It is in this sense that we say that the phase velocity is the speed at which the wave propagates.5 5It is worth noting here is that “velocity” is technically a vector; i.e., speed in a given direction. Nevertheless, this quantity is actually just a speed, and this particular abuse of terminology is generally accepted. p(x,t) x λ p(x,t) x p(x,t) x c⃝Y. Qin CC BY 4.0 Figure 1.3: The differential pressure p(x, t) for (a) t = 0, (b) t = 1/4f for “−βx,” as indicated in Equa- tion 1.4 (wave traveling to right); and (c) t = 1/4f for “+βx” (wave traveling to left).	Electromagnetics_Vol1_Page_22_Chunk1322
8 CHAPTER 1. PRELIMINARY CONCEPTS Phase velocity vp = λf is the speed at which a point of constant phase in a sinusoidal waveform travels. Recall that in Equation 1.2 we declared that βx is subtracted from the argument of the sinusoidal function. To understand why, let’s change the sign of βx and see if it still satisﬁes the wave equation – one ﬁnds that it does. Next, we repeat the previous experiment and see what happens. The result is shown in Figure 1.3(c). Note that points of constant phase have traveled an equal distance, but now in the −x direction. In other words, this alternative choice of sign for βx within the argument of the cosine function represents a wave that is propagating in the opposite direction. This leads us to the following realization: If the phase of the wave is decreasing with βx, then the wave is propagating in the +x direction. If the phase of the wave is increasing with βx, then the wave is propagating in the −x direction. Since the prospect of sound traveling toward the trumpet is clearly nonsense in the present situation, we may neglect the latter possibility. However, what happens if there is a wall located in the distance, behind your friend? Then, we expect an echo from the wall, which would be a second wave propagating in the reverse direction and for which the argument of the cosine function would contain the term “+βx.” Finally, let us return to electromagnetics. Electromagnetic waves satisfy precisely the same wave equation (i.e., Equation 1.1) as do sound waves, except that the phase velocity is much greater. Interestingly, though, the frequencies of electromagnetic waves are also much greater than those of sound waves, so we can end up with wavelengths having similar orders of magnitude. In particular, an electromagnetic wave with λ = 77.3 cm (the wavelength of the “A” note in the preceding example) lies in the radio portion of the electromagnetic spectrum. An important difference between sound and electromagnetic waves is that electromagnetic waves are vectors; that is, they have direction as well as magnitude. Furthermore, we often need to consider multiple electromagnetic vector waves (in particular, both the electric ﬁeld and the magnetic ﬁeld) in order to completely understand the situation. Nevertheless the concepts of wavenumber, wavelength, phase velocity, and direction of propagation apply in precisely the same manner to electromagnetic waves as they do to sound waves. Additional Reading: • “Wave Equation” on Wikipedia. • “Acoustic Wave Equation” on Wikipedia.	Electromagnetics_Vol1_Page_23_Chunk1323
1.4. GUIDED AND UNGUIDED WAVES 9 1.4 Guided and Unguided Waves [m0040] Broadly speaking, waves may be either guided or unguided. Unguided waves include those that are radiated by antennas, as well as those that are unintentionally radiated. Once initiated, these waves propagate in an uncontrolled manner until they are redirected by scattering or dissipated by losses associated with materials. Examples of guided waves are those that exist within structures such as transmission lines, waveguides, and optical ﬁbers. We refer to these as guided because they are constrained to follow the path deﬁned by the structure. Antennas and unintentional radiators emit un- guided waves. Transmission lines, waveguides, and optical ﬁbers carry guided waves. 1.5 Phasors [m0033] In many areas of engineering, signals are well-modeled as sinusoids. Also, devices that process these signals are often well-modeled as linear time-invariant (LTI) systems. The response of an LTI system to any linear combination of sinusoids is another linear combination of sinusoids having the same frequencies.6 In other words, (1) sinusoidal signals processed by LTI systems remain sinusoids and are not somehow transformed into square waves or some other waveform; and (2) we may calculate the response of the system for one sinusoid at a time, and then add the results to ﬁnd the response of the system when multiple sinusoids are applied simultaneously. This property of LTI systems is known as superposition. The analysis of systems that process sinusoidal waveforms is greatly simpliﬁed when the sinusoids are represented as phasors. Here is the key idea: A phasor is a complex-valued number that repre- sents a real-valued sinusoidal waveform. Specif- ically, a phasor has the magnitude and phase of the sinusoid it represents. Figures 1.4 and 1.5 show some examples of phasors and the associated sinusoids. It is important to note that a phasor by itself is not the signal. A phasor is merely a simpliﬁed mathematical representation in which the actual, real-valued physical signal is represented as a complex-valued constant. Here is a completely general form for a physical (hence, real-valued) quantity varying sinusoidally with angular frequency ω = 2πf: A(t; ω) = Am(ω) cos (ωt + ψ(ω)) (1.7) where Am(ω) is magnitude at the speciﬁed frequency, ψ(ω) is phase at the speciﬁed frequency, and t is time. Also, we require ∂Am/∂t = 0; that is, that the time 6 A “linear combination” of functions fi(t) where i = 1, 2, 3, ... is P i aifi(t) where the ai’s are constants.	Electromagnetics_Vol1_Page_24_Chunk1324
10 CHAPTER 1. PRELIMINARY CONCEPTS Ame j magnitude Im{C} Re{C} Am phase (	Electromagnetics_Vol1_Page_25_Chunk1325
1.5. PHASORS 11 dependence: C = Amejψ (1.10) This does not normally cause any confusion since the deﬁnition of a phasor requires that values of C and ψ are those that apply at whatever frequency is represented by the suppressed sinusoidal dependence ejωt. Table 1.4 shows mathematical representations of the same phasors demonstrated in Figure 1.4 (and their associated sinusoidal waveforms in Figure 1.5). It is a good exercise is to conﬁrm each row in the table, transforming from left to right and vice-versa. It is not necessary to use a phasor to represent a sinusoidal signal. We choose to do so because phasor representation leads to dramatic simpliﬁcations. For example: • Calculation of the peak value from data representing A(t; ω) requires a time-domain search over one period of the sinusoid. However, if you know C, the peak value of A(t) is simply \|C\|, and no search is required. • Calculation of ψ from data representing A(t; ω) requires correlation (essentially, integration) over one period of the sinusoid. However, if you know C, then ψ is simply the phase of C, and no integration is required. Furthermore, mathematical operations applied to A(t; ω) can be equivalently performed as operations on C, and the latter are typically much easier than the former. To demonstrate this, we ﬁrst make two important claims and show that they are true. Claim 1: Let C1 and C2 be two complex-valued constants (independent of t). Also, Re C1ejωt = Re C2ejωt for all t. Then, C1 = C2. Proof: Evaluating at t = 0 we ﬁnd Re {C1} = Re {C2}. Since C1 and C2 are constant with respect to time, this must be true for all t. At t = π/(2ω) we ﬁnd Re C1ejωt = Re {C1 · j} = −Im {C1} and similarly Re C2ejωt = Re {C2 · j} = −Im {C2} therefore Im {C1} = Im {C2}. Once again: Since C1 and C2 are constant with respect to time, this must be true for all t. Since the real and imaginary parts of C1 and C2 are equal, C1 = C2. What does this mean? We have just shown that if two phasors are equal, then the sinusoidal waveforms that they represent are also equal. Claim 2: For any real-valued linear operator T and complex-valued quantity C, T (Re {C}) = Re {T (C)}. Proof: Let C = cr + jci where cr and ci are real-valued quantities, and evaluate the right side of the equation: Re {T (C)} = Re {T (cr + jci)} = Re {T (cr) + jT (ci)} = T (cr) = T (Re {C}) What does this mean? The operators that we have in mind for T include addition, multiplication by a constant, differentiation, integration, and so on. Here’s an example with differentiation: Re ∂ ∂ω C = Re ∂ ∂ω (cr + jci) = ∂ ∂ω cr ∂ ∂ω Re {C} = ∂ ∂ω Re {(cr + jci)} = ∂ ∂ω cr In other words, differentiation of a sinusoidal signal can be accomplished by differentiating the associated phasor, so there is no need to transform a phasor back into its associated real-valued signal in order to perform this operation.	Electromagnetics_Vol1_Page_26_Chunk1326
12 CHAPTER 1. PRELIMINARY CONCEPTS A(t) C Am cos (ωt) Am Am cos (ωt + ψ) Amejψ Am sin (ωt) = Am cos	Electromagnetics_Vol1_Page_27_Chunk1327
1.6. UNITS 13 Summarizing: Phasor analysis does not limit us to sinusoidal waveforms. Phasor analysis is not only applica- ble to sinusoids and signals that are sufﬁciently narrowband, but is also applicable to signals of arbitrary bandwidth via Fourier analysis. Additional Reading: • “Phasor” on Wikipedia. • “Fourier analysis” on Wikipedia. 1.6 Units [m0072] The term “unit” refers to the measure used to express a physical quantity. For example, the mean radius of the Earth is about 6,371,000 meters; in this case the unit is the meter. A number like “6,371,000” becomes a bit cumbersome to write, so it is common to use a preﬁx to modify the unit. For example, the radius of the Earth is more commonly said to be 6371 kilometers, where one kilometer is understood to mean 1000 meters. It is common practice to use preﬁxes, such as “kilo-,” that yield values in the range 0.001 to 10, 000. A list of standard preﬁxes appears in Table 1.5. Writing out the names of units can also become tedious. For this reason, it is common to use standard abbreviations; e.g., “6731 km” as opposed to “6371 kilometers,” where “k” is the standard abbreviation for the preﬁx “kilo” and “m” is the standard abbreviation for “meter.” A list of commonly-used base units and their abbreviations are shown in Table 1.6. To avoid ambiguity, it is important to always indicate the units of a quantity; e.g., writing “6371 km” as opposed to “6371.” Failure to do so is a common source of error and misunderstandings. An example is Preﬁx Abbreviation Multiply by: exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 milli m 10−3 micro µ 10−6 nano n 10−9 pico p 10−12 femto f 10−15 atto a 10−18 Table 1.5: Preﬁxes used to modify units.	Electromagnetics_Vol1_Page_28_Chunk1328
14 CHAPTER 1. PRELIMINARY CONCEPTS Unit Abbreviation Quantiﬁes: ampere A electric current coulomb C electric charge farad F capacitance henry H inductance hertz Hz frequency joule J energy meter m distance newton N force ohm Ω resistance second s time tesla T magnetic ﬂux density volt V electric potential watt W power weber Wb magnetic ﬂux Table 1.6: Some units that are commonly used in elec- tromagnetics. the expression: l = 3t where l is length and t is time. It could be that l is in meters and t is in seconds, in which case “3” really means “3 m/s”. However, if it is intended that l is in kilometers and t is in hours, then “3” really means “3 km/h” and the equation is literally different. To patch this up, one might write “l = 3t m/s”; however, note that this does does not resolve the ambiguity we just identiﬁed – i.e., we still don’t know the units of the constant “3.” Alternatively, one might write “l = 3t where l is in meters and t is in seconds,” which is unambiguous but becomes quite awkward for more complicated expressions. A better solution is to write instead: l = (3 m/s) t or even better: l = at where a = 3 m/s since this separates this issue of units from the perhaps more-important fact that l is proportional to t and the constant of proportionality (a) is known. The meter is the fundamental unit of length in the International System of Units, known by its French acronym “SI” and sometimes informally referred to as the “metric system.” In this work, we will use SI units exclusively. Although SI is probably the most popular for engineering use overall, other systems remain in common use. For example, the English system, where the radius of the Earth might alternatively be said to be about 3959 miles, continues to be used in various applications and to a lesser or greater extent in various regions of the world. An alternative system in common use in physics and material science applications is the CGS (“centimeter-gram-second”) system. The CGS system is similar to SI, but with some signiﬁcant differences. For example, the base unit of energy is the CGS system is not the “joule” but rather the “erg,” and the values of some physical constants become unitless. Therefore – once again – it is very important to include units whenever values are stated. SI deﬁnes seven fundamental units from which all other units can be derived. These fundamental units are distance in meters (m), time in seconds (s), current in amperes (A), mass in kilograms (kg), temperature in kelvin (K), particle count in moles (mol), and luminosity in candela (cd). SI units for electromagnetic quantities such as coulombs (C) for charge and volts (V) for electric potential are derived from these fundamental units. A frequently-overlooked feature of units is their ability to assist in error-checking mathematical expressions. For example, the electric ﬁeld intensity may be speciﬁed in volts per meter (V/m), so an expression for the electric ﬁeld intensity that yields units of V/m is said to be “dimensionally correct” (but not necessarily correct), whereas an expression that cannot be reduced to units of V/m cannot be correct. Additional Reading: • “International System of Units” on Wikipedia. • “Centimetre-gram-second system of units” on Wikipedia.	Electromagnetics_Vol1_Page_29_Chunk1329
1.7. NOTATION 15 1.7 Notation [m0005] The list below describes notation used in this book. • Vectors: Boldface is used to indicate a vector; e.g., the electric ﬁeld intensity vector will typically appear as E. Quantities not in boldface are scalars. When writing by hand, it is common to write “E” or “−→ E ” in lieu of “E.” • Unit vectors: A circumﬂex is used to indicate a unit vector; i.e., a vector having magnitude equal to one. For example, the unit vector pointing in the +x direction will be indicated as ˆx. In discussion, the quantity “ˆx” is typically spoken “x hat.” • Time: The symbol t is used to indicate time. • Position: The symbols (x, y, z), (ρ, φ, z), and (r, θ, φ) indicate positions using the Cartesian, cylindrical, and polar coordinate systems, respectively. It is sometimes convenient to express position in a manner which is independent of a coordinate system; in this case, we typically use the symbol r. For example, r = ˆxx + ˆyy + ˆzz in the Cartesian coordinate system. • Phasors: A tilde is used to indicate a phasor quantity; e.g., a voltage phasor might be indicated as eV , and the phasor representation of E will be indicated as eE. • Curves, surfaces, and volumes: These geometrical entities will usually be indicated in script; e.g., an open surface might be indicated as S and the curve bounding this surface might be indicated as C. Similarly, the volume enclosed by a closed surface S may be indicated as V. • Integrations over curves, surfaces, and volumes will usually be indicated using a single integral sign with the appropriate subscript. For example: Z C · · · dl is an integral over the curve C Z S · · · ds is an integral over the surface S Z V · · · dv is an integral over the volume V. • Integrations over closed curves and surfaces will be indicated using a circle superimposed on the integral sign. For example: I C · · · dl is an integral over the closed curve C I S ··· ds is an integral over the closed surface S A “closed curve” is one which forms an unbroken loop; e.g., a circle. A “closed surface” is one which encloses a volume with no openings; e.g., a sphere. • The symbol “∼=” means “approximately equal to.” This symbol is used when equality exists, but is not being expressed with exact numerical precision. For example, the ratio of the circumference of a circle to its diameter is π, where π ∼= 3.14. • The symbol “≈” also indicates “approximately equal to,” but in this case the two quantities are unequal even if expressed with exact numerical precision. For example, ex = 1 + x + x2/2 + ... as a inﬁnite series, but ex ≈1 + x for x ≪1. Using this approximation e0.1 ≈1.1, which is in good agreement with the actual value e0.1 ∼= 1.1052. • The symbol “∼” indicates “on the order of,” which is a relatively weak statement of equality indicating that the indicated quantity is within a factor of 10 or so the indicated value. For example, µ ∼105 for a class of iron alloys, with exact values being being larger or smaller by a factor of 5 or so. • The symbol “≜” means “is deﬁned as” or “is equal as the result of a deﬁnition.” • Complex numbers: j ≜√−1. • See Appendix C for notation used to identify commonly-used physical constants. [m0026]	Electromagnetics_Vol1_Page_30_Chunk1330
16 CHAPTER 1. PRELIMINARY CONCEPTS Image Credits Fig. 1.1 c⃝V. Blacus, https://commons.wikimedia.org/wiki/File:Electromagnetic-Spectrum.svg, CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Fig. 1.2: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0074 fClap.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Fig. 1.3: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0074 fTrumpet.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).	Electromagnetics_Vol1_Page_31_Chunk1331
Chapter 2 Electric and Magnetic Fields 2.1 What is a Field? [m0001] A ﬁeld is the continuum of values of a quantity as a function of position and time. The quantity that the ﬁeld describes may be a scalar or a vector, and the scalar part may be either real- or complex-valued. In electromagnetics, the electric ﬁeld intensity E is a real-valued vector ﬁeld that may vary as a function of position and time, and so might be indicated as “E(x, y, z, t),” “E(r, t),” or simply “E.” When expressed as a phasor, this quantity is complex-valued but exhibits no time dependence, so we might say instead “eE(r)” or simply “eE.” An example of a scalar ﬁeld in electromagnetics is the electric potential, V ; i.e., V (r, t). A wave is a time-varying ﬁeld that continues to exist in the absence of the source that created it and is therefore able to transport energy. 2.2 Electric Field Intensity [m0002] Electric ﬁeld intensity is a vector ﬁeld we assign the symbol E and has units of electrical potential per distance; in SI units, volts per meter (V/m). Before offering a formal deﬁnition, it is useful to consider the broader concept of the electric ﬁeld. Imagine that the universe is empty except for a single particle of positive charge. Next, imagine that a second positively-charged particle appears; the situation is now as shown in Figure 2.1. Since like charges repel, the second particle will be repelled by the ﬁrst particle and vice versa. Speciﬁcally, the ﬁrst particle is exerting force on the second particle. If the second particle is free to move, it will do so; this is an expression of kinetic energy. If the second particle is somehow held in place, we say the second particle possesses an equal amount of potential energy. This potential energy is no less “real,” since we can convert it to kinetic energy simply by releasing the particle, thereby allowing it to move. Now let us revisit the original one particle scenario. F c⃝M. Goldammer CC BY SA 4.0 Figure 2.1: A positively-charged particle experiences a repulsive force F in the presence of another particle which is also positively-charged. Electromagnetics Vol 1. c⃝2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1	Electromagnetics_Vol1_Page_32_Chunk1332
18 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS F c⃝M. Goldammer CC BY SA 4.0 Figure 2.2: A map of the force that would be expe- rienced by a second particle having a positive charge. Here, the magnitude and direction of the force is indi- cated by the size and direction of the arrow. In that scenario, we could make a map in which every position in space is assigned a vector that describes the force that a particle having a speciﬁed charge q would experience if it were to appear there. The result looks something like Figure 2.2. This map of force vectors is essentially a description of the electric ﬁeld associated with the ﬁrst particle. There are many ways in which the electric ﬁeld may be quantiﬁed. Electric ﬁeld intensity E is simply one of these ways. We deﬁne E(r) to be the force F(r) experienced by a test particle having charge q, divided by q; i.e., E(r) ≜lim q→0 F(r) q (2.1) Note that it is required for the charge to become vanishingly small (as indicated by taking the limit) in order for this deﬁnition to work. This is because the source of the electric ﬁeld is charge, so the test particle contributes to the total electric ﬁeld. To accurately measure the ﬁeld of interest, the test charge must be small enough not to signiﬁcantly perturb the ﬁeld. This makes Equation 2.1 awkward from an engineering perspective, and we’ll address that later in this section. According the deﬁnition of Equation 2.1, the units of E are those of force divided by charge. The SI units for force and charge are the newton (N) and coulomb (C) respectively, so E has units of N/C. However, we typically express E in units of V/m, not N/C. What’s going on? The short answer is that 1 V/m = 1 N/C: N C = N · m C · m = J C · m = V m where we have used the fact that 1 N·m = 1 joule (J) of energy and 1 J/C = 1 V. Electric ﬁeld intensity (E, N/C or V/m) is a vec- tor ﬁeld that quantiﬁes the force experienced by a charged particle due to the inﬂuence of charge not associated with that particle. The analysis of units doesn’t do much to answer the question of why we should prefer to express E in V/m as opposed to N/C. Let us now tackle that question. Figure 2.3 shows a simple thought experiment that demonstrates the concept of electric ﬁeld intensity in terms of an electric circuit. This circuit consists of a parallel-plate capacitor in series with a 9 V battery.1 The effect of the battery, connected as shown, is to force an accumulation of positive charge on the upper plate, and an accumulation of negative charge on the lower plate. If we consider the path from the position labeled “A,” along the wire and through the battery to the position labeled “B,” the change in electric potential is +9 V. It must also be true that the change in electric potential as we travel from B to A through the capacitor is −9 V, since the sum of voltages over any closed loop in a circuit is zero. Said differently, the change in electric potential between the plates of the capacitor, starting from node A and ending at node B, is +9 V. Now, note that the spacing between the plates in the capacitor is 1 mm. Thus, the rate of change of the potential between the plates is 9 V divided by 1 mm, which is 9000 V/m. This is literally the electric ﬁeld intensity between the plates. That is, if one places a particle with an inﬁnitesimally-small charge between the plates (point “C”), and then measures the ratio of force to charge, one ﬁnds it is 9000 N/C pointing toward A. We come to the following remarkable conclusion: 1It is not necessary to know anything about capacitors to get to the point, so no worries!	Electromagnetics_Vol1_Page_33_Chunk1333
