Contents
1. Fundamentals of Electric Field
1.4.2 Comparison between Electrostatic and Gravitational Forces
1.4.3 Effect of Departure from Electrical Neutrality
1.4.4 Force due to a System of Discrete Charges
1.4.5 Force due to Continuous Charge Distribution
1.6 Electric Flux and Electric Flux Density
1.7.1 Equipotential vis-à-vis Electric Fieldline
1.7.2 Electric Potential of the Earth Surface
1.7.3 Electric Potential Gradient
1.7.4 Electric Potential Gradient and Electric Field Intensity
1.9 Field due to a Uniformly Charged Line
1.10 Field due to a Uniformly Charged Ring
1.11 Field due to a Uniformly Charged Disc
2. Gauss’s Law and Related Topics
2.2 Useful Definitions and Integrals
2.2.1 Electric Flux through a Surface
2.2.2 Charge within a Closed Volume
2.3 Integral Form of Gauss’s Law
2.4 Differential Form of Gauss’s Law
2.6 Poisson’s and Laplace’s Equations
2.7 Field due to a Continuous Distribution of Charge
2.8 Steps to Solve Problems Using Gauss’s Law
3. Orthogonal Coordinate Systems
3.1.3 Differential Distance and Metric Coefficient
3.2 Cartesian Coordinate System
3.3 Cylindrical Coordinate System
3.4 Spherical Coordinate System
3.5 Generalized Orthogonal Curvilinear Coordinate System
3.6.5.1 Curl of Electric Field
4. Single-Dielectric Configurations
4.3.1 Energy Stored in a Parallel Plate Capacitor
4.4 Energy Stored in Electric Field
4.5 Two Concentric Spheres with Homogeneous Dielectric
4.6 Two Co-Axial Cylinders with Homogeneous Dielectric
5.2 Field due to an Electric Dipole and Polarization Vector
5.2.1 Electric Dipole and Dipole Moment
5.2.2 Field due to an Electric Dipole
5.3.1 Non-Polar and Polar Molecules
5.3.2 Electronic Polarizability of an Atom
5.3.3.1 Electronic Polarizability
5.3.3.3 Orientational or Dipolar Polarizability
5.3.3.4 Interfacial Polarizability
5.4 Field due to a Polarized Dielectric
5.4.1 Bound Charge Densities of Polarized Dielectric
5.4.1.1 Bound Volume Charge Density
5.4.1.2 Bound Surface Charge Density
5.4.3 Field due to a Narrow Column of Uniformly Polarized Dielectric
5.4.4 Field within a Sphere Having Uniformly Polarized Dielectric
5.4.5 Sphere Having Constant Radial Distribution of Polarization
5.5 Electric Displacement Vector
5.5.3 Relationship between Free Charge Density and Bound Volume Charge Density
5.6 Classification of Dielectrics
5.6.1 Molecular Polarizability of Linear Dielectric
5.7 Frequency Dependence of Polarizabilities
5.8 Mass-Spring Model of Fields in Dielectrics
5.8.1 Dielectric Permittivity from Mass-Spring Model
6. Electrostatic Boundary Conditions
6.2 Boundary Conditions between a Perfect Conductor and a Dielectric
6.2.1 Boundary Condition for Normal Component of Electric Flux Density
6.2.2 Boundary Condition for Tangential Component of Electric Field Intensity
6.2.3 Field Just Off the Conductor Surface
6.3 Boundary Conditions between Two Different Dielectric Media
6.3.1 Boundary Condition for Normal Component of Electric Flux Density
6.3.2 Boundary Condition for Tangential Component of Electric Field Intensity
6.3.3 Boundary Condition for Charge-Free Dielectric – Dielectric Interface
7. Multi-Dielectric Configurations
7.2.1 Dielectrics in Parallel between the Plates
7.2.2 Dielectrics in Series between the Plates
7.2.4 Impregnation of Porous Solid Insulation
7.3 Co-Axial Cylindrical Configurations
8. Electrostatic Pressures on Boundary Surfaces
8.2 Mechanical Pressure on a Conductor–Dielectric Boundary
8.2.1 Electric Field Intensity Exactly on the Conductor Surface
8.2.2 Electrostatic Forces on the Plates of a Parallel Plate Capacitor
8.3 Mechanical Pressure on a Dielectric–Dielectric Boundary
8.3.1 Mechanical Pressure due to Dielectric Polarization
8.3.2 Mechanical Pressure on Surface Film at the Dielectric–Dielectric Boundary
