Electric Field Analysis

Electric Field Analysis

by Sivaji Chakravorti
Released December 2017
Publisher(s): CRC Press
ISBN: 9781351831161

Read it now on the O’Reilly learning platform with a 10-day free trial.

O’Reilly members get unlimited access to books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Start your free trial

Book description

This text begins by introducing the physical and mathematical fundamentals of electric fields, presenting problems from power and dielectric engineering to show how the theories are put into practice. It then discusses finite difference, finite element, charge simulation, and surface charge simulation methods for the numerical computation of electric fields, providing case studies for electric field distribution and exploring numerical field calculation for electric field optimization. The book also explains how high-voltage field optimization studies are carried out to meet the desired engineering needs.

Table of contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Table of Contents
  6. Foreword
  7. Preface
  8. Author
  9. 1. Fundamentals of Electric Field
    1. 1.1 Introduction
    2. 1.2 Electric Charge
    3. 1.3 Electric Fieldlines
    4. 1.4 Coulomb’s Law
      1. 1.4.1 Coulomb’s Constant
      2. 1.4.2 Comparison between Electrostatic and Gravitational Forces
      3. 1.4.3 Effect of Departure from Electrical Neutrality
      4. 1.4.4 Force due to a System of Discrete Charges
      5. 1.4.5 Force due to Continuous Charge Distribution
    5. 1.5 Electric Field Intensity
    6. 1.6 Electric Flux and Electric Flux Density
    7. 1.7 Electric Potential
      1. 1.7.1 Equipotential vis-à-vis Electric Fieldline
      2. 1.7.2 Electric Potential of the Earth Surface
      3. 1.7.3 Electric Potential Gradient
      4. 1.7.4 Electric Potential Gradient and Electric Field Intensity
    8. 1.8 Field due to Point Charge
    9. 1.9 Field due to a Uniformly Charged Line
    10. 1.10 Field due to a Uniformly Charged Ring
    11. 1.11 Field due to a Uniformly Charged Disc
    12. Objective Type Questions
    13. Bibliography
  10. 2. Gauss’s Law and Related Topics
    1. 2.1 Introduction
    2. 2.2 Useful Definitions and Integrals
      1. 2.2.1 Electric Flux through a Surface
      2. 2.2.2 Charge within a Closed Volume
      3. 2.2.3 Solid Angle
    3. 2.3 Integral Form of Gauss’s Law
      1. 2.3.1 Gaussian Surface
    4. 2.4 Differential Form of Gauss’s Law
    5. 2.5 Divergence Theorem
    6. 2.6 Poisson’s and Laplace’s Equations
    7. 2.7 Field due to a Continuous Distribution of Charge
    8. 2.8 Steps to Solve Problems Using Gauss’s Law
    9. Objective Type Questions
  11. 3. Orthogonal Coordinate Systems
    1. 3.1 Basic Concepts
      1. 3.1.1 Unit Vector
      2. 3.1.2 Right-Handed Convention
      3. 3.1.3 Differential Distance and Metric Coefficient
      4. 3.1.4 Choice of Origin
    2. 3.2 Cartesian Coordinate System
    3. 3.3 Cylindrical Coordinate System
    4. 3.4 Spherical Coordinate System
    5. 3.5 Generalized Orthogonal Curvilinear Coordinate System
    6. 3.6 Vector Operations
      1. 3.6.1 Gradient
      2. 3.6.2 Del Operator
      3. 3.6.3 Divergence
      4. 3.6.4 Laplacian
      5. 3.6.5 Curl
        1. 3.6.5.1 Curl of Electric Field
    7. Objective Type Questions
  12. 4. Single-Dielectric Configurations
    1. 4.1 Introduction
    2. 4.2 Displacement Current
    3. 4.3 Parallel Plate Capacitor
      1. 4.3.1 Energy Stored in a Parallel Plate Capacitor
    4. 4.4 Energy Stored in Electric Field
