Nonlinear permanent magnet hysteresis loop description method based on magnetic domain energy minimization

Abstract

The invention discloses a nonlinear permanent magnet hysteresis loop description method based on magnetic domain energy minimization. The method comprises the following steps: step 1, forming a state equation by using total energy obtained by adding magnetic domain energy of six magnetic domains along easy axis directions of different crystals in permanent magnet crystal grains; 2, the minimum value of magnetic domain energy is solved, and then the optimal direction of each magnetic domain unit magnetization vector in the crystal grains is obtained; and step 3, describing a magnetic domain occurrence probability by using a Boltzmann function, and determining a Preisach model distribution function by using a probability density function so as to obtain a macroscopic hysteresis characteristic. According to the method, micro-magnetic microcosmic characterization and macroscopic magnetic performance are associated from a micromechanism, a magnetization curve of a nonlinear permanent magnet can be accurately established, and the method has an important supporting effect on research work such as electromagnetic multi-physics field simulation and the like.

Description

Technical Field

[0001] The present invention belongs to the field of nonlinear permanent magnet magnetization characteristics, and relates to a nonlinear permanent magnet hysteresis loop description method, and in particular to a nonlinear permanent magnet hysteresis loop description method based on magnetic domain energy minimization. Background Art

[0002] The fabrication process for permanent magnets involves uneven microscopic grain diffusion, and the cold extrusion of areas near the cut surface during processing can have significant side effects. Furthermore, the overall magnetization method for nonlinear permanent magnets in electrical appliances is subject to the influence of a complex soft magnetic environment, such as contact and proximity between magnetic materials due to the magnetic circuit structure. This results in localized variations in the magnetization effects of permanent magnet magnetization, demagnetization, and magnetic stabilization techniques. These multi-dimensional influencing factors exacerbate the complexity of permanent magnet magnetization characteristics. Therefore, it is crucial to establish a bridge between microscopic magnetic characterization and macroscopic magnetic properties, starting from microscopic mechanisms, and accurately constructing nonlinear permanent magnet magnetization curves.

[0003] Determining the distribution function of the Preisach classical model has always been a focus of research by scholars both domestically and internationally. Currently, there are two main methods for determining it. The first is to calculate the Everett function based on experimentally measured data to obtain the model output value, and the second is to use an analytical function to approximate the distribution function. The first method generally uses measured first-order rotation curve data to calculate the Everett function. However, first-order rotation curve data is difficult to measure, and the calculation process requires derivative operations, which can easily lead to large errors. In contrast, the second method uses an analytical function to directly approximate the distribution function, which is relatively simple to implement. Commonly used function forms include Gaussian functions, Lorentz functions, super-Lorentz functions, and Cauchy functions. However, the disadvantage of this method is that the process of determining the unknown parameters in the function is relatively cumbersome and consumes a lot of computing resources. Therefore, there is still room for improvement in the process of determining the distribution function of the Preisach model. Summary of the Invention

[0004] The present invention provides a method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy. Starting from the microscopic mechanism, this method links the microscopic characterization of micromagnetism with the macroscopic magnetic properties, and can accurately establish the magnetization curve of the nonlinear permanent magnet. It plays an important supporting role in research work such as electromagnetic multi-physics field simulation.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] A method for describing a nonlinear permanent magnet hysteresis loop based on magnetic domain energy minimization comprises the following steps:

[0007] Step 1, using the total energy obtained by adding the magnetic domain energies of six magnetic domains along different crystal easy axes in the permanent magnet grains to form a state equation, and determining the parameter representation of the saturation magnetization, demagnetization factor, magnetocrystalline anisotropy constant, initial magnetic susceptibility, and free energy of the magnetic domain at the mesoscopic scale in the Preisach model;

[0008] Step 2: Calculate the minimum value of the magnetic domain energy, and then obtain the optimal direction of the unit magnetization vector of each magnetic domain in the grain;

[0009] Step 3: Use the Boltzmann function to describe the probability of occurrence of magnetic domains, determine the Preisach model distribution function from the probability density function, and then obtain the macroscopic hysteresis characteristics.

