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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009
Dual Three-Winding Transformer Equivalent
Circuit Matching Leakage Measurements
Francisco de León, Senior Member, IEEE, and Juan A. Martinez, Member, IEEE
Abstract—An equivalent circuit for the leakage inductance of
three-winding transformers is presented. The model is derived
from the principle of duality (between electric and magnetic
circuits) and matches terminal-leakage inductance measurements.
The circuit consists of a set of mutually coupled inductances and
does not contain negative inductances. Each inductance can be
computed from both: the geometrical information of the windings
and from terminal-leakage measurements taking two windings at
a time. The new model is suitable for steady state, electromechanical transients, and electromagnetic transient studies. The circuit
can be assembled in any circuit simulation program, such as
EMTP, PSPICE, etc. programs, using standard mutually coupled
inductances.
Index Terms—Electromagnetic transients, Electromagnetic
Transients Program (EMTP), equivalent circuit, negative inductance, principle of duality, three-winding transformers,
transformer leakage inductance.
I. INTRODUCTION
T
HE CURRENTLY used equivalent circuit for a threewinding transformer was obtained by Boyajian in 1924
[1]; see Fig. 1. The circuit often contains a negative inductance.
Notwithstanding that the negative inductance is not realizable,
it has not presented problems with frequency-domain studies
using phasors [1]–[4]. The equivalent has been successfully
used for many years for the study of power flow, short circuit,
transient stability, etc. However, when computing EM transients (time-domain modeling), the negative inductance has
been identified as the source of spurious oscillations [5]–[9].
There are several alternatives that eliminate the numerical oscillations, but none of the suggested solutions fully satisfies all
physical interpretation concerns. In [7], an autotransformer is
introduced to eliminate the negative inductance. Reference [8]
proposes a modification of the circuit structure shifting the magnetizing branch. In [9], the unstable condition is eliminated by
neglecting the magnetizing losses. These “fixes” point to the existence of a physical inconsistency.
To correct the inconsistency, a circuit derived from the principle of duality is presented in this paper. The model can be constructed by using mutually coupled inductances readily available in any time-domain simulation program, such as the Electromagentic Transients Program (EMTP). This is in full agreeManuscript received February 08, 2008; revised June 20, 2008. Current version published December 24, 2008. Paper no. TPWRD-00091-2008.
F. de León is with the Polytechnic Institute of New York University,
Brooklyn, NY 11201 USA (e-mail: fdeleon@poly.edu).
J. A. Martinez is with the Departament d’Enginyeria Elèctrica, Universitat
Politècnica de Catalunya, Barcelona 08028, Spain (e-mail: jamv@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2008.2007012
Fig. 1. Traditional equivalent circuit for the leakage inductance of
three-winding transformers.
ment with the Boyajian physical interpretation of the negative
inductance as being the result of magnetic mutual couplings [3].
However, no available model for three-winding transformers explicitly represents the mutual coupling of flux in air.
The parameters of the equivalent circuit proposed in this
paper can be obtained in two ways: 1) from the design data
and 2) from terminal-leakage inductance measurements of two
windings at a time. Therefore, the model of this paper is useful
to both: transformer designers and system analysts.
II. ORIGIN OF THE NEGATIVE INDUCTANCE IN THE EQUIVALENT
CIRCUIT OF A THREE-WINDING TRANSFORMER
The conventional model for the leakage inductance of a
three-winding transformer is a star-connected circuit [1]–[4];
,
, and
are commonly referred to as the
see Fig. 1.
windings’ leakage inductances. However, this point of view is
can also be seen as the mutual
neither absolute nor exclusive.
inductance between windings 2 and 3; similar interpretations
and
[3].
can also be seen as the mutual
exist for
inductance between circuits 1–2 and 1–3. The interpretation
,
, and
are simply
advocated in this paper is that
elements of an equivalent circuit that represent the terminal
behavior of the transformer accurately only for steady-state
simulations.
, , and
do not correspond to leakage flux paths as
the components of duality derived models. It has been demonstrated that numerical instabilities when simulating transients
are due to the nonphysical negative inductance [5]–[9]. Additionally, note that a negative resistance may appear in the star
equivalent network. In [3], there is an interpretation of this “virtual resistance.” However, our model does not require nonrealizable circuit components.
