238 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
Analysis of Very Fast Transients in
Layer-Type Transformer Windings
Marjan Popov, Senior Member, IEEE, Lou van der Sluis, René Peter Paul Smeets, Senior Member, IEEE, and
Jose Lopez Roldan, Senior Member, IEEE
Abstract—This paper deals with the measurement, modeling,
and simulation of very fast transient overvoltages in layer-type
distribution transformer windings. Measurements were per-
formed by applying a step impulse with 50-ns rise time on a
single-phase test transformer equipped with measuring points
along the winding. Voltages along the transformer windings were
computed by applying multiconductor transmission-line theory
for transformer layers and turns. Interturn voltage analysis has
also been performed. Computations are performed by applying an
inductance matrix determined in two different ways; by making
use of the inverse capacitance matrix and by making use of the
well known Maxwell formulas. The modeling of the transformer
and the computations are verified by measurements.
Index Terms—Fourier analysis, high-frequency model, interturn
voltages, transformer, very fast transients.
I. INTRODUCTION
T
HE problem of very fast transient overvoltages has been
widely studied and many publications have appeared on
the behavior of the electrical components at high and very high
frequencies [1]–[12]. Also, several CIGRE working groups and
two IEEE working groups (Switchgear Committee and Trans-
former Committee) that deal with the problem of fast transients
addressed the subject [13] and pointed out that it was sometimes
difficult to identify specific transformer failures related to fast
transients. The short rise time of a surge prompted by a lightning
or a switching impulse can cause deterioration in the insulation
and ultimately lead to a dielectric breakdown. The severity of
this process depends on several factors, such as the frequency
at which the transformer is exposed to this type of surge, the
system configuration, the specific application of the component
etc. Large power transformers are exposed to very fast transient
overvoltages (VFTOs) by atmospheric discharges or gas-insu-
lated substation (GIS) switching. Distribution transformers and
motors are exposed to fast surges if they are switched by circuit
Manuscript received November 29, 2005; revised April 12, 2006. This work
was supported by the Dutch Scientific Foundation NWO-STW under Grant
VENI, DET.6526. Paper no. TPWRD-00679-2005.
M. Popov and L. van der Sluis are with the Power Systems Laboratory,
Delft University of Technology, Delft 2628CD, The Netherlands (e-mail:
M.Popov@ieee.org; L.vanderSluis@ewi.tudelft.nl).
R. P. P. Smeets is with KEMA T&D Testing, Arnhem 6812 AR, The Nether-
lands, and also with the Department of Electrical Engineering, Eindhoven Uni-
versity of Technology, Eindhoven 5612 AZ, The Netherlands (e-mail: Rene.
Smeets@kema.com).
J. Lopez Roldan is with Pauwels Trafo Belgium N.V., Mechelen B-2800,
Belgium (e-mail: jose.lopez-roldan@pauwels.com).
Digital Object Identifier 10.1109/TPWRD.2006.881605
breakers (CBs). The occurrence of VFTO in a large shell-type
transformer was reported in [8] and [9], where it was demon-
strated that internal resonances occur and that interturn voltages
can rise to such a high value that an insulation breakdown can
take place. Multiple reignitions can occur during the switching
of transformers and motors with vacuum CBs (VCBs), because
of the ability of VCBs to interrupt high-frequency currents. The
development process of multiple reignitions has been traced in
detail [14], [15]. It has been shown that the problem is not caused
by the VCB or the transformer, but by an interaction of both
[10]. It is therefore imperative to ascertain the speed at which
transient oscillations propagate inside the windings and the coils
and to identify the possible reason for a potential transformer
failure.
In order to study the propagation of transients, a model is
needed which is able to simulate the voltage distribution along
the transformer winding. In [1]–[4], techniques of lumped
parameter models are presented. Recent publications have re-
vealed that the type of transformer winding is important for the
choice of transformer model. In [8], it was demonstrated that a
hybrid model based on multiconductor transmission-line theory
could be successfully applied to describe the wave propagation
in large shell-type transformers. In [11] and [12], two types
of models were presented for transformers with interleaved
windings; one was based on multiconductor transmission-line
theory, while another was based on coupled inductances and
capacitances. The last one uses a modified modal approach that
is described in [7]. The advantage of the latter model is that it
lends itself to the use of existing simulation software such as the
Electromagnetic Transients Program (EMTP). Models based
on multiconductor transmission-line theory can be applied if
frequency analysis is used. This model is purely numerical and
the losses and proximity effects, normally represented in a wide
frequency range can be easily taken into account.