2.2. ELECTRIC FIELD INTENSITY 19 A B + - 9V 1mm c⃝Y. Qin CC BY 3.0 Figure 2.3: A simple circuit used to describe the con- cept of electric ﬁeld intensity. In this example, E at point C is 9000 V/m directed from B toward A. E points in the direction in which electric poten- tial is most rapidly decreasing, and the magnitude of E is the rate of change in electric potential with distance in this direction. The reader may have noticed that we have deﬁned the electric ﬁeld in terms of what it does. We have have not directly addressed the question of what the electric ﬁeld is. This is the best we can do using classical physics, and fortunately, this is completely adequate for the most engineering applications. However, a deeper understanding is possible using quantum mechanics, where we ﬁnd that the electric ﬁeld and the magnetic ﬁeld are in fact manifestations of the same fundamental force, aptly named the electromagnetic force. (In fact, the electromagnetic force is found to be one of just four fundamental forces, the others being gravity, the strong nuclear force, and the weak nuclear force.) Quantum mechanics also facilitates greater insight into the nature of electric charge and of the photon, which is the fundamental constituent of electromagnetic waves. For more information on this topic, an excellent starting point is the video “Quantum Invariance & The Origin of The Standard Model” referenced at the end of this section. Additional Reading: • “Electric ﬁeld” on Wikipedia. • PBS Space Time video “Quantum Invariance & The Origin of The Standard Model,” on YouTube.	Electromagnetics_Vol1_Page_34_Chunk1334
20 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS 2.3 Permittivity [m0008] Permittivity describes the effect of material in determining the electric ﬁeld in response to electric charge. For example, one can observe from laboratory experiments that a particle having charge q gives rise to the electric ﬁeld E = ˆR q 1 4πR2 1 ǫ (2.2) where R is distance from the charge, ˆR is a unit vector pointing away from the charge, and ǫ is a constant that depends on the material. Note that E increases with q, which makes sense since electric charge is the source of E. Also note that E is inversely proportional to 4πR2, indicating that E decreases in proportion to the area of a sphere surrounding the charge – a principle commonly known as the inverse square law. The remaining factor 1/ǫ is the constant of proportionality, which captures the effect of material. Given units of V/m for E and C for Q, we ﬁnd that ǫ must have units of farads per meter (F/m). (To see this, note that 1 F = 1 C/V.) Permittivity (ǫ, F/m) describes the effect of ma- terial in determining the electric ﬁeld intensity in response to charge. In free space (that is, a perfect vacuum), we ﬁnd that ǫ = ǫ0 where: ǫ0 ∼= 8.854 × 10−12 F/m (2.3) The permittivity of air is only slightly greater, and usually can be assumed to be equal to that of free space. In most other materials, the permittivity is signiﬁcantly greater; that is, the same charge results in a weaker electric ﬁeld intensity. It is common practice to describe the permittivity of materials relative to the permittivity of free space. This relative permittivity is given by: ǫr ≜ǫ ǫ0 (2.4) Values of ǫr for a few representative materials is given in Appendix A.1. Note that ǫr ranges from 1 (corresponding to a perfect vacuum) to about 60 or so in common engineering applications. Also note that relative permittivity is sometimes referred to as dielectric constant. This term is a bit misleading, however, since permittivity is a meaningful concept for many materials that are not dielectrics. Additional Reading: • “Permittivity” on Wikipedia. • Appendix A.1 (“Permittivity of Some Common Materials”). • “Inverse square law” on Wikipedia.	Electromagnetics_Vol1_Page_35_Chunk1335
2.4. ELECTRIC FLUX DENSITY 21 2.4 Electric Flux Density [m0011] Electric ﬂux density, assigned the symbol D, is an alternative to electric ﬁeld intensity (E) as a way to quantify an electric ﬁeld. This alternative description offers some actionable insight, as we shall point out at the end of this section. First, what is electric ﬂux density? Recall that a particle having charge q gives rise to the electric ﬁeld intensity E = ˆR q 1 4πR2 1 ǫ (2.5) where R is distance from the charge and ˆR points away from the charge. Note that E is inversely proportional to 4πR2, indicating that E decreases in proportion to the area of a sphere surrounding the charge. Now integrate both sides of Equation 2.5 over a sphere S of radius R: I S E(r) · ds = I S ˆR q 1 4πR2 1 ǫ · ds (2.6) Factoring out constants that do not vary with the variables of integration, the right-hand side becomes: q 1 4πR2 1 ǫ I S ˆR · ds Note that ds = ˆRds in this case, and also that ˆR · ˆR = 1. Thus, the right-hand side simpliﬁes to: q 1 4πR2 1 ǫ I S ds The remaining integral is simply the area of S, which is 4πR2. Thus, we ﬁnd: I S E(r) · ds = q ǫ (2.7) The integral of a vector ﬁeld over a speciﬁed surface is known as ﬂux (see “Additional Reading” at the end of this section). Thus, we have found that the ﬂux of E through the sphere S is equal to a constant, namely q/ǫ. Furthermore, this constant is the same regardless of the radius R of the sphere. Said differently, the ﬂux of E is constant with distance, and does not vary as E itself does. Let us capitalize on this observation by making the following small modiﬁcation to Equation 2.7: I S [ǫE] · ds = q (2.8) In other words, the ﬂux of the quantity ǫE is equal to the enclosed charge, regardless of the radius of the sphere over which we are doing the calculation. This leads to the following deﬁnition: The electric ﬂux density D = ǫE, having units of C/m2, is a description of the electric ﬁeld in terms of ﬂux, as opposed to force or change in electric potential. It may appear that D is redundant information given E and ǫ, but this is true only in homogeneous media. The concept of electric ﬂux density becomes important – and decidedly not redundant – when we encounter boundaries between media having different permittivities. As we shall see in Section 5.18, boundary conditions on D constrain the component of the electric ﬁeld that is perpendicular to the boundary separating two regions. If one ignores the “ﬂux” character of the electric ﬁeld represented by D and instead considers only E, then only the tangential component of the electric ﬁeld is constrained. In fact, when one of the two materials comprising the boundary between two material regions is a perfect conductor, then the electric ﬁeld is completely determined by the boundary condition on D. This greatly simpliﬁes the problem of ﬁnding the electric ﬁeld in a region bounded or partially bounded by materials that can be modeled as perfect conductors, including many metals. In particular, this principle makes it easy to analyze capacitors. We conclude this section with a warning. Even though the SI units for D are C/m2, D describes an electric ﬁeld and not a surface charge density. It is certainly true that one may describe the amount of charge distributed over a surface using units of C/m2. However, D is not necessarily a description of actual charge, and there is no implication that the source of the electric ﬁeld is a distribution of surface charge. On the other hand, it is true that D can be interpreted as an equivalent surface charge density that would give rise to the observed electric ﬁeld, and in some cases, this equivalent charge density turns out to be the actual charge density.	Electromagnetics_Vol1_Page_36_Chunk1336
22 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS Additional Reading: • “Flux” on Wikipedia. 2.5 Magnetic Flux Density [m0003] Magnetic ﬂux density is a vector ﬁeld which we identify using the symbol B and which has SI units of tesla (T). Before offering a formal deﬁnition, it is useful to consider the broader concept of the magnetic ﬁeld. Magnetic ﬁelds are an intrinsic property of some materials, most notably permanent magnets. The basic phenomenon is probably familiar, and is shown in Figure 2.4. A bar magnet has “poles” identiﬁed as “N” (“north”) and “S” (“south”). The N-end of one magnet attracts the S-end of another magnet but repels the N-end of the other magnet and so on. The existence of a vector ﬁeld is apparent since the observed force acts at a distance and is asserted in a speciﬁc direction. In the case of a permanent magnet, the magnetic ﬁeld arises from mechanisms occurring at the scale of the atoms and electrons comprising the material. These mechanisms require some additional explanation which we shall defer for now. Magnetic ﬁelds also appear in the presence of current. For example, a coil of wire bearing a current is found to inﬂuence permanent magnets (and vice versa) in the same way that permanent magnets affect each other. This is shown in Figure 2.5. From this, we infer that the underlying mechanism is the same – i.e., the vector ﬁeld generated by a current-bearing coil is the same phenomenon as the vector ﬁeld associated with a permanent magnet. Whatever the source, we are S N c⃝Y. Qin CC BY 3.0 Figure 2.4: Evidence of a vector ﬁeld from observa- tions of the force perceived by the bar magnets on the right in the presence of the bar magnets on the left.	Electromagnetics_Vol1_Page_37_Chunk1337
2.5. MAGNETIC FLUX DENSITY 23 c⃝Y. Qin CC BY 4.0 Figure 2.5: Evidence that current can also create a magnetic ﬁeld. now interested in quantifying its behavior. To begin, let us consider the effect of a magnetic ﬁeld on a electrically-charged particle. First, imagine a region of free space with no electric or magnetic ﬁelds. Next, imagine that a charged particle appears. This particle will experience no force. Next, a magnetic ﬁeld appears; perhaps this is due to a permanent magnet or a current in the vicinity. This situation is shown in Figure 2.6 (top). Still, no force is applied to the particle. In fact, nothing happens until the particle in set in motion. Figure 2.6 (bottom) shows an example. Suddenly, the particle perceives a force. We’ll get to the details about direction and magnitude in a moment, but the main idea is now evident. A magnetic ﬁeld is something that applies a force to a charged particle in motion, distinct from (in fact, in addition to) the force associated with an electric ﬁeld. Now, it is worth noting that a single charged particle in motion is the simplest form of current. Remember also that motion is required for the magnetic ﬁeld to inﬂuence the particle. Therefore, not only is current the source of the magnetic ﬁeld, the magnetic ﬁeld also exerts a force on current. Summarizing: B B v > F = F c⃝Y. Qin CC BY 4.0 Figure 2.6: The force perceived by a charged particle that is (top) motionless and (bottom) moving with ve- locity v = ˆzv, which is perpendicular to the plane of this document and toward the reader. The magnetic ﬁeld describes the force exerted on permanent magnets and currents in the presence of other permanent magnets and currents. So, how can we quantify a magnetic ﬁeld? The answer from classical physics involves another experimentally-derived equation that predicts force as a function of charge, velocity, and a vector ﬁeld B representing the magnetic ﬁeld. Here it is: The force applied to a particle bearing charge q is F = qv × B (2.9) where v is the velocity of the particle and “×” denotes the cross product. The cross product of two vectors is in a direction perpendicular to each of the two vectors, so the force exerted by the magnetic ﬁeld is perpendicular to both the direction of motion and the direction in which the magnetic ﬁeld points. The reader would be well-justiﬁed in wondering why the force exerted by the magnetic ﬁeld should perpendicular to B. For that matter, why should the force depend on v? These are questions for which classical physics provides no obvious answers. Effective answers to these questions require concepts from quantum mechanics, where we ﬁnd that the magnetic ﬁeld is a manifestation of the fundamental	Electromagnetics_Vol1_Page_38_Chunk1338
24 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS and aptly-named electromagnetic force. The electromagnetic force also gives rises to the electric ﬁeld, and it is only limited intuition, grounded in classical physics, that leads us to perceive the electric and magnetic ﬁelds as distinct phenomena. For our present purposes – and for most commonly-encountered engineering applications – we do not require these concepts. It is sufﬁcient to accept this apparent strangeness as fact and proceed accordingly. Dimensional analysis of Equation 2.9 reveals that B has units of (N·s)/(C·m). In SI, this combination of units is known as the tesla (T). We refer to B as magnetic ﬂux density, and therefore tesla is a unit of magnetic ﬂux density. A fair question to ask at this point is: What makes this a ﬂux density? The short answer is that this terminology is somewhat arbitrary, and in fact is not even uniformly accepted. In engineering electromagnetics, the preference for referring to B as a “ﬂux density” is because we frequently ﬁnd ourselves integrating B over a mathematical surface. Any quantity that is obtained by integration over a surface is referred to as “ﬂux,” and so it becomes natural to think of B as a ﬂux density; i.e., as ﬂux per unit area. The SI unit for magnetic ﬂux is the weber (Wb). Therefore, B may alternatively be described as having units of Wb/m2, and 1 Wb/m2 = 1 T. Magnetic ﬂux density (B, T or Wb/m2) is a de- scription of the magnetic ﬁeld that can be deﬁned as the solution to Equation 2.9. When describing magnetic ﬁelds, we occasionally refer to the concept of a ﬁeld line, deﬁned as follows: A magnetic ﬁeld line is the curve in space traced out by following the direction in which the mag- netic ﬁeld vector points. This concept is illustrated in Figure 2.7 for a permanent bar magnet and Figure 2.8 for a current-bearing coil. Magnetic ﬁeld lines are remarkable for the following reason: A magnetic ﬁeld line always forms a closed loop. c⃝Y. Qin CC BY 4.0 Figure 2.7: The magnetic ﬁeld of a bar magnet, illus- trating ﬁeld lines. c⃝Y. Qin CC BY 4.0 Figure 2.8: The magnetic ﬁeld of a current-bearing coil, illustrating ﬁeld lines.	Electromagnetics_Vol1_Page_39_Chunk1339
2.6. PERMEABILITY 25 This is true in a sense even for ﬁeld lines which seem to form straight lines (for example, those along the axis of the bar magnet and the coil in Figures 2.7 and 2.8), since a ﬁeld line that travels to inﬁnity in one direction reemerges from inﬁnity in the opposite direction. Additional Reading: • “Magnetic Field” on Wikipedia. 2.6 Permeability [m0009] Permeability describes the effect of material in determining the magnetic ﬂux density. All else being equal, magnetic ﬂux density increases in proportion to permeability. To illustrate the concept, consider that a particle bearing charge q moving at velocity v gives rise to a magnetic ﬂux density: B(r) = µ qv 4πR2 × ˆR (2.10) where ˆR is the unit vector pointing from the charged particle to the ﬁeld point r, R is this distance, and “×” is the cross product. Note that B increases with charge and speed, which makes sense since moving charge is the source of the magnetic ﬁeld. Also note that B is inversely proportional to 4πR2, indicating that \|B\| decreases in proportion to the area of a sphere surrounding the charge, also known as the inverse square law. The remaining factor, µ, is the constant of proportionality that captures the effect of material. We refer to µ as the permeability of the material. Since B can be expressed in units of Wb/m2 and the units of v are m/s, we see that µ must have units of henries per meter (H/m). (To see this, note that 1 H ≜1 Wb/A.) Permeability (µ, H/m) describes the effect of ma- terial in determining the magnetic ﬂux density. In free space, we ﬁnd that the permeability µ = µ0 where: µ0 = 4π × 10−7 H/m (2.11) It is common practice to describe the permeability of materials in terms of their relative permeability: µr ≜µ µ0 (2.12) which gives the permeability relative to the minimum possible value; i.e., that of free space. Relative permeability for a few representative materials is given in Appendix A.2. Note that µr is approximately 1 for all but a small class of materials. These are known as magnetic	Electromagnetics_Vol1_Page_40_Chunk1340
26 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS materials, and may exhibit values of µr as large as ∼106. A commonly-encountered category of magnetic materials is ferromagnetic material, of which the best-known example is iron. Additional Reading: • “Permeability (electromagnetism)” on Wikipedia. • Section 7.16 (“Magnetic Materials”). • Appendix A.2 (“Permeability of Some Common Materials”). 2.7 Magnetic Field Intensity [m0012] Magnetic ﬁeld intensity H is an alternative description of the magnetic ﬁeld in which the effect of material is factored out. For example, the magnetic ﬂux density B (reminder: Section 2.5) due to a point charge q moving at velocity v can be written in terms of the Biot-Savart Law: B = µ qv 4πR2 × ˆR (2.13) where ˆR is the unit vector pointing from the charged particle to the ﬁeld point r, R is this distance, “×” is the cross product, and µ is the permeability of the material. We can rewrite Equation 2.13 as: B ≜µH (2.14) with: H = qv 4πR2 × ˆR (2.15) so H in homogeneous media does not depend on µ. Dimensional analysis of Equation 2.15 reveals that the units for H are amperes per meter (A/m). However, H does not represent surface current density,2 as the units might suggest. While it is certainly true that a distribution of current (A) over some linear cross-section (m) can be described as a current density having units of A/m, H is associated with the magnetic ﬁeld and not a particular current distribution. Said differently, H can be viewed as a description of the magnetic ﬁeld in terms of an equivalent (but not actual) current. The magnetic ﬁeld intensity H (A/m), deﬁned us- ing Equation 2.14, is a description of the mag- netic ﬁeld independent from material properties. It may appear that H is redundant information given B and µ, but this is true only in homogeneous media. The concept of magnetic ﬁeld intensity becomes important – and decidedly not redundant – when we encounter boundaries between media having different permeabilities. As we shall see in Section 7.11, 2The concept of current density is not essential to understand this section; however, a primer can be found in Section 6.2.	Electromagnetics_Vol1_Page_41_Chunk1341
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS 27 boundary conditions on H constrain the component of the magnetic ﬁeld which is tangent to the boundary separating two otherwise-homogeneous regions. If one ignores the characteristics of the magnetic ﬁeld represented by H and instead considers only B, then only the perpendicular component of the magnetic ﬁeld is constrained. The concept of magnetic ﬁeld intensity also turns out to be useful in a certain problems in which µ is not a constant, but rather is a function of magnetic ﬁeld strength. In this case, the magnetic behavior of the material is said to be nonlinear. For more on this, see Section 7.16. Additional Reading: • “Magnetic ﬁeld” on Wikipedia. • “Biot-Savart law” on Wikipedia. 2.8 Electromagnetic Properties of Materials [m0007] In electromagnetic analysis, one is principally concerned with three properties of matter. These properties are quantiﬁed in terms of constitutive parameters, which describe the effect of material in determining an electromagnetic quantity in response to a source. Here are the three principal constitutive parameters: • Permittivity (ǫ, F/m) quantiﬁes the effect of matter in determining the electric ﬁeld in response to electric charge. Permittivity is addressed in Section 2.3. • Permeability (µ, H/m) quantiﬁes the effect of matter in determining the magnetic ﬁeld in response to current. Permeability is addressed in Section 2.6. • Conductivity (σ, S/m) quantiﬁes the effect of matter in determining the ﬂow of current in response to an electric ﬁeld. Conductivity is addressed in Section 6.3. The electromagnetic properties of most common materials in most common applications can be quantiﬁed in terms of the constitutive parameters ǫ, µ, and σ. To keep electromagnetic theory from becoming too complex, we usually require the constitutive parameters to exhibit a few basic properties. These properties are as follows: • Homogeneity. A material that is homogeneous is uniform over the space it occupies; that is, the values of its constitutive parameters are constant at all locations within the material. A counter-example would be a material that is composed of multiple chemically-distinct compounds that are not thoroughly mixed, such as soil.	Electromagnetics_Vol1_Page_42_Chunk1342
