High-precision analytical method for stray magnetic field of outer rotor permanent magnet motor

Abstract

A kind of high-precision outer rotor permanent magnet motor stray field analytical method, it is related to outer rotor permanent magnet motor magnetic field calculation field, including the following steps: S1 in two-dimensional polar coordinate system, according to the structure equivalence principle, outer rotor permanent magnet motor is divided into 5+n sub-domain;S2 the Laplace equation or Poisson equation of each sub-domain is established, the boundary condition of the interface of ferromagnetic material and non-ferromagnetic material is solved by separation of variables method, the vector magnetic potential of each sub-domain;S3 according to boundary condition column write constraint equation, solve the harmonic coefficient of each vector magnetic potential;S4 sub-domain model and iterative convergence algorithm are integrated, and the rotor unsaturated permeability of motor at different radiuses of core under various operating conditions is automatically iterated and updated;S5 the partial derivative of the vector magnetic potential of outer air domain is solved, and the radial component and tangential component of stray magnetic field are obtained.The present application can accurately predict the core unsaturated permeability of motor at different radiuses of core under different operating conditions, and improve the calculation accuracy of stray magnetic field.

Description

Technical Field

[0001] The present invention relates to the field of magnetic field calculation of outer rotor permanent magnet motors, and in particular to a high-precision stray magnetic field analysis method for outer rotor permanent magnet motors. Background Art

[0002] With the widespread adoption of permanent magnet motors, data-driven methods for diagnosing motor faults using stray magnetic fields as signals are becoming increasingly popular. Acquiring large amounts of accurate stray magnetic field data in a short period of time is fundamental to motor fault diagnosis, as exemplified by patents such as ZL202110367107.X and CN113985282A. Compared to numerical methods, analytical methods offer the advantages of faster computation speed and higher efficiency, as exemplified by patents such as CN116244847A, CN116629167A, and CN117574636A. Analytical calculations of stray magnetic fields can be divided into attenuation coefficient methods and subdomain methods. The subdomain method, as demonstrated by patents published or announced with the following numbers: CN116629167A, CN114006559A, and CN108875168B, offers relatively high accuracy.

[0003] However, for external rotor permanent magnet motors, existing technologies only consider the saturation magnetic permeability of the core, specifying the magnetic permeability of the rotor core when it is unsaturated as a constant, ignoring the impact of the unsaturated magnetic permeability on the stray magnetic field. This is evident in patent documents such as those published or announced with the following numbers: CN108563912A, CN113343171A, CN108875168B, and CN 112949146 A. Under different motor operating conditions, the current and the remanent magnetization of the permanent magnets jointly affect the magnetic flux density at the same position on the rotor core, thereby affecting the magnetic permeability. Even under fixed operating conditions, the magnetic flux density varies at different rotor radii, causing the magnetic permeability to vary with radius. The magnetic flux density affects the magnetic permeability of the core, and the magnetic permeability determines the magnetic resistance of the core, which affects the attenuation of the magnetic field in the radial direction. This means that the magnetic field and magnetic permeability interact with each other. Therefore, even without considering the changes in various operating conditions of the motor, the change of the unsaturated magnetic permeability with the rotor radius will reduce the accuracy of the stray magnetic field analytical calculation; under the further influence of the operating condition changes, the accuracy of the stray magnetic field analytical calculation is low and the error is large. Summary of the Invention

[0004] In order to overcome the technical problems raised in the background technology, the present invention provides a high-precision outer rotor permanent magnet motor stray magnetic field analysis method that can accurately consider the influence of the unsaturated magnetic permeability of the rotor core and improve the accuracy of the stray magnetic field analysis calculation.

[0005] The technical solutions of the present invention are as follows:

[0006] A high-precision stray magnetic field analysis method for an outer rotor permanent magnet motor, characterized by comprising the following steps:

[0007] S1: In a two-dimensional polar coordinate system, according to the principle of structural equivalence, the outer rotor permanent magnet motor is divided into 5+n subdomains;

[0008] S2: Establish the Laplace equation or Poisson equation for each subdomain, and solve the vector magnetic potential of each subdomain by separation of variables method and boundary conditions at the interface between ferromagnetic material and non-ferromagnetic material;

[0009] S3: Write constraint equations based on boundary conditions and solve the harmonic coefficients of each vector magnetic potential;

[0010] S4: The subdomain model is integrated with the iterative convergence algorithm to automatically iteratively update the rotor unsaturated permeability at different core radii under various motor operating conditions.

