This document presents a new method for analyzing transient voltage distributions in transformer windings. It models the electric, magnetic, and current fields as equivalent circuits consisting of interconnected electric and magnetic networks. The electric network models the voltage distribution within each winding section using resistances and emfs derived from the geometry. The magnetic network models the flux paths using reluctances, which is then converted to an equivalent inductance network. Capacitances model the electric fields within and between sections. The electric, magnetic, and capacitance networks are coupled through ideal transformers to form a single overall mathematical model without mutual inductances. The model can analyze transient voltages inside the windings resulting from any applied terminal overvoltage waveform. An example application

1. New approach to the analysis of impulse voltage
distribution in transformer windings
0.Honorati
E. Santini
Indexing terms: Transformers, Modelling
and their degree of flexibility is rather limited. Inversion
Abstract: A new calculation method for the of the inductance matrices can produce marked errors,
analysis of transient voltage phenomena in HV while poor reliability in the assumed values for the induc-
power transformers is presented. Consideration is tive and resistive parameters can lead to uncertain
given to the electric, magnetic and current fields in results.
the windings. An equivalent circuit is derived; this In this paper a new approach is presented which may
consists in the connection of two networks, one be considered an alternative to those described above.
magnetic the other electric, whose components The model is again of the lumped parameters variety, but
with their numerical values are deduced directly is derived through the theory of magnetic networks,
from the geometry of the transformer and from its extended here to account for the capacitive phenomena.
electrical parameters. Application of the ensuing The model provides the transient voltages within the
mathematical model to a real case of a 132 kV, windings, and its main advantage is that there are no
50,000 kVa power transformer is discussed. mutual inductances in the equivalent circuit. The model’s
inductive and capacitive parameters can be calculated
through the geometry of the windings and the magnetic
structures. The approach is suitable for any configuration
1 Introduction of the windings and core, and permits analysis of the
transient voltage inside the windings (open or short-
The matter of overvoltage in windings has been a basic circuited) resulting from any waveform overvoltage
problem in the design of transformers for many decades applied at the terminals.
[l]. Many important studies have been carried out with The approach adopted is first defined, then the algo-
the aim of determining the distribution of the impulse rithms for studying the electrical and magnetic pheno-
overvoltages and the resonance frequencies of the wind- mena are presented. A mathematical description is given
ings [2, 3, 41. The question is still far from being solved. of the problem in the state space, and finally an applica-
The relevant phenomena are so complex that accurate tion to a real case is detailed.
modelling is very difficult. As power and voltage increase,
the electrical stresses become so high that greater accu-
racy is required in the analysis of the space and time 2 Derivation of the model
variation of the impulse overvoltages, as well as the volt-
ages induced in the other windings [S, 61. 2.1 Approach and schematisarion of the system
The analysis of impulse overvoltages has usually been The following points are the basis for establishing the
conducted on a distributed capacitance model, so that mathematical model:
transmission line theory is applicable [7]. This allows (a) The phenomena considered include a current field
qualitative evaluation of the phenomena under consider- within the windings, an electric field within the dielectrics
ation; series or derived capacities being the parameters space and a magnetic field throughout the entire space.
which upgrade or degrade the non-uniform distribution (b) The continuous systems represented by the fields
of the voltage in the windings. This model provides are reduced to lumped parameters systems, consisting of
simple formulas that are used for the design of many interlinked networks of electrical and magnetic two-
transformers. However, this method suffers from a dis- terminal circuits: the relevant approximations are left to
advantage in that it cannot represent the voltage dis- the designer’s ingenuity, depending on the required accu-
tribution at times following the application of the racy.
impulse, and therefore can only provide the initial dis- (c) Each winding is divided into sections; each section*
tribution. is replaced by a two-terminal electrical circuit, made up
More sophisticated approaches have recently been by a resistance and an emf produced by the time change
introduced. In these, turns or groups of turns are rep- of the magnetic flux in the branches linked with the
resented by finite cells, each defmed by the values of its section. The sections should be chosen so that a linear
resistance, self inductance and mutual inductance to the variation of the voltage from the beginning to the end of
other cells, as well by the capacity to earth and to the the section may be assumed. The two-terminal electric
other cells [8, 91. The solution programs are very large circuits constitute the electric network.
