A method for calculating square wave hysteresis loop considering the influence of high-frequency harmonic components

Abstract

This invention relates to square wave hysteresis loop analysis and calculation techniques, and in particular discloses a method for calculating square wave hysteresis loops considering the influence of high-frequency harmonic components. The method includes the following steps: Step 1, analyzing the analytical relationship between average magnetic flux density and magnetic field strength at different excitation frequencies, and using complex permeability to characterize the magnetization properties of nanocrystalline materials at different frequencies; Step 2, establishing time-domain analytical expressions for eddy current magnetic fields under sinusoidal excitation at different frequencies; Step 3, establishing a square wave dynamic hysteresis loop prediction model based solely on sinusoidal measurement data using Fourier analysis. The beneficial effect of this invention is that it can effectively improve the prediction accuracy of dynamic hysteresis loops under high-frequency sinusoidal and square wave excitation.

Description

Technical Field

[0001] This invention relates to the field of hysteresis loop calculation, and in particular to a method for calculating square wave hysteresis loop considering the influence of high-frequency harmonic components. Background Technology

[0002] With the rapid development of power electronics technology and the needs of smart grids and the energy internet, nanocrystalline materials, with their advantages of high permeability, high flux density, and low coercivity, are widely used in power electronic transformers. The high-frequency operation of transformers helps to increase their power density and effectively reduce their size, thus meeting the requirements of miniaturization and lightweighting of various electronic devices. However, changes in excitation frequency and waveform significantly affect the dynamic hysteresis characteristics of nanocrystalline materials, which play a decisive role in other characteristics such as power loss and saturation flux density of soft magnetic materials. Therefore, accurate prediction of the dynamic hysteresis loop of nanocrystalline materials under high-frequency sinusoidal and square wave excitation is crucial for the macroscopic electromagnetic characteristic characterization, magnetic loss calculation, and heat analysis of electromagnetic devices.

[0003] To date, hysteresis loop prediction models proposed by scholars both domestically and internationally can be broadly classified into two categories: (1) numerical hysteresis loop prediction models; and (2) analytical hysteresis loop prediction models. Among these, numerical hysteresis loop prediction models couple the hysteresis model with the electromagnetic field diffusion equation, employing numerical calculation methods such as the finite element method or finite difference method to simulate the hysteresis loop of ferromagnetic materials, achieving high simulation accuracy. However, this method consumes significant computation time and storage space during operation and suffers from non-convergence of calculation results. Compared to numerical methods, analytical hysteresis loop prediction models are mostly based on field separation theory, independently calculating and superimposing the hysteresis effect, eddy current effect, and residual effect to predict the hysteresis loop of soft magnetic materials. This type of prediction model is highly convenient and practical, and is currently the most studied type of hysteresis loop prediction model.

[0004] Analytical hysteresis models are mainly divided into Preisach models, JA models, and Energetic models. To make these hysteresis models applicable to the simulation of hysteresis loops of soft magnetic materials under different operating conditions, domestic and international experts and scholars have conducted extensive research. Addressing the issue that traditional Energetic hysteresis models are only suitable for hysteresis loop simulation under quasi-static conditions, those skilled in the art have considered the influence of dynamic eddy currents and residual losses of magnetic materials under AC excitation on the hysteresis loop, establishing a novel analytical dynamic Energetic model based on field separation technology and loss statistics theory, realizing the prediction of hysteresis loops of silicon steel sheets under harmonic conditions. Those skilled in the art have combined simulated annealing with the Levenberg-Marquardt intelligent algorithm to propose a new global optimization algorithm, achieving accurate and rapid identification of hysteresis model parameters. Those skilled in the art have also combined the Preisach and JA hysteresis models with loss separation theory to solve for the losses of silicon steel sheets under DC bias conditions, achieving good computational accuracy. Based on loss separation theory, hysteresis models can also be combined with circuit models to establish hysteresis loop prediction models under square wave and rectangular wave excitation. Furthermore, those skilled in the art have applied fractional-order theory to eddy current magnetic field modeling, achieving dynamic hysteresis loop simulation of ultrathin silicon steel under high-frequency sinusoidal excitation. Compared to integer-order theory, fractional-order theory possesses the characteristics of time memory and global correlation, giving it unique advantages in modeling high-frequency eddy current magnetic fields.

