A method for calculating end effect of surface-mounted permanent magnet motor
By defining the leakage flux coefficient and introducing a fitting function, combined with Maxwell's tensor method, the problem of calculating the end effect and armature reaction in the design of surface-mounted permanent magnet motors was solved, and fast, efficient and accurate calculations were achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NANJING NORMAL UNIVERSITY
- Filing Date
- 2023-02-13
- Publication Date
- 2026-07-07
AI Technical Summary
In the design of surface-mount permanent magnet motors, existing technologies struggle to accurately account for end effects and armature reactions while maintaining low computation time, resulting in insufficient computational accuracy.
By defining the leakage flux coefficient and introducing the leakage flux fitting function, combined with Maxwell's tensor method and geometric parameters, the end leakage flux can be calculated quickly and accurately, taking into account the end effects under no-load and loaded conditions.
It enables rapid and accurate simulation of end effects and armature reaction in a two-dimensional model, with fast calculation speed and high accuracy, approaching the results of three-dimensional finite element methods.
Smart Images
Figure CN116257724B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the numerical calculation of the end magnetic field of a surface-mounted permanent magnet motor, and more particularly to a method for calculating the end effect of a surface-mounted permanent magnet motor. Background Technology
[0002] Surface-mounted permanent magnet motors, including axial flux permanent magnet motors and radial flux permanent magnet motors, are widely used in various fields due to their compact mechanical structure and high power density. During the motor design and optimization phases, to balance computation time and accuracy, two-dimensional models are often solved instead of three-dimensional finite element models. Common methods include analytical methods and two-dimensional finite element methods. However, the main assumptions of these methods neglect end effects, which severely impacts the computational accuracy of the model. Therefore, to maintain low computation time while obtaining high-accuracy results during the design phase, how to consider end effects in a simple and quick way is currently a research hotspot.
[0003] Some scholars have used analytical methods to discuss the impact of end winding inductance and end leakage flux on motor performance, and to analyze leakage flux by calculating permeability using finite element method coupled with magnetic equivalent circuit. However, these methods are computationally complex and difficult to implement in the motor optimization stage. In recent years, scholars have suggested using an end leakage flux function to quantify the end leakage flux of permanent magnet motors, and then using this function to correct the two-dimensional analysis results to avoid using three-dimensional methods. A correction g(r) function based on curve fitting method is introduced to consider this, and based on this g(r) function, the end effect is considered in combination with the analysis model and finite element method. However, this will take a lot of time because the coefficients in the correction function are obtained from simplified three-dimensional finite element method. Some scholars have combined the solution of Maxwell's equations and magnetic equivalent circuit to determine the no-load magnetic field distribution of slotted axial flux permanent magnet motor considering end effect; by introducing end function, the influence of end effect on radial air gap magnetic field distribution is considered. Other scholars have proposed an analysis method based on the definition and superposition of appropriate field functions, which can accurately simulate different singularities of the prototype: PM field distribution, slotting effect, and radial end effect. Recently, researchers have proposed a new end leakage flux function that can be used to accurately quantify the end leakage flux of fractional slot concentrated winding surface permanent magnet motors. The drawback is that it does not take armature reaction into account. Summary of the Invention
[0004] Purpose of the invention: The purpose of this invention is to provide a method for calculating the end effect magnetic field of a surface-mounted permanent magnet motor that can take into account no-load and armature reaction, and quickly and accurately solve the leakage magnetic field within various parameter ranges.
[0005] Technical solution: The method for calculating the end effect of a surface-mount permanent magnet motor according to the present invention includes the following steps:
[0006] S1, under given conditions, the ratio of magnetic flux density at the end to that at the center length is defined as the leakage flux coefficient, and an expression for the leakage flux fitting function is proposed to calculate the leakage flux coefficient;
[0007] S2, the permanent magnet thickness H, stacking length L and air gap length g of the prototype are extracted as variables of the leakage magnetic field fitting function;
[0008] S3. Multiply the leakage flux coefficient Gl by the air gap magnetic flux density to obtain the magnetic flux density considering the end leakage flux in the three-dimensional case; obtain the torque considering the end effect through Maxwell's tensor method; integrate the leakage flux coefficient Gl to obtain the weighting factor G, and multiply the weighting factor G by the two-dimensional model result to obtain the quasi-three-dimensional result.
