Estimation of Magnetostrictive Hysteresis Properties of Electrical Steel Sheet Under External Stress Using Multi-Scale Domain Energy Model

Abstract

I. IntroductionThe iron losses, permeability and magnetostriction of electrical steel sheet are sensitive to mechanical stress, and the presence of mechanical stress may bring out the deformation of the magnetostrictive hysteresis loops, the increase of core loss, and the aggravation of vibration and noise in electrical machines or transformers. The accurate prediction of magnetic characteristics under stress is significant for optimum design of high power density motor. The multi-scale domain energy models [1-2], such as the assembled domain structure model (ADSM) have been conducted to represent the magnetic hysteretic behavior in an electrical steel sheet based on the minimization principle of total magnetic energy. In comparison with some phenomenon models such as Preisach, the domain energy model interprets the magnetic domain wall movements and rotation with the magnetic field applied and builds a relevance between the microstructure and magnetic hysteresis properties. However, due to the complexity of the parameter identification, the establishment of domain model is at the cost of large computation time and its modeling accuracy to the magnetostrictive loops is also to be improved.In this paper, a multi-scale domain energy model is improved to estimate the magnetostrictive characteristics under different external stresses, in which the hysteresis energy function is introduced instead of pinning energy in order to reduce the computation time. To identify the parameters in the model, the magnetic domain wall movements in an electrical steel sheet during the magnetization process are observed by a magnetic domain observation microscope, and the magnetostrictive hysteresis loops under several external stresses are measured. The validity of proposed model is verified by comparing the measured magnetostrictive loops with computed ones.II. Experiments and ApproachesA. ExperimentsWith the presence of external magnetic field intensity H, the balance state with the total inner energy in the ferromagnetic materials keeping the minimum is disturbed, and then the wall movements and magnetic moment rotation in the magnetic domains is occurring. This magnetization process in an electrical steel sheet is observed with a polished sample of with the size of 8mm width and 8mm length, as shown in Fig.1(a), by means of a magnetic domain observation microscope, BH-782PI-SHG, Japan. Fig.1(b) represents a domain figure under a 0.5Oe field in a non-oriented silicon steel, in which the difference between grey levels indicates different magnetization direction domains. Further, the magnetic hysteresis and magnetostrictive loops with different mechanical stresses applied are measured, as shown in Fig.1(b), by a laser magnetostrictive tester from Brockhaus, Germany, in which the shape and magnitude and area of magnetostrictive loops occur to increase after applying a 4Mpa compressive stress.B. Improved Multi-Scale Domain Energy ModelThe magnetic domain moment is alignment with the easy directions of the crystal in a grain when the material is in the demagnetized state. According to the ADSM model each grain is assumed to be composed of six magnetic domains along the easy directions, and the unit magnetization vector m of each domain is denoted by the variables θ and Φ as a vector [sinθsinΦ, sinθcosΦ, cosθ]. The θi ,Φi and the existence probability ri ( i=1,2,…,6) of magnetization vector for each domain can be determined according to the principle of local energy minimum. The total magnetic domain energy is expressed as the summation of Zeeman energy, the crystalline anisotropic energy, the magnetostatic energy and magnetoelastic energy in the ADSM. Further in order to model the magnetic hysteresis effect, the pinning energy have been added into the total energy.In this paper, in order to estimate the behavior of magnetostrictive hysteresis behavior, a hysteresis energy density function in (1) instead of the pinning energy is employed into the total domain energy in terms of simplifying the parameter identification process.Eh=-μ0Ms/χ0(α1,imx-p+α2,i+my-p+α3,imz-p) (1)where [α1,i, α2,i, α3,i] is direction cosine of the unit magnetization vector of the i-th domain with respect to the three easy axes; (mx-p, my-p, mz-p) is three components of magnetization vector at previous moment, MS is the saturation magnetization. The hysteresis energy density function can take the effect of the magnetization history, and explain the hysteresis phenomenon as an inertia.Further, the effect of external stress on the magnetostriction in the magnetoelastic energy is described as:Eσ=-3/2λ100σ(α21,iγ21+α22,iγ23+α23,iγ23)-3λ111σ(α1,iα2,iγ1γ2+α2,iα3,iγ2γ3+α3,iα1,iγ3γ1) (2)where (γ1, γ2, γ3) is the cosine of the angle between the stress and the three easy axes; λ100 and λ111 are saturation magnetostrictive coefficients along the [100] direction and the [111] direction; σ is the stress applied on the sample;Fig. 2 shows the measured and modeled results of magnetostriction loops, and more discussion will be presented in detail in the extended paper. **