An Improved Preisach Model for Magnetic Hysteresis of Grain-Oriented Silicon Steel under PWM Excitation

Abstract

In this paper, the Preisach model for magnetic hysteresis of grain-oriented silicon steel under PWM excitation is improved. First, an improved Preisach model for the magnetic hysteresis of grain-oriented silicon steel under PWM excitation is proposed. Second, the experimental platform for grain-oriented silicon steel sheets under PWM excitation is established. Finally, by comparative analysis, it is concluded that the error of the improved model is far less than that of the classical model (the error here refers to the discrepancy between experimental results and theoretical model predictions). The improved model is 1.4% to 9% more accurate than the classical model. A more accurate model can provide more accurate material parameters for the calculation of the magnetic field in the transformer core, which is of great significance to the production and design of the transformer.

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]. The above five common magnetic hysteresis models have their own advantages and disadvantages in engineering applications, summarized as shown in

Table 1. Comparison of magnetic hysteresis models.

Magnetic Hysteresis Models	Advantages	Disadvantages
Preisach Model	Based on the original model, the proposed simplified Preisach model is easy to compute and has a wide range of applications	Interactions between particles and specific magnetization processes are not considered
J-A Model	The principle is simple and easy to improve	The number of unknowns is high, and the parameter identification process is complex and computationally intensive
S-W Model	Analyzed from the energy point of view, it can reflect the nature of the magnetic hysteresis phenomenon	Limited scope of application
E&S Model	Vectors B and H and gives the relation to establish technological vector magnetic properties. Considers the anisotropic properties of ferromagnetic materials	Relies on a large amount of experimental data

. Hybrid magnetic hysteresis models are usually a combination of the Preisach model and S-W model.

However, when most magnetic hysteresis models and the improved models are used for PWM excitation, the value of the relative error will be increased (here, as in the abstract, the error refers to the discrepancy between experimental results and theoretical model predictions). Therefore, combined with Preisach model, an improved modeling method suitable for PWM excitation is proposed.

In this paper, an improved Preisach model is proposed, introducing correction coefficients and improving the approximation formula based on the classical Preisach model. At the same time, the method takes into account the wiping-out property in the magnetization process. The meaning of the wiping-out property is that when the input magnetic field strength H value is greater than certain historical maximum values or the input magnetic field strength H value is less than certain historical minimum values, the input voltage value will wipe these historical extreme values, and the historical processes related to these historical extreme values will also be wiped, no longer affecting future output. By comparing the principles of measuring magnetic properties of different grain-oriented silicon steel and selecting the appropriate measurement method, relying on the basis of the ring-sample test method, we built the experimental platform. Based on the LabVIEW2018 platform and the data acquisition and data processing functions of the data acquisition card, the magnetic properties of the grain-oriented silicon steel specimens were measured. Combined with the experimental data, by comparing the simulation results of the traditional Preisach model and the improved Preisach model proposed in this paper, the results show that the error of the improved model is smaller than that of the traditional model, i.e., the results of the improved hysteresis modeling are more accurate.

2. Magnetic Hysteresis Model

2.1. Classical Preisach Model

The Preisach model is a model commonly used for silicon steel materials, and its definition has many types. In this paper, we use the following methods. Firstly, the Preisach model considers the magnetic hysteresis of a magnetic material via an infinite set of magnetic dipoles, which have rectangular magnetic hysteresis loops, as shown in Figure 1 [21]. According to the classical Preisach model [22], the magnetization M due to the magnetic field H is calculated in accordance with Equation (1):

$$M={\displaystyle \underset{s}{\int }\mu \prime \left(\alpha ,\beta \right)}{\gamma }_{\alpha \beta }\left(H\right)\mathrm{d}\alpha \mathrm{d}\beta ={\displaystyle \underset{S+}{\int }\mu \prime \left(\alpha ,\beta \right)}\mathrm{d}\alpha \mathrm{d}\beta -{\displaystyle \underset{S-}{\int }\mu \prime \left(\alpha ,\beta \right)}\mathrm{d}\alpha \mathrm{d}\beta$$

(1)

where μ′(α, β) is the distribution function; α and β are threshold value in the increasing positive and negative directions, respectively; and γ_αβ is the magnetization of a single rectangular magnetic hysteresis operator. The γ_αβ property depends on α and β. γ_αβ is 1 when H is greater than α and −1 when H is less than β. When α < H < β, according to Figure 1, H decreases from α to β, and γ_αβ jumps from 1 to −1 when H is less than β. When the rectangular magnetic hysteresis operator is located in the S+ area, the magnetization currently displayed by the magnetic dipole operator is +1; when the rectangular magnetic hysteresis operator is located in the S− area, the magnetization currently displayed by the magnetic hysteresis operator is −1.