Paper 7201C (P7), received 10th October 1989 * Here ‘section’ stands for ‘element’: one or more pairs of discs, one or
The authors are with the Dipartmento di Ingegneria Elettrica Uni- more layers or some turns of a helix can be considered as a winding
versitl di Roma ‘La Sapienza’,Via Eudossiana 18,00184 Rome, ltaly section.
IEE PROCEEDINGS, Vol. 137, PI. C,No. 4, JULY I990 283

2. (4 magnetic field is replaced by branches of flux
The (ii) nonlinearity and the hysteresis of the magnetic
paths, each of which is considered as a two-terminal mag- materials
netic circuit characterised by its reluctance. These two (iii) losses due to eddy currents in the conductor
terminal magnetic circuits constitute the magnetic materials
network, where the mmf generators due to the currents
flowing in the windings are to be inserted. In order to simplify the model, the losses causing
(e) The electric and magnetic networks are interlinked. damping were taken into account by an adequate
The linkage determines the mmf in the magnetic network increase in the ohmic resistances of the windings. For the
and the emf (or magnetic flux linkages) in the electric transformer examined in the example, calculation of the
network. The generators need not be present in both net- frequency variation and the experimental pointers indi-
works and the networks consist of resistances and reluc- cated that a one hundredfold increase would be advis-
tances. able.
cf)The phenomena due to the electric field:
(i) those within each winding section (capacitive coup- 2.3 Capacitance network
lings between internal turns) For the phenomena associated with the electric field, the
(ii) those between each section and the adjacent sec- continuous system is replaced by a set of capacitors
tions which is equivalent within the prescribed approximation
(iii) those between the core and the facing sections limits. On the basis of point (c) above, it can be assumed
can be replaced by capacitances derived from the termin- that the voltage distribution is linear in each winding
als of each two-terminal circuit. This results in a capac- section, so the potentials of each two terminal electric cir-
itance network the nodes of which are the same as the cuits corresponding to the section unequivocally define
nodes of the electric network the field distribution within the section.
(g) In accordance with well known theorems, the mag- To analyse the influence of the electric field between
netic network can be replaced by its dual network, with the different sections, the electric field is replaced by a set
parameters interpreted as inductances (each correspond- of capacitors shunted between the two terminals of each
ing to the permeance of a branch of flux path). This electric circuit and between adjacent circuits (electrically
network is hooked up to nodes of the electric network coupled) according to the following procedure.
through ideal transformers.
Electric field within each winding section and series
In this manner a single network of resistances, induc- capacitance
tances and capacitances (without mutual inductances) is If there is a voltage difference between terminals A and B
obtained; this can be analysed by means of a specific of a winding section, then there is an electric field within
approach. it and an energy WEassociated with it; this can be calcu-
lated on the basis of the geometry of the system assuming
a linear distribution of the potential. The value of the
2.2 The magnetic network and its equivalent series equivalent capacitance C , to be shunted between
inductance network the terminals A and B can be determined on the basis of
Modelling a magnetic field through time-invariant flux the equation:
paths may seem impossible, but in the case of trans-
former structures (presence of ferromagnetic core and
concentric windings with small separation spaces) some
n
paths can be identified for the magnetic flux and the
problem becomes easier. where V, and V, are the voltages of the terminals A and
It is known that an accurate analysis of the magnetic B, E the permittivity and E the electric field within the
field through numerical methods (finite differences or winding region (instead of the volume integral it is
finite elements) is equivalent to replacing the field with an common practice to add the energies in the capacitive
invariant network of permeances. It is therefore reason- couplings between adjacent turns).
able to assume that a grid with a limited number of
nodes can represent the magnetic field with sufficient Electric field towards earth and capacitances shunted to
accuracy, automatically taking into account the different earth
distribution of the flux among the paths, as magneto- The coupling between the two terminal circuits represent-
motive forces vary in time. ing a winding section and the core is modelled as illus-
On the basis of the principle of duality, a magnetic trated in Fig. 1, where it is assumed that the earth is at
network can be replaced by an equivalent circuit
(consisting of inductances and accessible on the outside
through the insertion of ideal transformers). This consti-
tutes the model for the flux-linkage/current relations to
be introduced into the electric network so as to represent
the emf and to account for the effects of the magnetic
field.
