Multi-Scale FEM and Magnetic Vector Potential A for 3D Eddy Currents in Laminated Media K. Hollaus, and J. Schöberl Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-14 Vienna, Austria E-mail: karl.hollaus@tuwien.ac.at Abstract An accurate simulation of the eddy current losses by the finite element method in laminated cores of electrical devices is still a challenging task. Modeling each laminate individually is not an appropriate solution. Many finite elements have to be used in such a model leading to large systems of equations. A higher order multi-scale finite element method with the magnetic vector potential has been developed to cope with 3D problems considering edge effects directly and not in a post-processing step. Numerical simulations demonstrate a remarkable accuracy and low computational costs. Index Terms Laminated media, magnetic vector potential, multi-scale FEM, three dimensional eddy currents. I. Introduction The dimensions of the iron core and the thickness of the laminates are very different. Thus, finite element models considering each laminate require many finite elements leading to extremely large systems of equations. Thus, an efficient and accurate simulation of the eddy current losses in laminated iron cores is still a challenging task, see for instance [1] and [2]. Brute force methods apply either an anisotropic electric conductivity, [3], [4] and [5], or prescribe a current vector potential having a single component normal to the lamination [6] in finite element (FE) models. Considering a decomposition of the total magnetic flux into a main magnetic flux parallel to the lamination and a magnetic stray flux perpendicular to the lamination, the solution obtained by the above methods is frequently corrected in a second step exploiting different approaches, for example [7] and [8] for 3D problems. A multi-scale finite element method (MSFEM) seems to be very promising to obtain the solution in one step [9]. To improve the local approximation the main magnetic flux density parallel to the lamination is expanded into orthogonal even polynomials, so-called skin effect subbasis functions, in [1] and higher order corrector terms were determined solving the associated cell problems in [11]. The higher order MSFEM in [12] and [9] based on the magnetic vector potential A is extended appropriately into three dimensions considering simultaneously also the edge effects in this work. Eddy current losses obtained by the new higher order MSFEM have been compared with those obtained by reference solutions of finite element models considering each laminate individually. The high accuracy of the losses with respect to the penetration depth in the frequency domain and that in the time domain for linear material properties as well as the improvement in the computational costs are shown. II. Eddy Current Problem A. Eddy Current Problem in the Time Domain: The eddy current problem to be solved consists of a conducting material c enclosed by air, i.e., = c with an outer boundary Γ, see Fig. 1. The boundary value problem in the time domain with the magnetic vector potential A reads as curl µ 1 curl A + σ t A = in = c (1) A n = α on Γ, (2) where µ is the magnetic permeability, σ is the electric conductivity and t is the time. Dirichlet boundary conditions are prescribed on Γ. Figure 1: Eddy current problem model, 2D sketch. B. Variational Formulation: Find A h U h,α := {A h U h : A h n = α on Γ}, such that µ 1 curl A h curl v h d + σa h v h d = (3) t for all v h U h,, where U h is a finite element subspace of H(curl, ). The index h stands for the finite element discretization.
To get a unique solution the penalty term σ A h v h d (4) t is added to (3) in air, where σ = [13]. The conductivity in air is chosen as < σ << σ. III. Multi-Scale Finite Element Method MSFEM A. Higher Order MSFEM Approach with A for 3D: The feasible three-dimensional higher order multiscale approach à = A + φ 1 A 12 + φ 3 A 32 A 13 A 33 + w 1 φ 1x + w 3 φ 3x +... (5) with respect to Cartesian coordinates, where the normal vector of the lamination n l points in x-direction has been assumed. Since the magnetic flux density of the main field is an even function across the laminates only odd terms are considered in approach (5). The extension into 3D is surprisingly simple, but not straightforward comparing the 2D approach [9]. Approach (5) is based on the fact that the problem can be observed as a macro-structure with the large dimensions of the iron bulk, on the one hand, and on the other, the micro-structure with the very small thickness