While magnetic materials have different magnetization processes, different magnetization processes correspond to different diagrams. For example, when the magnetic field H increases or decreases, the Preisach diagram will change.

For this manuscript, since the measurements are in B = μ₀ × (H + M), and since M is much greater than H for the range of magnetic fields used in this work, one can rewrite Equation (1) with B.

$$B={\displaystyle \underset{s}{\int }\mu \left(\alpha ,\beta \right)}{\gamma }_{\alpha \beta }\left(H\right)\mathrm{d}\alpha \mathrm{d}\beta ={\displaystyle \underset{S+}{\int }\mu \left(\alpha ,\beta \right)}\mathrm{d}\alpha \mathrm{d}\beta -{\displaystyle \underset{S-}{\int }\mu \left(\alpha ,\beta \right)}\mathrm{d}\alpha \mathrm{d}\beta$$

(2)

Within the limiting magnetic hysteresis loop [23] (the magnetic hysteresis loop intersects the H-axis and the B-axis, and the limiting magnetic hysteresis loop is formed by the four intersections), the magnetization process can be divided into processes of increasing and decreasing magnetic field strength. Figure 2 shows the rise and fall of the magnetization process:

Inside the limit magnetic hysteresis loop, the magnetic field H decreases after the turning point:

$$B\left(H\right)=B\left(H_n\right)-2T\left(H_n,H\right)$$

(3)

The magnetic field H increases after the turning point:

$$B\left(H\right)=B\left(H_n\right)+2T\left(H,H_n\right)$$

(4)

where H_n is the magnetic field at the turning point, and B(H_n) is the magnetic induction strength corresponding to the turning point. T is the area integration over the small triangle, which can be superimposed by the diagram of the limiting magnetic hysteresis loop:

$$T\left(\alpha ,\beta \right)=\frac{B_u\left(\alpha \right)-B_d\left(\beta \right)}{2}+F\left(\alpha \right)F\left(-\beta \right)$$

(5)

where

$$\{\begin{array}{c}F\left(\alpha \right)=\frac{B_d\left(\alpha \right)-B_u\left(\alpha \right)}{2\sqrt{B_d\left(\alpha \right)}} \left(\alpha \ge 0\right) \\ F\left(\alpha \right)=\sqrt{B_d\left(-\alpha \right)} \left(\alpha <0\right) \end{array}$$

(6)

where B_d(α) and B_u(α) respectively represent the magnetic flux density corresponding to the rising and falling branches of the limit magnetic hysteresis loop when the magnetic field is α [24,25].

2.2. Improved Preisach Model

The above introduces the numerical calculation formula of the classical Preisach model. The classical Preisach method is suitable for sinusoidal excitation, and it is convenient to combine with experiments. Because the high-order harmonics are included in PWM excitation, this phenomenon leads to a more complex relationship between the variation of the two quantities of magnetic flux density and magnetic field strength, as shown in Figure 3. Figure 4 shows the harmonic analysis in H for both excitations. The plotting sequence for Figure 3 is in Appendix A. It is calculated by Matlab based on the experimental results. Due to the complexity of PWM excitation, an improved model for PWM excitation needs to be proposed.

In fact, the principle of magnetic domain movement is very complicated. In the Preisach model, the results of domain magnetization can be simulated, but the specific internal process is ignored.

Consistent with the assumptions of the Preisach model, it includes the initial state and final state, but unlike the original model, the intermediate process is also included. In particular, when the external excitation is non-sinusoidal (such as in PWM excitation), the classical Preisach model will produce large model errors. Therefore, the correction coefficients should be introduced to compensate for the error caused by PWM excitation [26].

On this basis, the correction coefficients γ₁, γ₂, and γ₃ are introduced to complete the derivation of the formula. For the improved Preisach model, the expression of magnetic flux density B under different conditions is shown in Formulas (7)–(9).