28 CHAPTER 2. ELECTRIC AND MAGNETIC FIELDS • Isotropy. A material that is isotropic behaves in precisely the same way regardless of how it is oriented with respect to sources and ﬁelds occupying the same space. A counter-example is quartz, whose atoms are arranged in a uniformly-spaced crystalline lattice. As a result, the electromagnetic properties of quartz can be changed simply by rotating the material with respect to the applied sources and ﬁelds. • Linearity. A material is said to be linear if its properties are constant and independent of the magnitude of the sources and ﬁelds applied to the material. For example, capacitors have capacitance, which is determined in part by the permittivity of the material separating the terminals (Section 5.23). This material is approximately linear when the applied voltage V is below the rated working voltage; i.e., ǫ is constant and so capacitance does not vary signiﬁcantly with respect to V . When V is greater than the working voltage, the dependence of ǫ on V becomes more pronounced, and then capacitance becomes a function of V . In another practical example, it turns out that µ for ferromagnetic materials is nonlinear such that the precise value of µ depends on the magnitude of the magnetic ﬁeld. • Time-invariance. An example of a class of materials that is not necessarily time-invariant is piezoelectric materials, for which electromagnetic properties vary signiﬁcantly depending on the mechanical forces applied to them – a property which can be exploited to make sensors and transducers. Linearity and time-invariance (LTI) are particularly important properties to consider because they are requirements for superposition. For example, in a LTI material, we may calculate the ﬁeld E1 due to a point charge q1 at r1 and calculate the ﬁeld E2 due to a point charge q2 at r2. Then, when both charges are simultaneously present, the ﬁeld is E1 + E2. The same is not necessarily true for materials that are not LTI. Devices that are nonlinear, and therefore not LTI, do not necessarily follow the rules of elementary circuit theory, which presume that superposition applies. This condition makes analysis and design much more difﬁcult. No practical material is truly homogeneous, isotropic, linear, and time-invariant. However, for most materials in most applications, the deviation from this ideal condition is not large enough to signiﬁcantly affect engineering analysis and design. In other cases, materials may be signiﬁcantly non-ideal in one of these respects, but may still be analyzed with appropriate modiﬁcations to the theory. [m0054]	Electromagnetics_Vol1_Page_43_Chunk1343
2.8. ELECTROMAGNETIC PROPERTIES OF MATERIALS 29 Image Credits Fig. 2.1: c⃝M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fTwoChargedParticles.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 2.2: c⃝M. Goldammer, https://commons.wikimedia.org/wiki/File:M0002 fForceMap.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 2.3: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0002 fBatCap.svg, CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/). Fig. 2.4: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fBarMagnet.svg, CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/). Fig. 2.5: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fCoilBarMagnet.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Fig. 2.6: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fFqvB.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Fig. 2.7: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:M0003 fFieldLinesBarMagnet.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Fig. 2.8: c⃝Y. Qin, https://commons.wikimedia.org/wiki/File:0003 fFieldLinesCoil.svg, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).	Electromagnetics_Vol1_Page_44_Chunk1344
Chapter 3 Transmission Lines 3.1 Introduction to Transmission Lines [m0028] A transmission line is a structure intended to transport electromagnetic signals or power. A rudimentary transmission line is simply a pair of wires with one wire serving as a datum (i.e., a reference; e.g., “ground”) and the other wire bearing an electrical potential that is deﬁned relative to that datum. Transmission lines having random geometry, such as the test leads shown in Figure 3.1, are useful only at very low frequencies and when loss, reactance, and immunity to electromagnetic interference (EMI) are not a concern. by Dmitry G Figure 3.1: These leads used to connect test equip- ment to circuits in a laboratory are a very rudimentary form of transmission line, suitable only for very low frequencies. However, many circuits and systems operate at frequencies where the length or cross-sectional dimensions of the transmission line may be a signiﬁcant fraction of a wavelength. In this case, the transmission line is no longer “transparent” to the circuits at either end. Furthermore, loss, reactance, and EMI are signiﬁcant problems in many applications. These concerns motivate the use of particular types of transmission lines, and make it necessary to understand how to properly connect the transmission line to the rest of the system. In electromagnetics, the term “transmission line” refers to a structure which is intended to support a guided wave. A guided wave is an electromagnetic wave that is contained within or bound to the line, and which does not radiate away from the line. This condition is normally met if the length and cross-sectional dimensions of the transmission line are small relative to a wavelength – say λ/100 (i.e., 1% of the wavelength). For example, two randomly-arranged wires might serve well enough to carry a signal at f = 10 MHz over a length l = 3 cm, since l is only 0.1% of the wavelength λ = c/f = 30 m. However, if l is increased to 3 m, or if f is increased to 1 GHz, then l is now 10% of the wavelength. In this case, one should consider using a transmission line that forms a proper guided wave. Preventing unintended radiation is not the only concern. Once we have established a guided wave on a transmission line, it is important that power applied to the transmission line be delivered to the circuit or device at the other end and not reﬂected back into the source. For the random wire f = 10 MHz, l = 3 cm example above, there is little need for concern, since we expect a phase shift of roughly Electromagnetics Vol 1. c⃝2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1	Electromagnetics_Vol1_Page_45_Chunk1345
3.2. TYPES OF TRANSMISSION LINES 31 0.001 · 360◦= 0.36◦over the length of the transmission line, which is about 0.72◦for a round trip. So, to a good approximation, the entire transmission line is at the same electrical potential and thus transparent to the source and destination. However, if l is increased to 3 m, or if f is increased to 1 GHz, then the associated round-trip phase shift becomes 72◦. In this case, a reﬂected signal traveling in the opposite direction will add to create a total electrical potential, which varies in both magnitude and phase with position along the line. Thus, the impedance looking toward the destination via the transmission line will be different than the impedance looking toward the destination directly. (Section 3.15 gives the details.) The modiﬁed impedance will depend on the cross-sectional geometry, materials, and length of the line. Cross-sectional geometry and materials also determine the loss and EMI immunity of the transmission line. Summarizing: Transmission lines are designed to support guided waves with controlled impedance, low loss, and a degree of immunity from EMI. 3.2 Types of Transmission Lines [m0144] Two common types of transmission line are coaxial line (Figure 3.2) and microstrip line (Figure 3.3). Both are examples of transverse electromagnetic (TEM) transmission lines. A TEM line employs a single electromagnetic wave “mode” having electric and magnetic ﬁeld vectors in directions perpendicular to the axis of the line, as shown in Figures 3.4 and 3.5. TEM transmission lines appear primarily in radio frequency applications. TEM transmission lines such as coaxial lines and microstrip lines are designed to support a single electromagnetic wave that propagates along the length of the transmission line with electric and magnetic ﬁeld vectors perpendicular to the direc- tion of propagation. Not all transmission lines exhibit TEM ﬁeld structure. In non-TEM transmission lines, the electric and magnetic ﬁeld vectors that are not necessarily perpendicular to the axis of the line, and the structure of the ﬁelds is complex relative to the ﬁeld structure of TEM lines. An example of a transmission line that exhibits non-TEM ﬁeld structure is the waveguide (see example in Figure 3.6). Waveguides are most prevalent at radio frequencies, and tend to appear in applications where it is important to achieve very low loss or where power levels are very high. Another example is common “multimode” optical ﬁber (Figure 3.7). Optical ﬁber exhibits complex ﬁeld c⃝Tkgd2007 CC BY 3.0 (modiﬁed) Figure 3.2: Structure of a coaxial transmission line.	Electromagnetics_Vol1_Page_46_Chunk1346
32 CHAPTER 3. TRANSMISSION LINES g ound plane dielectric slab metallic trace c⃝SpinningSpark CC BY SA 3.0 (modiﬁed) Figure 3.3: Structure of a microstrip transmission line. Figure 3.4: Structure of the electric and magnetic ﬁelds within coaxial line. In this case, the wave is propagating away from the viewer. Figure 3.5: Structure of the electric and magnetic ﬁelds within microstrip line. (The ﬁelds outside the line are possibly signiﬁcant, complicated, and not shown.) In this case, the wave is propagating away from the viewer. c⃝Averse CC BY SA 2.0 Germany Figure 3.6: A network of radio frequency waveguides in an air trafﬁc control radar. structure because the wavelength of light is very small compared to the cross-section of the ﬁber, making the excitation and propagation of non-TEM waves difﬁcult to avoid. (This issue is overcome in a different type of optical ﬁber, known as “single mode” ﬁber, which is much more difﬁcult and expensive to manufacture.) Higher-order transmission lines, including radio- frequency waveguides and multimode optical ﬁber, are designed to guide waves that have rela- tively complex structure. Additional Reading: • “Coaxial cable” on Wikipedia. • “Microstrip” on Wikipedia. • “Waveguide (electromagnetism)” on Wikipedia. • “Optical ﬁber” on Wikipedia.	Electromagnetics_Vol1_Page_47_Chunk1347
3.3. TRANSMISSION LINES AS TWO-PORT DEVICES 33 c⃝BigRiz CC BY SA 3.0 Unported Figure 3.7: Strands of optical ﬁber. 3.3 Transmission Lines as Two-Port Devices [m0077] Figure 3.8 shows common ways to represent transmission lines in circuit diagrams. In each case, the source is represented using a Th´evenin equivalent circuit consisting of a voltage source VS in series with an impedance ZS.1 In transmission line analysis, the source may also be referred to as the generator. The termination on the receiving end of the transmission line is represented, without loss of generality, as an impedance ZL. This termination is often referred to as the load, although in practice it can be any circuit that exhibits an input impedance of ZL. The two-port representation of a transmission line is completely described by its length l along with some combination of the following parameters: • Phase propagation constant β, having units of rad/m. This parameter also represents the wavelength in the line through the relationship λ = 2π/β. (See Sections 1.3 and 3.8 for details.) • Attenuation constant α, having units of 1/m. 1For a refresher on this concept, see “Additional Reading” at the end of this section. This parameter quantiﬁes the effect of loss in the line. (See Section 3.8 for details.) • Characteristic impedance Z0, having units of Ω. This is the ratio of potential (“voltage”) to current when the line is perfectly impedance-matched at both ends. (See Section 3.7 for details.) These parameters depend on the materials and geometry of the line. Note that a transmission line is typically not transparent to the source and load. In particular, the load impedance may be ZL, but the impedance presented to the source may or may not be equal to ZL. (See Section 3.15 for more on this concept.) Similarly, the source impedance may be ZS, but the impedance presented to the load may or may not be equal to ZS. The effect of the transmission line on the source and load impedances will depend on the parameters identiﬁed above. ZL ZS VS VS VS ZS ZS ZL ZL l l c⃝Omegatron CC BY SA 3.0 Unported (modiﬁed) Figure 3.8: Symbols representing transmission lines: Top: As a generic two-conductor direct connection. Middle: As a generic two-port “black box.” Bottom: As a coaxial cable.	Electromagnetics_Vol1_Page_48_Chunk1348
34 CHAPTER 3. TRANSMISSION LINES Additional Reading: • “Th´evenin’s theorem” on Wikipedia. 3.4 Lumped-Element Model [m0029] It is possible to ascertain the relevant behaviors of a transmission line using elementary circuit theory applied to a differential-length lumped-element model of the transmission line. The concept is illustrated in Figure 3.9, which shows a generic transmission line aligned with its length along the z axis. The transmission line is divided into segments having small but ﬁnite length ∆z. Each segment is modeled as an identical two-port having the equivalent circuit representation shown in Figure 3.10. The equivalent circuit consists of 4 components as follows: • The resistance R′∆z represents the series-combined ohmic resistance of the two conductors. This should account for both conductors since the current in the actual transmission line must ﬂow through both conductors. The prime notation reminds us that R′ is resistance per unit length; i.e., Ω/m, and it is only after multiplying by length that we get a resistance in Ω. • The conductance G′∆z represents the leakage of current directly from one conductor to the other. When G′∆z > 0, the resistance between the conductors is less than inﬁnite, and therefore, current may ﬂow between the conductors. This amounts to a loss of power separate from the loss associated with R′ above. G′ has units of S/m. Further note that G′ is not equal to 1/R′ as deﬁned above. G′ and R′ are describing entirely different physical mechanisms (and in principle either could be deﬁned as either a resistance or a conductance). • The capacitance C′∆z represents the capacitance of the transmission line structure. Capacitance is the tendency to store energy in electric ﬁelds and depends on the cross-sectional geometry and the media separating the conductors. C′ has units of F/m. • The inductance L′∆z represents the inductance of the transmission line structure. Inductance is the tendency to store energy in magnetic ﬁelds, and (like capacitance) depends on the	Electromagnetics_Vol1_Page_49_Chunk1349
3.5. TELEGRAPHER’S EQUATIONS 35 z z z z z Figure 3.9: Interpretation of a transmission line as a cascade of discrete series-connected two-ports. R'Δz L'Δz G'Δz C'Δz c⃝Omegatron CC BY SA 3.0 Unported (modiﬁed) Figure 3.10: Lumped-element equivalent circuit model for each of the two-ports in Figure 3.9. cross-sectional geometry and the media separating the conductors. L′ has units of H/m. In order to use the model, one must have values for R′, G′, C′, and L′. Methods for computing these parameters are addressed elsewhere in this book. 3.5 Telegrapher’s Equations [m0079] In this section, we derive the equations that govern the potential v(z, t) and current i(z, t) along a transmission line that is oriented along the z axis. For this, we will employ the lumped-element model developed in Section 3.4. To begin, we deﬁne voltages and currents as shown in Figure 3.11. We assign the variables v(z, t) and i(z, t) to represent the potential and current on the left side of the segment, with reference polarity and direction as shown in the ﬁgure. Similarly we assign the variables v(z + ∆z, t) and i(z + ∆z, t) to represent the potential and current on the right side of the segment, again with reference polarity and direction as shown in the ﬁgure. Applying Kirchoff’s voltage law from the left port, through R′∆z and L′∆z, and returning via the right port, we obtain: v(z, t) −(R′∆z) i(z, t) −(L′∆z) ∂ ∂ti(z, t) −v(z + ∆z, t) = 0 (3.1) Moving terms referring to current to the right side of the equation and then dividing through by ∆z, we obtain −v(z + ∆z, t) −v(z, t) ∆z = R′ i(z, t) + L′ ∂ ∂ti(z, t) (3.2) Then taking the limit as ∆z →0: −∂ ∂z v(z, t) = R′ i(z, t) + L′ ∂ ∂ti(z, t) (3.3) R'Δz L'Δz G'Δz C'Δz v(z,t) i(z+Δz,t) i(z,t) + _ + _ v(z+Δz,t) c⃝Omegatron CC BY SA 3.0 Unported (modiﬁed) Figure 3.11: Lumped-element equivalent circuit transmission line model, annotated with sign conven- tions for potentials and currents.	Electromagnetics_Vol1_Page_50_Chunk1350
36 CHAPTER 3. TRANSMISSION LINES Applying Kirchoff’s current law at the right port, we obtain: i(z, t)−(G′∆z) v(z+∆z, t)−(C′∆z) ∂ ∂tv(z+∆z, t) −i(z + ∆z, t) = 0 (3.4) Moving terms referring to potential to the right side of the equation and then dividing through by ∆z, we obtain −i(z + ∆z, t) −i(z, t) ∆z = G′ v(z + ∆z, t) + C′ ∂ ∂tv(z + ∆z, t) (3.5) Taking the limit as ∆z →0: −∂ ∂z i(z, t) = G′ v(z, t) + C′ ∂ ∂tv(z, t) (3.6) Equations 3.3 and 3.6 are the telegrapher’s equa- tions. These coupled (simultaneous) differential equations can be solved for v(z, t) and i(z, t) given R′, G′, L′, C′ and suitable boundary con- ditions. The time-domain telegrapher’s equations are usually more than we need or want. If we are only interested in the response to a sinusoidal stimulus, then considerable simpliﬁcation is possible using phasor representation.2 First we deﬁne phasors eV (z) and eI(z) through the usual relationship: v(z, t) = Re n eV (z) ejωto (3.7) i(z, t) = Re n eI(z) ejωto (3.8) Now we see: ∂ ∂z v(z, t) = ∂ ∂z Re n eV (z) ejωto = Re ∂ ∂z eV (z) ejωt 2For a refresher on phasor analysis, see Section 1.5. In other words, ∂v(z, t)/∂z expressed in phasor representation is simply ∂eV (z)/∂z; and ∂ ∂ti(z, t) = ∂ ∂tRe n eI(z) ejωto = Re ∂ ∂t h eI(z)ejωti = Re nh jωeI(z) i ejωto In other words, ∂i(z, t)/∂t expressed in phasor representation is jωeI(z). Therefore, Equation 3.3 expressed in phasor representation is: −∂ ∂z eV (z) = [R′ + jωL′] eI(z) (3.9) Following the same procedure, Equation 3.6 expressed in phasor representation is found to be: −∂ ∂z eI(z) = [G′ + jωC′] eV (z) (3.10) Equations 3.9 and 3.10 are the telegrapher’s equations in phasor representation. The principal advantage of these equations over the time-domain versions is that we no longer need to contend with derivatives with respect to time – only derivatives with respect to distance remain. This considerably simpliﬁes the equations. Additional Reading: • “Telegrapher’s equations” on Wikipedia. • “Kirchhoff’s circuit laws” on Wikipedia.	Electromagnetics_Vol1_Page_51_Chunk1351
3.6. WAVE EQUATION FOR A TEM TRANSMISSION LINE 37 3.6 Wave Equation for a TEM Transmission Line [m0027] Consider a TEM transmission line aligned along the z axis. The phasor form of the Telegrapher’s Equations (Section 3.5) relate the potential phasor eV (z) and the current phasor eI(z) to each other and to the lumped-element model equivalent circuit parameters R′, G′, C′, and L′. These equations are −∂ ∂z eV (z) = [R′ + jωL′] eI(z) (3.11) −∂ ∂z eI(z) = [G′ + jωC′] eV (z) (3.12) An obstacle to using these equations is that we require both equations to solve for either the potential or the current. In this section, we reduce these equations to a single equation – a wave equation – that is more convenient to use and provides some additional physical insight. We begin by differentiating both sides of Equation 3.11 with respect to z, yielding: −∂2 ∂z2 eV (z) = [R′ + jωL′] ∂ ∂z eI(z) (3.13) Then using Equation 3.12 to eliminate eI(z), we obtain −∂2 ∂z2 eV (z) = −[R′ + jωL′] [G′ + jωC′] eV (z) (3.14) This equation is normally written as follows: ∂2 ∂z2 eV (z) −γ2 eV (z) = 0 (3.15) where we have made the substitution: γ2 = (R′ + jωL′) (G′ + jωC′) (3.16) The principal square root of γ2 is known as the propagation constant: γ ≜ p (R′ + jωL′) (G′ + jωC′) (3.17) The propagation constant γ (units of m−1) cap- tures the effect of materials, geometry, and fre- quency in determining the variation in potential and current with distance on a TEM transmission line. Following essentially the same procedure but beginning with Equation 3.12, we obtain ∂2 ∂z2 eI(z) −γ2 eI(z) = 0 (3.18) Equations 3.15 and 3.18 are the wave equations for eV (z) and eI(z), respectively. Note that both eV (z) and eI(z) satisfy the same linear homogeneous differential equation. This does not mean that eV (z) and eI(z) are equal. Rather, it means that eV (z) and eI(z) can differ by no more than a multiplicative constant. Since eV (z) is potential and eI(z) is current, that constant must be an impedance. This impedance is known as the characteristic impedance and is determined in Section 3.7. The general solutions to Equations 3.15 and 3.18 are eV (z) = V + 0 e−γz + V − 0 e+γz (3.19) eI(z) = I+ 0 e−γz + I− 0 e+γz (3.20) where V + 0 , V − 0 , I+ 0 , and I− 0 are complex-valued constants. It is shown in Section 3.8 that Equations 3.19 and 3.20 represent sinusoidal waves propagating in the +z and −z directions along the length of the line. The constants may represent sources, loads, or simply discontinuities in the materials and/or geometry of the line. The values of the constants are determined by boundary conditions; i.e., constraints on eV (z) and eI(z) at some position(s) along the line. The reader is encouraged to verify that the Equations 3.19 and 3.20 are in fact solutions to Equations 3.15 and 3.18, respectively, for any values of the constants V + 0 , V − 0 , I+ 0 , and I− 0 .	Electromagnetics_Vol1_Page_52_Chunk1352