The primary terminals of the ideal transformers, with
the resistances in series, must then be connected in series
or in parallel, exactly like the considered winding sec-
tions. Reference 10 should be consulted for a complete ‘L L/ I
discussion of the subject. This approach allows the intro- o
duction into the model of: Fig. 1 Circuit sirnulatiom
(i) effects of the partial linkage which occur within the a Capacitive coupling of section lo earth
windings b Voltage distribution across section
284 IEE PROCEEDINGS, Vol. 137, Pt. C, No. 4, JULY 1990

3. zero potential and the terminals A and B have different causes no alteration in the circuit simulation or in the
potentials. solution of the final system.
The numerical value of the capacitances C,, C, and C,
can be determined by imposing equality between the 3.1 Mathematical model of the capacitive network
energies stored and is obtained by means of analysis of The capacitive network and its N constituent nodes are
the electric field. considered. Once the reference node has been chosen
Let CG denote the capacity to earth of the whole (normally the earth), the matrix equation describing the
section under uniform voltage conditions between the equilibrium for the remaining N - 1 nodes is:
terminals, and let V and V denote the potentials at the
, ,
section terminals: the energy associated with the electric pcv= I + U (4)
field is then .
where D = dldt:
, ,
C is a square matrix of order (N - 1, N - I), con-
W = tcG(fv:
E + + fv' v,) (2) structed according to the classic rule of formation for
for the capacitors of Fig. 1 this is nodal matrices;
V i s an N - 1 vector whose components represent the
= +
WE tc,v: tc, v: tC,(V' - V,)2 + (3) node voltages:
By imposing equality of the two previous expressions for I and ri are N - 1 vectors and represent the known
any value of V and V ,then
' , terms of eqn. 4, i.e. the currents in the primaries of the
ideal transformers and those injected by current gener-
c , = c, = t c , ; c, = - fc, ators outside the transformer, respectively.
No physical meaning is attributed to the negative capac-
itance C, since it is the result of a process of synthesis. 32 Mathematical model of inductive network
The mesh-method algorithm is used for writing the equa-
Electricfield between two winding sections and capacitance tions of the inductive network. To apply this method, the
between sections number and composition of the independent meshes
The electric field in the insulation space between two must be determined; from them the number of the N ,
winding sections is approximated by the circuit shown in mesh currents is obtained. The meshes are then chosen in
Fig. 2. Through a procedure analogous to the one above, such a way that the secondaries of the ideal transformers
appear in one mesh only.
The descriptive equation is of the type:
pLIM = EM (5)
where p = d/dt;
L is a square matrix of (N,, N,) order, formed in
accordance with the classic rule of mesh-method analysis;
I is the (N,) vector of the mesh currents;
,
E, is the ( N M )vector of the known terms (the second-
U
1 b ary voltages of the ideal transformers).
Fig. 2 Circuit simulations By transferring the resistances of the winding sections
(1Capacitive coupling between two sections to the secondaries of the ideal transformers, the mesh
b Voltage distribution across two sections equation is modified as follows:
pLIM + RI, = E, (6)
equality of the energy of the electric field and that stored
in the capacitors yields the following capacitance values:
where matrix is a diagonal 'quare matrix Of Order
( N ? , N M ) ,which only has positive values in the elements
C1 - c '
'- -
2 - - i C D which refer to meshes where the secondary of an ideal
transformer is present.
c'-C'-
5 - 6-fCD 4 Global mathematical model
where CD denotes the shunted capacitance between two At this point eqns. 4 and 6 written for the capacitive and
adjacent winding sections, in uniform potential condi- inductive networks must be assembled. The components
tions in each section. representing the connection between the two networks
are ideal transformers, which are described by the equa-
3 Integral model
tions:
EL = KEM (7)
The final network consists of the coupling of the induc-
tance, capacitance and resistance networks, through the IM = KIL
ideal transformers described in the construction of the where E , and IL are the primary quantities, and the
inductive network. The structure of the final network transformation matrix K is diagonal of order ( N , , N M ) .