On the initial magnetization curve, the expression of magnetic flux density is shown by Formula (7).

$$B\left(H\right)={\gamma }_1{\left[F(-H)-F(H)\right]}^2$$

(7)

Inside the limiting magnetic hysteresis loop, the magnetic field H decreases after the turning point:

$$B\left(H\right)=B\left(H_n\right)-2{\gamma }_{2}T\left(H_n,H\right)$$

(8)

The magnetic field H increases after the turning point:

$$B\left(H\right)=B\left(H_n\right)+2{\gamma }_{3}T\left(H,H_n\right)$$

(9)

Since the correction coefficients γ₁, γ₂, and γ₃ are unknown, we need to know the conditions to solve them. The correction coefficients are obtained by using the turning point magnetic field H_n1 and H_n2 measured in the experiment, and the solution formula is given by Formula (10). In fact, the experimental results can not only provide data for solving coefficients but also verify the results of the model. Therefore, the corresponding experimental platform for magnetic hysteresis of grain-oriented silicon steel sheet under PWM excitation needs to be designed.

$$\begin{array}{l}{\gamma }_1=\frac{B(H_{n1})}{{\left[F(-H)-F(H)\right]}^2}\\ {\gamma }_2={\gamma }_3=\frac{B(H_{n1})-B(H_{n2})}{2T(H_{n1},H_{n2})}\end{array}$$

(10)

The improved Preisach model is described by Equations (7)–(10). Theoretically, combined with the experimental setup (correcting errors in classical models with measurements from experimental setups), the magnetic hysteresis loop of any grain-oriented silicon steel sheet excited by PWM can be obtained.

During the magnetization process, magnetic materials meet two basic properties, namely the wiping-out property and congruence property. The wiping-out property is explained in Section 1. 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 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 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 α and β, 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, h(m) is inserted into the increasing minimum array based on its size, and the minimum value larger than the current voltage value is discarded to ensure that the current voltage value is at the end of the increasing minimum array. Then, the number of elements, n, in the increasing minimum array is obtained, and the first 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. In addition, this model still has the advantages of the Preisach model. Based on the values of the limiting magnetic hysteresis loop and turning point, it can simulate the changes of any local magnetic hysteresis curve. It has a wide application range and can better describe the local magnetic hysteresis properties of ferromagnetic materials under complex excitation. The simulation of the magnetic properties of grain-oriented silicon steel under PWM excitation has obvious advantages.

In the improved Preisach numerical implementation method, when the magnetic field intensity transitions between the rising and falling stages, the maximum and minimum values are stored in the maximum and minimum arrays, respectively. The improved Preisach numerical calculation method has memory due to the increasing array length; at the same time, during the storage process, the maximum values that are smaller than the present voltage value and the minimum values that are larger than the current voltage value are continuously cleared, thus granting the Preisach numerical implementation the wiping-out property.

3. Experimental Verification

3.1. Experimental Principles and Platform Construction

In the study of magnetic properties, the design of the experimental platform is the basic link. Nowadays, experimental measurement devices for magnetic properties continue to receive a lot of attention from scholars, and the design of the experimental platform for different working conditions is a key research area. Starting from the actual needs, the magnetic property measurement method of grain-oriented silicon steel can be roughly divided into the DC measurement method and AC measurement method, and each method can be divided into one-dimensional, two-dimensional, and three-dimensional measurement methods according to consideration of the anisotropy and measurement dimension. Due to the actual service of power transformers, more reflect the AC magnetic properties, and orientation of silicon steel according to rolling direction permeability is much more prominent than the non-rolling direction; that is, the magnetic circuit is mainly along the rolling direction. Therefore, this paper only considers a one-dimensional AC magnetic property measurement method. Presently, the international standard methods are the Epstein test method [6], the single sheet test method [7] and the ring-sample test method [8].

For experimental measurements using the ring-sample test method, the sample is made into a rectangular ring or two opposing “C”-shaped samples with a rectangular cross-section, as shown in Figure 6. Because of the different lengths of the inner and outer diameters of the ring samples, the lengths of the magnetic circuits are also different, so the values measured by the ring-sample test method are average values. The ring-sample test method also needs to use ring-shaped samples according to the actual situation of their own winding, applying the appropriate excitation signal, respectively, in the primary and secondary windings measured in the H and B signals, so as to obtain the magnetic hysteresis line under AC excitation and the various magnetic properties of the parameters.

