160 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 ; 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 –. 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 –. There are several alternatives that eliminate the numerical oscillations, but none of the suggested solutions fully satisfies all physical interpretation concerns. In , an autotransformer is introduced to eliminate the negative inductance. Reference  proposes a modification of the circuit structure shifting the magnetizing branch. In , 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: [email protected]). J. A. Martinez is with the Departament d’Enginyeria Elèctrica, Universitat Politècnica de Catalunya, Barcelona 08028, Spain (e-mail: [email protected]). 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 . 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 –; , , 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 . 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 –. Additionally, note that a negative resistance may appear in the star equivalent network. In , 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 , . 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 : (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. 162 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, , , and . 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 . 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  for the turn-to-turn modeling of transformer windings and recently used for the representation of entire windings in . 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 164 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  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 . 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  with our model. The rated transformer data are given in Table III (note that winding numbers 2 and 3 are switched with respect to ). 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  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, 166 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) 168 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009 REFERENCES  A. Boyajian, “Theory of three-circuit transformers,” AIEE Trans., pp. 208–528, Feb. 1924.  F. Starr, “Equivalent circuits -I,” AIEE Trans., vol. 57, pp. 287–298, Jun. 1932.  L. F. Blume, Transformer Engineering. New York: Wiley, 1951.  Electrical Transmission and Distribution Reference Book. U.S.: Elect. Syst. Technol. Inst., (Westinghouse T&D Book), ABB, 1997.  H. W. Dommel, Electromagnetic Transients Program Reference Manual (EMTP Theory Book). Portland, OR: BPA, 1986.  W. S. Meyer and T.-H. Liu, “Unstable saturable transformer,” Can/Am EMTP News, vol. 93, no. 2, pp. 15–16, Apr. 1993.  P. S. Holenarsipur, N. Mohan, V. D. Albertson, and J. 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Power Del., vol. 7, no. 1, pp. 376–383, Jan. 1992.  R. M. Del Vecchio, “Applications of a multiterminal transformer model using two winding leakage inductances,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1300–1308, Jul. 2006.  J. A. Martinez, R. Walling, B. A. Mork, J. Martin-Arnedo, and D. Durbak, “Parameter determination for modeling system transients-part III: Transformers,” IEEE Trans. Power Del., vol. 20, no. 3, pp. 2051–2062, Jul. 2005. 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.