[0009] Furthermore, in step S1, the given conditions are as follows:
[0010] a) Edge leakage flux is unrelated to the main magnetic flux;
[0011] b) The leakage magnetic field is the same on both the inner and outer sides, so only the leakage magnetic field on one side is considered;
[0012] c) The leakage flux of the end effect only considers the reduction of the radial component of the magnetic flux;
[0013] d) Magnetic circuit saturation is not considered;
[0014] e) There is no magnetic leakage at the center length, which serves as a reference point.
[0015] Furthermore, in step S1, the expression for the leakage coefficient Gl′ is as follows:
[0016]
[0017] The expression for the magnetic flux leakage fitting function is as follows:
[0018]
[0019] In the formula, l is the coordinate on the calculation line in the magnetic flux leakage analysis model, and the calculation line extends from the center to the end; K1 and K2 are parameters related to the permanent magnet thickness H, the stacking length L, and the air gap length g.
[0020] Furthermore, in step S3, the radial component B of the air gap magnetic flux density at the end position rend The expression is as follows:
[0021] B rend =Gl·B r
[0022] Where Gl is the leakage magnetic coefficient, B r This represents the radial component of the air gap magnetic flux density.
[0023] Furthermore, in step S3, the expression for the torque considering end effects is obtained using Maxwell's tensor method as follows:
[0024]
[0025] Where μ0 is the free permeability; B r B θ These are the radial and tangential components of the magnetic flux density, respectively; L stk It is the stacking length of the motor.
[0026] Furthermore, in step S3, by calculating the integral of Gl over the stacking length, the curve is converted into a single value, and the expression for the weighting factor G is as follows:
[0027]
[0028] Among them, L stk It is the stacking length of the motor.
[0029] Compared with the prior art, the significant advantages of this invention are as follows:
[0030] 1. Based on the assumption that the end leakage flux is independent of the main magnetic circuit, a leakage flux analysis model for the permanent magnet and winding was designed; sensitivity analysis was performed to extract the key geometric parameters affecting the end leakage flux and obtain the variation characteristics of the end leakage flux.
[0031] 2. A fitting function Gl that accurately represents end leakage flux is proposed to quantify end leakage flux of surface-mounted permanent magnet motors, making it obtainable through calculation. The axial correlation of radial magnetic flux density is converted into a function calculated by geometric parameters. By fitting the motor geometric parameters to the function, leakage flux can be directly calculated using motor parameters. The calculation is fast, accurate, and does not require the use of finite element method.
[0032] 3. Based on the leakage flux analysis model of permanent magnet and winding, the end effects of no-load and armature reaction are considered respectively. The leakage flux coefficient in the no-load state is equal to the leakage flux coefficient of the permanent magnet model, and the leakage flux coefficient in the load state is equal to the product of the leakage flux coefficient of the permanent magnet model and the leakage flux coefficient of the winding model. Thus, the end leakage flux calculation of the motor under no-load and load conditions is realized. Attached Figure Description
[0033] Figure 1 This is a schematic diagram of the permanent magnet leakage magnetic field model of the present invention;
[0034] Figure 2 This is a schematic diagram of the winding leakage flux model of the present invention;
[0035] Figure 3 (a) in the diagram is a schematic diagram of the magnetic flux density curve.
[0036] (b) is a schematic diagram of the leakage magnetic flux coefficient curve;
[0037] Figure 4 (a) in the figure is a schematic diagram of the function Gl with different values of K1 in this invention.
[0038] (b) is a schematic diagram of the function Gl with different values of K2 in this invention;
[0039] Figure 5 This is a flowchart of the calculation method of the present invention;
[0040] Figure 6 This is a schematic diagram of a three-dimensional finite element model of the prototype of the present invention;
[0041] Figure 7 This is a comparison chart of the no-load leakage magnetic flux coefficient of the prototype in this invention;
[0042] Figure 8 This is a comparison chart of the leakage flux coefficients of the prototype under load in this invention;
[0043] Figure 9 This is a schematic diagram of the two-dimensional finite element model of the prototype in this invention;
[0044] Figure 10 Comparison of radial magnetic flux density obtained by different methods at the air gap center of the prototype in this invention at 2.5 mm;
[0045] Figure 11 This is a torque comparison diagram of the prototype in this invention;
[0046] Figure 12 A comparison diagram of the back electromotive force of A under no-load rated speed;
[0047] Figure 13 A comparison diagram of the line back electromotive force at different speeds under no-load conditions;
[0048] Figure 14 This is a current-torque ratio diagram for the low-speed range. Detailed Implementation
[0049] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.