Among the three methods, the ring-sample test method has the advantages of simple principle and convenient design, and it is suitable laboratory tests. The ring-sample test method can meet the experimental needs of this paper. Therefore, this paper is based on the principle of the ring-sample test method to carry out the design and construction of the magnetic characteristic platform applicable to grain-oriented silicon steel under PWM excitation conditions. In order to verify the accuracy of the improved model, the test principle in Figure 7 was used.

The sample tested in this article is a grain-oriented silicon steel sample (B27R085). Its structure diagram and related parameters are shown in Figure 8 and

Table 2. Sample size of grain-oriented silicon steel.

A	B	C	D	E	F	Area S	Magnetic Circuit l
18.0 mm	25.0 mm	84.0 mm	30.0 mm	62.0 mm	122.0 mm	5.4 cm2	264 mm

and

Table 3. Parameters of grain-oriented silicon steel samples.

Technical Parameters	Model/Value
Silicon steel grade	B27R085
Manufacturer	BAOSHAN IRON & STEEL Co., Ltd.
Expected Bs/T	1.5
Conductivity of silicon steel/S·m−1	2.22 × 106
Density of silicon steel/kg·dm−3	7.65
N1/N2	10
Area S/cm2	A × D
Stacking factor/%	97.5
Type of Coating	Inorganic
Hardness (HV1)	200

HV1: A type of Vickers hardness.

.

Before measurement, based on the relevant information about the saturation magnetic induction strength and corresponding magnetic field strength, and after estimation and calculation, 10 turns were selected as the winding turns of grain-oriented silicon steel and evenly wound on the ring sample. In order to reduce the impact of magnetic leakage during measurement on the measurement results, the primary excitation winding and the secondary induction winding were selected, with the same number of turns for winding. In order to solve the problems of excessive winding loss and high internal temperature during the measurement process, enameled wire was selected for winding.

The simulator used for the measurement system built in this article is the LabVIEW2018 programming platform developed by the NI Company in the Austin, TX, USA; thus, the data acquisition card uses the latest high-speed acquisition card NI-USB−6363, provided by the NI Company. In the graphical development environment of NI LabVIEW, there is no need to write lines of text code; rather, a data collection system is developed through drag and drop icons. The signals that need to be collected in the experiment include analog signals of the voltage at both ends of the sampling resistor connected in series with the primary winding (primary winding), used to generate the magnetic field strength H signal, and analog signals of the voltage at both ends of the secondary winding, used to calculate the magnetic induction strength B signal. Both B and H signals adopt the differential measurement mode, meaning that each signal requires two input channels for collection. The DAQ assistant module of NI LabVIEW can help users quickly initialize and set output and acquisition channels. Therefore, when using this platform, it is necessary to correctly set the input and output channels in the LabVIEW platform to complete the signal output and acquisition process correctly.

In addition, the functions of signal amplification and isolation are completed by a power amplifier and isolation amplifier, respectively, to reduce error and ensure safety. The role of the isolation amplifier can be simply understood as “isolation” and “amplification”. The “isolation” refers to electrical isolation of the sample to be measured from the host computer, to prevent abnormal signals or step signals of the sample from harming the host computer and the data acquisition card; “amplification” refers to the external signals collected to amplify a certain number of times in order to improve the accuracy of the experiment and reduce the experimental error. The power amplifier plays the role of power amplification of the excitation signal, and the waveform output from the output channel of the data acquisition card cannot be used directly for the excitation function but needs to be amplified by the power amplifier to form a stable excitation signal. The equipment models are shown in

Table 4. Physical parameters of grain-oriented silicon steel samples.

Equipment Name	Model Number
Isolation Amplifier	Nanjing Hongbin Weak Signal Inspection Co., Nanjing, China. HB-814A
Power Amplifier	Aigtek, Xi’an, China. ATA-308

. The experimental equipment diagram is shown in Figure 9.

It should be noted here that both the power amplifier and the isolation amplifier used in the experiment do not affect the results of the experiment, and their amplification of the input signal is linear. The role of the power amplifier is observed through an experiment: the power amplifier is set to ×10 the voltage amplification ratio, the input signal is an industrial frequency, the sinusoidal signal of amplitude is 1 V, and the signal at its output is monitored, as shown in Figure 10a. When the frequency of the input signal is 1.55 kHZ (the highest number of harmonics monitored in Figure 4 is 31), the output image is as shown in Figure 10b. It can be seen that the output signal is linearly amplified. This proves that the use of the amplifier does not affect the experimental results.