[0005] In summary, existing models primarily simulate the hysteresis loop of silicon steel sheets under sinusoidal, harmonic, and DC bias conditions, while research on hysteresis loop prediction models for nanocrystalline materials under high-frequency sinusoidal and square wave excitation is limited. Furthermore, most dynamic hysteresis loop prediction models are established under the assumption of uniform magnetic flux density distribution within ferromagnetic materials, thus failing to account for the influence of high-frequency harmonic components in square waves on the dynamic hysteresis loop. Therefore, the classical dynamic hysteresis loop modeling method exhibits poor prediction accuracy under high-frequency sinusoidal and square wave excitation. Summary of the Invention

[0006] The purpose of this invention is to solve the above problems by designing a method for calculating square wave hysteresis loops that takes into account the influence of high-frequency harmonic components.

[0007] The technical solution of the present invention to achieve the above objectives is a method for calculating square wave hysteresis loops considering the influence of high-frequency harmonic components, the method comprising the following steps:

[0008] Step 1: Analyze the analytical relationship between average magnetic flux density and magnetic field strength at different excitation frequencies, and use the ratio of average magnetic flux density to magnetic field strength as the relative complex permeability to characterize the magnetization characteristics of nanocrystalline materials at different frequencies.

[0009] Step 2: Combine the elliptic equivalence theory with the eddy current loss calculation formula considering the high-frequency skin effect to establish the time-domain analytical expression of the eddy current magnetic field under different frequency excitations.

[0010] Step 3: Analyze the influence of high-frequency harmonic components in square wave excitation on the dynamic hysteresis loop. Fourier analysis is used to establish a prediction model for the dynamic hysteresis loop under square wave excitation, thereby enabling the prediction of the dynamic hysteresis loop under square wave excitation based solely on sinusoidal measurement data.

[0011] The process of analyzing the analytical relationship between average magnetic flux density and magnetic field strength at different excitation frequencies in step one is as follows:

[0012] The relationship between magnetic flux density and magnetic field strength inside ferromagnetic materials is characterized using the analytical expression of the static Energetic hysteresis model:

[0013] (1)

[0014] (2)

[0015] (3)

[0016] Equation (1) characterizes the linear characteristics of ferromagnetic materials; Equation (2) characterizes the nonlinear characteristics of materials; Equation (3) describes the hysteresis effect of materials; where, H This represents the actual magnetic field strength of the ferromagnetic material. H d The demagnetizing field strength, H r The strength of the reversible magnetic field. H i The strength of the irreversible magnetic field;

[0017] The magnetic field strength inside the ferromagnetic material H It consists of three parts: the magnetic field strength corresponding to hysteresis loss. H hys Magnetic field strength corresponding to eddy current loss H eddy The magnetic field strength corresponding to abnormal loss H ex :

[0018] (4)

[0019] In the formula H hys It can be calculated analytically from the static Energetic hysteresis model; under the assumptions of neglecting the skin effect and uniform magnetic flux density distribution inside the ferromagnetic material, the eddy current magnetic field strength can be established based on Maxwell's equations. H eddy( t ) and average magnetic flux density B av ( t The relationship expression between )

[0020] (5)

[0021] In the formula, σ For the electrical conductivity of the material, d The thickness of the material laminations;

[0022] Based on Bertotti's loss statistics theory and the assumption of average distribution of internal resistance field, an abnormal effect prediction model is established, as shown in Equation (6):

[0023] (6)

[0024] In the formula, G = 0.1356, which is a dimensionless constant. S The cross-sectional area of ​​the magnetic core. V 0( B m ) represents statistical parameters related to the internal magnetic units of the material. δ B =sgn(d B / d t ) represents a symbolic parameter;

[0025] Based on field separation theory, the magnetic field strength generated by eddy current effect and residual effect can be introduced into the static Energetic model to establish the characterization equation of the dynamic Energetic hysteresis model:

[0026] (7).

