Introduction to Numerical Electrostatics Using MATLAB E-Book

Lawrence N. Dworsky

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley - IEEE
• Sprache: Englisch

Seitenzahl: 502

Veröffentlichungsjahr: 2014

Beschreibung

Readers are guided step by step through numerous specific problems and challenges, covering all aspects of electrostatics with an emphasis on numerical procedures. The author focuses on practical examples, derives mathematical equations, and addresses common issues with algorithms. Introduction to Numerical Electrostatics contains problem sets, an accompanying web site with simulations, and a complete list of computer codes. * Computer source code listings on accompanying web site * Problem sets included with book * Readers using MATLAB or other simulation packages will gain insight as to the inner workings of these packages, and how to account for their limitations * Example computer code is provided in MATLAB * Solutions Manual * The first book of its kind uniquely devoted to the field of computational electrostatics

Leseprobe

Contents

Cover

Title page

Copyright page

Preface

Introduction

Acknowledgments

1 A Review of Basic Electrostatics

1.1 Charge, Force, and the Electric Field

1.2 Electric Flux Density and Gauss’s Law

1.3 Conductors

1.4 Potential, Gradient, and Capacitance

1.5 Energy in the Electric Field

1.6 Poisson’s and Laplace’s Equations

1.7 Dielectric Interfaces

1.8 Electric Dipoles

1.9 The Case for Approximate Numerical Analysis

Problems

2 The Uses of Electrostatics

2.1 Basic Circuit Theory

2.2 Radio Frequency Transmission Lines

2.3 Vacuum Tubes and Cathode Ray Tubes

2.4 Field Emission and the Scanning Electron Microscope

2.5 Electrostatic Force Devices

2.6 Gas Discharges and Lighting Devices

3 Introduction to the Method of Moments Technique for Electrostatics

3.1 Fundamental Equations

3.2 A Working Equation Set

3.3 The Single-Point Approximation for Off-Diagonal Terms

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

3.5 Approximating L

i,j

Problems

4 Examples Using the Method of Moments

4.1 A First Modeling Program

4.2 Input Data File Preparation for the First Modeling Program

4.3 Processing the Input Data

4.4 Generating the Li,j Array

4.5 Solving the System and Examining Some Results

4.6 Limits of Resolution

4.7 Voltages and Fields

4.8 Varying the Geometry

Problems

5 Symmetries, Images, and Dielectrics

5.1 Symmetries

5.2 Images

5.3 Multiple Images and the Symmetric Stripline

5.4 Dielectric Interfaces

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

5.6 Charge Profiles and Current Bunching

5.7 Cylinder Between Two Planes

Problems

6 Triangles

6.1 Introduction to Triangular Cells

6.2 Right Triangles

6.3 Calculating L

i,i

(Self) Coefficients

6.4 Calculating L

i,j

FOR i ≠j

6.5 Basic Meshing and Data Formats for Triangular Cell MoM Programs

6.6 Using MATLAB to Generate Triangular Meshings

6.7 Calculating Voltages

6.8 Calculating the Electric Field

6.9 Three-Dimensional Structures

6.10 Charge Profiles

Problems

7 Summary and Overview

7.1 Where We Were, Where We’re Going

8 The Finite Difference Method

8.1 Introduction and a Simple Example

8.2 Setting up and Solving a Basic Problem

8.3 The Gauss–Seidel (Relaxation) Solution Technique

8.4 Charge, Gauss’s Law, and Resolution

8.5 Voltages and Fields

8.6 Stored Energy and Capacitance

Problems

9 Refining the Finite Difference Method

9.1 Refined Grids

9.2 Arbitrary Conductor Shapes

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

9.4 Example: Structure with a Dielectric Interface

9.5 Axisymmetric Cylindrical Coordinates

9.6 Symmetry Boundary Condition

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

9.8 Extrapolation

9.9 Three-Dimensional Grids

Problems

10 Multielectrode Systems

10.1 Multielectrode Structures

10.2 Utilizing Superposition

10.3 Utilizing Symmetry

