Index
Note: Locators followed by “f” and “t” denote figures and tables in the text
2D system
capacitive-resistive field computation in, 441–442
FEM formulation with multi-dielectric media, 333–334
surface charge elements in
contribution of nodal charge densities to coefficient matrix, 432–434, 432f
electric field intensity exactly on electrode surface, 435–436
method of integration over, 434–435
types of elements for, 340–343
3D system
capacitive-resistive field computation in, 442–446, 444f
FDM equations
for multi-dielectric medium, 293–301, 293f
for single-dielectric medium, 282–286, 282f
FEM formulation in
isoparametric element, 353
linear hexahedral element, 351–353
natural coordinates of linear tetrahedral element, 349–350, 349f
A
Air flux, 11
Anisotropic dielectrics, 139–143, 140f
polarization characteristics, 141f
Arbitrary line segment charge, 382–384
Arbitrary ring segment charge, 384–385, 384f
Artificial neural network (ANN), 492
aided optimization of 3D electrode and insulator, 512–515
assembly for optimization, 515f
flowchart, 514f
based optimization of electrode and insulator, 507–512
actual electric field intensity, 510, 510f
conical support insulator, 510, 511f
optimized end profile, 509, 509f
optimized insulator contour, 512, 512f
parallel disc electrode, 508f
tangential field intensity, 511, 511f
features, 507
Assignment factor (λ), 401, 401f
Asymmetric sphere gaps, 415–416, 415f
Axi-symmetric configurations
field mapping in, 264–266, 265f
surface charge elements in
contribution of nodal charge densities to coefficient matrix, 432–434, 432f
electric field intensity on electrode surface, 435–436
method of integration over, 434–435
Axi-symmetric systems
capacitive-resistive field computation, 395, 441–442
configurations. See Axi-symmetric configurations
FDM equations in
for multi-dielectric medium, 301–313
for single-dielectric medium, 287–293
FEM formulation in, 334–335, 334f
triangular element for, 334
types of elements for, 340–343
B
Benchmark models for numerical code validation
cylinder in uniform external field, 460
dielectric sphere coated with thin conducting layer in uniform external field, 461
sphere in uniform external field, 460–461
Biaxial material, 143
Boundary conditions
conductor and dielectric, 150–153
on conductor surfaces, 247
field just off conductor surface, 153
for normal component of electric flux density, 150–151, 150f
for tangential component of electric field intensity, 152–153, 152f
conformal mapping
of non-co-axial cylinders, 249
of unequal parallel cylinders, 251
defined, 149
different dielectric media, 153–159
for charge-free dielectric–dielectric interface, 158, 158f
for normal component of electric flux density, 154–155, 154f
for tangential component of electric field intensity, 156–157, 156f
first, 376
solving complex potential, 244
computation of surface resistance on, 444f
linear triangular, 440f
Boundary value problems, 272
Bound surface charge density, 121–123
Bound volume charge density
defined, 121
free charge density and, relationship, 129
of polarized dielectric, 122–123
Brick, 351
Bushing elements, field optimization, 495
C
equipotential lines for, 466f
geometry and boundaries of, 463f
problem of design, 462
relative dielectric permittivities, 463
stress control in, 462
Cartesian coordinate systems, 62–64
constant coordinate surfaces, 62f
differential area and volume elements, 63, 64f
differential line element, 63, 63f
divergence function, 80
electric field intensity, 198–199, 209
gradient function, 77
Cauchy–Riemann equations, 238, 239, 244, 258
C-domain, 408
Charge simulation method (CSM), 427
accuracy criteria
solution of system of equations in CSM, 402