    5. 4.5 Two Concentric Spheres with Homogeneous Dielectric
    6. 4.6 Two Co-Axial Cylinders with Homogeneous Dielectric
    7. 4.7 Field Factor
    8. Objective Type Questions
  13. 5. Dielectric Polarization
    1. 5.1 Introduction
    2. 5.2 Field due to an Electric Dipole and Polarization Vector
      1. 5.2.1 Electric Dipole and Dipole Moment
      2. 5.2.2 Field due to an Electric Dipole
      3. 5.2.3 Polarization Vector
    3. 5.3 Polarizability
      1. 5.3.1 Non-Polar and Polar Molecules
      2. 5.3.2 Electronic Polarizability of an Atom
      3. 5.3.3 Types of Polarizability
        1. 5.3.3.1 Electronic Polarizability
        2. 5.3.3.2 Ionic Polarizability
        3. 5.3.3.3 Orientational or Dipolar Polarizability
        4. 5.3.3.4 Interfacial Polarizability
    4. 5.4 Field due to a Polarized Dielectric
      1. 5.4.1 Bound Charge Densities of Polarized Dielectric
        1. 5.4.1.1 Bound Volume Charge Density
        2. 5.4.1.2 Bound Surface Charge Density
      2. 5.4.2 Macroscopic Field
      3. 5.4.3 Field due to a Narrow Column of Uniformly Polarized Dielectric
      4. 5.4.4 Field within a Sphere Having Uniformly Polarized Dielectric
      5. 5.4.5 Sphere Having Constant Radial Distribution of Polarization
    5. 5.5 Electric Displacement Vector
      1. 5.5.1 Electric Susceptibility
      2. 5.5.2 Dielectric Permittivity
      3. 5.5.3 Relationship between Free Charge Density and Bound Volume Charge Density
    6. 5.6 Classification of Dielectrics
      1. 5.6.1 Molecular Polarizability of Linear Dielectric
      2. 5.6.2 Piezoelectric Materials
      3. 5.6.3 Ferroelectric Materials
      4. 5.6.4 Electrets
    7. 5.7 Frequency Dependence of Polarizabilities
    8. 5.8 Mass-Spring Model of Fields in Dielectrics
      1. 5.8.1 Dielectric Permittivity from Mass-Spring Model
    9. 5.9 Dielectric Anisotropy
      1. 5.9.1 Tensor of Rank
      2. 5.9.2 Permittivity Tensor
    10. Objective Type Questions
  14. 6. Electrostatic Boundary Conditions
    1. 6.1 Introduction
    2. 6.2 Boundary Conditions between a Perfect Conductor and a Dielectric
      1. 6.2.1 Boundary Condition for Normal Component of Electric Flux Density
      2. 6.2.2 Boundary Condition for Tangential Component of Electric Field Intensity
      3. 6.2.3 Field Just Off the Conductor Surface
    3. 6.3 Boundary Conditions between Two Different Dielectric Media
      1. 6.3.1 Boundary Condition for Normal Component of Electric Flux Density
      2. 6.3.2 Boundary Condition for Tangential Component of Electric Field Intensity
      3. 6.3.3 Boundary Condition for Charge-Free Dielectric – Dielectric Interface
    4. Objective Type Questions
  15. 7. Multi-Dielectric Configurations
    1. 7.1 Introduction
    2. 7.2 Parallel Plate Capacitor
      1. 7.2.1 Dielectrics in Parallel between the Plates
      2. 7.2.2 Dielectrics in Series between the Plates
      3. 7.2.3 Void in Insulation
      4. 7.2.4 Impregnation of Porous Solid Insulation
    3. 7.3 Co-Axial Cylindrical Configurations
    4. Objective Type Questions
  16. 8. Electrostatic Pressures on Boundary Surfaces
    1. 8.1 Introduction
    2. 8.2 Mechanical Pressure on a Conductor–Dielectric Boundary
      1. 8.2.1 Electric Field Intensity Exactly on the Conductor Surface
      2. 8.2.2 Electrostatic Forces on the Plates of a Parallel Plate Capacitor
    3. 8.3 Mechanical Pressure on a Dielectric–Dielectric Boundary
      1. 8.3.1 Mechanical Pressure due to Dielectric Polarization
      2. 8.3.2 Mechanical Pressure on Surface Film at the Dielectric–Dielectric Boundary
