1. Hubei Provincial Engineering Technology Research Center for Power Transmission Line China Three Gorges University Yichang 443002 China; 2. College of Electrical Engineering and New Energy China Three Gorges University Yichang 443002 China; 3. Huawei Beijing R&D Centre Beijing 100085 China; 4. State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power University Beijing 102206 China
Abstract：The complex nonlinear hysteresis characteristics of soft magnetic materials significantly impact the energy losses, multi-physics field, and other properties of electrical equipment. Thus, it is crucial to accurately simulate the hysteresis loops of the soft magnetic materials. Under this circumstance, many hysteresis models, such as the well-known Preisach and Jiles-Atherton (J-A) models, have been developed. As the FEA problem of the electrical equipment is usually formulated by magnetic vector potential, making the magnetic flux density B the output of the differential equations, the inverse hysteresis models are more in demand. The inverse Preisach model is usually viewed as the most accurate among all the inverse hysteresis models. However, it involves complex integral operations, so its computation cost is high. As a result, it is not practical in electrical engineering, which makes the accurate analytical inverse Preisach model inheriting low computation cost more desired. In this paper, to our knowledge, an analytical inverse Preisach model, the only one among all of the Preisach models, is developed and proposed. Firstly, the analytical expressions of permeability of the initial magnetization curve, ascending and descending segments of the hysteresis loop are obtained using our proposed analytical forward Preisach hysteresis model based on the analytical Everett function. The analytical inverse Preisach hysteresis model, as shown in Eq.(1), is derived with the difference method. A grain-oriented silicon steel sample and a non-oriented silicon steel sample are selected to measure their hysteresis loops at different flux density levels. The simulated hysteresis loops and the measured ones are compared. It is shown that the predicted hysteresis loops agree with the measured ones. Additionally, each calculated hysteresis loss (the area enclosed by the corresponding simulated hysteresis loop) is compared with the measured one, which also tests the accuracy of our proposed model. Besides, the average relative error of the analytical inverse Preisach model is smaller than that of the inverse J-A model by more than 10%, and its computation speed is approximately 2.67 times faster than that of the inverse J-A model. $H(t+\Delta t)=\left\{ \begin{array}{*{35}{l}} H(t)+\frac{B(t+\Delta t)-B(t)}{\sum\limits_{i=1}^{n}{\frac{2\alpha _{i}^{2}{{\text{e}}^{-{{\beta }_{i}}H(t)}}}{{{\beta }_{i}}{{\gamma }_{i}}{{\left( 1+{{\gamma }_{i}}{{\text{e}}^{-{{\beta }_{i}}H(t)}} \right)}^{2}}}}\left( \frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{-{{\beta }_{i}}H(t)}}}-\frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{{{\beta }_{i}}H(t)}}} \right)+{{k}_{1}}+\frac{{{k}_{2}}}{{{k}_{3}}}\left[ 1-{{\tanh }^{2}}\left( \frac{H(t)}{{{k}_{3}}} \right) \right]} & {{H}_{\text{m}}}\text{=0 and }{{B}_{\text{m}}} & \text{=0} \\ H(t)+\frac{B(t+\Delta t)-B(t)}{\sum\limits_{i=1}^{n}{\frac{-2\alpha _{i}^{2}{{\text{e}}^{{{\beta }_{i}}H(t)}}}{{{\beta }_{i}}{{\gamma }_{i}}{{\left( 1+{{\gamma }_{i}}{{\text{e}}^{{{\beta }_{i}}H(t)}} \right)}^{2}}}}\left( \frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{-{{\beta }_{i}}H(t)}}}-\frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{-{{\beta }_{i}}{{H}_{\text{m}}}}}} \right)+{{k}_{1}}+\frac{{{k}_{2}}}{{{k}_{3}}}\left[ 1-{{\tanh }^{2}}\left( \frac{H(t)}{{{k}_{3}}} \right) \right]} & H\le {{H}_{\text{m}}} \\ H(t)+\frac{B(t+\Delta t)-B(t)}{\sum\limits_{i=1}^{n}{\frac{2\alpha _{i}^{2}{{\text{e}}^{-{{\beta }_{i}}H(t)}}}{{{\beta }_{i}}{{\gamma }_{i}}{{\left( 1+{{\gamma }_{i}}{{\text{e}}^{-{{\beta }_{i}}H(t)}} \right)}^{2}}}}\left( \frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{{{\beta }_{i}}{{H}_{\text{m}}}}}}-\frac{1}{1+{{\gamma }_{i}}{{\text{e}}^{{{\beta }_{i}}H(t)}}} \right)+{{k}_{1}}+\frac{{{k}_{2}}}{{{k}_{3}}}\left[ 1-{{\tanh }^{2}}\left( \frac{H(t)}{{{k}_{3}}} \right) \right]} & H{{H}_{\text{m}}} \\ \end{array} \right.$ (A1) where αi, βi, γi, k1 ,k2, k3 are the model coefficients, n is the number of items in Everett function, Hm and Bm are the last reversal magnetic field intensity and magnetic flux density, respectively.
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