38 CHAPTER 3. TRANSMISSION LINES 3.7 Characteristic Impedance [m0052] Characteristic impedance is the ratio of voltage to current for a wave that is propagating in single direction on a transmission line. This is an important parameter in the analysis and design of circuits and systems using transmission lines. In this section, we formally deﬁne this parameter and derive an expression for this parameter in terms of the equivalent circuit model introduced in Section 3.4. Consider a transmission line aligned along the z axis. Employing some results from Section 3.6, recall that the phasor form of the wave equation in this case is ∂2 ∂z2 eV (z) −γ2 eV (z) = 0 (3.21) where γ ≜ p (R′ + jωL′) (G′ + jωC′) (3.22) Equation 3.21 relates the potential phasor eV (z) to the equivalent circuit parameters R′, G′, C′, and L′. An equation of the same form relates the current phasor eI(z) to the equivalent circuit parameters: ∂2 ∂z2 eI(z) −γ2 eI(z) = 0 (3.23) Since both eV (z) and eI(z) satisfy the same linear homogeneous differential equation, they may differ by no more than a multiplicative constant. Since eV (z) is potential and eI(z) is current, that constant can be expressed in units of impedance. Speciﬁcally, this is the characteristic impedance, so-named because it depends only on the materials and cross-sectional geometry of the transmission line – i.e., things which determine γ – and not length, excitation, termination, or position along the line. To derive the characteristic impedance, ﬁrst recall that the general solutions to Equations 3.21 and 3.23 are eV (z) = V + 0 e−γz + V − 0 e+γz (3.24) eI(z) = I+ 0 e−γz + I− 0 e+γz (3.25) where V + 0 , V − 0 , I+ 0 , and I− 0 are complex-valued constants whose values are determined by boundary conditions; i.e., constraints on eV (z) and eI(z) at some position(s) along the line. Also, we will make use of the telegrapher’s equations (Section 3.5): −∂ ∂z eV (z) = [R′ + jωL′] eI(z) (3.26) −∂ ∂z eI(z) = [G′ + jωC′] eV (z) (3.27) We begin by differentiating Equation 3.24 with respect to z, which yields ∂ ∂z eV (z) = −γ V + 0 e−γz −V − 0 e+γz (3.28) Now we use this this to eliminate ∂eV (z)/∂z in Equation 3.26, yielding γ V + 0 e−γz −V − 0 e+γz = [R′ + jωL′] eI(z) (3.29) Solving the above equation for eI(z) yields: eI(z) = γ R′ + jωL′ V + 0 e−γz −V − 0 e+γz (3.30) Comparing this to Equation 3.25, we note I+ 0 = γ R′ + jωL′ V + 0 (3.31) I− 0 = −γ R′ + jωL′ V − 0 (3.32) We now make the substitution Z0 = R′ + jωL′ γ (3.33) and observe V + 0 I+ 0 = −V − 0 I− 0 ≜Z0 (3.34) As anticipated, we have found that coefﬁcients in the equations for potentials and currents are related by an impedance, namely, Z0. Characteristic impedance can be written entirely in terms of the equivalent circuit parameters by substituting Equation 3.22 into Equation 3.33, yielding: Z0 = s R′ + jωL′ G′ + jωC′ (3.35)	Electromagnetics_Vol1_Page_53_Chunk1353
3.8. WAVE PROPAGATION ON A TEM TRANSMISSION LINE 39 The characteristic impedance Z0 (Ω) is the ratio of potential to current in a wave traveling in a single direction along the transmission line. Take care to note that Z0 is not the ratio of eV (z) to eI(z) in general; rather, Z0 relates only the potential and current waves traveling in the same direction. Finally, note that transmission lines are normally designed to have a characteristic impedance that is completely real-valued – that is, with no imaginary component. This is because the imaginary component of an impedance represents energy storage (think of capacitors and inductors), whereas the purpose of a transmission line is energy transfer. Additional Reading: • “Characteristic impedance” on Wikipedia. 3.8 Wave Propagation on a TEM Transmission Line [m0080] In Section 3.6, it is shown that expressions for the phasor representations of the potential and current along a transmission line are eV (z) = V + 0 e−γz + V − 0 e+γz (3.36) eI(z) = I+ 0 e−γz + I− 0 e+γz (3.37) where γ is the propagation constant and it assumed that the transmission line is aligned along the z axis. In this section, we demonstrate that these expressions represent sinusoidal waves, and point out some important features. Before attempting this section, the reader should be familiar with the contents of Sections 3.4, 3.6, and 3.7. A refresher on fundamental wave concepts (Section 1.3) may also be helpful. We ﬁrst deﬁne real-valued quantities α and β to be the real and imaginary components of γ; i.e., α ≜Re {γ} (3.38) β ≜Im {γ} (3.39) and subsequently γ = α + jβ (3.40) Then we observe e±γz = e±(α+jβ)z = e±αz e±jβz (3.41) It may be easier to interpret this expression by reverting to the time domain: Re e±γzejωt = e±αz cos (ωt ± βz) (3.42) Thus, e−γz represents a damped sinusoidal wave traveling in the +z direction, and e+γz represents a damped sinusoidal wave traveling in the −z direction. Let’s deﬁne eV +(z) and eI+(z) to be the potential and current associated with a wave propagating in the +z direction. Then: eV +(z) ≜V + 0 e−γz (3.43)	Electromagnetics_Vol1_Page_54_Chunk1354
40 CHAPTER 3. TRANSMISSION LINES z v+(z,t=0) Figure 3.12: The potential v+(z, t) of the wave travel- ing in the +z direction at t = 0 for ψ = 0. or equivalently in the time domain: v+(z, t) = Re n eV +(z) ejωto = Re V + 0 e−γzejωt = V + 0 e−αz cos (ωt −βz + ψ) (3.44) where ψ is the phase of V + 0 . Figure 3.12 shows v+(z, t). From fundamental wave theory we recognize β ≜ Im {γ} (rad/m) is the phase propaga- tion constant, which is the rate at which phase changes as a function of distance. Subsequently the wavelength in the line is λ = 2π β (3.45) Also we recognize: α ≜Re {γ} (1/m) is the attenuation constant, which is the rate at which magnitude diminishes as a function of distance. Sometimes the units of α are indicated as “Np/m” (“nepers” per meter), where the term “neper” is used to indicate the units of the otherwise unitless real-valued exponent of the constant e. Note that α = 0 for a wave that does not diminish in magnitude with increasing distance, in which case the transmission line is said to be lossless. If α > 0 then the line is said to be lossy (or possibly “low loss” if the loss can be neglected), and in this case the rate at which the magnitude decreases with distance increases with α. Next let us consider the speed of the wave. To answer this question, we need to be a bit more speciﬁc about what we mean by “speed.” At the moment, we mean phase velocity; that is, the speed at which a point of constant phase seems to move through space. In other words, what distance ∆z does a point of constant phase traverse in time ∆t? To answer this question, we ﬁrst note that the phase of v+(z, t) can be written generally as ωt −βz + φ where φ is some constant. Similarly, the phase at some time ∆t later and some point ∆z further along can be written as ω (t + ∆t) −β (z + ∆z) + φ The phase velocity vp is ∆z/∆t when these two phases are equal; i.e., when ωt −βz + φ = ω (t + ∆t) −β (z + ∆z) + φ Solving for vp = ∆z/∆t, we obtain: vp = ω β (3.46) Having previously noted that β = 2π/λ, the above expression also yields the expected result vp = λf (3.47) The phase velocity vp = ω/β = λf is the speed at which a point of constant phase travels along the line. Returning now to consider the current associated with the wave traveling in the +z direction: eI+(z) = I+ 0 e−γz (3.48) We can rewrite this expression in terms of the characteristic impedance Z0, as follows: eI+(z) = V + 0 Z0 e−γz (3.49)	Electromagnetics_Vol1_Page_55_Chunk1355
3.9. LOSSLESS AND LOW-LOSS TRANSMISSION LINES 41 Similarly, we ﬁnd that the current eI−(z) associated with eV −(z) for the wave traveling in the −z direction is eI−(z) = −V − 0 Z0 e−γz (3.50) The negative sign appearing in the above expression emerges as a result of the sign conventions used for potential and current in the derivation of the telegrapher’s equations (Section 3.5). The physical signiﬁcance of this change of sign is that wherever the potential of the wave traveling in the −z direction is positive, then the current at the same point is ﬂowing in the −z direction. It is frequently necessary to consider the possibility that waves travel in both directions simultaneously. A very important case where this arises is when there is reﬂection from a discontinuity of some kind; e.g., from a termination which is not perfectly impedance-matched. In this case, the total potential eV (z) and total current eI(z) can be expressed as the general solution to the wave equation; i.e., as the sum of the “incident” (+z-traveling) wave and the reﬂected (−z-traveling) waves: eV (z) = eV +(z) + eV −(z) (3.51) eI(z) = eI+(z) + eI−(z) (3.52) The existence of waves propagating simultaneously in both directions gives rise to a phenomenon known as a standing wave. Standing waves and the calculation of the coefﬁcients V − 0 and I− 0 due to reﬂection are addressed in Sections 3.13 and 3.12 respectively. 3.9 Lossless and Low-Loss Transmission Lines [m0083] Quite often the loss in a transmission line is small enough that it may be neglected. In this case, several aspects of transmission line theory may be simpliﬁed. In this section, we present these simpliﬁcations. First, recall that “loss” refers to the reduction of magnitude as a wave propagates through space. In the lumped-element equivalent circuit model (Section 3.4), the parameters R′ and G′ of the represent physical mechanisms associated with loss. Speciﬁcally, R′ represents the resistance of conductors, whereas G′ represents the undesirable current induced between conductors through the spacing material. Also recall that the propagation constant γ is, in general, given by γ ≜ p (R′ + jωL′) (G′ + jωC′) (3.53) With this in mind, we now deﬁne “low loss” as meeting the conditions: R′ ≪ωL′ (3.54) G′ ≪ωC′ (3.55) When these conditions are met, the propagation constant simpliﬁes as follows: γ ≈ p (jωL′) (jωC′) = p −ω2L′C′ = jω √ L′C′ (3.56) and subsequently α ≜Re {γ} ≈0 (low-loss approx.) (3.57) β ≜Im {γ} ≈ω √ L′C′ (low-loss approx.) (3.58) vp = ω/β ≈ 1 √ L′C′ (low-loss approx.) (3.59) Similarly: Z0 = s R′ + jωL′ G′ + jωC′ ≈ r L′ C′ (low-loss approx.) (3.60)	Electromagnetics_Vol1_Page_56_Chunk1356
42 CHAPTER 3. TRANSMISSION LINES Of course if the line is strictly lossless (i.e., R′ = G′ = 0) then these are not approximations, but rather the exact expressions. In practice, these approximations are quite commonly used, since practical transmission lines typically meet the conditions expressed in Inequalities 3.54 and 3.55 and the resulting expressions are much simpler. We further observe that Z0 and vp are approximately independent of frequency when these conditions hold. However, also note that “low loss” does not mean “no loss,” and it is common to apply these expressions even when R′ and/or G′ is large enough to yield signiﬁcant loss. For example, a coaxial cable used to connect an antenna on a tower to a radio near the ground typically has loss that is important to consider in the analysis and design process, but nevertheless satisﬁes Equations 3.54 and 3.55. In this case, the low-loss expression for β is used, but α might not be approximated as zero. s σs b a Figure 3.13: Cross-section of a coaxial transmission line, indicating design parameters. 3.10 Coaxial Line [m0143] Coaxial transmission lines consists of metallic inner and outer conductors separated by a spacer material as shown in Figure 3.13. The spacer material is typically a low-loss dielectric material having permeability approximately equal to that of free space (µ ≈µ0) and permittivity ǫs that may range from very near ǫ0 (e.g., air-ﬁlled line) to 2–3 times ǫ0. The outer conductor is alternatively referred to as the “shield,” since it typically provides a high degree of isolation from nearby objects and electromagnetic ﬁelds. Coaxial line is single-ended3 in the sense that the conductor geometry is asymmetric and the shield is normally attached to ground at both ends. These characteristics make coaxial line attractive for connecting single-ended circuits in widely-separated locations and for connecting antennas to receivers and transmitters. Coaxial lines exhibit TEM ﬁeld structure as shown in Figure 3.14. Expressions for the equivalent circuit parameters C′ and L′ for coaxial lines can be obtained from basic electromagnetic theory. It is shown in Section 5.24 that the capacitance per unit length is C′ = 2πǫs ln (b/a) (3.61) where a and b are the radii of the inner and outer conductors, respectively. Using analysis shown in 3The references in “Additional Reading” at the end of this section may be helpful if you are not familiar with this concept.	Electromagnetics_Vol1_Page_57_Chunk1357
3.10. COAXIAL LINE 43 Figure 3.14: Structure of the electric and magnetic ﬁelds within coaxial line. In this case, the wave is propagating away from the viewer. Section 7.14, the inductance per unit length is L′ = µ0 2π ln b a (3.62) The loss conductance G′ depends on the conductance σs of the spacer material, and is given by G′ = 2πσs ln (b/a) (3.63) This expression is derived in Section 6.5. The resistance per unit length, R′, is relatively difﬁcult to quantify. One obstacle is that the inner and outer conductors typically consist of different materials or compositions of materials. The inner conductor is not necessarily a single homogeneous material; instead, the inner conductor may consist of a variety of materials selected by trading-off between conductivity, strength, weight, and cost. Similarly, the outer conductor is not necessarily homogeneous; for a variety of reasons, the outer conductor may instead be a metal mesh, a braid, or a composite of materials. Another complicating factor is that the resistance of the conductor varies signiﬁcantly with frequency, whereas C′, L′, and G′ exhibit relatively little variation from their electro- and magnetostatic values. These factors make it difﬁcult to devise a single expression for R′ that is both as simple as those shown above for the other parameters and generally applicable. Fortunately, it turns out that the low-loss conditions R′ ≪ωL′ and G′ ≪ωC′ are often applicable,4 so that R′ and G′ are important only if it is necessary to compute loss. 4See Section 3.9 for a reminder about this concept. Since the low-loss conditions are often met, a convenient expression for the characteristic impedance is obtained from Equations 3.61 and 3.62 for L′ and C′ respectively: Z0 ≈ r L′ C′ (low-loss) = 1 2π rµ0 ǫs ln b a (3.64) The spacer permittivity can be expressed as ǫs = ǫrǫ0 where ǫr is the relative permittivity of the spacer material. Since p µ0/ǫ0 is a constant, the above expression is commonly written Z0 ≈60 Ω √ǫr ln b a (low-loss) (3.65) Thus, it is possible to express Z0 directly in terms of parameters describing the geometry (a and b) and material (ǫr) used in the line, without the need to ﬁrst compute the values of components in the lumped-element equivalent circuit model. Similarly, the low-loss approximation makes it possible to express the phase velocity νp directly in terms the spacer permittivity: νp ≈ 1 √ L′C′ (low-loss) = c √ǫr (3.66) since c ≜1/√µ0ǫ0. In other words, the phase velocity in a low-loss coaxial line is approximately equal to the speed of electromagnetic propagation in free space, divided by the square root of the relative permittivity of the spacer material. Therefore, the phase velocity in an air-ﬁlled coaxial line is approximately equal to speed of propagation in free space, but is reduced in a coaxial line using a dielectric spacer. Example 3.1. RG-59 Coaxial Cable. RG-59 is a very common type of coaxial line. Figure 3.15 shows a section of RG-59 cut away so as to reveal its structure. The radii are a ∼= 0.292 mm and b ≈1.855 mm (mean), yielding L′ ≈370 nH/m. The spacer material is polyethylene having ǫr ∼= 2.25, yielding	Electromagnetics_Vol1_Page_58_Chunk1358
44 CHAPTER 3. TRANSMISSION LINES C′ ≈67.7 pF/m. The conductivity of polyethylene is σs ∼= 5.9 × 10−5 S/m, yielding G′ ≈200 µS/m. Typical resistance per unit length R′ is on the order of 0.1 Ω/m near DC, increasing approximately in proportion to the square root of frequency. From the above values, we ﬁnd that RG-59 satisﬁes the low-loss criteria R′ ≪ωL′ for f ≫43 kHz and G′ ≪ωC′ for f ≫470 kHz. Under these conditions, we ﬁnd Z0 ≈ p L′/C′ ∼= 74 Ω. Thus, the ratio of the potential to the current in a wave traveling in a single direction on RG-59 is about 74 Ω. The phase velocity of RG-59 is found to be vp ≈1/ √ L′C′ ∼= 2 × 108 m/s, which is about 67% of c. In other words, a signal that takes 1 ns to traverse a distance l in free space requires about 1.5 ns to traverse a length-l section of RG-59. Since vp = λf, a wavelength in RG-59 is 67% of a wavelength in free space. Using the expression γ = p (R′ + jωL′) (G′ + jωC′) (3.67) with R′ = 0.1 Ω/m, and then taking the real part to obtain α, we ﬁnd α ∼0.01 m−1. So, for example, the magnitude of the potential or current is decreased by about 50% by traveling a distance of about 70 m. In other words, e−αl = 0.5 for l ∼70 m at relatively low frequencies, and increases with increasing frequency. Additional Reading: • “Coaxial cable” on Wikipedia. Includes descriptions and design parameters for a variety of commonly-encountered coaxial cables. • “Single-ended signaling” on Wikipedia. • Sec. 8.7 (“Differential Circuits”) in S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press, 2016. c⃝Arj CC BY SA 3.0 Figure 3.15: RG-59 coaxial line. A: Insulating jacket. B: Braided outer conductor. C: Dielectric spacer. D: Inner conductor. h W l r t 0 c⃝7head7metal7 CC BY SA 3.0 (modiﬁed) Figure 3.16: Microstrip transmission line structure and design parameters. 3.11 Microstrip Line [m0082] A microstrip transmission line consists of a narrow metallic trace separated from a metallic ground plane by a slab of dielectric material, as shown in Figure 3.16. This is a natural way to implement a transmission line on a printed circuit board, and so accounts for an important and expansive range of applications. The reader should be aware that microstrip is distinct from stripline, which is a very different type of transmission line; see “Additional Reading” at the end of this section for disambiguation of these terms. A microstrip line is single-ended5 in the sense that the 5The reference in “Additional Reading” at the end of this section may be helpful if you are not familiar with this concept.	Electromagnetics_Vol1_Page_59_Chunk1359
3.11. MICROSTRIP LINE 45 conductor geometry is asymmetric and the one conductor – namely, the ground plane – also normally serves as ground for the source and load. The spacer material is typically a low-loss dielectric material having permeability approximately equal to that of free space (µ ≈µ0) and relative permittivity ǫr in the range 2 to about 10 or so. A microstrip line nominally exhibits TEM ﬁeld structure. This structure is shown in Figure 3.17. Note that electric and magnetic ﬁelds exist both in the dielectric and in the space above the dielectric, which is typically (but not always) air. This complex ﬁeld structure makes it difﬁcult to describe microstrip line concisely in terms of the equivalent circuit parameters of the lumped-element model. Instead, expressions for Z0 directly in terms of h/W and ǫr are typically used instead. A variety of these expressions are in common use, representing different approximations and simpliﬁcations. A widely-accepted and broadly-applicable expression is:6 Z0 ≈42.4 Ω √ǫr + 1 × ln " 1 + 4h W ′ Φ + r Φ2 + 1 + 1/ǫr 2 π2 !# (3.68) where Φ ≜14 + 8/ǫr 11 4h W ′ (3.69) and W ′ is W adjusted to account for the thickness t of the microstrip line. Typically t ≪W and t ≪h, for which W ′ ≈W. Simpler approximations for Z0 are also commonly employed in the design and analysis of microstrip lines. These expressions are limited in the range of h/W for which they are valid, and can usually be shown to be special cases or approximations of Equation 3.68. Nevertheless, they are sometimes useful for quick “back of the envelope” calculations. Accurate expressions for wavelength λ, phase propagation constant β, and phase velocity vp are similarly difﬁcult to obtain for waves in microstrip line. An approximate technique employs a result from 6This is from Wheeler 1977, cited in “Additional Reading” at the end of this section. h W Figure 3.17: Structure of the electric and magnetic ﬁelds within microstrip line. (The ﬁelds outside the line are possibly signiﬁcant, complicated, and not shown.) In this case, the wave is propagating away from the viewer. the theory of uniform plane waves in unbounded media (Equation 9.38 from Section 9.2): β = ω√µǫ (3.70) It turns out that the electromagnetic ﬁeld structure in the space between the conductors is well-approximated as that of a uniform plane wave in unbounded media having the same permeability µ0 but a different relative permittivity, which we shall assign the symbol ǫr,eff (for “effective relative permittivity”). Then β ≈ω√µ0 ǫr,eff ǫ0 (low-loss microstrip) = β0√ǫr,eff (3.71) In other words, the phase propagation constant in a microstrip line can be approximated as the free-space phase propagation β0 ≜ω√µ0ǫ0 times a correction factor √ǫr,eff. Then ǫr,eff may be crudely approximated as follows: ǫr,eff ≈ǫr + 1 2 (3.72) i.e., ǫr,eff is roughly the average of the relative permittivity of the dielectric slab and the relative permittivity of free space. The assumption employed here is that ǫr,eff is approximately the average of these values because some fraction of the power in the guided wave is in the dielectric, and the rest is above the dielectric. Various approximations are available to improve on this approximation; however, in practice variations in the value of ǫr for the dielectric due to manufacturing processes typically make a more precise estimate irrelevant.	Electromagnetics_Vol1_Page_60_Chunk1360