      3. 8.3.3 Total Mechanical Pressure on the Dielectric– Dielectric Boundary
    4. 8.4 Two Dielectric Media in Series between a Parallel Plate Capacitor
    5. 8.5 Two Dielectric Media in Parallel between a Parallel Plate Capacitor
      1. 8.5.1 Electrostatic Pump
    6. Objective Type Questions
  17. 9. Method of Images
    1. 9.1 Introduction
    2. 9.2 Image of a Point Charge with Respect to an Infinitely Long Conducting Plane
      1. 9.2.1 Point Charge between Two Conducting Planes
    3. 9.3 Image of a Point Charge with Respect to a Grounded Conducting Sphere
      1. 9.3.1 Method of Successive Images
      2. 9.3.2 Conducting Sphere in a Uniform Field
    4. 9.4 Image of an Infinitely Long Line Charge with Respect to an Infinitely Long Conducting Plane
    5. 9.5 Two Infinitely Long Parallel Cylinders
    6. 9.6 Salient Features of Method of Images
    7. Objective Type Questions
  18. 10. Sphere or Cylinder in Uniform External Field
    1. 10.1 Introduction
    2. 10.2 Sphere in Uniform External Field
      1. 10.2.1 Conducting Sphere in Uniform Field
      2. 10.2.2 Dielectric Sphere in Uniform Field
    3. 10.3 Cylinder in Uniform External Field
      1. 10.3.1 Conducting Cylinder in Uniform Field
      2. 10.3.2 Dielectric Cylinder in Uniform Field
    4. Objective Type Questions
  19. 11. Conformal Mapping
    1. 11.1 Introduction
    2. 11.2 Basic Theory of Conformal Mapping
      1. 11.2.1 Mapping of Shapes
      2. 11.2.2 Preservation of Angles in Conformal Mapping
    3. 11.3 Concept of Complex Potential
    4. 11.4 Procedural Steps in Solving Problems Using Conformal Mapping
    5. 11.5 Applications of Conformal Mapping in Electrostatic Potential Problems
      1. 11.5.1 Conformal Mapping of Co-Axial Cylinders
      2. 11.5.2 Conformal Mapping of Non-Co-Axial Cylinders
      3. 11.5.3 Conformal Mapping of Unequal Parallel Cylinders
        1. 11.5.3.1 Conformal Mapping of Equal Parallel Cylinders
    6. Objective Type Questions
  20. 12. Graphical Field Plotting
    1. 12.1 Introduction
    2. 12.2 Experimental Field Mapping256
    3. 12.3 Field Mapping Using Curvilinear Squares
      1. 12.3.1 Foundations of Field Mapping
      2. 12.3.2 Sketching of Curvilinear Squares
      3. 12.3.3 Construction of Curvilinear Square Field Map
      4. 12.3.4 Capacitance Calculation from Field Map
    4. 12.4 Field Mapping in Multi-Dielectric Media
    5. 12.5 Field Mapping in Axi-Symmetric Configuration
    6. Objective Type Questions
    7. Bibliography
  21. 13. Numerical Computation of Electric Field
    1. 13.1 Introduction
    2. 13.2 Methods of Determination of Electric Field Distribution
    3. 13.3 Uniqueness Theorem
    4. 13.4 Procedural Steps in Numerical Electric Field Computation
    5. Objective Type Questions
  22. 14. Numerical Computation of High-Voltage Field by Finite Difference Method
    1. 14.1 Introduction
    2. 14.2 FDM Equations in 3D System for Single-Dielectric Medium
    3. 14.3 FDM Equations in Axi-Symmetric System for Single-Dielectric Medium
      1. 14.3.1 FDM Equation for a Node Lying Away from the Axis of Symmetry
      2. 14.3.2 FDM Equation for a Node Lying on the Axis of Symmetry
    4. 14.4 FDM Equations in 3D System for Multi-Dielectric Media
    5. 14.5 FDM Equations in Axi-Symmetric System for Multi-Dielectric Media
      1. 14.5.1 For Series Dielectric Media
        1. 14.5.1.1 For the Node on the Dielectric Interface Lying Away from the Axis of Symmetry
        2. 14.5.1.2 For the Node on the Dielectric Interface Lying on the Axis of Symmetry