A. Model Derived From Terminal Measurements
The parameters of the circuit can be obtained by matching the
inductive network (Fig. 1) to leakage inductances measured at the
0885-8977/$25.00 © 2008 IEEE
DE LEÓN AND MARTINEZ: DUAL THREE-WINDING TRANSFORMER EQUIVALENT
161
TABLE I
LEAKAGE INDUCTANCE TESTS FOR A THREE-WINDING TRANSFORMER
Fig. 2. Geometrical arrangement of windings in a transformer window.
For a standard transformer design
and,
becomes negative. It will be shown in Section II-B
therefore,
that the negative inductance appears in the traditional model
(Fig. 1) because it does not consider the mutual couplings that
take place in the region of the middle winding.
B. Model Derived From Design Information
terminals. The measurements are performed by taking two windings at a time. One winding is energized with a second winding
that is short-circuited while keeping the third winding open.
,
, and
can be obtained
Three inductances
from terminal measurements as follows.
is obtained when energizing winding number 1 while
winding 2 is short-circuited and winding 3 is open.
is obtained by energizing winding 2 and short-circuiting
winding 3 with winding 1 left open.
is obtained when winding 1 is energized and winding 3
is short-circuited with winding 2 open.
Table I describes the testing setup. For convenience, in this
paper, all inductances are referred to a common number of turns
. The inductances of the network of Fig. 1 are computed by
solving a set of equations such that the terminal measurements
are matched. In the calculation, no consideration is given to the
physical meaning of the inductances.
By inspection of the circuit of Fig. 1, the following relations
can be obtained:
The leakage inductances for a pair of windings can be computed from the design parameters assuming a trapezoidal flux
distribution [10], [11]. For the arrangement of windings and
dimensions depicted in Fig. 2, the magnetic flux distributions
during tests used to measure leakage inductances would be those
shown in Fig. 3. One can see that most of the leakage flux exits
from the core in the region between the windings under test.
Given the flux distribution of Fig. 4, we obtain the following
expressions [11]:
(3)
where is the common (or base) number of turns and is the
mean length of the winding turn. Substituting (3) into (2), we
obtain
(1)
,
, and
are computed to match leakage measurements and, therefore, one should not assign a physical meaning
to them. Solving (1) we obtain
(4)
(2)
Clearly, the value of
is always negative since the thickness
is always positive. It is interesting
of the middle winding
,
,
, ,
and )
to note that all inductances (
are functions of the thickness of the winding in the center.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009
Fig. 4. Trapezoidal flux distribution for leakage inductance tests.
Fig. 5. Duality derived model for a three-winding transformer.
the duality model, the following relation holds (neglecting mag. One can obtain from (3)
netizing):
Fig. 3. Magnetic flux distribution during leakage inductance tests.
(5)
III. DUALITY MODEL
Duality models are obtained from the geometrical arrangement of windings in the transformer window. No attention is
paid to the terminal-leakage measurements and, therefore, there
is frequently an inconsistency between duality models and terminal-leakage measurements. Duality models have been largely
discussed in the literature; see, for example, [11], [12], and [15].
The easiest way to build a duality model is to establish the flux
paths in the transformer window and assign an inductance to
each one [11]. The process is illustrated in Fig. 5.
The magnetizing flux is represented by the nonlinear induc,
, and
) and the leakage flux by the two
tances (
and
). The values of these two induclinear inductances (
and
.
tances match the measured leakage inductances
. In
However, there is no match for the leakage inductance
Accordingly, the duality model wrongly accounts for the
leakage inductance between the internal and the external windings.
is short by
(6)
Compare the magnetic flux distribution (for the region of )
and
with the one for
in
between the test for
Fig. 4. The flux in the center winding when computing
is smaller than that for
. This explains why a duality
derived model does not properly account for
. One can also
and
share a common path in the resee that the fluxes
and
are magnetigion of the central winding. Thus,
cally coupled.
DE LEÓN AND MARTINEZ: DUAL THREE-WINDING TRANSFORMER EQUIVALENT
163
Fig. 7. Duality-derived model for a three-winding transformer, including magnetizing branches and winding resistances.