[0050] (I) Implementation Method
[0051] This invention provides a method for calculating the end effect of a surface-mounted permanent magnet motor based on geometric parameters, comprising the following steps:
[0052] Step 1: State the assumptions, specifically as follows:
[0053] a) Edge leakage flux is unrelated to the main magnetic flux;
[0054] b) The leakage magnetic field is the same on both the inner and outer sides, so only the leakage magnetic field on one side is considered;
[0055] c) The end effect mainly affects the radial component of the magnetic flux density, and has little effect on the tangential component. The leakage flux calculation in this invention only considers the reduction of the radial component of the magnetic flux.
[0056] d) Magnetic circuit saturation is not considered;
[0057] e) There is no magnetic leakage at the center length, which serves as a reference point;
[0058] Since the unloaded magnetic field is equal to the magnetic field generated by the permanent magnet, and the loaded magnetic field is equal to the magnetic field generated by the permanent magnet plus the magnetic field generated by the winding; therefore, when considering both unloaded and loaded conditions, this invention proposes a finite element leakage magnetic field model for the permanent magnet that considers the end effects of the permanent magnet magnetic field, such as... Figure 1 As shown, the finite element leakage magnetic field model of the winding considers the end effect of the armature magnetic field, as follows. Figure 2 As shown.
[0059] First, the definition of the leakage flux coefficient Gl′ is given, which is the ratio of the magnetic flux density at the end to that at the center length. It is a curve that starts at 1 and gradually decreases (see attached figure), and the expression is as follows:
[0060]
[0061] In equation (1), the variable l is the coordinate on the calculation line in the leakage magnetic field analysis model. The calculation line extends from the center to the end, such as... Figure 1 , Figure 2 As shown; B(l) is the magnetic flux density at position l, B(R) mean The value at the center length point is 1. The initial part of the curve is equal to 1, meaning there is no magnetic leakage near the center length point; the descending part of the curve represents the magnetic leakage at the ends, and its value indicates a decrease in magnetic flux. For example... Figure 3 As shown, (a) is the magnetic flux density data obtained from the calculation line, and (b) is the corresponding leakage flux coefficient.
[0062] The leakage flux fitting function representing the leakage flux coefficient is given below:
[0063]
[0064] This fitting function transforms the leakage coefficient curve into a function, converting the solution for the leakage coefficient into solving for variables K1 and K2. When l = K1, the leakage coefficient Gl reaches half of its maximum value. The higher the value of parameter K1, the shorter the descending portion of the leakage coefficient Gl curve, i.e., the shorter the end where leakage exists. Parameter K2 determines the slope of the leakage coefficient Gl relative to length l. The higher the value of parameter K2, the steeper the leakage coefficient Gl curve, such as... Figure 4As shown in (a) and (b) in the figure. Both the leakage flux of the permanent magnet and the leakage flux of the winding can be represented by the leakage flux fitting function of equation (2). The fitting range is the length of the calculation line, which is equal to half of the stacking length. The key geometric parameters of permanent magnet thickness H, stacking length L, and air gap length g are extracted, and the variable value ranges are set as shown in Table 1.
[0065] Table 1. Geometric variables of the function and their range of variation
[0066]
[0067] Equations (3), (4), (5), and (6) are expressions for solving K1 and K2 using geometric parameters H, L, and g, where K1 pm K2 pm The parameters K1, K2, and K1 are obtained from the permanent magnet leakage magnetic field model. arm K2 arm These represent parameters K1 and K2 obtained from the winding leakage flux model. 1a K 1b K 1c K 1d K 2a K 2b K 2c K 2d K 2e K 2f K 2g K 2h K 1A K 1B K 1C K 2A K 2B K 2C K 2D K 2E These are constants obtained by fitting different geometric parameters.