[0027] The derivation of the complex permeability in step one assumes that the electric fields inside the material move only along the direction of the magnetic field. y The axial direction changes, and based on the one-dimensional model of nanocrystalline materials, a system of Maxwell's equations is established. H ( y , t The diffusion equation is as follows:

[0028] (8)

[0029] Temporal boundary conditions H (± d , t )= H 0e jωt Substituting into equation (8) and solving, we get:

[0030] (9)

[0031] In the formula, γ Let be the propagation constant, which is related to the skin depth, i.e.:

[0032] (10)

[0033] (11)

[0034] In the formula, μ z ( B m ( ) represents the relative permeability in the rolling direction under different magnetic flux densities. σ Let be the electrical conductivity of the material, and the average magnetic flux density inside the nanocrystalline material be:

[0035] (12)

[0036] Therefore, the relative complex permeability of the ferromagnetic material exhibiting frequency-varying characteristics along the rolling direction can be obtained as follows:

[0037] (13).

[0038] The process of establishing the time-domain analytical expression of the eddy current magnetic field under different frequency excitations in step two is as follows:

[0039] Equation (7) is improved by introducing a skin effect factor. k sin To consider the impact of the high-frequency skin effect on eddy current loss, the calculation formula is as follows:

[0040] (14)

[0041] In the formula, k sin The value of and δ z / d It is related to the ratio, when δ z / d When >1, k sin =1, when δ z / d <1 hour, k sin Let be a function that varies with frequency, and its relationship is shown in the following equation:

[0042] (15)

[0043] (16)

[0044] High frequency excitation H eddy ( t )and B av ( t The expression for ) can be represented as:

[0045] (17)

[0046] In the formula, θ The phase difference between the average magnetic flux density and the eddy current magnetic field strength can be calculated using equation (18):

[0047] (18)

[0048] Integrating the area of ​​equation (15) yields the eddy current loss after elliptic equivalent:

[0049] (19)

[0050] Following the principle that the eddy current losses are equal before and after the equivalent equation, the amplitude of the eddy current magnetic field strength after the equivalent equation can be obtained by combining equations (14) and (19). H eddy for:

[0051] (20).

[0052] In step three, the expression for the average magnetic flux density function inside the nanocrystalline material under square wave excitation can be derived using Fourier analysis:

[0053] n =1,3,5… (21)

[0054] In the formula, f 0 represents the square wave excitation frequency; based on the modeling of the high-frequency sinusoidal eddy current magnetic field intensity, the time-domain equation for the eddy current magnetic field intensity under square wave excitation is derived as follows:

[0055] (twenty two)

[0056] The derivative of the average magnetic flux density under square wave excitation is:

[0057] n =1,3,5… (23)

[0058] The dynamic hysteresis model under square wave excitation can be established from equations (22), (23), and (7):

[0059] (twenty four)

[0060] In the formula, H hys ( t It can be calculated from equation (7).

[0061] Compared with the prior art, the present invention has the following beneficial effects:

[0062] 1. This application considers the influence of high-frequency harmonic components in square wave excitation on dynamic hysteresis loop, and realizes the prediction of dynamic hysteresis loop under square wave excitation based solely on sinusoidal measurement data;

[0063] 2. This application takes into account the hysteresis effect between the average magnetic flux density and the magnetic field strength under different excitation frequencies, and the prediction accuracy of the dynamic hysteresis loop under high-frequency sinusoidal and square wave excitation is significantly improved. Attached Figure Description

[0064] Figure 1 This is a flowchart of a square wave hysteresis loop calculation method considering the influence of high-frequency harmonic components, as described in this invention.