Fig. 6. New duality derived model for a three-winding transformer.
IV. NEW DUALITY-DERIVED MODEL
AND TERMINAL MEASUREMENTS
In this section, a new model derived from the principle of
duality is proposed. The model consists of a network of mutually coupled inductors. The model simultaneously: matches
with terminal-leakage measurements, does not have negative inductances, and each element can be identified with a leakage
flux path.
The circuit of this paper is the evolution of the matrix model
originally presented in [13] for the turn-to-turn modeling of
transformer windings and recently used for the representation
of entire windings in [14].
From the analysis of the magnetic flux distribution during
the leakage test (see Figs. 3 and 4) and from the fact that the
expressions for all inductances are functions of , we postulate
and
must be mutually coupled.
that
The proposed leakage inductance circuit is derived from the
between
duality model by adding a mutual coupling
and
as shown in Fig. 6. This allows for compensating the
missing factor (6). The dot marks have been selected in such a
.
way that the total inductance increases for the test of
Applying the three tests depicted in Table I to the circuit of
Fig. 6, we obtain
(7)
can be determined from (7) in a straightforward manner
yielding
(8)
Equation (8) describes the computation of the compensating
mutual inductance directly from the leakage inductance tests.
is positive in most cases because
for
standard designs. By substituting (5) into (8), one can obtain an
as a function of the design parameters as
expression for
(9)
is half the factor (6) because
Note that, as expected,
enters twice in the total series inductance calculation. Additionally, note that
(10)
The complete dual equivalent circuit, including the magnetizing branches and the winding resistances, is shown in Fig. 7.
Fig. 8. Duality model for the iron core of a core-type transformer.
This circuit has the magnetic and electric elements separated by
three ideal transformers. Three magnetizing branches represent
the leg, the yokes, and the flux return (dually) connected at the
terminals of the ideal transformers.
gives the magnetic coupling of the
The mutual inductance
in (10) does
leakage fields between windings (flux in air).
not have any relationship with the commonly used mutual inductance between windings governed by the flux in the core.
,
, and
The latter is represented in the dual sense by
in the circuit of Fig. 7.
V. MAGNETIZING PARAMETERS
The inductances representing the leakage flux are computed
from tests (7) and (8) or from the design parameters (3) and
,
, and
(9). To compute the magnetizing inductances
, one must know the construction and dimensions of the
core. However, this is rarely available. Here, we will show that
regardless of the core construction (shell type or core type), the
equivalent circuit of Fig. 7 applies.
A. Single-Phase Three-Winding Transformers
Figs. 8 and 9 show the iron cores of single-phase core-type
and single-phase shell-type transformers, respectively. Note that
the structure of the equivalent circuit is the same for both core
geometries. The leakage part of the circuit and the shunt resis,
, and
have been omitted for clarity.
tances
The magnetizing inductance and its associated shunt resistance (used to represent hysteresis and eddy current losses) of
a transformer are obtained from the open-circuit test. Only one
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009
TABLE II
RELATIONSHIPS BETWEEN THE MODEL (FIG. 7) LEAKAGE INDUCTANCES
AND RESISTANCES TO THOSE OBTAINED FROM TERMINAL TESTS
By substituting (13) in (11) and after some algebra, we obtain
(14)
Substituting (14) in (13), we obtain
Fig. 9. Duality model for the iron core of a shell-type transformer.
(15)
magnetizing inductance
and one resistance
are determined from the measurements.
From Fig. 7, one can see that during an open-circuit test, the
three magnetizing branches are in parallel since the voltage drop
,
and are negligible. Then,
in the leakage inductances
the relationship between the measured magnetizing values (
and
) and the model values is given by
(11)
A sensible approximation for
,
,
,
,
,
can be obtained from the values of
and
when
and
the geometrical information is not known by assuming that the
window approximates a square. Thus, the length of the yokes
. From
is the same as the length of the legs
the visual analysis of Fig. 3, one can realize that the leakage
flux leaves the core at about 1/3 and 2/3 of the yoke length.
Therefore, we divide the yoke into thirds; see Figs. 8 and 9. The
and
(and
and
) becomes
flux length path for
(and
) is 2/3
5/3 while the flux length path for
(there are two yokes represented by the pair
,
).