suggests the elaboration of a mathematical model which The linkage between the nodal voltages of the capac-
allows the networks to be modelled separately and subse- itive network and the voltages on the primaries of the
quently joined up. ideal transformers is expressed by the relationship :
The parameters representing the winding resistances
should be inserted in series with the primaries of the ideal Et= -M,V (9)
transformers. To facilitate the numerical solution it is in which the matrix M is a connection matrix of order
advisable to consider the resistive elements as being (N - 1, N M ) ;the values of its coefficients can be 1, 0, or
inserted in the secondary of the ideal transformers; this - 1.
IEE PROCEEDINGS, Vol. 137, Pt. C,No. 4, JULY 1990 285

4. A similar equation can be written for the currents: The schematisation involves two different levels of
approximation :
I = MIL (10) (a) every whole winding of the transformer is con-
Starting from eqns. 4 and 5 and taking into account eqns. sidered as a section
9-12 then: (b) windings are divided into three sections, to take
account of the presence of flux paths which transverse to
pCV = MIL (1 1) the windings
pKLKrM = -M , V - KRKr, (12) For approximation a the previously presented
and letting: approach and the modelling with electric and magnetic
networks leads to the Fig. 4 circuit; with appropriate
KLK = L , KRK= R
the system can be written in the final form:
A%= BX + U'; %= d X / d t (13)
For the time integration of this system of ordinary differ-
ential equations, a fourth-order multi-step method based
on Runge-Kutta's algorithm has been set up; this allows
a check of local error based on the distance in the state Fig. 4 Division of windings in single section
space between the tips of the vector solution calculated
with step h and that calculated with step h/2. simplification, this is derived by schematisation of the
magnetic field indicated in Fig. 3. The constraints inher-
5 Application of method ent in the configuration of the winding connections must
be duly introduced. The ensuing circuit has been
The method described has been applied to calculating the analysed by means of a computer program for the
time behaviour and internal distribution of the voltage in various connection configurations. In all the tests, the
a transformer subjected to an impulse voltage, using a free terminal of the regulating winding was supplied with
dedicated computer program. The features of the 3-phase the impulse voltage described by the function
transformer under consideration are given in the Appen-
dix. v,(t)= V2,(e-" - e-@') (14)
Computer simulation has permitted determination of with a = 1.72 x lo4 and fi = 1.54 x lo6.
the configuration which is most affected by the over- In the following, V, ( t ) denotes the voltage difference at
voltages, as well as the time variation of the voltage gra- the terminals of the HV, and V,(t) the voltage across the
dient in the windings. LV terminals (if not short-circuited).
The transformer has concentric windings. Fig. 3 shows The simulation was carried out up to a maximum time
the column of the core around which they are wound. of 80 ps, for all the cases described above.
Starting from the iron core, there is the low-voltage
winding (LV), the high-voltage winding (HV) and the
5.1 Division of windings into one section
high-voltage regulating winding (HVR). The study is con-
fined to just one phase, since this does not lead to any Configuration with H V R in addition and LV
significant errors. short-circuited
Figs. 5-11 illustrate the voltage in the windings for
several conditions as a function of time.
Fig. 5a illustrates the voltage of nodes 7 and 9 of the
I
circuit in Fig. 4. These nodes represent the terminal of the
regulating winding directly impulsed by the overvoltage
and the other terminal of the regulating winding, coincid-
ing with the HV terminal, respectively.
I In Fig. 5a it can be seen that the voltage on the HV is
I characterised by an alternating component with a fre-
II
~~ $"
- quency of about 50KHz superimposed on a unidirec-
tional damped component. The ratio between the peak
value of the voltage across the terminals of HV, V,, ,and
the peak value of the input voltage, V,,, is about 1.36.
t Fig. 5b shows the traces of VI and V, in the experimental
I test.