capacitive-resistive field computation by
surface resistance, 393–397, 394f, 395f
volume resistance, 388–393, 390f, 391f
with complex fictitious charges, 385–387, 386f
development in
least square error, 402–403, 402f
optimized, 403
disadvantages of, 407
examples of
asymmetric sphere gaps, 415–416
post-type insulator, 414–415, 414f
single-core cable termination with stress cone, 411–414, 413f
three-core belted cable, 410–411, 410f
field computation under transient voltage by
formulation for
multi-dielectric medium, 375–377
single-dielectric medium, 372–375, 372f
hybrid method involving FEM and, 408, 409f
Circulation of vector, 81
Clausius–Mossotti relations, 132
Closed-type boundary, 428, 433, 436
Co-axial cylinders
electric field intensity, 169, 169f
solid-type oil-impregnated paper bushing, 166, 167f, 168
conformal mapping of, 246–248, 247f
with homogeneous dielectric medium, 103–106, 104f, 107
Combined surface, 408
Complex potential, concept of, 244–245
Concentric spheres with homogeneous dielectric medium, 101–103, 102f, 107
Condenser bushing of transformer, 450, 451f
Conducting cylinder in uniform field, 232–233
Conducting sphere in uniform field, 206–207, 207f, 224–226
Conductor
boundary conditions for dielectric and, 150–153
field just off conductor surface, 153
for normal component of electric flux density, 150–151, 150f
for tangential component of electric field intensity, 152–153, 152f
-dielectric boundary, mechanical pressure, 178–181
electric field intensity on conductor surface, 179
electrostatic forces on parallel plate capacitor, 179–180
simulation of boundary elements, 450
Conductor surface
boundary conditions, 247
electric field intensity on, 179
electric stresses along high voltage, 464f
field just off, 153
Conformal mapping
applications in electrostatic potential problems, 246–252
of co-axial cylinders, 246–248, 247f
of equal parallel cylinders, 252
of non-co-axial cylinders, 248–250, 248f
of unequal parallel cylinders, 250–251, 250f
mapping of shapes, 239–240, 239f
preservation of angles in, 241–244, 241f
for linear fractional transformation, 248, 250
procedural steps in solving problems using, 245, 245f
reason for using, 240
Conical insulator in GIS, 447–448, 448f
Constant stress element (CST), 329, 346
Continuous charge distribution, 50–51, 98
Contour correction techniques
electrode and insulator optimization, 489–491
field optimization using, 496–507
electrode and insulator with approximation of corrected contour, 500–504, 501f
insulator contour by simultaneous displacement, 496–500, 497f, 498f, 500f
parametric optimization of insulator profile, 504–507
principle, 501f
Contour points, 373
Control points, 314
Conventional CSM, 403
Coordinate systems, 59
constant coordinate surfaces, 62f
differential area and volume elements, 63, 64f
differential line element, 63, 63f
divergence function, 80
gradient function, 77
constant coordinate surfaces, 65, 66f
depiction, 65f
differential area element, 67, 67f, 68f
differential line element, 66, 67f
differential volume element, 68, 68f
divergence function, 80
gradient function, 77
differential distance and metric coefficient, 61
characterization, 75t
constant coordinate surfaces, 73, 74f
differential line, area and volume elements, 74, 75f
right-handed convention, 61
signature of, 61
constant coordinate surfaces, 69, 70f
depiction, 69f
differential area element, 70, 72f
differential line element, 70, 71f
differential volume element, 72, 73f
divergence function, 80
gradient function, 77
effect of departure from electrical neutrality, 7–8
electrostatic and gravitational forces, comparison, 6–7
force due to continuous charge distribution, 9–10
force due to system of discrete charges, 9, 9f
Curl, 77