[0068] K1 pm =K 1a +K 1b H+K 1c L+K 1d g (3)
[0069]
[0070] K1 arm =K 1A +K 1B L+K 1C g (5)
[0071] K2 arm =K 2A +K 2B L+K 2C g+K2D L / g+K 2E g -2 (6)
[0072] By solving the permanent magnet leakage magnetic model and the winding leakage magnetic model using equations (3), (4), (5), (6) and Table 1 respectively, the parameters K1 and K2 in the fitting function of equation (2) are obtained, and then the leakage magnetic coefficient Gl is obtained.
[0073] Step 2: Extract the geometric parameters of the prototype as variables.
[0074] The geometric parameters of the prototype are as follows: permanent magnet thickness H, stacking length L, and air gap length g. Substituting these into equations (3), (4), (5), and (6) yields K1 and K2. The Gl curve is calculated based on the leakage magnetic field fitting function given in equation (2), which is the leakage magnetic field coefficient curve. Finally, the leakage magnetic field coefficient Gl curve is flipped along the y-axis to obtain the leakage magnetic field coefficient distribution along the axial length of the complete motor. The overall flowchart is as follows. Figure 5 As shown.
[0075] Taking a prototype as an example, its three-dimensional finite element model is as follows: Figure 6 As shown, the permanent magnetization directions of the Halbach array are 0° and 90°, which can improve the air gap magnetic flux density and power density. Due to its short axial length, the end leakage flux has a significant impact on the output performance, making it necessary to consider the end effect.
[0076] The parameters H=6, L=45, and g=1 were extracted and substituted into the leakage magnetic flux fitting function to calculate the leakage magnetic flux coefficient over the complete axial length. Figure 7 This section compares the leakage flux curve of the permanent magnet with that of the finite element method under no-load conditions. Under no-load conditions, only the permanent magnet leakage flux needs to be considered, and the leakage flux coefficient Gl of the motor in the no-load state is... noload The leakage magnetic coefficient Gl calculated with the permanent magnet model pm same.
[0077] Gl noload =Gl pm (7)
[0078] Figure 8 This section compares the leakage flux curves of the permanent magnet and windings with those under load using finite element analysis. Under load, the magnetic flux density equals the sum of the magnetic flux density generated by the permanent magnet and the windings. (Treatment method) Motor load-state leakage flux coefficient Gl load With the permanent magnet model leakage coefficient Gl pm and the leakage flux coefficient Gl of the winding model coil The products are the same, and the expression is as follows:
[0079] Gl load =Gl pm ·Gl coil (8)
[0080] Depend on Figure 7 , Figure 8 It can be seen that the leakage magnetic field fitting curves have good consistency.
[0081] Step 3: Multiply the leakage magnetic coefficient Gl obtained from the leakage magnetic fitting function with the two-dimensional air gap magnetic flux density result to obtain the quasi-three-dimensional result magnetic flux density considering the end effect; the obtained leakage magnetic coefficient Gl can also be used to correct the two-dimensional back EMF and torque results.
[0082] As shown in equation (9), the radial component B of the air gap magnetic flux density obtained in two dimensions is... r Multiplying by the leakage flux coefficient Gl yields the radial component B of the air gap magnetic flux density at the end position. rend .
[0083] B rend =Gl·B r (9)
[0084] Cogging torque can be calculated using the Maxwell tensor method combined with the multi-section method, as follows:
[0085]
[0086] Where μ0 is the free permeability, and B r B θ These are the radial and tangential components of the magnetic flux density, respectively. stk θ is the stacking length of the motor, and θ is the tangential angle of the motor.
[0087] To more conveniently and directly consider end-effects, a weighting factor G is introduced as follows:
[0088]
[0089] That is, calculate the integral of the leakage magnetic coefficient Gl over the stacking length and convert the curve into a single value.
[0090] Multiplying the weighting factor G directly with the two-dimensional magnetic flux density back electromotive force or torque result yields a quasi-three-dimensional result that is very close to the three-dimensional finite element result.