[0010] Compared with the prior art, the present invention has the following advantages:

[0011] The present invention analyzes and describes the magnetization process of AlNiCo permanent magnets from a microscopic perspective. The material model is more refined, can meet the requirements of precise optimization design, has a wider range of applications, and is convenient for accurately modeling electromagnetic systems containing nonlinear permanent magnets. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 Flowchart for establishing a nonlinear permanent magnet hysteresis loop model based on magnetic domain energy minimization;

[0013] Figure 2 It is a curve diagram of energy changes in the process of calculating the minimum value of total energy;

[0014] Figure 3 This is the r change curve during the calculation of the total energy minimum;

[0015] Figure 4 The diagram of the fitting process of the dominant magnetic domain energy term in order to obtain the analytical expression of the probability of the dominant magnetic domain appearing;

[0016] Figure 5 The diagram of the fitting process of the energy terms of the remaining magnetic domains in order to obtain the analytical expression of the probability of the dominant magnetic domain appearing;

[0017] Figure 6 Schematic diagram of the Preisach distribution function obtained after the parameters of the analytical expression for the probability of the dominant magnetic domain appearing are determined;

[0018] Figure 7 The figure is a comparison chart of the results obtained by fitting the model with the actual magnetization curve of a certain permanent magnet 1;

[0019] Figure 8 The following is a comparison chart of the results obtained by fitting this model with two actual magnetization curves of a certain permanent magnet 2;

[0020] Figure 9 For use Figure 8 Corresponding model, demagnetization curves and excitation signal diagrams under different remanence obtained by cyclically charging and demagnetizing the permanent magnet. DETAILED DESCRIPTION

[0021] The technical solution of the present invention is further described below with reference to the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention that does not depart from the spirit and scope of the technical solution of the present invention should be included in the scope of protection of the present invention.

[0022] The present invention provides a method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy, such as Figure 1 As shown, the method includes the following steps:

[0023] Step 1: Use the total energy obtained by adding the magnetic domain energies of the six magnetic domains (six magnetic domains along different crystal easy axis directions) in the permanent magnet grains to form a state equation, and determine the parameter representation of the saturation magnetization, demagnetization factor, magnetocrystalline anisotropy constant, initial magnetic susceptibility, and free energy of the magnetic domain at the mesoscopic scale in the Preisach model. The specific steps are as follows:

[0024] Step 1-1: In ADSM, the unit magnetization vector m of the i-th magnetic domain i (i.e. the direction of the magnetic moment of the magnetic domain) can be expressed as:

[0025] (1)

[0026] Where: θ i is the angle between the unit magnetization vector and the crystal easy axis

[001] ; φ i is the angle between the projection of the unit magnetization vector on the (110) plane and the crystal easy axis

[010] ; i=1,...,6.

[0027] Step 1-2: The free energy of each magnetic domain at the mesoscopic scale can be obtained by adding the four magnetic domain energies given by formula (2): Zeeman energy, magnetocrystalline anisotropy, static magnetoenergy, and hysteresis energy:

[0028] (2)

[0029] Where: W total is the total free energy; W zeeman is Zeeman energy; W an is the magnetocrystalline anisotropy energy; W st is the static magnetoelectric energy; W h is the hysteresis energy.

[0030] Formula (3) is the static magnetic energy considering the change of magnetic domain area The energy is a necessary condition for the formation of magnetic domains:

[0031] (3)

[0032] Formula (4) is the magnetocrystalline anisotropy The energy makes the magnetic domain moment align toward the easy axis of the crystal:

[0033] (4)

[0034] Formula (5) is the Zeeman energy Expression, which tends to align the magnetic domain moments with the direction of the external magnetic field:

[0035] (5)

[0036] Formula (6) is the hysteresis energy Expression, which is the energy introduced to describe the hysteresis phenomenon:

[0037] (6)

[0038] In the above expressions: μ0 is the vacuum permeability; M s is the saturation magnetization; k x 、k y and k z is the demagnetization factor; (m x , m y , m z ) is the sum of the unit magnetization vectors of the magnetic domain; α1, α2, α3 represent the components of the unit magnetization vector of the magnetic domain on the easy axis of the crystal; K is the magnetocrystalline anisotropy constant; χ0 is the initial magnetic susceptibility; (m hx , m hy , m hz ) is the unit magnetization vector sum corresponding to the previous moment.

