Homogenization Approaches for Laminated Magnetic Cores using the Example of Transient 3D Transformer Modeling

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1 Homogeniation Approaches for Laminated Magnetic Cores using the Eample of Transient 3D Transformer Modeling H.Neuert *1, J. Ziske 1, T. Heimpold 1,and R. Disselnkötter 2 1 Technische Universität Dresden, German, Institute of Electromechanical and Electronic Design, 2 ABB AG, Corporate Research Center German, Ladenurg, German *Corresponding author: D Dresden, German, holger.neuert@tu-dresden.de Astract: A specific issue in transformer modeling using the finite element method is the consideration of electric sheets or other laminated core materials which are used to reduce edd currents. It would e impractical to eplicitl model a large numer of sheets as this would lead to a large numer of elements and hence to unacceptale computational costs. Homogeniation procedures overcome this prolem. The sustitute the laminated structure a solid having nearl the same electromagnetical ehavior. In our stud, we have implemented several of them in a transformer model using COMSOL Multiphsics. Simulation results otained with the different homogeniation approaches are compared to eperimental test results. This reveals the accurac and the application limits of the investigated homogeniation approaches. Kewords: transformer model, laminated core, electric sheets, nonlinear ferromagnetic material, transient analsis. 1. Introduction Proper modeling and simulation of transformers with finite element methods is a challenge due to the involved nonlinearities and coupling effects etween different phsical domains. Primaril, there are the highl nonlinear magnetic characteristics of the core materials including ferromagnetic hsteresis, edd currents in the windings and in the core, and temperaturedependent material properties like the electrical conductivit. With nonlinear magnetic material or a non-harmonic ecitation, simulations need to e performed in transient rather than in timeharmonic mode. The transformer models presented in former pulications solve most of the mentioned prolems in a satisfactor manner. This was demonstrated for the eample of a current transformer [1, 2] which has a primar winding of onl one turn and is short-circuited on the secondar side. A specific issue in transformer modeling is the consideration of magnetic cores ased on stacked electric sheets or strip wound or other laminated core materials which are used to reduce edd currents and thus to minimie the dnamic hsteresis losses and to improve the dnamic ehavior of magnetic actuators. Normall, the thickness of the individual sheets is small as compared to the core thickness. Therefore, it would e impractical to eplicitl model a large numer of sheets with a high dimensional aspect ratio, as this would lead to a large numer of elements and hence to unacceptal high computational costs. This is even more the case as the transition oundar condition is not applicale etween domains in time-dependent studies. It would e therefore also necessar to model the insulation laers etween the sheets. To overcome this prolem, several homogeniation approaches for laminated cores have een proposed in the past [3-7]. The replace the laminated structure a single domain of an electricall orthotropic material which ehiits the same macroscopic ehavior in a certain range of conditions. Thus, the computational effort can e significantl reduced. Figure 1. Transformers with laminated soft magnetic ring core for static and dnamic hsteresis measurements.

2 In our stud, we have implemented these approaches in a 3D-transformer model. The oained solutions are compared to eperimental test results. This comparison is made oth for ring core arrangements similar to those which are used for magnetic material characteriation according to DIN EN and for configurations when the core is operated as a current transformer. 2. Transient Electromagnetic Transformer Model from the solution in the transformer model. Therefore, the secondar winding is not modeled in this case. 2.2 Multi-Turn Coil Modeling and Coupling to Eternal Circuitr We did not use the multi-turn coil domains provided the AC/DC Module due to their strict requirements concerning the geometr of the electricall conductive cross section area. 2.1 Principle of Modeling Figure 1 depicts the transformer samples which which were oth eperimentall investigated (s. Sec. 4) and simulated. The core consists of stacked electric sheets which wear a closel wound secondar coil with N 2 turns at one leg and a primar coil with N 1 turns equall distriuted over all legs of the closed ring core. In order to evaluate modeling approaches for laminated magnetic cores, measured and simulated hsteresis loops B(H) of the core material have to e compared. The can e measured and simulated as follows. A primar current i 1 fed the current source generates a magnetic field H(t) in the core material according to Ampère s circuital law. For a closed core with a constant cross section area A c and a mean length of the ferromagnetic path L c a simplified relation can e derived: i1( t) N1 H t = (1) ( ) L c According to the law of induction a voltage occurs in the secondar winding. The change of the mean flu densit B in the core in the time interval t 2 - t 1 can e calculated integrating the secondar voltage u 2 : B 1 t2 = u2 dt N t1 2A (2) c Equations (1) and (2) are used to measure the static and dnamic magnetic hsteresis [8, 9]. Figure 2 shows the parametric transformer model. It allows to compute the interactions etween the electrical and the magnetic susstems. An eternal circuitr (a) including a current source and an eternal resistor feeds the finite element transformer model (). In contrast to the measurement, B can directl e derived Figure 2. Model of the transformer with laminated core; (a) eternal circuitr, () finite element model; 1 magnetic core, 2 primar windings, 3 measuring volume (surrounding air domain is not shown). Instead, we modeled the coils as homogenied, electricall poorl conductive domains. A current source as part of the eternal circuit generates i 1 and feeds the coil domains with an eternal current densit. The primar voltage V prim is computed integrating the induced electrical field over the coil domains and considering the ohmic voltage drop across the winding resistance [2]. Edd currents in the windings can e neglected in the investigated frequenc range. Finite element transformer model and primar eternal circuit are i-directionall coupled with an IvsU element which is an inherent component of the cir application mode. If it is necessar to uild a complete transformer model, a secondar eternal circuit can e coupled to the finite element transformer model in the same wa. The transformer model can also e fed a voltage source in the primar circuit instead of a current source. In this case, the correct current value must e iterativel found in each time step for which the sum of the ohmic and induced voltage of the primar winding equals the source voltage. This increases computing time and ma cause convergence prolems if nonlinear magnetic material is applied.

