Iron Loss Modeling of Anisotropic Soft Magnetic Steels in FEM Simulation Environment

Abstract

Due to their texture, the magnetic characteristics of grain oriented (GO) steels depend on the angle between the applied magnetic field and the rolling direction RD. These GO steels exhibit advantageous magnetic characteristics, such as high permeability in the RD and relatively poor properties in the transverse direction TD. In order to obtain a better electromagnetic modeling of the electrical energy conversion systems using GO steels, the dissipative phenomena and the anisotropy must be reliably represented. One example is the case of the GO electrical steels used in stators of turbo-generators. In order to create an appropriate model, the associated magnetic losses have to be taken into account for any angle between the applied field and the RD. However, the problem of computing iron losses whatever the angle between the applied field and the RD has remained unsolved especially within the finite element method (FEM) framework.In this paper, a phenomenological iron loss model, recently proposed in [1], taking into account the anisotropy and nonlinearity of GO sheets has been successfully implemented in the L2EP FEM software baptized code_Carmel [2]. The model relies on the ‘loss separation’ approach. In this approach, which is based on the traditional decomposition proposed by Bertotti [3], the total losses are decomposed into three terms (hysteresis, classical and excess losses). The originality of the model concerns the first loss term i.e., the quasi-static losses dissipated per magnetization cycle, depending on the polarization level Jp and on the angle θ between the applied field and RD. The proposed quasi-static loss model considers the magnetization process and the evolution of the domain wall structure (dws) with the angle θ [4]. It is built from the two sets of W-Jp quasi-static experimental data obtained in the principal directions RD and TD and is based on the interactions between the main classes of magnetic domains, 180° and 90° dws, known as “Néel phases” [5] through a single analytical relationship involving the volumic fraction occupied, for any direction of the magnetic field, by these classes. This same principle is applied for the determination of the excess losses. At this stage, the corresponding input data are the excess losses measured in both RD and TD. Concerning the classical loss term, it simply takes the expression resulting from the classical eddy current theory which is independent of the angle θ. Indeed, the case of an infinite sheet of thickness e, with a linear constitutive law (µ independent of H) and conductivity σ, under a sinusoidal field of frequency f, allows an entirely analytical resolution of the magnetic diffusion equation. Finally, we get a description of the total losses W(f,Jp) as a function of the angle θ, without any adjustable parameter. The model has been validated on industrial sheets of EDF turbo-generators by means of simple study cases under periodic (cyclic) waveforms. Thus, we dispose of an efficient and reliable tool for modeling the magnetic losses of GO sheets within the FEM simulation environment.In order to validate the implementation of the developed model, a finite element analysis (FEA) has been performed using code_Carmel. A 3D linear magnetostatic simulation has been done using the magnetic vector potential formulation. A diagonal permeability tensor has been used to represent the anisotropic properties of the GO material. This tensor is formed assuming non-diagonal terms. The values of the tensor for the axes x and y were obtained from the characterization of the industrial GO steel C150-35S in the RD and TD, while for the axis z a very low relative permeability of ≈ 30 is assumed, as measured in [6].Furthermore, the analyzed geometry corresponds to a GO sheet with dimensions of 10 cm x 10 cm and a thickness of 0.35 mm, surrounded by a coil along the x-axis. The iron loss calculation is performed in a section at the center of the sheet with an area of 1 cm x 1 cm. The elements at the center are subjected to a sinusoidal waveform of Jp of 0.5 T. Thirteen values of θ have been evaluated in this study (0°, 5°, 10°, 15°, 20°, 25°, 30°, 42°, 54.7°, 65°, 75°, 82.5° and 90°). They correspond to the angles characterized experimentally using the Epstein frame method with Epstein strips cut at different angles from the industrial GO steel sheet. Finally, from the distribution of H and J obtained in the FEA, the iron losses have been calculated in post-treatment. The main entries for the iron loss model were the hysteresis and excess losses in the RD and TD. In Table 1, an example comparing the static and dynamic losses obtained from the FEA against the experimental losses is given for some angles at 0.5 T. Further validations, analysis and discussions are envisaged in the extended version of the paper. **