Transient Oscillations in Transformer Primary Winding.

Prof. C. S. Indulkar, Fellow

The paper uses an analytical formula to determine the transient voltage distribution in a transformer winding and presents the results for the grounded neutral and isolated neutral cases.

Keywords :  Transient voltage; transformer winding; analytical formula; grounded neutral; isolated neutral

### NOTATIONS

L : inductance p.u length of winding (including the partial interlinkages)

C : shunt capacitance to ground p.u length of winding

K : series capacitance p.u length along the winding

g : shunt conductance to ground p.u length of winding

G :shunt conductance p.u length  along the winding

r :series resistance p.u length of winding

n : turns p.u length

e :potential to ground any point of the winding.

t : time

x : points along the winding measured from the neutral end.

l : length of the winding.

E : amplitude of the step voltage applied at x=l.

# Formerly, Professor atIndian Institute of Technology, New Delhi. Present address: B3 Gokul Society, Vasna Road, Baroda, 390015, Gujarat

E-mail: c.indulkar@ieee.org

(mlt) : mean length of turn.

2h : length of the leakage path.

### INTRODUCTION

This work describes a method for calculating the transient voltage waveforms at different points of a transformer winding following the impact of a step function of voltage at the input terminal of the winding. This investigation is necessary for the protection of extra high voltage transformers against traveling waves produced by the charging of uncharged transmission lines and transformers or by lightning. Lovass-Nagy and Szendy1 describe a matrix method for calculating the transient voltage distribution in a cascade consisting of a transmission line and a transformer winding, subjected to a voltage surge at the sending end of the line. The present author in previous papers2-3 applies the eigenvalue analysis technique to the determination of transient voltage distribution in a transformer winding and at various points of a cascade of transmission line and transformer, when energized from one end. In the eigenvalue approach3, expressions for the steady-state voltage at any node of the ladder in terms of the eigenvalues and eigenvectors of the chain matrix of the ladder section are used. Numerical integration of the Fourier inverse integral4 then calculates the transient voltages across the various nodes of the ladder. Gardner and Barnes5 use the difference equations approach in conjunction with the Laplace transform and its inverse to develop an analytical expression for the transient voltage distribution in a ladder network of a transformer winding. Since there are two independent variables n, the node number, and t, the time, there is a need for two direct Laplace transformations, one with respect to n and the other with respect to t, and the corresponding two inverse Laplace transformations. The winding of a transformer is actually a distributed–constant system, but for an approximate analysis of the winding’s behaviour, it is convenient to use in place of the distributed-constant system an approximate equivalent network having lumped constants and a finite number of loops. When the number of loops taken is large, however, analysis by ordinary procedure becomes too cumbersome to carry out and becomes practically prohibitive when the number of loops is ten or more. Gardner and Barnes therefore  attack the problem by use of difference equations and give the complete details of the development of an analytical expression for the transient voltage at any node of the ladder representing the transformer winding. The magnitude of each space harmonic and such functions as the space distribution at any instant and the time variation at any point of the voltage to ground are obtainable with the aid of the analytical expression. The analytical expression for the transient voltages at any node of the ladder network however considers only the inductance and capacitance parameters of the transformer winding and ignores the resistance parameter and hence the winding loss. The analytical expression gives the transient voltage at any node of the ladder   for only the grounded neutral case. The analytical expression cannot handle the isolated neutral case where the amplitude of oscillations is greater. Furthermore no numerical results of the transient voltage waveforms are given. Bewley6 has given analytical expressions for the transient voltage distribution of a transformer winding considering the winding loss for the neutral grounded case and ignoring the winding loss for the neutral isolated case. Bewley has shown that ignoring the secondary entirely and using an “effective inductance” in the analytical expressions can calculate the primary internal oscillations of a two –winding transformer quite accurately. Bewley has derived the analytical expressions using the following procedure:

·        Derive the differential equations of the circuit.

·        Specify the terminal conditions.

·        Determine the initial distributions at the instant of impact of   a step function of voltage.

·        Determine the final distributions.

·        Obtain the complete solution to satisfy the above, making use of the identity:

Initial distribution= Final distribution + Transient terms at first instant.

·        In order to express the two sides of the above equation in common terms, carry out Fourier expansions of the initial and final distributions.