[0039] Step 2: Calculate the minimum value of the magnetic domain energy, and then obtain the optimal direction of the unit magnetization vector of each magnetic domain in the grain. The specific steps are as follows:

[0040] Step 2-1: Obtain the minimum value of the total energy of the magnetic domain through the genetic algorithm, and obtain the θ corresponding to the unit magnetization vector of the i-th magnetic domain in the grain. i ,φ i .

[0041] Step 2-2: The total magnetic domain energy involved in solving the energy minimum is the sum of the energies of the six magnetic domains in a grain, as shown in Equation (7):

[0042] (7)

[0043] Where: W total is the total free energy, that is, the sum of the energies of the six magnetic domains, Wzeeman,i is the Zeeman energy of the i-th magnetic domain; W an,i is the magnetocrystalline anisotropy energy of the i-th magnetic domain; W st,i is the magnetostatic energy of the i-th magnetic domain; W h,i is the hysteresis energy of the i-th magnetic domain.

[0044] Step 3: Use the Boltzmann function to determine the probability of occurrence of magnetic domains. The unit magnetization vectors and occurrence probabilities of the six magnetic domains are used to determine the Preisach model distribution function from the probability density function, and then the macroscopic hysteresis characteristics are obtained. The specific steps are as follows:

[0045] Step 3-1: Obtain θ through the magnetic domain energy minimization process i ,φ i 12 unknown variables. On this basis, starting from the total energy of each magnetic domain in the grain, the probability of each magnetic domain appearing in the grain is expressed by the Boltzmann function. The probability of each magnetic domain appearing r i The expression is shown in formula (8), and it must satisfy Constraints:

[0046] (8)

[0047] Where: A s Used to measure the regularity of the crystal, the expression is as follows:

[0048]

[0049] Step 3-2: Select one of the magnetic domains as the dominant magnetic domain. According to the size of the dominant magnetic domain in the magnetization direction, use its occurrence probability r or 1-r as the reference of the Preisach model distribution function. First, fit the discrete r or 1-r. Then, transform the two-dimensional r or 1-r into a three-dimensional function in the rotating coordinate system, that is, the distribution function required by the Preisach model, where p is the offset of the center point of the distribution function, which is used to adjust the squareness of the hysteresis loop. After obtaining the distribution function required by the Preisach model, the hysteresis loop of this permanent magnet can be obtained.

[0050] Example:

[0051] This embodiment takes the 759C relay as an example, in which the permanent magnet used is AlNiCo. AlNiCo alloys are high-performance magnets with preferred grain orientation. The magnetic structure of cast AlNiCo magnets is affected by their crystal structure. <100> For easy magnetization direction, the developed <100> Crystal texture (columnar crystals) gives magnets improved magnetic properties. Commercial magnets are classified as partially oriented and fully oriented based on the degree of perfection of directional crystallization. Both high remanence and high coercivity AlNiCo alloys can be made into directional crystallized magnets.

[0052] Alnico is a cubic crystal, and its three cubic axes are all easy magnetization directions. Therefore, the three easy axes are taken as the three coordinate axes in the three-dimensional coordinate system, and the components of the unit magnetization vector of the magnetic domain on the easy axis of the crystal α1, α2, and α3 are also the components of the magnetization vector on the coordinate axis. The anisotropy constants of equiaxed Alnico, semi-columnar Alnico, and columnar Alnico are 304×10 5 ergcm -3 , 297×10 5 ergcm -3 ,322×10 5 ergcm -3 Here we take columnar AlNiCo as an example. The vacuum magnetic permeability μ0 is 4π×10 -7 N.A. -2 The formula for calculating the demagnetization factor of macroscopic materials is:

[0053]

[0054]

[0055]

[0056] Where a, b, and c are the length, width, and height of the macro material.

[0057] The calculation formula of initial magnetic susceptibility χ0 is:

[0058]

[0059] Substituting the above parameters into the total energy expression of the magnetic domains, the genetic algorithm is used to solve the minimum total energy and obtain the unit magnetization vectors corresponding to the six magnetic domains. The Boltzmann function is then used to solve the corresponding occurrence probability of each magnetic domain. The magnetization vector of each magnetic domain is multiplied by the corresponding occurrence probability and the saturation magnetization intensity and then added together to obtain the magnetization intensity under the current magnetic field intensity. By inputting an excitation signal with a changing magnetic field intensity, the corresponding MH curve can be obtained.