8.3.3 Total Mechanical Pressure on the Dielectric– Dielectric Boundary
8.4 Two Dielectric Media in Series between a Parallel Plate Capacitor
8.5 Two Dielectric Media in Parallel between a Parallel Plate Capacitor
9.2 Image of a Point Charge with Respect to an Infinitely Long Conducting Plane
9.2.1 Point Charge between Two Conducting Planes
9.3 Image of a Point Charge with Respect to a Grounded Conducting Sphere
9.3.1 Method of Successive Images
9.3.2 Conducting Sphere in a Uniform Field
9.4 Image of an Infinitely Long Line Charge with Respect to an Infinitely Long Conducting Plane
9.5 Two Infinitely Long Parallel Cylinders
9.6 Salient Features of Method of Images
10. Sphere or Cylinder in Uniform External Field
10.2 Sphere in Uniform External Field
10.2.1 Conducting Sphere in Uniform Field
10.2.2 Dielectric Sphere in Uniform Field
10.3 Cylinder in Uniform External Field
10.3.1 Conducting Cylinder in Uniform Field
10.3.2 Dielectric Cylinder in Uniform Field
11.2 Basic Theory of Conformal Mapping
11.2.2 Preservation of Angles in Conformal Mapping
11.3 Concept of Complex Potential
11.4 Procedural Steps in Solving Problems Using Conformal Mapping
11.5 Applications of Conformal Mapping in Electrostatic Potential Problems
11.5.1 Conformal Mapping of Co-Axial Cylinders
11.5.2 Conformal Mapping of Non-Co-Axial Cylinders
11.5.3 Conformal Mapping of Unequal Parallel Cylinders
11.5.3.1 Conformal Mapping of Equal Parallel Cylinders
12.2 Experimental Field Mapping256
12.3 Field Mapping Using Curvilinear Squares
12.3.1 Foundations of Field Mapping
12.3.2 Sketching of Curvilinear Squares
12.3.3 Construction of Curvilinear Square Field Map
12.3.4 Capacitance Calculation from Field Map
12.4 Field Mapping in Multi-Dielectric Media
12.5 Field Mapping in Axi-Symmetric Configuration
13. Numerical Computation of Electric Field
13.2 Methods of Determination of Electric Field Distribution
13.4 Procedural Steps in Numerical Electric Field Computation
14. Numerical Computation of High-Voltage Field by Finite Difference Method
14.2 FDM Equations in 3D System for Single-Dielectric Medium
14.3 FDM Equations in Axi-Symmetric System for Single-Dielectric Medium
14.3.1 FDM Equation for a Node Lying Away from the Axis of Symmetry
14.3.2 FDM Equation for a Node Lying on the Axis of Symmetry
14.4 FDM Equations in 3D System for Multi-Dielectric Media
14.5 FDM Equations in Axi-Symmetric System for Multi-Dielectric Media
14.5.1 For Series Dielectric Media
14.5.1.1 For the Node on the Dielectric Interface Lying Away from the Axis of Symmetry
14.5.1.2 For the Node on the Dielectric Interface Lying on the Axis of Symmetry
14.5.2 For Parallel Dielectric Media
14.6.2 Simulation of an Unbounded Field Region
14.7.1 Transmission Line Parallel Conductors
15. Numerical Computation of High-Voltage Field by Finite Element Method
15.4 Variational Approach towards FEM Formulation
15.4.1 FEM Formulation in a 2D System with Single-Dielectric Medium
15.4.2 FEM Formulation in 2D System with Multi-Dielectric Media
15.4.3 FEM Formulation in Axi-Symmetric System
15.4.4 Shape Function, Global and Natural Coordinates
15.4.5 Derivation of Field Variables Using Natural Coordinates
15.4.6 Other Types of Elements for 2D and Axi-Symmetric Systems
15.4.6.1 Quadratic Triangular Element
15.4.6.2 Linear Quadrilateral Element
15.4.6.3 Quadratic Quadrilateral Element
15.4.7 FEM Formulation in 3D System
15.4.7.1 Natural Coordinates of Linear Tetrahedral Element
15.4.7.2 Linear Hexahedral Element
15.4.7.3 Isoparametric Element
15.4.8 Mapping of Finite Elements
15.5 Features of Discretization in FEM
15.5.2 Acceptability of Element after Discretization
15.6 Solution of System of Equations in FEM
15.6.1 Sources of Error in FEM
