Electrostatic Fields
SUMMARY
Quantification of Electric Fields:
• Analytical solutions
• Graphical solutions
• Numerical methods
Electric Fields:
• Series/parallel
• Homogeneous/inhomogeneous
Dielectrics in Electric Fields:
• Polarization
EFFECT OF ELECTRIC FIELD ON INSULATORS
An atom consists of a positive core (nucleus)
surrounded by negative electrons forming an
electron shell
In an insulator,
the outermost
electron shell is
full
In a conductor,
the outer electron
shell usually have
empty spaces
• Lots of energy
required to release
a free electron from
this shell to transport
current
• Very low energy
(weak external
electric field)
required to release
charge carriers
HIGH VOLTAGE ENGINEERING
THE ART OF INSULATION DESIGN
An insulator, also called a dielectric, is a material that resists
the flow of electric current.
An insulating material has atoms with tightly bonded valence
electrons (outermost electrons of an atom). The valence (en
yüksek değer) band containing the highest energy electrons is
full, and a large energy gap separates this band from the
conduction band above it.
n=3
There is always some voltage (breakdown
voltage) that will give the electrons enough
energy to be excited into the conduction band.
Once this voltage is exceeded, the material
come to end being an insulator, and charge
begin to pass through it.
n=2
n=1
+Ze
∆E = hv
HIGH VOLTAGE ENGINEERING
THE ART OF INSULATION DESIGN
*During electrical breakdown, any free charge carrier being
accelerated by the strong electric field has sufficient velocity to
ionize any atom it strikes (liberate electrons).
*These freed electrons and ions are in turn accelerated and
strike other atoms, creating more charge carriers, in a
chain reaction(zincirleme reaksiyonda).
*Rapidly the insulator becomes filled with mobile
carriers, and its resistance drops to a low level.
INSULATORS vs. DIELECTRICS
A dielectric is electrical
insulation that can be polarized by
an applied electric field.
• When a dielectric is placed in an
electric field, electric charges do
not flow through the material, as in
a conductor, but only slightly
shift from their average balance
positions causing dielectric
polarization.
• Positive charges are displaced
along the field and negative
charges shift in the opposite
direction.
This creates an internal electric
field that partly compensates
the external field inside the
dielectric.
The term "insulator"
refers to a low degree
of electrical
conduction.
The term "dielectric"
is typically used to
describe materials
with high
polarizability.
(expressed by the
dielectric constant εr)
Analyzing Electric Fields
Analytical Solutions
Graphical Representation
Numerical Methods
ELECTRIC FIELD
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ELECTRIC FIELD
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find the compound force on charge
Q ‘’if charge Q is put at point p’’
ELECTRIC FIELD
linear load density
surface charge density
volumetric charge density
ELECTRIC FIELD
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Differential force
ELECTRIC FIELD
Let our Q load be at the point (x, 0)
ELECTRIC FIELD
Insulators are NOT IDEAL – Leakage Current
Further requirements for static field conditions:
Homogeneous
:Characteristics are constant
Isotropic
:Electric flux and electric field
are in the same direction
Susceptibility(hassaslık) :
Φ
cannot depend on electric field strength.
E
Measure of how easily a dielectric
un
polarizes in response to an electric field
Φ   E  un dA 
A
Q

ELECTRIC FIELD
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ELECTRIC FIELD
Example: Field of Loaded
conductive spheres
+
+
+
+
+
R
+
+
+
Gauss kanununun uygulamaları
Example: Field of Loaded conductive spheres
Electric Field Lines
Electric Field Lines
Electric Field Lines
Analyzing Electric Fields
Analytical Solutions
Graphical Representation
Numerical Methods
ELECTRIC FIELD
ANALYTIC SOLUTION
ɛ
1. PLANE
• Electric flux direction is always from positive
charge towards negative charge
Q
 E = EA 

U = Ed 
Q
d
A
A
+Q
Q
E=
A
C=
d
Q A

U
d
Capacitance C increases with permittivity ε
Multiple
layers:
A
C n
di

i 1  i
E:
U:
-Q
Solid Dielectrics
XLPE
cable
2. CYLINDER
(cylinder insulator)
• Charge can be viewed as a line charge at the center
axis with a charge density per distance of q
Total charge Q = ql
• The line charge causes an electric flux through its
surrounding cylinder
 E = AE  2rlE 
Q


r
Er
ql

l
• Electric field decreases when travelling from the
inner radius ri to outer radius ro
•
as radius r increases, surface area increases through
which the electric flux travels
q
Q
Er =

2π  r 2π l r
ri
ro
The potential difference between cylinders can be derived as:
ro
ro
q
q
Q
dr =
ln r o 
ln r o
2π  r
2π  r i 2π l r i
ri
U =  Er dr = 
ri
Electric field at any distance r can be expressed as:
U
Er =
r lnro ri 
U
[r  ri]
Maximum value at inner radius ri : Emax =
U
ri lnro ri 
2π l
Capacitance can be expressed as C =
lnr o r i 
E
r4
E
r3
Emax
r2
r
r1
ɛ1
ɛ2
ɛ3
Emax =
r1
r2
U
ri lnro ri 
r3
r4
r
When the maximum electric field Emax at the inner radius
exceeds the dielectric strength Eb of the insulator,
U
breakdown or partial discharge may occur. E =
max
The voltage at which breakdown
may
occur
in
a
cylinder
construction can be given as,
U b  Eb ri lnro ri 
The cylinder insulator is optimized to
withstand the maximum breakdown
voltage Ubmax when the inner
radius ri = ro/e
U b max  Eb ri
Euler Number =e =2,71
ri lnro ri 
1,2
Ub/Ubmax
1
0,8
0,6
0,4
0,2
1/e
0
0
0,2
0,4
0,6
ri / ro
0,8
1