3.2. Experimental Principles and Platform Construction

The functions of a complete magnetic characteristic measurement system include the demagnetization function, excitation signal generation function, and data processing function, which are all realized by the graphical programming language of the LabVIEW platform.

3.2.1. Demagnetization Signal Generation

The first step of the experiment to measure the magnetic hysteresis line of grain-oriented silicon steel is demagnetization. The role of demagnetization is to ensure that the starting working point of the oriented silicon steel is located in the origin; if the starting working point is not fixed to the origin, it will lead to a large experimental measurement of the random error. Therefore, the sample should be demagnetized before performing the magnetic property measurement experiment. The demagnetization signal is generated by the host computer, which generates a sinusoidal attenuation signal and outputs it through the data acquisition card, so that the starting point of the magnetic property measurement is near the origin. The numerical expression for the demagnetization signal is sin(2πf_dt) × e^−mt, where f_d is the frequency of the demagnetization signal, t is the time, and m is the attenuation factor. The waveform of the demagnetization signal output is shown in Figure 11. During the experiment, it was found that a demagnetization effect less than 0.3 is effectively sufficient and greater than 0.3 is not effectively sufficient, so the attenuation coefficient can be set to within 0.3, as needed.

3.2.2. Excitation Signal Generation Function

In the actual working process of the power transformer, the common excitation signals are the sinusoidal signal, square wave signal, PWM signal, etc. In order to simulate the actual service process of the transformer, it is necessary to consider the generation of the above three signals. The sine signal can be generated directly by the LabVIEW sinusoidal signal control, as a square wave for each cycle of the duty cycle of a certain PWM wave. This paper focuses on the study of PWM signal excitation with regard to grain-oriented silicon steel specimens and the magnetic properties of the change rule. Therefore, this section only introduces the generation of PWM excitation in the measurement platform.

PWM is a very effective technique to achieve control of analog circuits by using the digital output of microprocessors, which is explained in detail in [29]. The carrier ratio is the ratio of the carrier signal frequency f_v to the modulated signal frequency f_s, and the mathematical expression is shown in Equation (11). Generally speaking, the frequency of the carrier signal in PWM technology is much higher than the modulated signal frequency; this paper, for the convenience of research, uses a fixed carrier ratio of 20.

$$N=f_v/f_s$$

(11)

Another important parameter in PWM technology is the modulation index (also known as the amplitude modulation ratio), defined as the ratio of the modulated waveform amplitude to the carrier waveform amplitude V_v, with the mathematical expression as shown in Equation (12). In practical engineering applications, K > 1 is over-modulation and K < 1 is under-modulation.

$$K=V_s/V_v$$

(12)

PWM wave can be divided into unipolar PWM and bipolar PWM; the different modulation methods will lead to differences between the two. In this paper, we use the simple and efficient modulation principle of bipolar PWM, which can be generated by inputting the modulated wave and carrier wave into the comparison function. When the modulation index changes, the generation process is as shown in Figure 12. The PWM excitation signal is generated by the host computer.

3.2.3. Data Acquisition and Processing

The main elements of data acquisition and processing are more complex and can be divided into the following three parts: acquisition and calculation of the magnetic induction and magnetic field strength signals, selection of the reference magnetic induction waveform, and waveform iteration of the output voltage.

(a)

Acquisition and calculation of magnetic induction and magnetic field strength signals

According to Ampère’s circuital law and Faraday’s Law of Electromagnetic Induction, the magnetic field strength H and magnetic induction strength B signals are calculated from the collected primary side current and secondary side voltage signals.

$$H=\frac{N_1{u}_1}{LR}$$

(13)

$$B=\frac{\varphi }{S}=\frac{1}{N_{2}S}{\displaystyle \int u_{2}dt}$$

(14)

where N₁ is the number of turns of the excitation winding; N₂ is the number of turns of the measurement winding; ϕ is the magnetic flux in the sample; S is the cross-section of the sample magnetic circuit for the product; u₁ is the primary voltage; u₂ is the secondary side of the induced electromotive force; L is the length of the magnetic circuit; and R is the primary side of the resistance value of the series resistance. It can be seen that u₁ and u₂ in the formula are the voltage signals to be collected, and after the data acquisition card is set up, AI0 and AI1 are used as the data acquisition channels to collect u₁ and u₂, respectively, so that the data can be processed in the host computer.