New developments in EMTP and Matlab have opened up pos-
sibilities for simulating very large circuits of coupled elements.
The disadvantage is that resistances must be constant with the
frequency. Usually, they are estimated for a constant frequency
so that they give the same power factor for an
– circuit with
constant resistance
compared with an – circuit where
is frequency dependent [16].
This disadvantage is precluded when frequency analysis
is used. The resistance can be calculated for each frequency.
However, the drawback of frequency analysis is the high order
of the inductance and capacitance matrices that describe the
transformer coils. Apart from that, the inverse Fourier trans-
form is normally conducted at discrete frequencies by applying
0885-8977/$20.00 © 2006 IEEE
POPOV et al.: ANALYSIS OF VERY FAST TRANSIENTS 239
fast Fourier transform (FFT) analysis. An accurate inversion
to time domain is achieved by applying a continuous inverse
Fourier transform. This generally requires long computation
time or may even be unrealizable because of the large frequency
spectrum, which requires operation with very large matrices.
The transmission-line theory has been reported as efficient for
the analysis of transients in motor windings [17]–[19].
This paper presents a model based on multiconductor trans-
mission-line theory for a 15-kVA single-phase test transformer
with layer-type windings. The results of the voltage transients
computed at the end of the first and the second layer were com-
pared with laboratory measurements. The method is also applied
for the analysis of the interturn voltages.
II. M
ODEL FOR
DETERMINATION OF THE
LINE-END
VOLTAGES
ON THE
HIGH-VOLTAGE
WINDING
The origin of multiconductor transmission-line modeling
(MTLM) is described through the theory of natural modes in
[11] and [20]. When a network of
coupled lines exists, and
when
and are the impedance and admittance matrices,
which are the self and mutual impedances and admittances
between the lines, then
(1)
where
and are incident voltage and current vectors of the
line. Note that
. Applying the modal analysis, the
system can be represented by the following two-port network:
(2)
where
(3)
In (2) and (3)
, current vectors at the sending and the
receiving end of the line;
, voltage vectors at the sending and the
receiving end of the line;
matrix of eigenvectors of the matrix ;
eigenvalues of the matrix ;
length of the line.
The system representation in (2) was applied for the compu-
tation of transients in transformer windings.
Distribution transformers are normally constructed with
a large number of turns, and it would be ideal to compute
voltages in every turn by representing each turn as a separate
line. This implies that the model has to operate with matrices
that contain a huge number of elements, which is too large to
Fig. 1. Windings or turns represented by transmission lines.
be stored in the average memory of presently available desktop
computers. A practical solution is to reduce the order of the
matrices. This can be achieved by grouping a number of turns
as a single line so that the information at the end of the line
remains unchanged, as in the case when separate lines are used
[8], [9]. This approach is used for layer-type modeling. Fig. 1
shows the representation of the windings by transmission lines.
At the end, the line is terminated by impedance
. This means
that only a group of turns can be examined and the other turns
of the transformer winding can be represented by equivalent
impedance. As the equivalent impedance has a significant in-
fluence, it must be calculated accurately for each frequency.
In [17], a method is proposed for estimating this impedance
accurately. Hybrid modeling gives a good approximation for
layer-type windings. The transformer is therefore modeled on
a layer-to-layer basis instead of a turn-to-turn basis. Applying
(2) to Fig. 1 results in the following equation:
.
.
.
.
.
.
.
.
.
.
.
.
(4)
In (4),
and are square matrices of the th order calcu-
lated by (3). The following equations hold for Fig. 1:
(5)
By using these equations and making some matrix operations
(see Appendix A), (4) can be expressed as
.
.
.
.
.
.
(5)
240 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
When we observe the model on a layer-to-layer basis, then
; hence, (5) can be rewritten as
.
