An Improved Anisotropic Vector Preisach Model for Nonoriented Electrical Steel Sheet Based on Iron Loss Separation Theory

An improved anisotropic vector Preisach model is proposed in this paper to describe the hysteresis properties of nonoriented (NO) electrical steel sheet (ESS) under 50 Hz rotating magnetic fields. The proposed model consists of three components, static hysteresis component, eddy current component, and excess component, which is based on the iron loss separation theory. The static hysteresis component is constructed by the static vector Preisach model. The proposed model is identified by the measured hysteresis properties under 1 Hz and 50 Hz magnetic fields. Finally, the experimental results prove the effectiveness of the proposed anisotropic vector hysteresis model.


Introduction
With the rapid development of the electrical motor drive technology, different types of electrical motors are designed to meet various needs. And, more and more attention has been paid to the performance and efficiency of the motor. However, to improve the performance and the efficiency of the electrical motor in the simulation design stage, accurate magnetic field distribution and iron loss analysis are extremely significant. Until now, finite element analyses (FEA) is one of the most popular and accuracy numerical methods to calculate the magnetic field distribution and the iron loss of electrical motor. However, accuracy and speed describe the hysteresis properties between magnetic field strength (H) and magnetic flux density (B) is quite essential to calculation accuracy and speed of FEA.
On the one hand, the magnet filed in the teeth of the motor is alternating field, and the magnet filed in the yoke is rotating filed. According to the experimental experience, the hysteresis properties under rotating field are quite different from that under alternating fields. However, until now, most of the commercial FEA software only can consider the alternating fields. us, the calculation accuracy is not satisfied, and it cannot be used for motor design and optimization directly. In additional, the nonoriented (NO) electric steel sheet (ESS) also presents anisotropic properties under both alternating and rotating magnetic fields in practice. erefore, an anisotropic vector hysteresis model should be developed to describe the vector hysteresis property of ESS and combined with FEA to analyse the performance and efficiency of the motor.
Until now, to describe the vector hysteresis property of ESS and improve the accuracy of FEA, many versions of hysteresis models have been developed, such as vector Preisach model, vector Jiles-Atherton (JA) model, vector E&S model, and vector Play models [1][2][3][4].
e Preisach model has been widely concerned by many researchers because of its perfect modelling results, and the Preisach model is constructed from the physical point of view [5]. e classical Preisach model is proposed by Preisach firstly [6], and it is applied to describe the hysteresis property by many researchers [5,[7][8][9][10][11].
e original vector version of the Preisach model is developed by using the superposition of the classical scalar Preisach models along different azimuthal direction [12][13][14]. And, because the anisotropic properties cannot consider in the original vector Preisach model, it is an isotropic model. To apply the vector Preisach model to FEA and analysis the iron loss, a few frequencies dependent vector Preisach models have been developed based on the iron loss separation theory [15][16][17]. However, the above models are limited pay attention to the anisotropic property. erefore, some versions of the vector Preisach model which can consider the anisotropic properties have been proposed for NO ESS. However, the model results only can present weakly anisotropic property, and they do not match the experimentally measured data well.
In this paper, an improved anisotropic vector Preisach model is proposed to describe hysteresis behaviour of NO ESS, which has low anisotropic property. e proposed model consists of three components, static hysteresis component, eddy current component, and excess component, which is based on the iron loss separation. e static hysteresis component is developed by the vector Preisach model which is identified by using the experimental data from static magnetic field. e average static scalar property is measured from the ring-type core under 1 Hz magnetic field. And, the parameter in the proposed model is identified by using the purely rotating magnetic fields under 50 Hz magnetic fields whose magnitude is limited to 1.6 T. e identification and validation of the model are performed by using measurement data of the NO ESS, 35PN440, which is obtained from a two-dimensional (2D) single sheet tester (SST).