46 CHAPTER 3. TRANSMISSION LINES Using this concept, we obtain λ = 2π β = 2π β0√ǫr,eff = λ0 √ǫr,eff (3.73) where λ0 is the free-space wavelength c/f. Similarly the phase velocity vp, can be estimated using the relationship vp = ω β = c √ǫr,eff (3.74) i.e., the phase velocity in microstrip is slower than c by a factor of √ǫr,eff. Example 3.2. 50 ΩMicrostrip in FR4 Printed Circuit Boards. FR4 is a low-loss ﬁberglass epoxy dielectric that is commonly used to make printed circuit boards (see “Additional Reading” at the end of this section). FR4 circuit board material is commonly sold in a slab having thickness h ∼= 1.575 mm with ǫr ∼= 4.5. Let us consider how we might implement a microstrip line having Z0 = 50 Ωusing this material. Since h and ǫr are ﬁxed, the only parameter remaining to set Z0 is W. A bit of experimentation with Equation 3.68 reveals that h/W ≈1/2 yields Z0 ≈50 Ωfor ǫr = 4.5. Thus, W should be about 3.15 mm. The effective relative permittivity is ǫr,eff ≈(4.5 + 1)/2 = 2.75 so the phase velocity for the wave guided by this line is about c/ √ 2.75; i.e., 60% of c. Similarly, the wavelength of this wave is about 60% of the free space wavelength. Additional Reading: • “Microstrip” on Wikipedia. • “Printed circuit board” on Wikipedia. • “Stripline” on Wikipedia. • “Single-ended signaling” on Wikipedia. • Sec. 8.7 (“Differential Circuits”) in S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press, 2016. • H.A. Wheeler, “Transmission Line Properties of a Strip on a Dielectric Sheet on a Plane,” IEEE Trans. Microwave Theory & Techniques, Vol. 25, No. 8, Aug 1977, pp. 631–47. • “FR-4” on Wikipedia.	Electromagnetics_Vol1_Page_61_Chunk1361
3.12. VOLTAGE REFLECTION COEFFICIENT 47 3.12 Voltage Reﬂection Coefﬁcient [m0084] We now consider the scenario shown in Figure 3.18. Here a wave arriving from the left along a lossless transmission line having characteristic impedance Z0 arrives at a termination located at z = 0. The impedance looking into the termination is ZL, which may be real-, imaginary-, or complex-valued. The questions are: Under what circumstances is a reﬂection – i.e., a leftward traveling wave – expected, and what precisely is that wave? The potential and current of the incident wave are related by the constant value of Z0. Similarly, the potential and current of the reﬂected wave are related by Z0. Therefore, it sufﬁces to consider either potential or current. Choosing potential, we may express the incident wave as eV +(z) = V + 0 e−jβz (3.75) where V + 0 is determined by the source of the wave, and so is effectively a “given.” Any reﬂected wave must have the form eV −(z) = V − 0 e+jβz (3.76) Therefore, the problem is solved by determining the value of V − 0 given V + 0 , Z0, and ZL. Considering the situation at z = 0, note that by ZL Z0 I =0 V + L L z z V + V Figure 3.18: A wave arriving from the left incident on a termination located at z = 0. deﬁnition we have ZL ≜ eVL eIL (3.77) where eVL and eIL are the potential across and current through the termination, respectively. Also, the potential and current on either side of the z = 0 interface must be equal. Thus, eV +(0) + eV −(0) = eVL (3.78) eI+(0) + eI−(0) = eIL (3.79) where eI+(z) and eI−(z) are the currents associated with eV +(z) and eV −(z), respectively. Since the voltage and current are related by Z0, Equation 3.79 may be rewritten as follows: eV +(0) Z0 − eV −(0) Z0 = eIL (3.80) Evaluating the left sides of Equations 3.78 and 3.80 at z = 0, we ﬁnd: V + 0 + V − 0 = eVL (3.81) V + 0 Z0 −V − 0 Z0 = eIL (3.82) Substituting these expressions into Equation 3.77 we obtain: ZL = V + 0 + V − 0 V + 0 /Z0 −V − 0 /Z0 (3.83) Solving for V − 0 we obtain V − 0 = ZL −Z0 ZL + Z0 V + 0 (3.84) Thus, the answer to the question posed earlier is that V − 0 = ΓV + 0 , where (3.85) Γ ≜ZL −Z0 ZL + Z0 (3.86) The quantity Γ is known as the voltage reﬂection coefﬁcient. Note that when ZL = Z0, Γ = 0 and therefore V − 0 = 0. In other words,	Electromagnetics_Vol1_Page_62_Chunk1362
48 CHAPTER 3. TRANSMISSION LINES If the terminating impedance is equal to the char- acteristic impedance of the transmission line, then there is no reﬂection. If, on the other hand, ZL ̸= Z0, then \|Γ\| > 0, V − 0 = ΓV + 0 , and a leftward-traveling reﬂected wave exists. Since ZL may be real-, imaginary-, or complex-valued, Γ too may be real-, imaginary-, or complex-valued. Therefore, V − 0 may be different from V + 0 in magnitude, sign, or phase. Note also that Γ is not the ratio of I− 0 to I+ 0 . The ratio of the current coefﬁcients is actually −Γ. It is quite simple to show this with a simple modiﬁcation to the above procedure and is left as an exercise for the student. Summarizing: The voltage reﬂection coefﬁcient Γ, given by Equation 3.86, determines the magnitude and phase of the reﬂected wave given the incident wave, the characteristic impedance of the trans- mission line, and the terminating impedance. [m0085] We now consider values of Γ that arise for commonly-encountered terminations. Matched Load (ZL = Z0). In this case, the termination may be a device with impedance Z0, or the termination may be another transmission line having the same characteristic impedance. When ZL = Z0, Γ = 0 and there is no reﬂection. Open Circuit. An “open circuit” is the absence of a termination. This condition implies ZL →∞, and subsequently Γ →+1. Since the current reﬂection coefﬁcient is −Γ, the reﬂected current wave is 180◦ out of phase with the incident current wave, making the total current at the open circuit equal to zero, as expected. Short Circuit. “Short circuit” means ZL = 0, and subsequently Γ = −1. In this case, the phase of Γ is 180◦, and therefore, the potential of the reﬂected wave cancels the potential of the incident wave at the open circuit, making the total potential equal to zero, as it must be. Since the current reﬂection coefﬁcient is −Γ = +1 in this case, the reﬂected current wave is in phase with the incident current wave, and the magnitude of the total current at the short circuit non-zero as expected. Purely Reactive Load. A purely reactive load, including that presented by a capacitor or inductor, has ZL = jX where X is reactance. In particular, an inductor is represented by X > 0 and a capacitor is represented by X < 0. We ﬁnd Γ = −Z0 + jX +Z0 + jX (3.87) The numerator and denominator have the same magnitude, so \|Γ\| = 1. Let φ be the phase of the denominator (+Z0 + jX). Then, the phase of the numerator is π −φ. Subsequently, the phase of Γ is (π −φ) −φ = π −2φ. Thus, we see that the phase of Γ is no longer limited to be 0◦or 180◦, but can be any value in between. The phase of reﬂected wave is subsequently shifted by this amount. Other Terminations. Any other termination, including series and parallel combinations of any number of devices, can be expressed as a value of ZL which is, in general, complex-valued. The associated	Electromagnetics_Vol1_Page_63_Chunk1363
3.13. STANDING WAVES 49 value of \|Γ\| is limited to the range 0 to 1. To see this, note: Γ = ZL −Z0 ZL + Z0 = ZL/Z0 −1 ZL/Z0 + 1 (3.88) Note that the smallest possible value of \|Γ\| occurs when the numerator is zero; i.e., when ZL = Z0. Therefore, the smallest value of \|Γ\| is zero. The largest possible value of \|Γ\| occurs when ZL/Z0 →∞(i.e., an open circuit) or when ZL/Z0 = 0 (a short circuit); the result in either case is \|Γ\| = 1. Thus, 0 ≤\|Γ\| ≤1 (3.89) 3.13 Standing Waves [m0086] A standing wave consists of waves moving in op- posite directions. These waves add to make a distinct magnitude variation as a function of dis- tance that does not vary in time. To see how this can happen, ﬁrst consider that an incident wave V + 0 e−jβz, which is traveling in the +z axis along a lossless transmission line. Associated with this wave is a reﬂected wave V − 0 e+jβz = ΓV + 0 e+jβz, where Γ is the voltage reﬂection coefﬁcient. These waves add to make the total potential eV (z) = V + 0 e−jβz + ΓV + 0 e+jβz = V + 0	Electromagnetics_Vol1_Page_64_Chunk1364
50 CHAPTER 3. TRANSMISSION LINES z V + 0 2 V + 0 V(z) / z=0 (a) Potential. I(z) ! " z=0 z I + 0 2 I + 0 (b) Current. Figure 3.19: Standing wave associated with an open- circuit termination at z = 0 (incident wave arrives from left). eI(z) = \|V + 0 \| Z0 p 2 −2 cos (2βz + φ) (3.98) where φ = 0 for an open circuit and φ = π for a short circuit. The result for an open circuit termination is shown in Figure 3.19(a) (potential) and 3.19(b) (current). The result for a short circuit termination is identical except the roles of potential and current are reversed. In either case, note that voltage maxima correspond to current minima, and vice versa. Also note: The period of the standing wave is λ/2; i.e., one- half of a wavelength. This can be conﬁrmed as follows. First, note that the frequency argument of the cosine function of the standing wave is 2βz. This can be rewritten as 1.25 1.00 0.75 0.50 0.00 0 # 2 $ 3% &z 1.50 2.00 ' 2 3 ( 2 5) 2 1.75 0.25 \|V\| 0 / V [rad] Γ*, -. Γ 0 1 23 4 Γ=1 Γ 5 6 78 9 Γ:; by Inductiveload (modiﬁed) Figure 3.20: Standing waves associated with loads exhibiting various reﬂection coefﬁcients. In this ﬁgure the incident wave arrives from the right. 2π (β/π) z, so the frequency of variation is β/π and the period of the variation is π/β. Since β = 2π/λ, we see that the period of the variation is λ/2. Furthermore, this is true regardless of the value of Γ. Mismatched loads. A common situation is that the termination is neither perfectly-matched (Γ = 0) nor an open/short circuit (\|Γ\| = 1). Examples of the resulting standing waves are shown in Figure 3.20. Additional Reading: • “Standing Wave” on Wikipedia.	Electromagnetics_Vol1_Page_65_Chunk1365
3.14. STANDING WAVE RATIO 51 3.14 Standing Wave Ratio [m0081] Precise matching of transmission lines to terminations is often not practical or possible. Whenever a signiﬁcant mismatch exists, a standing wave (Section 3.13) is apparent. The quality of the match is commonly expressed in terms of the standing wave ratio (SWR) of this standing wave. Standing wave ratio (SWR) is deﬁned as the ratio of the maximum magnitude of the standing wave to minimum magnitude of the standing wave. In terms of the potential: SWR ≜maximum \|eV \| minimum \|eV \| (3.99) SWR can be calculated using a simple expression, which we shall now derive. In Section 3.13, we found that: eV (z) = \|V + 0 \| q 1 + \|Γ\|2 + 2 \|Γ\| cos (2βz + φ) (3.100) The maximum value occurs when the cosine factor is equal to +1, yielding: max eV = \|V + 0 \| q 1 + \|Γ\|2 + 2 \|Γ\| (3.101) Note that the argument of the square root operator is equal to (1 + \|Γ\|)2; therefore: max eV = \|V + 0 \| (1 + \|Γ\|) (3.102) Similarly, the minimum value is achieved when the cosine factor is equal to −1, yielding: min eV = \|V + 0 \| q 1 + \|Γ\|2 −2 \|Γ\| (3.103) So: min eV = \|V + 0 \| (1 −\|Γ\|) (3.104) Therefore: SWR = 1 + \|Γ\| 1 −\|Γ\| (3.105) 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Γ <? @ Figure 3.21: Relationship between SWR and \|Γ\|. This relationship is shown graphically in Figure 3.21. Note that SWR ranges from 1 for perfectly-matched terminations (Γ = 0) to inﬁnity for open- and short-circuit terminations (\|Γ\| = 1). It is sometimes of interest to ﬁnd the magnitude of the reﬂection coefﬁcient given SWR. Solving Equation 3.105 for \|Γ\| we ﬁnd: \|Γ\| = SWR −1 SWR + 1 (3.106) SWR is often referred to as the voltage standing wave ratio (VSWR), although repeating the analysis above for the current reveals that the current SWR is equal to potential SWR, so the term “SWR” sufﬁces. SWR < 2 or so is usually considered a “good match,” although some applications require SWR < 1.1 or better, and other applications are tolerant to SWR of 3 or greater. Example 3.3. Reﬂection coefﬁcient for various values of SWR. What is the reﬂection coefﬁcient for the above-cited values of SWR? Using Equation 3.106, we ﬁnd: • SWR = 1.1 corresponds to \|Γ\| = 0.0476.	Electromagnetics_Vol1_Page_66_Chunk1366
52 CHAPTER 3. TRANSMISSION LINES • SWR = 2.0 corresponds to \|Γ\| = 1/3. • SWR = 3.0 corresponds to \|Γ\| = 1/2. 3.15 Input Impedance of a Terminated Lossless Transmission Line [m0087] Consider Figure 3.22, which shows a lossless transmission line being driven from the left and which is terminated by an impedance ZL on the right. If ZL is equal to the characteristic impedance Z0 of the transmission line, then the input impedance Zin will be equal to ZL. Otherwise Zin depends on both ZL and the characteristics of the transmission line. In this section, we determine a general expression for Zin in terms of ZL, Z0, the phase propagation constant β, and the length l of the line. Using the coordinate system indicated in Figure 3.22, the interface between source and transmission line is located at z = −l. Impedance is deﬁned at the ratio of potential to current, so: Zin(l) ≜ eV (z = −l) eI(z = −l) (3.107) Now employing expressions for eV (z) and eI(z) from Section 3.13 with z = −l, we ﬁnd: Zin(l) = V + 0	Electromagnetics_Vol1_Page_67_Chunk1367
3.15. INPUT IMPEDANCE OF A TERMINATED LOSSLESS TRANSMISSION LINE 53 e−jβl: Zin(l) = Z0 1 + Γe−j2βl 1 −Γe−j2βl (3.109) Recall that Γ in the above expression is: Γ = ZL −Z0 ZL + Z0 (3.110) Summarizing: Equation 3.109 is the input impedance of a lossless transmission line having characteristic impedance Z0 and which is terminated into a load ZL. The result also depends on the length and phase propagation constant of the line. Note that Zin(l) is periodic in l. Since the argument of the complex exponential factors is 2βl, the frequency at which Zin(l) varies is β/π; and since β = 2π/λ, the associated period is λ/2. This is very useful to keep in mind because it means that all possible values of Zin(l) are achieved by varying l over λ/2. In other words, changing l by more than λ/2 results in an impedance which could have been obtained by a smaller change in l. Summarizing to underscore this important idea: The input impedance of a terminated lossless transmission line is periodic in the length of the transmission line, with period λ/2. Not surprisingly, λ/2 is also the period of the standing wave (Section 3.13). This is because – once again – the variation with length is due to the interference of incident and reﬂected waves. Also worth noting is that Equation 3.109 can be written entirely in terms of ZL and Z0, since Γ depends only on these two parameters. Here’s that version of the expression: Zin(l) = Z0 ZL + jZ0 tan βl Z0 + jZL tan βl (3.111) This expression can be derived by substituting Equation 3.110 into Equation 3.109 and is left as an exercise for the student. Finally, note that the argument βl appearing Equations 3.109 and 3.111 has units of radians and is referred to as electrical length. Electrical length can be interpreted as physical length expressed with respect to wavelength and has the advantage that analysis can be made independent of frequency.	Electromagnetics_Vol1_Page_68_Chunk1368
54 CHAPTER 3. TRANSMISSION LINES 3.16 Input Impedance for Open- and Short-Circuit Terminations [m0088] Let us now consider the input impedance of a transmission line that is terminated in an open- or short-circuit. Such a transmission line is sometimes referred to as a stub. First, why consider such a thing? From a “lumped element” circuit theory perspective, this would not seem to have any particular application. However, the fact that this structure exhibits an input impedance that depends on length (Section 3.15) enables some very useful applications. First, let us consider the question at hand: What is the input impedance when the transmission line is open- or short-circuited? For a short circuit, ZL = 0, Γ = −1, so we ﬁnd Zin(l) = Z0 1 + Γe−j2βl 1 −Γe−j2βl = Z0 1 −e−j2βl 1 + e−j2βl (3.112) Multiplying numerator and denominator by e+jβl we obtain Zin(l) = Z0 e+jβl −e−jβl e+jβl + e−jβl (3.113) Now we invoke the following trigonometric identities: cos θ = 1 2 e+jθ + e−jθ (3.114) sin θ = 1 j2 e+jθ −e−jθ (3.115) Employing these identities, we obtain: Zin(l) = Z0 j2 (sin βl) 2 (cos βl) (3.116) and ﬁnally: Zin(l) = +jZ0 tan βl (3.117) Figure 3.23(a) shows what’s going on. As expected, Zin = 0 when l = 0, since this amounts to a short 0 0 0.25 0.5 0.75 1 imaginary part of input impedance length [wavelengths] (a) Short-circuit termination (ZL = 0). 0 0 0.25 0.5 0.75 1 imaginary part of input impedance length [wavelengths] (b) Open-circuit termination (ZL →∞). Figure 3.23: Input reactance (Im{Zin}) of a stub. Re{Zin} is always zero. circuit with no transmission line. Also, Zin varies periodically with increasing length, with period λ/2. This is precisely as expected from standing wave theory (Section 3.13). What is of particular interest now is that as l →λ/4, we see Zin →∞. Remarkably, the transmission line has essentially transformed the short circuit termination into an open circuit! For an open circuit termination, ZL →∞, Γ = +1, and we ﬁnd Zin(l) = Z0 1 + Γe−j2βl 1 −Γe−j2βl = Z0 1 + e−j2βl 1 −e−j2βl (3.118)	Electromagnetics_Vol1_Page_69_Chunk1369
3.17. APPLICATIONS OF OPEN- AND SHORT-CIRCUITED TRANSMISSION LINE STUBS 55 Following the same procedure detailed above for the short-circuit case, we ﬁnd Zin(l) = −jZ0 cot βl (3.119) Figure 3.23(b) shows the result for open-circuit termination. As expected, Zin →∞for l = 0, and the same λ/2 periodicity is observed. What is of particular interest now is that at l = λ/4 we see Zin = 0. In this case, the transmission line has transformed the open circuit termination into a short circuit. Now taking stock of what we have determined: The input impedance of a short- or open- circuited lossless transmission line is com- pletely imaginary-valued and is given by Equa- tions 3.117 and 3.119, respectively. The input impedance of a short- or open-circuited lossless transmission line alternates between open- (Zin →∞) and short-circuit (Zin = 0) conditions with each λ/4-increase in length. Additional Reading: • “Stub (electronics)” on Wikipedia. 3.17 Applications of Open- and Short-Circuited Transmission Line Stubs [m0145] The theory of open- and short-circuited transmission lines – often referred to as stubs – was addressed in Section 3.16. These structures have important and wide-ranging applications. In particular, these structures can be used to replace discrete inductors and capacitors in certain applications. To see this, consider the short-circuited line (Figure 3.23(b) of Section 3.16). Note that each value of l that is less than λ/4 corresponds to a particular positive reactance; i.e., the transmission line “looks” like an inductor. Also note that lengths between λ/4 and λ/2 result in reactances that are negative; i.e., the transmission line “looks” like a capacitor. Thus, it is possible to replace an inductor or capacitor with a short-circuited transmission line of the appropriate length. The input impedance of such a transmission line is identical to that of the inductor or capacitor at the design frequency. The variation of reactance with respect to frequency will not be identical, which may or may not be a concern depending on the bandwidth and frequency response requirements of the application. Open-circuited lines may be used in a similar way. This property of open- and short-circuited transmission lines makes it possible to implement impedance matching circuits (see Section 3.23), ﬁlters, and other devices entirely from transmission lines, with fewer or no discrete inductors or capacitors required. Transmission lines do not suffer the performance limitations of discrete devices at high frequencies and are less expensive. A drawback of transmission line stubs in this application is that the lines are typically much larger than the discrete devices they are intended to replace. Example 3.4. Emitter Induction Using Short-Circuited Line. In the design of low-noise ampliﬁers using	Electromagnetics_Vol1_Page_70_Chunk1370
56 CHAPTER 3. TRANSMISSION LINES bipolar transistors in common-emitter conﬁguration, it is often useful to introduce a little inductance between the emitter and ground. This is known as “inductive degeneration,” “emitter induction,” or sometimes by other names. It can be difﬁcult to ﬁnd suitable inductors, especially for operation in the UHF band and higher. However, a microstrip line can be used to achieve the desired inductive impedance. Determine the length of a stub that implements a 2.2 nH inductance at 6 GHz using microstrip line with characteristic impedance 50 Ωand phase velocity 0.6c. Solution. At the design frequency, the impedance looking into this section of line from the emitter should be equal to that of a 2.2 nH inductor, which is +jωL = +j2πfL = +j82.9 Ω. The input impedance of a short-circuited stub of length l which is grounded (thus, short-circuited) at the opposite end is +jZ0 tan βl (Section 3.16). Setting this equal to +j82.9 Ωand noting that Z0 = 50 Ω, we ﬁnd that βl ∼= 1.028 rad. The phase propagation constant is (Section 3.8): β = ω vp = 2πf 0.6c ∼= 209.4 rad/m (3.120) Therefore, the length of the microstrip line is l = (βl) /β ∼= 4.9 mm. Additional Reading: • “Stub (electronics)” on Wikipedia. 3.18 Measurement of Transmission Line Characteristics [m0089] This section presents a simple technique for measuring the characteristic impedance Z0, electrical length βl, and phase velocity vp of a lossless transmission line. This technique requires two measurements: the input impedance Zin when the transmission line is short-circuited and Zin when the transmission line is open-circuited. In Section 3.16, it is shown that the input impedance Zin of a short-circuited transmission line is Z(SC) in = +jZ0 tan βl and when a transmission line is terminated in an open circuit, the input impedance is Z(OC) in = −jZ0 cot βl Observe what happens when we multiply these results together: Z(SC) in · Z(OC) in = Z2 0 that is, the product of the measurements Z(OC) in and Z(SC) in is simply the square of the characteristic impedance. Therefore Z0 = q Z(SC) in · Z(OC) in (3.121) If we instead divide these measurements, we ﬁnd Z(SC) in Z(OC) in = −tan2 βl Therefore: tan βl = " −Z(SC) in Z(OC) in #1/2 (3.122) If l is known in advance to be less than λ/2, then the electrical length βl can be determined by taking the inverse tangent. If l is of unknown length and longer than λ/2, one must take care to account for the periodicity of tangent function; in this case, it may not	Electromagnetics_Vol1_Page_71_Chunk1371
