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
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.
Email: c.indulkar@ieee.org
(mlt) : mean length of turn.
2h : length of the leakage path.
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. LovassNagy and Szendy^{1} 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 papers^{23 }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 approach^{3}, expressions for the steadystate 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 integral^{4 }then calculates the transient voltages across the various nodes of the ladder. Gardner and Barnes^{5} 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 distributedconstant 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. Bewley^{6} 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 expressions^{6} which have not been used previously to obtain such numerical results.
Bewley^{6} 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
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
E = E (sinh bx/ sinh bl) + E A_{s }e^{}^{gs t}cosw_{s}t sin (spx/l) (3)
s= 1
where
A_{s }= 2(1)^{s} sp(a^{2 }b^{2}) l^{2}
(a^{2} l^{2} +s^{2}p^{2}) (b^{2} l^{2} +s^{2}p^{2})
a^{2 }= C/K
b^{2 }=RG
The computer program first calculates the following to obtain gs and w_{s }respectively.
l = sp/l
a = (+/)jl or l^{2} = a^{2}
gs = rKa^{4} –(rC +gL)a^{2} +LG
2L(C – Ka^{2})
H = (rKa^{4} –rCa^{2} –gLa^{2} +LG)^{2} 4L(C –Ka^{2})(a^{4} +gra^{4 }–rGa^{2})
w_{s }=_{ }_{Ö H}
2L (C – Ka^{2})
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:
The complete solution of equation (1) for isolated neutral and for no losses is
µ
E = E + E S B_{s }cos(spx/2l)_{.} cosW_{s}t (4)
s= 1
where
B_{s} = 16a^{2} l^{2} sin s p/2
s p(4a^{2} l^{2} +s^{2}p^{2})
W_{s }= s^{2}p^{2}/ 4 l^{2}
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. Bewley^{6 }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 opencircuited or
shortcircuited transmission line provided a zero value is used for the
capacitance K.
NUMERICAL
RESULTS





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 shortcircuited or an opencircuited
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 interturn
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 monitoring^{7} of the transformer
to diagnose power transformer failures.
1.
V. LovassNagy 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 38,1985
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 ^{–11}F.
Capacitance of winding to ground, lC= 10^{9}F.
r = 0.1 ohm p. u length of winding
L is calculated using the formula, L = 0.4 p n^{2} (mlt) (Henries)
h 10^{8}
g=3 x 10^{4} S.
G = 10^{8}S.
^{ }
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
l
dx x
G/dx
Ldx
K/dx
Cdx Gdx
Figure1 Ideal complete circuit for a winding
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 onetenth
of winding from the sending end