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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
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LECTURE NOTES
ON
HIGH VOLTAGE ENGINEERING

UNIT-I
INTRODUCTION TO HIGH VOLTAGE TECHNOLOGY AND APPLICATIONS
INTRODUCTION
The potential at a point plays an important role in obtaining any information regarding the electrostatic
field at that point. The electric field intensity can be obtained from the potential by gradient operation
on the potential
i.e. E = – ∇ V ...(1)
which is nothing but differentiation and the electric field intensity can be used to find electric flux
density using the relation
D = εE
...(2)
The divergence of this flux density which is again a differentiation results in volume charge
density.
∇ . D = ρv
...(3)
Therefore, our objective should be to evaluate potential which of course can be found in terms
of, charge configuration. However it is not a simple job as the exact distribution of charges for a
particular potential at a point is not readily available. Writing εE = D in equation (3) we have
εE = ρv
or
or
– ∇ ε ∇ V = ρv
2
ε ∇ V = – ρv
ρv
2
∇ V=– ε
...(4)
This is known as Poisson‘s equation. However, in most of the high voltage equipments, space
charges are not present and hence ρv = 0 and hence equation (4) is written as
2
∇ V=0
...(5)
Equation (5) is known as Laplace‘s equation
or
If ρv = 0, it indicates zero volume charge density but it allows point charges, line charge, ring
charge and surface charge density to exist at singular location as sources of the field.
Here ∇ is a vector operator and is termed as del operator and expressed mathematically in
cartesian coordinates as
∇=
∂
ax
∂
ay
∂
az
∂x
∂y
∂z
where a x , ay and az are unit vectors in the respective increasing directions.
...(6)

Hence Laplace‘s equation in cartesian coordinates is given as
2
2
2
∂ V ∂ V ∂ V
2
∇ V = ∂x 2 ∂y 2 ∂z 2 = 0
...(7)
Since ∇ . ∇ is a dot produce of two vectors, it is a scalar quantity. Following methods are
normally used for determination of the potential distribution
(i) Numerical methods
(ii) Electrolytic tank method.
Some of the numerical methods used are
(a) Finite difference method (FDM)
(b) Finite element method (FEM)
(c) Charge simulation method (CSM)
(d) Surface charge simulation method (SCSM).
FINITE DIFFERENCE METHOD
Let us assume that voltage variations is a two dimensional problem i.e. it varies in x-y plane and it
does not vary along z-co-ordinate and let us divide the interior of a cross section of the region where
the potential distribution is required into squares of length h on a side as shown in Fig. 0.1.
y
V2
b
V3
c
a
V0
V1
x
d
V
4
Fig. 0.1 A portion of a region containing a two-dimensional potential field divided
into square of side h .
Assuming the region to be charge free
∇ . D = 0 or ∇ . E = 0
and for a two-dimensional situation
∂Ex
∂E
y =0
∂x
∂y
and from equation (7) the Laplace equation is
2
2
∂ V
∂ V
2
2
∂x ∂y
=0
...(8)
Approximate values for these partial derivatives may be obtained in terms of the assumed
values (Here V0 is to be obtained when V1, V2, V3 and V4 are known Fig. 1.

∂V
∂x
a
V1 − V0
h
∂V
∂x
and
c
V0 − V3
h
...(9)
From the gradients
∂ V
2
∂x
∂x
a
c
V1 − V0 −2 V0 V3
h
0
h
0
V2 − V0 − V0 V4
2
h
2
∂ V
2
∂y
Similarly
− ∂V
∂V
∂x
2
...(10)
Substituting in equation (8) we have
2
∂ V
∂x
2
∂
2
V V1 V2 V3 V4 − 4V0 = 0
∂y
2
h2
1
(V1 + V2 + V3 + V4)
...(11)
4
As mentioned earlier the potentials at four corners of the square are either known through computations or at start, these correspond to boundary potentials which are known a priori. From equation
(11) it is clear that the potential at point O is the average of the potential at the four neighbouring
points. The iterative method uses equation (11) to determine the potential at the corner of every square
sub-division in turn and then the process is repeated over the entire region until the difference in values is
or
V0 =
less than a prespecified value.
The method is found suitable only for two dimensional symmetrical field where a direct solution is
possible. In order to work for irregular three dimensional field so that these nodes are fixed upon
boundaries, becomes extremely difficult. Also to solve for such fields as very large number of V(x, y)
values of potential are required which needs very large computer memory and computation time and hence
this method is normally not recommended for a solution of such electrostatic problems.
FINITE ELEMENT METHOD
This method is not based on seeking the direct solution of Laplace equation as in case of FDM, instead
in Finite element method use is made of the fact that in an electrostatic field the total energy enclosed
in the whole field region acquires a minimum value. This means that this voltage distribution under
given conditions of electrode surface should make the enclosed energy function to be a minimum for a
given dielectric volume v.
We know that electrostatic energy stored per unit volume is given as
2
W= 1 ∈E
...(12)
2
For a situation where electric field is not uniform, and if it can be assumed uniform for a differential volume δv, the electric energy over the complete volume is given as
1
W= 2
z
V
1
2 ∈ ( − ∇V ) dv
...(13)

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