of electric field, 85
orthogonal curvilinear coordinate system, 81, 82f
Curved quadrilateral element, mapping of, 355f
Curvilinear field map, 258
Curvilinear squares
construction of field map, 261
Cylinders
supported on wedge, 446–447, 447f
in uniform external field, 228–235, 229f
Cylindrical coordinate systems, 64–68
constant coordinate surfaces, 65, 66f
depiction, 65f
differential area element, 67, 67f, 68f
differential line element, 66, 67f
differential volume element, 68, 68f
divergence function, 80
gradient function, 77
Cylindrical insulator, 360, 361f
D
del operator, 19
Deviation angle, 315
Dielectric–dielectric boundary, 428, 445, 450, 472–473
free charge density on, 437–438
due to dielectric polarization, 181–183
on surface film, 183–185, 184f
total mechanical pressure on, 185–186
Dielectric polarization
bound charge densities, 121–123
macroscopic field, 123
narrow column of uniformly, 123–124, 124f
sphere having constant radial distribution of polarization, 125–126
sphere having uniformly, 124–125, 125f
mechanical pressure due to, 181–183
Dielectrics (medium), 111
polarization characteristics, 141f
boundary conditions for conductor and, 150–153
field just off conductor surface, 153
for normal component of electric flux density, 150–151, 150f
for tangential component of electric field intensity, 152–153, 152f
boundary conditions for different, 153–159
for charge-free dielectric–dielectric interface, 158, 158f
for normal component of electric flux density, 154–155, 154f
for tangential component of electric field intensity, 156–157, 156f
ferroelectric materials, 134–135, 134f
molecular polarizability of linear dielectric, 131–133, 131f
piezoelectric materials, 133–134
co-axial cylindrical configurations, 166–173
electric field intensity, 169, 169f
solid-type oil-impregnated paper bushing, 166, 167f, 168
homogeneous, 131
co-axial cylinders with, 103–106
isotropic, 130
mass-spring model of fields in, 137–139, 137f
multi-. See Multi-dielectric medium
parallel plate capacitor
parallel between, 189–192, 189f
in parallel between plates, 162–163, 162f
in series between plates, 163–165, 163f
permittivity
in electric displacement vector, 129
in mass-spring model of fields, 138–139
single-. See Single-dielectric medium
in uniform field
Differential distance, 61
Dirichlet’s condition, 376
in 3D system, 313
error, 357
Disc-type insulator, 318, 318f
of capacitor, 90f
defined, 90
sinusoidal voltage waveform, 91, 91f
Divergence function, 77
Cartesian coordinate system, 80
cylindrical coordinate system, 80
orthogonal curvilinear coordinate system, 77, 78f
spherical coordinate systems, 80
energy stored in electric field, 99
polarized dielectric, 121
vector identity, 275
Dry band, effect of, 469–471, 470f
E
Electric charge, 2, 7, 92, 135, 261
field due to continuous distribution of, 50–51
types, 2
Electric dipole, 112
and dipole moment, 112–113, 113f
image charges, 200f
Electric displacement vector, 127–130
defined, 128
dielectric permittivity, 129
electric susceptibility, 128–129
free and bound volume charge density, relationship, 129
Electric field, 3, 15, 39, 46, 93, 137, 211, 256–258, 261, 427, 462
curl of, 85
electric potential in, 15
energy stored in, 96–101, 96f, 98f
equations of, 12
numerical computation, 411–412, 459, 466, 471, 476, 476f
procedural steps, 276–277, 277f
problem formulation, 272
symmetry of, 45
Electric field distribution, 196
around GIS spacer, 475–482, 476f
around post-type insulator
effect of partial surface pollution, 468–469
effect of uniform surface pollution, 466–468
impulse field distribution, 471
potential distribution, 469f
in condenser bushing, 471–475, 472f
methods of determination, 272–274, 273f