15.7.1 Using FEM in the Design Cycle
15.8.1 Circuit Breaker Contacts
15.8.3 Porcelain Bushing of Transformer
16. Numerical Computation of High-Voltage Field by Charge Simulation Method
16.2 CSM Formulation for Single-Dielectric Medium
16.2.1 Formulation for Floating Potential Electrodes
16.3 CSM Formulation for Multi-Dielectric Media
16.4 Types of Fictitious Charges
16.4.2 Infinite Length Line Charge
16.4.3 Finite Length Line Charge
16.4.5 Arbitrary Line Segment Charge
16.4.6 Arbitrary Ring Segment Charge
16.5 CSM with Complex Fictitious Charges
16.6 Capacitive-Resistive Field Computation by CSM
16.6.1 Capacitive-Resistive Field Computation Including Volume Resistance
16.6.2 Capacitive-Resistive Field Computation Including Surface Resistance
16.7 Field Computation by CSM under Transient Voltage
16.7.1 Transient Field Computation Including Volume Resistance
16.7.2 Transient Field Computation Including Surface Resistance
16.8.1 Factors Affecting Simulation Accuracy
16.8.2 Solution of System of Equations in CSM
16.10 Comparison of CSM with FEM
16.11 Hybrid Method Involving CSM and FEM
16.12.1 Three-Core Belted Cable
16.12.3 Single-Core Cable Termination with Stress Cone
16.12.5 Asymmetric Sphere Gaps
17. Numerical Computation of High-Voltage Field by Surface Charge Simulation Method
17.2 SCSM Formulation for Single-Dielectric Medium
17.3 Surface Charge Elements in 2D and Axi-Symmetric Configurations
17.3.3 Contribution of Nodal Charge Densities to Coefficient Matrix
17.3.4 Method of Integration over a Surface Charge Element
17.3.5 Electric Field Intensity Exactly on the Electrode Surface
17.4 SCSM Formulation for Multi-Dielectric Media
17.5 SCSM Formulation in 3D System
17.6 Capacitive-Resistive Field Computation by SCSM
17.6.1 Capacitive-Resistive Field Computation in 2D and Axi-Symmetric Systems
17.6.2 Capacitive-Resistive Field Computation in 3D System
17.7.1 Cylinder Supported on Wedge
17.7.2 Conical Insulator in Gas-Insulated System
17.7.3 Metal Oxide Surge Arrester
17.7.4 Condenser Bushing of Transformer
18. Numerical Computation of Electric Field in High-Voltage System – Case Studies
18.2 Benchmark Models for Validation
18.2.1 Cylinder in Uniform External Field
18.2.2 Sphere in Uniform External Field
18.2.3 Dielectric Sphere Coated with a Thin Conducting Layer in Uniform External Field
18.3 Electric Field Distribution in the Cable Termination
18.4 Electric Field Distribution around a Post-Type Insulator
18.4.1 Effect of Uniform Surface Pollution
18.4.2 Effect of Partial Surface Pollution
18.4.4 Impulse Field Distribution
18.5 Electric Field Distribution in a Condenser Bushing
18.6 Electric Field Distribution around a Gas-Insulated Substation Spacer
19. Electric Field Optimization
19.2 Review of Published Works
19.2.1 Conventional Contour Correction Techniques for Electrode and Insulator Optimization
19.2.2 Optimization of High-Voltage System Elements
19.2.3 Soft-Computing Techniques for Electrode and Insulator Optimization
19.2.4 Optimization of Switchgear Elements
19.2.5 Optimization of Bushing Elements
19.2.6 User-Friendly Optimization Environment
19.3 Field Optimization Using Contour Correction Techniques
19.3.1 Insulator Contour Optimization by Simultaneous Displacement
19.3.1.1 Contour Correction Keeping Potential Difference Constant
19.3.1.2 Contour Correction Keeping Distance Constant
19.3.2 Electrode and Insulator Contour Correction with Approximation of Corrected Contour
19.3.3 Parametric Optimization of Insulator Profile
19.4 ANN-Based Optimization of Electrode and Insulator Contours
19.4.1 ANN-Based Optimization of Electrode Contour
19.4.2 ANN-Based Optimization of Insulator Contour
19.5 ANN-Aided Optimization of 3D Electrode–Insulator Assembly