10.4 Circuital Relations and a Caveat

10.5 Floating Electrodes

Problems

11 Probabilistic Potential Theory

11.1 Random Walks and the Diffusion Equation

11.2 Voltage at a Point from Random Walks

11.3 Diffusion

11.4 Variable-Step-Size Random Walks

11.5 Three-Dimensional Structures

Problems

12 The Finite Element Method (FEM)

12.1 Introduction

12.2 Solving Laplace’s Equation by Minimizing Stored Energy

12.3 A Simple One-Dimensional Example

12.4 A Very Simple Finite Element Approximation

12.5 Arbitrary Number of Lines Approximation

12.6 Mixed Dielectrics

12.7 A Quadratic Approximation

12.8 A Simple Two-Dimensional FEM Program

Problems

13 Triangles and Two-Dimensional Unstructured Grids

13.1 Introduction

13.2 Aside: The Area of a Triangle

13.3 The Coefficient Matrix

13.4 A Simple Example

13.5 A Two-Dimensional Triangular Mesh Program

Problems

14 A Zoning System and Some Examples

14.1 General Introduction

14.2 Introduction to gmsh

14.3 Translating the gmsh.msh File

14.4 Running the FEM Analysis

14.5 More gmsh Features and Examining the Electric Field

14.6 Multiple Electrodes

Problems

15 Some FEM Topics

15.1 Symmetries

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

15.3 Axisymmetric Structures

15.4 The Graded-Potential Boundary Condition

15.5 Unbounded Regions

15.6 Dielectric Materials

Problems

16 FEM in Three Dimensions

16.1 Creating Three-Dimensional Meshes

16.2 The FEM Coefficient Matrix in Three Dimensions

16.3 Parsing the gmsh Files and Setting Boundary Conditions

16.4 Open Boundaries and Cylinders in Space

Problems

17 Electrostatic Forces

17.1 Introduction

17.2 Electron Beam Acceleration and Control

17.3 The Electrostatic Relay (Switch)

17.4 Electrets and Piezoelectricity: an Overview

17.5 Points on a Sphere

Problems

A Interfacing with Other Languages

Index

Eula

List of Tables

c03

TABLE 3.1 Accuracy of Empirical Formula versus Accuracy of MATLABInteger2 Calculation

c04

TABLE 4.1 Complete Data File for a Parallel Plate Capacitor

TABLE 4.2 Arrays for Dataset in Table 4.1

TABLE 4.3 Data File for Capacitor Shown in Figure 4.18

TABLE 4.4 Data File for Capacitor Shown in Figure 4.19

TABLE 4.5 Data File for Capacitor Shown in Figure 4.22

c08

TABLE 8.1 Results of fd3.m.

c13

TABLE 13.1Node Information

TABLE 13.2Triangle Information

TABLE 13.3 Boundary Condition Information

List of Illustrations

c01

FIGURE 1.1 Electric field lines for point charges at (-1,0,0) and (1,0,0).

FIGURE 1.2 Ey(0,y,0) for two identical positive charges.

FIGURE 1.3 Ex(0,y,0) for two charges of opposite sign.

FIGURE 1.4 Sphere of uniform charge density ρ.

FIGURE 1.5 E(r) for a sphere of radius a, charge density ρ.

FIGURE 1.6 Two concentric spherical shells.

FIGURE 1.7 Electric field between two concentric opposite-charge conductive shells.

FIGURE 1.8 Electric field between two large parallel plates, near the center.

FIGURE 1.9 Electric field lines and equipotential surfaces about a point charge.

FIGURE 1.10 Repeat of Figure 1.1 with equipotential lines superimposed.

FIGURE 1.11 Circular coaxial transmission line and a cross section of the line.

FIGURE 1.12 Three examples of dielectric interfaces.

FIGURE 1.13 Boundary between two different dielectric materials.

FIGURE 1.14 Two-dielectric-layer parallel plate capacitor.

FIGURE 1.15 Structure of an electric dipole.

FIGURE 1.16 Electric dipole in a uniform field.

FIGURE 1.17 Section of concentric circles demonstrating a nonuniform field.

FIGURE 1.18 A common conductor–dielectric structure.

FIGURE P1.1 Two suspended charged spheres, Problem 1.1.

FIGURE P1.3 Layout for Problem 1.3.

FIGURE P1.5 A multilayer chip capacitors.

FIGURE P1.6 A quadrupole charge configuration.

c02

FIGURE 2.1Four basic circuit elements.

FIGURE 2.2 Some simple circuit examples using voltage sources.

FIGURE 2.3 Some simple circuit examples using current sources.