(b)

Reference Magnetic Induction Waveform Selection

In order to ensure the accuracy of measurement, it is critical to select the reference magnetic flux density waveform. In the process of determining the reference waveform, it is necessary to determine the proportion of harmonics in the reference magnetic induction intensity waveform. The harmonic proportion properties of the reference waveform are related to the excitation properties of PWM. Compared with the fundamental wave, the impedance of higher-order harmonics is relatively large; therefore, the increase in the amplitude of the fundamental component of the power supply is the main factor leading to saturation of the tested sample. To avoid the effect of core saturation, in the experiments under PWM excitation, the reference waveform of magnetic induction intensity is selected, with the fundamental waveform as a specified sinusoidal waveform, and the proportions of each harmonic component are kept the same as the proportions of each harmonic component to the fundamental waveform in the case where B is not saturated. The harmonics are selected to be extracted and superimposed to finally obtain the reference magnetic induction intensity waveform. The reference waveform B_ref can be expressed in the form of Equation (15).

$$B_{\mathrm{ref}}\left(t\right)={\displaystyle \sum _{i=1}^n{B}_n}\cos \left(2\pi nft+{\phi }_n\right)$$

(15)

where n is the number of harmonics selected for superposition (in this paper the first 30 harmonics are selected for superposition), f is the fundamental frequency of the magnetic induction intensity, and φ_n is the phase angle value of the nth harmonic of the magnetic induction intensity.

(c)

Waveform Iteration of Output Voltage

In this paper, in the PWM excitation magnetic characteristic measurement experiment, the waveform selected for control is the waveform of magnetic induction strength B. Since the NI data acquisition card not only completes the output of the PWM excitation signal but also realizes the acquisition function of the B and H signals, the whole system can form a feedback mode; thus, the closed-loop control system is selected for the control of the waveform iteration.

In order to measure the magnetism of silicon steel sheet under PWM excitation, it is necessary to iterate the measured waveform until it is close to the reference waveform. The method can be expressed as follows:

$$u_1^{\left(n+1\right)}=u_1^{\left(n\right)}+\frac{k\left(B_{ref}-B_{{}_{mea}}^{\left(n\right)}\right)}{B_{ref}}\left|u_1^{\left(n\right)}\right|$$

(16)

where u₁ is the primary side voltage of the sample, B_ref is the reference flux density waveform, B_mea is the measured flux density waveform, n is the number of iterations, and k is the iteration coefficient.

During the experiment, the voltage waveform is iterated according to the process in Figure 13, and the waveform after iteration will no longer be the standard PWM waveform; the difference between the PWM waveform before iterating the voltage and after iterating the voltage is shown in Figure 14. From the figure, it can be seen that compared to the PWM waveform before iteration, the PWM waveform after iteration introduces higher harmonics, which makes the controlled magnetic induction intensity waveform consistent with the selected reference waveform, which is one of the key tasks to accomplish accurate measurement.

At this point, after the hardware design of the magnetic characterization experimental platform and the implementation of the functions of the modules in LabVIEW, all the preparatory work for the magnetic hysteresis loop measurement has been completed. Before the start of each experiment, it is necessary to run the demagnetization program, so that the initial working point of the grain-oriented silicon steel specimen is near the origin. Then, the appropriate PWM excitation waveform is set in the host computer, and after running the program and outputting it, the magnetic hysteresis loop obtained from the measurement can be displayed in the host computer.

4. Results and Discussion

By modeling with the improved model, a range of γ values were obtained. Figure 15 exhibits the trend of γ as H and K vary. It can be observed from Figure 15 that γ is directly proportional to H and inversely proportional to K over the experimental range.

With the help of the experimental platform, the magnetic hysteresis loop measurement results are obtained. Furthermore, the classical model and the improved model are used to model, and the magnetic hysteresis loops can be obtained. The results are compared and analyzed as follows. Figure 16, Figure 17, Figure 18 and Figure 19 show the measurement results of bipolar PWM excitation with modulation ratios of 1.3, 1.1, 0.9, and 0.7 (K = 1.3, 1.1, 0.9, and 0.7) respectively, as well as a fundamental frequency of 50 Hz and 11 pulses, and give the comparison of modeling results before and after improvement, where the blue line is the experimental value of B and H, the orange line is the modeling value of B and H. The plotting sequence for Figure 16, Figure 17, Figure 18 and Figure 19 is in Appendix A.

