An analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation

Abstract

This invention discloses an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation, comprising the following steps: constructing an analytical Everett function based on the hyperbolic tangent function; calculating the reversible magnetization susceptibility of the hysteresis loop to characterize the reversible magnetization part, the expression of which is fitted by an exponential function; calculating the irreversible magnetization component of the hysteresis loop, obtaining the calculated value of the Everett function from the descending branch of the concentric hysteresis loop to determine the fitness parameter of the analytical Everett function; replacing the magnetic field strength with the effective magnetic field strength, the feedback coefficient being calculated by averaging the feedback coefficients at the negative saturation points of the descending branches of different hysteresis loops; and using the magnetic field strength... H Using the time-step difference method, we can consider the inverse form of the analytical Preisach model and solve for d. B / d H The present invention verifies the accuracy and efficiency of the model by comparing the simulated hysteresis loop with the measured data of B30P105 grain-oriented silicon steel and the computational efficiency with the traditional model.

Description

Technical Field

[0001] This invention relates to the field of magnetic hysteresis characteristic analysis technology for soft magnetic materials, specifically to an analytical modeling method for the magnetic hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation. Background Technology

[0002] Accurate modeling of the hysteresis characteristics of silicon steel is crucial for magnetic field analysis and efficiency optimization of electrical equipment such as motors and transformers. Numerous scholars have studied different hysteresis models to more efficiently solve electromagnetic field calculation problems considering hysteresis characteristics. Among them, the classical Preisach model is one of the most popular, based on the physical assumption of a magnetic dipole and possessing relatively high simulation accuracy. However, in the design and manufacturing of electrical equipment, finite element analysis typically uses the curl of the vector magnetic potential A to calculate the magnetic field distribution. In this case, coupled positive Preisach models lead to exceptionally difficult convergence and a significantly increased number of iterations, hindering the rapid and efficient resolution of engineering problems. Therefore, the inverse Preisach model is more suitable for finite element hysteresis calculations.

[0003] Traditional discrete Everett functions exhibit poor convergence performance and require significant memory, increasing the overall runtime of the finite element analysis process. Furthermore, discrete Everett functions can only calculate values ​​falling within the measured limiting hysteresis loop data range. When the magnetic flux density exceeds the measured limiting hysteresis loop during finite element iteration, singular values ​​can occur, causing the entire calculation process to diverge. Therefore, further research is needed on generalized inverse Preisach models with analytic Everett functions. Existing methods use closed-form Everett functions to improve the computational efficiency of hysteresis models; however, these methods do not consider feedback coefficients during the magnetization process. For example:

[0004] 1) Reference 1: Liu Ren, Du Yingxue, Li Lin, et al. Analytical inverse Preisach hysteresis model [J]. Journal of Electrical Engineering, 2023, 38(10):2567-2576.

[0005] 2) Document 2: S. Hussain, DALowther. An Efficient Implementation of the Classical Preisach Model[J]. IEEE Transactions on Magnetics, 2018, 54(3):7300204.

[0006] Therefore, a generalized analytical inverse Preisach model needs to be established, and the irreversible magnetization is modeled using the analytical Everett function. By using the time-step difference method, the reversible hysteresis component and the magnetization-related hysteresis component are coupled in the analytical inverse model, and an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation is proposed. Summary of the Invention

[0007] The purpose of this invention is to overcome the above-mentioned shortcomings and provide an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation, so as to solve the problems mentioned in the background art.

[0008] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation, which includes the following steps:

[0009] Step 1: Construct an analytic Everett function based on the hyperbolic tangent function;

[0010] Step 2: Calculate the reversible magnetic susceptibility χ rev , is used to characterize the reversible magnetization part; the reversible magnetic susceptibility is equal to the slope dM / dH of the point of rotation of the concentric hysteresis loop, and the expression is fitted by an exponential function;

[0011] Step 3: Calculate the irreversible magnetization component of the hysteresis loop, which is equal to the measured magnetization intensity of the hysteresis loop minus the reversible magnetization component; based on the irreversible part of the quasi-static hysteresis loop, obtain the calculated value of the Everett function from the descending branch of the concentric hysteresis loop, and determine the fitness parameter of the analytical Everett function.