FIGURE 2.4 Simple RC circuit.

FIGURE 2.5 Voltage–current response of simple RC circuit.

FIGURE 2.6 Some common transmission line cross sections: (a) coaxial cable; (b) stripline; (c) open-wire line; (d) open microstripline; (e) boxed microstripline. Note that there is nothing necessarily micro about the microstrip line –– the name is historical.

FIGURE 2.7 The vacuum tube diode.

FIGURE 2.8 The vacuum tube triode.

FIGURE 2.9 The electrostatic cathode ray tube.

c03

FIGURE 3.1A parallel plate capacitor.

FIGURE 3.2 A parallel plate capacitor made up of two subelectrodes.

FIGURE 3.3 An asymmetric parallel plate capacitor.

FIGURE 3.4 Rectangle of dimensions (2a,2b) showing triangle used in developing equation (3.26).

FIGURE 3.5 Steps in evaluating in-plane coefficient for a rectangle along the X axis.

FIGURE 3.6 Notation for an in-plane rectangle along the Y axis.

FIGURE 3.7 Steps in evaluating an in-plane coefficient for a rectangle not crossing either axis.

FIGURE 3.8 1/Li,j and the single-point approximation versus zi.

FIGURE 3.9 Figure 3.8 with the empirical function superimposed.

FIGURE P3.3 Diagram for Problem 3.3.

FIGURE P3.4 Diagram for Problem 3.4.

c04

FIGURE 4.1Basic rectangular modeling elements.

FIGURE 4.2 Rectangular electrode rotated in plane about an arbitrary point.

FIGURE 4.3 Capacitor layout described by Table 4.1.

FIGURE 4.4 Charge distribution on a simple parallel plate capacitor.

FIGURE 4.5 Capacitance C versus nx = ny for the simple two-electrode capacitor.

FIGURE 4.6 High-resolution plot of charge distribution on a parallel plate capacitor.

FIGURE 4.7 Refined dataset for simple capacitor example.

FIGURE 4.8 Example of ratio of capacitance to ideal parallel plate capacitance as height h decreases.

FIGURE 4.9 Example capacitance as h increases.

FIGURE 4.10 Equipotential surfaces for capacitor with h = 1.

FIGURE 4.11 Electric field magnitude lines superimposed on equipotentials of Figure 4.10.

FIGURE 4.12 Equipotentials for a tightly spaced parallel plate capacitor.

FIGURE 4.13 Equipotentials for widely spaced parallel plate capacitor.

FIGURE 4.14 Cross section of capacitor with one electrode offset.

FIGURE 4.15 Parallel plate capacitance versus electrode offset.

FIGURE 4.16 Capacitor with different-size square electrodes.

FIGURE 4.17 Capacitance of the capacitor shown in Figure 4.16.

FIGURE 4.18 Capacitor with two different connecting leads.

FIGURE 4.19 Capacitor with an electrode rotated about its center.

FIGURE 4.20 Capacitance as a function of rotation angle θ for capacitor shown in Figure 4.19.

FIGURE 4.21 Capacitor with an electrode rotated about an arbitrary point.

FIGURE 4.22 Capacitance as a function of rotation angle θ for capacitor shown in Figure 4.21.

c05

FIGURE 5.1Twelve-cell parallel plate capacitor.

FIGURE 5.2 Edge view of structure shown in Figure 5.1.

FIGURE 5.3 Air dielectric microstripline with length variable L.

FIGURE 5.4 Capacitance versus length of microstripline shown in Figure 5.3.

FIGURE 5.5 Multiple-image placement for symmetric stripline: (a) Stripline cross section; (b) First-pass image charge placement; (c) Final image charge placement.

FIGURE 5.6 Equipotential surface approximation for multiple-image charges.

FIGURE 5.7 Two air–dielectric striplines with length variable L.

FIGURE 5.8 Junction of two dissimilar striplines.

FIGURE 5.9 Dielectric interface shown in cross section.

FIGURE 5.10 An open microstrip transmission line section.

FIGURE 5.11 Multiple-image pattern for the open microstripline.

FIGURE 5.12 Cross section of an infinite line of charge and a Gaussian surface.

FIGURE 5.13 Layout for calculating Li,j.

FIGURE 5.14 Results of air dielectric microstripline calculation.

FIGURE 5.15 Charge profile in upper conductor of air microstrip, with w/h = 1.

