Low-Frequency Electromagnetic Modeling for Electrical and Biological Systems Using MATLAB

PREFACE xi

ACKNOWLEDGMENTS xv

ABOUT THE COMPANION WEBSITE xvii

PART I LOW-FREQUENCY ELECTROMAGNETICS.COMPUTATIONAL MESHES.COMPUTATIONAL PHANTOMS 1

1 Classification of Low-Frequency Electromagnetic Problems. Poisson and Laplace Equations in Integral Form 3

Introduction 3

1.1 Classification of Low-Frequency Electromagnetic Problems 4

1.2 Poisson and Laplace Equations Boundary Conditions and Integral Equations 18

References 30

2 Triangular Surface Mesh Generation and Mesh Operations 35

Introduction 35

2.1 Triangular Mesh and its Quality 36

2.2 Delaunay Triangulation. 3D Volume and Surface Meshes 46

2.3 Mesh Operations and Transformations 56

2.4 Adaptive Mesh Refinement and Mesh Decimation 75

2.5 Summary of MATLAB® Scripts 81

References 85

3 Triangular Surface Human Body Meshes for Computational Purposes 89

Introduction 89

3.1 Review of Available Computational Human Body Phantoms and Datasets 92

3.2 Triangular Human Body Shell Meshes Included with the Text 96

3.3 VHP-F Whole-Body Model Included with the Text 108

References 126

PART II ELECTROSTATICS OF CONDUCTORS AND DIELECTRICS. DIRECT CURRENT FLOW 131

4 Electrostatics of Conductors. Fundamentals of the Method of Moments. Adaptive Mesh Refinement 133

Introduction 133

4.1 Electrostatics of Conductors. MoM (Surface Charge Formulation) 134

4.2 Gaussian Quadratures. Potential Integrals. Adaptive Mesh Refinement 147

4.3 Summary of MATLAB® Modules 162

References 167

5 Theory and Computation of Capacitance. Conducting Objects in External Electric Field 169

Introduction 169

5.1 Capacitance Definitions: Self-Capacitance 170

5.2 Capacitance of Two Conducting Objects 180

5.3 Systems of Three Conducting Objects 188

5.4 Isolated Conducting Object in an External Electric Field 196

5.5 Summary of MATLAB® Modules 204

References 212

6 Electrostatics of Dielectrics and Conductors 215

Introduction 215

6.1 Dielectric Object in an External Electric Field 216

6.2 Combined Metal-Dielectric Structures 229

6.3 Application Example: Modeling Charges in Capacitive Touchscreens 239

6.4 Summary of MATLAB® Modules 245

References 253

7 Transmission Lines: Two-Dimensional Version of the Method of Moments 257

Introduction 257

7.1 Transmission Lines: Value of the Electrostatic Model-Analytical Solutions 258

7.2 The 2D Version of the MoM for Transmission Lines 273

7.3 Summary of MATLAB® Modules 284

References 287

8 Steady-State Current Flow 289

Introduction 289

8.1 Boundary Conditions. Integral Equation. Voltage and Current Electrodes 290

8.2 Analytical Solutions for DC Flow in Volumetric Conducting Objects 300

8.3 MoM Algorithm for DC Flow. Construction of Electrode Mesh 311

8.4 Application Example: EIT 320

8.5 Application Example: tDCS 327

8.6 Summary of MATLAB® Modules 336

References 341

PART III LINEAR MAGNETOSTATICS 347

9 Linear Magnetostatics: Surface Charge Method 349

Introduction 349

9.1 Integral Equation of Magnetostatics: Surface Charge Method 350

9.2 Analytical versus Numerical Solutions: Modeling Magnetic Shielding 358

9.3 Summary of MATLAB® Modules 367

References 369

10 Inductance. Coupled Inductors. Modeling of a Magnetic Yoke 371

Introduction 371

10.1 Inductance 372

10.2 Mutual Inductance and Systems of Coupled Inductors 385

10.3 Modeling of a Magnetic Yoke 404

10.4 Summary of MATLAB® Modules 415

References 421

PART IV THEORY AND APPLICATIONS OF EDDY CURRENTS 423

11 Fundamentals of Eddy Currents 425

Introduction 425

11.1 Three Types of Eddy Current Approximations 426

11.2 Exact Solution for Eddy Currents without Surface Charges Created by Horizontal Loops of Current 440

11.3 Exact Solution for a Sphere in an External AC Magnetic Field 453

11.4 A Simple Approximate Solution for Eddy Currents in a Weakly Conducting Medium 460

11.5 Summary of MATLAB® Modules 464

References 470

12 Computation of Eddy Currents via the Surface Charge Method 473

Introduction 473

12.1 Numerical Solution in a Weakly Conducting Medium with External Magnetic Field 474

12.2 Comparison with FEM Solutions from Maxwell 3D of ANSYS: Solution Convergence 481

12.3 Eddy Currents Excited by a Coil 488

12.4 Summary of MATLAB® Modules 497

References 504

PART V NONLINEAR ELECTROSTATICS 507

13 Electrostatic Model of a pn-Junction: Governing Equations and Boundary Conditions 509

Introduction 509

13.1 Built-in Voltage of a pn-Junction 510

13.2 Complete Electrostatic Model of a pn-Junction 533

References 545

14 Numerical Simulation of pn-Junction and Related Problems: Gummel's Iterative Solution 547

Introduction 547

14.1 Iterative Solution for Zero Bias Voltage 548

14.2 Numerical Solution for the Electric Field Region 560

14.3 Analytical Solution for the Diffusion Region: Shockley Equation 579

14.4 Summary of MATLAB® Modules 587

References 588

INDEX 591