Electrostatic Field and Field Control

1
Electrostatic Field and Field Control

• Background
• Fundamental equations
• Field in homogeneous medium
• Field in multi-dielectric media
• Field-strength control
• Numerical field calculation methods
• Triple-junction electric field

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
2
Why is the field analysis important?

• Field efficiency (utilization) factor,
• Dependency of the dielectric strength on electric
  field and its distribution.
• In insulation design.
• In problem analysis.
• Note difference between AC and DC.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
3
According to my professor

• Most people think
• Calculation of Electric field, we solve
• E
• Calculation of Magnetic field we solve
• H

ASY
ARD
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
4
Fundamentals

• Scalar fields potential, temperature.
• Vector fields electric field, magnetic field.
• Units,
• charge C
• potential V
• electric field V/m, flux density C/m2
• capacitance F
• permittivity F/m, e0 8.854x10-12 F/m.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
5
Electric field

• Definition force per unit charge placed under
  the field (N/C, V/m).
• Eb of air 3x106 N/C (approx.),
• of SF6 (43p 38)x105 N/C, ppressure
  (atm).
• Photocopier-drum field 100x103 N/C (approx.).
• Charged comb 1x103 N/C.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
6
Electric-field line or force line

• Start from a positive charge.
• Terminate at a negative charge.
• No crossing.
• Density of force lines.
• Iso-potential line.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
7
Point-charge electric field
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
8
Example 1

• Q Compute the electric field in free space at a
  distance
• 1 cm, (b) 5 cm, (c) 10 cm, (d) 50 cm from a point
  charge of 10-6 C.
• A
• 
  

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
9
Example 2

• Q Two charges of 6x10-8 and -12x10-8 C are
  located in free space by a separation of 10 cm
  (a) Calculate the electric field at the middle
  point between the charges. (b) Find the point of
  zero electric field.
• A
• (a)
  
• (b)

Q1
Q2
P
x
10 cm
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
10
Point-dipole electric field

• Definition M dipole moment (Cm).
• Coordinates (r, q, f).
• Applications spherical objects in uniform field.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
11
Line-charge electric field

• q line-charge density C/m.
• Coordinates (r, q, z).
• Note why V12 ?

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
12
Line-dipole electric field

• M line-dipole moment (C).
• Coordinates (r, q, z).

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
13
Electric field in homogeneous media The method
of images

• Concentric spheres
• Coaxial cylinders
• Sphere gaps
• Parallel cylinders
• Conductor sphere under a uniform field

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
14
Example 3
Q Two parallel-plates with a separation d. If
the charge density on each plate is equal to s
(C/m2). Determine the electric field between the
plate and the potential difference if the
permittivity of the medium inside is e. A


B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
15
Example 4
Q A conductor sphere having an electric field of
3kV/mm on its surface. Determine the potential
of the sphere if the sphere diameter is 1 m. A


B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
16
Concentric spheres

• Use a point charge, Q0.
• Real charge?
• Maximal field position?
• Field in the exterior?

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
17
Example 5

• Q Concentric-sphere conductors in the figure
  with the dimensions a, b, and c equal to 3, 7,
  and 8 cm, respectively.
• If Q 10-8 C on the inner sphere and 0 on the
  outer one, determine the potentials on the
  conductors and the electric field in each area.
• If Q 10-8 C on the inner sphere and -10-8 C on
  the outer one, repeat (a).

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
18
Example 5 (cont.)
A (a) (b)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
19
Coaxial cylinders

• Use a line charge, q0.
• Maximal field position?
• Optimal diameter-ratio

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
20
Example 6
Q Determine the surface charge densities on two
coaxial cylinders where rI 5 cm, rE 10 cm,
and er 1 if the potential difference is 100
V. A

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
21
Comparison between two similar arrangements of
2D/AS
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
22
Sphere gaps (I)

• Image charge for a grounded conductor sphere.
• Potential due to Q (R0/d) cancels that of Q
  every point on the sphere surface.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
23
Method of Images

• Iterative calculation for more than two objects.

f1
f2
boonchai.t_at_chula.ac.th , Department of Electrical
Engineering, Faculty of Engineering,
Chulalongkorn University, Thailand
24
Sphere gaps (II)