[0012] Step 4: Construct the analytical form of the generalized inverse Preisach model; In order to suppress the congruence characteristics of the classical model, the influence of magnetization feedback is considered, and the effective magnetic field strength is used instead of the magnetic field strength. The feedback coefficient can be calculated by finding the average value of the feedback coefficient at the negative saturation point of the descending branch of different hysteresis loops.

[0013] Step 5: Use the time-step difference method of magnetic field strength H to consider the inverse form of the analytical Preisach model and solve for the dB / dH expression;

[0014] Furthermore, the analytic Everett function based on the hyperbolic tangent function constructed in step one is as follows:

[0015]

[0016] In the formula, a to e are constants, which are the fitness parameters of the analytical Everett function; α is the threshold for the magnetic dipole to flip in the forward direction, and β is the threshold for the magnetic dipole to flip in the reverse direction.

[0017] Furthermore, in step two, the reversible magnetic susceptibility χrev It is equal to the slope dM / dH of the point of rotation of the concentric hysteresis loop, where M is the magnetization and H is the magnetic field strength;

[0018] Reversible magnetic susceptibility χ rev The expression for (H) is fitted by an exponential function as follows:

[0019]

[0020] In the formula, g1 and g2 are parameters to be determined. The fitting result shows that the parameter values ​​g1 are 5026 and g2 are -0.0307.

[0021] Furthermore, after calculating the reversible magnetic susceptibility in step two, the reversible magnetization M... rev (H) is:

[0022]

[0023] Furthermore, in step three, the irreversible magnetization component of the hysteresis loop is obtained by subtracting the reversible magnetization component from the magnetization intensity of the hysteresis loop, specifically:

[0024] M irr =M meas (H)-M rev (H)

[0025] In the formula, M irr It is the irreversible magnetization component, M meas It is the measured magnetization of the hysteresis loop, M. rev It is a reversible magnetization component.

[0026] Furthermore, in step three, based on the irreversible portion of the quasi-static hysteresis loop, the calculated value of the Everett function is obtained from the descending branch of the concentric hysteresis loop:

[0027]

[0028] In the formula, M αα M is the amplitude of the magnetic flux density of the concentric hysteresis loop. αβ These are the points on the descending branch, used to determine the fitness parameters a to e of the analytic Everett function.

[0029] Furthermore, considering the influence of magnetization feedback in step four, the effective magnetic field strength is used instead of the magnetic field strength:

[0030] H e =H+KM

[0031] In the formula, H e Where K is the effective magnetic field strength, and K is the feedback coefficient, which can be calculated by averaging the feedback coefficients at the negative saturation points of the descending branch of different hysteresis loops. Its expression is:

[0032]

[0033] In the formula, M e χ is the measured magnetization, M is the magnetization calculated using the classical Preisach model, and χ is the magnetic susceptibility of the principal hysteresis loop.

[0034] Furthermore, the time-step difference expression for the magnetic field strength H in step five is as follows:

[0035]

[0036] In the formula, H i This is the current magnetic field strength value, H. i+1 This is the magnetic field strength value in the next step, B. i This is the current magnetic flux density value, B. i+1 This is the magnetic flux density value (dB / dH) for the next step. i It is magnetic flux density B i With magnetic field strength H i The derivative of .

[0037] Furthermore, the derivation of the dB / dH expression in step five is as follows:

[0038]

[0039] Considering the effect of magnetization feedback, the effective magnetic field strength is used instead of the magnetic field strength, and dM / dH is converted to:

[0040]

[0041] in,

[0042]

[0043]

[0044] In the formula, dM / dH e for:

[0045]

[0046] In the formula, dM rev / dH e Through Calculate dM irr / dH e The expression is:

[0047]

[0048] In the formula, h kIt is the magnetic field strength of the hysteresis loop at the previous point of rotation, and h is the current calculated operating point; when h ≤ h k When h > h, it is a downward magnetization process; when h > h k At this time, it is an upward magnetization process.

