## 2 Magnetic Hysteresis Model
### Improved Preisach Model
The meaning of the congruence property is that when ferromagnetic materials are excited by a series of input signals with the same maximum and minimum values, all closed curves formed are congruent with each other, where congruence refers to the shape of the curve being similar. When using the basic formula of the classical Preisach method, the basic calculation formula itself has self-consistency with the congruence property, but it does not have the wiping-out property that should be possessed during the magnetization process. When the excitation signal appears, when a certain magnetic field intensity \(\mathbf{H}\) is greater than the maximum value of a certain historical magnetic field intensity or less than the minimum value of a certain historical magnetic field intensity, the basic method, relying solely on the Preisach numerical formula, cannot meet the conditions. Especially in the process of PWM excitation, due to the complex waveform changes of magnetic field intensity \(\mathbf{H}\), it is common to meet the conditions of the wiping-out phenomenon during signal loading. On the other hand, when the excitation signal is complex, especially when the excitation is a PWM signal, the Preisach model itself has a certain model error. Therefore, in this paper, an improved Preisach model is proposed to resolve the aforementioned problems, so that it inherits both congruence properties and wiping-out properties, while reducing the errors brought by classical Preisach models [27; 28]. The specific process is shown in Figure 5. First, we define and initialize two dynamic arrays \(\alpha\) and \(\beta\), storing the decreasing maximum and increasing minimum sequences that occur when the magnetic field intensity changes. When the magnetic field intensity is in an increasing state, the magnetic field intensity array \(h(m)\) is inserted into the decreasing maximum value sequence based on its size, and the maximum value smaller than the input magnetic field intensity value is cleared to ensure that the current voltage value is at the end of the decreasing maximum value sequence. Then, the number of elements in the decreasing maximum value sequence \(n\) is obtained, and the first \(n{-}1\) elements in the increasing minimum value array are taken to form a new increasing minimum value array. Finally, the turning point data are assigned to \(H_{n}\). We substitute the data into Equation (8) to obtain the magnetic induction intensity during the process of increasing magnetic field intensity. When the voltage drops, \(hfirst n elements of the maximum array are taken to form a new decreasing maximum array. Finally, the turning point data are assigned to \(H_{n}\). We substitute the data into Equation (9) to obtain the magnetic induction intensity during the decrease of magnetic field intensity. From a numerical calculation perspective, the improved Preisach model combines data from magnetic characteristic experiments to ensure the accuracy of each local loop turning point, thus making the magnetic characteristic curve modeled by this method closer to the true value. From the analysis of the properties of the Preisach model itself, in the simulation of classical Preisach models, there is a significant deviation in the simulation results of the initial stage, and the simulation values are larger than the measured values. There is also a certain deviation in the simulation results of the descending and ascending stages. This is because the mathematical model used is based on the rectangular magnetic hysteresis properties of the magnetic hysteresis operator, without considering the specific change process. Therefore, when loading pulse excitation, the complex changes in the magnetic properties of grain-oriented silicon steel caused by the excitation were not reflected in the model, resulting in deviations in the simulation results. The improved Preisach model incorporates correction coefficients to compensate for the process, which is not reflected in the original model.