[0065] Figure 2 This is a schematic diagram of a one-dimensional model of the nanocrystalline material described in this invention;

[0066] Figure 3 This refers to the relative permeability of 1K107B under different magnetic flux densities as described in this invention. μ z A curve graph;

[0067] Figure 4 This is the invention described B m Curves showing the variation of complex permeability amplitude and phase difference with frequency at 0.5T and 1T;

[0068] Figure 5 This is a physical image of the soft magnetic material magnetic property measurement platform described in this invention;

[0069] Figure 6 This is a comparison diagram of the simulated hysteresis loop and the measured hysteresis loop described in this invention;

[0070] Figure 7 This is a graph showing the variation of the residual loss coefficient with magnetic flux density as described in this invention.

[0071] Figure 8 This is a comparison chart of the simulation results and experimental results of the hysteresis loop under sinusoidal excitation described in this invention;

[0072] Figure 9 This is a comparison chart of the simulation results and experimental results of the hysteresis loop under square wave excitation as described in this invention. Detailed Implementation

[0073] The present invention will now be described in detail with reference to the accompanying drawings, such as... Figure 1-9 As shown;

[0074] The Energetic hysteresis model has advantages such as fast solution speed, simple parameter identification, and high accuracy in simulating local small hysteresis loops. Therefore, this application uses this model as a basis to study the dynamic hysteresis loop modeling of nanocrystalline materials. The Energetic hysteresis model was proposed by Austrian scholar H. Hauser in 1994. This model is based on the intrinsic energy conservation and magnetic domain statistics theory of ferromagnetic materials, and uses a concise analytical expression to characterize the relationship between the magnetic flux density and magnetic field strength inside ferromagnetic materials:

[0075] (1)

[0076] (2)

[0077] (3)

[0078] Equation (1) characterizes the linear characteristics of ferromagnetic materials; Equation (2) characterizes the nonlinear characteristics of materials; Equation (3) describes the hysteresis effects of materials, such as remanence and coercivity; where, H This represents the actual magnetic field strength of the ferromagnetic material. H d The demagnetizing field strength; H r The strength of the reversible magnetic field; H i The strength of the irreversible magnetic field.

[0079] Studies on dynamic hysteresis models are typically based on loss separation theory, which considers the magnetic field strength within ferromagnetic materials. H It consists of three parts: the magnetic field strength corresponding to hysteresis loss. H hys Magnetic field strength corresponding to eddy current loss H eddy The magnetic field strength corresponding to abnormal loss H ex :

[0080] (4)

[0081] In the formula H hys The results are obtained from the static hysteresis model (i.e., equations (1), (2), and (3)). The calculation of eddy current magnetic field strength often employs the classical eddy current effect prediction model. Under the assumptions of neglecting the skin effect and uniform magnetic flux density distribution within the ferromagnetic material, the eddy current magnetic field strength can be established based on Maxwell's equations. H eddy( t ) and average magnetic flux density B av ( t The relationship expression between )

[0082] (5)

[0083] In the formula, σ The electrical conductivity of the material; d The thickness of the material laminations.

[0084] Research on the prediction of abnormal effects currently mostly adopts the abnormal effect prediction model established by Fiorillo F. based on Bertotti loss statistics theory and the assumption of average distribution of internal resistance field, as shown in Equation (6).

[0085] (6)

[0086] In the formula, G = 0.1356, which is a dimensionless constant; S The cross-sectional area of ​​the magnetic core; V 0( B m ) is a statistical parameter related to the magnetic units inside the material, and is related to the peak value of the magnetic flux density; δ B =sgn(d B / d t ) is a symbolic parameter.

[0087] Based on field separation theory, the magnetic field strength generated by eddy current effect and residual effect can be introduced into the static Energetic model to establish the characterization equation of the dynamic Energetic hysteresis model:

[0088] (7)

[0089] The dynamic hysteresis prediction model based on field separation theory has clear physical meaning, but the assumption that the magnetic flux density inside the ferromagnetic material is uniformly distributed and the skin effect is negligible makes it applicable only in the low frequency range. The model of equation (7) is also known as the classical model.