[0049] The beneficial effects of this invention are:

[0050] This invention proposes an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation. An analytical Everett function is used to model the irreversible magnetization component, while reversible magnetization depends only on current magnetization. The reversible magnetization component is characterized by reversible magnetic susceptibility. To suppress the congruence characteristics of the classical model, the influence of magnetization feedback is considered, and the effective magnetic field strength is used instead of the magnetic field strength. The feedback coefficient is calculated by averaging the feedback coefficients at the negative saturation points of different hysteresis loops. This invention has clear physical meaning, few unknown parameters, considers the magnetization mechanism of the material, and is highly practical. The accuracy and efficiency of the proposed model are verified by comparing the simulated hysteresis loop with measured data of B30P105 grain-oriented silicon steel and the computational efficiency with traditional models. This invention employs the time-step difference method for magnetic field strength H, considering the inverse form of the analytical Preisach model, which facilitates coupling with finite element numerical calculation programs that consider hysteresis characteristics, improving the calculation speed. This method is of great significance for considering the operating status, loss assessment, and residual magnetism prediction of electrical equipment. Attached Figure Description

[0051] Figure 1 Flowchart of an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation;

[0052] Figure 2 Magnetization curve of B30P105 electrical steel sheet;

[0053] Figure 3 A comparison chart of measured and calculated values ​​of the hysteresis loop under sinusoidal excitation;

[0054] Figure 4 A comparison of measured and calculated values ​​of the hysteresis loop when the harmonic phase angle is 0° and the excitation contains 30% third harmonics.

[0055] Figure 5 This is a comparison chart of the measured and calculated values ​​of the hysteresis loop under PWM excitation with a duty cycle of 0.9. Detailed Implementation

[0056] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0057] Example 1: As Figure 1As shown, an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation is proposed, which includes the following steps:

[0058] Step 1: Unlike the discrete form of the classic Preisach model, construct an analytical Everett function based on the hyperbolic tangent function.

[0059] The classic Preisach model involves complex double integral operations on the distribution function:

[0060]

[0061] B = μ0(H + M(H))

[0062] In the formula, μ(α,β) is the distribution function of the Preisach model, representing the distribution density of the magnetic dipole; α is the threshold for the magnetic dipole to flip in the forward direction, and β is the threshold for the magnetic dipole to flip in the reverse direction; γ αβ The hysteresis operator is controlled by α and β, and only outputs 1 and -1.

[0063] To avoid the double integral problem, a discrete Everett function E(α,β) is introduced into the classic Preisach model:

[0064]

[0065] In the formula, M k H is the magnetization of the hysteresis loop at the previous point of rotation. k H is the inflection point of the first-order hysteresis loop curve, and H is the current operating point.

[0066] The constructed analytic Everett function based on the hyperbolic tangent function is as follows:

[0067]

[0068] In the formula, a to e are constants, which are the fitness parameters of the analytical Everett function.

[0069] Step 2: Calculate the reversible magnetic susceptibility χ rev The reversible magnetic susceptibility is used to characterize the reversible magnetization. The reversible magnetic susceptibility is equal to the slope dM / dH of the point of rotation of the concentric hysteresis loop, where M is the magnetization intensity and H is the magnetic field intensity.

[0070] Reversible magnetic susceptibility χ rev The expression for (H) is fitted by an exponential function as follows:

[0071]

[0072] In the formula, g1 and g2 are parameters to be determined. The fitting result shows that the parameter values ​​g1 are 5026 and g2 are -0.0307.

[0073] After calculating the reversible magnetic susceptibility, the reversible magnetization M rev (H) is:

[0074]

[0075] Step 3: Calculate the irreversible magnetization component of the hysteresis loop, which is equal to the measured magnetization of the hysteresis loop minus the reversible component. The expression is:

[0076] M irr =M meas (H)-M rev (H)

[0077] In the formula, M irr It is the irreversible magnetization component, M meas It is the measured magnetization of the hysteresis loop, M. rev It is a reversible magnetization component. The magnetization curve of B30P105 electrical steel sheet is shown below. Figure 2 As shown.

[0078] Based on the irreversible portion of the quasi-static hysteresis loop, the Everett function is calculated from the descending branch of the concentric hysteresis loop.

[0079]

[0080] In the formula, M αα M is the amplitude of the magnetic flux density of the concentric hysteresis loop. αβ These are the points on the descent branch, used to determine the fitness parameters a to e of the analytic Everett function. The values ​​are as follows:

[0081] Table 1 analyzes the fitness parameter values ​​of the Everett function.