I
I Confguration with H V R in addition and LV
I open-circuited
I
Fig. 6 illustrates the results obtained in the analysis of
this case. It is noted that the ratio between the impulse
Fig. 3 Transformer schematisation maximum and the maximum value of the voltage reached
286 IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY 1990

5. by the end of the main winding attains a value of 1.40 in 40 kHz. Fig. 7 b shows the traces of VI and V, in the
this case. The ratio between the peak value of the voltage experimental test.
applied to the line terminal and the maximum value
reached on the LV, V,, , is about 1 1 , which is thus less
than the transformer turns-ratio (about 20).
I .
0 IO 20 30 40 50 60 70 80
t . P
a
b
Fig. 7 Comparison o uoltages
f
HVR in subtraction,LV short-circuited
a Calculated
b Expenmental ( I hor. div. = IO ps)
Configuration with H V R in subtraction and LV
b
open-circuited
Fig. 5 Comparison o voltages
f
Fig. 8 gives the results of the numerical simulation. The
HVR in addition,LV short-circuitted
(1Calculated frequency of the oscillation is about 44 kHz in this case.
b Experimental (1 hor. div. = lops) The V,,/V,, ratio is about 1.76, which is the highest
value. The ratio between V I , and V,, has also increased
with respect to the case of HVR in addition, reaching a
value of about 12.
N
0
i
- 50 80
t.15
t.P
Fig. 8 Calculated voltage to earth
Fig. 6 Calaculated uoltage to earth HVR in subtraction,LV open-circuited
HVR in addition,LV open-circuited
5.2 Division of windings into three sections:
ConJguration with H V R in subtraction and LV configuration with HVR in addition, LV
short-circuited short-circuited
Fig. 7 a shows the time variation of the voltages. The By dividing each winding of the transformer into three
ratio V,,/V,, in this case attains a value of 1.67, and the sections, the magnetic field is schematized as indicated in
frequency of the alternating component of VI is about Fig. 9a and the inductive network of 9b is obtained. The
1EE PROCEEDINGS, Vol. 137, Pt. C, No. 4, JULY 19W 287

6. inductances deriving from branches of flux paths in iron be noted that the V,,/V,, ratio in Fig. 11 is approx-
have been shown in black, the inductances which come imately 1.39, the difference being about 2% compared
from flux paths in air are in white and the inductances with the same ratio determined by considering each
which come from flux paths inside the windings are indi- winding as a single section. Fig. 12 shows the voltage gra-
cated by hatched lines.
N
0
>
a
11 1
0
t.PS
Fig. 11 Calculated voltage to earth
HVR in addition, LV short-circuited
Winding divided into three sections
18 17 17 16 16 15
b
Fig. 9 Diuision ofwindings
(I Flux paths b Inductive network
Fig. 10 shows the capacitive network, which must be
connected to the inductive network of Fig. 9b at nodes
bearing the same number.
The constraints deriving from winding connections
must be imposed on the network. For the case of HVR in
addition and LV in short circuit, Fig. 11 shows the time
variation of the voltages at the HVR and the HV termin-
als, practically coinciding with those of Fig. 5a. It should
501
Fig. 12 Calculated uoltages in H V R
D Section a e Smtian c
Fig. 10 Capacitive network b Section b
288 IEE PROCEEDINGS, Vol. 137, Pi.C, No. 4, JULY 1990

7. dient in the three HVR sections, and Fig. 13 the voltage sents greater difficulty, but its permeance has less influ-
gradient in the HV sections. The magnitude of the greater ence ;
voltage gradient in the first section of the main winding is (c) The capacitance network has parameters which
particularly evident in the latter. depend on the geometry of the transformer, in particular
on the distance between windings and between turns;
(4 Damping elements, which are very important for
analysing the time behaviour of the phenomena, can be
introduced into the model, the easiest way being to
ensure an adequate increase in the winding ohmic resist-
ance. The approach also allows a more accurate simula-
tion of losses by introducing other damping parameters;
(e) The model can be used for any case of rapidly
varying electromagnetic transients, so as to include
capacitive phenomena, core magnetisation phenomena
N (non linearity) and effects due to the other windings (open
or short circuited); for these windings the model deter-
>
v
mines the internal voltage distribution.
The computer program proposed to solve the network
was designed to suit the structure itself and proves to be
-25- particularly eficient, providing results very close to those
experimentally measured.
