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LOW-FREQUENCYELECTROMAGNETICMODELING FORELECTRICAL ANDBIOLOGICAL SYSTEMSUSING MATLABreg


SERGEY N MAKAROV

GREGORY M NOETSCHER

ARA NAZARIAN

Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

MATLAB and Simulink are registered trademarks of The MathWorks Inc See wwwmathworkscomtrademarks for alist of additional trademarks The MathWorks Publisher Logo identifies books that contain MATLABreg contentUsed with permission The MathWorks does not warrant the accuracy of the text or exercises in this book orin the software downloadable from httpwwwwileycomWileyCDAWileyTitleproductCd-047064477Xhtml andhttpwwwmathworkscom matlabcentralfileexchangeterm=authorid3A80973 The bookrsquos or downloadablesoftwarersquos use or discussion of MATLABreg software or related products does not constitute endorsement orsponsorship by The MathWorks of a particular use of the MATLABreg software or related products

For MATLABreg and Simulinkreg product information or information on other related products please contactThe MathWorks Inc3 Apple Hill DriveNatick MA 01760-2098 USATel 508-647-7000Fax 508-647-7001E-mail infomathworkscomWeb wwwmathworkscomHow to buy wwwmathworkscomstore

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Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparingthis book they make no representations or warranties with respect to the accuracy or completeness of the contentsof this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purposeNo warranty may be created or extended by sales representatives or written sales materials The advice and strategiescontained herein may not be suitable for your situation You should consult with a professional where appropriateNeither the publisher nor author shall be liable for any loss of profit or any other commercial damages includingbut not limited to special incidental consequential or other damages

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MATLABreg is a trademark of The MathWorks Inc and is used with permission The MathWorks does not warrantthe accuracy of the text or exercises in this book This bookrsquos use or discussion of MATLABreg software or relatedproducts does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approachor particular use of the MATLABreg software

Library of Congress Cataloging-in-Publication Data

Makarov Sergey NLow-frequency electromagnetic modeling for electrical and biological systems using MATLABreg Sergey N Makarov Gregory M Noetscher Ara Nazarian

pages cmIncludes bibliographical references and indexISBN 978-1-119-05256-2 (cloth)

1 ELF electromagnetic fieldsndashMathematical models 2 Electromagnetic devicesndashComputersimulation 3 ElectromagnetismndashComputer simulation 4 BioelectromagnetismndashComputersimulation 5 MATLAB I Noetscher Gregory M 1978ndash II Nazarian Ara 1971ndash III TitleTK78672M34 2016621382 24028553ndashdc23

2015004420

Cover image courtesy of the authors

Set in 1012pt Times by SPi Global Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

1 2016

To Natasha Yen and Rosalynn

CONTENTS

PREFACE xi

ACKNOWLEDGMENTS xv

ABOUT THE COMPANION WEBSITE xvii

PART I LOW-FREQUENCY ELECTROMAGNETICSCOMPUTATIONAL MESHESCOMPUTATIONAL PHANTOMS 1

1 Classification of Low-Frequency Electromagnetic ProblemsPoisson and Laplace Equations in Integral Form 3

Introduction 311 Classification of Low-Frequency Electromagnetic Problems 412 Poisson and Laplace Equations Boundary Conditions

and Integral Equations 18References 30

2 Triangular Surface Mesh Generation and Mesh Operations 35

Introduction 3521 Triangular Mesh and its Quality 3622 Delaunay Triangulation 3D Volume and Surface Meshes 4623 Mesh Operations and Transformations 5624 Adaptive Mesh Refinement and Mesh Decimation 7525 Summary of MATLABreg Scripts 81References 85

3 Triangular Surface Human Body Meshes for Computational Purposes 89

Introduction 8931 Review of Available Computational Human Body Phantoms

and Datasets 9232 Triangular Human Body Shell Meshes Included with the Text 9633 VHP-F Whole-Body Model Included with the Text 108References 126

