Introduction to Numerical Electrostatics Using MATLAB

LAWRENCE N. DWORSKY earned his PhD and BEE in Electrical Engineering at New York University and his Masters of Electrical Engineering at Princeton University. He was an Assistant Professor of Mathematics at Union Community College and an Assistant Professor of Electrical Engineering at Columbia University. From 1974 until 2002, Dr Dworsky served on the technical staff of Motorola, Inc. He has 39 issued U.S. Patents and has authored numerous technical articles. This is his fourth book.

Preface xi

Introduction xiii

Acknowledgments xv

1 A Review of Basic Electrostatics 1

1.1 Charge, Force, and the Electric Field 1

1.2 Electric Flux Density and Gauss’s Law 5

1.3 Conductors 7

1.4 Potential, Gradient, and Capacitance 10

1.5 Energy in the Electric Field 16

1.6 Poisson’s and Laplace’s Equations 18

1.7 Dielectric Interfaces 20

1.8 Electric Dipoles 24

1.9 The Case for Approximate Numerical Analysis 27

Problems 29

2 The Uses of Electrostatics 33

2.1 Basic Circuit Theory 33

2.2 Radio Frequency Transmission Lines 41

2.3 Vacuum Tubes and Cathode Ray Tubes 44

2.4 Field Emission and the Scanning Electron Microscope 47

2.5 Electrostatic Force Devices 48

2.6 Gas Discharges and Lighting Devices 49

3 Introduction to the Method of Moments Technique for Electrostatics 51

3.1 Fundamental Equations 51

3.2 A Working Equation Set 55

3.3 The Single-Point Approximation for Off-Diagonal Terms 56

3.4 Exact Solutions for the Diagonal Term and In-Plane Terms 57

3.5 Approximating Li,j 61

Problems 64

4 Examples using the Method of Moments 67

4.1 A First Modeling Program 67

4.2 Input Data File Preparation for the First Modeling Program 68

4.3 Processing the Input Data 71

4.4 Generating the Li,j Array 73

4.5 Solving the System and Examining Some Results 73

4.6 Limits of Resolution 76

4.7 Voltages and Fields 78

4.8 Varying the Geometry 82

Problems 87

5 Symmetries, Images and Dielectrics 89

5.1 Symmetries 89

5.2 Images 90

5.3 Multiple Images and the Symmetric Stripline 95

5.4 Dielectric Interfaces 102

5.5 Two-Dimensional Cross Sections of Uniform Three-Dimensional Structures 108

5.6 Charge Profiles and Current Bunching 113

5.7 Cylinder between Two Planes 116

Problems 121

6 Triangles 123

6.1 Introduction to Triangular Cells 123

6.2 Right Triangles 124

6.3 Calculating Li,i (Self ) Coefficients 125

6.4 Calculating Li,j for i ≠ j 127

6.5 Basic Meshing and Data Formats for Triangular Cell MoM Programs 127

6.6 Using MATLAB to Generate Triangular Meshings 135

6.7 Calculating Voltages 139

6.8 Calculating the Electric Field 141

6.9 Three-Dimensional Structures 143

6.10 Charge Profiles 152

Problems 156

7 Summary and Overview 159

7.1 Where We Were, Where We’re Going 159

8 The Finite Difference Method 163

8.1 Introduction and a Simple Example 163

8.2 Setting Up and Solving a Basic Problem 165

8.3 The Gauss–Seidel (Relaxation) Solution Technique 172

8.4 Charge, Gauss’s Law, and Resolution 175

8.5 Voltages and Fields 177

8.6 Stored Energy and Capacitance 178

Problems 181

9 Refining the Finite Difference Method 183

9.1 Refined Grids 183

9.2 Arbitrary Conductor Shapes 189

9.3 Mixed Dielectric Regions and a New Derivation of the Finite Difference Equation 194

9.4 Example: Structure with a Dielectric Interface 195

9.5 Axisymmetric Cylindrical Coordinates 196

9.6 Symmetry Boundary Condition 205

9.7 Duality, and Upper and Lower Bounds to Solutions for Transmission Lines 207

9.8 Extrapolation 214

9.9 Three-Dimensional Grids 217

Problems 223

10 Multielectrode Systems 227

10.1 Multielectrode Structures 227

10.2 Utilizing Superposition 229

10.3 Utilizing Symmetry 230

10.4 Circuital Relations and a Caveat 230

10.5 Floating Electrodes 232

Problems 234

11 Probabilistic Potential Theory 237

11.1 Random Walks and the Diffusion Equation 237

11.2 Voltage at a Point from Random Walks 239

11.3 Diffusion 246

11.4 Variable-Step-Size Random Walks 249

11.5 Three-Dimensional Structures 260

Problems 261

12 The Finite Element Method (FEM) 265

12.1 Introduction 265

12.2 Solving Laplace’s Equation by Minimizing Stored Energy 266

12.3 A Simple One-Dimensional Example 267

12.4 A Very Simple Finite Element Approximation 271

12.5 Arbitrary Number of Lines Approximation 274

12.6 Mixed Dielectrics 278

12.7 A Quadratic Approximation 279

12.8 A Simple Two-Dimensional FEM Program 282

Problems 287

13 Triangles and Two-Dimensional Unstructured Grids 289

13.1 Introduction 289

13.2 Aside: The Area of a Triangle 290

13.3 The Coefficient Matrix 291

13.4 A Simple Example 293

13.5 A Two-Dimensional Triangular Mesh Program 296

Problems 300

14 A Zoning System and Some Examples 303

14.1 General Introduction 303

14.2 Introduction to gmsh 304

14.3 Translating the gmsh.msh File 308

14.4 Running the FEM Analysis 319

14.5 More gmsh Features and Examining the Electric Field 320

14.6 Multiple Electrodes 324

Problems 327

15 Some FEM Topics 329

15.1 Symmetries 329

15.2 A Symmetry Example, Including a Two-Sided Capacitance Estimate 330

15.3 Axisymmetric Structures 337

15.4 The Graded-Potential Boundary Condition 348

15.5 Unbounded Regions 352

15.6 Dielectric Materials 364

Problems 371

16 FEM in Three Dimensions 375

16.1 Creating Three-Dimensional Meshes 375

16.2 The FEM Coefficient Matrix in Three Dimensions 384

16.3 Parsing the gmsh Files and Setting Boundary Conditions 386

16.4 Open Boundaries and Cylinders in Space 392

Problems 396

17 Electrostatic Forces 399

17.1 Introduction 399

17.2 Electron Beam Acceleration and Control 400

17.3 The Electrostatic Relay (Switch) 408

17.4 Electrets and Piezoelectricity: An Overview 414

17.5 Points on a Sphere 415

Problems 419

Appendix Interfacing with Other Languages 423

Index 431