.
.
.
.
.
.
.
.
(6)
where
(7)
and
is the inverse matrix of the matrix . is a square
matrix of order
that contains the values of
(7). One can note that the
element in (5) is the terminal
admittance of the transformer.
The voltages at the end of each layer can be calculated when
the voltage at the input is known and the corresponding transfer
functions are calculated. The time-domain solution results from
the inverse Fourier transform
real
(8)
In (8), the interval
, the smoothing constant , and the
step frequency length
must be properly chosen in order to
arrive at an accurate time-domain response [20]. The modified
transformation requires the input function
to be filtered
by an
window function. To compute the voltages in
separate turns, the same procedure can be applied.
III. T
EST TRANSFORMER
A. Transformer Description
To calculate the voltage transients in transformer windings,
it is important to determine the transformer parameters with
higher accuracy. These parameters are the inductances, the ca-
pacitances, and the frequency-dependent losses. The modeling
approach depends heavily on the transformer construction and
the type of windings. The test transformer in this case is a single-
phase layer-type oil transformer. Fig. 2 shows the transformer
during production in the factory.
The primary transformer winding consists of layers with a
certain number of turns; the secondary winding is made of foil-
type layers. The transformer is equipped with special measuring
points in the middle and at the end of the first layer of the trans-
former high-voltage side, and also at the end of the second layer.
All measuring points can be reached from the outside of the
transformer and measurements can be performed directly at the
layers. Table I shows the transformer data.
B. Determination of the Transformer Parameters
1) Capacitance: Fig. 3 shows the capacitances that are nec-
essary for the computation of the fast transients inside the wind-
ings.
Fig. 2. Test transformer during production in the factory.
TABLE I
T
RANSFORMER
DATA
Fig. 3. Description of the capacitances inside a transformer.
These were calculated by using the basic formulas for plate
and cylindrical capacitors. This is allowed because the layers
and turns are so close to each other that the influence of the
edges is negligible.
The capacitances
between the turns are important for the
computation of transients in the turns. However, since the very
POPOV et al.: ANALYSIS OF VERY FAST TRANSIENTS 241
Fig. 4. Network for layer-to-layer static voltage distribution.
Fig. 5. Computed static voltage distribution for different grounding capaci-
tances.
large dimensions of the matrix prevent the voltages in each turn
from being solved at one and the same time, a matrix reduction
can be applied [21], [22] so that the order of matrices corre-
sponds not to a single turn but to a group of turns. In this way,
the voltages at the end of the observed group of turns remain un-
changed. Later, these voltages can be used for the computation
of the voltage transients inside a group of turns. Capacitances
between layers and capacitance between the primary
and the secondary winding were calculated straightforwardly by
treating the layers as a cylindrical capacitor.
The capacitances
to the ground are small in this case and
are estimated at less than 1 pF. These are the capacitances from
the layers to the core. We can see in Fig. 2 that only a part of the
surface of the layers is at a short distance from the core and that
it is mostly the geometry of the surface that influences the value
of
. This is explained in Appendix B. Another method is
based on the extension of the width of the layer halfway into the
barrier on either side of the layer [4]. The capacitances to ground
are the capacitances that govern the static voltage distribution.
Fig. 5 shows the calculated static voltage distribution of each
layer for a unit input voltage. When the ground capacitance is
between 1 and 100 pF, the voltage distribution is more or less
linear.
The equivalent input capacitance in Fig. 4 is approximately
the same as the terminal phase-to-ground capacitance. The fact
that the ground capacitances have a small value means that the
phase-to-ground capacitance at the high-voltage side can be cal-
culated as a series connection of the interlayer capacitances
. Table II shows the calculated interlayer capacitances. The
equivalent value that results from these capacitances is 1.21
nF. The value of the phase-to-ground capacitance at the high-
voltage side is measured in two ways. An average value of 1.25
nF is measured by an impedance analyzer. The other method is
the voltage divider method described in [23]. The transformer
high-voltage winding is connected in series with a capacitor of
a known capacitance. A square impulse voltage is injected at the
input and the voltage is measured at both sides. The transformer
phase-to-ground capacitance is determined with a voltage divi-
sion formula. Applying this method, an average value of 1.14 nF
was measured.