PART II ELECTROSTATICS OF CONDUCTORS ANDDIELECTRICS DIRECT CURRENT FLOW 131

4 Electrostatics of Conductors Fundamentals of the Methodof Moments Adaptive Mesh Refinement 133

Introduction 13341 Electrostatics of Conductors MoM (Surface Charge

Formulation) 13442 Gaussian Quadratures Potential Integrals Adaptive Mesh

Refinement 14743 Summary of MATLABreg Modules 162References 167

5 Theory and Computation of Capacitance ConductingObjects in External Electric Field 169

Introduction 16951 Capacitance Definitions Self-Capacitance 17052 Capacitance of Two Conducting Objects 18053 Systems of Three Conducting Objects 18854 Isolated Conducting Object in an External Electric Field 19655 Summary of MATLABreg Modules 204References 212

6 Electrostatics of Dielectrics and Conductors 215

Introduction 21561 Dielectric Object in an External Electric Field 21662 Combined MetalndashDielectric Structures 22963 Application Example Modeling Charges in Capacitive

Touchscreens 23964 Summary of MATLABreg Modules 245References 253

viii CONTENTS

7 Transmission Lines Two-Dimensional Version of theMethod of Moments 257

Introduction 25771 Transmission Lines Value of the Electrostatic

ModelmdashAnalytical Solutions 25872 The 2D Version of the MoM for Transmission Lines 27373 Summary of MATLABreg Modules 284References 287

8 Steady-State Current Flow 289

Introduction 28981 Boundary Conditions Integral Equation Voltage and Current

Electrodes 29082 Analytical Solutions for DC Flow in Volumetric Conducting

Objects 30083 MoM Algorithm for DC Flow Construction of

Electrode Mesh 31184 Application Example EIT 32085 Application Example tDCS 32786 Summary of MATLABreg Modules 336References 341

PART III LINEAR MAGNETOSTATICS 347

9 Linear Magnetostatics Surface Charge Method 349

Introduction 34991 Integral Equation of Magnetostatics Surface Charge

Method 35092 Analytical versus Numerical Solutions Modeling Magnetic

Shielding 35893 Summary of MATLABreg Modules 367References 369

10 Inductance Coupled Inductors Modeling of a Magnetic Yoke 371

Introduction 371101 Inductance 372102 Mutual Inductance and Systems of Coupled Inductors 385103 Modeling of a Magnetic Yoke 404104 Summary of MATLABreg Modules 415References 421

ixCONTENTS

PART IV THEORY AND APPLICATIONS OF EDDY CURRENTS 423

11 Fundamentals of Eddy Currents 425

Introduction 425111 Three Types of Eddy Current Approximations 426112 Exact Solution for Eddy Currents without Surface Charges

Created by Horizontal Loops of Current 440113 Exact Solution for a Sphere in an External AC Magnetic Field 453114 A Simple Approximate Solution for Eddy Currents in a

Weakly Conducting Medium 460115 Summary of MATLABreg Modules 464References 470

12 Computation of Eddy Currents via the Surface Charge Method 473

Introduction 473121 Numerical Solution in a Weakly Conducting Medium

with External Magnetic Field 474122 Comparison with FEM Solutions from Maxwell 3D

of ANSYS Solution Convergence 481123 Eddy Currents Excited by a Coil 488124 Summary of MATLABreg Modules 497References 504

PART V NONLINEAR ELECTROSTATICS 507

13 Electrostatic Model of a pn-Junction Governing Equationsand Boundary Conditions 509

Introduction 509131 Built-in Voltage of a pn-Junction 510132 Complete Electrostatic Model of a pn-Junction 533References 545

14 Numerical Simulation of pn-Junction and Related ProblemsGummelrsquos Iterative Solution 547

Introduction 547141 Iterative Solution for Zero Bias Voltage 548142 Numerical Solution for the Electric Field Region 560143 Analytical Solution for the Diffusion Region

Shockley Equation 579144 Summary of MATLABreg Modules 587References 588

INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT

This text provides a systematic detailed and design-oriented course on elec-tromagnetic modeling at low frequencies for electrical and biological systemsLow-frequency electromagnetic modeling which is also known as a static orquasistatic approximation is a well-established theoretical subject Today the roleof low-frequency electromagnetic modeling in system design and testing is dominantin many disciplines Examples include capacitive touchscreens in cellphonesthe near-field wireless link between two cellphones or in implanted devices powerelectronics various bioelectromagnetic stimulation setups modern biomolecularelectrostatics and many others The text is divided into five parts