FIGURE 5.16 Definition of terms for ground-plane charge calculation.

FIGURE 5.17 Charge distribution on ground plane of microstripline.

FIGURE 5.18 Cross section of two-line microstrip.

FIGURE 5.19 Capacitance of two-line microstrip versus Outside Width.

FIGURE 5.20 Three-line neutral charge structure.

FIGURE 5.21 Multiline structure stacking to create equipotentials.

c06

FIGURE 6.1 Right triangles.

FIGURE 6.2 Layout for calculating Li,i for a right triangle.

FIGURE 6.3 Simple hand meshing of a square planar electrode.

FIGURE 6.4 Meshing of a planar square electrode with an off-center hole.

FIGURE 6.5 Uniformly meshed sphere.

FIGURE 6.6 Triangles described in mom_tri_1.m.

FIGURE 6.7 Arbitrary triangle split into two right triangles.

FIGURE 6.8 Triangular cell structure created by mesh_gen_1.

FIGURE 6.9 Voltage profiles of mom_tri_2.m example.

FIGURE 6.10 Ez profiles of mom_tri_2.m example.

FIGURE 6.11 Ez profiles of mom_tri_2.m example, z near top electrode.

FIGURE 6.12 Parallel plate with a Gaussian “lump” on one plate.

FIGURE 6.13 Structure shown in Figure 6.12 with increased resolution of Gaussian tip.

FIGURE 6.14 Results of the distmesh sample code distmesh_example.m.

FIGURE 6.15 Combined electrode structure created by mesh_gen_3.

FIGURE 6.16 Voltage profile for mesh_gen_3 structure.

FIGURE 6.17 Two concentric spheres, radii 1 and 2.

FIGURE 6.18 V(r) between the spheres: model and analytic results.

FIGURE 6.19 Parallel plate capacitor with one electrode composed of randomly generated cells.

FIGURE 6.20 Charge density versus X at Y = 0 approximated by using triangles with |Y| < 0.02.

c08

FIGURE 8.1A rectangular grid of points.

FIGURE 8.2 Stripline cross section.

FIGURE 8.3 Example of node (a) and variable numbering (b) with (0,0) at the center.

FIGURE 8.4 A practical example geometry for FD modeling.

FIGURE 8.5 Repeat of Figure 8.4 with Gaussian surfaces for charge calculation shown.

FIGURE 8.6 Capacitance C from Gauss's law as a function of resolution parameter ns.

FIGURE 8.7 Definitions for the linear interpolation of voltage values.

FIGURE 8.8 Repeat of Figure 8.6 with total energy capacitance calculation added.

FIGURE 8.9 Example of an inherently inaccurate model: (a) actual structure; (b) modeled structure.

FIGURE P8.2a

FIGURE P8.3(a) Structure of Problem 8.1 rotated 45° on grid.

FIGURE P8.4a

c09

FIGURE 9.1 Stripline example with a refined grid layout.

FIGURE 9.2 (a) hi and (b) hj grid layouts for example program.

FIGURE 9.3 Stripline capacitance for various center conductor thicknesses.

FIGURE 9.4 A boundary condition circle on a square grid.

FIGURE 9.5 Capacitance of a circular conductor in a square box.

FIGURE 9.6 Square grid shows dielectric surface.

FIGURE 9.7 Boxed microstrip transmission line.

FIGURE 9.8 Conical metal tip between parallel metal plates.

FIGURE 9.9 Equipotential surfaces near the sharp tip.

FIGURE 9.10 Spindt tip field emitter structure.

FIGURE 9.11 Ez near ToT of a Spindt tip structure.

FIGURE 9.12 Sharp needle tip between two electrodes.

FIGURE 9.13 Peak field versus post height and post diameter for a sharp tip.

FIGURE 9.14 Symmetry example with row of nodes on the line of symmetry.

FIGURE 9.15 Symmetry example with line of symmetry between rows of nodes.

FIGURE 9.16 Cross section of a square transmission line on a low-resolution grid.

FIGURE 9.17 Upper right quarter of structure in Figure 9.16; (a) th of original line; (b) dual of (a).

FIGURE 9.18 High and low boundary capacitances for the square transmission line.

FIGURE 9.19 Repeat of example from Chapter 8.

FIGURE 9.20 Transmission line cross section with no symmetry axes.