of the laminates d and the width of the air gaps d in between (Fig. 1). The mean value A considers the large scale variations of the macro-structure and the scalar quantities A 12, A 13, A 32, A 33, w 1 and w 3 and the periodic micro-shape functions φ 1 and φ 3, see Fig. 2, and their derivatives φ 1x and φ 3x, respectively, the rough variations of the micro-structure. An extension to an approach of order five is indicated by the additional micro-shape function φ 5 in Fig. 2. The case with φ 5 or an even higher one has not been studied because it is practically beyond the scope of engineers as shown in [9]. An approach with an arbitrarily vector n l is straightforward. Since the solutions of A, A 12, A 13, A 32, A 33, w 1 and w 3 are smooth standard finite element basis functions [14] have been used to represent them. B. Homogenization: For the sake of convenience homogenization is only demonstrated by means of the stiffness term for the first order MSFEM approach in this section. Similar considerations hold for the mass term and for higher order MSFEM approaches of both the stiffness and the mass term. Replacing A h in the bilinear form of (3) by (5) and accordingly also the test function v h leads to the Figure 2: Micro-shape functions. symmetric bilinear µ 1[ curl ( A h + φ 1 (, A 12h, A 13h ) T + w 1h (φ 1x,, ) ) T curl ( v h + φ 1 (, v 12h, v 13h ) T + q 1h (φ 1x,, ) ) ] T d + σ [ ( A h + φ 1 (, A 12h, A 13h ) T + w 1h (φ 1x,, ) ) T t (v h + φ 1 (, v 12h, v 13h ) T + q 1h (φ 1x,, ) T ) ] d =, (6) where the test functions v 12h, v 13h and q 1h vanish in. Numerical experiments have shown that neglecting the derivatives of (, A 12h, A 13h ) T yields a more accurate solution. Simple manipulations and neglecting the derivative of (, A 12h, A 13h ) T, the first integral in (6) can be written as A(A, A 12h, A 13h, w 1 ; v, v 12h, v 13h, q 1h ) = T (curl A h ) x (curl v h ) x (curl A h ) y (curl v h ) y (curl A h ) z (curl v h ) z A 12h S v 12h A 13h v 13h w 1yh w 1zh q 1yh q 1zh d. (7) The detailed stiffness matrix S can be found in Appendix A. The entries of S were averaged over a period p = d+d of the lamination as shown for instance in [15]: µ 1 = 1 p µ 1 φ 1x = 1 p = 1 p p p p µ 1 (x)dx = µ 1 Fe d + µ 1 d p µ 1 (x)φ 1x (x)dx = 2 µ 1 Fe µ 1 p µ 1 (x)φ 1x (x)φ 1x (x)dx = 4 p (µ 1 Fe d + µ 1 ) d In (8), µ Fe represents the magnetic permeability of iron an µ that of air. Averaged coefficients and quantities arising from them are indicated by the bar. (8)
The mass term in (6) yields B(A, A 12h, A 13h, w 1 ; v, v 12h, v 13h, q 1h ) = A T M v d, (9) t after similar manipulations as for (7) along with (8), where A stands for ( Ah, A 12h, A 13h, w 1h ) T and v for ( vh, v 12h, v 13h, q 1h ) T, respectively. Figure 3: Laminated iron with dimensions in mm. C. Variational Formulation: Using (7) and (9) the variational formulation for the homogenized MSFEM considering third order approximation reads as: Find (A h, A 12h, A 13h, A 32h, A 33h, w 1h, w 3h ) V h,α := {(A h, A 12h, A 13h, A 32h, A 33h, w 1h, w 3h ) : A h U h, A 12h, A 13h, A 32h and A 33h V h, w 1h and w 3h W h and A h n = α on Γ}, such that A(A, A 12h, A 13h, A 32h, A 33h, w 1h, w 3h ; v, v 12h, v 13h, v 32h, v 33h, q 1h, q 3h ) + B(A, A 12h, A 13h, A 32h, A 33h, w 1h, w 3h ; v, v 12h, v 13h, v 32h, v 33h, q 1h, q 3h ) = (1) Figure 4: FE model for the reference solution (RS). for all (v h, v 12h, v 13h, v 32h, v 33h, q 1h, q 3h ) V h,, where U h is a finite element subspace of H(curl, ), V h of L 2 ( m ) and W h of H 1 ( m ), respectively, and span{φ 1, φ 3 } is a subspace of H 1 per( m ). The subdomain m comprises the laminates and the air gaps in between (see Fig. 1). Natural boundary conditions hold on the interface Γ m. The same regularization as for the mass term in (6), in section II. B., is applied to mass term of (1) in. IV. Numerical Example The stack of 1 laminates, shown in Fig. 3, immersed in a homogeneous time harmonic magnetic field which is prescribed by inhomogeneous Dirichlet boundary conditions has been studied. The simple but demanding problem exhibits a large main magnetic field, a significant magnetic stray field and pronounced boundary layers. A thickness of both, iron layer and air gap, of d +d =.25mm, an unfavorable fill factor of c f =.9, a conductivity of σ = 2 1 6 S/m and a relative permeability of µ r = 5, were selected. To ensure a fair comparison between the reference solution and the solution obtained by the MSFEM approach finite element grids have been used which differ only in the direction perpendicular to the lamination as shown in Fig. 4 and 5. A. Results: Figure 5: FE model for the MSFEM. The relative error of the eddy current losses is very small for a wide range of penetration depths and vanishes for a specific frequency of both curves because the sign changes as can easily be seen in Fig. 6. Eddy current losses with respect to time are shown in Fig. 7 for a time periodic excitation with a frequency of f = 15Hz. The steady state is almost achieved after one period of the excitation. A comparison of the required computer resources are summarized in Table 1, Appendix B. The number of