TABLE II
L
AYER-TO-LAYER CAPACITANCE
(
1
10 F)
The capacitances matrix was formed as follows:
capacitance of layer to ground and the sum of all
other capacitances connected to layer
;
capacitance between layers and taken with the
negative sign
.
The capacitance matrix has the diagonal, upper diagonal, and
lower diagonal elements nonzero values and all other elements
are zeros.
Dividing these values with the length of a turn, the capaci-
tance per-unit length can be calculated.
2) Inductances: The easiest way to determine the inductance
matrix
is to calculate the elements from the capacitance matrix
(9)
where the velocity of the wave propagation
is calculated by
(10)
and
and are the speed of light in vacuum and the equiva-
lent dielectric constant of the transformer insulation, and N is
the number of turns in a layer. Matrix
that results from (9)
should be multiplied by the vector
, the elements of which
are squares of the lengths of the turns in all layers. We have to
point out that if matrices
and are given in this form, then
the length of the turn in (3) should be set to one. When using
telegraphists equations, it is a common practice to represent the
matrices
and with their distributed parameters. Therefore,
when the capacitance matrix
contains the distributed capaci-
tances of the layers, the vector
in (9) should be omitted. But
regarding the reduction of the order of matrices and applying
other formulas for computation of inductances, which are more
convenient to calculate the inductances in [H] and not in [H/m],
it is shown that it is not necessary to represent the parameters
with their distributed values.
Equation (9) is justified for very fast transients when the flux
does not penetrate into the core, and when only the first a few
microseconds are observed [17], [24]. The inductances can also
be calculated by using the basic formulas for self- and mutual
inductances of the turns [22], the so-called Maxwell formulas.
For turns as represented in Fig. 6, the self-inductance can be
calculated as [25]
(11)
242 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 1, JANUARY 2007
Fig. 6. Representation of circular turns for calculating inductances.
Fig. 7. Inductance matrix reduction method.
where and are the radius and the diameter of the turn. Radius
is calculated as a geometrical mean distance of the turn. The
mutual inductances between turns
and in Fig. 6 are obtained
considering the two conductors as two ring wires
(12)
where
; , , and are the
positions shown in Fig. 6; and
and are complete
elliptic integrals of the first and second kind.
In this case, it is assumed that the flux does not penetrate in-
side the core and a zero-flux region exists. Therefore, the ob-
tained self- and mutual inductances are compensated
and (13)
The
and are fictitious ring currents at the zero-flux region
with radius
with directions opposite to those of turns and .
The method applied here holds for inductances on a turn-to-turn
basis. The large matrix can be reduced by applying a matrix
reduction method based on the preservation of the same flux
in the group of turns [21]. The reduction process is simply the
addition of elements in the new matrix as shown in Fig. 7.
Formulas such as those published in [5], [25], and [26] can
also be used. The computed matrix according to (11)–(13) is
shown in the Appendix. The values of the matrix
computed
by (9) are lower than the values computed by the accurate for-
mulas (11)–(13). Applying (5), the frequency characteristics of
the transformer can be calculated.
3) Copper and Dielectric Losses: Losses play an essential
role in an accurate computation of the distributed voltages. The
losses were calculated from the inductance matrix
and the
TABLE III
M
EASURING
EQUIPMENT
Fig. 8. Recording equipment for the measurement of fast transient oscillations.
Fig. 9. Impedance analyzer for measuring the transformer impedance charac-
teristic.
capacitance matrix [12]. The impedance and admittance ma-
trices
and are then
(14)
In (14), the second term in the first equation corresponds
to the Joule losses taking into account the skin effect in the
copper conductor and the proximity effect. The second term in
the second equation represents the dielectric losses. In (14),
is the distance between layers; is the conductor conductivity;
and
is the loss tangent of the insulation.
IV. M
EASUREMENTS AND SIMULATIONS
A. Test Equipment
The equipment used for measuring the fast transients in the
transformer and impedance characteristics is listed in Table III.
The equipment itself is shown in Figs. 8 and 9.