An examination of the results shows how the schema-
tisation of the windings in one single section allows the
time variation of the voltage at the terminals to be deter-
mined with sufficient accuracy. If the variation of the
voltage gradients in the most stressed zones is required,
these must be divided up, albeit into only a limited
number of sections.
I
7 Acknowledgments
-25 The authors would like to express their thanks to O.E.L.
for the experimental tests and the features of the trans-
former presented in the example.
50 I 8 References
1 HELLER, B., and VEVERKA, A.: ‘Surge phenomena in electrical
machines’ (London, 1968)
2 McWHIRTER, J.H., FAHRNKOPF, C.D., and STEELE, J.H.:
‘Determination of impulse stresses within transformer windings by
5 20 10 20 30 40 50 60 70 80 computer’, AIEE Trans., 1957, PAS-75, pp. 1267-1273
3 McNUTT, W.J., BLALOCK, T.J., and HINTON, R.A.: ‘Response
of transformer windings to system transient voltage’, IEEE Trans.,
1974, PAS-93, (2), pp. 4 5 7 4 6
4 DEGENEFF, R.C., McNUTT, W.J., NEUGEBAUER, W.,
Fig. 13 Calculated Doltages in H V PANEK, I. McCALLUM, M.E., and HONEY, C.C.: ‘Transformer
LIscctiona - -
resuonse to system switchinn voltaaes’. IEEE Trans.. 1982, PAS-
b Section b lOi, ( ) pp. 1457-1465
6.
e Seciionc 5 MUSIL, R.J., PREININGER, G., SCHOPPPER, E.., and
. ~.
WENGER. S.: ‘Voltage stresses oroduced bv amriodic and oscil-
lating system overvoltages in transformer windings’, IEEE Trans.,
0 Conclusions 1981, PAS-100, pp. 431438
6 ADIELSON, T., CARLSON, A., MARGOLIS, H.B., and HALL-
The main advantages offered by the approach described ADAY, J.A.: ‘Resonant overvoltages in EHV transformers’, IEEE
Trans., 1981, PAS-100, pp. 3563-3570
are : 7 STEIN, G.M.: ‘A study of the initial surge distribution in concentric
(a) The number, type and dimensions of the sections transformer windings’, IEEE Trans., 1964, PAS-83, pp. 877-892
into which the windings are divided can be varied as 8 DEGENEFF, R.C. : ‘A general method for determining resonances
desired, depending on the ingenuity of the designer, in in transformer windings’, IEEE Trans., 1977, PAS-%, pp. 423-430
order to increase the precision of the calculation and con- 9 MIKI, A., HOSOYA, T., and OKUYAMA, K.: ‘A calculation
method for impulse voltage distribution and transferred voltage in
sequently the complexity of the model; transformer windings’, IEEE Trans., 1978, PAS91, pp. 93&939
(b) The network of inductances derived through the 10 HONORATI, 0.. and SANTINI, E.: ‘Response of transformer
theory of magnetic networks avoids introducing mutual windings to transient voltages: The magnetic-electric network
inductances; the inductive parameters depend on the per- method, modeling and application’, to be published.
11 CREPAZ, S., DOGLIO, G., PESCU, G., and UBALDINI, M.:
meance of the branches of flux paths and can therefore be ‘Ripanizione della tensione ad impulso nei trasformatori con avvol-
directly computed on the basis of the geometric features gimento di regolazione di linea’, L’Energia Elettrica, 1982, 11, pp.
of the windings and of the core; the flux in the tank pre- 455473
IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY I990 289

8. 9 Appendix Capacitances to earth, pF LV 3250
HVR 800
Features of the transformer on which the simulation was
Binary short circuit HVR-HV 0.6867
carried out. inductances, H HVR-LV 1.2655
Power, KVA 50 OOO HV-LV 0.4269
Rated voltage, KV 132 10.1% 140
Number of turns HVR
Series capacitances, pF HVR 400 HV 1372
HV 370 LV 12
LV 10
Capacitances between HV-HVR 1200 The experimental test was obtained by applying waves
windings, pF HV-LV 1300 from a recurrent surge generator to the transformer.
IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY 1990