[0011] S5: Calculate the partial derivative of the vector magnetic potential in the external air domain to calculate the radial and tangential components of the stray magnetic field. Figure 1 shown.

[0012] The S1 comprises the following steps:

[0013] S1.1 Take a circular cross section of the motor and establish a two-dimensional polar coordinate system with the center point of the cross section as the coordinate origin.

[0014] S1.2 According to the equivalence principle, the motor's external air, rotor core, permanent magnets, and air gap are equivalent to an annular domain in the polar coordinate system, and the slot opening and stator slot are equivalent to a sector domain in the polar coordinate system, as shown in the following example: Figure 2 shown.

[0015] The permanent magnet motor has an outer rotor and inner stator structure, and the magnetic field propagates radially through the permanent magnets and the rotor core. The difference in unsaturated magnetic permeability at different radii of the rotor core has a non-negligible effect on magnetic field attenuation. In order to consider the difference in unsaturated magnetic permeability at different positions of the rotor, the rotor core is evenly divided into n annular domains along the radial direction. The larger n is, the more accurately the difference in unsaturated magnetic permeability at different positions of the rotor is considered, but the longer the calculation time is. Taking into account both calculation accuracy and calculation efficiency, n can usually be 3-6. In the present invention, n is 3.

[0016] After subdividing the rotor core into n annular domains, the subdomain names are uniformly defined. From the outside in, the subdomains are named: the outer air domain is subdomain 1, rotor core domain 1 is subdomain 2, rotor core domain 2 is subdomain 3, ..., rotor core domain n is subdomain n+1, the permanent magnet domain is subdomain n+2, the air gap domain is subdomain n+3, the slot opening domain is subdomain n+4, and the slot subdomain is subdomain n+5.

[0017] The S2 comprises the following steps:

[0018] S2.1 According to the Ampere circuit law of Maxwell's equations and the constitutive relationship of isotropic linear media, the governing equation of the subdomain is:

[0019]

[0020] Where A is the vector magnetic potential. r 、M θ are the radial and tangential components of the residual magnetization of the permanent magnet, respectively. r and θ represent the radial radius and tangential angle of the polar coordinate system. J is the current density. μ0, μ r represent the vacuum permeability and the permeability of other subdomain materials respectively;

[0021] S2.2 Determine the boundary conditions for the interface between the rotor core and non-ferromagnetic material, and the interface between the stator core and non-ferromagnetic material:

[0022] Complex boundary conditions are used at the interface between the rotor core, permanent magnets and external air:

[0023]

[0024]

[0025] Where A1, A2, H 1θ 、H 2θ 、A n+1 、H (n+1)θ 、A n+2 、H (n+2)θ Respectively represent the vector magnetic potential of the external air domain, the vector magnetic potential of the rotor core domain 1, the tangential magnetic field intensity of the external air domain, the tangential magnetic field intensity of the rotor core domain 1, the vector magnetic potential of the rotor core domain n, the tangential magnetic field intensity of the rotor core domain n, the vector magnetic potential of the permanent magnet domain, and the tangential magnetic field intensity of the permanent magnet domain. When n takes different values, just substitute them into the above general formula. R ro 、R r They represent the radius of the interface between the external air domain and the rotor core domain 2, and the radius of the interface between the rotor core domain n and the permanent magnet domain, respectively.

[0026] Since the stator structure is located inside the permanent magnet, the magnetic field does not pass through the stator when leaking radially to the outside of the motor. To simplify the calculation process, homogeneous boundary conditions are still used at the intersection of the stator slots, slot openings, and the stator core:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] Where A (n+4)i 、A (n+5)i 、B (n+5)iθ Represent the vector magnetic potential of the i-th slot opening domain, the vector magnetic potential of the i-th slot subdomain, and the tangential magnetic flux density of the i-th slot subdomain respectively. i Refers to the center position angle of the i-th slot. i represents the number of the slot subdomain and slot opening domain, i ranges from 1 to N s . N s Refers to the total number of slot sub-domains and slot opening domains of the motor. In this invention, N s Take 51. β sa , β oa Represent the width of the slot subdomain and slot opening domain respectively. sb Represents the groove bottom radius of the groove subdomain.

