dual three-winding transformer equivalent circuit matching

160 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009

Dual Three-Winding Transformer EquivalentCircuit Matching Leakage Measurements

Francisco de León, Senior Member, IEEE, and Juan A. Martinez, Member, IEEE

Abstract—An equivalent circuit for the leakage inductance ofthree-winding transformers is presented. The model is derivedfrom the principle of duality (between electric and magneticcircuits) and matches terminal-leakage inductance measurements.The circuit consists of a set of mutually coupled inductances anddoes not contain negative inductances. Each inductance can becomputed from both: the geometrical information of the windingsand from terminal-leakage measurements taking two windings ata time. The new model is suitable for steady state, electromechan-ical transients, and electromagnetic transient studies. The circuitcan be assembled in any circuit simulation program, such asEMTP, PSPICE, etc. programs, using standard mutually coupledinductances.

Index Terms—Electromagnetic transients, ElectromagneticTransients Program (EMTP), equivalent circuit, negative in-ductance, principle of duality, three-winding transformers,transformer leakage inductance.

I. INTRODUCTION

T HE CURRENTLY used equivalent circuit for a three-winding 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 studiesusing phasors [1]–[4]. The equivalent has been successfullyused for many years for the study of power flow, short circuit,transient stability, etc. However, when computing EM tran-sients (time-domain modeling), the negative inductance hasbeen identified as the source of spurious oscillations [5]–[9].

There are several alternatives that eliminate the numerical os-cillations, but none of the suggested solutions fully satisfies allphysical interpretation concerns. In [7], an autotransformer isintroduced to eliminate the negative inductance. Reference [8]proposes a modification of the circuit structure shifting the mag-netizing branch. In [9], the unstable condition is eliminated byneglecting the magnetizing losses. These “fixes” point to the ex-istence of a physical inconsistency.

To correct the inconsistency, a circuit derived from the prin-ciple of duality is presented in this paper. The model can be con-structed by using mutually coupled inductances readily avail-able in any time-domain simulation program, such as the Elec-tromagentic Transients Program (EMTP). This is in full agree-

Manuscript received February 08, 2008; revised June 20, 2008. Current ver-sion 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: [email protected]).

J. A. Martinez is with the Departament d’Enginyeria Elèctrica, UniversitatPolitècnica de Catalunya, Barcelona 08028, Spain (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2008.2007012

Fig. 1. Traditional equivalent circuit for the leakage inductance ofthree-winding transformers.

ment with the Boyajian physical interpretation of the negativeinductance as being the result of magnetic mutual couplings [3].However, no available model for three-winding transformers ex-plicitly represents the mutual coupling of flux in air.

The parameters of the equivalent circuit proposed in thispaper can be obtained in two ways: 1) from the design dataand 2) from terminal-leakage inductance measurements of twowindings at a time. Therefore, the model of this paper is usefulto 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 athree-winding transformer is a star-connected circuit [1]–[4];see Fig. 1. , , and are commonly referred to as thewindings’ leakage inductances. However, this point of view isneither absolute nor exclusive. can also be seen as the mutualinductance between windings 2 and 3; similar interpretationsexist for and [3]. can also be seen as the mutualinductance between circuits 1–2 and 1–3. The interpretationadvocated in this paper is that , , and are simplyelements of an equivalent circuit that represent the terminalbehavior of the transformer accurately only for steady-statesimulations.

, , and do not correspond to leakage flux paths asthe components of duality derived models. It has been demon-strated that numerical instabilities when simulating transientsare due to the nonphysical negative inductance [5]–[9]. Addi-tionally, note that a negative resistance may appear in the starequivalent network. In [3], there is an interpretation of this “vir-tual resistance.” However, our model does not require nonreal-izable circuit components.

A. Model Derived From Terminal Measurements

The parameters of the circuit can be obtained by matching theinductive network (Fig. 1) to leakage inductances measured at the

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DE LEÓN AND MARTINEZ: DUAL THREE-WINDING TRANSFORMER EQUIVALENT 161

TABLE ILEAKAGE INDUCTANCE TESTS FOR A THREE-WINDING TRANSFORMER

terminals. The measurements are performed by taking two wind-ings at a time. One winding is energized with a second windingthat is short-circuited while keeping the third winding open.