[0090] Under high-frequency excitation, the magnetic field generated by the hysteresis effect accounts for a very small proportion of the total magnetic field compared to the eddy current magnetic field. Therefore, this application does not consider the influence of the skin effect on the hysteresis magnetic field. Furthermore, existing literature generally does not consider the influence of the skin effect on anomalous magnetic fields. In summary, the influence of the skin effect on the dynamic hysteresis model is mainly reflected in the eddy current magnetic field term.

[0091] The shape of the hysteresis loop varies significantly at different frequencies. The main reason for this frequency variation is the skin effect, which leads to different constitutive relations between magnetic flux density and magnetic field strength at different frequencies. As the frequency increases, due to the influence of the skin effect on the eddy current magnetic field strength, a phase difference will exist between the average magnetic flux density and magnetic field strength within the material. Therefore, the permeability under high-frequency excitation should vary with frequency. f Varying complex permeability. Accurate simulation of the dynamic hysteresis loop of soft magnetic materials under high-frequency excitation requires consideration of the relationship between complex permeability and frequency. This application derives the relative complex permeability as a function of frequency based on Maxwell's equations.

[0092] The thickness of the stack of nanocrystalline materials d It is 0.022mm, which is much smaller than the material width. w Therefore, it can be assumed that the field quantities inside the material only move along the direction of the material. y Axial direction variation. A one-dimensional model of nanocrystalline materials is as follows: Figure 2 As shown, based on Maxwell's equations, we can establish... H ( y , t The diffusion equation for ) is:

[0093] (8)

[0094] Temporal boundary conditions H (± d , t )= H 0e jωt Substituting into equation (8) and solving, we get:

[0095] (9)

[0096] In the formula, γ Let be the propagation constant, which is related to the skin depth, i.e.:

[0097] (10)

[0098] (11)

[0099] In the formula, μ z ( B m () represents the relative permeability in the rolling direction under different magnetic flux densities; σ The electrical conductivity of the material is given by: The average magnetic flux density inside the material is given by:

[0100] (12)

[0101] Therefore, the relative complex permeability of the ferromagnetic material exhibiting frequency-varying characteristics along the rolling direction can be obtained as follows:

[0102] (13)

[0103] The relative permeability of nanocrystalline 1K107B at a frequency of 100Hz was experimentally measured. μ z ( B m The curve showing the variation of magnetic flux density peak value is as follows: Figure 3 As shown:

[0104] Will Figure 3 middle B m Relative permeability at 0.5T and 1T μ z Substituting into equation (13), the amplitude of the complex permeability and the average magnetic flux density of the nanocrystalline 1K107B under this magnetic flux density can be obtained. B With magnetic field strength H The trend of phase difference between them as a function of frequency, such as Figure 4 As shown.

[0105] Depend on Figure 4 It can be seen that the amplitude of complex permeability is constant at low frequencies, and is related to the relative permeability. μ z When the average magnetic flux density and magnetic field strength are equal, the phase difference between them is zero. Under high-frequency excitation, the amplitude of the complex permeability gradually decreases, while the phase initially increases and then stabilizes with frequency. Furthermore, at the same frequency, the relative permeability of soft magnetic materials differs under different magnetic flux densities, leading to differences in the amplitude and phase of their corresponding complex permeability.

[0106] As the above analysis shows, the classical dynamic hysteresis loop modeling method cannot characterize the hysteresis effect between magnetic flux density and magnetic field strength under high-frequency excitation. However, square wave excitation is composed of superimposed sine waves of different frequencies and amplitudes. Therefore, modeling the eddy current magnetic field under sinusoidal excitation of different frequencies is the primary prerequisite for predicting hysteresis loops under square wave excitation. Under high-frequency excitation, due to the skin effect, the non-uniform distribution of magnetic flux density inside ferromagnetic materials renders the classical loss separation theory invalid. Therefore, this application improves the classical eddy current loss calculation formula by introducing a skin effect factor. k sin To consider the impact of the high-frequency skin effect on eddy current loss, the calculation formula is as follows:

[0107] (14)

[0108] In the formula, k sinThe value of and δ z / d It is related to the ratio. When δ z / d When >1, k sin =1; when δ z / d <1 hour, k sin It is a function that varies with frequency.