[0033] S2.3 Use the separation of variables method and the interface conditions between the stator and rotor cores and the non-ferromagnetic material to calculate the vector magnetic potential of each subdomain:

[0034] Since the homogeneous boundary conditions at the interface between the rotor core and the non-ferromagnetic material are abandoned, the vector magnetic potentials of the permanent magnet domain and the external air domain cannot be simplified. At this time, the vector magnetic potentials of the two are:

[0035]

[0036]

[0037] in,

[0038]

[0039]

[0040] Where k is the harmonic order, ω r is the angular velocity of the motor rotor, t is the time, α0 is the initial angle of the motor rotor, B r is the residual magnetic strength of the motor permanent magnet, α p is the pole arc coefficient of the motor, p is the number of motor pole pairs, and μ0 is the vacuum permeability. r 、R m 、R a are the radius of the interface between the rotor core and the permanent magnet, the radius of the interface between the permanent magnet and the air gap, and the outer radius of the external air. (n+2)k 、B (n+2)k 、C (n+2)k 、D (n+2)k is the desired harmonic coefficient of the vector magnetic potential of the permanent magnet domain. A 1k 、B1k 、C 1k 、D 1k is the desired harmonic coefficient of the vector magnetic potential in the outer air domain.

[0041] The vector magnetic potentials of other subdomains all satisfy the following formula, which can be further simplified by boundary conditions:

[0042]

[0043] Where A v0 、B v0 、A vx 、B vx 、C vx 、D vx Represents the harmonic coefficient of the vector magnetic potential of the vth solution domain. R vou 、R vin Represent the outer radius and inner radius of the vth solution domain respectively. x represents the harmonic order of the vth subdomain. r represents the solution radius. γ v is the period of the Fourier series of the vth solution domain. v0 is the initial phase angle of the vth solution domain.

[0044] The S3 comprises the following steps:

[0045] S3.1 Write the constraint equations according to the boundary conditions and solve the harmonic coefficients of the vector magnetic potential. The boundary conditions between each subdomain are: the vector magnetic potential is equal, the tangential magnetic field strength is equal,

[0046] A x | r＝R =A y | r＝R

[0047] H xθ | r＝R =H yθ | r＝R

[0048] In order to more intuitively show the importance of the unsaturated magnetic permeability at different rotor radii, the relationship between the magnetic field strength and magnetic permeability of the rotor core is given:

[0049] H θ|rotor =μ Fe B θ|rotor

[0050] Where,

[0051]

[0052] x and y represent the numbers of two adjacent subdomains, respectively, and R represents the radius of the interface between two adjacent subdomains; B θ|rotor 、H θ|rotor、A rotor 、μ Fe They represent the tangential flux density, tangential magnetic field intensity, vector magnetic potential and magnetic permeability of the rotor core respectively.

[0053] The above formula shows that the boundary condition of "equal tangential magnetic field strength" at the interface between the rotor core and the non-ferromagnetic material is related to the core's unsaturated permeability. Accurately predicting the permeability at different core radii is key to improving the accuracy of boundary condition calculations. Methods for predicting permeability at different core radii are detailed in Section S4.

[0054] S3.2 When the Fourier series periods of two adjacent subdomains are different, the Fourier series index variables of the two are unified by the Fourier expansion method. Take the vector magnetic potential at the interface between the slot opening domain and the slot subdomain as an example:

[0055]

[0056] The two expand into:

[0057]

[0058]

[0059] Where,

[0060]

[0061]

[0062] R s is the radius of the interface between the slot opening domain and the air gap domain; m and f are the harmonic coefficients of the slot opening domain and the slot subdomain respectively; B (n+4)i0 、A (n+4)im 、B (n+4)im is the desired harmonic coefficient of the vector magnetic potential in the slot opening region; B (n+5)if is the desired harmonic coefficient of the vector magnetic potential in the slot subdomain.

[0063] In order to unify the Fourier series index variables of the two subdomains, the vector magnetic potential of the slot subdomain is Fourier expanded:

[0064]

[0065] Where,

[0066]

[0067]

[0068] in,

[0069]

[0070] The boundary conditions after unifying the Fourier series indicator variables can be expressed as follows:

[0071]

[0072] Where A (n+5)Fou. is the vector magnetic potential of the slot subdomain after Fourier expansion.