Three inductances , , and can be obtainedfrom terminal measurements as follows.

is obtained when energizing winding number 1 whilewinding 2 is short-circuited and winding 3 is open.

is obtained by energizing winding 2 and short-circuitingwinding 3 with winding 1 left open.

is obtained when winding 1 is energized and winding 3is 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 measurementsare matched. In the calculation, no consideration is given to thephysical meaning of the inductances.

By inspection of the circuit of Fig. 1, the following relationscan be obtained:

(1)

, , and are computed to match leakage measure-ments and, therefore, one should not assign a physical meaningto them. Solving (1) we obtain

(2)

Fig. 2. Geometrical arrangement of windings in a transformer window.

For a standard transformer design and,therefore, becomes negative. It will be shown in Section II-Bthat the negative inductance appears in the traditional model(Fig. 1) because it does not consider the mutual couplings thattake place in the region of the middle winding.

B. Model Derived From Design Information

The leakage inductances for a pair of windings can be com-puted from the design parameters assuming a trapezoidal fluxdistribution [10], [11]. For the arrangement of windings anddimensions depicted in Fig. 2, the magnetic flux distributionsduring tests used to measure leakage inductances would be thoseshown in Fig. 3. One can see that most of the leakage flux exitsfrom the core in the region between the windings under test.

Given the flux distribution of Fig. 4, we obtain the followingexpressions [11]:

(3)

where is the common (or base) number of turns and is themean length of the winding turn. Substituting (3) into (2), weobtain

(4)

Clearly, the value of is always negative since the thicknessof the middle winding is always positive. It is interestingto note that all inductances ( , , , , and )are functions of the thickness of the winding in the center.


Fig. 3. Magnetic flux distribution during leakage inductance tests.

III. DUALITY MODEL

Duality models are obtained from the geometrical arrange-ment of windings in the transformer window. No attention ispaid to the terminal-leakage measurements and, therefore, thereis frequently an inconsistency between duality models and ter-minal-leakage measurements. Duality models have been largelydiscussed in the literature; see, for example, [11], [12], and [15].The easiest way to build a duality model is to establish the fluxpaths in the transformer window and assign an inductance toeach one [11]. The process is illustrated in Fig. 5.

The magnetizing flux is represented by the nonlinear induc-tances ( , , and ) and the leakage flux by the twolinear inductances ( and ). The values of these two induc-tances match the measured leakage inductances and .However, there is no match for the leakage inductance . In

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-netizing): . One can obtain from (3)

(5)

Accordingly, the duality model wrongly accounts for theleakage inductance between the internal and the external wind-ings. is short by

(6)

Compare the magnetic flux distribution (for the region of )between the test for and with the one for inFig. 4. The flux in the center winding when computing

is smaller than that for . This explains why a dualityderived model does not properly account for . One can alsosee that the fluxes and share a common path in the re-gion of the central winding. Thus, and are magneti-cally coupled.


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 ofduality is proposed. The model consists of a network of mu-tually coupled inductors. The model simultaneously: matcheswith terminal-leakage measurements, does not have negative in-ductances, and each element can be identified with a leakageflux path.

The circuit of this paper is the evolution of the matrix modeloriginally presented in [13] for the turn-to-turn modeling oftransformer windings and recently used for the representationof entire windings in [14].

From the analysis of the magnetic flux distribution duringthe leakage test (see Figs. 3 and 4) and from the fact that theexpressions for all inductances are functions of , we postulatethat and must be mutually coupled.

The proposed leakage inductance circuit is derived from theduality model by adding a mutual coupling betweenand as shown in Fig. 6. This allows for compensating themissing factor (6). The dot marks have been selected in such away that the total inductance increases for the test of .

Applying the three tests depicted in Table I to the circuit ofFig. 6, we obtain

(7)

can be determined from (7) in a straightforward manneryielding

(8)

Equation (8) describes the computation of the compensatingmutual inductance directly from the leakage inductance tests.

is positive in most cases because forstandard designs. By substituting (5) into (8), one can obtain anexpression for as a function of the design parameters as

(9)

Note that, as expected, is half the factor (6) becauseenters twice in the total series inductance calculation. Addition-ally, note that

(10)

The complete dual equivalent circuit, including the magne-tizing branches and the winding resistances, is shown in Fig. 7.