[0060] The excitation signal of the magnetic field strength is set to: first from Hmax to -Hmax, then from -Hmax to Hmax, forming a complete limiting hysteresis loop. The energy of each item in the expression of the total energy of the magnetic domain, the curve of r changing with h is as follows Figure 2 and Figure 3 shown.

[0061] Throughout the magnetization process, each change in the direction of the magnetization intensity often results in a gradual increase in the probability of one of the six magnetic domains appearing. This means that this domain gradually becomes dominant, and the direction of the magnetization intensity becomes the direction of its magnetic moment. Meanwhile, the probabilities of the other five domains appearing are equal. In some cases, two dominant magnetic domains with the same orientation appear during the calculation process, while the probabilities of the other four domains appearing are equal. In these cases, the two dominant magnetic domains can be considered to be the same domain. Therefore, the magnetization process along one coordinate axis can be considered the relative change in the probabilities of the appearance of two magnetic domains with opposite orientations. This means that the Boltzmann function directly reflects the change in magnetization intensity.

[0062] Using the above method, parameters such as the magnetocrystalline anisotropy constant are determined, the state equation of the total energy of the magnetic domain is listed, the minimum value of the magnetic domain energy is obtained, and then the corresponding energy terms are used to calculate the probability of occurrence of each magnetic domain. The probability of occurrence of the dominant magnetic domain under different magnetic field intensities is obtained through observation.

[0063] Due to the existence of various anisotropic energy terms in the total energy state equation, the direction of the magnetization vector of the magnetic domain can only be one of the six directions of the positive and negative directions of the three coordinate axes. If the magnetization vector of the dominant magnetic domain is on the magnetization direction coordinate axis, its occurrence probability directly reflects the change law of the magnetization direction and is directly used in subsequent calculations; if the component of the magnetization vector of the dominant magnetic domain on the magnetization direction coordinate axis is 0, the corresponding value used in subsequent calculations is 1-r.

[0064] For ease of use and stability considerations, it is necessary to fit the discrete r obtained in the previous step. The calculation results show that the probability of occurrence of the five magnetic domains other than the dominant magnetic domain is equal. According to Equation (8), as long as the three energy terms (Zeeman energy, magnetocrystalline anisotropy energy, and static magnetostatic energy) of the dominant magnetic domain and the other magnetic domains are obtained, r can be directly calculated. Therefore, it is only necessary to fit the sum of the three energy terms of the two magnetic domains.

[0065] The rising and falling branches of the hysteresis loop are symmetrical. Here, we only need to calculate one of them. Take the part where the probability of the dominant magnetic domain appears from 0 to 1, and use the quadratic function ax 2 +bx+c and the linear function dx+g fit the three energy terms of the dominant magnetic domain and other magnetic domains. The fitting results are as follows Figure 4 , Figure 5 As shown, it can be seen that the fit is very good, and five coefficients a, b, c, d, and g are extracted.

[0066] Express r using analytical expression α ,get:

[0067]

[0068] In order to further simplify the model, the above formula is transformed into the final form:

[0069]

[0070] in:

[0071]

[0072]

[0073]

[0074] In this calculation, the component of the dominant domain magnetization vector on the magnetization direction coordinate axis is 0, so its occurrence probability is opposite to the change law of the magnetization direction, and the value actually used in the distribution function is 1-r.

[0075] After reversing r, the image is rotated along the vertical axis to become a three-dimensional coordinate image. Correspondingly, x in the formula becomes ,in p is the center point offset of the distribution function, which is used to adjust the square of the hysteresis loop to obtain the distribution function required by the Preisach model, such as Figure 6 shown.

[0076] In actual use, you only need to make sure as well as p These four parameters can be used to obtain the BH curve of a permanent magnet. When the limiting hysteresis loop of a permanent magnet is known, the model is first fitted with the actual data through the simplex search algorithm (Nelder-Mead method) to obtain as well as p These four parameters can be used to obtain the BH curve under different magnetic field excitations and the local hysteresis loop.

[0077] Figure 7 and Figure 8 The results of fitting the model with the actual magnetization curve are shown in Figure 2. The fitting is very good. Figure 8 Corresponding model, the permanent magnet is cyclically charged and demagnetized, and the demagnetization curves under different remanence can be obtained, such as Figure 9 shown.

[0078] In summary, the nonlinear permanent magnet hysteresis loop description method based on magnetic domain energy minimization provided by the present invention can accurately establish the magnetization curve of the nonlinear permanent magnet, which plays an important supporting role in research work such as electromagnetic multi-physics field simulation.