[0091] Two-dimensional models, such as Figure 9 The magnetic flux density at the center of the air gap at a stacking length of 2.5 mm was measured using the sampling machine, and the radial and tangential components were analyzed. The end magnetic flux density was obtained by multiplying the magnetic flux density at the center length by the leakage flux coefficient Gl value at the corresponding position. The leakage flux coefficient Gl value at r = 2.5 mm was 0.85, meaning that the magnetic flux density at this location was only 85% of that at the center.
[0092] Figure 10The graph shows a comparison of magnetic flux density at 2.5 mm. The three lines represent the magnetic flux density curves obtained from the 2D model, the 2D model considering end effects, and the 3D model, respectively. Clearly, the 2D model considering end effects helps improve the accuracy of the flux waveform.
[0093] Figure 11 To output torque, the two-dimensional result T is used. 2D Multiplying by the weighting factor G yields the quasi-three-dimensional result T. 3D* The expression is as follows:
[0094] T 3D* =T 2D ·G (12)
[0095] It can be observed that after considering the end effect using the given leakage magnetic field fitting function, the results of the two-dimensional model are very close to those of the three-dimensional model.
[0096] (II) Verification Experiment
[0097] To verify the correctness of the method of the present invention, experiments on no-load characteristics and low-speed motor output characteristics were completed based on the prototype and its experimental platform, and the experimental results were analyzed.
[0098] The no-load back electromotive force waveform of the motor at its rated speed is as follows: Figure 12 As shown, the calculation results of the given model are basically consistent with the experimental results. The measured values of the line back electromotive force at different rotational speeds are compared, for example... Figure 13 As shown, the calculated back electromotive force after considering leakage flux is very close to the experimental results, with a very small error.
[0099] Output torque tests were conducted while maintaining the prototype's operating speed below 500 rpm. With the direct-axis current at 0 A, different magnitudes of orthogonal-axis currents were applied to obtain the motor torque (0–150 A) under different orthogonal-axis currents, resulting in the current-torque curves shown below. Figure 14 As shown, the torque calculation results are poor under high current input conditions, which may be due to magnetic circuit saturation. It can be observed that good consensus has been achieved, which also demonstrates the accuracy of the proposed method.
Claims
1. A method for calculating the end effect of a surface-mounted permanent magnet motor, characterized in that, The steps include the following: S1, under given conditions, the ratio of magnetic flux density at the end to that at the center length is defined as the leakage flux coefficient, and an expression for the leakage flux fitting function is proposed to calculate the leakage flux coefficient; S2, the permanent magnet thickness H, stacking length L and air gap length g of the prototype are extracted as variables of the leakage magnetic field fitting function; S3, multiply the leakage flux coefficient Gl by the air gap magnetic flux density to obtain the magnetic flux density considering the end leakage flux in the three-dimensional case; obtain the torque considering the end effect through Maxwell's tensor method; integrate the leakage flux coefficient Gl to obtain the weighting factor G, and multiply the weighting factor G by the two-dimensional model result to obtain the quasi-three-dimensional result; In step S1, the given conditions are as follows: a) Edge leakage flux is unrelated to the main magnetic flux; b) The leakage magnetic field is the same on both the inner and outer sides, so only the leakage magnetic field on one side is considered; c) The leakage flux due to end effect only considers the reduction of the radial component of the magnetic flux; d) Magnetic circuit saturation is not considered; e) There is no magnetic leakage at the center length, which serves as a reference point; In step S1, the expression for the magnetic flux leakage fitting function is as follows: ， In the formula, Here are the coordinates on the calculation line in the magnetic flux leakage analysis model, which extends from the center point to the end; K1 and K2 are parameters related to the permanent magnet thickness H, the stacking length L, and the air gap length g; Gl is the magnetic flux leakage coefficient.
2. The method for calculating the end effect of a surface-mounted permanent magnet motor according to claim 1, characterized in that, In step S3, the radial component B of the air gap magnetic flux density at the end position rend The expression is as follows: ， Among them, B r The radial component of the air gap magnetic flux density obtained in two dimensions.
3. The method for calculating the end effect of a surface-mounted permanent magnet motor according to claim 2, characterized in that, In step S3, the expression for the torque considering end effects is obtained using Maxwell's tensor method as follows: ， Where μ0 is the free permeability; B θ L is the tangential component of the magnetic flux density, and θ is the tangential angle of the motor. stk It is the stacking length of the motor.