Fig. 7. Duality-derived model for a three-winding transformer, including mag-netizing branches and winding resistances.

Fig. 8. Duality model for the iron core of a core-type transformer.

This circuit has the magnetic and electric elements separated bythree ideal transformers. Three magnetizing branches representthe leg, the yokes, and the flux return (dually) connected at theterminals of the ideal transformers.

The mutual inductance gives the magnetic coupling of theleakage fields between windings (flux in air). in (10) doesnot have any relationship with the commonly used mutual in-ductance between windings governed by the flux in the core.The latter is represented in the dual sense by , , and

in the circuit of Fig. 7.

V. MAGNETIZING PARAMETERS

The inductances representing the leakage flux are computedfrom tests (7) and (8) or from the design parameters (3) and(9). To compute the magnetizing inductances , , and

, one must know the construction and dimensions of thecore. However, this is rarely available. Here, we will show thatregardless of the core construction (shell type or core type), theequivalent circuit of Fig. 7 applies.

A. Single-Phase Three-Winding Transformers

Figs. 8 and 9 show the iron cores of single-phase core-typeand single-phase shell-type transformers, respectively. Note thatthe structure of the equivalent circuit is the same for both coregeometries. The leakage part of the circuit and the shunt resis-tances , , and have been omitted for clarity.

The magnetizing inductance and its associated shunt resis-tance (used to represent hysteresis and eddy current losses) ofa transformer are obtained from the open-circuit test. Only one


Fig. 9. Duality model for the iron core of a shell-type transformer.

magnetizing inductance and one resistance are deter-mined from the measurements.

From Fig. 7, one can see that during an open-circuit test, thethree magnetizing branches are in parallel since the voltage dropin the leakage inductances , and are negligible. Then,the relationship between the measured magnetizing values (and ) and the model values is given by

(11)

A sensible approximation for , , , , ,and can be obtained from the values of and whenthe geometrical information is not known by assuming that thewindow approximates a square. Thus, the length of the yokesis the same as the length of the legs . Fromthe visual analysis of Fig. 3, one can realize that the leakageflux 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. Theflux length path for and (and and ) becomes5/3 while the flux length path for (and ) is 2/3(there are two yokes represented by the pair , ).

Resistances are directly proportional to the path length whileinductances are inversely proportional to length. The resistanceand inductance associated with a leg length are

(12)

Then, we have

(13)

TABLE IIRELATIONSHIPS 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

(15)

When the transformer is tall and slim with a small leakage in-ductance, then . As an extreme case, we can assumethat . Consequently, the flux length path for ,

, , and is 4/3 while the flux length path forand is 1/3 . When the transformer is short and wide witha large leakage inductance, then . As the other ex-treme, consider that . Now, the length of the fluxpath for , , , and is 7/3 while it is 4/3 for

and . Table II summarizes the standard, maximum,and minimum values for the magnetizing inductances andresistances as a function of and .

B. Three-Phase Three-Winding Transformers

Figs. 10 and 11 show the models for three-phase three-winding transformers (core type and shell type, respectively).The magnetizing losses, the ideal transformers, and thewinding’s resistance can be added in a similar fashion as forsingle-phase transformers (see Fig. 7).

The magnetizing parameters are more difficult to obtain fora three-phase transformer than for a single-phase transformer.There are no standardized tests that would allow for accuratedetermination of the parameters. One needs to find a way toenergize one of the windings in every limb with all other coilsin the transformer opened. Although the tests can always beperformed at the factory, it may not be possible to test whenonly the transformer terminals (after connections) are availablein the field. Cooperation from transformer manufacturers willbe most probably needed to properly determine the magnetizingparameters of a dual equivalent circuit. Leakage parameters canbe computed using the procedures for single-phase transformersdescribed before.