Regarding the calculation of the curves, a period of 0.02 s is taken for a period of 2000 points. That is, the integration interval is 10⁻⁵ s. In order to compare the effects of the two methods, the results of the above two situations were analyzed.

Table 5. Residual and saturated magnetic flux error comparison between the two models.

Modulation Ratio K	Method	Br/T	eBr%	Bm/T	eBm%
K = 1.5	Experimental results	1.087	/	1.479	/
	Classical Preisach model	1.119	2.94	1.548	4.67
	Improved Preisach model	1.103	1.472	1.514	2.366
K = 1.3	Experimental results	1.115	/	1.456	/
	Classical Preisach model	1.149	3.050	1.524	4.67
	Improved Preisach model	1.132	1.525	1.491	2.404
K = 1.1	Experimental results	1.171	/	1.415	/
	Classical Preisach model	1.228	4.867	1.481	4.664
	Improved Preisach model	1.183	1.024	1.448	2.332
K = 0.9	Experimental results	1.185	/	1.365	/
	Classical Preisach model	1.251	5.495	1.524	11.60
	Improved Preisach model	1.201	1.332	1.403	2.564
K = 0.7	Experimental results	1.216	/	1.359	/
	Classical Preisach model	1.148	5.592	1.427	11.11
	Improved Preisach model	1.204	1.002	1.397	2.796

shows the values and errors of residual and saturated magnetic induction intensities under different modulation ratios. It can be seen that when the excitation is PWM, the error of the improved model is much smaller than that of the conventional model. The improved model is 1.4% to 9% more accurate than the classical model.

The error of the improved model and the classical model changes with the change of K. Figure 20 shows the change in error of both models for different values of K. The reference value is the corresponding experimental result as stated in Table 5. It shows that the error of the improved model is always less than the traditional model, i.e., the improved model is superior. The error values of the two models fluctuate with the change of K. When K > 1.3, the change of error of the two models tends to stabilize.

From the point of view of numerical calculation, the improved Preisach model combines the data from the magnetic property experiments to ensure the accuracy of the turning point of each local loop, which makes the magnetic property curves obtained by this method of modeling closer to the real values. From the analysis of the properties of the Preisach model itself, in the simulation of the classical Preisach model, the initial magnetization curve segment has a large deviation from the experimental measurements, and there is also a certain error in the rising and falling segments inside the limiting magnetic hysteresis loop. The reason lies in the fact that the basic theory of the adopted classical Preisach model is to equate the magnetic domains to the magnetic hysteresis operator, which has only step properties and does not reflect the intermediate states of the magnetization process in the model. Therefore, when the external excitation signal is complex, it will produce a large error. In contrast, the improved Preisach model adds correction coefficients to offset the process not embodied in the model. In addition, the model is improved from the original method but still has the advantages of the classical method, which can simulate the change of the internal magnetic hysteresis loop by only measuring the values of the limiting magnetic hysteresis loop and the turning point, and its simulation of the magnetic properties of the grain-oriented silicon steel under the PWM excitation has obvious advantages.

In future work, the model will be further improved by combining the physical significance of magnetic domain science and the change of the Preisach plane with finite element calculations of power transformers and designing experiments to verify the calculation results. The universality of the experimental platform will be increased by measuring different working conditions, including many different sizes of grain-oriented silicon steel or more grades of ferromagnetic materials; at the same time, more flexible waveform control programs will be designed, and the convenience of experimental software development will be increased. Measurement of multiple hysteresis loop data will be accomplished, and a hysteresis model with prediction function will be realized.

5. Conclusions

In this paper, an improved magnetic hysteresis model for PWM excitation is presented, which can consider the local magnetic hysteresis loop and calculate the magnetic flux density waveform. On this basis, a grain-oriented silicon steel sheet measurement platform is designed. According to the experimental data, the accuracy of the magnetic hysteresis model is verified. By comparing the two methods, we find that the improved model error is lower than the classical model, that is, the improved magnetic hysteresis modeling result is more accurate.