3 3. Laminated Magnetic Core with Nonlinear Magnetiation Curve 3.1 Approimation of the Nonlinear Magnetiation Curve The magnetic core material in the model is descried magnetic and electrical properties. The nonlinear magnetic ehavior of the ferromagnetic transformer core needs to e considered. In order to avoid circular variale definitions in the constitutive relations, the H = f( B )e B form is used. The used approach does not account for the static magnetic hsteresis. However, the resulting error is acceptale in man practical cases, especiall if high qualit magnetic material is used. At least two tpes of implementations are possile. Firstl, the commutation curve can e implemented in the form of an interpolation tale. Piecewise cuic interpolation, additional data points at higher field strengths, and linear etrapolation are the optimum settings for fast computation and improved convergence [2]. A monotonousl increasing sequence of the data points must e ensured a good qualit of the measurement data. Secondl, a gloal approimation function valid for B = [0; ] and derived from measurements fitting, can also e used. material, e. g. grain-oriented sheets cannot et e adequatel considered. 3.2 Homogeniation Approaches for Laminated Magnetic Cores Edd currents in laminated structures would e correctl represented in a transient simulation model when the electrical and magnetic material properties and the geometr of the magnetic core are eplicitl modeled (Figure 4.a). However, in most cases, this is practicall impossile. As mentioned aove, homogeniation approaches replace the laminated structure a single solid domain of an electricall anisotropic material which ehiits nearl the same macroscopic ehavior in a certain range of conditions (Figure 4.). This is achieved adapting the conductivit such that the Ohmic resistance in the homogenied core is equal to that of the current path in the laminated structure.. Figure 4. Edd currents in a transformer core with laminated electric sheets (a) and homogenied core (). Figure 3. Implemented commutation curves from measurements of the investigated core materials. B using appropriate setup functions for fitting H(B), strictl increasing values, differentiailit and high determination are achieved even from qualitativel poor data [2]. Currentl, oth approaches are implemented onl for isotropic magnetic material. Anisotropic In each of the approaches, an orthogonal electrical conductivit is proposed to adapt the ehavior as desired. With the direction normal to the sheet plane, is = (3)