Measurement System for the Scalar Preisach Model.
e vector static Preisach model consisted of the scalar static Preisach model. erefore, the scalar version should be constructed firstly. To construct the scalar Preisach model, the static scalar hysteresis behaviour should be measured. However, the pure static hysteresis behaviour of ESS without any eddy current and excess effects is quite difficult to measure. erefore, in this paper, the hysteresis behaviour under 1 Hz exciting current is considered as the static hysteresis behaviour, and the corresponding measurement system is developed shown in Figure 1. In Figure 1, the exciting coil and the B-coil are wound around the ring-type core. e exciting coil is used to generate the magnetic fields, and the H-waveform can be obtained from the current of the exciting coil. And, there are 40 and 600 turns of the exciting coil for lower and higher H value measurement, respectively. e B-coil of 20 turns is used to measure the B-waveform. e ring-type core is the lamination core, which is made up of the toroidal NO ESS,35PN440, and it is used to obtain the average scalar B-H property of the specimen [18]. e inner and outer diameters of the toroidal ESS are 40 mm and 50 mm, respectively. e hysteresis minor loops with different maximum value of B under 1 Hz alternating magnetic fields can be measured by this experimental device, and the measured results can be used to identify the static Preisach model.

Scalar Preisach Model and Its Identification.
In FEA method, the B-waveform is required to calculate the Hwaveform. erefore, an inverse version of the scalar Preisach model, which outputs H-waveform, can be calculated from the input B-swaveform by a double integral defined over the Preisach triangle [4] as follows: where c αβ is a hysteresis operator [4], which is similar with a relay as shown in Figure 2 and given as (2), controlled by the increasing α and decreasing β values of the input B and can only have +1 and − 1 values, μ is the distribution function of the hysteresis operators, and it should be identified by the experimental data: e double integral operation in (1) will consume a lot of time when the model is applied to FEM. To improve the identification accuracy and the calculation efficiency of the Preisach model, an Everett function is defined as follows: where (α 0 , β 0 ) is the coordinate of a point within the integral zone and B m is the maximum value of B. erefore, equation (1) can be instead by addition and subtraction of the Everett function as follows: where B k are the extreme points of the input B-waveform, and they are stored in the extreme point memory as shown in Figure 3 [7]. In the traditional identification method, the distribution function μ or the Everett function should be identified by using the first-order reverse curve (FORC) [7]. However, the FORC is quite difficult to measure, and the accuracy of the measurement results is not satisfied. And, negative values will appear in the distribution function because of the measurement errors.
In this paper, the Everett function is defined from experimentally measured symmetric minor B-H loops as follows: where h + (B a , B) and h − (B a , B) are, respectively, the ascending and descending branches of the symmetric B-H loop with maximum amplitude B a . In this paper, totally 16 symmetric minor B-H loops are measured from the measurement system in Figure 1, and the range of B a is from 0.1 T to 1.6 T; the step is 0.1 T. And, then the Everett function is identified by the measurement data and (5). e result of the Everett function which is identified by (5) is shown in Figure 4(a), and Figure 4(b) shows the comparison results between the modelling and measurement results under alternating 1 Hz magnet fields with different values of B a . From the results, the modelling results can match well with the measured ones.

Vector Preisach Model.
e traditional vector Preisach model consists of the superposition of classical scalar Preisach models along different azimuthal directions as follows: where H φi is the output H-waveform of the classical scalar Preisach model along the direction of φ i , having the input Bwaveform along the direction of B φi as input, and η is the vector distribution function corresponding to the vector Preisach model.
According to the scalar Everett function E(α, β), the vector Everett function F(α′, β′) is proposed to implement the vector Preisach model, and it can be calculated from the classical scalar Everett function as follows: To implement the improved anisotropic vector Preisach model, equation (5) should be rewritten to its discrete format as where ω i is the Gauss integral weight, and (6a) and (6b) can be solved by using the method which is explained in [7].
To consider the anisotropic property, the input B-waveform along the azimuthal direction φ i can be calculated as where δ � sign(cos(θ B − φ i + ψ)), ω is a parameter to consider the anisotropic property, and ψ is a parameter which is used to control the initial phase of the H-waveform. If the parameter ω � 1, the output of H-waveform is purely circle under the circle rotating magnetic field. And, when ω > 1, the result is flower shape and it can simulate weakly anisotropic behavior.

Measurement System.
In this paper, a round-type twodirectional single sheet tester (R-2D-SST), as shown in Figure 5 is developed to measure the vector hysteresis properties for NO ESS under both alternating and rotating 50 Hz magnetic field conditions [19][20][21]. Its specifications are shown in Table 1.