      2. 14.5.2 For Parallel Dielectric Media
    6. 14.6 Simulation Details
      1. 14.6.1 Discretization
      2. 14.6.2 Simulation of an Unbounded Field Region
      3. 14.6.3 Accuracy Criteria
      4. 14.6.4 System of FDM Equation
    7. 14.7 FDM Examples
      1. 14.7.1 Transmission Line Parallel Conductors
      2. 14.7.2 Post-Type Insulator
      3. 14.7.3 Disc-Type Insulator
    8. Objective Type Questions
    9. Bibliography
  23. 15. Numerical Computation of High-Voltage Field by Finite Element Method
    1. 15.1 Introduction
    2. 15.2 Basics of FEM
    3. 15.3 Procedural Steps in FEM
    4. 15.4 Variational Approach towards FEM Formulation
      1. 15.4.1 FEM Formulation in a 2D System with Single-Dielectric Medium
      2. 15.4.2 FEM Formulation in 2D System with Multi-Dielectric Media
      3. 15.4.3 FEM Formulation in Axi-Symmetric System
      4. 15.4.4 Shape Function, Global and Natural Coordinates
      5. 15.4.5 Derivation of Field Variables Using Natural Coordinates
      6. 15.4.6 Other Types of Elements for 2D and Axi-Symmetric Systems
        1. 15.4.6.1 Quadratic Triangular Element
        2. 15.4.6.2 Linear Quadrilateral Element
        3. 15.4.6.3 Quadratic Quadrilateral Element
      7. 15.4.7 FEM Formulation in 3D System
        1. 15.4.7.1 Natural Coordinates of Linear Tetrahedral Element
        2. 15.4.7.2 Linear Hexahedral Element
        3. 15.4.7.3 Isoparametric Element
      8. 15.4.8 Mapping of Finite Elements
    5. 15.5 Features of Discretization in FEM
      1. 15.5.1 Refinement of FEM Mesh
      2. 15.5.2 Acceptability of Element after Discretization
    6. 15.6 Solution of System of Equations in FEM
      1. 15.6.1 Sources of Error in FEM
    7. 15.7 Advantages of FEM
      1. 15.7.1 Using FEM in the Design Cycle
    8. 15.8 FEM Examples
      1. 15.8.1 Circuit Breaker Contacts
      2. 15.8.2 Cylindrical Insulator
      3. 15.8.3 Porcelain Bushing of Transformer
    9. Objective Type Questions
    10. Bibliography
  24. 16. Numerical Computation of High-Voltage Field by Charge Simulation Method
    1. 16.1 Introduction
    2. 16.2 CSM Formulation for Single-Dielectric Medium
      1. 16.2.1 Formulation for Floating Potential Electrodes
    3. 16.3 CSM Formulation for Multi-Dielectric Media
    4. 16.4 Types of Fictitious Charges
      1. 16.4.1 Point Charge
      2. 16.4.2 Infinite Length Line Charge
      3. 16.4.3 Finite Length Line Charge
      4. 16.4.4 Ring Charge
      5. 16.4.5 Arbitrary Line Segment Charge
      6. 16.4.6 Arbitrary Ring Segment Charge
    5. 16.5 CSM with Complex Fictitious Charges
    6. 16.6 Capacitive-Resistive Field Computation by CSM
      1. 16.6.1 Capacitive-Resistive Field Computation Including Volume Resistance
      2. 16.6.2 Capacitive-Resistive Field Computation Including Surface Resistance
    7. 16.7 Field Computation by CSM under Transient Voltage
      1. 16.7.1 Transient Field Computation Including Volume Resistance
      2. 16.7.2 Transient Field Computation Including Surface Resistance
    8. 16.8 Accuracy Criteria
      1. 16.8.1 Factors Affecting Simulation Accuracy
      2. 16.8.2 Solution of System of Equations in CSM
    9. 16.9 Other Development in CSM
      1. 16.9.1 Least Square Error CSM
      2. 16.9.2 Optimized CSM
      3. 16.9.3 Region-Oriented CSM
    10. 16.10 Comparison of CSM with FEM
    11. 16.11 Hybrid Method Involving CSM and FEM
    12. 16.12 CSM Examples
      1. 16.12.1 Three-Core Belted Cable
      2. 16.12.2 Sphere Gap
      3. 16.12.3 Single-Core Cable Termination with Stress Cone