4 Based on different assumptions and appling different simplifications, the homogeniation approaches listed in Tale 1 have een derived. is the isotropic conductivit of the asic material, n the numer and the width of the stacked sheets, and D the overall thickness of the laminated core (Figure 4). Tale 1. Homogeniation approaches for laminated magnetic core materials SILVA [3] HAHNE [4] KIWITT [5] KÜHNER [6] WANG [7] Solid core = = = 0 = = = n D 2δ L ( + d 2δ ) δ L = 1/ πµ f 1 = = n 2 = = = 1 n = = = 2 d = = = = 2 L 2 4. Simulation Results and Measurements We investigated laminated ring cores of different sies oth with simulations and tests (Tale 2). Depending on the core sie (Figure 3), significant differences in the static magnetic ehavior of the core material was found in the tests. The originate from edge-effects caused the laser cutting process performed on the electric sheets. Therefore, the influences of the geometr and the material cannot e strictl separated in the tests. Tale 2. Investigated soft magnetic ring cores Core width Mean length of the flu path Core material Sheet thickness d Numer of sheets n 5 Numer of primar turns 60 Numer of secondar turns 22 5, 10, 15 mm 150 mm M400-50A 0,5 mm In a strict sense, the HAHNE model [4] cannot e used with non-harmonic ecitations or nonlinear magnetic materials due to its eplicit dependence on the signal frequenc and the core permeailit. We simplified this model using the maimum permeailit and investigated onl harmonic ecitations. All these homogeniation procedures do not adjust the core permeailit, apart from a filling factor which considers the thickness of the insulation laers etween the sheets if necessar. With the simplification of [4], the conductivit is depending onl on geometric parameters in all of the approaches.. Figure 5. Simulated dnamic hsteresis curves calculated with different homogeniation models; core width = 15 mm We used harmonic current ecitation with a peak field of 1000 A/m in a frequenc range from 1 H up to 2 kh oth in the tests and in the simulations. Additionall, static hsteresis measurements were performed to derive commutation curves in H(B) form for implementation in the models (s. Sec. 3.1). Figure 5 depicts eamples of simulated dnamic hsteresis curves for a core width of = 15 mm and a frequenc of 100 H in compa-

5 rison to measured data. Figure 6, Figure 7 and Figure 8 compare the measured and simulated coercivit for all core sies and frequencies. the measured dnamic coercivit was reduced the measured static one. Oviousl, the approaches according to Tale 1 calculate quite different dnamic hsteresises framed the Silva and the solid core model. Even in the validit range of the models elow f g considerale deviations occur. 5. Conclusions and Outlook Figure 6. Simulated and measured coercivit of the dnamic hsteresis, core width = 5 mm Figure 7. Simulated and measured coercivit of the dnamic hsteresis, core width = 10 mm It has een shown that homogeniation procedures known from literature do not precisel predict the form of dnamic hsteresis in the investigated parameter range. The comparison of simulation results and measurements ased on the coercivit of the hsteresis curves for the same ecitation reveal significant deviations. These results do not encourage an uncritical use of the homogeniation procedures according to Tale 1 if precise modeling of laminated magnetic cores is required. Especiall a reliale prediction of dnamic losses is not to e epected. Focus of future work will e on models with eplicitl modeled core sheets to compare their results with those from homogenied models and measurements. Additional parameters should e included to improve comparailit of measured and simulation results. Further, homogeniation procedures which ase on additional terms in the magnetic material equation depending on the flu densit rate should e included and compared to the linear approaches we have involved up to now. 6. References Figure 8. Simulated and measured coercivit of the dnamic hsteresis, core width = 15 mm The coercivit ma e used to compare the approimation qualit of the different homogeniation procedures. For etter comparailit, 1. T. Bödrich, H. Neuert, R. Disselnkötter: Transient Finite Element Analsis of a SPICEcoupled Transformer with COMSOL Multiphsics. Proc. of the 4th European COMSOL Conference, Paris (FR), (2010) 2. H. Neuert, T. Bödrich, R. Disselnkötter: Transient Electromagnetic-Thermal FE-Model of a SPICE-Coupled Transformer Including Edd Currents with COMSOL Multiphsics 4.2. Proc. of the 5th European COMSOL Conference, Stuttgart (D), (2011) 3. V. Silva, G. Meunier, A. Foggia: A 3-D finiteelement computation of edd currents and losses in laminated iron cores allowing for electric and

6 magnetic anisotrop, IEEE Trans. on Magn (1995) 4. P. Hahne, R. Diet, B. Rieth, T. Weiland: Determination of anisotropic equivalent conductivit of laminated cores for numerical computation, IEEE Trans. on Magn (1996) 5. J.E. Kiwitt, A. Huer, K. Reiß: Modellierung gelechter Eisenkerne durch homogene anisotrope Kerne für dnamische Magnetfelderechnungen. Electrical Engineering (Archiv fur Elektrotechnik) (1999) 6. A. Kühner: Dreidimensionale FEM- Modellierung von Hochspannungsleistungstransformatoren ur Untersuchung ihres transienten Verhaltens. Univ. Fridericiana Karlsruhe, Fakultät für Elektrotechnik (1999) 7. J. Wang, H. Lin, Y. Huang, X. Sun: A New Formulation of Anisotropic Equivalent Conductivit in Laminations. IEEE Trans. on Magn (2011) 8. DIN EN Magnetic materials Part 4: Methods of measurement of d.c. magnetic properties of magneticall soft materials (2009) 9. DIN EN Methods of measurement of the magnetic properties of magneticall soft metallic and powder materials at frequencies in the range of 20 H to 200 kh the use of ring specimens (2004)