Traditional Vector Preisach Model.
Based on the theory of iron loss separation, Dlala [10] proposed a simple hysteresis model. And, the model assumes that the total H is composed of three components as follows: where hysteresis is calculated from scalar Preisach model, B φi is calculated by (9), d is the thickness of ESS, σ is the conductivity, and the parameters r, p, ψ, and ω can be fitted α β +1 -1 Figure 2: A hysteresis operator with the increasing value α and decreasing value β. erefore, an improved model is proposed as follows:  where z i is used to describe the anisotropic properties of ESS. In the model, the parameters r, p, and ψ are assumed as piecewise linear shape functions as And, the parameter z is assumed as a 2-D piecewise linear shape functions as where ΔB � 0.1 T and B j � jΔB, j � 4, 5, . . ., 16. Δφ � 20°and φ i � i Δφ, i � 0, 1, . . ., 8. And, μ ψ,j , μ r,j , μ p,j , and λ z,i can be identified from the measured vector B-H locus under circle rotating fields whose magnitude is B j by using the particle swarm optimization (PSO) method.
e objective function is defined as the root-meansquared error between the measured values and the calculated values of one B-H loop under circle rotating fields whose magnitude is B j , as follows: where n is the number of sampling points of the vector hysteresis locus, H exper is the measure magnetic field strength values on the sampling points, and H model is the modelling magnetic field strength values on the sampling points. In addition, the effective applied range of the proposed model depends on the maximum B value of the measurement B-H locus which is used to identify the model. Mathematical Problems in Engineering magnet field for the range of 0.4 ≤ B max ≤ 1.6 (T), which step is 0.1 T, is used to identify the parameters in the proposed vector Preisach model by using PSO method. e identification results of parameter z, r, p, and ψ are shown in Figure 6. e measured, original, and proposed modelling H locus under circular rotating magnetic field conditions when B max � 1.6 T and B max � 1.48 T are compared and shown in Figure 7. From the figure, the accuracy of the proposed model is much higher than the original model under the both rotating magnetic fields. In addition, the measured H locus when B max � 1.6 T is used to identify the parameter in the proposed model, while when B max � 1.48 T it is not used. e measured and improved modelling H locus under circular rotating magnetic field conditions with different B max value is compared and shown in Figure 8. In the figure, the H locus for B max � 1.3 T, 1.4 T, and 1.5 T is used in the identification of parameters z, r, p, and ψ. While those for B max � 1.25 T, 1.35 T, 1.45 T, and 1.48 T, it is not used. From the figure, the proposed vector hysteresis model can predict the vector hysteresis properties very accurately under various rotating magnetic fields. However, the accuracy when B max is 1.45 T and 1.48 T is lower than others because of the interpretation method is used to obtain the parameter in the model. e H locus predicted by the proposed model for alternating magnetic fields of B m � 1.6 T along rolling and transverse direction is shown in Figure 9 together with that from measured data. From the figure, the proposed model also can simulate the hysteresis properties for unidirectional scalar hysteresis properties. And, the deviation d H between the measured and modelling results along rolling and transverse direction is 3.47% and 4.51%, respectively. And, the deviation, d H , is defined as   Mathematical Problems in Engineering where N is the number of sampling points of the B-H loop, τ stands for ωt ∈ [0, 2π] with as the angular frequency of a Bwaveform, and H mea and H mod are the measured and modelling H-waveforms, respectively. Figure 10 shows the modelling performance of the proposed vector hysteresis model when it is applied to elliptically rotating magnetic field conditions. e predicted and measured results match well each other, and it is shown that the proposed vector hysteresis model can be successfully applied to various rotating magnetic field conditions.

Conclusion
In this paper, an improved anisotropic vector Preisach hysteresis model is proposed to describe the vector and weakly anisotropic hysteresis behavior for NO ESS under 50 Hz rotating magnetic fields. e proposed model consists of three components, static hysteresis component, eddy current component, and excess component, and coefficients z, r, p, and ψ are introduced to increase the anisotropic property and the accuracy. e proposed model is identified by the measured hysteresis properties under 1 Hz and 50 Hz alternating and rotating magnetic fields. And, because maximum B value of the measured data for identification is 1.6 T, the limitation of the model is only up to the magnetic field whose B max � 1.6 T. rough applications to NO ESS,   the proposed vector hysteresis model is proven to describe the anisotropic properties under various rotating magnetic fields.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.