Part I Low-Frequency Electromagnetics Computational MeshesComputational Phantoms

Part II Electrostatics of Conductors and Dielectrics Direct Current FlowPart III Linear MagnetostaticsPart IV Theory and Applications of Eddy CurrentsPart V Nonlinear Electrostatics

DISTINCT FEATURES

A unique feature of this text is the combination of fundamental electromagnetictheory and application-oriented computation algorithms realized in the form of dis-tinct MATLABreg modules The modules are stand alone open-source simulators

which have a user-friendly and intuitive GUI and a highly visualized interactive out-put They are accessible to all MATLAB users No additional MATLAB toolboxesare necessary The modules may be either employed along with this text or usedand modified independently for both research and demonstration purposes

Yet another unique feature of the text is a large collection of computationalhuman phantoms including segmentation of the Visible Human Projectreg datasetperformed over the last four years In 2014 this model was evaluated and acceptedby the IEEE International Committee on Electromagnetic Safety for the calculationof specific absorption rates The computational human phantoms are an integralpart of the present text Simultaneously they can be imported into major commercialelectromagnetic software packages such as ANSYS COMSOL and CST

AUDIENCE

The text is intended for use in courses on computational electromagnetics and incourses covering general electromagnetics and bioelectromagnetics The targetedaudience includes electrical and biomedical engineering students at the graduate orsenior undergraduate levels as well as practicing researchers engineers and medicaldoctors working on sensor and bioelectromagnetic applications The MATLAB mod-ules can be used for demonstration purposes in any undergraduate classes

NUMERICAL ALGORITHM

The three-dimensional method of moments (MoM) which is the surface chargeboundary element method is studied and utilized throughout the text It is applicableto all linear static and quasistatic problems considered herein including heterogeneousobjects such as human tissues The major development steps of the MoM approachare not collected in a single chapter They are specifically quantified and explainedfor each distinct physical problem pertaining to quasistatic electromagnetics Thesesteps include

bull Poisson and Laplace equations in one two and three dimensions (Chapters 1 4 6and 8ndash14)

bull boundary conditions and the corresponding integral equations in terms of therelevant surface charge density (Chapters 1 and 4ndash12)

bull construction of the MoM matrix direct and iterative solutions (Chapters 4ndash8)

bull methods for computations of potential integrals and their implementationGaussian quadratures (Chapters 4 and 6)

bull Adaptive mesh refinement (Chapters 2 4 and 7)

bull MoM basis functions (Chapter 4)

bull Cancellation error (Chapter 9)

xii PREFACE

For nonlinear problems the situation complicates An iterative solution may beapplied Chapters 13 and 14 study an important example of a nonlinear solution inone dimension for semiconductor pn-junction modeling

APPLICATION EXAMPLES

Application examples included in this text are related to both electrical and biologicalproblems They cover all major subjects of low-frequency electromagnetic theoryThese computational examples are applicable to many practical engineering problemsand are designed to gain readerrsquos interest and motivation in the subject matter Theexamples include

bull self-capacitance of a human body modeling of ESDmdashelectrostatic discharge(Chapter 5)

bull capacitances of two or three arbitrary conductorsmdashcapacitive sensors (Chapter 5)

bull human body computational phantom under a power line (Chapter 5)

bull dielectric objects subjected to an applied electric field (Chapter 6)

bull metal-dielectric capacitors (Chapter 6)

bull modeling charges in cellphone capacitive touchscreens (Chapter 6)

bull modeling two-dimensional single-ended and differential transmission lines(Chapter 7)

bull electric impedance tomographymdashsimple shapes and human tissues (Chapter 8)

bull transcranial direct current stimulation of human tissues (Chapter 8)

bull magnetic objects subjected to an applied magnetic field (Chapter 9)

bull static magnetic shielding with a magnetic material (Chapter 9)

bull computation of self- and mutual-coil inductances withwithout magnetic core(Chapter 10)

bull wireless inductive power transfer between two arbitrary inductors (Chapter 10)

bull modeling gap field and leakage flux of a magnetic yoke (Chapter 10)

bull eddy currents created by loop(s) of current in a conducting specimen (Chapter 11)

bull upper estimate of eddy currents (Chapter 11)

bull eddy currents excited in a human body (Chapter 12)

bull nonlinear electrostatic modeling of a pn-junctionmdashjunction capacitance(Chapter 14)