• Consider a sphere having a potential V0 and a
  grounded plane.
• Image charges are repetitively applied until the
  solutions converge.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
25
Example 7
Q Determine the maximal electric field in the
configuration of sphere gap where the conductor
radius is 1 cm, the gap is 0.4 cm, and the
potentials on the conductors are 1 and -1 V. A


B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
26
Example 7 (cont.)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
27
Two parallel cylinders (I)

• Constant ratio of distance from a pair of line
  charges.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
28
Two parallel cylinders (II)

• Now, consider distance D d2x.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
29
Two parallel cylinders (III)

• Now, consider distance D d2x.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
30
Example 8
Q Determine the maximal electric field in the
configuration of two parallel-cylinder conductors
where the radius is 1 cm, the separation D is 2.4
cm, and the potentials on the conductors are 1
and -1 kV. A

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
31
Comparison between 2D/AS Potential
Parallel-cylinders
Sphere-gap
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
32
Comparison between 2D/AS Field
Parallel-cylinders
Sphere-gap
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
33
Conductor sphere in a uniform field

• Use a line-pole for solution.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
34
Conductor sphere in a uniform field

• Use a point-pole for solution.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
35
Electric field in multi-dielectric media
boundary conditions

• Boundary conditions.
• Parallel plates with multiple dielectric layers.
• Coaxial cable with multiple dielectric layers.
• Dielectric sphere under a uniform field.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
36
Boundary conditions for complex-dielectric systems

• Conditions of electric field.

• Field directions.

• If e2gte1, then q2gtq1, except q1 q2 0o or 90o.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
37
Parallel plates with multiple dielectric layers
(I)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
38
Parallel plates with multiple dielectric layers
(II)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
39
Coaxial cable with multiple dielectric layers (I)

• Obtain Emax const. by
• e1r1 e2r2 enrn.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
40
Coaxial cable with multiple dielectric layers
(II)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
41
Coaxial cable with multiple dielectric layers
(III)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
42
Dielectric sphere under a uniform field
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
43
Concept of capacitance

• Definition C Q/V. (F)
• Parallel plates
• Coaxial cylinders
• Capacitance in multi-conductor systems

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
44
Field control capacitor bushing (I)

• Dielectric sheets of different l with conducting
  foils inserted between layers.
• Treated as coaxial cylinders of complex
  dielectrics.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
45
Field control capacitor bushing (II)

• Approx.
• Const.-field obtained by l0R0 l1R1
  ln-1Rn-1.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
46
Numerical field calculation methods

• Domain subdivision methods
• Finite element method (FEM)
• Finite difference method (FDM)
• Boundary subdivision methods
• Charge simulation method (CSM)
• Surface charge method (SCM)
• Boundary element method (BEM)
• Comments
• Commercial software
• In-house program

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
47
Example Optimization of GIS-spacer (I)
Structure and conditions of calculation.
Results cylindrical spacer
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
48
Example Optimization of GIS-spacer (II)
Results elliptical cross-section spacer.
Results summary for various types of spacer.
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
49
Triple-junction electric field

• What is the triple junction?
• A point where three media (including
  conductor) meet each other.
• What is its important in HV insulation?
• Weak point due to field intensification.

B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
50
Triple-junction qc 0 to 90 deg.
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
51
Triple-junction qc 0 deg.
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
52
Stress control in practice
Various types of spacer designed to reduce field
stress near the contact points.
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
53
Charge relaxation

• Insulation system
• Conductive system
• Transient system?

Apply a step voltage V0 to the HV electrode.
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
54
Electric field analysis for biological cells (I)
Transient analysis of fusing cells under an
external electric field for two cases (1) Cells
of equal radii (2) Cells of different
radii Calculation Electric field / Charge
relation by the BEM Iteration of charge buildup
by RUNGE-KUTTA method
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
55
Electric field analysis for biological cells (II)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th
56
Electric field analysis for biological cells (III)
B Techaumnat Dept. of Electr. Eng., Chulalongkorn
Univ. Boonchai.t_at_chula.ac.th