[0073] From the above formula, the constraint equation can be derived:

[0074]

[0075]

[0076] Similarly, by deriving the constraint equations for each adjacent subdomain using the boundary conditions (equal vector magnetic potential and equal tangential magnetic field strength) described in step S3.1, and solving all constraint equations simultaneously, we can obtain the harmonic coefficients for each subdomain. Substituting these harmonic coefficients into the vector magnetic potential from step S2.3 and taking the partial derivative of the vector magnetic potential yields the magnetic flux density for each subdomain.

[0077] The S4 step is as follows:

[0078] S4.1 Interpolate the B(H) curve inherent to the rotor ferromagnetic material and estimate the function between magnetic permeability and magnetic flux density to obtain μ Fe (B) curve, such as Figure 4 shown.

[0079] S4.2 Solve the partial derivatives of the vector magnetic potential of n rotor core domains to obtain the magnetic flux density at different radii of the rotor core.

[0080]

[0081]

[0082]

[0083] Where B 2r 、B 3r 、B (n+1)r Represent the radial magnetic flux density of subdomain 2, the radial magnetic flux density of subdomain 3, and the radial magnetic flux density of subdomain n+1 respectively. n+1 Represent the solution radius of subdomains 2, 3, and n+1 respectively. ro1 、R r1 A represents the radius of the interface between subdomain 2 and subdomain 3, and the radius of the interface between subdomain 3 and subdomain n+1. 2k 、B 2k 、C 2k 、D 2k is the harmonic coefficient of the vector magnetic potential of subdomain 2.3k 、B 3k 、C 3k 、D 3k is the harmonic coefficient of the vector magnetic potential of subdomain 3. (n+1)k 、B (n+1)k 、C (n+1)k 、D (n+1)k is the harmonic coefficient of the vector magnetic potential of subdomain n+1.

[0084] S4.3 combines the subdomain method in steps S1-S3 with the iterative convergence algorithm to automatically iteratively update the values ​​of the magnetic permeability at different radii of the rotor core under various working conditions of the motor. The process is as follows Figure 5 As shown. Combining the subdomain method with the iterative convergence algorithm means first assigning an initial value to the magnetic permeability at different radii of the rotor core through empirical values, and then solving the magnetic flux density at different radii of the rotor core through the subdomain method. Find the B(H) curve of the material used for the rotor core, and calculate μ by interpolation of the B(H) curve. Fe (B) Curve. The rotor core magnetic flux B calculated by the subdomain method is used as the independent variable, and μ Fe (B) The corresponding magnetic permeability μ is calculated from the curve Fe . Determine the initial assigned magnetic permeability and the μ Fe (B) Whether the difference in magnetic permeability calculated from the curve is within the tolerance range. If not, the magnetic permeability is updated according to the magnetic permeability range and the iterative algorithm is used to iterate again until the magnetic permeability meets the tolerance.

[0085] S4.3.1 Based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, ..., and subdomain n+1 are respectively Feb Assign an initial value; b takes 2, 3, ..., n+1, representing the unsaturated permeability μ of subdomains 2, 3, and subdomain n+1 respectively Fe2 、μ Fe3 、μ Fen+1 ;

[0086] S4.3.2 Substitute the initial value of magnetic permeability into the subdomain model. The rotor flux density formula in S4.2 can be obtained by the subdomain method to obtain the flux density at different rotor radii. In order to improve the fault tolerance of the model, the flux density value range of the three solution domains is calculated based on 20% of the maximum value of the flux density in each subdomain. The flux density value is used as the function curve μ Fe The independent variable of (B) can be used to obtain the range of magnetic permeability.

[0087] S4.3.3 Take the magnetic flux density at different rotor radii as the independent variable and use the function curve μ Fe (B) The corresponding μ can be obtained Fetb , judge μ Fetb and μ FebWhether the error requirements are met, the error formula is as follows:

[0088]

[0089] Where μ Feb is the magnetic permeability in the subdomain model, which is automatically assigned through the iterative loop program; μ Fetb is the magnetic permeability obtained from the B(H) curve.

[0090] If the error requirements are not met, the iterative algorithm is combined with the subdomain method to continue the iteration. In addition, the magnetic permeabilities at different rotor radii will affect each other, so the three magnetic permeabilities in the iterative procedure must meet the error requirements simultaneously.