Every application example is demonstrated with a distinct stand-alone MATLABmodule which can be extended and modified for relevant research purposes

ANALYTICAL SOLUTIONS

Complete or summarized analytical solutions to a large number of quasistatic electro-magnetic problems are presented throughout the text These solutions provide a

xiiiPREFACE

foundation for understanding the physics behind electromagnetic equations andas benchmarks against which numerical solutions have been tested The analyticalsolutions include

bull self-capacitance of a conducting sphere and a conducting circle (Chapter 5)

bull capacitance of two non-concentric conducting spheres (Chapter 5)

bull a conducting sphere in an external uniform electric field (Chapter 5)

bull a dielectric sphere in an external uniform electric field (Chapter 6)

bull characteristic impedances of wire transmission lines (Chapter 7)

bull characteristic impedancespropagation speed of printed transmission lines(Chapter 7)

bull circular voltage electrode on top of an infinite half-space (Chapter 8)

bull circular current electrode(s) on top of an infinite half-space with an intermediatelayer (Chapter 8)

bull magnetic sphere in an external uniform magnetic field (Chapter 9)

bull magnetic hollow sphere in an external uniform magnetic field (Chapter 9)

bull formulas for inductance of simple geometriesmdashshortlong solenoids loops(Chapter 10)

bull planar skin layer (Chapter 11)

bull conducting specimen in a uniform rotating magnetic field (Chapter 11)

bull eddy currents generated in a conducting half-space by horizontal loop(s) of excit-ing current (or coils of such loops) (Chapter 11)

bull eddy currents in a conducting sphere subject to a uniform external AC magneticfield (Chapter 11)

bull built-in voltage of a semiconductor pn-junction Boltzmann statistics for carrierconcentrations (Chapters 13 and 14)

ORGANIZATION OF THE TEXT

Since different sections in the chapters often cover separate subjects we presentseparate lists of problems for each section Each chapter concludes with a summaryof the corresponding MATLAB modules All MATLAB modules have been madeavailable on wwwwileycomgolowfrequencyelectromagneticmodeling

OTHER COMPUTATIONAL SOFTWARE

Where appropriate numerical solutions given in the text are compared with finite-element solutions generated by the commercial low-frequency simulator Maxwell 3Dof ANSYS

xiv PREFACE

ACKNOWLEDGMENTS

The authors are thankful to Professor Reinhold Ludwig (ECE Worcester Poly-technic Institute) Professor Alexander E Emanuel (ECE Worcester PolytechnicInstitute) Professor Alvaro Pascual-Leone (Harvard University Medical School)Professor Michael W Prairie (ECE Norwich University) Dr Mikhail Kozlov(Neurophysics Department Max Plank Institute for Human Cognitive and BrainSciences Leipzig Germany) Professor Dr-Ing Martin Ochmann (Beuth Universityof Technology Berlin Germany) Dr Hans Steyskal (Air Force retired) Dr MarcThompson (Thompson Consulting Inc Harvard MA) Dr Sara Louie (ANSYSInc) Dr Aghogho Obi (Biomedical and Electronic Systems Laboratory GEGlobal Research) Dr Farid U Dowla (Lawrence Livermore National Laboratory)Dr Vishwanath Iyer (MathWorks Inc) Dr Shashank Kulkarni (MathWorksInc) Dr Mark Reichelt (MathWorks Inc) Dr Charles H Moses (National GridMA) Mr Jeremy Carson (Natick Soldier Research Development and EngineeringCenter) and Dr Alexander Prokop (CST-Computer Simulation Technology AGDarmstadt Germany) who provided useful insight and stimulated discussionsleading to the present text

The VHP-Female full-body computational phantom and some MATLABreg

processing algorithms included with the text are a joint effort of NEVA Electro-magnetics LLC Yarmouth Port MA and the ECE Department at WorcesterPolytechnic Institute MA The authors thank former and current students andemployees including Mr Xingchi Dai Mr Aung Thu Htet Mr Anh Le TranMr Vishal Kumar Rathi Mr Htay AungWinMsMariyaM ZagalskayaMr JeffreyM Elloian Ms Niang Suan Thang and Mr Xavier Jamel Jackson We trulyappreciate the tremendous help and various contributions of Mr JanakinadhYanamadala a PhD candidate at Worcester Polytechnic Institute Finally we arethankful to Dr Glenn H Larsen of National Science Foundation for his supportof the development of computational human phantoms for FEM modeling

