transformer engineering design and practice 1_phần 9 pdf

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7

Surge Phenomena in Transformers

For designing the insulation of a transformer suitable for all kinds of overvoltages,

the voltage stresses within the windings need to be determined. For this purpose,

voltage distributions within the transformer windings for the specific test voltages

are calculated. For AC test voltages of power frequency, the voltage distribution is

linear with respect to the number of turns and can be calculated exactly. For the

calculation of the impulse voltage distribution in the windings, they are required

to be simulated in terms of an equivalent circuit consisting of lumped R, L and C

elements. There are a number of accurate methods described in the literature for

computation of winding response to impulse voltages, some of which are

discussed in this chapter. Electric stresses in the insulation within and outside the

windings are obtained by analytical or numerical techniques which are described

in the next chapter.

7.1 Initial Voltage Distribution

When a step voltage impinges on the transformer winding terminals, the initial

distribution in the winding depends on the capacitances between turns, between

windings, and those between windings and ground. The winding inductances have

no effect on the initial voltage distribution since the magnetic field requires a finite

time to build up (current in an inductance cannot be established instantaneously).

Thus, the inductances practically do not carry any current, and the voltage

distribution is predominantly decided by the capacitances in the network, and the

problem can be considered as entirely electrostatic without any appreciable error.

In other words, the presence of series capacitances between winding sections

causes the transformer to respond to abrupt impulses as a network of capacitances

for all frequencies above its lower natural frequencies of oscillations. When the

applied voltage is maintained for a sufficient time (50 to 100 microseconds),

Copyright © 2004 by Marcel Dekker, Inc.

278 Chapter 7

appreciable currents begin to flow in the inductances eventually leading to the

uniform voltage distribution. Since there is difference between the initial and final

voltage distributions, as shown in figure 7.1, a transient phenomenon takes place

during which the voltage distribution readjusts itself from the initial to final value.

During this transient period, there is continual interchange of energy between

electric and magnetic fields. On account of a low damping factor of the

transformer windings, the transient is oscillatory. The voltage at any point in the

winding oscillates about the final voltage value, reaching a maximum as shown by

curve c. It is obvious that the strength of the transformer windings to lightning

voltages can be significantly increased if the difference between the initial and

final distributions can be minimized. This not only reduces the excessive stresses

at the line end but also mitigates the oscillations thereby keeping voltage to ground

at any point in the winding insignificantly higher than the final voltage

distribution.

The differential equation governing the initial voltage distribution u0=u(x,0),

for the representation of a winding shown in figure 7.2 (and ignoring inductive

effects), is [1]

(7.1)

In figure 7.2, Ls, cg and cs denote self inductance per unit length, shunt capacitance

per unit length to ground and series capacitance per unit length between adjacent

turns respectively.

Figure 7.1 Impulse voltage distribution

Copyright © 2004 by Marcel Dekker, Inc.

Surge Phenomena in Transformers 279

Solution of the above equation is given by

µ0=A1ekx+A2e-kx (7.2)

where

(7.3)

The constants of integration A1 and A2 can be obtained from the boundary

conditions at the line and neutral ends of the winding. For the solidly grounded

neutral, we have µ0=0 for x=0. Putting these values in equation 7.2 we get

A1+A2=0 or A1=-A2

Whereas at the line end, x=L (L is the winding axial length) and u0=U (amplitude

of the step impulse voltage) giving

(7.4)

Substituting the above expression in equation 7.2 we get

(7.5)

The initial voltage gradient at the line end of the winding is given by

Figure 7.2 Representation of a transformer winding

Copyright © 2004 by Marcel Dekker, Inc.

280 Chapter 7

(7.6)

The initial voltage gradient is maximum at the line end. Since kL>3 in practice,

coth giving the initial gradient at the line end for a unit amplitude surge

(U=1)as

(7.7)

The uniform gradient for the unit amplitude surge is 1/L.

(7.8)

where CG and CS are the total ground capacitance and series capacitance of the

transformer winding respectively. The ratio has been denoted by the

distribution constant α. Thus, the maximum initial gradient at the line end is α

times the uniform gradient. The higher the value of ground capacitance, the higher

are the values of α and voltage stress at the line end.

For the isolated neutral condition, the boundary conditions,

give the following expression for the initial voltage distribution:

(7.9)

For the isolated neutral condition, the maximum initial gradient at the line end can

be written as

(7.10)

For a unit amplitude surge and (α=kL)>3, Hence, the initial gradient

becomes

(7.11)

Copyright © 2004 by Marcel Dekker, Inc.