Electric field factor, 106–107
Electric field intensity, 10–11, 13, 31, 93, 125, 233, 256, 258–262, 265–266, 346, 471
along spacer surface, 477f, 478f, 479f
area related to, 12f
boundary conditions for tangential component of
conductor and dielectric, 152–153, 152f
different dielectric media, 156–157, 156f
Cartesian coordinates, 198–199, 209
co-axial cylindrical configurations, 169, 169f
coefficients, expressions for, 379–385
comparison of resultant, 479, 480f, 481f, 482f
on conductor surface, 179
due to uniformly charged
electric dipole, 114
electric potential gradient and, 17–21, 17f
error in, 315
parallel plate capacitor, 94–95, 94f
point charge, due to, 21–26, 22f
sphere having uniformly polarized dielectric, 124–125
Electric fieldlines, 3–4, 255–257
due to positive/negative charges, 3, 4f
equipotential vis-à-vis, 16, 16f
for tracing, 257
Electric field theory, 39
Electric flux, 11–13, 261, 263
electric charge density, relationship, 49
Electric flux density, 11–13, 256
boundary conditions for normal component of, 234
conductor and dielectric, 150–151, 150f
different dielectric media, 154–155, 154f
co-axial cylinders, 103
concentric spheres, 101
electric charge density, relationship, 49
Neumann condition for normal component, 437
at radial distance
co-axial cylinders, 103
concentric spheres, 101
Gaussian surface, 168
Electric flux lines, 187, 258, 290, 462
Electric lines of force. See Electric fieldlines
Electric potential, 13–21, 48–50, 123, 246, 281–283, 288, 293, 343
annular strip of charge, 29–30
definition of, 14f
drop, 18
elementary charge
uniformly charged disc, 29
uniformly charged line, 27
uniformly charged ring, 28
energy, 329, 334, 338, 340, 341–343, 346, 347
entire line charge, 27
entire ring charge, 28
equipotential vis-à-vis electric fieldline, 16, 16f
gradient, 17
gradient and electric field intensity, 17–19, 17f, 18f
integral form of electric field intensity and, 14
Electric stress
comparison of resultant, 481
control, 462
distribution, 468f, 469f, 473–474, 473f, 474f, 475f
Electrode and insulator optimization
ANN-aided optimization of 3D, 512–515
assembly for optimization, 515f
flowchart, 514f
actual electric field intensity, 510, 510f
conical support insulator, 510, 511f
optimized end profile, 509, 509f
optimized insulator contour, 512, 512f
parallel disc electrode, 508f
tangential field intensity, 511, 511f
assembly, 466
contour correction techniques for, 489–491
with approximation of corrected contour, 500–504, 503f, 504f
parametric optimization of insulator profile, 504–507
principle, 501f
shielding electrode, 504, 505f
by simultaneous displacement, 496–500, 497f, 498f, 500f
soft-computing techniques for, 492–494
Electrode boundary, 428, 432, 441, 446, 450
Electronic polarizability, 116–117, 117f
Electrostatic forces
and gravitational force, comparison, 6–7
lifting of copper sphere due to, 8, 8f
on plates of parallel plate capacitor, 179–180
Electrostatic Green’s function, 51
Electrostatic pump, 190–191, 190f
Elements, 326
Elliptic integrals, 382
Equal and opposite separation constant solution
cylinder in uniform external field, 231–232
sphere in uniform external field, 223–224
Equal parallel cylinders, conformal mapping of, 252
Equipotentials, 15, 15f, 258, 261
properties, 16
vis-à-vis electric fieldline, 16, 16f
Experimental field mapping
analogy of stationary current field with static electric field, 256, 256t
electrolytic tank setup, 257
F
FDM equations
in 3D system
for multi-dielectric medium, 293–301
for single-dielectric medium, 282–286, 282f
in axi-symmetric system
for multi-dielectric medium, 301–313
for single-dielectric medium, 287–293
development in 2D system, 284f
system of, 315
F-domain, 408