ABOUT THE COMPANION WEBSITE

This book is accompanied by a companion website

wwwwileycomgolowfrequencyelectromagneticmodeling

The website includes downloads or links for

bull MATLABreg modules for individual chapters intended for modeling basicelectrostatic direct-current magnetostatic and eddy-current problems

bull Sample project files introducing various electromagnetic sensor configurationspower electronics problems and bioelectromagnetic problems

bull Basic mesh generators and major mesh processing tools

bull Nineteen human body shells from real subjects in the form of triangular surfacemeshes optimized for FEM modeling

bull The full-body computational human phantom VHP-Female with over 120individual tissues in the form of non-intersecting 2-manifold triangular surfacemeshes optimized for FEM modeling in STL and NASTRAN formats

bull Full body computational human phantom VHP-Female in ANSYSMAXWELL3D format with material properties from 50 Hz to 1 MHz (uniformstep size 1 kHz)

bull Full body computational human phantom VHP-Female in ANSYS HFSS formatwith material properties from 10 MHz to 60 GHz (non-uniform step size)

bull Different fat shells for the VHP-Female phantom reflecting variable BMI(body mass index)

bull Separate regions of the VHP-Female phantom (head head and shoulders thoraxabdomen pelvic region thighs etc)

bull Color versions of all figures

PART I

LOW-FREQUENCYELECTROMAGNETICSCOMPUTATIONAL MESHESCOMPUTATIONAL PHANTOMS

1CLASSIFICATION OF LOW-FREQUENCY ELECTROMAGNETICPROBLEMS POISSON AND LAPLACEEQUATIONS IN INTEGRAL FORM

INTRODUCTION

The first section of this chapter starts with a physical model of an electric circuit Thisexample allows us to introduce and visualize the following primary research areas ofstatic and quasistatic analyses

bull Electrostatics

bull Magnetostatics

bull Direct current (DC) flow

bull Eddy current quasistatic approximation

Next we quantify the necessary physical conditions that justify static and quasi-static approximations of Maxwellrsquos equations Three major dimensionless parametersencountered in static and quasistatic approximations are as follows

bull The ratio of problem dimensions to the wavelength

bull The ratio of charge relaxation time to the wave period

bull The ratio of problem dimensions to the skin depth

The end of the first section is devoted to nonlinear electrostatics which is an impor-tant part of semiconductor device analysis with critical analogues to the subject ofbimolecular research

Low-Frequency Electromagnetic Modeling for Electrical and Biological Systems Using MATLABregFirst Edition Sergey N Makarov Gregory M Noetscher and Ara Nazariancopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons IncCompanion website wwwwileycomgolowfrequencyelectromagneticmodeling

The second section introduces the Poisson and Laplace equations along with thefree-space Greenrsquos function and briefly outlines the Greenrsquos function technique Wespecify Dirichlet Neumann and mixed boundary conditions and demonstrate practi-cal examples of each Special attention is paid to the integral form of the Poisson andLaplace equations which present the foundation for the boundary element method(BEM) We consider the surface charge density at boundaries as the unknown func-tion and thus utilize the surface charge method (SCM)

We establish the continuity of the potential function at boundaries and mathemat-ically derive the discontinuity condition for the normal potential derivative This con-dition provides the framework of almost all specific integral equations for individualstatic and quasistatic problems of various types studied in the main text

11 CLASSIFICATION OF LOW-FREQUENCYELECTROMAGNETIC PROBLEMS

Low-frequency electromagnetics finds its applications in many areas of electricalengineering including the fields of power electronics and power lines [1ndash4] semicon-ductor devices and integrated circuits [5 6] alternative energy [7] and nondestructivetesting and evaluation [8 9] Major biomedical applications include EEG ECG andEMG (cf [10 11]) biomedical impedance tomography [12ndash18] and rather new fieldssuch as biomolecular electrostatics [19ndash22] and magnetic [23ndash25] and DC [26ndash30]brain stimulation among many others