[0109] (15)

[0110] (16)

[0111] Under low-frequency excitation, the magnetic flux density inside the nanocrystalline material is uniformly distributed, and the phase difference between it and the eddy current magnetic field strength is zero. If the waveforms of the magnetic flux density and magnetic field strength inside the ferromagnetic material are not distorted, the hysteresis loop generated by the eddy current effect can be characterized as about y An axisymmetric ellipse. With increasing frequency, due to the skin effect, there is a certain phase difference between the average magnetic flux density and the eddy current magnetic field strength. At this point, the hysteresis loop generated by the eddy current effect is an ellipse offset by a certain angle, and this angle varies with frequency. Therefore, under high-frequency excitation... H eddy ( t )and B av ( t The expression for ) can be represented as:

[0112] (17)

[0113] In the formula, θ The phase difference between the average magnetic flux density and the eddy current magnetic field strength can be calculated using equation (18):

[0114] (18)

[0115] Integrating the area of ​​equation (15) yields the eddy current loss after elliptic equivalent:

[0116] (19)

[0117] Following the principle that the eddy current losses are equal before and after the equivalent equation, the amplitude of the eddy current magnetic field strength after the equivalent equation can be obtained by combining equations (14) and (19). H eddy for:

[0118] (20)

[0119] This modeling method not only considers the influence of the skin effect on eddy current loss, but also accurately characterizes the hysteresis effect between the average magnetic flux density and the magnetic field strength at different frequencies and magnetic flux densities, thereby realizing the prediction of dynamic hysteresis loops under high-frequency sinusoidal excitation.

[0120] Any excitation waveform can be formed by superimposing sine waves of different frequencies and amplitudes. Therefore, using Fourier analysis, the expression for the average magnetic flux density function inside nanocrystalline materials under square wave excitation can be derived as follows:

[0121] n =1,3,5… (21)

[0122] In the formula, f 0 represents the square wave excitation frequency. As shown in equation (21), under square wave excitation, the magnetic flux density waveform contains high-frequency high-order harmonic components. These high-frequency harmonic components will cause a certain phase difference between the average magnetic flux density and the eddy current magnetic field strength inside the nanocrystalline material, while the classical model does not consider the influence of this factor on the dynamic hysteresis loop.

[0123] Based on the modeling of high-frequency sinusoidal eddy current magnetic field intensity, this application derives the time-domain equation for the eddy current magnetic field intensity under square wave excitation as follows:

[0124] (twenty two)

[0125] The derivative of the average magnetic flux density under square wave excitation is:

[0126] n =1,3,5… (23)

[0127] The dynamic hysteresis model under square wave excitation can be established from equations (22), (23), and (7):

[0128] (twenty four)

[0129] In the formula, H hys ( t ) was calculated using the Energetic model. Example

[0130] This application established a system for measuring the magnetic properties of soft magnetic materials under high-frequency sinusoidal and square wave excitation. The ring measurement method was used to measure the magnetic properties of 1K107B nanocrystals under sinusoidal and square wave excitation at different frequencies: 100Hz-40kHz and 1kHz-25kHz. The dimensions and electromagnetic parameters of the 1K107B nanocrystal ring sample are shown in Table 1. The experimental testing system is as follows: Figure 6 As shown in Table 1, which lists the dimensions and electromagnetic parameters of the nanocrystalline 1K107B ring sample, the main contents of which are as follows:

[0131]

[0132] This application employs the whale optimization algorithm to identify the Energetic model parameters of the hysteresis loop under quasi-static (100Hz) conditions. The parameter identification results are shown in Table 2. A comparison between the simulated and measured hysteresis loops is also provided. Figure 6 As shown: Table 2 is the parameter table for the Energetic model, and its specific contents are as follows:

[0133]

[0134] The coefficients in the skin effect factor of eddy current magnetic field strength can be identified by measuring the hysteresis loops at different frequencies. k e =67.95, γ =-0.396. Distributional characteristic parameter of residual loss per unit period. V 0( B m) can be identified by measuring the core loss under low-frequency excitation, where the skin effect is negligible. This application identifies the distribution characteristic parameters of the residual loss based on experimental measurements of low-frequency (5kHz-15kHz) losses under different magnetic flux densities. V 0( B m ) value, the result is as follows Figure 7 As shown.

[0135] The prediction accuracy of the nanocrystalline dynamic hysteresis model established in this application is compared with that of the classical model. A comparison is made between the simulated hysteresis loop and the measured hysteresis loop under sinusoidal excitation in different frequency and magnetic flux density ranges. Figure 8 As shown:

[0136] Depend on Figure 8It is known that, because the classical model does not consider the influence of the high-frequency skin effect, the simulated hysteresis loop under high-frequency sinusoidal excitation differs from the measured hysteresis loop by a certain angle. The high-frequency sinusoidal dynamic hysteresis model established in this application considers the influence of the high-frequency skin effect on eddy current loss and the phase difference between magnetic flux density and magnetic field strength under different excitation frequencies. Compared with the classical dynamic hysteresis model, the simulation accuracy is significantly improved. Therefore, this model can be used for hysteresis loop prediction under square wave excitation.

[0137] Based on the model parameters identified under sinusoidal excitation, a dynamic hysteresis loop prediction model under square wave excitation can be established using Fourier analysis. A comparison of simulated and measured hysteresis loops under square wave excitation with different magnetic flux densities at a frequency of 25kHz is provided. Figure 9 As shown:

[0138] Depend on Figure 9 It can be seen that, compared with the classical dynamic hysteresis model, the prediction accuracy of the square wave dynamic hysteresis loop prediction model established in this application is significantly improved because it considers the influence of the phase difference between magnetic flux density and magnetic field strength caused by high-frequency harmonic components on the dynamic hysteresis characteristics.

[0139] It should be noted that in this application, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, the phrase "comprising an element defined as..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0140] The above technical solutions only embody the preferred technical solutions of the present invention. Any modifications that may be made by those skilled in the art to certain parts thereof embody the principles of the present invention and fall within the protection scope of the present invention.