However, Bewley also has not given any numerical results for the transient voltage waveforms in the transformer winding. The  present paper determines the transient voltages at various points of the transformer winding using Bewley’s analytical expressions6 which have not been used previously to obtain such numerical results.

### TRANSFORMER WINDING MODEL

Bewley6 shows that the essential characteristics of the oscillations in the transformer primary winding are essentially the same as obtain when the secondary winding  is ignored.  Referring to Figure1, the partial differential equation of the system is

rK5e      + (1+gr) 4e  - LK4e        -(rC +gL) 3e     -rG 2e      +LC 2e      +LG e      = 0

### ¶x4¶t                   ¶x4        ¶x2¶t2                    ¶x2¶t         ¶x2     ¶t2               ¶t

(1)

Neglecting losses, equation (1) reduces to

4e  - LK4e    + LC 2e       = 0                                                                                             (2)

### Solution for grounded neutral

The complete solution of equation (1) for grounded neutral that satisfies the initial conditions at x =0 and x= l, the initial and final potential distributions at t=o and t =µ respectively, is

µ

E = E (sinh bx/ sinh bl)      + E            As e-gs tcoswst sin (spx/l)                                            (3)

s= 1

where

As =         2(-1)s sp(a2 -b2) l2

(a2 l2 +s2p2) (b2 l2 +s2p2)

a2 = C/K

b2 =RG

The computer program first calculates the following to obtain -gs and ws  respectively.

l = sp/l

a = (+/-)jl or l2 = -a2

gs = rKa4 –(rC +gL)a2 +LG

2L(C – Ka2)

H =  (rKa4 –rCa2 –gLa2 +LG)2- 4L(C –Ka2)(a4 +gra4 –rGa2)

ws  =  Ö H

2L (C – Ka2)

For zero losses, b = 0 and the solution for the potential at any point is obtained by taking zero

values for r, g and G respectively.

The effect of the losses is four –fold:

1. Damping of oscillations by exponential decrement factors.

1. Decrease in the amplitudes of oscillations.

1. Decrease in the frequencies of oscillation.

1. Shift in the final distribution or axis of oscillation.

### Solution for isolated neutral

The complete solution of equation (1) for isolated neutral and for no losses is

µ

E = E + E   S  Bs cos(spx/2l). cosWst                                                                                        (4)

s= 1

where

Bs = -16a2 l2 sin s p/2

s p(4a2 l2 +s2p2)

Ws =            s2p2/ 4 l2

ÖP

where

P = L(C + K s2p2/4 l2  )

The analytical expressions (3) and (4) give the voltage to ground at any point along the length of the winding having one end grounded or isolated and the other subjected to a unit step voltage, the winding being initially without currents or charges. Bewley6 gives the complete derivation of the analytical expressions for the winding grounded or isolated at the far end for an infinite rectangular applied voltage wave. These analytical expressions for the transient voltages are directly applicable to a machine winding and may also be used for determining the voltage transients in an open-circuited or short-circuited transmission line provided a zero value is used for the capacitance K.

NUMERICAL RESULTS

The Appendix gives the transformer winding data. Figure 2 shows the voltage waveforms at the mid-point of the transformer winding, and at one-tenth of winding from the grounded neutral end, for an applied unit step voltage at the beginning of the winding. The graph shows that there is a slight overvoltage at the mid-point and the voltage oscillates around 0.5 p.u. The transient voltage near the grounded neutral end, at one-tenth of the winding length, oscillates around 0 p. u voltage. Figure 3 shows the transient voltage at the midpoint of the winding for the isolated neutral case. It is worth noting that the analytical formula that is used is for a lossless winding in the isolated neutral case. The peak transient overvoltage at the mid-point is more than 2 p.u and the peak amplitude of the oscillations would stabilize around 2 p.u if the calculations continue beyond 500 ms.  Figures 2 and 3 respectively show that the oscillation frequency for the transient voltage at the mid-point of the winding for the lossless winding in the isolated neutral case is much higher than that for the lossy winding in the grounded neutral case. Figures 4 and 5 show respectively the transient voltages for the isolated neutral case at the open end of the winding and at one-tenth of winding length from the sending end. The transient voltage at one-tenth of the winding from the sending end oscillates around 1 p.u as expected.