Figure 6: Comparison of the eddy current losses in the frequency domain. Figure 7: Eddy current losses with respect to time. Reference solution RS, the penetration depth was δ = =.13mm at a frequency of f = 5Hz. 1 π f σµ unknowns required by the higher order MSFEM is only about 1% that of the reference solution. [2] S. Nogawa, M. Kuwata, T. Nakau, D. Miyagi, and N. Takahashi, Study of modeling method of lamination of reactor core, IEEE Trans. Magn., vol. 42, no. 4, pp. 1455 1458, Apr. 26. [3] V. C. Silva, G. Meunier, and A. Foggia, A 3d finite-element computation of eddy currents and losses in laminated iron cores allowing for electric and magnetic anisotropy, IEEE Trans. Magn., vol. 31, no. 3, pp. 2139 2141, May 1995. [4] K. Hollaus and O. Bíró, Estimation of 3-d eddy currents in conducting laminations by an anisotropic conductivity and a 1-d analytical solution, COMPEL, vol. 18, pp. 494 53, 1999. [5] H. Kaimori, A. Kameari, and K. Fujiwara, FEM computation of magnetic field and iron loss in laminated iron core using homogenization method, IEEE Trans. Magn., vol. 43, no. 4, pp. 145 148, Apr. 27. [6] A. Jack and B. Mecrow, Calculation of three-dimensional electromagnetic fields involving laminar eddy currents, IEE Proc., Pt. A, vol. 134, no. 8, pp. 663 671, september 1987. [7] K. Hollaus and O. Bíró, A FEM formulation to treat 3d eddy currents in laminations, IEEE Trans. Magn., vol. 36, no. 4, pp. 1289 1292, 2. [8] O. Bíró, K. Preis, and I. Ticar, A FEM method for eddy current analysis in laminated media, COMPEL, vol. 24, no. 1, pp. 241 248, 25. [9] K. Hollaus and J. Schöberl, A higher order multi-scale FEM with a for 2d eddy current problems in laminated iron, IEEE Trans. Magn., 214, accepted for publication. [1] P. Dular, J. Gyselinck, and L. Krähenbühl, A time-domain finite element homogenization technique for lamination stacks using skin effect sub-basis functions, COMPEL, vol. 25, no. 1, pp. 6 16, 26. [11] O. Bottauscio, M. Chiampi, and A. Manzin, Computation of higher order spatial derivatives in the multiscale expansion of electromagnetic-field problems, IEEE Trans. Magn., vol. 44, no. 6, pp. 1194 1197, 28. [12] K. Hollaus, A. Hannukainen, and J. Schöberl, Two-scale homogenization of the nonlinear eddy current problem with FEM, IEEE Trans. Magn., vol. 5, no. 2, pp. 413 416, Feb 214. [13] P. Ledger and S. Zaglmayr, hp-finite element simulation of three-dimensional eddy current problems on multiply connected domains, Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 49-52, pp. 3386 341, 21. [14] J. Schöberl and S. Zaglmayr, High order Nédélec elements with local complete sequence properties, COMPEL, vol. 24, no. 2, pp. 374 384, 25. [15] K. Hollaus and J. Schöberl, Homogenization of the eddy current problem in 2d, ser. 14th Int. IGTE Symp., Graz, Austria, Sep. 21, pp. 154 159. V. Conclusions A 3D approach for a higher order MSFEM with the magnetic vector potential for the eddy current problem in laminated material considering directly the edge effects has been presented. The accuracy of the approximate method is very satisfactory and the computational costs are small. In the next step a problem with nonlinear material properties in terms of a magnetization curve will be solved. Acknowledgment This work was supported by the Austrian Science Fund (FWF) under Project P 2728-N15. References [1] H. De Gersem, S. Vanaverbeke, and G. Samaey, Threedimensional - two-dimensional coupled model for eddy currents in laminated iron cores, IEEE Trans. Magn., vol. 48, no. 2, pp. 815 818, 212.
A. Stiffness Matrix: Appendix The stiffness matrix S is symmetric indicated by the star: µ 1 µ 1 µ 1 φ 1x µ 1 φ 1x µ 1 µ 1 φ 1 µ 1 φ 1 S = B. Numerical Data: Table 1: Number of degrees of freedom. Total No. H(curl, ) L 2 ( m ) H 1 ( m ) RS 4 442,592 a ) 4 442,592 - - MSFEM with span{φ 1 } 388,889 a ) 323,196 26,991 b ) 11,711 b ) MSFEM with span{φ 1, φ 3 } 454,582 a ) 323,196 26,991 b ) 11,711 b ) a ) For 2 nd order finite elements. b ) Holds for one quantity in approach (4).