FIGURE 9.21 Results of extrapolation curve fit.

FIGURE 9.22 Square plate in a cubic box.

FIGURE 9.23 Symmetry choice used (a) and a possible better choice (b).

FIGURE 9.24 Illustration of boundary condition ambiguity in FD technique.

FIGURE P9.1 Two parallel plates inside a box and a rectangular dielectric slab.

FIGURE P9.3 A periodic structure.

c10

FIGURE 10.1 Three-conductor structure (a) and its circuital equivalent (b).

FIGURE 10.2 Two lines on either side of a dielectric board.

FIGURE 10.3 Two-electrode symmetric microstrip structure.

FIGURE 10.4 Two ideal-capacitor series circuit.

FIGURE 10.5 Example of a floating electrode in a grid.

FIGURE P10.1 Three-capacitor circuit.

FIGURE P10.5 Symmetric three-electrode structure.

c11

FIGURE 11.1 Four fixed-step two-dimensional random walks.

FIGURE 11.2 Two-dimensional structure example for PPT analysis.

FIGURE 11.3 Graph of a normal distribution; μ = 5, σ = 1.

FIGURE 11.4 Repeated runs of program ppt1.m showing convergence of results.

FIGURE 11.5 Structure with several electrodes and electrode voltages.

FIGURE 11.6 Convergence of the probability as calculated by ppt2.m.

FIGURE 11.7 Two-dimensional potential problem with no grid.

FIGURE 11.8 Further possible random steps, following Figure 11.7.

FIGURE 11.9 Typical result of running ppt3.m.

FIGURE 11.10 Structure using rectangles and a circle as electrodes.

FIGURE 11.11 ppt4.m results for data1.txt dataset.

FIGURE 11.12 Examples of various degrees of sharpness near a field point.

FIGURE 11.13 Example of three-dimensional gridless random-walk step generation.

FIGURE P11.6 Geometry for Problem 11.6.

FIGURE P11.7 Structure for Problem 11.7.

c12

FIGURE 12.1 Concentric circle example structure.

FIGURE 12.2 V(r) for concentric circular electrodes and a straight-line approximation to V(r).

FIGURE 12.3 Ratio of C using a linear approximation to exact C versus b/a.

FIGURE 12.4 Two-line voltage approximation function.

FIGURE 12.5 Exact single-line and two-line voltage approximations.

FIGURE 12.6 Capacitance for single- and two-line voltage approximations.

FIGURE 12.7 Node and region (element) layout for one-dimensional (1d) example.

FIGURE 12.8 Capacitance ration versus number of nodes for concentric circle structure.

FIGURE 12.9 Voltage profile produced by fed1d1.m for n = 12.

FIGURE 12.10 Basis functions for one element using a quadratic approximation.

FIGURE 12.11 Section of a rectangular grid for FEM modeling.

FIGURE 12.12 Notation for equation (12.61).

FIGURE P12.1 Square coaxial cable cross section discussed in Section 9.7.

FIGURE P12.3 Structure for Problem 12.4.

c13

FIGURE 13.1 Arbitrary triangle inscribed in a rectangle for area calculation.

FIGURE 13.2 General triangle for analysis.

FIGURE 13.3 A very simple triangular mesh structure.

FIGURE 13.4 Sketch of structure and boundary conditions as understood by the computer.

FIGURE 13.5 Sketch of structure and grayscale map of voltage profile.

FIGURE P13.4 Structure for Problem 13.2.

FIGURE P13.1 Structure for Problem 13.1.

FIGURE P13.5 Structure for Problem 13.3.

c14

FIGURE 14.1 The gmsh structure described by struct14_1.geo.

FIGURE 14.2 struct14_1.geo showing 2d meshing (lc = 1).

FIGURE 14.3 struct14_1.geo showing finer meshing (lc = 0.25).

FIGURE 14.4 Structure defined by struct14_2.geo.

FIGURE 14.5 Two-dimensional meshing of struct14_2.geo.

FIGURE 14.6 gmsh zoning of struct14_2.geo showing boundary conditions.

FIGURE 14.7 Graphical output of fem_2d_1 showing voltage profile.

FIGURE 14.8 Structure produced by struct14_3.geo.

FIGURE 14.9 Voltage distribution of struct14_3.geo.

FIGURE 14.10 Electric field magnitude profile for structure shown in struct14_3.geo.

