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U(x , t)
  
U(
x , t)
V
U(
x , t)
U(
x , t)
(15)
1
1
1
1
 
V
U( x ,t)
U(x ,t)
(16)
1
1
The compensation source of value  V is applied in the point x 1 , but its effect must be
propagated throughout the line, since the non existence of line in the right of x 1 alters the
induced voltage values along whole line.
In order to compute the voltage at any point x , we can sum the induced voltage computed
for the point x , assuming an infinite line, to the value of the compensation source applied in
x 1 .
However, the compensation source located at x 1 suffers a time delay during the trajectory
between x and x 1 , that is:
(17)
V(x,t)
U(x,t)
  
U( x,t)
[ U( x ,t)
U(x ,t)]u(t
t )
1
1
f
The function u ( t - t f ) is a unit step function and t f is the travel time between the point x and
x 1 , i.e.:
xx
1
t
(18)
f
v
0
where v 0 is the propagation velocity, which for the simulations in question will be assumed
as being equal to the velocity of light.
The same procedure can be achieved supposing a discontinuity at the left of x . Then,
assuming a finite line, with the origin at the point x 0 and termination in x f , the induced
voltage at a point x along the distribution line can be estimated according to the following
expression:
V(x,t)
U(x,t)
  
U( x,t)
[
U( x ,t)
U(x ,t)] u(t

t )
f
f
f
f
(19)

[
U(x , t)
 
U(
x , t)] u(t
 
t )
0
0
0
0
The replacement of the line discontinuity effect by a voltage compensation source is an
effective procedure, mainly when is desired to produce computational algorithms.
4. Expert system for specification of transformers and equipments protection
against atmospheric discharges
The expert system proposed in this work, which was developed in order to help in the
arresters specification for equipments and distribution transformer protection, has its
implementation aspects based on the studies about induced voltages presented in previous
sections. Besides using those suggested modifications, the expert system incorporates in a
integrated way the databases referent to equipments installed on the distribution lines of
Bandeirante Energy, as well as the databases involved with arresters and atmospheric
discharge characteristics incident in its concession region.
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