Additional complications are the facts that 1) all limbs aremutually coupled and 2) the iron core is highly nonlinear. Thus,during tests, the different components of the core could beexcited at a different flux density than during normal operation,therefore rendering the tests meaningless. The subject hasbeen extensively treated in [15] for two-winding three-phasetransformers. The magnetizing part of the equivalent circuits


Fig. 10. Model derived from the principle of duality for three-winding core-type transformers matching-terminal-leakage measurements.

Fig. 11. Model derived from the principle of duality for three-winding shell-type transformers matching-terminal-leakage measurements.

of Fig. 10 (three-winding three-phase transformers) closelyresembles the circuits of [15]. One needs to only add the extrainductances connected to the middle winding.

VI. EXAMPLE

As an illustration and validation example, we have simulatedthe unstable case presented in [8] with our model. The ratedtransformer data are given in Table III (note that winding num-bers 2 and 3 are switched with respect to [8]). The values of the

TABLE IIITEST TRANSFORMER DATA

magnetizing pair, resistance, and inductance, derived from anopen-circuit test and referred to the low-voltage side (13.8 kV)are and . The short-circuit testshave 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)

The leakage inductance values can be computed for the low-voltage side using the impedance and inductance base

, yielding mH. Thus, we havemH, mH, and mH. The model

is shown in Fig. 12. The test consists in energizing the high-voltage winding with a cosinusoidal function at whilekeeping the other two windings open. Fig. 13 shows that thevoltage 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 beseen 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 thesame fashion as in [8] by looking at the eigenvalues of the statematrix. 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,


Fig. 12. Equivalent circuit of the three-winding transformer without negativeinductance.

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 thecircuit of Fig. 14 is given in the Appendix. The five eigenvaluesare

(17)

The circuit is stable because none of the eigenvalues are posi-tive and the zero modes are never excited. The circuit of Fig. 14was tested yielding stable results under all conditions. An inves-tigation of the singularities showed that they were caused by notconsidering the damping effects due to the resistances of wind-ings 1 and 3. When the resistances are included in the analysis(referred to the high-voltage side ; ),the eigenvalues become

(18)

Fig. 15. Equivalent circuit for the study of numerical stability.

Therefore, the state matrix is stiff, but not singular. Since alleigenvalues are real and negative, the circuit is always stable, aspreviously noted with the simulations.

VIII. CONCLUSION

In this paper, a solution to a long-standing problem withmodels for three-winding transformers has been found, uni-fying the two available modeling methodologies. On one hand,there are models obtained from terminal measurements that payno consideration to the physical meaning of the inductancesand frequently rely on a negative inductance. On the other hand,there are duality-derived models which pay no attention to theterminal-leakage measurements, and mismatches with terminalmeasurements frequently occur.

The new equivalent circuit, proposed in this paper, is a du-ality-derived model applicable to single-phase and three-phasetransformers. The proposed circuit matches with terminal-leakage measurements and does not have negative inductances.Each element can be identified with a leakage flux path andcan be computed from the geometrical information of thewindings and the terminal-leakage measurements taking twowindings at a time. Therefore, the model of this paper is usefulto transformer designers and to system analysts as well. Addi-tionally, the model can be built with readily available elementsin EMTP-type programs.

APPENDIX

STATE EQUATION FOR THE CIRCUIT OF FIG. 14

Fig. 15 shows the circuit of Fig. 14 in a suitable shape for an-alytical investigation. Applying Kirchhoff Voltage Law (KVL)to each parallel magnetizing branch, we have

(19)

(20)

(21)


For the leakage portion of the model, we can write

(22)

Kirchhoff’s current law (KCL) for each node gives

(23)

(24)

(25)

Substituting from (20) in (24) and rearranging,we obtain

(26)

From (20), we obtain

(27)

Substituting (23) in (19), (26) in (20), and (25) in (21), we obtainthe differential equations for the magnetizing inductances as

(28)

(29)

(30)

To obtain the differential equations for the leakage inductances,we develop (22) as

(31)

(32)

Substituting (19), (23), (21), (25), and (27) in (31) and (32), weobtain

(33)

(34)

where

(35)

Equations (33), (34), (28), (27), and (30) comprise a set of statelinear equations of the form

(36)

With

(37)

one can build the state matrix as shown in (38) at the bottom ofthe page where

(39)

ACKNOWLEDGMENT

The authors would like to thank S. Magdaleno, undergraduatestudent of Universidad Michoacana (Mexico), for performingthe finite-element simulations of Fig. 3 and X. Xu, graduate stu-dent at the Polytechnic Institute of New York University, for per-forming the ATP simulations presented in Fig. 13.