[0091] In the present invention, the accurately predicted magnetic permeability is updated to the boundary condition "equal tangential magnetic field strength" of step S3.1, thereby improving the accuracy of the calculation of the tangential magnetic field strength and thus improving the calculation accuracy of the stray magnetic field.

[0092] The S5 step is as follows:

[0093] S5.1 Take the partial derivative of the vector magnetic potential in the external air domain to obtain the radial and tangential components of the stray magnetic field.

[0094]

[0095]

[0096] The analytical calculation method for the stray magnetic field of an outer rotor permanent magnet motor provided by the present invention takes into account the rotor's unsaturated magnetic permeability distribution and has the following beneficial effects:

[0097] (1) The important influence of the unsaturated magnetic permeability of the rotor core on the stray magnetic field is taken into account, and the automatic update of the unsaturated magnetic permeability at different rotor radii under different current magnitudes and permanent magnet operating points is achieved through the collaborative integration of subdomain models and iterative convergence algorithms.

[0098] (2) The magnetic permeability in the boundary conditions at the inner and outer radii of the rotor core is distinguished and predicted, which improves the calculation accuracy of the tangential magnetic field strength at the interface between the non-ferromagnetic material and the rotor core.

[0099] (3) While meeting the computational accuracy requirements, the analytical model is 15.2 times faster than the finite element method and occupies only 4% of the computer memory. The computational accuracy of the analytical model proposed in this invention is similar to that of the finite element method, but the computational efficiency is much higher than that of the finite element method.

[0100] (4) The stray magnetic field analytical calculation method of the present invention can not only accurately solve the stray magnetic field, but also analyze the spatial order-amplitude-frequency characteristics of the stray magnetic field of the outer rotor permanent magnet motor, study the influence mechanism of the rotor unsaturated magnetic permeability on the stray magnetic field, and provide a reference for the analytical calculation of the stray magnetic field of other types of motors.

[0101] To address the issue of how the unsaturated magnetic permeability of a motor varies with rotor radius and operating conditions, this method divides the rotor core into multiple solution domains and integrates a subdomain method with an iterative convergence algorithm. This method accurately predicts the unsaturated magnetic permeability of the core under different motor operating conditions and at different core radii. Based on this predicted permeability, it improves the accuracy of the boundary conditions at the interface between the rotor core and non-ferromagnetic materials, thereby accurately calculating the stray magnetic field. BRIEF DESCRIPTION OF THE DRAWINGS

[0102] Figure 1 is a flow chart of the present invention;

[0103] Figure 2 This is a schematic diagram of the sub-domain division of the stray magnetic field of the outer rotor permanent magnet motor;

[0104] Figure 3 This is a schematic diagram of the boundary conditions of the stray magnetic field subdomain model;

[0105] Figure 4 is the B(H) curve of the motor rotor ferromagnetic material and μ obtained by interpolation Fe (B) Graph;

[0106] Figure 5 It is a flow chart of the iterative convergence algorithm for updating the non-saturated magnetic permeability of the rotor core at different radii.

[0107] Figure 6 is a schematic diagram of the stray magnetic field calculated by the present invention;

[0108] Figure 7 It is a schematic diagram comparing the stray magnetic field calculated by the method of the present invention and the finite element method with the stray magnetic field measured experimentally. DETAILED DESCRIPTION

[0109] The present invention will be described in detail below with reference to the accompanying drawings and specific examples. The embodiment provides a detailed implementation method and specific operating steps based on the technical method of the present invention. It should be pointed out that the protection scope of the present invention is not limited to the examples described below. Taking a certain outer rotor permanent magnet motor as an example, its rated speed is 440rpm, it has 46 poles and 51 slots, the remanence of the permanent magnet is 0.35T, and the relative magnetic permeability of the permanent magnet is 1.1. According to the method of the present invention, the load stray magnetic field of the prototype is calculated, and the process is as follows: Figure 1 The specific steps are as follows:

[0110] S1: Represent an outer rotor permanent magnet motor in a two-dimensional polar coordinate system. Based on the principle of structural equivalence, the outer rotor permanent magnet motor is divided into 5+n subdomains. Here, n represents the number of subdomains into which the rotor core is evenly divided along the radial direction and is a positive integer. A larger n indicates a more accurate consideration of the unsaturated permeability at different radii of the rotor core, but the calculation time increases. Considering both accuracy and time, n is typically 3-6; in this embodiment, n is 3.