Resistances are directly proportional to the path length while
inductances are inversely proportional to length. The resistance
and inductance associated with a leg length are
(12)
Then, we have
(13)
When the transformer is tall and slim with a small leakage in. As an extreme case, we can assume
ductance, then
that
. Consequently, the flux length path for
,
,
, and
is 4/3 while the flux length path for
and
is 1/3 . When the transformer is short and wide with
. As the other exa large leakage inductance, then
treme, consider that
. Now, the length of the flux
,
,
, and
is 7/3 while it is 4/3 for
path for
and
. Table II summarizes the standard, maximum,
and
and minimum values for the magnetizing inductances
resistances
as a function of
and
.
B. Three-Phase Three-Winding Transformers
Figs. 10 and 11 show the models for three-phase threewinding transformers (core type and shell type, respectively).
The magnetizing losses, the ideal transformers, and the
winding’s resistance can be added in a similar fashion as for
single-phase transformers (see Fig. 7).
The magnetizing parameters are more difficult to obtain for
a three-phase transformer than for a single-phase transformer.
There are no standardized tests that would allow for accurate
determination of the parameters. One needs to find a way to
energize one of the windings in every limb with all other coils
in the transformer opened. Although the tests can always be
performed at the factory, it may not be possible to test when
only the transformer terminals (after connections) are available
in the field. Cooperation from transformer manufacturers will
be most probably needed to properly determine the magnetizing
parameters of a dual equivalent circuit. Leakage parameters can
be computed using the procedures for single-phase transformers
described before.
Additional complications are the facts that 1) all limbs are
mutually coupled and 2) the iron core is highly nonlinear. Thus,
during tests, the different components of the core could be
excited at a different flux density than during normal operation,
therefore rendering the tests meaningless. The subject has
been extensively treated in [15] for two-winding three-phase
transformers. The magnetizing part of the equivalent circuits
DE LEÓN AND MARTINEZ: DUAL THREE-WINDING TRANSFORMER EQUIVALENT
165
Fig. 10. Model derived from the principle of duality for three-winding core-type transformers matching-terminal-leakage measurements.
TABLE III
TEST TRANSFORMER DATA
magnetizing pair, resistance, and inductance, derived from an
open-circuit test and referred to the low-voltage side (13.8 kV)
and
. The short-circuit tests
are
have given the following per unit leakage reactances:
0.10, and
0.84, and
0.96.
The model leakage inductances are computed in per unit from
(7) and (8) as follows:
(16)
Fig. 11. Model derived from the principle of duality for three-winding shelltype transformers matching-terminal-leakage measurements.
of Fig. 10 (three-winding three-phase transformers) closely
resembles the circuits of [15]. One needs to only add the extra
inductances connected to the middle winding.
VI. EXAMPLE
As an illustration and validation example, we have simulated
the unstable case presented in [8] with our model. The rated
transformer data are given in Table III (note that winding numbers 2 and 3 are switched with respect to [8]). The values of the
The leakage inductance values can be computed for the lowvoltage side using the impedance and inductance base
, yielding
mH. Thus, we have
mH,
mH, and
mH. The model
is shown in Fig. 12. The test consists in energizing the highwhile
voltage winding with a cosinusoidal function at
keeping the other two windings open. Fig. 13 shows that the
voltage on the low-voltage terminal is stable, while [8, Fig. 2]
shows numerical instability for the same case.
We have varied the integration time step over a wide range
(from 0.1 to 1.0 ms). Although numerical oscillations can be
seen at the start of the simulations when using large time steps,
all simulations are always stable for short or long study times;
we tested simulation times longer than 1000 s.
VII. NUMERICAL STABILITY ANALYSIS
The numerical stability of the new model is analyzed in the
same fashion as in [8] by looking at the eigenvalues of the state
matrix. We start by referring the circuit to the high-voltage side,
as shown in Fig. 14. The values of the circuit elements are
,
mH,
295 mH,
3.5 mH,
,
,
,
168 H,
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009
Fig. 12. Equivalent circuit of the three-winding transformer without negative
inductance.
Fig. 15. Equivalent circuit for the study of numerical stability.