[0082]

[0083] Step 4: Construct the analytical form of the generalized inverse Preisach model. To suppress the congruence characteristics of the classical model, the influence of magnetization feedback is considered, and the effective magnetic field strength is used instead of the magnetic field strength:

[0084] H e =H+KM

[0085] In the formula, H e Where K is the effective magnetic field strength, and K is the feedback coefficient, which can be calculated by averaging the feedback coefficients at the negative saturation points of the descending branch of different hysteresis loops. Its expression is:

[0086]

[0087] In the formula, M eχ is the measured magnetization, M is the magnetization calculated using the classical Preisach model, and χ is the magnetic susceptibility of the principal hysteresis loop. In this invention, the magnetic flux density amplitude B is used. p The feedback coefficient K is calculated using hysteresis loops of 0.4T, 0.8T, and 1.2T, and is -5.694 × 10⁻⁶. -7 .

[0088] Step 5: Use the time-step difference method of magnetic field strength H to consider the inverse form of the analytical Preisach model:

[0089]

[0090] In the formula, H i This is the current magnetic field strength value, H. i+1 This is the magnetic field strength value in the next step, B. i This is the current magnetic flux density value, B. i+1 This is the magnetic flux density value (dB / dH) for the next step. i It is magnetic flux density B i With magnetic field strength H i The derivative of .

[0091] Based on B = μ0(H + M(H)), the expression for dB / dH is derived as follows:

[0092]

[0093] Considering the effect of magnetization feedback, the effective magnetic field strength is used instead of the magnetic field strength, and dM / dH is converted to:

[0094]

[0095] in,

[0096]

[0097]

[0098] In the formula, dM / dH e for:

[0099]

[0100] In the formula, dM rev / dH e pass Calculate dM irr / dH e The expression is:

[0101]

[0102] In the formula, h kis the magnetic field strength of the hysteresis loop at the previous point of rotation, and h is the current calculated operating point. When h ≤ h k When h > h, it is a downward magnetization process; when h > h k At this time, it is an upward magnetization process.

[0103] The accuracy of the model was verified by comparing the simulated hysteresis loop with measured data of B30P105 grain-oriented silicon steel. Figure 3 A comparison chart of measured and calculated values ​​of the hysteresis loop under sinusoidal excitation; Figure 4 A comparison of measured and calculated values ​​of the hysteresis loop when the harmonic phase angle is 0° and the excitation contains 30% third harmonics. Figure 5 The figure shows a comparison between the measured and calculated values ​​of the hysteresis loop under PWM excitation with a duty cycle of 0.9. It can be seen that the analytical inverse Preisach model proposed in this invention can effectively predict asymmetric small loops caused by harmonics and PWM excitation, demonstrating broad prospects from an engineering perspective.

[0104] The accuracy and computational efficiency of the analytical model proposed in this invention are compared with those of the classical model, and the time required to run both models is recorded. 1000 data points are taken for each magnetization cycle, and the simulation results are shown in Table 2 on a computer with a 2.50GHz Intel Core i5-7300 CPU.

[0105] Table 2 Comparison of accuracy and computational efficiency between the analytical model of this invention and the classical model

[0106]

[0107] In the table, B p Let be the magnetic flux density amplitude, T be the corresponding running time, and σ be the error between the calculated and experimental hysteresis loop, calculated using the absolute value of the average percentage error.

[0108]

[0109] In the formula, H cal and H meas These are the calculated and measured values ​​of the magnetic field strength, and N is the number of magnetic flux density sampling points in the magnetization period.

[0110] It can be seen that the analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation proposed in this invention has a much shorter computation time compared to the classical model. This facilitates coupling with finite element numerical calculation programs that consider hysteresis characteristics, thereby improving the computation speed. This method is of great significance for considering the operating status, loss assessment, and residual magnetism prediction of electrical equipment. At the same time, it also achieves a significant improvement in accuracy compared to the classical model.

[0111] This invention proposes an analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation by considering the analytical Everett function of the irreversible magnetization component. Research results show that this model has high accuracy for both sinusoidal and non-sinusoidal excitations, and its average computation time is faster than classical methods. It is also beneficial for coupling with finite element numerical calculation programs that consider hysteresis characteristics, and has broad engineering application prospects. This method is of great significance for refined electromagnet design, reasonable prediction of the operating status of electrical equipment, loss assessment, and accurate calculation of the transient electromagnetic characteristics of transformer closing residual magnetism.