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE 57 be possible to unambiguously determine βl. Although we shall not present the method here, it is possible to resolve this ambiguity by making multiple measurements over a range of frequencies. Once βl is determined, it is simple to determine l given β, β given l, and then vp. For example, the phase velocity may be determined by ﬁrst ﬁnding βl for a known length using the above procedure, calculating β = (βl) /l, and then vp = ω/β. 3.19 Quarter-Wavelength Transmission Line [m0091] Quarter-wavelength sections of transmission line play an important role in many systems at radio and optical frequencies. The remarkable properties of open- and short-circuited quarter-wave line are presented in Section 3.16 and should be reviewed before reading further. In this section, we perform a more general analysis, considering not just open- and short-circuit terminations but any terminating impedance, and then we address some applications. The general expression for the input impedance of a lossless transmission line is (Section 3.15): Zin(l) = Z0 1 + Γe−j2βl 1 −Γe−j2βl (3.123) Note that when l = λ/4: 2βl = 2 · 2π λ · λ 4 = π Subsequently: Zin(λ/4) = Z0 1 + Γe−jπ 1 −Γe−jπ = Z0 1 −Γ 1 + Γ (3.124) Recall that (Section 3.15): Γ = ZL −Z0 ZL + Z0 (3.125) Substituting this expression and then multiplying numerator and denominator by ZL + Z0, one obtains Zin(λ/4) = Z0 (ZL + Z0) −(ZL −Z0) (ZL + Z0) + (ZL −Z0) = Z0 2Z0 2ZL (3.126) Thus, Zin(λ/4) = Z2 0 ZL (3.127) Note that the input impedance is inversely proportional to the load impedance. For this reason, a	Electromagnetics_Vol1_Page_72_Chunk1372
58 CHAPTER 3. TRANSMISSION LINES l= /4 Figure 3.24: Impedance-matching using a quarter- wavelength transmission line. transmission line of length λ/4 is sometimes referred to as a quarter-wave inverter or simply as a impedance inverter. Quarter-wave lines play a very important role in RF engineering. As impedance inverters, they have the useful attribute of transforming small impedances into large impedances, and vice-versa – we’ll come back to this idea later in this section. First, let’s consider how quarter-wave lines are used for impedance matching. Look what happens when we solve Equation 3.127 for Z0: Z0 = p Zin(λ/4) · ZL (3.128) This equation indicates that we may match the load ZL to a source impedance (represented by Zin(λ/4)) simply by making the characteristic impedance equal to the value given by the above expression and setting the length to λ/4. The scheme is shown in Figure 3.24. Example 3.5. 300-to-50 Ωmatch using an quarter-wave section of line. Design a transmission line segment that matches 300 Ωto 50 Ωat 10 GHz using a quarter-wave match. Assume microstrip line for which propagation occurs with wavelength 60% that of free space. Solution. The line is completely speciﬁed given its characteristic impedance Z0 and length l. The length should be one-quarter wavelength with respect to the signal propagating in the line. The free-space wavelength λ0 = c/f at 10 GHz is ∼= 3 cm. Therefore, the wavelength of the signal in the line is λ = 0.6λ0 ∼= 1.8 cm, and the length of the line should be l = λ/4 ∼= 4.5 mm. The characteristic impedance is given by Equation 3.128: Z0 = √ 300 Ω· 50 Ω∼= 122.5 Ω (3.129) This value would be used to determine the width of the microstrip line, as discussed in Section 3.11. It should be noted that for this scheme to yield a real-valued characteristic impedance, the product of the source and load impedances must be a real-valued number. In particular, this method is not suitable if ZL has a signiﬁcant imaginary-valued component and matching to a real-valued source impedance is desired. One possible workaround in this case is the two-stage strategy shown in Figure 3.25. In this scheme, the load impedance is ﬁrst transformed to a real-valued impedance using a length l1 of transmission line. This is accomplished using Equation 3.123 (quite simple using a numerical search) or using the Smith chart (see “Additional Reading” at the end of this section). The characteristic impedance Z01 of this transmission line is not critical and can be selected for convenience. Normally, the smallest value of l1 is desired. This value will always be less than λ/4 since Zin(l1) is periodic in l1 with period λ/2; i.e., there are two changes in the sign of the imaginary component of Zin(l1) as l1 is increased from zero to λ/2. After eliminating the imaginary component of ZL in this manner, the real component of the resulting impedance may then be transformed using the quarter-wave matching technique described earlier in this section. Example 3.6. Matching a patch antenna to 50 Ω. A particular patch antenna exhibits a source impedance of ZA = 35 + j35 Ω. (See “Microstrip antenna” in “Additional Reading” at the end of this section for some optional reading on patch antennas.) Interface this antenna to 50 Ωusing the technique described above. For the section of transmission line adjacent to the	Electromagnetics_Vol1_Page_73_Chunk1373
3.19. QUARTER-WAVELENGTH TRANSMISSION LINE 59 L D/4 (completely real-valued) Figure 3.25: Impedance-matching a complex-valued load impedance using quarter-wavelength transmis- sion line. patch antenna, use characteristic impedance Z01 = 50 Ω. Determine the lengths l1 and l2 of the two segments of transmission line, and the characteristic impedance Z02 of the second (quarter-wave) segment. Solution. The length of the ﬁrst section of the transmission line (adjacent to the antenna) is determined using Equation 3.123: Z1(l1) = Z01 1 + Γe−j2β1l1 1 −Γe−j2β1l1 (3.130) where β1 is the phase propagation constant for this section of transmission line and Γ ≜ZA −Z01 ZA + Z01 ∼= −0.0059 + j0.4142 (3.131) We seek the value of smallest positive value of β1l1 for which the imaginary part of Z1(l1) is zero. This can determined using a Smith chart (see “Additional Reading” at the end of this section) or simply by a few iterations of trial-and-error. Either way we ﬁnd Z1(β1l1 = 0.793 rad) ∼= 120.719 −j0.111 Ω, which we deem to be close enough to be acceptable. Note that β1 = 2π/λ, where λ is the wavelength of the signal in the transmission line. Therefore l1 = β1l1 β1 = β1l1 2π λ ∼= 0.126λ (3.132) The length of the second section of the transmission line, being a quarter-wavelength transformer, should be l2 = 0.25λ. Using Equation 3.128, the characteristic impedance Z02 of this section of line should be Z02 ∼= p (120.719 Ω) (50 Ω) ∼= 77.7 Ω (3.133) Discussion. The total length of the matching structure is l1 + l2 ∼= 0.376λ. A patch antenna would typically have sides of length about λ/2 = 0.5λ, so the matching structure is nearly as big as the antenna itself. At frequencies where patch antennas are commonly used, and especially at frequencies in the UHF (300–3000 MHz) band, patch antennas are often comparable to the size of the system, so it is not attractive to have the matching structure also require a similar amount of space. Thus, we would be motivated to ﬁnd a smaller matching structure. Although quarter-wave matching techniques are generally effective and commonly used, they have one important contraindication, noted above – They often result in structures that are large. That is, any structure which employs a quarter-wave match will be at least λ/4 long, and λ/4 is typically large compared to the associated electronics. Other transmission line matching techniques – and in particular, single stub matching (Section 3.23) – typically result in structures which are signiﬁcantly smaller. The impedance inversion property of quarter-wavelength lines has applications beyond impedance matching. The following example demonstrates one such application: Example 3.7. RF/DC decoupling in transistor ampliﬁers. Transistor ampliﬁers for RF applications often receive DC current at the same terminal which delivers the ampliﬁed RF signal, as shown in Figure 3.26. The power supply typically has a low output impedance. If the power supply is directly connected to the transistor, then the RF	Electromagnetics_Vol1_Page_74_Chunk1374
60 CHAPTER 3. TRANSMISSION LINES RF out (moderate Z) DC power in (low E) Figure 3.26: Use of an inductor to decouple the DC input power from the RF output signal at the output of a common-emitter RF ampliﬁer. will ﬂow predominantly in the direction of the power supply as opposed to following the desired path, which exhibits a higher impedance. This can be addressed using an inductor in series with the power supply output. This works because the inductor exhibits low impedance at DC and high impedance at RF. Unfortunately, discrete inductors are often not practical at high RF frequencies. This is because practical inductors also exhibit parallel capacitance, which tends to decrease impedance. A solution is to replace the inductor with a transmission line having length λ/4 as shown in Figure 3.27. A wavelength at DC is inﬁnite, so the transmission line is essentially transparent to the power supply. At radio frequencies, the line transforms the low impedance of the power supply to an impedance that is very large relative to the impedance of the desired RF path. Furthermore, transmission lines on printed circuit boards are much cheaper than discrete inductors (and are always in stock!). Additional Reading: • “Quarter-wavelength impedance transformer” on Wikipedia. • “Smith chart” on Wikipedia. • “Microstrip antenna” on Wikipedia. RF out (moderate F) DC power in (low G) H/4 Figure 3.27: Decoupling of DC input power and RF output signal at the output of a common-emitter RF ampliﬁer, using a quarter-wavelength transmission line. 3.20 Power Flow on Transmission Lines [m0090] It is often important to know the power associated with a wave on a transmission line. The power of the waves incident upon, reﬂected by, and absorbed by a load are each of interest. In this section we shall work out expressions for these powers and consider some implications in terms of the voltage reﬂection coefﬁcient (Γ) and standing wave ratio (SWR). Let’s begin by considering a lossless transmission line that is oriented along the z axis. The time-average power associated with a sinusoidal wave having potential v(z, t) and current i(z, t) is Pav(z) ≜1 T Z t0+T t0 v(z, t) i(z, t) dt (3.134) where T ≜2π/f is one period of the wave and t0 is the start time for the integration. Since the time-average power of a sinusoidal signal does not change with time, t0 may be set equal to zero without loss of generality. Let us now calculate the power of a wave incident from z < 0 on a load impedance ZL at z = 0. We may express the associated potential and current as follows: v+(z, t) = V + 0 cos (ωt −βz + φ) (3.135)	Electromagnetics_Vol1_Page_75_Chunk1375
3.20. POWER FLOW ON TRANSMISSION LINES 61 i+(z, t) = V + 0 Z0 cos (ωt −βz + φ) (3.136) And so the associated time-average power is P + av(z) = 1 T Z T 0 v+(z, t) i+(z, t) dt = V + 0 2 Z0 · 1 T Z T 0 cos2 (ωt −βz + φ) dt (3.137) Employing a well-known trigonometric identity: cos2 θ = 1 2 + 1 2 cos 2θ (3.138) we may rewrite the integrand as follows cos2 (ωt −βz + φ) = 1 2 + 1 2 cos (2 [ωt −βz + φ]) (3.139) Then integrating over both sides of this quantity Z T 0 cos2 (ωt −βz + φ) dt = T 2 + 0 (3.140) The second term of the integral is zero because it is the integral of cosine over two complete periods. Subsequently, we see that the position dependence (here, the dependence on z) is eliminated. In other words, the power associated with the incident wave is the same for all z < 0, as expected. Substituting into Equation 3.137 we obtain: P + av = V + 0 2 2Z0 (3.141) This is the time-average power associated with the incident wave, measured at any point z < 0 along the line. Equation 3.141 gives the time-average power as- sociated with a wave traveling in a single direc- tion along a lossless transmission line. Using precisely the same procedure, we ﬁnd that the power associated with the reﬂected wave is P − av = ΓV + 0 2 2Z0 = \|Γ\|2 V + 0 2 2Z0 (3.142) or simply P − av = \|Γ\|2 P + av (3.143) Equation 3.143 gives the time-average power associated with the wave reﬂected from an impedance mismatch. Now, what is the power PL delivered to the load impedance ZL? The simplest way to calculate this power is to use the principle of conservation of power. Applied to the present problem, this principle asserts that the power incident on the load must equal the power reﬂected plus the power absorbed; i.e., P + av = P − av + PL (3.144) Applying the previous equations we obtain: PL = 1 −\|Γ\|2 P + av (3.145) Equations 3.145 gives the time-average power transferred to a load impedance, and is equal to the difference between the powers of the incident and reﬂected waves. Example 3.8. How important is it to match 50 Ωto 75 Ω? Two impedances which commonly appear in radio engineering are 50 Ωand 75 Ω. It is not uncommon to ﬁnd that it is necessary to connect a transmission line having a 50 Ωcharacteristic impedance to a device, circuit, or system having a 75 Ωinput impedance, or vice-versa. If no attempt is made to match these impedances, what fraction of the power will be delivered to the termination, and what fraction of power will be reﬂected? What is the SWR? Solution. The voltage reﬂection coefﬁcient going from 50 Ωtransmission line to a 75 Ωload is Γ = 75 −50 75 + 50 = +0.2 The fraction of power reﬂected is \|Γ\|2 = 0.04, which is 4%. The fraction of power transmitted is 1 −\|Γ\|2, which is 96%. Going from a 50 Ω transmission line to a 75 Ωtermination changes only the sign of Γ, and therefore, the fractions of reﬂected and transmitted power remain 4% and	Electromagnetics_Vol1_Page_76_Chunk1376
62 CHAPTER 3. TRANSMISSION LINES 96%, respectively. In either case (from Section 3.14): SWR = 1 + \|Γ\| 1 −\|Γ\| = 1.5 This is often acceptable, but may not be good enough in some particular applications. Sufﬁce it to say that it is not necessarily required to use an impedance matching device to connect 50 Ω to 75 Ωdevices. 3.21 Impedance Matching: General Considerations [m0092] “Impedance matching” refers to the problem of transforming a particular impedance ZL into a modiﬁed impedance Zin. The problem of impedance matching arises because it is not convenient, practical, or desirable to have all devices in a system operate at the same input and output impedances. Here are just a few of the issues: • It is not convenient or practical to market coaxial cables having characteristic impedance equal to every terminating impedance that might be encountered. • Different types of antennas operate at different impedances, and the impedance of most antennas vary signiﬁcantly with frequency. • Different types of ampliﬁers operate most effectively at different output impedances. For example, ampliﬁers operating as current sources operate most effectively with low output impedance, whereas ampliﬁers operating as voltage sources operate most effectively with high output impedances. • Independently of the above issue, techniques for the design of transistor ampliﬁers rely on intentionally mismatching impedances; i.e., matching to an impedance different than that which maximizes power transfer or minimizes reﬂection. In other words, various design goals are met by applying particular impedances to the input and output ports of the transistor.7 For all of these reasons, electrical engineers frequently ﬁnd themselves with the task of transforming a particular impedance ZL into a modiﬁed impedance Zin. The reader is probably already familiar with many approaches to the impedance matching problem that 7For a concise introduction to this concept, see Chapter 10 of S.W. Ellingson, Radio Systems Engineering, Cambridge Univ. Press, 2016.	Electromagnetics_Vol1_Page_77_Chunk1377
3.22. SINGLE-REACTANCE MATCHING 63 employ discrete components and do not require knowledge of electromagnetics.8 To list just a few of these approaches: transformers, resistive (lossy) matching, single-reactance matching, and two-reactance (“L” network) matching. However, all of these have limitations. Perhaps the most serious limitations pertain to the performance of discrete components at high frequencies. Here are just a few of the most common problems: • Practical resistors actually behave as ideal resistors in series with ideal inductors • Practical capacitors actually behave as ideal capacitors in series with ideal resistors • Practical inductors behave as ideal inductors in parallel with ideal capacitors, and in series with ideal resistors. All of this makes the use of discrete components increasingly difﬁcult with increasing frequency. One possible solution to these types of problems is to more precisely model each component, and then to account for the non-ideal behavior by incorporating the appropriate models in the analysis and design process. Alternatively, one may consider ways to replace particular troublesome components – or, in some cases, all discrete components – with transmission line devices. The latter approach is particularly convenient in circuits implemented on printed circuit boards at frequencies in the UHF band and higher, since the necessary transmission line structures are easy to implement as microstrip lines and are relatively compact since the wavelength is relatively small. However, applications employing transmission lines as components in impedance matching devices can be found at lower frequencies as well. 8For an overview, see Chapter 9 of S.W. Ellingson, Radio Sys- tems Engineering, Cambridge Univ. Press, 2016. 3.22 Single-Reactance Matching [m0093] An impedance matching structure can be designed using a section of transmission line combined with a discrete reactance, such as a capacitor or an inductor. In the strategy presented here, the transmission line is used to transform the real part of the load impedance or admittance to the desired value, and then the reactance is used to modify the imaginary part to the desired value. (Note the difference between this approach and the quarter-wave technique described in Section 3.19. In that approach, the ﬁrst transmission line is used to zero the imaginary part.) There are two versions of this strategy, which we will now consider separately. The ﬁrst version is shown in Figure 3.28. The purpose of the transmission line is to transform the load impedance ZL into a new impedance Z1 for which Re{Z1} = Re{Zin}. This can be done by solving the equation (from Section 3.15) Re {Z1} = Re Z0 1 + Γe−j2βl 1 −Γe−j2βl (3.146) for l, using a numerical search, or using the Smith chart.9 The characteristic impedance Z0 and phase propagation constant β of the transmission line are independent variables and can be selected for convenience. Normally, the smallest value of l that satisﬁes Equation 3.146 is desired. This value will be ≤λ/4 because the real part of Z1 is periodic in l with period λ/4. After matching the real component of the impedance in this manner, the imaginary component of Z1 may then be transformed to the desired value (Im{Zin}) by attaching a reactance Xs in series with the transmission line input, yielding Zin = Z1 + jXS. Therefore, we choose Xs = Im {Zin −Z1} (3.147) The sign of Xs determines whether this reactance is a capacitor (Xs < 0) or inductor (Xs > 0), and the value of this component is determined from Xs and the design frequency. 9For more about the Smith chart, see “Additional Reading” at the end of this section.	Electromagnetics_Vol1_Page_78_Chunk1378
64 CHAPTER 3. TRANSMISSION LINES ZL Z1 Z0 I jXS Zin Figure 3.28: Single-reactance matching with a series reactance. Example 3.9. Single reactance in series. Design a match consisting of a transmission line in series with a single capacitor or inductor that matches a source impedance of 50Ωto a load impedance of 33.9 + j17.6 Ωat 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are 50Ωand 0.6c respectively. Solution. From the problem statement: Zin ≜ZS = 50 Ωand ZL = 33.9 + j17.6 Ωare the source and load impedances respectively at f = 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are Z0 = 50 Ωand vp = 0.6c respectively. The reﬂection coefﬁcient Γ (i.e., ZL with respect to the characteristic impedance of the transmission line) is Γ ≜ZL −Z0 ZL + Z0 ∼= −0.142 + j0.239 (3.148) The length l of the primary line (that is, the one that connects the two ports of the matching structure) is determined using the equation: Re {Z1} = Re Z0 1 + Γe−j2βl 1 −Γe−j2βl (3.149) where here Re {Z1} = Re {ZS} = 50 Ω. So a more-speciﬁc form of the equation that can be solved for βl (as a step toward ﬁnding l) is: 1 = Re 1 + Γe−j2βl 1 −Γe−j2βl (3.150) By trial and error (or using the Smith chart if you prefer) we ﬁnd βl ∼= 0.408 rad for the primary line, yielding Z1 ∼= 50.0 + j29.0 Ωfor the input impedance after attaching the primary line. We may now solve for l as follows: Since vp = ω/β (Section 3.8), we ﬁnd β = ω vp = 2πf 0.6c ∼= 52.360 rad/m (3.151) Therefore l = (βl) /β ∼= 7.8 mm. The impedance of the series reactance should be jXs ∼= −j29.0 Ωto cancel the imaginary part of Z1. Since the sign of this impedance is negative, it must be a capacitor. The reactance of a capacitor is −1/ωC, so it must be true that − 1 2πfC ∼= −29.0 Ω (3.152) Thus, we ﬁnd the series reactance is a capacitor of value C ∼= 3.7 pF. The second version of the single-reactance strategy is shown in Figure 3.29. The difference in this scheme is that the reactance is attached in parallel. In this case, it is easier to work the problem using admittance (i.e., reciprocal impedance) as opposed to impedance; this is because the admittance of parallel reactances is simply the sum of the associated admittances; i.e., Yin = Y1 + jBp (3.153) where Yin = 1/Zin, Y1 = 1/Z1, and Bp is the discrete parallel susceptance; i.e., the imaginary part of the discrete parallel admittance. So, the procedure is as follows. The transmission line is used to transform YL into a new admittance Y1 for which Re{Y1} = Re{Yin}. First, we note that Y1 ≜1 Z1 = Y0 1 −Γe−j2βl 1 + Γe−j2βl (3.154) where Y0 ≜1/Z0 is characteristic admittance. Again, the characteristic impedance Z0 and phase	Electromagnetics_Vol1_Page_79_Chunk1379
