## 1 Introduction  
In recent years, with the rapid development of the power industry, the demand for electrical equipment such as power transformers has gradually increased. Power transformers play an extremely important role in urban and modern construction, which puts forward higher requirements for the safe operation and design of electrical equipment. With the continuous development and improvement of PWM (Pulse Width Modulation), its use in the control of power equipment has expanded. The signal generated by PWM supply contains a large number of high-order harmonics, and the voltage waveform changes many times in one cycle, resulting in obvious changes in the magnetic hysteresis properties and loss properties of the transformer core material. The core is the main magnetic circuit part in the power transformer, which is usually made of grain-oriented silicon steel sheets with high silicon content. Grain-oriented silicon steel has been widely used in the power transformer core manufacturing industry due to its advantages of low iron loss, high magnetic induction strength, smooth surface, and strong punchability. The core characteristics of power transformers are mainly characterized by the magnetic properties of grain-oriented silicon steel. One of the most important magnetic properties of grain-oriented silicon steel is the magnetic hysteresis phenomenon [1; 2; 3]. Since the Preisach model used in this paper is based on the theory of ferromagnetism, a brief explanation is necessary here. According to [4], magnetic hysteresis refers to the phenomenon that in the process of magnetization and demagnetization of ferromagnetic physical materials, the change of magnetic induction strength of magnetic materials lags behind magnetic field strength, and on a micro level, when the magnetic field is changing, the domain wall inside the magnetic materials moves continuously, and the magnetic domain changes and rotates. Numerous studies have shown that the magnetic properties of ferromagnetic materials under PWM excitation are very different from those under sinusoidal excitation [5]. There are standard methods of measuring the magnetic properties of grain-oriented silicon steel sheets under sinusoidal excitation [6; 7; 8], which can give the measurement parameters of hysteresis loops of electrical steel sheets. However, the magnetic hysteresis of magnetic materials is a complex, nonlinear problem, and it is very difficult to achieve accurate simulation. The traditional electromagnetic field numerical calculation and most of the existing commercial software for electromagnetic calculation do not consider the influence of the magnetic hysteresis of materials, but only use a single value magnetization curve to approximate the magnetic properties of materials. This has brought significant errors to the design process and calculation results of the operating properties of power transformers [9; 10]. Therefore, it is necessary to establish a magnetic hysteresis model suitable for numerical calculation of the transformer core material, accurately simulate and predict the magnetic hysteresis loop of the material, and provide an accurate material parameter model for the magnetic field distribution calculation of the product-level transformer laminated core, thereby improving the efficiency of power equipment, reducing energy loss, and playing an important role in the safety and stability of the power grid and the design of power equipment [11; 12; 13]. Therefore, the measurement, modeling, and magnetic hysteresis loop prediction of grain-oriented silicon steel under PWM excitation have important practical significance and theoretical value. In terms of magnetic hysteresis modeling, the classical magnetic hysteresis models mainly include the Preisach model [14], the Jiles-Atherton (J-A) model [15], the Stoner-Wohlfarth (S-W) model [16], the Enokizono-Soda (E&S) model [17], the hybrid magnetic hysteresis model [18], etc. On this basis, a series of improvements and advancements through in-depth research are presented in [19; 20].