[0112] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The embodiments and features described in these embodiments can be arbitrarily combined without conflict. The scope of protection of the present invention should be limited to the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. An analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation, characterized in that: It includes the following steps: Step 1: Construct an analytic Everett function based on the hyperbolic tangent function; Step 2: Calculate the reversible magnetic susceptibility χ rev , used to characterize the reversible magnetization; the reversible magnetic susceptibility is equal to the slope d of the point of rotation of the concentric hysteresis loop. M / d H The expression is fitted by an exponential function; Step 3: Calculate the irreversible magnetization component of the hysteresis loop, which is equal to the measured magnetization intensity of the hysteresis loop minus the reversible magnetization component. Based on the irreversible part of the quasi-static hysteresis loop, the calculated value of the Everett function is obtained from the descending branch of the concentric hysteresis loop, so as to determine the fitness parameter of the analytical Everett function. Step 4: Construct the analytical form of the generalized inverse Preisach model. In order to suppress the congruence characteristics of the classical model, the influence of magnetization feedback is considered. The effective magnetic field strength is used instead of the magnetic field strength. The feedback coefficient can be calculated by finding the average value of the feedback coefficient at the negative saturation point of the descending branch of different hysteresis loops. Step 5: Using magnetic field strength H Using the time-step difference method, we can consider the inverse form of the analytical Preisach model and solve for d. B / d H expression; The analytic Everett function based on the hyperbolic tangent function constructed in step one is as follows: ； In the formula, a arrive e It is a constant, representing the fitness parameter of the analytical Everett function; α The threshold for the positive flipping of the magnetic dipole. β The threshold for the magnetic dipole to reverse flip; Reversible magnetic susceptibility in step two χ rev The slope d equals the point of rotation of the concentric hysteresis loop M / d H ,in, M The magnetization intensity, H The magnetic field strength; Reversible magnetic susceptibility χ rev ( H The expression for ) is fitted by the exponential function as follows: ； In the formula, g 1 and g 2 represents the parameter to be determined, and the parameter values ​​in the fitting result are... g 1 is 5026, g 2 is -0.0307; In step four, considering the influence of magnetization feedback, the effective magnetic field strength is used instead of the magnetic field strength: ； In the formula, H e It is the effective magnetic field strength. K It is the feedback coefficient, which can be calculated by averaging the feedback coefficients at the negative saturation points of the descending branch of different hysteresis loops. Its expression is: ； In the formula, M e It is the measured magnetization intensity. M The magnetization was calculated using the classical Preisach model. χ It is the magnetic susceptibility of the principal hysteresis loop.

2. The analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation according to claim 1, characterized in that: After calculating the reversible magnetic susceptibility in step two, reversible magnetization... M rev ( H )for: 。 3. The analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation according to claim 1, characterized in that: In step three, the irreversible magnetization component of the hysteresis loop is obtained by subtracting the reversible magnetization component from the magnetization intensity of the hysteresis loop, specifically: ； In the formula, M irr It is an irreversible magnetization component. M meas It is the measured magnetization of the hysteresis loop. M rev It is a reversible magnetization component.

4. The analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation according to claim 1, characterized in that: In step three, based on the irreversible portion of the quasi-static hysteresis loop, the Everett function is calculated from the descending branch of the concentric hysteresis loop. ； In the formula, M αα It is the amplitude of the magnetic flux density of the concentric hysteresis loop. M αβ These are the points on the descent branch, used to determine the fitness parameter of the analytic Everett function. a arrive e .

5. The analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation according to claim 1, characterized in that: Magnetic field strength in step five H The expression for the time-step difference method is: ； In the formula, H i This is the current magnetic field strength value. H i+1 This is the magnetic field strength value for the next step. B i This is the current magnetic flux density value. B i+1 This is the magnetic flux density value in the next step, (d B / d H ) i It is magnetic flux density B i With magnetic field strength H i The derivative of .

6. The analytical modeling method for the hysteresis characteristics of grain-oriented silicon steel under non-sinusoidal excitation according to claim 1, characterized in that: Derivation of d in step five B / d H The expression is: ； Considering the influence of magnetization feedback, the effective magnetic field strength is used instead of the magnetic field strength, d M / d H Convert to: ； in, ； ； In the formula, d M / d H e for: ； In the formula, d M rev / d H e pass Calculate d M irr / d H e The expression is: ； In the formula, h k It is the magnetic field strength of the hysteresis loop at the previous point of rotation. h This is the current running point of the calculation; when When, it is a downward magnetization process; when At this time, it is an upward magnetization process.

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