Simulation of the stress dependence of hysteresis loss using an energy-based domain model

The assembled domain structure model (ADSM) is a multiscale magnetization model that can be used to simulate the magnetic properties of a core material. This paper reveals the mechanism of the hysteresis loss increase due to compressive stress applied to a silicon steel sheet by conducting a simulation using the ADSM. A simple method of adjusting the simulated hysteresis loss to the measured loss is also proposed. By adjusting the hysteresis loss under a stress-free condition, the stress dependence of the hysteresis loss of a non-oriented silicon steel sheet is quantitatively reconstructed using the ADSM, where the stress-induced anisotropy strengthens the pinning effect along the stress direction. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.4993661


I. INTRODUCTION
The magneto-mechanical interaction in iron-core materials has been studied intensively to take into account magnetic deterioration due to compressive stressing of the motor core. [5][6][7] In representing the magneto-mechanical interaction, phenomenological modeling of the magnetization process is not useful because it requires parameter fitting of the measured property, yet a magnetic measurement under an arbitrary vector/tensor combination of magnetization and stress directions is practically difficult. Accordingly, a physical magnetization model is required to predict the stress dependence of hysteresis loss without a magnetic measurement under mechanical stress and the following model parameter fitting.
Several physical multiscale models 2,3,8 have successfully been used to predict the permeability decrease dependent on mechanical stress. However, the prediction of the stress dependence of the hysteresis-loss property 1,2,11 remains a challenging task because the physical modeling of the pinning field is an open problem.
The assembled domain structure model (ADSM) 3,4 is a physical magnetization model that includes the pinning effect on the crystal-grain scale. The model parameters are given by material constants, such as the anisotropy constant and magnetostriction constants. The magneto-elastic energy causes magneto-mechanical interaction in the core material, yielding stress-dependent magnetic properties. 3 The pinning field is simulated under an assumption of a statistical distribution of pinning sites. 3 The main purpose of this paper is to reveal the mechanism of the loss increase due to the compressive stressing of a silicon steel sheet on the basis of simulation results obtained using the ADSM. A simple method of adjusting the simulated hysteresis loss using the ADSM to the measured loss is also proposed.

II. ADSM WITH A PINNING FIELD
The ADSM 3 (II.A) and pinning model 4 (II.B and C) are briefly explained.

A. ADSM
The ADSM is a multiscale model for which macroscopic magnetization is constructed by assembling mesoscopic cells called simplified domain structure models (SDSMs) (Fig. 1). An SDSM has six domains corresponding to the three easy axes of cubic anisotropy. The magnetization state in each cell is represented by the volume ratios r i and the magnetization vectors m i = (sinθ i cosφ i , sinθ i sinφ i , cosθ i ) (i = 1 . . . 6) of the six domains. The variable vector in a cell j is denoted x j = (θ 1 , . . . , θ 6 , φ 1 , . . . , φ 6 , r 1 , . . . , r 5 ) (r 6 = 1 r 1 . . . r 5 ). The variable vectors x j ( j = 1, 2, . . . ) are determined so as to give a local minimum of the total magnetic energy e, which consists of the Zeeman energy, crystalline anisotropy energy, magnetostatic energy, and magnetoelastic energy. For convenience of formulation, the energy components are normalized by the crystalline anisotropy constant while the magnetic field is normalized by the anisotropy field. For example, the normalized magnetoelastic energy is given as where σ is the stress, λ 100 and λ 111 are the magnetostriction constants, K is the crystalline anisotropy constant, (α 1,i , α 2,i , α 3,i ) and (γ 1 , γ 2 , γ 3 ) are the direction cosines of the magnetization vectors of domain i and the stress σ with respect to the three easy axes of cubic anisotropy.

B. Distribution of the pinning field
Suppose that the macroscopic relationship between the normalized average magnetization m and the applied field h is represented as where h ah (m) represents the anhysteretic magnetization curve and h p (m) is the pinning field. The anhysteretic field is determined by Zeeman energy, crystalline anisotropic energy, magnetostatic energy, and magnetoelastic energy. The pinning field is additionally required to move a domain wall against friction generated by the pinning sites. The distribution of pinning sites is defined by a density function f (p) satisfying where p is the pinning strength. The magnetization proceeds with domain wall motions passing through pinning sites. In the case that the magnetization proceeds from the demagnetized state, the domain wall moves when h p = h h ah exceeds p and accordingly the magnetization is described as The density function f (p) is determined from the density of impurities or defects in the grains. This paper simply uses a Gaussian distribution as the density function. For convenience of simulation, relation (3) between h p and m is reformulated using the scalar stop model 9 having input m and output is a hysteretic function described by the stop model. The stop model is identified from the MH loops generated by the pinning model above.