Ferroelectric materials of dielectrics, 134–135, 134f
Fictitious charges
contour points
for CSM formulation, 372, 372f
CSM with complex, 385–387, 386f, 388, 397, 400, 407, 410, 410f
determining, 376
for multi-dielectric media, 376f
arbitrary line segment charge, 382–384, 383f
arbitrary ring segment charge, 384–385, 384f
finite length line charge, 380–381, 380f
infinite length line charge, 379–380, 379f
Field factor. See Electric field factor
Field mapping
in axi-symmetric configuration, 264–266, 265f
experimental method of, 256–257, 256t
methods, 255
in multi-dielectric media, 263–264, 264f
between two co-axial cylinders, 259f
using curvilinear squares, 257–263, 258f
capacitance calculation from, 261–263
construction of, 261
isolated curvilinear rectangle, 262
Field optimization
of bushing elements, 495
flowchart of conventional, 513f
high-voltage system elements, 491–492
of switchgear elements, 494–495
user-friendly optimization environment, 495
using contour correction techniques, 496–507
electrode and insulator with approximation of corrected contour, 500–504, 503f, 504f
insulator contour by simultaneous displacement, 496–500, 497f, 498f, 500f
parametric optimization of insulator profile, 504–507
Field utilization factor, 107
Finite difference method (FDM), 358
equations in. See FDM equations
grid, 316–318, 316f, 317f, 318f
Finite element method (FEM), 325
approach towards formulation
in 2D system with multi-dielectric media, 333–334
in 2D system with single-dielectric medium, 328–333
in axi-symmetric system, 334–335
derivation of field variables using natural coordinates, 338–340
mapping of finite elements, 353–354
natural coordinates of, 337–338, 337f
relationship between global and natural coordinates, 337
shape functions of, 336–337, 336f
types of elements for 2D and axi-symmetric systems, 340–345
comparison of CSM with, 407–408
depiction formulation for, 326f
examples of
circuit breaker contacts, 359–360, 360f
cylindrical insulator, 360, 361f
porcelain bushing of transformer, 360–362, 362f
features of discretization, 354–356
acceptability of element after, 356
refinement of mesh, 355–356, 357f
hybrid method involving CSM and, 408–409
principle of, 371
procedural steps in, 327
solution of system of equations in, 356–358
Finite elements, 325
Finite length line charge, 380–381, 380f
Floating potential electrodes, 374–375
Formulation error, 357
Fringing of flux, 94, 95f, 179
Functionally graded material (FGM) spacer, 494–495
G
Gas-insulated substation (GIS) spacer, 408
axi-symmetric configuration of, 476f
conical insulator in, 447–448, 448f
electric field distribution, 475–482, 476f
high-voltage DC (HVDC), 480–481
Gas-insulated transmission line (GIL), 105–106, 105f, 493
Gaussian elimination technique, 402
Gaussian pillbox, 150–151, 154
Gaussian surface, 39, 45–46, 45f, 46f, 93–94, 150, 162, 164, 262
Gauss–Legendre Quadrature rule, 434–436, 435t
Gauss’s law, 101, 103, 127–128, 151, 154, 162–164, 262
cylindrical Gaussian surface, 93–94
divergence theorem, 48
electric flux/charge density, relationship, 49
Gaussian surface, 45–46, 45f, 46f
steps to solve problems, 51–55
Global equation system, 327
Gradient
Cartesian coordinate system, 77
defined, 77
scalar function U, 76
spherical coordinate system, 77
Grading ring, 450
Graphical field plotting, 255
Gravitation/gravitational force, 2
capacitor plates, liquid column, 191
electrostatic and, comparison, 6–7
Newton’s law for, 6
H
Harmonic function, 239, 244–246
Hexahedral element, mapping of, 355f
High gradient insulators (HGIs), 492
High-voltage (HV) conductor surface, 463, 476
High-voltage DC (HVDC) GIS, 480–481
High-voltage system elements, field optimization, 491–492
Homogeneous dielectric medium, 131