Claims

1. A method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy, characterized in that The method comprises the following steps: Step 1, using the total energy obtained by adding the magnetic domain energies of six magnetic domains along different crystal easy axes in the permanent magnet grains to form a state equation, and determining the parameter representation of the saturation magnetization, demagnetization factor, magnetocrystalline anisotropy constant, initial magnetic susceptibility, and free energy of the magnetic domain at the mesoscopic scale in the Preisach model; Step 2: Calculate the minimum value of the magnetic domain energy, and then obtain the optimal direction of the unit magnetization vector of each magnetic domain in the grain; Step 3: Use the Boltzmann function to describe the probability of occurrence of magnetic domains, determine the Preisach model distribution function from the probability density function, and then obtain the macroscopic hysteresis characteristics.

2. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 1, characterized in that The specific steps of step 1 are as follows: Step 1-1: In ADSM, the unit magnetization vector m of the i-th magnetic domain i Expressed as: Where: θ i is the angle between the unit magnetization vector and the crystal easy axis [001]; φ i is the angle between the projection of the unit magnetization vector on the (110) plane and the crystal easy axis [010]; i=1,...,6; Step 1-2: The free energy of each magnetic domain at the mesoscopic scale is obtained by adding the four magnetic domain energies: Zeeman energy, magnetocrystalline anisotropy, magnetostatic energy, and hysteresis energy: Where: W total is the total free energy; W zeeman is Zeeman energy; W an is the magnetocrystalline anisotropy energy; W st is the static magnetoelectric energy; W h is the hysteresis energy.

3. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 2, characterized in that The expressions for magnetostatic energy, magnetocrystalline anisotropy, Zeeman energy, and hysteresis energy are as follows: Where: μ0 is the magnetic permeability of vacuum; M s is the saturation magnetization; k x 、k y and k z is the demagnetization factor; (m x , m y , m z ) is the sum of the unit magnetization vectors of the magnetic domain; α1, α2, α3 represent the components of the unit magnetization vector of the magnetic domain on the easy axis of the crystal; K is the magnetocrystalline anisotropy constant; χ0 is the initial magnetic susceptibility; (m hx , m hy , m hz ) is the unit magnetization vector sum corresponding to the previous moment.

4. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 2, characterized in that The specific steps of step 2 are as follows: Step 2-1: Obtain the minimum value of the total energy of the magnetic domain through the genetic algorithm, and obtain the θ corresponding to the unit magnetization vector of the i-th magnetic domain in the grain. i ,φ i ; Step 2-2: The total magnetic domain energy involved in solving the energy minimum is the sum of the energies of the six magnetic domains in a grain, expressed as follows: Where: W total is the total free energy, W zeeman,i is the Zeeman energy of the i-th magnetic domain; W an,i is the magnetocrystalline anisotropy energy of the i-th magnetic domain; W st,i is the magnetostatic energy of the i-th magnetic domain; W h,i is the hysteresis energy of the i-th magnetic domain; i=1,...,6.

5. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 4, characterized in that The specific steps of step 3 are as follows: Step 3-1: Obtain θ through the magnetic domain energy minimization process i ,φ i 12 unknown variables. On this basis, starting from the total energy of each magnetic domain in the grain, the probability of each magnetic domain appearing in the grain is expressed by the Boltzmann function. The probability of each magnetic domain appearing r i : Where: A s Used to measure crystal regularity; Step 3-2: Select one of the magnetic domains as the dominant magnetic domain. According to the size of the dominant magnetic domain in the magnetization direction, use its occurrence probability r or 1-r as the reference of the Preisach model distribution function. First, fit the discrete r or 1-r. Then, transform the two-dimensional r or 1-r into a three-dimensional function in the rotating coordinate system, that is, the distribution function required by the Preisach model, where p is the offset of the center point of the distribution function, which is used to adjust the squareness of the hysteresis loop. After obtaining the distribution function required by the Preisach model, the hysteresis loop of this permanent magnet can be obtained.

6. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 5, characterized in that In the step 3-1, .

7. The method for describing the hysteresis loop of a nonlinear permanent magnet based on minimization of magnetic domain energy according to claim 5, characterized in that In step 3-1, the expression of As is as follows: 。

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