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 xiIntroduction xiiiAcknowledgments xv1 A Review of Basic Electrostatics 11.1 Charge, Force, and the Electric Field 11.2 Electric Flux Density and Gauss’s Law 51.3 Conductors 71.4 Potential, Gradient, and Capacitance 101.5 Energy in the Electric Field 161.6 Poisson’s and Laplace’s Equations 181.7 Dielectric Interfaces 201.8 Electric Dipoles 241.9 The Case for Approximate Numerical Analysis 27Problems 292 The Uses of Electrostatics 332.1 Basic Circuit Theory 332.2 Radio Frequency Transmission Lines 412.3 Vacuum Tubes and Cathode Ray Tubes 442.4 Field Emission and the Scanning Electron Microscope 472.5 Electrostatic Force Devices 482.6 Gas Discharges and Lighting Devices 493 Introduction to the Method of Moments Technique for Electrostatics 513.1 Fundamental Equations 513.2 A Working Equation Set 553.3 The Single-Point Approximation for Off-Diagonal Terms 563.4 Exact Solutions for the Diagonal Term and In-Plane Terms 573.5 Approximating Li,j 61Problems 644 Examples using the Method of Moments 674.1 A First Modeling Program 674.2 Input Data File Preparation for the First Modeling Program 684.3 Processing the Input Data 714.4 Generating the Li,j Array 734.5 Solving the System and Examining Some Results 734.6 Limits of Resolution 764.7 Voltages and Fields 784.8 Varying the Geometry 82Problems 875 Symmetries, Images and Dielectrics 895.1 Symmetries 895.2 Images 905.3 Multiple Images and the Symmetric Stripline 955.4 Dielectric Interfaces 1025.5 Two-Dimensional Cross Sections of Uniform Three-Dimensional Structures 1085.6 Charge Profiles and Current Bunching 1135.7 Cylinder between Two Planes 116Problems 1216 Triangles 1236.1 Introduction to Triangular Cells 1236.2 Right Triangles 1246.3 Calculating Li,i (Self ) Coefficients 1256.4 Calculating Li,j for i ≠ j 1276.5 Basic Meshing and Data Formats for Triangular Cell MoM Programs 1276.6 Using MATLAB to Generate Triangular Meshings 1356.7 Calculating Voltages 1396.8 Calculating the Electric Field 1416.9 Three-Dimensional Structures 1436.10 Charge Profiles 152Problems 1567 Summary and Overview 1597.1 Where We Were, Where We’re Going 1598 The Finite Difference Method 1638.1 Introduction and a Simple Example 1638.2 Setting Up and Solving a Basic Problem 1658.3 The Gauss–Seidel (Relaxation) Solution Technique 1728.4 Charge, Gauss’s Law, and Resolution 1758.5 Voltages and Fields 1778.6 Stored Energy and Capacitance 178Problems 1819 Refining the Finite Difference Method 1839.1 Refined Grids 1839.2 Arbitrary Conductor Shapes 1899.3 Mixed Dielectric Regions and a New Derivation of the Finite Difference Equation 1949.4 Example: Structure with a Dielectric Interface 1959.5 Axisymmetric Cylindrical Coordinates 1969.6 Symmetry Boundary Condition 2059.7 Duality, and Upper and Lower Bounds to Solutions for Transmission Lines 2079.8 Extrapolation 2149.9 Three-Dimensional Grids 217Problems 22310 Multielectrode Systems 22710.1 Multielectrode Structures 22710.2 Utilizing Superposition 22910.3 Utilizing Symmetry 23010.4 Circuital Relations and a Caveat 23010.5 Floating Electrodes 232Problems 23411 Probabilistic Potential Theory 23711.1 Random Walks and the Diffusion Equation 23711.2 Voltage at a Point from Random Walks 23911.3 Diffusion 24611.4 Variable-Step-Size Random Walks 24911.5 Three-Dimensional Structures 260Problems 26112 The Finite Element Method (FEM) 26512.1 Introduction 26512.2 Solving Laplace’s Equation by Minimizing Stored Energy 26612.3 A Simple One-Dimensional Example 26712.4 A Very Simple Finite Element Approximation 27112.5 Arbitrary Number of Lines Approximation 27412.6 Mixed Dielectrics 27812.7 A Quadratic Approximation 27912.8 A Simple Two-Dimensional FEM Program 282Problems 28713 Triangles and Two-Dimensional Unstructured Grids 28913.1 Introduction 28913.2 Aside: The Area of a Triangle 29013.3 The Coefficient Matrix 29113.4 A Simple Example 29313.5 A Two-Dimensional Triangular Mesh Program 296Problems 30014 A Zoning System and Some Examples 30314.1 General Introduction 30314.2 Introduction to gmsh 30414.3 Translating the gmsh.msh File 30814.4 Running the FEM Analysis 31914.5 More gmsh Features and Examining the Electric Field 32014.6 Multiple Electrodes 324Problems 32715 Some FEM Topics 32915.1 Symmetries 32915.2 A Symmetry Example, Including a Two-Sided Capacitance Estimate 33015.3 Axisymmetric Structures 33715.4 The Graded-Potential Boundary Condition 34815.5 Unbounded Regions 35215.6 Dielectric Materials 364Problems 37116 FEM in Three Dimensions 37516.1 Creating Three-Dimensional Meshes 37516.2 The FEM Coefficient Matrix in Three Dimensions 38416.3 Parsing the gmsh Files and Setting Boundary Conditions 38616.4 Open Boundaries and Cylinders in Space 392Problems 39617 Electrostatic Forces 39917.1 Introduction 39917.2 Electron Beam Acceleration and Control 40017.3 The Electrostatic Relay (Switch) 40817.4 Electrets and Piezoelectricity: An Overview 41417.5 Points on a Sphere 415Problems 419Appendix Interfacing with Other Languages 423Index 431

“The author well organized fundamental theories on electrostatics and also presented numerical examples, in which typical numerical methods, e.g., finite difference method, finite element method, and method of moment, are introduced and demonstrated by Matlab.”  (Zentralblatt MATH, 1 October 2014)