two co-axial cylinders with, 103–106, 104f, 107
two concentric spheres with, 101–103, 102f, 107
I
electric dipole, 200f
location, 203
magnitude, 203
Image of true charge distribution, 195
Images, method of, 195
features of, 216
infinitely long line charge
infinitely long conducting plane, 208–211, 208f, 211f
infinitely long parallel cylinders replaced, 211, 212f
infinitely long parallel cylinders, 211–216, 212f
point charge
grounded conducting sphere, 201–207, 202f
infinitely long conducting plane, 196–201, 196f, 201f
Impulse field distribution, 471
Indirect boundary element method, 427
Infinite length line charge, 379–380, 379f
Infinitely long conducting plane
infinitely long line charge, 208–211, 208f, 211f
point charge, 196–201, 196f, 201f
Infinitely long line charge
Gaussian surface due to, 46, 46f
infinitely long conducting plane, 208–211, 208f, 211f
infinitely long parallel cylinders replaced, 211, 212f
Insulator contour optimization, 491, 493, 494. See also Electrode and insulator optimization
ANN-aided optimization of 3D, 512–515
assembly for optimization, 515f
flowchart, 514f
actual electric field intensity, 510, 510f
conical support insulator, 510, 511f
optimized end profile, 509, 509f
optimized insulator contour, 512, 512f
parallel disc electrode, 508f
tangential field intensity, 511, 511f
contour correction techniques for, 489–491
with approximation of corrected contour, 500–504, 503f, 504f
parametric optimization of insulator profile, 504–507
principle, 501f
shielding electrode, 504, 505f
by simultaneous displacement, 496–500
distance constant, 498–500, 498f, 500f
potential difference constant, 496–497, 497f
soft-computing techniques for, 492–494
tangential field intensity, 501f, 511, 511f
Interfacial polarizability, 119, 119f
Ionic polarizability, 117–118, 117f
Isoparametric element, 353
Isoparametric hexahedral coordinates, 352
Isotropic dielectrics, 130
J
Jacobian matrix, 338, 350, 440
L
Laplace’s equation, 49–50, 237–239, 244, 246, 287, 294, 302, 304, 306, 308, 313
axis of symmetry, 290
Cartesian coordinates, 284
conditions
infinitely long line charge, 208
point charge with infinitely long conducting plane, 197
cylindrical coordinates, 209, 228–229, 287–288
in multi-dielectric media, 296, 304, 306, 308
in spherical coordinates, 220–221
Laplacian, 49
Cartesian coordinate system, 80, 84
cylindrical coordinate system, 81, 84
spherical coordinate system, 81, 84
Least square error CSM (LSECSM), 402–403, 402f
L’Hospital’s rule, 290
Linear charge density distribution, 429
Linear dielectrics
molecular polarizability of, 131–133, 131f
Linear hexahedral element, 351–353, 352f
Linear, isotropic and homogeneous (LIH) dielectrics, 111, 130–131
Linear quadrilateral element, 341–342, 342f, 354f
Linear stress triangle, 340
Linear tetrahedron element, 345, 346f
Linear triangular element
boundary, 440f
mapping of, 354f
modelling of HV insulator using, 359f
simplest 2D element, 329, 329f, 358
Long-range interaction, 2
M
Mass-spring model of fields in dielectrics, 137–139
of atom, 137f
dielectric permittivity, 138–139
Mechanical pressure
conductor–dielectric boundary, 178–181
electric field intensity on conductor surface, 179
electrostatic forces on parallel plate capacitor, 179–180
dielectric–dielectric boundary, 181–187
due to dielectric polarization, 181–183
on surface film, 183–185, 184f
total mechanical pressure on, 185–186
Metal oxide surge arresters, 449–450, 449f
Metric coefficients, 61
Molecular polarizability of linear dielectric, 131–133, 131f
Multi-dielectric medium
2D system with
arrangement of, 437f
elemental discretization for, 333, 333f
FDM equations in 3D system for, 293–301, 293f
FDM equations in axi-symmetric system for