C. Application to the ADSM
In the ADSM, the pinning field is added cell by cell. The pinning field h p is additionally required to change the volume ratio r i in a cell corresponding to the domain wall motion. The energy minimization procedure requires terms of ∂e/∂r i to decrease the total energy by changing r i . To consider the pinning effect, the pinning energy e p in every cell is added to e, where ∂e p /∂r i is the pinning field h p in the cell. Accordingly, the effective field ∂e p /∂r i refers to the force of friction acting on the domain wall.
For simplicity, suppose that the magnetization proceeds with 180 • domain wall motion with two domains i and i in a cell. The normalized magnetization along direction i in the cell is then given by m cell = 2r i -1. Accordingly, the pinning field generated by the domain wall motion i is given as h p (m cell ) = h p (2r i 1). Thus, ∂e p /∂r i is expressed as Terms ∂e p /∂φ i and ∂e p /∂θ i are set to zero.

D. Adjustment of hysteresis loss
This paper proposes a method for adjusting the simulated hysteresis loss to the measured loss using a weighting function of the stop model. 10 The stop model with weighting function is given as where w(m) is the weighting function. Ref. 10 introduced an iterative method to determine the weighting function as where k is the iteration number, L simulated is the hysteresis loss for amplitude m computed using the ADSM with w k (m)S(m) and L measured is the measured loss. For simplicity, this paper sets w(m) = w 1 (m), which means that the adjustment of loss is not very accurate. Using w(m), the pinning field is described as This adjustment is a kind of parameter fitting using loss measurements. This paper aims to predict the hysteresis loss under mechanical stress from the hysteresis loss without stress. Hence, the weighting function is determined from the measured hysteresis loss without stress.

A. Stress-dependent magnetic property
The magnetization process of a NO silicon steel sheet is simulated using 8 × 8 × 1 cells where the unit cell dimension ratio is 1:1:  Figure 2 shows the simulated hysteresis loss at 50 Hz with and without mechanical stress along the rolling direction (RD). The weighting function improves the representation of hysteresis loss over 1 T under a stress-free condition. A quantitative agreement of the loss increase due to compressive stress is obtained. Figure 3 shows BH loops with and without stress, where the simulated loops roughly agree with measured loops and the decrease in permeability due to compressive stress is predicted using the ADSM. Figure 4 shows the dependence of hysteresis loss on the applied stress along the RD. The weighting function improves the loss property representation at 1.5 T. If the simulated hysteresis loss without stress agrees with the measured loss, the loss increase due to compressive stress is accurately represented. The tensile stress does not affect the loss appreciably.

B. Dependence of hysteresis loss on mechanical loss
Equation (9) has unidirectional anisotropy, where the stress direction of compressive or tensile stress becomes the hard or easy axis of magnetization, respectively. Figure 5 shows the distribution of magnetization directions in the domains of 8 × 8 cells with/without stress, where the distribution of the angle difference β between the magnetization direction and the RD is shown. At the demagnetizing state (m = 0), the distribution of β is almost symmetric with respect to β = π/2 to cancel the average magnetization by pairs of domains having anti-parallel magnetization When compressive stress is applied along the RD, the magnetization is nearly perpendicular to the RD in the demagnetizing state because of the stress-induced anisotropy. Under compressive stress, the magnetization also proceeds with 180 • domain wall motion roughly within 60 • ≤ β ≤ 120 • when the magnetization is small. Figure 6 plots the difference of β between µ 0 M = 0 and 0.4 T, where the distribution is almost point-symmetric with respect to (π/2, 0) showing the shift of the distribution from β to π β . When the magnetization becomes large, there is a shift of the distribution from β to nearly β π/2, which corresponds to the 90 • domain wall motion. The compressive stress thus increases the 180 • domain wall motion roughly within π/3 ≤ β ≤ 2π/3 and the 90 • domain wall motion, which strengthens the pinning field as follows.
If the magnetization proceeds with 180 • domain wall motion between two domains, the magnetization along the RD is where the volume ratio of the two domains are r and 1 r having the magnetization angles of β and β + π. The domain motion ∆r = ∆m cell /(2cosβ) increases if β is near π/2, which results in a strong pinning field. If the magnetization proceeds with 90 • domain wall motion between two domains having magnetization angles of β and β ± π/2, for example, the magnetization along the RD is m cell = r cosβ + (1 − r) cos(β ± π/2) = r cosβ ± (r − 1) sinβ.
The domain motion ∆r = ∆m cell /(cosβ ± sinβ) exceeds 180 • domain wall motion with β ≈ 0. The ADSM can predict the loss increase due to the compressive stress because it describes the effect of stress on the magnetization state in the crystal-grain scale depending on the distributed crystal orientations.

IV. CONCLUSION
The mechanism of hysteresis loss increase due to compressive stress was discussed according to simulation results obtained using the ADSM. The dependence of hysteresis loss on the applied stress was predicted using the ADSM, and agrees with the measured loss property quantitatively. The simulated distribution of magnetization angles shows that the compressive stress suppresses the development of magnetic domains having magnetization nearly parallel/antiparallel to the stress direction. As a result, the compressive stress causes mechanically induced anisotropy that increases the hysteresis loss by strengthening the pinning field along the compressed direction in the process of domain wall motion.

ACKNOWLEDGMENTS
This work was supported in part by the Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (C) Grant No. 26420232.