FIGURE 14.11 Two-electrode structure described by struct 14_14.geo.

c15

FIGURE 15.1 Example of a simple symmetric structure.

FIGURE 15.2 Sample structure with two symmetry lines.

FIGURE 15.3 Structure shown in Figure 15.2 after meshing.

FIGURE 15.4 Structure from Figure 15.2 showing voltage distribution.

FIGURE 15.5 Structure from Figure 15.2 showing field distribution.

FIGURE 15.6 Structure with eightfold symmetry.

FIGURE 15.7 Cross section choices for concentric square transmission line: (a) Full cross section: (b)symmetry; (c)symmetry; (d)symmetry.

FIGURE 15.8 Perspective view of concentric square transmission line.

FIGURE 15.9 (a) Cross section between r and z and (b) perspective view of concentric cylinder structures.

FIGURE 15.10 Infinite-length concentric cylinder structure.

FIGURE 15.11 Structure defined using struct15_2.geo.

FIGURE 15.12 Results of FEM analysis of struct8.geo.

FIGURE 15.13 Structure from Figure 15.11 showing Z symmetry.

FIGURE 15.14 A simplified CRT outline.

FIGURE 15.15 fem_2d.3.m figure showing CRT boundary condition setup.

FIGURE 15.16 The r = 0 voltage profile for CRT FEM model.

FIGURE 15.17 Cross section of a parallel wire transmission line.

FIGURE 15.18 FEM analysis of parallel wire line versus outside boundary circle radius.

FIGURE 15.19 Section of struct15_4.geo near the outer circle.

FIGURE 15.20 Results of open boundary condition approximation.

FIGURE 15.21 mesh file struct15_5.jpg.

FIGURE 15.22 gmsh structure file with dielectric slab edge defined.

FIGURE P15.2 Sample subsection of concentric coaxial line.

c16

FIGURE 16.1 A meshed cube.

FIGURE 16.2 Two concentric cubes.

FIGURE 16.3 gmsh concentric spheres prior to meshing.

FIGURE 16.4 Concentric spheres after meshing.

FIGURE 16.5 Voltage distribution between the spheres.

FIGURE 16.6 Wire frame model for two cylinders in a sphere.

FIGURE P16.1 Structure and dimensions for Problem 16.1.

c17

FIGURE 17.1Simple structure demonstrating electron beam steering and focusing.

FIGURE 17.2 Electron trajectories for several focusing voltages: (a) Vf  =  220 V; (b) Vf  =  140 V; (c) Vf  =  120 V.

FIGURE 17.3 Deflected electron beam.

FIGURE 17.4 Typical electromagnetic relay.

FIGURE 17.5 Essential structure of simple MEMM relay.

FIGURE 17.6 Plot of equation (17.19).

FIGURE 17.7 Net force versus position for several voltages (normalized).

FIGURE 17.8 MEM relay with separate control electrodes.

FIGURE 17.9 Electrostatic comb structure capacitor.

FIGURE 17.10 Diagram showing 79 points on sphere distribution example.

FIGURE P17.1 CRT shell with bc voltages specified.

FIGURE P17.4a Structure for Problem 17.3.

Guide

Cover

Table of Contents

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Introduction to Numerical Electrostatics Using MATLAB®

Lawrence N. Dworsky

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Dworsky, Lawrence N., 1943– author. Introduction to numerical electrostatics using MATLAB® / Lawrence N. Dworsky  pages cm Includes index.

 ISBN 978-1-118-44974-5 (hardback)1. Electromagnetism–Data processing. 2. Electrostatics–Data processing. I. Title.  QC760.54.D86 2014  537′.20151–dc23

2013018019

Preface

My graduate work was in the area of microwave oscillation mechanisms in semiconductor devices. My contribution was the prediction, analysis, and verification of yet another mode of semiconductor microwave oscillation. After graduation, I taught for a brief period (2 years) and worked part-time for a local electronics firm designing positive–intrinsic–negative (PIN) diode attenuators for their line of microwave signal generators. Then, in 1974, I went to work for Motorola, Inc., in southern Florida.

The part of Motorola that I joined designed and manufactured two-way portable radios (for police, fire department, etc.) and radio pagers, along with the supporting infrastructure systems. The developmental push at the time was to extend the product base up to the (then new) 900-MHz bands.