111 Physical Model of an Electric Circuit

The bulk of low-frequency electromagnetic problems may be visualized with the helpof a static or a quasistatic model of an electric circuit as shown in Figure 11 Themodel includes three elements

1 A voltage power source that in the direct current (DC) case generates a constantvoltage between its terminals

2 An electric load that consumes electric power The load may be modeled as aresistant material of low conductivity

3 Two finite-conductivity conductors that extend from the source to the loadThese wires form a transmission line In the laboratory both wires may be arbi-trarily bent However this is not the case in power electronics and high-frequency circuits

Figure 11a shows the (computed) electric field or electric field intensity E every-where in space The subject of electrostatics is the computation ofE and the associatedquantities (surface charges capacitances) when there is no load attached to the sourceIn other words there is no DC flow in the conductors In this case the field distributionaround the transmission line might be somewhat different from that shown inFigure 11a However the difference becomes negligibly small when the wires in

4 CLASSIFICATION OF LOW-FREQUENCY ELECTROMAGNETIC PROBLEMS

+ndash1V

Lines of force (E)

Equipotentialcontours

Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)

Lines of magnetic field (H)

Electriccurrent

Poynting vector

Load

Electrostatics direct current flow(a)

(b)Magnetostatics direct current flow

Lines ofmagneticfield (H)

Sʹ

Sʹ

FIGURE 11 Physical model of an electric circuit depicting (a) Electrostatics and(b) Magnetostatics scenarios produced by direct current flow Note that the electric fieldbetween the two wires decreases when moving from the source to the load This is not thecase when the wires have the infinite conductivity resulting in zero potential drop Thisfigure was generated using numerical modeling tools developed in the text

Figure 11a are close to idealmdashpossessing a very large conductivity The situationbecomes more complicated when a dielectric material which alters the electric fieldboth inside and outside is present

Exercise 11 How would the voltage (or potential) of two wires in Figure 11achange under open-circuit conditions (the electrostatic model)

Answer Both wire surfaces will become strictly equipotential surfaces say at1 and 0 V There will be no electric field within the wires themselves

The subject of DC computations is the evaluation of the electric field in conductorsthemselves and in the surrounding space This is exactly the problem shown inFigure 11a After the electric field E is found the current density J in the con-ductors is obtained as E multiplied by the conductivity (see Fig 11b) DC com-putations deal with finite-conductivity conductors whereas in electrostatics anyconductor is ideal At the same time electrostatics models dielectric materials orinsulators DC computations are typically not intended to do so since there is nocurrent present in insulators DC computations may deal with quite complicatedcurrent distributions in heterogeneous conducting media for example humantissues

Exercise 12 As far as DC flow is concerned Figure 11a and b has a few sim-plifications What is the most significant one

Answer The electric field distribution and the associated currentdistribution within the load may be highly nonuniform at least close to the loadterminals

The subject of magnetostatics is the computation of the magnetic field or mag-netic field intensity H and the associated quantities (mutual and self-induc-tances) The magnetic field is due to currents flowing in conductors as shownin Figure 11b Magnetostatics typically deals with external current excitationswhich are known a priori (eg from DC analysis) The situation complicateswhen a magnetic material which alters the magnetic field both inside and outsideis present

Exercise 13 After the magnetic fieldH and the electric field E in Figure 11b arefound a vector P=EtimesH (also shown in Fig 11b) may be constructed everywherein space What is the intuitive feel of this vector


Answer This is the Poynting vector a density-of-power flux with the units ofWm2 Its integral over the entire circuit cross section shown in Figure 11b willgive us the total power delivered to the load

The subject of eddy current theory (or quasistatic theory) is the effect of a time-varying magnetic field producing alternating currents According to Faradayrsquos lawof induction this magnetic field will create a secondary electric field in conductorsIn its turn the secondary electric field will result in certain currents known as eddycurrents These eddy currents may be excited in a conductor without immediate elec-trode contacts (which is to say in a wireless manner) Theymay also affect the originalalternating current distribution (via the skin layer effect) The situation greatly com-plicates for arbitrary geometries and in heterogeneous conducting media where eddycurrents have to cross boundaries between different materials

