HIGH VOLTAGE TECHNIQUES electrostatic field analysis methods

HIGH VOLTAGE TECHNIQUES
electrostatic field analysis methods
Assistant Professor Suna BOLAT
Eastern Mediterranean University
Department of Electric & Electronic Engineering
Electrostatic field analysis methods
1. Analytical calculations
2. Analog methods
3. Numerical methods
Analytical calculations
•
•
•
•
Analytical solution of differential equations (Laplace, Poisson)
Conform transform
Schwarz – Christoffel transform
...
Analog methods
• Graphical methods
• Experimental methods
– On the model
– On a real system
Numerical methods
•
•
•
•
•
•
Finite difference method
Finite element method
Boundary element method
Charge simulation method
Monte – Carlo method
Moment method
Experimental methods
•
•
•
•
Electrolytic tank experiment
Semi-conductor paper method
Resistance simulation method
Grass seed method
Electrolytic tank experiment
Principle: static electric field has an analogy with current field.
Application:
• Create a scaled model of electrode system
• Replace the dielectric with a conductive material
• Determine the current field lines on conductive media
• Draw the electric field lines perdendicular to them
Current lines
analogy
Flux lines
To voltage source
Model electrodes
Electrolitic liquid
(?)
Experimental setup
Measurement bridge
• 𝐼=
𝑈
𝑅1 +𝑅2
• 𝑈2 = 𝐼𝑅2 =
𝑅2
𝑈
𝑅1 +𝑅2
= 𝑈𝐴
Numerical methods
• Finite difference method
Principle: it leans on finite difference operations
All the derivatives are substituted by numerical representations.
𝜕2𝑉 𝜕2𝑉
+ 2=0
2
𝜕𝑥
𝜕𝑦
𝑑2 𝑉 𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦
≅
2
𝑑𝑥
ℎ2
𝑑 2 𝑉 𝑉 𝑥, 𝑦 + 𝑘 − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − 𝑘)
≅
2
𝑑𝑦
𝑘2
𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦
ℎ2
𝑉 𝑥, 𝑦 + 𝑘 − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − 𝑘)
+
=0
2
𝑘
Letting k = h, (square grids)
𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦
𝑉 𝑥, 𝑦 + ℎ − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − ℎ)
+
=0
ℎ2
ℎ2
1
𝑉 𝑥, 𝑦 = 𝑉 𝑥 + ℎ, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦 + 𝑉 𝑥, 𝑦 + ℎ + 𝑉(𝑥, 𝑦 − ℎ)
4
Example
Numerical methods
• Charge Simulation method
Principle: simulating the field between condutors by using
simulation charges
Q1
+
X
B1
+
Q4
Q2
X B2
+
+
Q3
XB3
+
XB4
+
+
+ V1
Ṽ
-
-
-
-
-
-
V2
-
Steps
• Place simulation charges outside of the region to be analyzed
• Determine boundary points
• Solve potential equation to calculate simulation charges for
boundary points
• Control the value of charges
• Calculate potential and electric field values for the desired
point using determined simulation charges
Accuracy of this method depends on
1.
2.
3.
4.
5.
Type of the simulation charges
Number of simulation charges
Location of simulation charges
Number of boundary points
Location of boundary points
Types of simulation charges
• Point charge
• Line charge
• ...
Point charge
For spherical systems
𝑉=
𝑞
4𝜋ε𝑟
𝑟=
potential factor: 𝑃 =
𝑥𝑞 − 𝑥𝑝
2
+ 𝑦𝑞 − 𝑦𝑝
1
4𝜋ε𝑟
(𝑋𝑞 , 𝑌𝑞 )
q
P (𝑋𝑝 , 𝑌𝑝 )
2
q1
q2
electrode
V
X
B1
X
B2
q3
X
B1 Boundary points
Voltages at the boundaries
𝑉𝐵1
𝑞1
𝑞2
𝑞3
=
+
+
4𝜋 ε 𝑟11 4𝜋 ε 𝑟12 4𝜋 ε 𝑟13
𝑉𝐵2
𝑞1
𝑞2
𝑞3
=
+
+
4𝜋 ε 𝑟21 4𝜋 ε 𝑟22 4𝜋 ε 𝑟23
𝑉𝐵3
𝑞1
𝑞2
𝑞3
=
+
+
4𝜋 ε 𝑟31 4𝜋 ε 𝑟32 4𝜋 ε 𝑟33
In general...
[P] [q] = [V]
Simulation charges
vector
Potential factor matrix
Potential vector
• After finding simulation charges, the value of the charges
should be controlled
• Choose control points on known potentials&
Potential of any point
𝑃11
𝑃21
𝑃31
𝑃12
𝑃22
𝑃32
𝑃13
𝑃23
𝑃33
𝑞1
𝑉1
𝑞2 = 𝑉2
𝑞3
𝑉3
Potential of any K point in the region:
𝑞1
𝑞2
𝑞3
𝑉𝐾 =
+
+
4𝜋 ε 𝑟1𝐾 4𝜋 ε 𝑟1𝐾 4𝜋 ε 𝑟1𝐾
𝑟𝑖𝑘 =
(𝑥𝑖 − 𝑥𝑘 )2 −(𝑦𝑖 − 𝑦𝑘 )2
Electric field at any point
𝑑𝑉
𝐸=−
=−
𝑑𝑟
𝑑
𝑞
4𝜋ε𝑟
𝑑𝑟
𝑞
1
𝐸=
. 2
4𝜋ε𝑟 𝑟
Infinite line charge
For cylindrical systems
𝜆=
𝑉𝑃 =
l
r0
V=0
P
𝑄
𝑙
𝜆
𝑟0
ln
2𝜋𝜀 𝑟
r0: the distance between line
charge and the point with 0
potential
r: the distance between
charge and the point P
Potential at my heart
if I stand under a high voltage line
r
q
rHP
VL
h
𝑉ℎ𝑒𝑎𝑟𝑡
Suna
V=0
Ground (earth)
𝑞
ℎ
=
ln
2𝜋𝜀 𝑟𝐻𝑃
𝑞
ℎ
𝑉𝐿 =
ln
2𝜋𝜀 𝑟
Chapter 1 is over...