Claims

1. A method for calculating square wave hysteresis loops considering the influence of high-frequency harmonic components, characterized in that, The method includes the following steps: Step 1: Analyze the analytical relationship between average magnetic flux density and magnetic field strength at different excitation frequencies, and use the ratio of average magnetic flux density to magnetic field strength as complex permeability to characterize the magnetization characteristics of nanocrystalline materials at different frequencies. Step 2: Combine the elliptic equivalent theory with the eddy current loss calculation formula considering the high-frequency skin effect to establish the time-domain analytical expression of the eddy current magnetic field under sinusoidal excitation at different frequencies. Step 3: Analyze the influence of high-frequency harmonic components in square wave excitation on the dynamic hysteresis loop. Fourier analysis is used to establish a prediction model for the dynamic hysteresis loop under square wave excitation, thereby enabling the prediction of the dynamic hysteresis loop under square wave excitation based solely on sinusoidal measurement data. The process of analyzing the analytical relationship between average magnetic flux density and magnetic field strength at different excitation frequencies in step one is as follows: The relationship between magnetic flux density and magnetic field strength inside ferromagnetic materials is characterized using the analytical expression of the static Energetic hysteresis model: (1) (2) (3) Equation (1) characterizes the linear characteristics of ferromagnetic materials; Equation (2) characterizes the nonlinear characteristics of materials; Equation (3) describes the hysteresis effect of the material; where H is the actual magnetic field strength of the ferromagnetic material. d H represents the demagnetizing field strength. r H is the reversible magnetic field strength. i The strength of the irreversible magnetic field; The magnetic field strength H inside the ferromagnetic material is divided into three terms: the magnetic field strength H corresponding to hysteresis loss. hys The magnetic field strength H corresponding to eddy current loss eddy The magnetic field strength H corresponding to abnormal loss ex : (4) In the formula H hys It can be calculated analytically from the static Energetic hysteresis model; under the assumptions of neglecting the skin effect and uniform magnetic flux density distribution inside the ferromagnetic material, the eddy current magnetic field strength H can be established based on Maxwell's equations. eddy (t) and average magnetic flux density B av The relationship expression between (t) is: (5) In the formula, σ is the electrical conductivity of the material, and d is the thickness of the material stack; Based on Bertotti's loss statistics theory and the assumption of average distribution of internal resistance field, an abnormal effect prediction model is established, as shown in Equation (6): (6) In the formula, G = 0.1356, which is a dimensionless constant, S is the cross-sectional area of ​​the magnetic core, and V0(B) is the cross-sectional area of ​​the magnetic core. m δ represents a statistical parameter related to the internal magnetic units of the material. B =sgn(dB / dt) is a symbolic parameter; Based on field separation theory, the magnetic field strength generated by eddy current effect and residual effect can be introduced into the static Energetic model to establish the characterization equation of the dynamic Energetic hysteresis model: (7)； The process of establishing the time-domain analytical expression of the eddy current magnetic field under different frequency excitations in step two is as follows: Equation (7) is improved by introducing the skin effect factor k. sin To consider the impact of the high-frequency skin effect on eddy current loss, the calculation formula is as follows: (14) In the formula, k sin The value of δ z The ratio of / d is related, when δ z When / d>1, k sin =1, when δ z When / d<1, k sin Let be a function that varies with frequency, and its relationship is shown in the following equation: (15) (16) H under high frequency excitation eddy (t) and B av The expression for (t) can be represented as: (17) In the formula, θ is the phase difference between the average magnetic flux density and the eddy current magnetic field strength, which can be calculated by equation (18): (18) Integrating the area of ​​equation (15) yields the eddy current loss after elliptic equivalent: (19) Following the principle that the eddy current losses are equal before and after the equivalence, the amplitude H of the eddy current magnetic field strength after the equivalence can be obtained by combining equations (14) and (19). eddy for: (20)； In step three, the expression for the average magnetic flux density function inside the nanocrystalline material under square wave excitation can be derived using Fourier analysis: n=1,3,5… (21) In the formula, f0 is the square wave excitation frequency; based on the modeling of the high-frequency sinusoidal eddy current magnetic field intensity, the time-domain equation of the eddy current magnetic field intensity under square wave excitation is derived as follows: (22) The derivative of the average magnetic flux density under square wave excitation is: n=1,3,5… (23) The dynamic hysteresis model under square wave excitation can be established from equations (22), (23), and (7): (24) In the formula, H hys (t) can be calculated from equation (7).

2. The method for calculating square wave hysteresis loop considering the influence of high-frequency harmonic components according to claim 1, characterized in that, The derivation of complex permeability in step one assumes that all field quantities inside the material vary only along the y-axis. Based on the one-dimensional model of nanocrystalline materials, a diffusion equation for H(y,t) is established according to Maxwell's equations. This diffusion equation is: (8) The time-domain boundary condition H(±d,t)=H0e jωt Substituting into equation (8) and solving, we get: (9) In the formula, γ is the propagation constant, which is related to the skin depth, i.e.: (10) (11) In the formula, μ z (B m Let σ be the relative permeability along the rolling direction under different magnetic flux densities, σ be the electrical conductivity of the material, and the average magnetic flux density inside the nanocrystalline material be: (12) Therefore, the relative complex permeability of the ferromagnetic material exhibiting frequency-varying characteristics along the rolling direction can be obtained as follows: (13)。

Images