(38)


REFERENCES

[1] A. Boyajian, “Theory of three-circuit transformers,” AIEE Trans., pp.208–528, Feb. 1924.

[2] F. Starr, “Equivalent circuits -I,” AIEE Trans., vol. 57, pp. 287–298,Jun. 1932.

[3] L. F. Blume, Transformer Engineering. New York: Wiley, 1951.[4] Electrical Transmission and Distribution Reference Book. U.S.:

Elect. Syst. Technol. Inst., (Westinghouse T&D Book), ABB, 1997.[5] H. W. Dommel, Electromagnetic Transients Program Reference

Manual (EMTP Theory Book). Portland, OR: BPA, 1986.[6] W. S. Meyer and T.-H. Liu, “Unstable saturable transformer,” Can/Am

EMTP News, vol. 93, no. 2, pp. 15–16, Apr. 1993.[7] P. S. Holenarsipur, N. Mohan, V. D. Albertson, and J. Cristofersen,

“Avoiding the use of negative inductances and resistances in modelingthree-winding transformers for computer simulations,” in Proc. IEEEPower Eng, Soc. Winter Meeting, New York, Jan. 1999, pp. 1025–1030.

[8] X. Chen, “Negative inductance and numerical instability of the sat-urable transformer component in EMTP,” IEEE Trans. Power Del., vol.15, no. 4, pp. 1199–1204, Oct. 2000.

[9] T. Henriksen, “How to avoid unstable time domain responses causedby transformer models,” IEEE Trans. Power Del., vol. 17, no. 2, pp.516–522, Apr. 2002.

[10] K. Karsai, D. Kerenyi, and L. Kiss, Large Power Transformers. NewYork: Elsevier, 1987.

[11] G. Slemon, Electric Machines and Drives. Reading, MA: Addison-Wesley, 1992.

[12] J. A. Martinez and B. A. Mork, “Transformer modeling for low- andmid-frequency transients—a review,” IEEE Trans. Power Del., vol. 20,no. 2, pt. 2, pp. 1625–1632, Apr. 2005.

[13] F. de Leon and A. Semlyen, “Efficient calculation of elementary pa-rameters of transformers,” IEEE Trans. Power Del., vol. 7, no. 1, pp.376–383, Jan. 1992.

[14] R. M. Del Vecchio, “Applications of a multiterminal transformer modelusing two winding leakage inductances,” IEEE Trans. Power Del., vol.21, no. 3, pp. 1300–1308, Jul. 2006.

[15] J. A. Martinez, R. Walling, B. A. Mork, J. Martin-Arnedo, and D.Durbak, “Parameter determination for modeling system transients-partIII: Transformers,” IEEE Trans. Power Del., vol. 20, no. 3, pp.2051–2062, Jul. 2005.

Francisco de León (S’86–M’92–SM’02) was bornin Mexico City, Mexico, in 1959. He received theB.Sc. and the M.Sc. (Hons.) degrees in electricalengineering from the National Polytechnic Institute,Mexico City, Mexico, in 1983 and 1986, respec-tively, and the Ph.D. degree from the University ofToronto, Toronto, ON, Canada, in 1992.

He has held several academic positions in Mexicoand has worked for the Canadian electric industry.Currently, he is an Associate Professor at the Poly-technic Institute of New York University, Brooklyn,

NY. His research interests include the analysis of power definitions under non-sinusoidal conditions, the transient and steady-state analyses of power systems,the thermal rating of cables, and the calculation of electromagnetic fields ap-plied to machine design and modeling.

Juan A. Martinez (M’83) was born in Barcelona,Spain.

He is Professor Titular at the Department d’En-ginyeria Elèctrica of the Universitat Politècnica deCatalunya, Barcelona. His teaching and research in-terests include transmission and distribution, powersystem analysis, and EMTP applications.

dual three-winding transformer equivalent circuit matching

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