for node on dielectric interface lying away from, 301–304, 302f
for node on dielectric interface lying on, 304–306, 305f
for parallel dielectric media, 306–313, 307f
for series dielectric media, 301
fictitious charges for, 375, 376f
field mapping in, 263–264, 264f
system of equations for CSM in, 377
Multi-electrode arrangement, 437f
N
Natural coordinate system, 337, 341
Negative mass, 7
Neumann boundary condition, 437, 445
Nodal connectivity, 332
Nodes, 313
Non-co-axial cylinders, conformal mapping of, 248–250, 248f
Numerical code validation, benchmark models for
cylinder in uniform external field, 460
dielectric sphere coated with thin conducting layer in uniform external field, 461
sphere in uniform external field, 460–461
Numerical computation, 411–412, 459, 466, 471, 476, 476f
procedural steps, 276–277, 277f
simulation
system of FDM equation, 315
of unbounded field region, 314
O
Open-type boundary, 428, 434, 436
Optimized CSM (OCSM), 403
Orientational or dipolar polarizability, 118–119, 118f
Orthogonal curvilinear coordinate systems, 72–75
characterization, 75t
constant coordinate surfaces, 73, 74f
defined, 73
differential line, area and volume elements, 74, 75f
P
Parallel plate capacitor, 92–96
dielectrics in
parallel between plates, 162–163, 162f, 189–192, 189f
series between plates, 163–165, 163f, 187–188, 187f
electric field intensity, 94–95, 94f
electrostatic forces on plates of, 179–180
field due to infinitely long-charged sheet, 93f
impregnation of porous solid insulation, 166
void in insulation, 165–166, 165f
Partial discharge (PD), 166
Partial surface pollution, effect of, 468–469
Permittivity tensor of anisotropic dielectric, 141–143
Piezoelectricity, 133
Piezoelectric materials of dielectrics, 133–134
electric flux density, 11
field due to, 21–26, 22f, 113, 113f
Gauss’s law for discrete, 43–44, 43f
images, method of
grounded conducting sphere, 201–207, 202f
infinitely long conducting plane, 196–201, 196f, 201f
sphere simulated by, 411, 411f
Poissonian field, 437
Poisson’s equation, 39, 48–49, 274
Polar axis, 64
defined, 115
electronic polarizability of atom, 116–117, 116f
frequency dependence of, 136–137, 136f
of linear dielectric, molecular, 131–133, 131f
non-polar and polar molecules, 115–116, 116f
types
electronic, 117
orientational or dipolar, 118–119, 118f
Polarization, 112
in anisotropic dielectric, 140, 140f, 141f
mechanical pressure due to dielectric, 181–183
Polarization vector, 112, 114–115
dielectric–vacuum boundary, 183f
Polarized dielectric, field due to, 119–127, 120f
bound charge densities, 121–123
bound surface charge density, 122–123
bound volume charge density, 122
equivalent charge distribution, 122f
macroscopic field, 123
narrow column of uniformly, 123–124, 124f
sphere having constant radial distribution of polarization, 125–126
sphere having uniformly, 124–125, 125f
Pole, 64
Polymeric insulation, 449
Positive definite matrices, 356–357
Post-type insulator, 414–415, 466, 467f
by conventional CSM, 414, 414f
electric field distribution around
effect of partial surface pollution, 468–469
effect of uniform surface pollution, 466–468
impulse field distribution, 471
potential distribution, 469f
Potential discrepancy, 401
Potential drop, 18
Potential energy
in 3D electric field, 345
electric
in tetrahedral element, 347
in triangular element, 334
principle of minimum, 328
Potential theory, 244
Q
Quadratic quadrilateral element, 342–343, 342f
Quadratic triangular element, 340–341, 341f
Quantum electromagnetism, 7
Quarks, 2
R
Region of interest (ROI), 272, 276, 276f, 281–283, 288, 313, 372, 374–375, 407–408
Region-oriented CSM (ROCSM), 403–406, 404f–406f
Riemann mapping theorem, 237–238
