## 2 Magnetic Hysteresis Model
### Improved Preisach Model  
The above introduces the numerical calculation formula of the classical Preisach model. The classical Preisach method is suitable for sinusoidal excitation, and it is convenient to combine with experiments. Because the high-order harmonics are included in PWM excitation, this phenomenon leads to a more complex relationship between the variation of the two quantities of magnetic flux density and magnetic field strength, as shown in Figure 3. Figure 4 shows the harmonic analysis in H for both excitations. The plotting sequence for Figure 3 is in Appendix A. It is calculated by Matlab based on the experimental  
Figure 2: Rise and fall of magnetization process. results. Due to the complexity of PWM excitation, an improved model for PWM excitation needs to be proposed. In fact, the principle of magnetic domain movement is very complicated. In the Preisach model, the results of domain magnetization can be simulated, but the specific internal process is ignored. Consistent with the assumptions of the Preisach model, it includes the initial state and final state, but unlike the original model, the intermediate process is also included. In particular, when the external excitation is non-sinusoidal (such as in PWM excitation), the classical Preisach model will produce large model errors. Therefore, the correction coefficients should be introduced to compensate for the error caused by PWM excitation [26]. On this basis, the correction coefficients \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\) are introduced to complete the derivation of the formula. For the improved Preisach model, the expression of magnetic flux density \(B\) under different conditions is shown in Formulas (7)-(9). On the initial magnetization curve, the expression of magnetic flux density is shown by Formula (7). \[B(H)=\gamma_{1}[F(-H)-F(H)]^{2} \tag{7}\]  
Inside the limiting magnetic hysteresis loop, the magnetic field \(H\) decreases after the turning point:  
\[B(H)=B(H_{\pi})-2\gamma_{2}T(H_{\pi},H) \tag{8}\]  
Figure 4: Harmonic analysis of two excitations. (**a**) Harmonic analysis of magnetic field strength under PWM excitation. (**b**) Harmonic analysis of magnetic field strength under sinusoidal excitation. Figure 3: Comparison of magnetic flux density and magnetic field intensity waveforms under different excitation: (**a**) magnetic flux density waveforms; (**b**) magnetic field intensity waveforms. The magnetic field \(\mathbf{H}\) increases after the turning point:  
\[B(H)=B(H_{n})+2\gamma_{3}T(H,H_{n}) \tag{9}\]  
Since the correction coefficients \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\) are unknown, we need to know the conditions to solve them. The correction coefficients are obtained by using the turning point magnetic field \(\mathbf{H_{n1}}\) and \(\mathbf{H_{n2}}\) measured in the experiment, and the solution formula is given by Formula (10). In fact, the experimental results can not only provide data for solving coefficients but also verify the results of the model. Therefore, the corresponding experimental platform for magnetic hysteresis of grain-oriented silicon steel sheet under PWM excitation needs to be designed. \[ \begin{array}{l}\gamma_{1}=\frac{B(H_{\text{nt}})}{[F(-H)-F(H)]^{2}}\\ \gamma_{2}=\gamma_{3}=\frac{B(H_{\text{nt}})-B(H_{\text{nt}})}{2T(H_{\text{nt} },H_{\text{nt}2})}\end{array}  \tag{10}\]  
The improved Preisach model is described by Equations (7)-(10). Theoretically, combined with the experimental setup (correcting errors in classical models with measurements from experimental setups), the magnetic hysteresis loop of any grain-oriented silicon steel sheet excited by PWM can be obtained. During the magnetization process, magnetic materials meet two basic properties, namely the wiping-out property and congruence property. The wiping-out property is explained in Section 1.