[0111] S1.1 Take a circular cross section of the motor and establish a two-dimensional polar coordinate system with the center point of the cross section as the coordinate origin.

[0112] S1.2 Based on the equivalence principle, the motor's external air, rotor core, permanent magnets, and air gap are equated to an annular domain in a polar coordinate system, and the slot openings and stator slots are equated to sector-shaped domains in a polar coordinate system. Considering both computational complexity and accuracy, n is set to 3 in this invention, dividing the rotor core into three subdomains.

[0113] After the rotor core is subdivided into n annular domains, rotor core domain 1, rotor core domain 2, ..., rotor core domain n are obtained; for the convenience of description, the names of the subdomains are uniformly defined as: the outer air domain is subdomain 1, the rotor core domain 1 is subdomain 2, the rotor core domain 2 is subdomain 3, ..., the rotor core domain n is subdomain n+1, the permanent magnet domain is subdomain n+2, the air gap domain is subdomain n+3, the slot opening domain is subdomain n+4, and the slot subdomain is subdomain n+5. The model diagram of the motor subdomain division is shown in the figure below. Figure 2 shown.

[0114] S2: Establish the Laplace equation or Poisson equation for each subdomain, and solve the vector magnetic potential of each subdomain through the separation of variables method and the boundary conditions at the interface between ferromagnetic materials and non-ferromagnetic materials.

[0115] S2.1 According to Ampere's circuit law of Maxwell's equations and the constitutive relation of isotropic linear media, the governing equation of the subdomain is:

[0116]

[0117] Since the permanent magnet domain and slot subdomain contain magnetic field excitation sources, the governing equations for them are Poisson's equations, while the governing equations for the remaining subdomains are Laplace equations.

[0118] The governing equations for the permanent magnet domain are:

[0119]

[0120] Where,

[0121]

[0122]

[0123] in,

[0124]

[0125] M θk =0

[0126] The governing equation of the slot subdomain is:

[0127]

[0128] Where,

[0129]

[0130] J i1 、J i2 Represent the upper and lower currents in the slot subdomain respectively, S is the cross-sectional area of ​​the slot, and f is the harmonic order of the slot subdomain. i0 is the fundamental amplitude of current density, J if is the harmonic amplitude of current density f, θ i is the position of the i-th slot.

[0131] The governing equations for the other subdomains are:

[0132]

[0133] S2.2 Determine the boundary conditions for the interface between the rotor core and non-ferromagnetic material, and for the interface between the rotor core and non-ferromagnetic material. The prototype is an outer rotor permanent magnet motor. The magnetic field decays outward from the permanent magnets through the rotor, not the stator. To ensure accuracy and minimize the complexity of the derivation, complex boundary conditions are used for the interface between the rotor and non-ferromagnetic material, while homogeneous boundary conditions are used for the interface between the stator and non-ferromagnetic material.

[0134] The interface conditions between the rotor core, permanent magnets and external air are:

[0135]

[0136]

[0137]

[0138]

[0139] Where R ro =110.6mm; R m =100.8mm.

[0140] The interface conditions between the stator slots, slot openings and stator core are:

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] Where, β sa =2.5999deg;β oa =2.2985deg; R sb =86.6mm.

[0147] S2.3 The vector magnetic potential of each subdomain is calculated by the separation of variables method, combined with the interface conditions between the stator and rotor cores and the non-ferromagnetic material.

[0148] The vector magnetic potential of subdomain 1:

[0149]

[0150] Where R a =2212mm, R ro =110.6mm.

[0151] The vector magnetic potential of subdomain 2:

[0152]

[0153] Where R ro1 =107.9mm.

[0154] The vector magnetic potential of subdomain 3:

[0155]

[0156] Where R r1 =105.2mm.

[0157] The vector magnetic potential of subdomain 4:

[0158]

[0159] Where R r =102.5mm.

[0160] The vector magnetic potential of subdomain 5:

[0161]

[0162] Where R m =100.8mm.

[0163] The vector magnetic potential of subdomain 6:

[0164]

[0165] Where R s =100.0mm.

[0166] The vector magnetic potential of subdomain 7:

[0167]

[0168] Where, R t =98.02mm, m is the harmonic order.