Therefore, the state matrix is stiff, but not singular. Since all
eigenvalues are real and negative, the circuit is always stable, as
previously noted with the simulations.
VIII. CONCLUSION
In this paper, a solution to a long-standing problem with
models for three-winding transformers has been found, unifying the two available modeling methodologies. On one hand,
there are models obtained from terminal measurements that pay
no consideration to the physical meaning of the inductances
and frequently rely on a negative inductance. On the other hand,
there are duality-derived models which pay no attention to the
terminal-leakage measurements, and mismatches with terminal
measurements frequently occur.
The new equivalent circuit, proposed in this paper, is a duality-derived model applicable to single-phase and three-phase
transformers. The proposed circuit matches with terminalleakage measurements and does not have negative inductances.
Each element can be identified with a leakage flux path and
can be computed from the geometrical information of the
windings and the terminal-leakage measurements taking two
windings at a time. Therefore, the model of this paper is useful
to transformer designers and to system analysts as well. Additionally, the model can be built with readily available elements
in EMTP-type programs.
Fig. 13. Stable voltage at the low-voltage terminals (13.8 kV).
Fig. 14. Equivalent circuit for the study of numerical stability.
421 H,
168 H. The derivation of the state equation for the
circuit of Fig. 14 is given in the Appendix. The five eigenvalues
are
(17)
The circuit is stable because none of the eigenvalues are positive and the zero modes are never excited. The circuit of Fig. 14
was tested yielding stable results under all conditions. An investigation of the singularities showed that they were caused by not
considering the damping effects due to the resistances of windings 1 and 3. When the resistances are included in the analysis
;
),
(referred to the high-voltage side
the eigenvalues become
(18)
APPENDIX
STATE EQUATION FOR THE CIRCUIT OF FIG. 14
Fig. 15 shows the circuit of Fig. 14 in a suitable shape for analytical investigation. Applying Kirchhoff Voltage Law (KVL)
magnetizing branch, we have
to each parallel
(19)
(20)
(21)
DE LEÓN AND MARTINEZ: DUAL THREE-WINDING TRANSFORMER EQUIVALENT
For the leakage portion of the model, we can write
167
Substituting (19), (23), (21), (25), and (27) in (31) and (32), we
obtain
(22)
(33)
Kirchhoff’s current law (KCL) for each node gives
(23)
(34)
(24)
(25)
Substituting
we obtain
where
from (20) in (24) and rearranging,
(35)
(26)
Equations (33), (34), (28), (27), and (30) comprise a set of state
linear equations of the form
From (20), we obtain
(27)
Substituting (23) in (19), (26) in (20), and (25) in (21), we obtain
the differential equations for the magnetizing inductances as
(36)
With
(37)
(28)
one can build the state matrix as shown in (38) at the bottom of
the page where
(29)
(39)
(30)
To obtain the differential equations for the leakage inductances,
we develop (22) as
(31)
(32)
ACKNOWLEDGMENT
The authors would like to thank S. Magdaleno, undergraduate
student of Universidad Michoacana (Mexico), for performing
the finite-element simulations of Fig. 3 and X. Xu, graduate student at the Polytechnic Institute of New York University, for performing the ATP simulations presented in Fig. 13.
(38)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009
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Francisco de León (S’86–M’92–SM’02) was born
in Mexico City, Mexico, in 1959. He received the
B.Sc. and the M.Sc. (Hons.) degrees in electrical
engineering from the National Polytechnic Institute,
Mexico City, Mexico, in 1983 and 1986, respectively, and the Ph.D. degree from the University of
Toronto, Toronto, ON, Canada, in 1992.
He has held several academic positions in Mexico
and has worked for the Canadian electric industry.
Currently, he is an Associate Professor at the Polytechnic Institute of New York University, Brooklyn,
NY. His research interests include the analysis of power definitions under nonsinusoidal conditions, the transient and steady-state analyses of power systems,
the thermal rating of cables, and the calculation of electromagnetic fields applied to machine design and modeling.
Juan A. Martinez (M’83) was born in Barcelona,
Spain.
He is Professor Titular at the Department d’Enginyeria Elèctrica of the Universitat Politècnica de
Catalunya, Barcelona. His teaching and research interests include transmission and distribution, power
system analysis, and EMTP applications.