      4. 16.12.4 Post-Type Insulator
      5. 16.12.5 Asymmetric Sphere Gaps
    13. Objective Type Questions
    14. Bibliography
  25. 17. Numerical Computation of High-Voltage Field by Surface Charge Simulation Method
    1. 17.1 Introduction
    2. 17.2 SCSM Formulation for Single-Dielectric Medium
    3. 17.3 Surface Charge Elements in 2D and Axi-Symmetric Configurations
      1. 17.3.1 Straight Line Element
      2. 17.3.2 Elliptic Arc Element
      3. 17.3.3 Contribution of Nodal Charge Densities to Coefficient Matrix
      4. 17.3.4 Method of Integration over a Surface Charge Element
      5. 17.3.5 Electric Field Intensity Exactly on the Electrode Surface
    4. 17.4 SCSM Formulation for Multi-Dielectric Media
    5. 17.5 SCSM Formulation in 3D System
    6. 17.6 Capacitive-Resistive Field Computation by SCSM
      1. 17.6.1 Capacitive-Resistive Field Computation in 2D and Axi-Symmetric Systems
      2. 17.6.2 Capacitive-Resistive Field Computation in 3D System
    7. 17.7 SCSM Examples
      1. 17.7.1 Cylinder Supported on Wedge
      2. 17.7.2 Conical Insulator in Gas-Insulated System
      3. 17.7.3 Metal Oxide Surge Arrester
      4. 17.7.4 Condenser Bushing of Transformer
    8. Objective Type Questions
    9. Bibliography
  26. 18. Numerical Computation of Electric Field in High-Voltage System – Case Studies
    1. 18.1 Introduction
    2. 18.2 Benchmark Models for Validation
      1. 18.2.1 Cylinder in Uniform External Field
      2. 18.2.2 Sphere in Uniform External Field
      3. 18.2.3 Dielectric Sphere Coated with a Thin Conducting Layer in Uniform External Field
    3. 18.3 Electric Field Distribution in the Cable Termination
    4. 18.4 Electric Field Distribution around a Post-Type Insulator
      1. 18.4.1 Effect of Uniform Surface Pollution
      2. 18.4.2 Effect of Partial Surface Pollution
      3. 18.4.3 Effect of Dry Band
      4. 18.4.4 Impulse Field Distribution
    5. 18.5 Electric Field Distribution in a Condenser Bushing
    6. 18.6 Electric Field Distribution around a Gas-Insulated Substation Spacer
    7. Objective Type Questions
    8. Bibliography
  27. 19. Electric Field Optimization
    1. 19.1 Introduction
    2. 19.2 Review of Published Works
      1. 19.2.1 Conventional Contour Correction Techniques for Electrode and Insulator Optimization
      2. 19.2.2 Optimization of High-Voltage System Elements
      3. 19.2.3 Soft-Computing Techniques for Electrode and Insulator Optimization
      4. 19.2.4 Optimization of Switchgear Elements
      5. 19.2.5 Optimization of Bushing Elements
      6. 19.2.6 User-Friendly Optimization Environment
    3. 19.3 Field Optimization Using Contour Correction Techniques
      1. 19.3.1 Insulator Contour Optimization by Simultaneous Displacement
        1. 19.3.1.1 Contour Correction Keeping Potential Difference Constant
        2. 19.3.1.2 Contour Correction Keeping Distance Constant
      2. 19.3.2 Electrode and Insulator Contour Correction with Approximation of Corrected Contour
      3. 19.3.3 Parametric Optimization of Insulator Profile
    4. 19.4 ANN-Based Optimization of Electrode and Insulator Contours
      1. 19.4.1 ANN-Based Optimization of Electrode Contour
      2. 19.4.2 ANN-Based Optimization of Insulator Contour
    5. 19.5 ANN-Aided Optimization of 3D Electrode–Insulator Assembly
    6. Objective Type Questions
    7. References
  28. Index

Product information

  • Title: Electric Field Analysis
  • Author(s): Sivaji Chakravorti
  • Release date: December 2017
  • Publisher(s): CRC Press
  • ISBN: 9781351831161