IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 999 Thermal-Electromagnetic Coupling in the Finite-Element Simulation of Power Transformers Kurt Preis1 , Oszkár Bíró1 , Gerhard Buchgraber2 , and Igor Tičar3 Institute for Fundamentals and Theory in Electrical Engineering (IGTE), Graz University of Technology, A-8010 Graz, Austria VA TECH ELIN Transformatoren GmbH & Co., A-8160 Weiz, Austria Faculty of Electrical Engineering and Computer Science, University of Maribor, 2000 Maribor, Slovenia A finite element method is presented to compute the electromagnetic field and hence the temperature rise in complex three-dimensional structures such as power transformers. The temperature dependence of both the heat transfer coefficients and the electric conductivity is taken into account establishing a strong coupling between the thermal and electromagnetic equations. Index Terms—Electromagnetic coupling, finite element methods (FEM), power transformer thermal factors. I. INTRODUCTION HE eddy current losses in structural parts of power transformers cause overheating that may be harmful for the operation. It is essential to be able to predict those temperature rises accurately in order to find design versions avoiding undesirable overheating effects. The eddy current losses can be determined from an electromagnetic simulation. Finite element analyzes yielding the eddy current losses in three-dimensional power transformer models have been reported in [1], [2], and [3], for example. An integral approach has been used to solve the same problem in [4]. Once the losses are known, it is possible to tackle the thermal equations with the Joule losses acting as source terms. This has been done using finite elements, among others, in [3], [5] and, employing integral techniques, in [4]. The thermal problem is nonlinear, due to the dependence of the heat transfer coefficient on temperature [5], [6]. Furthermore, the electric conductivity also varies with temperature, i.e., there exists a strong coupling between the electromagnetic and the thermal phenomena. This latter is usually taken into account in induction heating problems, see, e.g., [7]. The aim of the present paper is to extend the method of [2] and [3] to take account of the temperature dependence of the heat transfer coefficient and of the conductivity. Since the electromagnetic analysis needs substantially higher computational efforts than the thermal one, the recomputation of the electromagnetic field is restricted to the eddy current domain. T II. ELECTROMAGNETIC ANALYSIS density in the windings and leads. In the models investigated, the is made up of the tank and the bushing eddy current domain where adapters. The rest of the problem region constitutes , since there are no eddy currents present there. The equations to be solved are (3) (4) is the conductivity depending on the temperature where and is the permeability depending on . The Galerkin equations corresponding to (3) and (4) are solved (5) (6) where and are the finite element approximations, is the number of edges in is the number of nodes in are the edge basis functions and are the nodal basis functions. The edge finite element method is applied to the potential formulation [8] with III. THERMAL ANALYSIS (1) (2) The temperature distribution in the domain with a heat conductivity is governed by the differential equation where is the current density, is the magnetic field intensity, is the current vector potential and is the magnetic scalar pois a current vector potential whose curl is the current tential. (7) is the power loss density. The apwhere propriate boundary condition on the bounding surface of the conducting domain is of Robin type (8) Digital Object Identifier 10.1109/TMAG.2006.871439 0018-9464/$20.00 © 2006 IEEE 1000 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 where is the heat flux, is the ambient temperature and is the heat transfer coefficient depending both on the heat flux and on temperature [5], [6], [10]. The Galerkin equations corresponding to the differential equation (7) and the boundary condition (8) are (9) is the finite element approximation, is the number where and are the nodal basis funcof nodes in tions. Due to the temperature dependence of , the equations in (9) are nonlinear. They are solved by applying a direct iteration method of updating the heat transfer coefficient at each iteration step. Fig. 1. Dependence of the heat transfer coefficient on the heat flux along steel-oil interface [10]. IV. ITERATIVE COUPLING The coupled system (5), (6), and (9) can be solved by successive iterations. Initially, (5) and (6) are solved with a conduccorresponding to a constant temperature. This is foltivity lowed by the thermal analysis described in Section III yielding a temperature distribution. Thereupon, the conductivity is updated in each integration point. In order to avoid having to carry out the computationally extremely expensive electromagnetic analysis anew, the recalculation of the eddy current field with the updated conductivity is restricted to . This can be achieved by obtained by the initial analfreezing the magnetic field in ysis using the conductivity . Two options are then open for setting the boundary conditions for on the interface : One is to prescribe the tangential component of by assigning the and freezing the value zero to the tangential component of on obtained from the solution of (5) and (6). The value of second possibility is to specify the normal component of using the normal derivative of on yielded by (5) and (6) as a Neumann boundary condition for the scalar potential. Adis set to zero again ditionally, the tangential component of to ensure that the normal component of the current density vanishes. The proof that the boundary value problem arising in by defining both and is unique is given in [9]. As the numerical examples of Section V show, the convergence of the iterative procedure is rapid. Furthermore, the