Right-handed convention, 61
S
Separation constant solution
equal and opposite
cylinder in uniform external field, 231–232
sphere in uniform external field, 223–224
zero
cylinder in uniform external field, 230–231
sphere in uniform external field, 222–223
Simultaneous displacement
insulator contour optimization by, 496–500
distance constant, 498–500, 498f, 500f
potential difference constant, 496–497, 497f
method of, 499
Simultaneous linear equations, 332, 356–357
Single-dielectric medium
fictitious charges and contour points, 372, 372f
floating potential electrodes, 374–375
FDM equations in 3D system for, 282–286
FDM equations in axi-symmetric system for
insulator geometry, 287f
for node lying away from, 288–289, 288f
for a node lying on, 290–293, 290f
FEM formulation in 2D system with, 328–333
Solid–gas dielectric interface, 414
Sphere-gap arrangements, 411, 412f
method of successive images, 204, 204f
Spheres
asymmetric gaps, 415–416, 415f
with constant radial distribution of polarization, 125–126
with homogeneous dielectric medium, concentric, 101–103, 102f, 107
image of point charge to grounded conducting, 201–207, 202f
successive images, method of, 204–206
in uniform external field, 220–228, 220f, 460–461
dielectric coated with thin conducting layer in, 461, 461f
dielectric sphere, 226–228, 228f
with uniformly polarized dielectric, 124–125, 125f
Spherical coordinate systems, 69–72
constant coordinate surfaces, 69, 70f
depiction, 69f
differential area element, 70, 72f
differential line element, 70, 71f
differential volume element, 72, 73f
divergence function, 80
gradient function, 77
Spontaneous polarization, 134
Steradian, 41
Streamlines. See Electric flux lines
Sub-parametric element, 353
Successive images, method of, 204–206
defined, 205
to sphere-gap arrangement, 204, 204f
Sulphur hexafluoride (SF6), 466
Super-parametric element, 353
Support vector machine (SVM), 494
Surface charge elements, 430–436
method of integration over, 434–435
and nodes on electrode boundary, 428f
Surface charge simulation method (SCSM)
capacitive-resistive field computation by
axi-symmetric systems, 441–442
examples of
condenser bushing of transformer, 450, 451f
conical insulator in GIS, 447–448, 448f
cylinder supported on wedge, 446–447, 447f
metal oxide surge arrester, 449–450
formulation for
multi-dielectric medium, 436–438
single-dielectric medium, 428–430
Surface-oriented CSM. See Conventional CSM
Switchgear elements, field optimization, 494–495
T
Tangential field intensity
along spacer surface, 477, 478f
boundary condition for
conductor–dielectric boundary, 152–153, 152f
dielectric–dielectric boundary, 156–158, 156f
electrode and insulator optimization, 489–490, 496
insulator contours, 501f, 504–505
non-optimized/optimized profiles, 507f
optimized insulator contour, 511–512, 511f
Taylor series
in 3D system, 283
expansion, 283, 288, 290, 294, 302, 305, 307
Thermoelectrets, 135
Three-dimensional (3D) system. See 3D system
Transmission line parallel conductors, 316–317, 316f
Triangular boundary elements, 439, 439f
Triple junction, 446
Two-dimensional (2D) system. See 2D system
U
Unequal parallel cylinders, conformal mapping of, 250–252
Uniaxial material, 143
Uniform external field
cylinder in, 228–235, 229f, 460
dielectric sphere coated with thin conducting layer in, 461, 461f
sphere in, 220–228, 220f, 460–461
conducting sphere, 206–207, 207f, 224–226
dielectric sphere, 226–228, 228f
Uniform surface pollution, effect of, 466–468, 467f
cylindrical coordinate system, 65–66, 66f
spherical coordinate system, 70, 71f
User-friendly optimization environment, 495
V
Vector operations
del operator, 77
W
Weighted residuals approach, 327
Z
Zero separation constant solution