This organization had no interest whatsoever in my semiconductor physics background. They were, however, keenly interested in the skills that I had acquired during my graduate and part-time work designing stripline and microstrip (transmission line) circuitry. Motorola needed capability with stripline and microstrip filters, interconnects, materials, and so on, and my job was to help develop this capability. I never returned to semiconductor physics, and 40 years after finishing graduate school, I think it's safe to say that I never will.

Transmission line circuit design consists of two parts: (1) the actual circuit design based on the transmission line parameters and (2) the relationships between these transmission line parameters and the physical structure and materials.

As will be explained in Chapter 2, transmission line parameters can be described in terms of their DC (direct current; i.e., electrostatic) capacitance. A significant part of my effort therefore was devoted to performing electrostatic analyses of stripline and microstrip structures to predict the electric fields and capacitances in these structures. I didn't realize it at the time, but I was developing a skill that I would continue using and improving for the rest of my career.

While working on stripline and microstrip circuits, I also worked on piezoelectric (quartz) resonator and filter technology. The piezoelectric device models involved coupling of mechanical motion to electric fields, but the very high ratio of acoustic to electric wave velocities in the materials of interest (approximately 105) allows the electrical part of the analysis to consist of electrostatic analysis.

The 1980s saw the introduction of analog cellular telephone technology in the United States. These cellular telephones and base stations required a complex two-filter system called a duplexer that would enable a cellular telephone to transmit and receive simultaneously on two nearby frequencies using the same antenna. (The challenge was to keep your own transmitter's signal out of your receiver.) These duplexers were realized using blocks of high-dielectric-constant ceramic, with partially metalized surfaces, acting as interconnected resonators. Once again, electrostatic modeling was required, and new modeling programs and approaches had to be perfected.

In the late 1980s we did exploratory work in the newly emerging field of micromachined electromechanical devices. Using semiconductor industry processing technologies, it was becoming possible to build extremely small accelerometers, switches, and resonators whose operation is based on electrostatic forces. This was a new area of electrostatic modeling for me. Everything I had done before had involved electrodes that stayed in place, and we never cared about the physical forces involved. Now, we had to calculate the forces and keep track of the fields and forces as the electrodes moved. Again, I was extending my experience base in electrostatic modeling.

In the early 1990s we became interested in vacuum microelectronics, particularly in a structure called the field emission display, an electronic display whose operation is based on electron emission from millions of very small, sharp, metal tips due to high local electric fields. Once again, I was extending my electrostatic modeling experience to include structures with vastly different scales (submicromillimeter resolution near the tips to millimeter resolution near the screen). Structural capacitances were of interest in that they could limit circuit switching speeds, but the principal issues were the magnitude and uniformity of the fields at the emitter tips and then the electron trajectory control (both desired and undesired) due to these fields as the electrons traveled to the screen, striking the light emitting phosphors when they arrived. In these models the electrodes remained immobile, so the fields didn't change; electron trajectories were the principal subject of interest.

Putting my history together, although I didn't realize it at the time, I have spent more time creating and working with electrostatic analyses than with any single other electrical engineering discipline. These analyses were never a goal unto themselves. They were an engineering tool. The simplest approach that could do the job was always the chosen approach.

The philosophy of this book follows from my personal experience. There is an incredibly long list of mathematical approaches to numerical electrostatic modeling, but in terms of learning the electrostatics and choosing a modeling approach to study a given situation, I try to avoid using more exotic schemes simply “because they're there.” This doesn't mean that all of the approaches in the literature aren't interesting, important, and valuable, but in any given circumstance the simplest tool that can do a job is probably the best tool for that job.

Lawrence N. Dworsky

Introduction

An introductory treatment of electrostatics usually begins with Coulomb's law, the concepts of charge, the electric field and energy stored in the field, potential, capacitance, and so on. Poisson's and Laplace's equations soon appear.

Unfortunately, almost no real world problem can be solved in closed form using the latter equations. The most basic of electrostatic device analysis, the electric fields surrounding and the capacitance of a simple parallel plate capacitor, cannot be found.

Typically, a few interesting solution techniques, such as the separation of variables, are presented. Then the author has to choose a path. Other solution techniques such as conformal mapping can be shown; if the book is to be more than an introductory text, more formal materials such as Greene's functions can be introduced. In any case, the practitioner with real world geometries to be analyzed has been abandoned.