Exercise 14 As far as the eddy current theory is concerned Figure 11a and b hasa few simplifications What is the most significant one

Answer The current distribution in thick metal wire conductors is nonuniformeven at 60 Hz The current density mostly concentrates within a skin layer closeto the conductorrsquos surface

Finally the load in Figure 11 may be a basic semiconductor element a diode forexample The internal diode behavior at reverse and small forward-bias voltages isstill modeled by electrostatic equations but those equations will be nonlinear At largeforward-bias voltages DC theory is applied which also becomes nonlinear

112 Starting Point of StaticQuasistatic Analysis

In order to quantitatively explain various static and quasistatic approximations weneed to start with the full set of Maxwellrsquos equations which include electric field Emeasured in Vm magnetic fieldH measured in Am volumetric electric current den-sity J of free chargeswith the units of Am2 and the (volume or surface) electric chargedensity ρ of free charges with the units of Cm3 or Cm2 Permittivity ε measured inFm and permeability μ measured in Hm may vary in space Maxwellrsquos equations arethen given as

Amperersquos law modified by displacement currents

εpartEpartt

=nablatimesHminusJ 1 1

Faradayrsquos law

μpartHpartt

= minusnablatimesE 1 2

7CLASSIFICATION OF LOW-FREQUENCY ELECTROMAGNETIC PROBLEMS

Gaussrsquo law for electric fields

nabla εE= ρ 1 3

Gaussrsquo law for magnetic fields (no magnetic charges)

nabla μH = 0 1 4

Continuity equation for the electric current

partρ

partt+nabla J = 0 1 5

The continuity equation (15) for the electric current is not independent it isdirectly obtained from Equations 13 and 11 keeping in mind that the divergenceof the curl of any vector field is always 0 and that the medium is locally homogeneousElectric current is related to the electric field by a local form of Ohmrsquos law

J= σE 1 6

where σ is the medium conductivity with the units of Sm

113 Electrostatic Magnetostatic and DC Approximations

Certain approximations can be made when analyzing objects subject to electromag-netic excitation Consider an object under study of a certain size D with a givenelectromagnetic excitation at a frequency of f =ω 2π and the corresponding wave-length λ as shown in Figure 12 we assume that λ is the shortest wavelength in theobject material The necessary condition for both electrostatic and magnetostaticapproximations and for the DC approximation is the condition [32]

D λ λequivc

f c=

1με

1 7

The time derivative in Equations 11 and 12 may be approximated as part partt f Thespatial derivatives may be approximated as part partx part party part partz 1 D Therefore

D

D ltlt λ

λ

ε μ σ

FIGURE 12 Illustration of electrostatic and magnetostatic approximations


Equation 17 rewritten in the form f c D suggests that the two terms with time deri-vatives in both Equations 11 and 12 are much smaller than the terms with spatialderivatives and can therefore be entirely neglected The local speed of light c playsthe role of a proportionality constant when such a comparison is made This processof neglecting all terms with time derivatives in Equations 11ndash15 is the electrostaticandor magnetostatic approximation

114 Static Versus Parametric Quasistatic Analysis

1141 Parametric Dependence on Time The word ldquostaticrdquo used in the previoustext is somewhat confusing It often means not only the true steady-state problem butalso a large number of problems where the time dependence is present only parame-trically through time-dependent excitation conditions or otherwise An example iscurrent in a thin wire subject to a time-varying voltage The current remains the samealong the wire (follows a static pattern) which is then simply multiplied by a time-varying factor At the same time the absolute operating frequency may still be quitehighmdashon the order of tens or hundreds of kHz or so Therefore the ldquostaticrdquo approx-imation often also implies a low-frequency parametric approximation

1142 Radiation Conditions Any oscillating electromagnetic system eventuallyemits radio waves They can be extremely weak but they do exist at large distancesof r with r ge λ This effect is not described by the parametric quasistatic analysis

Example 11 Two electrodes attached to a human body are separated byD= 37 2 cm The electrodes source and sink a total current of i t = I0 cos 2πftwhere f = 10kHz I0 = 1mA Determine whether or not Equation 17 is satisfiedthat is whether or not the solution for the current distribution within the body maybe approximately given by the product J(r) cos 2πft where J(r) is the solution ofthe steady-state problem with the injection current I0

