transformer engineering design and practice 1_phần 9

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), 277 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 Figure 7.2 Representation of a transformer winding 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 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. Surge Phenomena in Transformers 281 Figure 7.3 Initial voltage distribution Hence, the value of maximum initial gradient at the line end is the same for the grounded and isolated neutral conditions for abrupt impulses or very steep wave fronts. The initial voltage distribution for various values of a is plotted in figure 7.3 for the grounded and isolated neutral conditions. The total series capacitance (CS) and ground capacitance (CG) of the transformer winding predominantly decide the initial stresses in it for steep fronted voltage surges. The total series capacitance consists of capacitance between turns and capacitance between disks/ sections of the winding, whereas the total ground capacitance includes the capacitance between the winding and core/tank/other windings. Thus, the initial voltage distribution is characterized by the distribution constant, (7.12) This parameter indicates the degree of deviation of the initial voltage distribution from the final linear voltage distribution which is decided solely by winding inductances. The higher the value of α, the higher are the deviation and amplitudes of oscillations which occur between the initial and final voltage distributions. For a conventional continuous disk winding, the value of α may be in the range of 5 to 30. Any change in the transformer design, which decreases the distribution constant of the winding, results in a more uniform voltage distribution and reduces the voltage stresses between different parts of the winding. The initial voltage distribution of the winding can be made closer to the ideal linear distribution (α=0) by increasing its series capacitance and/ or reducing its capacitance to ground. If the ground capacitance is reduced, more current flows through the series capacitances, tending to make the voltage across the various winding sections more uniform. The (ideal) uniform initial impulse voltage distribution will be achieved if no current flows through the (shunt) ground capacitances. Usually, it is very difficult and less cost-effective to reduce the Copyright © 2004 by Marcel Dekker, Inc. ... - tailieumienphi.vn