eddy current distributions in obtained using any of the above two of boundary conditions on are practically the same. V. NUMERICAL EXAMPLES The two examples presented are taken from [2]. They involve bushing adapters carrying eddy currents due to the high current, low voltage leads. The dependence of the heat transfer coefficient along the steel-oil interface on the heat flux is shown in Fig. 1, and its temperature dependence in Fig. 2. The heat transfer coefficient on the steel-air interface has been taken to be 10 W/(m K). The temperature of oil has been taken to be 70 C and that of air is assumed to be 32 C. Fig. 2. Dependence of the heat transfer coefficient on temperature along steel oil interface (source: measurements by VA TECH ELIN). Fig. 3. Model of the bushing adapter and detail of the tank of a three-phase generator step up transformer with current filaments modeling leads. A. Generator Step Up Transformer The bushing adapter of the three-phase generator step up transformer introduced in [2] is made of nonmagnetic steel. The model with the leads taken into account by current filaments is shown in Fig. 3. The temperature dependence of the conductivity of nonmagnetic steel has been measured as (10) The current density distribution obtained by solving (5) and (6) using a constant conductivity corresponding to a temperature of 82 C and the resulting temperature distribution are shown in Figs. 4 and 5. The number of degrees of freedom in this analysis has been 285 825. Due to the relatively slight variation of the conductivity with temperature, practically the same distributions are obtained after the conductivity has been PREIS et al.: THERMAL-ELECTROMAGNETIC COUPLING IN FINITE-ELEMENT SIMULATION OF POWER TRANSFORMERS Fig. 4. Current density distribution in the bushing adapter and part of the tank of generator step up transformer. Fig. 5. Temperature distribution in the bushing adapter and part of the tank of generator step up transformer. 1001 Fig. 7. Model of the bushing adapter and detail of the tank of a single-phase autotransformer with current filaments modeling leads. Fig. 8. Current density distribution in the bushing adapter and part of the tank of autotransformer using   . = Fig. 9. Temperature distribution in the bushing adapter and part of the tank of autotransformer using the current density distribution of Fig. 8. B. Transformer With Tertiary System Fig. 6. Comparison of the measured and computed temperatures along a horizontal line just below the bushing adapters. updated using (10) and the problem region restricted to the conducting steel parts forming . This result is also independent of the boundary condition on the surface of . The eddy current analysis with the tangential component of prescribed has involved 30 845 and, with the normal component of given, 57 032 degrees of freedom. The temperature has been measured along a horizontal line just below the adapters. The comparison of the measured and computed temperatures is shown in Fig. 6. The bushing adapter of the single-phase autotransformer with tertiary system described in [2] is made of magnetic steel. The model analyzed is shown in Fig. 7. The measured temperature dependence of the conductivity of steel is (11) The current density distribution obtained by solving (5), (6) with the constant conductivity corresponding to the temperature of 73 C is plotted in Fig. 8 and the resulting temperature distribution in Fig. 9. The eddy current model requires 336 999 degrees of freedom. Updating the conductivity using (11) and re- 1002 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 Fig. 10. Current density distribution in the bushing adapter and part of the tank of autotransformer with the conductivity corresponding to the temperature is prescribed. Prescribing the shown in Fig. 9. The tangential component of normal component of , the change of the plot is not visible. B H Fig. 12. Temperature distribution in the bushing adapter and part of the tank of autotransformer using the current density distribution of Fig. 10 if the normal component of is prescribed. B ACKNOWLEDGMENT This work was supported in part by the Austrian FFG under Grant 809 934. REFERENCES Fig. 11. Temperature distribution in the bushing adapter and part of the tank of autotransformer using the current density distribution of Fig. 10 if the tangential component of is prescribed H stricting the problem domain to the eddy current region results in the slightly changed current density distribution shown in Fig. 10 if the tangential component of is prescribed. This eddy current analysis involves 123 373 unknowns. No appreciable change in the eddy current distribution occurs if the normal component of is given, leading to 142 752 degrees of freedom. The difference between the two current distributions can be observed by comparing the two resulting temperature distributions shown in Figs. 11 and 12. Further iterations do not bring any appreciable change in current density or temperature. VI. CONCLUSION In the method presented for taking account of the strong coupling between electromagnetic and thermal phenomena in power transformers it is sufficient to carry out the computationally demanding electromagnetic analysis in the entire problem domain once only. The iterations involve much simpler analyzes restricted to the eddy current region. [1] K. Tekletsadik and M. Saravolac, “Calculation of losses in structural parts of transformers by FE method,” in Proc. Inst. Elect. Eng. Colloq. Field Modeling: Applications to High Voltage Power Apparatus London, U.K., Jan., 17 1996, pp. 4/1–4/3. [2] O. Bíró, K. Preis, and G. Buchgraber, “Finite element model to compute transformer losses,” in Proce. 11th Int. IGTE Symp. Numerical Field Calculation in Electrical Engineering, Seggauberg (Graz), Austria, Sep. 13–15, 2004, pp. 330–333. [3] G. Buchgraber, O. Bíró, P. Kalcher, and K. 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