3.22. SINGLE-REACTANCE MATCHING 65 YL J1 K 0 M jO p Pin Figure 3.29: Single-reactance matching with a parallel reactance. propagation constant β of the transmission line are independent variables and can be selected for convenience. In the present problem, we aim to solve the equation Re {Y1} = Re Y0 1 −Γe−j2βl 1 + Γe−j2βl (3.155) for the smallest value of l, using a numerical search or using the Smith chart. After matching the real component of the admittances in this manner, the imaginary component of the resulting admittance may then be transformed to the desired value by attaching the susceptance Bp in parallel with the transmission line input. Since we desire jBp in parallel with Y1 to be Yin, the desired value is Bp = Im {Yin −Y1} (3.156) The sign of Bp determines whether this is a capacitor (Bp > 0) or inductor (Bp < 0), and the value of this component is determined from Bp and the design frequency. In the following example, we address the same problem raised in Example 3.9, now using the parallel reactance approach: Example 3.10. Single reactance in parallel. Design a match consisting of a transmission line in parallel with a single capacitor or inductor that matches a source impedance of 50Ωto a load impedance of 33.9 + j17.6 Ωat 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are 50Ωand 0.6c respectively. Solution. From the problem statement: Zin ≜ZS = 50 Ωand ZL = 33.9 + j17.6 Ωare the source and load impedances respectively at f = 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are Z0 = 50 Ωand vp = 0.6c respectively. The reﬂection coefﬁcient Γ (i.e., ZL with respect to the characteristic impedance of the transmission line) is Γ ≜ZL −Z0 ZL + Z0 ∼= −0.142 + j0.239 (3.157) The length l of the primary line (that is, the one that connects the two ports of the matching structure) is the solution to: Re {Y1} = Re Y0 1 −Γe−j2βl 1 + Γe−j2βl (3.158) where here Re {Y1} = Re {1/ZS} = 0.02 mho and Y0 = 1/Z0 = 0.02 mho. So the equation to be solved for βl (as a step toward ﬁnding l) is: 1 = Re 1 −Γe−j2βl 1 + Γe−j2βl (3.159) By trial and error (or the Smith chart) we ﬁnd βl ∼= 0.126 rad for the primary line, yielding Y1 ∼= 0.0200 −j0.0116 mho for the input admittance after attaching the primary line. We may now solve for l as follows: Since vp = ω/β (Section 3.8), we ﬁnd β = ω vp = 2πf 0.6c ∼= 52.360 rad/m (3.160) Therefore, l = (βl) /β ∼= 2.4 mm. The admittance of the parallel reactance should be jBp ∼= +j0.0116 mho to cancel the imaginary part of Y1. The associated impedance is 1/jBp ∼= −j86.3 Ω. Since the sign of this impedance is negative, it must be a capacitor. The reactance of a capacitor is −1/ωC, so it	Electromagnetics_Vol1_Page_80_Chunk1380
66 CHAPTER 3. TRANSMISSION LINES must be true that − 1 2πfC ∼= −86.3 Ω (3.161) Thus, we ﬁnd the parallel reactance is a capacitor of value C ∼= 1.2 pF. Comparing this result to the result from the series reactance method (Example 3.9), we see that the necessary length of transmission line is much shorter, which is normally a compelling advantage. The tradeoff is that the parallel capacitance is much smaller and an accurate value may be more difﬁcult to achieve. Additional Reading: • “Smith chart” on Wikipedia. 3.23 Single-Stub Matching [m0094] In Section 3.22, we considered impedance matching schemes consisting of a transmission line combined with a reactance which is placed either in series or in parallel with the transmission line. In many problems, the required discrete reactance is not practical because it is not a standard value, or because of non-ideal behavior at the desired frequency (see Section 3.21 for more about this), or because one might simply wish to avoid the cost and logistical issues associated with an additional component. Whatever the reason, a possible solution is to replace the discrete reactance with a transmission line “stub” – that is, a transmission line which has been open- or short-circuited. Section 3.16 explains how a stub can replace a discrete reactance. Figure 3.30 shows a practical implementation of this idea implemented in microstrip. This section explains the theory, and we’ll return to this implementation at the end of the section. Figure 3.31 shows the scheme. This scheme is usually implemented using the parallel reactance approach, as depicted in the ﬁgure. Although a series reactance scheme is also possible in principle, it is usually not as convenient. This is because most transmission lines use one of their two conductors as a local datum; e.g., the ground plane of a printed circuit board for microstrip line is tied to ground, and the outer conductor (“shield”) of a coaxial cable is usually tied to ground. This is contrast to a discrete reactance QR TUV Wtor X [ \] ^_ `a bc de Spinningspark CC BY SA 3.0 Figure 3.30: A practical implementation of a single- stub impedance match using microstrip transmission line. Here, the stub is open-circuited.	Electromagnetics_Vol1_Page_81_Chunk1381
3.23. SINGLE-STUB MATCHING 67 fL h1 i 01 k1 l m n oin stub characteristics: q 02 r2 2 stub may be open- or short-circuited stub Figure 3.31: Single-stub matching. (such as a capacitor or inductor), which does not require that either of its terminals be tied to ground. This issue is avoided in the parallel-attached stub because the parallel-attached stub and the transmission line to which it is attached both have one terminal at ground. The single-stub matching procedure is essentially the same as the single parallel reactance method, except the parallel reactance is implemented using a short- or open-circuited stub as opposed a discrete inductor or capacitor. Since parallel reactance matching is most easily done using admittances, it is useful to express Equations 3.117 and 3.119 (input impedance of an open- and short-circuited stub, respectively, from Section 3.16) in terms of susceptance: Bp = −Y02 cot (β2l2) short-circuited stub (3.162) Bp = +Y02 tan (β2l2) open-circuited stub (3.163) As in the main line, the characteristic impedance Z02 = 1/Y02 is an independent variable and is chosen for convenience. A ﬁnal question is when should you use a short-circuited stub, and when should you use an open-circuited stub? Given no other basis for selection, the termination that yields the shortest stub is chosen. An example of an “other basis for selection” that frequently comes up is whether DC might be present on the line. If DC is present with the signal of interest, then a short circuit termination without some kind of remediation to prevent a short circuit for DC would certainly be a bad idea. In the following example we address the same problem raised in Section 3.22 (Examples 3.9 and 3.10), now using the single-stub approach: Example 3.11. Single stub matching. Design a single-stub match that matches a source impedance of 50Ωto a load impedance of 33.9 + j17.6 Ω. Use transmission lines having characteristic impedances of 50Ωthroughout, and leave your answer in terms of wavelengths. Solution. From the problem statement: Zin ≜ZS = 50 Ωand ZL = 33.9 + j17.6 Ωare the source and load impedances respectively. Z0 = 50 Ωis the characteristic impedance of the transmission lines to be used. The reﬂection coefﬁcient Γ (i.e., ZL with respect to the characteristic impedance of the transmission line) is Γ ≜ZL −Z0 ZL + Z0 ∼= −0.142 + j0.239 (3.164) The length l1 of the primary line (that is, the one that connects the two ports of the matching structure) is the solution to the equation (from Section 3.22): Re {Y1} = Re Y01 1 −Γe−j2β1l1 1 + Γe−j2β1l1 (3.165) where here Re {Y1} = Re {1/ZS} = 0.02 mho and Y01 = 1/Z0 = 0.02 mho. Also note 2β1l1 = 2 2π λ l1 = 4π l1 λ (3.166) where λ is the wavelength in the transmission line. So the equation to be solved for l1 is: 1 = Re 1 −Γe−j4πl1/λ 1 + Γe−j4πl1/λ (3.167) By trial and error (or using the Smith chart; see “Additional Reading” at the end of this section) we ﬁnd l1 ∼= 0.020λ for the primary line,	Electromagnetics_Vol1_Page_82_Chunk1382
68 CHAPTER 3. TRANSMISSION LINES yielding Y1 ∼= 0.0200 −j0.0116 mho for the input admittance after attaching the primary line. We now seek the shortest stub having an input admittance of ∼= +j0.0116 mho to cancel the imaginary part of Y1. For an open-circuited stub, we need Bp = +Y0 tan 2πl2/λ ∼= +j0.0116 mho (3.168) The smallest value of l2 for which this is true is ∼= 0.084λ. For a short-circuited stub, we need Bp = −Y0 cot 2πl2/λ ∼= +j0.0116 mho (3.169) The smallest positive value of l2 for which this is true is ∼= 0.334λ; i.e., much longer. Therefore, we choose the open-circuited stub with l2 ∼= 0.084λ. Note the stub is attached in parallel at the source end of the primary line. Single-stub matching is a very common method for impedance matching using microstrip lines at frequences in the UHF band (300-3000 MHz) and above. In Figure 3.30, the top (visible) traces comprise one conductor, whereas the ground plane (underneath, so not visible) comprises the other conductor. The end of the stub is not connected to the ground plane, so the termination is an open circuit. A short circuit termination is accomplished by connecting the end of the stub to the ground plane using a via; that is, a plated-through that electrically connects the top and bottom layers. Additional Reading: • “Stub (electronics)” on Wikipedia. • “Smith chart” on Wikipedia. • “Via (electronics)” on Wikipedia. [m0151]	Electromagnetics_Vol1_Page_83_Chunk1383
3.23. SINGLE-STUB MATCHING 69 Image Credits Fig. 3.1: Dmitry G, https://en.wikipedia.org/wiki/File:Mastech test leads.JPG, public domain. Fig. 3.2: c⃝Tkgd2007, https://commons.wikimedia.org/wiki/File:Coaxial cable cutaway.svg, CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/). Minor modiﬁcations from the original. Fig. 3.3: c⃝SpinningSpark, https://en.wikipedia.org/wiki/File:Microstrip structure.svg, CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Minor modiﬁcations from the original. Fig. 3.6: c⃝Averse, https://en.wikipedia.org/wiki/File:Diplexer1.jpg, CC BY SA 2.0 Germany (https://creativecommons.org/licenses/by-sa/2.0/de/deed.en). Fig. 3.7: c⃝BigRiz, https://en.wikipedia.org/wiki/File:Fibreoptic.jpg, CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en). Fig. 3.8: c⃝Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line symbols.svg, CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en). Modiﬁed. Fig. 3.10: c⃝Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line element.svg, CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en). Modiﬁed. Fig. 3.11: c⃝Omegatron, https://commons.wikimedia.org/wiki/File:Transmission line element.svg, CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en). Modiﬁed. Fig. 3.15: c⃝Arj, https://en.wikipedia.org/wiki/File:RG-59.jpg, CC BY SA 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en). Fig. 3.16: c⃝7head7metal7, https://commons.wikimedia.org/wiki/File:Microstrip scheme.svg, CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Minor modiﬁcations from the original. Fig. 3.20: by Inductiveload, https://commons.wikimedia.org/wiki/File:Standing Wave Ratio.svg, public domain. Modiﬁcations from the original. Fig. 3.30: c⃝Spinningspark, https://commons.wikimedia.org/wiki/File:Stripline stub matching.svg, CC BY SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Minor modiﬁcations from the original: curly braces deleted.	Electromagnetics_Vol1_Page_84_Chunk1384
Chapter 4 Vector Analysis 4.1 Vector Arithmetic [m0006] A vector is a mathematical object that has both a scalar part (i.e., a magnitude and possibly a phase), as well as a direction. Many physical quantities are best described as vectors. For example, the rate of movement through space can be described as speed; i.e., as a scalar having SI base units of m/s. However, this quantity is more completely described as velocity; i.e., as a vector whose scalar part is speed and direction indicates the direction of movement. Similarly, force is a vector whose scalar part indicates magnitude (SI base units of N), and direction indicates the direction in which the force is applied. Electric and magnetic ﬁelds are also best described as vectors. In mathematical notation, a real-valued vector A is said to have a magnitude A = \|A\| and direction ˆa such that A = Aˆa (4.1) where ˆa is a unit vector (i.e., a real-valued vector having magnitude equal to one) having the same direction as A. If a vector is complex-valued, then A is similarly complex-valued. Cartesian Coordinate System. Fundamentals of vector arithmetic are most easily grasped using the Cartesian coordinate system. This system is shown in Figure 4.1. Note carefully the relative orientation of the x, y, and z axes. This orientation is important. For example, there are two directions that are perpendicular to the z = 0 plane (in which the x- and y-axes lie), but the +z axis is speciﬁed to be one of y x z s tu wx{ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.1: Cartesian coordinate system. these in particular. Position-Fixed vs. Position-Free Vectors. It is often convenient to describe a position in space as a vector for which the magnitude is the distance from the origin of the coordinate system and for which the direction is measured from the origin toward the position of interest. This is shown in Figure 4.2. These position vectors are “position-ﬁxed” in the sense that they are deﬁned with respect to a single point in space, which in this case is the origin. Position vectors can also be deﬁned as vectors that are deﬁned with respect to some other point in space, in which case they are considered position-ﬁxed to that position. Position-free vectors, on the other hand, are not deﬁned with respect to a particular point in space. An Electromagnetics Vol 1. c⃝2018 S.W. Ellingson CC BY SA 4.0. https://doi.org/10.21061/electromagnetics-vol-1	Electromagnetics_Vol1_Page_85_Chunk1385
4.1. VECTOR ARITHMETIC 71 \| x z r1 r } c⃝K. Kikkeri CC BY SA 4.0 Figure 4.2: Position vectors. The vectors r1 and r1 are position-ﬁxed and refer to particular locations. v ~ v c⃝K. Kikkeri CC BY SA 4.0 Figure 4.3: Two particles exhibiting the same veloc- ity. In this case, the velocity vectors v1 and v2 are position-free and equal. example is shown in Figure 4.3. Particles 1 m apart may both be traveling at 2 m/s in the same direction. In this case, the velocity of each particle can be described using the same vector, even though the particles are located at different points in space. Position-free vectors are said to be equal if they have the same magnitudes and directions. Position-ﬁxed vectors, on the other hand, must also be referenced to the same position (e.g., the origin) to be considered equal. Basis Vectors. Each coordinate system is deﬁned in terms of three basis vectors which concisely describe x z y x z c⃝K. Kikkeri CC BY SA 4.0 Figure 4.4: Basis vectors in the Cartesian coordinate system. all possible ways to traverse three-dimensional space. A basis vector is a position-free unit vector that is perpendicular to all other basis vectors for that coordinate system. The basis vectors ˆx, ˆy, and ˆz of the Cartesian coordinate system are shown in Figure 4.4. In this notation, ˆx indicates the direction in which x increases most rapidly, ˆy indicates the direction in which y increases most rapidly, and ˆz indicates the direction in which z increases most rapidly. Alternatively, you might interpret ˆx, ˆy, and ˆz as unit vectors that are parallel to the x-, y-, and z-axes and point in the direction in which values along each axis increase. Vectors in the Cartesian Coordinate System. In Cartesian coordinates, we may describe any vector A as follows: A = ˆxAx + ˆyAy + ˆzAz (4.2) where Ax, Ay, and Az are scalar quantities describing the components of A in each of the associated directions, as shown in Figure 4.5. This description makes it clear that the magnitude of A is: \|A\| = q A2x + A2y + A2z (4.3) and therefore, we can calculate the associated unit	Electromagnetics_Vol1_Page_86_Chunk1386
72 CHAPTER 4. VECTOR ANALYSIS x z x z A c⃝K. Kikkeri CC BY SA 4.0 Figure 4.5: Components of a vector A in the Carte- sian coordinate system. vector as ˆa = A \|A\| = A q A2x + A2y + A2z = ˆxAx	Electromagnetics_Vol1_Page_87_Chunk1387
4.1. VECTOR ARITHMETIC 73 from r1 to r2, the distance between these points, and the associated unit vector. Solution. The vector that points from r1 to r2 is R = r2 −r1 = (1 −2)ˆx + (−2 −3)ˆy + (3 −1)ˆz = −ˆx −5ˆy + 2ˆz (4.9) The distance between r1 and r2 is simply the magnitude of this vector: \|R\| = q (−1)2 + (−5)2 + (2)2 ∼= 5.48 m (4.10) The unit vector ˆR is simply R normalized to have unit magnitude: ˆR = R/ \|R\| ∼= (−ˆx −5ˆy + 2ˆz) /5.48 ∼= −0.182ˆx −0.913ˆy + 0.365ˆz (4.11) Multiplication of a Vector by a Scalar. Let’s say a particular force is speciﬁed by a vector F. What is the new vector if this force is doubled? The answer is simply 2F – that is, twice the magnitude applied in the same direction. This is an example of scalar multiplication of a vector. Generalizing, the product of the scalar α and the vector A is simply αA. Scalar (“Dot”) Product of Vectors. Another common task in vector analysis is to determine the similarity in the direction in which two vectors point. In particular, it is useful to have a metric which, when applied to the vectors A = ˆaA and B = ˆbB, has the following properties (see Figure 4.7): • If A is perpendicular to B, the result is zero. • If A and B point in the same direction, the result is AB. • If A and B point in opposite directions, the result is −AB. • Results intermediate to these conditions depend on the angle ψ between A and B, measured as if A and B were arranged “tail-to-tail” as shown in Figure 4.8. c⃝K. Kikkeri CC BY SA 4.0 Figure 4.7: Special cases of the dot product. In vector analysis, this operator is known as the scalar product (not to be confused with scalar multiplication) or the dot product. The dot product is written A · B and is given in general by the expression: A · B = AB cos ψ (4.12) Note that this expression yields the special cases previously identiﬁed, which are ψ = π/2, ψ = 0, and ψ = π, respectively. The dot product is commutative; i.e., A · B = B · A (4.13) The dot product is also distributive; i.e., A · (B + C) = A · B + A · C (4.14) The dot product has some other useful properties. For example, note: A · A = (ˆxAx + ˆyAy + ˆzAz) · (ˆxAx + ˆyAy + ˆzAz) = ˆx · ˆxA2 x + ˆx · ˆyAxAy + ˆx · ˆzAxAz + ˆy · ˆxAxAy + ˆy · ˆyA2 y + ˆy · ˆzAyAz + ˆz · ˆxAxAz + ˆz · ˆyAyAz + ˆz · ˆzA2 z (4.15) which looks pretty bad until you realize that ˆx · ˆx = ˆy · ˆy = ˆz · ˆz = 1 (4.16)	Electromagnetics_Vol1_Page_88_Chunk1388
74 CHAPTER 4. VECTOR ANALYSIS A B ¥ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.8: Calculation of the dot product. and any other dot product of basis vectors is zero. Thus, the whole mess simpliﬁes to: A · A = A2 x + A2 y + A2 z (4.17) This is the square of the magnitude of A, so we have discovered that A · A = \|A\|2 = A2 (4.18) Applying the same principles to the dot product of potentially different vectors A and B, we ﬁnd: A · B = (ˆxAx + ˆyAy + ˆzAz) · (ˆxBx + ˆyBy + ˆzBz) = AxBx + AyBy + AzBz (4.19) This is a particularly easy way to calculate the dot product, since it eliminates the problem of determining the angle ψ. In fact, an easy way to calculate ψ given A and B is to ﬁrst calculate the dot product using Equation 4.19 and then use the result to solve Equation 4.12 for ψ. Example 4.2. Angle between two vectors. Consider the position vectors C = 2ˆx + 3ˆy + 1ˆz and D = 3ˆx −2ˆy + 2ˆz, both expressed in units of meters. Find the angle between these vectors. Solution. From Equation 4.12 C · D = CD cos ψ (4.20) where C = \|C\|, D = \|D\|, and ψ is the angle we seek. From Equation 4.19: C · D = CxDx + CyDy + CzDz (4.21) = 2 · 3 + 3 · (−2) + 1 · 2 m2 (4.22) = 2 m2 (4.23) also C = q C2x + C2y + C2z ∼= 3.742 m (4.24) D = q D2x + D2y + D2z ∼= 4.123 m (4.25) so cos ψ = C · D CD ∼= 0.130 (4.26) Taking the inverse cosine, we ﬁnd ψ = 82.6◦. Cross Product. The cross product is a form of vector multiplication that results in a vector that is perpendicular to both of the operands. The deﬁnition is as follows: A × B = ˆnAB sin ψAB (4.27) As shown in Figure 4.9, the unit vector ˆn is determined by the “right hand rule.” Using your right hand, curl your ﬁngers to traverse the angle ψAB beginning at A and ending at B, and then ˆn points in the direction of your fully-extended thumb. It should be apparent that the cross product is not commutative but rather is anticommutative; that is, A × B = −B × A (4.28) You can conﬁrm this for yourself using either Equation 4.27 or by applying the right-hand rule. The cross product is distributive: A × (B + C) = A × B + A × C (4.29) There are two useful special cases of the cross product that are worth memorizing. The ﬁrst is the cross product of a vector with itself, which is zero: A × A = 0 (4.30) The second is the cross product of vectors that are perpendicular; i.e., for which ψAB = π/2. In this	Electromagnetics_Vol1_Page_89_Chunk1389
4.1. VECTOR ARITHMETIC 75 n B A AB ¦ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.9: The cross product A × B. x y z Figure 4.10: Cross products among basis vectors in the Cartesian system. The cross product of any two basis vectors is the third basis vector when the order of operands is counter-clockwise, as shown in the dia- gram, and is −1 times the third basis vector when the order of operands is clockwise with respect to the ar- rangement in the diagram. case: A × B = ˆnAB (4.31) Using these principles, note: ˆx × ˆx = ˆy × ˆy = ˆz × ˆz = 0 (4.32) whereas ˆx × ˆy = ˆz (4.33) ˆy × ˆz = ˆx (4.34) ˆz × ˆx = ˆy (4.35) A useful diagram that summarizes these relationships is shown in Figure 4.10. It is typically awkward to “manually” determine ˆn in Equation 4.27. However, in Cartesian coordinates the cross product may be calculated as: A × B = ˆx (AyBz −AzBy) + ˆy (AzBx −AxBz) + ˆz (AxBy −AyBx) (4.36) This may be easier to remember as a matrix determinant: A × B = ˆx ˆy ˆz Ax Ay Az Bx By Bz (4.37) Similar expressions are available for other coordinate systems. Vector analysis routinely requires expressions involving both dot products and cross products in different combinations. Often, these expressions may be simpliﬁed, or otherwise made more convenient, using the vector identities listed in Appendix B.3. Additional Reading: • “Vector ﬁeld” on Wikipedia. • “Vector algebra” on Wikipedia.	Electromagnetics_Vol1_Page_90_Chunk1390