Solution We model the human body as a mass of muscle tissue with parametersε = 2 6 times 104ε0 μ= μ0 at 10 kHz [31] where ε0 = 8 854 times 10minus12 F m and μ0 =1 257 times 10minus6 H m are permittivity and permeability of vacuum respectively Inthis case the wavelength λ within the body is 186 m The ratio Dλ is thusequal to 0002 This reasonably small quantity allows us to consider the productJ(r) cos 2πft to be a viable solution though a more accurate and sophisticated anal-ysis might be necessary or desired

Electrostatic and DC approximations are studied in Parts II and III of this text

115 Eddy Current (Quasistatic) Approximation

The eddy current approximation is a true quasistatic approximation which cannot beobtained through the multiplication of a static solution by the time-varying factor It


relates to media with a large or significant conductivity Typical examples are metalsseawater human body tissues soils and similar materials The eddy current approx-imation in its most general form only affects Amperersquos law (11) This approximationwill be explained with reference to Figure 13 In terms of the general eddy currentapproximation we assume that the conduction current from Equation 16 dominatesthe displacement current in Amperersquos law that is for a periodic excitation

J = σE = σ E εpartEpartt

asympεω E 1 8

It is seen from Equation 18 that the following inequality should be satisfied

εω

σ1 or ωτ 1 τ =

ε

σ1 9

where ω is the angular frequency of interest and the constant τ is known as the chargerelaxation time Inequality (19) means that only the displacement current is neglectedthat is only the time derivative in Amperersquos law is neglected Faradayrsquos law of induc-tion still remains intact

Neglecting displacement currents implies that the wave propagation mechanismis lost we no longer permit transmission of electromagnetic waves Instead a diffu-sion equation will be obtained with a formally infinite propagation speed of smallperturbations

Example 12 A human body is subject to a 20 kHz AC magnetic field generatedby an external coil (see Fig 13) Determine whether or not Equation 19 is satis-fied that is if the displacement current can be neglected compared to the conduc-tion current

Solution We model the human body as a mass of muscle tissue with parametersε = 1 6 times 104ε0 σ = 0 35 S m at 20 kHz [31] We use the value ε0 = 8 854 times10minus12 F m and obtain the following value of the ratio in Equation 19

ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ

FIGURE 13 Illustration of eddy current (quasistatic) approximation




SERGEY N MAKAROV

GREGORY M NOETSCHER

ARA NAZARIAN














2015004420




10 9 8 7 6 5 4 3 2 1

1 2016


CONTENTS

PREFACE xi

ACKNOWLEDGMENTS xv





















viii CONTENTS
















ixCONTENTS
















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ
















2015004420




10 9 8 7 6 5 4 3 2 1

1 2016


CONTENTS

PREFACE xi

ACKNOWLEDGMENTS xv





















viii CONTENTS
















ixCONTENTS
















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ




CONTENTS

PREFACE xi

ACKNOWLEDGMENTS xv





















viii CONTENTS
















ixCONTENTS
















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ
















viii CONTENTS
















ixCONTENTS
















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ


















ixCONTENTS
















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ


















INDEX 591

x CONTENTS

PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ



PREFACE

SUBJECT OF THE TEXT




DISTINCT FEATURES




AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ





AUDIENCE


NUMERICAL ALGORITHM









xii PREFACE

























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ



























xiiiPREFACE






















xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ
























xiv PREFACE

ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ



ACKNOWLEDGMENTS


















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ

















PART I



INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ




INTRODUCTION


bull Electrostatics

bull Magnetostatics




















+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ














+ndash1V

Lines of force (E)


Electric field (E)


005 V

04 V+

ndash

Load

+ndash1V

Lines of force (E)


Electriccurrent

Poynting vector

Load




Sʹ

Sʹ



















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ




















εpartEpartt


Faradayrsquos law

μpartHpartt




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ




nabla εE= ρ 1 3


nabla μH = 0 1 4


partρ



J= σE 1 6




D λ λequivc

f c=

1με

1 7


D

D ltlt λ

λ

ε μ σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ















asympεω E 1 8


εω

σ1 or ωτ 1 τ =

ε

σ1 9





ltlt 1

i(t) = I0 cos 2πft

εω

ε σ

σ



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