A book about numerical analysis techniques typically presents just that – numerical analysis techniques. The few examples presented are usually based as much on the ease of their presentation as on their ultimate usefulness.

My goal in writing this book is to present enough basic electrostatic theory as necessary to get into real world problems, then to present several of the available numerical techniques that are applicable to these problems, and finally to present numerous, detailed, examples showing how these techniques are applied. In other words, I am presenting the basics of electrostatics and several relevant numerical analysis techniques, with the emphasis on practical geometries.

The numerical analysis of problems in fields such as electrostatics typically have three distinct phases:

Pre-processing.

The conversion of the physical description of the problem to a data set that is meaningfully digestible to the numerical analysis program chosen for the job.

Numerical analysis.

The calculations, based upon the data set describing the geometry and the chosen boundary conditions, resulting (typically) in an approximate solution for the voltage distribution over the chosen space.

Post-processing.

Calculations and programming necessary to provide summary data such as capacitance and computer visualization of the results.

This book will concentrate on item 2, the numerical analysis. Three fundamental techniques – method of moments, finite difference, and finite elements – are introduced. Sample problems are presented and computer code that solves the problems is developed.

In order to accomplish the above, some pre-processing capability is necessary. Rather than develop this capability or simply request that the reader “get it done,” several freely available packages are introduced and basic tutorials on their usage are presented. These tutorials are not exhaustive; they introduce enough of the capabilities of these packages to allow the student to follow, replicate, and extensively modify the examples to the student’s own needs. No claims are being made that the packages chosen are the best possible choices for the job. They are, however, choices that work well.

Post-processing of numerical analysis results, in this book, is done on an ad-hoc basis. Calculating the capacitance of a structure is often very useful because, if the example structure has been analyzed by other techniques and the results published, accuracy of results can be compared. This comparison allows for a convenient, one number figure of merit for choices of resolution, approximate boundary conditions, and so on. Often, graphical interpretations of field, voltage, and/or charge distributions are presented. These are useful as a quick visual check on the boundary conditions and on gaining insight into the electrostatic properties (high field points, etc.) of the structure being studied.

With only one exception, all the numerical analysis, post-processing and graphics were created using MATLAB®. This type of work is what MATLAB is designed to do. The analysis techniques presented convert partial differential equations into sets of linear equations with coefficients and variables represented by matrices and vectors. MATLAB is a scientific programming language designed with the matrix as the fundamental data type. The language is expressive, the available function list extensive and the easily used graphics superb.

MATLAB is fundamentally an interpreted computer language. It achieves impressive processing speeds by providing a liberal assortment of precompiled functions. Preparing user-written code for these functions involves a procedure that MATLAB calls “vectorizing.” Vectorizing means not writing explicit loops to process array elements because the language itself allows processing of all of the array elements simultaneously. From an authoring point of view, this raises a question: Should demonstration code be vectorized as much as possible in the name of program execution speed or should demonstration code be written to explicitly parallel the derivations of the formulas in the text in the name of pedagogical clarity?

There is no absolutely right answer to the above question. In every case in this book judgment calls were made – vectorized code is shown when the algorithm seems “clear enough.” This compromise, like all compromises, won’t please everybody every time; ideally, it will please enough readers enough of the time.

The problems at the end of the chapters were written, as much as possible, to be extensions of the chapters, rather than “verify XXX or put numbers into YYYY.” Many of the problems involve modifying existing or writing new MATLAB code, leading to capabilities that were not presented in the chapters’ materials. When the thrust of the problem isn’t to extend the modeling capability, it is to make a point about the electrostatics issues involved in the example being treated. In all cases, solving the problems and reading the solution discussions will be an integral part of the learning process.

Solutions to end-of-chapter exercises may be found at the book companion site, www.wiley.com/go/numerelectrostatics. Additional resources may be found at www.lawrencedworsky.com.

Acknowledgments

I certainly could not have done this job alone. I thank Kari Capone and George Telecki (retired) at John Wiley & Sons: Kari for her very good natured help in teaching me how to prepare my manuscript then reviewing it, and George for his initial support and encouragement. The folks at MathWorks were very generous in providing me with MATLAB software and support. Without the love, patience, and support of my wife Suzanna, I might never have started this project and I certainly never would have finished it.

L. N. D.