76 CHAPTER 4. VECTOR ANALYSIS 4.2 Cartesian Coordinates [m0004] The Cartesian coordinate system is introduced in Section 4.1. Concepts described in that section – i.e., the dot product and cross product – are described in terms of the Cartesian system. In this section, we identify some additional features of this system that are useful in subsequent work and also set the stage for alternative systems; namely the cylindrical and spherical coordinate systems. Integration Over Length. Consider a vector ﬁeld A = ˆxA(r), where r is a position vector. What is the integral of A over some curve C through space? To answer this question, we ﬁrst identify a differential-length segment of the curve. Note that this segment of the curve can be described as dl = ˆxdx + ˆydy + ˆzdz (4.38) The contribution to the integral for that segment of the curve is simply A · dl. We integrate to obtain the result; i.e., Z C A · dl (4.39) For example, if A = ˆxA0 (i.e., A(r) is a constant) and if C is a straight line from x = x1 and x = x2 along some constant y and z, then dl = ˆxdx, A · dl = A0dx, and subsequently the above integral is Z x2 x1 A0dx = A0 (x2 −x1) (4.40) In particular, notice that if A0 = 1, then this integral gives the length of C. Although the formalism seems unnecessary in this simple example, it becomes very useful when integrating over paths that vary in more than one direction and with more complicated integrands. Note that the Cartesian system was an appropriate choice for preceding example because this allowed two of the three basis directions (i.e., y and z) to be immediately eliminated from the calculation. Said differently, the preceding example is expressed with the minimum number of varying coordinates in the Cartesian system. Here’s a counter-example. If C had been a circle in the z = 0 plane, then the problem would have required two basis directions to be considered – namely, both x and y. In this case, another system – namely, cylindrical coordinates (Section 4.3) – minimizes the number of varying coordinates (to just one, which is φ). Integration Over Area. Now we ask the question, what is the integral of some vector ﬁeld A over some surface S? The answer is Z S A · ds (4.41) We refer to ds as the differential surface element, which is a vector having magnitude equal to the differential area ds, and is normal (perpendicular) to each point on the surface. There are actually two such directions. We’ll return to clear up the ambiguity in a moment. Now, as an example, if A = ˆz and S is the surface bounded by x1 ≤x ≤x2, y1 ≤y ≤y2, then ds = ˆz dx dy (4.42) since dxdy is differential surface area in the z = 0 plane and ˆz is normal to the z = 0 plane. So A · ds = dxdy, and subsequently the integral in Equation 4.41 becomes Z x2 x1 Z y2 y1 dx dy = (x2 −x1) (y2 −y1) (4.43) Note that this has turned out to be a calculation of area. Once again, we see the Cartesian system was an appropriate choice for this example because this choice minimizes the number of varying coordinates; in the above example, the surface of integration is described by a constant value of z with variable values of x and y. If the surface had instead been a cylinder or a sphere, not only would all three basis directions be variable, but also the surface normal would be variable, making the problem dramatically more complicated. Now let’s return to the issue of the direction of ds. We chose +ˆz, but why not choose −ˆz – also a normal to the surface – as this direction? To answer simply, the resulting area would be negative. “Negative area” is the expected (“positive”) area, except with respect to the opposite-facing normal vector. In the present problem, the sign of the area is not important, but	Electromagnetics_Vol1_Page_91_Chunk1391
in	Electromagnetics_Vol1_Page_91_Chunk1392
4.3. CYLINDRICAL COORDINATES 77 some problems this sign becomes important. One example of a class of problems for which the sign of area is important is when the quantity of interest is a ﬂux. If A were a ﬂux density, then the integration over area that we just performed indicates the magnitude and direction of ﬂux, and so the direction chosen for ds deﬁnes the direction of positive ﬂux. Section 2.4 describes the electric ﬁeld in terms of a ﬂux (i.e., as electric ﬂux density D), in which case positive ﬂux ﬂows away from a positively-charged source. Integration Over Volume. Another common task in vector analysis is integration of some quantity over a volume. Since the procedure is the same for scalar or vector quantities, we shall consider integration of a scalar quantity A(r) for simplicity. First, we note that the contribution from a differential volume element dv = dx dy dz (4.44) is A(r) dv, so the integral over the volume V is Z V A(r) dv (4.45) For example, if A(r) = 1 and V is a cube bounded by x1 ≤x ≤x2, y1 ≤y ≤y2, and z1 ≤z ≤z2, then the above integral becomes Z x2 x1 Z y2 y1 Z z2 z1 dx dy dz = (x2 −x1) (y2 −y1) (z2 −z1) (4.46) i.e., this is a calculation of volume. The Cartesian system was an appropriate choice for this example because V is a cube, which is easy to describe in Cartesian coordinates and relatively difﬁcult to describe in any other coordinate system. Additional Reading: • “Cartesian coordinate system” on Wikipedia. 4.3 Cylindrical Coordinates [m0096] Cartesian coordinates (Section 4.2) are not convenient in certain cases. One of these is when the problem has cylindrical symmetry. For example, in the Cartesian coordinate system, the cross-section of a cylinder concentric with the z-axis requires two coordinates to describe: x and y. However, this cross section can be described using a single parameter – namely the radius – which is ρ in the cylindrical coordinate system. This results in a dramatic simpliﬁcation of the mathematics in some applications. The cylindrical system is deﬁned with respect to the Cartesian system in Figure 4.11. In lieu of x and y, the cylindrical system uses ρ, the distance measured from the closest point on the z axis,1 and φ, the angle measured in a plane of constant z, beginning at the +x axis (φ = 0) with φ increasing toward the +y direction. The basis vectors in the cylindrical system are ˆρ, ˆφ, and ˆz. As in the Cartesian system, the dot product of like basis vectors is equal to one, and the dot product of unlike basis vectors is equal to zero. The cross 1Note that some textbooks use “r” in lieu of ρ for this coordinate. § x ¨ z © ª z « z c⃝K. Kikkeri CC BY SA 4.0 Figure 4.11: Cylindrical coordinate system and asso- ciated basis vectors.	Electromagnetics_Vol1_Page_92_Chunk1393
78 CHAPTER 4. VECTOR ANALYSIS ¬ z Figure 4.12: Cross products among basis vectors in the cylindrical system. (See Figure 4.10 for instructions on the use of this diagram.) products of basis vectors are as follows: ˆρ × ˆφ = ˆz (4.47) ˆφ × ˆz = ˆρ (4.48) ˆz × ˆρ = ˆφ (4.49) A useful diagram that summarizes these relationships is shown in Figure 4.12. The cylindrical system is usually less useful than the Cartesian system for identifying absolute and relative positions. This is because the basis directions depend on position. For example, ˆρ is directed radially outward from the ˆz axis, so ˆρ = ˆx for locations along the x-axis but ˆρ = ˆy for locations along the y axis. Similarly, the direction ˆφ varies as a function of position. To overcome this awkwardness, it is common to set up a problem in cylindrical coordinates in order to exploit cylindrical symmetry, but at some point to convert to Cartesian coordinates. Here are the conversions: x = ρ cos φ (4.50) y = ρ sin φ (4.51) and z is identical in both systems. The conversion from Cartesian to cylindrical is as follows: ρ = p x2 + y2 (4.52) φ = arctan (y, x) (4.53) where arctan is the four-quadrant inverse tangent function; i.e., arctan(y/x) in the ﬁrst quadrant (x > 0, y > 0), but possibly requiring an adjustment · ˆρ ˆφ ˆz ˆx cos φ −sin φ 0 ˆy sin φ cos φ 0 ˆz 0 0 1 Table 4.1: Dot products between basis vectors in the cylindrical and Cartesian coordinate systems. for the other quadrants because the signs of both x and y are individually signiﬁcant.2 Similarly, it is often necessary to represent basis vectors of the cylindrical system in terms of Cartesian basis vectors and vice-versa. Conversion of basis vectors is straightforward using dot products to determine the components of the basis vectors in the new system. For example, ˆx in terms of the basis vectors of the cylindrical system is ˆx = ˆρ (ˆρ · ˆx) + ˆφ ˆφ · ˆx + ˆz (ˆz · ˆx) (4.54) The last term is of course zero since ˆz · ˆx = 0. Calculation of the remaining terms requires dot products between basis vectors in the two systems, which are summarized in Table 4.1. Using this table, we ﬁnd ˆx = ˆρ cos φ −ˆφ sin φ (4.55) ˆy = ˆρ sin φ + ˆφ cos φ (4.56) and of course ˆz requires no conversion. Going from Cartesian to cylindrical, we ﬁnd ˆρ = ˆx cos φ + ˆy sin φ (4.57) ˆφ = −ˆx sin φ + ˆy cos φ (4.58) Integration Over Length. A differential-length segment of a curve in the cylindrical system is described in general as dl = ˆρdρ + ˆφρdφ + ˆz dz (4.59) Note that the contribution of the φ coordinate to differential length is ρdφ, not simply dφ. This is because φ is an angle, not a distance. To see why the associated distance is ρdφ, consider the following. 2Note that this function is available in MATLAB and Octave as atan2(y,x).	Electromagnetics_Vol1_Page_93_Chunk1394
4.3. CYLINDRICAL COORDINATES 79 l ® x ¯ ° ± c⃝K. Kikkeri CC BY SA 4.0 Figure 4.13: Example in cylindrical coordinates: The circumference of a circle. The circumference of a circle of radius ρ is 2πρ. If only a fraction of the circumference is traversed, the associated arclength is the circumference scaled by φ/2π, where φ is the angle formed by the traversed circumference. Therefore, the distance is 2πρ · φ/2π = ρφ, and the differential distance is ρdφ. As always, the integral of a vector ﬁeld A(r) over a curve C is Z C A · dl (4.60) To demonstrate the cylindrical system, let us calculate the integral of A(r) = ˆφ when C is a circle of radius ρ0 in the z = 0 plane, as shown in Figure 4.13. In this example, dl = ˆφ ρ0 dφ since ρ = ρ0 and z = 0 are both constant along C. Subsequently, A · dl = ρ0dφ and the above integral is Z 2π 0 ρ0 dφ = 2πρ0 (4.61) i.e., this is a calculation of circumference. Note that the cylindrical system is an appropriate choice for the preceding example because the problem can be expressed with the minimum number of varying coordinates in the cylindrical system. If we had attempted this problem in the Cartesian system, we would ﬁnd that both x and y vary over C, and in a relatively complex way.3 3Nothing will drive this point home more ﬁrmly than trying it. It can be done, but it’s a lot more work... ² x d³ ´dµ ds c⃝K. Kikkeri CC BY SA 4.0 Figure 4.14: Example in cylindrical coordinates: The area of a circle. Integration Over Area. Now we ask the question, what is the integral of some vector ﬁeld A over a circular surface S in the z = 0 plane having radius ρ0? This is shown in Figure 4.14. The differential surface vector in this case is ds = ˆz (dρ) (ρdφ) = ˆz ρ dρ dφ (4.62) The quantities in parentheses are the radial and angular dimensions, respectively. The direction of ds indicates the direction of positive ﬂux – see the discussion in Section 4.2 for an explanation. In general, the integral over a surface is Z S A · ds (4.63) To demonstrate, let’s consider A = ˆz; in this case A · ds = ρ dρ dφ and the integral becomes Z ρ0 0 Z 2π 0 ρ dρ dφ = Z ρ0 0 ρ dρ Z 2π 0 dφ = 1 2ρ2 0 (2π) = πρ2 0 (4.64) which we recognize as the area of the circle, as expected. The corresponding calculation in the Cartesian system is quite difﬁcult in comparison. Whereas the previous example considered a planar surface, we might consider instead a curved surface.	Electromagnetics_Vol1_Page_94_Chunk1395
80 CHAPTER 4. VECTOR ANALYSIS y ¶ · ¸ ¹2 º1 dϕ dz 0 c⃝K. Kikkeri CC BY SA 4.0 Figure 4.15: Example in cylindrical coordinates: The area of the curved surface of a cylinder. Here we go. What is the integral of a vector ﬁeld A = ˆρ over a cylindrical surface S concentric with the z axis having radius ρ0 and extending from z = z1 to z = z2? This is shown in Figure 4.15. The differential surface vector in this case is ds = ˆρ (ρ0dφ) (dz) = ˆρρ0 dφ dz (4.65) The integral is Z S A · ds = Z 2π 0 Z z2 z1 ρ0 dφdz = ρ0 Z 2π 0 dφ Z z2 z1 dz = 2πρ0 (z2 −z1) (4.66) which is the area of S, as expected. Once again, the corresponding calculation in the Cartesian system is quite difﬁcult in comparison. Integration Over Volume. The differential volume element in the cylindrical system is dv = dρ (ρdφ) dz = ρ dρ dφ dz (4.67) For example, if A(r) = 1 and the volume V is a cylinder bounded by ρ ≤ρ0 and z1 ≤z ≤z2, then Z V A(r) dv = Z ρ0 0 Z 2π 0 Z z2 z1 ρ dρ dφ dz = Z ρ0 0 ρ dρ Z 2π 0 dφ Z z2 z1 dz = πρ2 0 (z2 −z1) (4.68) i.e., area times length, which is volume. Once again, the procedure above is clearly more complicated than is necessary if we are interested only in computing volume. However, if the integrand is not constant-valued then we are no longer simply computing volume. In this case, the formalism is appropriate and possibly necessary. Additional Reading: • “Cylindrical coordinate system” on Wikipedia.	Electromagnetics_Vol1_Page_95_Chunk1396
4.4. SPHERICAL COORDINATES 81 4.4 Spherical Coordinates [m0097] The spherical coordinate system is deﬁned with respect to the Cartesian system in Figure 4.16. The spherical system uses r, the distance measured from the origin;4 θ, the angle measured from the +z axis toward the z = 0 plane; and φ, the angle measured in a plane of constant z, identical to φ in the cylindrical system. Spherical coordinates are preferred over Cartesian and cylindrical coordinates when the geometry of the problem exhibits spherical symmetry. For example, in the Cartesian coordinate system, the surface of a sphere concentric with the origin requires all three coordinates (x, y, and z) to describe. However, this surface can be described using a single constant parameter – the radius r – in the spherical coordinate system. This leads to a dramatic simpliﬁcation in the mathematics in certain applications. 4Note that some textbooks use “R” in lieu of r for this coordi- nate. y » ¼ z r θ θ ϕ r c⃝K. Kikkeri CC BY SA 4.0 Figure 4.16: Spherical coordinate system and associ- ated basis vectors. θ ϕ r Figure 4.17: Cross products among basis vectors in the spherical system. (See Figure 4.10 for instructions on the use of this diagram.) The basis vectors in the spherical system are ˆr, ˆθ, and ˆφ. As always, the dot product of like basis vectors is equal to one, and the dot product of unlike basis vectors is equal to zero. For the cross-products, we ﬁnd: ˆr × ˆθ = ˆφ (4.69) ˆθ × ˆφ = ˆr (4.70) ˆφ × ˆr = ˆθ (4.71) A useful diagram that summarizes these relationships is shown in Figure 4.17. Like the cylindrical system, the spherical system is often less useful than the Cartesian system for identifying absolute and relative positions. The reason is the same: Basis directions in the spherical system depend on position. For example, ˆr is directed radially outward from the origin, so ˆr = ˆx for locations along the x-axis but ˆr = ˆy for locations along the y axis and ˆr = ˆz for locations along the z axis. Similarly, the directions of ˆθ and ˆφ vary as a function of position. To overcome this awkwardness, it is common to begin a problem in spherical coordinates, and then to convert to Cartesian coordinates at some later point in the analysis. Here are the conversions: x = r cos φ sin θ (4.72) y = r sin φ sin θ (4.73) z = r cos θ (4.74) The conversion from Cartesian to spherical	Electromagnetics_Vol1_Page_96_Chunk1397
82 CHAPTER 4. VECTOR ANALYSIS · ˆr ˆθ ˆφ ˆx sin θ cos φ cos θ cos φ −sin φ ˆy sin θ sin φ cos θ sin φ cos φ ˆz cos θ −sin θ 0 Table 4.2: Dot products between basis vectors in the spherical and Cartesian coordinate systems. coordinates is as follows: r = p x2 + y2 + z2 (4.75) θ = arccos (z/r) (4.76) φ = arctan (y, x) (4.77) (4.78) where arctan is the four-quadrant inverse tangent function.5 Dot products between basis vectors in the spherical and Cartesian systems are summarized in Table 4.2. This information can be used to convert between basis vectors in the spherical and Cartesian systems, in the same manner described in Section 4.3; e.g. ˆx = ˆr (ˆr · ˆx) + ˆθ ˆθ · ˆx + ˆφ ˆφ · ˆx (4.79) ˆr = ˆx (ˆx · ˆr) + ˆy (ˆy · ˆr) + ˆz (ˆz · ˆr) (4.80) and so on. Example 4.3. Cartesian to spherical conversion. A vector ﬁeld G = ˆx xz/y. Develop an expression for G in spherical coordinates. Solution: Simply substitute expressions in terms of spherical coordinates for expressions in terms of Cartesian coordinates. Use Table 4.2 and Equations 4.72–4.74. Making these substitutions and applying a bit of mathematical clean-up afterward, one obtains G = ˆr sin θ cot φ + ˆφ cos θ cot φ −ˆφ · r cos θ cos φ (4.81) 5Note that this function is available in MATLAB and Octave as atan2(y,x). dl ρ z θ a ½ ¾ ¿ À Á Â Ã Ä c⃝K. Kikkeri CC BY SA 4.0 Figure 4.18: Example in spherical coordinates: Pole- to-pole distance on a sphere. Integration Over Length. A differential-length segment of a curve in the spherical system is dl = ˆr dr + ˆθ r dθ + ˆφ r sin θ dφ (4.82) Note that θ is an angle, as opposed to a distance. The associated distance is r dθ in the θ direction. Note also that in the φ direction, distance is r dφ in the z = 0 plane and less by the factor sin θ for z <> 0. As always, the integral of a vector ﬁeld A(r) over a curve C is Z C A · dl (4.83) To demonstrate line integration in the spherical system, imagine a sphere of radius a centered at the origin with “poles” at z = +a and z = −a. Let us calculate the integral of A(r) = ˆθ, where C is the arc drawn directly from pole to pole along the surface of the sphere, as shown in Figure 4.18. In this example, dl = ˆθ a dθ since r = a and φ (which could be any value) are both constant along C. Subsequently, A · dl = a dθ and the above integral is Z π 0 a dθ = πa (4.84) i.e., half the circumference of the sphere, as expected. Note that the spherical system is an appropriate choice for this example because the problem can be	Electromagnetics_Vol1_Page_97_Chunk1398
4.4. SPHERICAL COORDINATES 83 expressed with the minimum number of varying coordinates in the spherical system. If we had attempted this problem in the Cartesian system, we would ﬁnd that both z and either x or y (or all three) vary over C and in a relatively complex way. Integration Over Area. Now we ask the question, what is the integral of some vector ﬁeld A over the surface S of a sphere of radius a centered on the origin? This is shown in Figure 4.19. The differential surface vector in this case is ds = ˆr (r dθ) (r sin θ dφ) = ˆr r2 sin θ dθ dφ (4.85) As always, the direction is normal to the surface and in the direction associated with positive ﬂux. The quantities in parentheses are the distances associated with varying θ and φ, respectively. In general, the integral over a surface is Z S A · ds (4.86) In this case, let’s consider A = ˆr; in this case A · ds = a2 sin θ dθ dφ and the integral becomes Z π 0 Z 2π 0 a2 sin θ dθ dφ = a2 Z π 0 sin θdθ Z 2π 0 dφ = a2 (2) (2π) = 4πa2 (4.87) which we recognize as the area of the sphere, as expected. The corresponding calculation in the Cartesian or cylindrical systems is quite difﬁcult in comparison. Integration Over Volume. The differential volume element in the spherical system is dv = dr (rdθ) (r sin θdφ) = r2dr sin θ dθ dφ (4.88) For example, if A(r) = 1 and the volume V is a a r sinθ dϕ y x z r dθ c⃝K. Kikkeri CC BY SA 4.0 Figure 4.19: Example in spherical coordinates: The area of a sphere. sphere of radius a centered on the origin, then Z V A(r) dv = Z a 0 Z π 0 Z 2π 0 r2dr sin θ dθ dφ = Z a 0 r2dr Z π 0 sin θ dθ Z 2π 0 dφ = 1 3a3 (2) (2π) = 4 3πa3 (4.89) which is the volume of a sphere. Additional Reading: • “Spherical coordinate system” on Wikipedia.	Electromagnetics_Vol1_Page_98_Chunk1399
84 CHAPTER 4. VECTOR ANALYSIS 4.5 Gradient [m0098] The gradient operator is an important and useful tool in electromagnetic theory. Here’s the main idea: The gradient of a scalar ﬁeld is a vector that points in the direction in which the ﬁeld is most rapidly increasing, with the scalar part equal to the rate of change. A particularly important application of the gradient is that it relates the electric ﬁeld intensity E(r) to the electric potential ﬁeld V (r). This is apparent from a review of Section 2.2 (“Electric Field Intensity”); see in particular, the battery-charged capacitor example. In that example, it is demonstrated that: • The direction of E(r) is the direction in which V (r) decreases most quickly, and • The scalar part of E(r) is the rate of change of V (r) in that direction. Note that this is also implied by the units, since V (r) has units of V whereas E(r) has units of V/m. The gradient is the mathematical operation that relates the vector ﬁeld E(r) to the scalar ﬁeld V (r) and is indicated by the symbol “∇” as follows: E(r) = −∇V (r) (4.90) or, with the understanding that we are interested in the gradient as a function of position r, simply E = −∇V (4.91) At this point we should note that the gradient is a very general concept, and that we have merely identiﬁed one application of the gradient above. In electromagnetics there are many situations in which we seek the gradient ∇f of some scalar ﬁeld f(r). Furthermore, we ﬁnd that other differential operators that are important in electromagnetics can be interpreted in terms of the gradient operator ∇. These include divergence (Section 4.6), curl (Section 4.8), and the Laplacian (Section 4.10). In the Cartesian system: ∇f = ˆx∂f ∂x + ˆy∂f ∂y + ˆz∂f ∂z (4.92) Example 4.4. Gradient of a ramp function. Find the gradient of f = ax (a “ramp” having slope a along the x direction). Solution. Here, ∂f/∂x = a and ∂f/∂y = ∂f/∂z = 0. Therefore ∇f = ˆxa. Note that ∇f points in the direction in which f most rapidly increases, and has magnitude equal to the slope of f in that direction. The gradient operator in the cylindrical and spherical systems is given in Appendix B.2. Additional Reading: • “Gradient” on Wikipedia.	Electromagnetics_Vol1_Page_99_Chunk1400

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