## 2 Magnetic Hysteresis Model  
### Classical Preisach Model  
The Preisach model is a model commonly used for silicon steel materials, and its definition has many types. In this paper, we use the following methods. Firstly, the Preisach model considers the magnetic hysteresis of a magnetic material via an infinite set of magnetic dipoles, which have rectangular magnetic hysteresis loops, as shown in Figure 1[21]. According to the classical Preisach model [22], the magnetization \(\mathbf{M}\) due to the magnetic field \(\mathbf{H}\) is calculated in accordance with Equation (1):  
\[M=\int\limits_{s}\mu^{\prime}(\alpha,\beta)\gamma_{\alpha\beta}(H)\mathrm{d} \alpha\mathrm{d}\beta=\int\limits_{S+}\mu^{\prime}(\alpha,\beta)\mathrm{d} \alpha\mathrm{d}\beta-\int\limits_{S-}\mu^{\prime}(\alpha,\beta)\mathrm{d} \alpha\mathrm{d}\beta \tag{1}\]  
where \(\mu^{\prime}(\alpha,\beta)\) is the distribution function; \(\alpha\) and \(\beta\) are threshold value in the increasing positive and negative directions, respectively; and \(\gamma_{\alpha\beta}\) is the magnetization of a single rectangular magnetic hysteresis operator. The \(\gamma_{\alpha\beta}\) property depends on \(\alpha\) and \(\beta\). \(\gamma_{\alpha\beta}\) is 1 when H is greater than \(\alpha\) and \(-1\) when H is less than \(\beta\). When \(\alpha<\) H \(<\beta\), according to Figure 1, H decreases from \(\alpha\) to \(\beta\), and \(\gamma_{\alpha\beta}\) jumps from 1 to \(-1\) when H is less than \(\beta\). When the rectangular magnetic hysteresis operator is located in the S+ area, the magnetization currently displayed by the magnetic dipole operator is \(+1\); when the rectangular magnetic hysteresis operator is located in the S\(-\) area, the magnetization currently displayed by the magnetic hysteresis operator is \(-1\).  
While magnetic materials have different magnetization processes, different magnetization processes correspond to different diagrams. For example, when the magnetic field \(\mathbf{H}\) increases or decreases, the Preisach diagram will change.  
For this manuscript, since the measurements are in B = \(\mu_{0}\times(\mathrm{H}+\mathrm{M})\), and since \(\mathbf{M}\) is much greater than \(\mathbf{H}\) for the range of magnetic fields used in this work, one can rewrite Equation (1) with \(\mathbf{B}\).  
\[B=\int\limits_{s}\mu(\alpha,\beta)\gamma_{\alpha\beta}(H)\mathrm{d}\alpha \mathrm{d}\beta=\int\limits_{S+}\mu(\alpha,\beta)\mathrm{d}\alpha\mathrm{d} \beta-\int\limits_{S-}\mu(\alpha,\beta)\mathrm{d}\alpha\mathrm{d}\beta \tag{2}\]  
Within the limiting magnetic hysteresis loop [23] (the magnetic hysteresis loop intersects the \(H\)-axis and the \(B\)-axis, and the limiting magnetic hysteresis loop is formed  
Figure 1: Elementary magnetic hysteresis loop and Preisach diagram.  
by the four intersections), the magnetization process can be divided into processes of increasing and decreasing magnetic field strength. Figure 2 shows the rise and fall of the magnetization process:  
Inside the limit magnetic hysteresis loop, the magnetic field \(\mathbf{H}\) decreases after the turning point:  
\[B(H)=B(H_{n})-2T(H_{n},H) \tag{3}\]  
The magnetic field \(\mathbf{H}\) increases after the turning point:  
\[B(H)=B(H_{n})+2T(H,H_{n}) \tag{4}\]  
where \(H_{n}\) is the magnetic field at the turning point, and \(B(H_{n})\) is the magnetic induction strength corresponding to the turning point. \(T\) is the area integration over the small triangle, which can be superimposed by the diagram of the limiting magnetic hysteresis loop:  
\[T(\alpha,\beta)=\frac{B_{u}(\alpha)-B_{d}(\beta)}{2}+F(\alpha)F(-\beta) \tag{5}\]  
where  
\[\left\{\begin{array}{ll}F(\alpha)=\frac{B_{d}(\alpha)-B_{u}(\alpha)}{2\sqrt {B_{d}(\alpha)}}&(\alpha\geq 0)\\ F(\alpha)=\sqrt{B_{d}(-\alpha)}&(\alpha<0)\end{array}\right. \tag{6}\]  
where \(B_{d}(\alpha)\) and \(B_{u}(\alpha)\) respectively represent the magnetic flux density corresponding to the rising and falling branches of the limit magnetic hysteresis loop when the magnetic field is \(\alpha\)[24, 25].