### CONCLUSIONS AND FURTHER WORK

The aim of the paper is to present the applications of available analytical expressions for determining the transient voltages in the transformer primary winding for the grounded neutral and the isolated neutral cases respectively. The determination of the transient voltages at various points in a short-circuited or an open-circuited transmission line using modified analytical formulae that take the capacitance parameter K along the winding as zero is also possible. The results of the analytical formulae and those obtained by the lattice diagram technique can be compared.  The present paper studies the transient voltages for the isolated neutral case using the available analytical formula for the lossless winding. Further development of the formula that considers the winding resistance parameter r for the neutral isolated case may also be attempted. The study can also be extended to show how the results of transient response analysis are sensitive to the transformer winding ground faults at different locations on the winding. For this purpose, a laboratory model of the transformer winding could help in the comparison of the numerical results of the transient study with the laboratory tests. The laboratory model of the transformer winding, consisting of 10 sections, with the given values of L, C and R for each section may be built up. The application of a square wave of 10 kHz at the sending end of the model could determine the transient voltages at different points of the winding or the line using an oscilloscope. The laboratory model should be flexible enough for introducing an inter-turn fault, an earth fault, or an impedance variation fault at any point of the winding. The numerical values of the transient responses at different points for both the faulted and unfaulted transformer windings can be stored in a database and used for the condition monitoring7 of the transformer to diagnose power transformer failures.

## REFERENCES

1.                  V. Lovass-Nagy and C. Szendy, ‘Calculation Of Transient Voltages In A Transformer Connected To A Transmission Line’, Proc. IEE, Vol.111, 1964, p.1133

2.                  C. S. Indulkar, S.N. Saha and D.P. Kothari, ‘ Transient Voltages In A Transformer Connected To A 500 Kv Transmission Line’, IEEE /PES Winter Meeting, New York, February 3-8,1985

3.                  C. S. Indulkar, ‘Eigenvalue Analysis Of Identical Ladder Networks’, Int. J. Elect. Enging Educ., Vol.17,1980, p.359.

4.                  N. Mullineaux, S.J. Day and J. N. Reed, ‘Developments In Obtaining Transient Response Using Fourier Transform- Use Of The Modified Fourier Transform’ Int. J. Elect. Enging Educ.,Vol. 4 ,1966, p.31.

5.                  M.F. Gardner J. L. Barnes, ‘Transients In Linear Systems’ , John Wiley, New York, 1942, p.320.

6.                  L. V. Bewley, ”Travelling Waves On Transmission Systems’ Dover, New York,1963, p.413.

7.                  C. Bengtsson, ‘Status And Trends In Transformer Monitoring’, IEEE Transactions on Power Delivery, Vol. 11, 1996, No.3, p.1379.

Appendix

Transformer data

Number of turns, nl = 1000.

Winding length, l = 1.5 m.

Mean length of turn of coil, (mlt) = 1.25 m.

Length of leakage path, h = 0.12 m.

Capacitance from end to end of winding, K/l = 10 –11F.

Capacitance of winding to ground, lC= 10-9F.

r = 0.1 ohm p. u length of winding

L is calculated using the formula, L = 0.4 p n2 (mlt)     (Henries)

h 108

g=3 x 10-4 S.

G = 10-8S.

Note: g = shunt conductance to ground p.u length of winding, and G =shunt conductance p.u length along the winding are both assumed zero in obtaining the numerical results of this paper.

LIST OF FIGURE CAPTIONS

Figure1 Ideal complete circuit for a winding

Figure2 Transient voltages for the neutral grounded case

Figure3 Transient voltage at midpoint of winding for isolated neutral case

Figure4 Transient voltage at the open end of   winding for isolated neutral case

Figure5 Transient voltage at one-tenth of winding length from the sending end

l

dx                                x

G/dx

Ldx

K/dx

Cdx               Gdx

Figure1 Ideal complete circuit for a winding

Figure2 Transient voltages for the neutral grounded case

Figure 3 Transient voltage at midpoint of  isolated neutral case

Figure 4 Transient voltage at the open end of winding for isolated neutral  case

Figure 5 Transient voltage at one-tenth  of  winding from the sending end