Dynamic Hysteresis Modeling Considering Skin Effect for Magnetic Circuit Analysis

Abstract

In recent years, quantitative analysis of iron loss taking magnetic hysteresis behavior into account is essential to development of high-efficiency electric machines. In a finite element method (FEM)-based design and analysis, iron loss of electric machines is approximately calculated based on Steinmetz’s equation [1], although its calculation accuracy is not necessarily sufficient in the case of PWM and dc-biased excitation conditions. Thus, various hysteresis models have been proposed so far [2]-[4], in order to accurately simulate the magnetic hysteresis behavior. However, most of the general-purpose FEM programs neglect magnetic hysteresis since its calculation takes a lot of time in general. Therefore, establishment of a simple and practical method for calculating iron loss including magnetic hysteresis behavior is strongly required.In a previous paper, a magnetic circuit model incorporating a play model [5], which is one of the phenomenological models of magnetic hysteresis behavior, was proposed [6]. It was clear that the proposed model can calculate the hysteresis loop of the magnetic reactor with high-speed and high-accuracy. However, there is a possibility for further improvement in the calculation accuracy of the minor loop generated from carrier harmonics under PWM excitation because the influence of the skin effect cannot be considered. Therefore, in this paper, the magnetic circuit model is improved to be that the dynamic hysteresis characteristics including the skin effect can be considered, so that the minor loop can be calculated with higher accuracy. The validity of the improved model is proved by using ring cores made of various materials, such as grain-oriented (GO) and non-oriented (NO) silicon steels, an amorphous alloy, and a soft magnetic composite (SMC).Fig. 1(a) shows the previously proposed magnetic circuit model incorporating the play model. As shown in the figure, the dc hysteresis is represented by the play model. The play model can express an arbitrary hysteresis loop by multiplying play hysterons with different widths by shape functions. To derive the shape functions, the play model generally requires a large number of measured dc hysteresis loops with different maximum flux densities [1]. On the other hand, the classical eddy current loss and the anomalous eddy current loss are denoted by the inductance and the controlled-source, respectively. Fig. 1 (b) shows the comparison of the measured and calculated hysteresis loops under PWM excitation for each core material. This figure reveals that the previous magnetic circuit model can calculate the major loop with high accuracy, while there are errors in the shape of the minor loops. The cause of the errors is not to consider the influence of the skin effect by carrier harmonics.Therefore, in order to calculate minor loops with higher accuracy, the magnetic circuit model is improved as shown in Fig. 2(a). In the previous model, the eddy current loss is given by single inductance element. On the contrary, in the proposed model, the influence of the skin effect is represented by a ladder circuit in which inductances and resistances are repeatedly connected in parallel and in series. This ladder circuit represents the frequency characteristic of the complex permeability at higher frequencies as the number of circuit stages is larger. The parameters Lm and Rm of the circuit elements are given by the following equations:Lm = (4/σd^2)×(l/S) (1)Rm = (1/μ)×(l/S) (2)where the magnetic path length is l, the cross-sectional area is S, the conductivity is σ, the thickness of the silicon steel is d, and the permeability is μ, respectively.The validity of the proposed model is proved by comparing the measured and calculated hysteresis loops under PWM excitation, as shown in Fig. 2(b). In the analysis, the ladder circuit is terminated by the second inductance Lm1. From this figure, it is clear that the minor loops can be calculated with much higher accuracy comparing with the previous model for GO and NO silicon steels and a SMC. On the other hand, the calculation accuracy hardly changes for an amorphous metal since the values of σ and d is quite small. The cause of the error of the minor loops is considered to be the overestimated anomalous eddy current loss, which is one of the issues to be solved in the future.In the future, we plan to apply the proposed method to the analysis of reactors and transformers, which are used for switching power supplies, in order to calculate the iron loss considering magnetic hysteresis behavior including the minor loop with higher accuracy.This work was supported by Grant-in-Aid for JSPS Fellows (JP19J20572). **