[0169] The vector magnetic potential of subdomain 8:

[0170]

[0171] Where, R sb =86.7mm.

[0172] S3: List some constraint equations based on the boundary conditions and solve the harmonic coefficients of each vector magnetic potential.

[0173] S3.1 Use boundary conditions to write constraint equations and solve the vector magnetic potential for the harmonic coefficients. The boundary conditions between subdomains are: equal vector magnetic potential and equal tangential magnetic field strength.

[0174] A x | r＝R =A y | r＝R

[0175] H xθ | r＝R =H yθ | r＝R

[0176] Where x and y represent the numbers of two adjacent subdomains, and R represents the radius of the interface between two adjacent subdomains.

[0177] The tangential magnetic field strength is related to the magnetic permeability. The expression of the rotor tangential magnetic field strength is:

[0178] H θ|rotor =μ Fe B θ|rotor

[0179] Where,

[0180]

[0181] In the present invention, the magnetic permeability of the rotor core at different radii is accurately predicted through step S4 and then updated to the boundary condition of "equal tangential magnetic field strength", thereby improving the accuracy of the tangential magnetic field strength calculation in the boundary condition and ultimately improving the accuracy of the stray magnetic field calculation.

[0182] S3.2 When the Fourier series periods of two subdomains are different, it is necessary to unify the Fourier series index variables. Take the case where the vector magnetic potential at the interface between the slot subdomain and the slot opening domain is equal as an example:

[0183]

[0184] The two expand into:

[0185]

[0186]

[0187] In order to unify the Fourier series index variables of the two subdomains, the vector magnetic potential of the slot subdomain is Fourier expanded:

[0188]

[0189] Where,

[0190]

[0191]

[0192] in,

[0193]

[0194] The boundary conditions after unifying the Fourier series indicator variables can be expressed as follows:

[0195]

[0196] From the above formula, the constraint equation can be derived:

[0197]

[0198]

[0199] Similarly, through Figure 3 The constraint equations are derived from the other boundary conditions given in step S3.1. Solving all the constraint equations simultaneously yields the harmonic coefficients. Substituting the harmonic coefficients into the vector magnetic potential equations in step S2.3 and taking the partial derivative yields the magnetic flux density for each subdomain.

[0200] S4: μ is obtained by interpolation based on the inherent B(H) curve of the ferromagnetic material used in the rotor. FeCurve (B) solves the rotor core magnetic flux density based on the subdomain model established in S1-3. Combining the subdomain model established in S1-S3 with the convergence algorithm automatically iterates and updates the rotor's unsaturated magnetic permeability at different core radii under different motor operating conditions.

[0201] S4.1 Query the inherent B(H) curve of the ferromagnetic material used in the rotor, interpolate the B(H) curve, estimate the function between magnetic permeability and magnetic flux density, and obtain μ Fe (B) curve, such as Figure 4 As shown, the present invention selects a one-dimensional linear interpolation method.

[0202] S4.2 Substitute the harmonic coefficients of the rotor core domain obtained in S3 into its vector magnetic potential, and then calculate the partial derivative of the vector magnetic potential to obtain the magnetic flux density at different radii of the rotor core.

[0203]

[0204]

[0205]

[0206] S4.3 combines the subdomain method in steps S1-S3 with the iterative convergence algorithm to automatically iteratively update the values ​​of the magnetic permeability at different radii of the rotor core under various working conditions of the motor. The process is as follows Figure 5 As shown. Combining the subdomain method with the iterative convergence algorithm means first assigning an initial value to the magnetic permeability at different radii of the rotor core through empirical values, and then solving the magnetic flux density at different radii of the rotor core through the subdomain method. Find the B(H) curve of the material used for the rotor core, and calculate μ by interpolation of the B(H) curve. Fe (B) Curve. The rotor core magnetic flux B calculated by the subdomain method is used as the independent variable, and μ Fe (μ) curve to calculate the corresponding magnetic permeability μ Fe . Determine the initial assigned magnetic permeability and the μ Fe (B) Whether the difference in magnetic permeability calculated from the curve is within the tolerance range. If not, the magnetic permeability is updated according to the magnetic permeability range and the iterative algorithm is used to iterate again until the magnetic permeability meets the tolerance.

[0207] S4.3.1 First, based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, and subdomain 4 are Feb Assign initial values ​​2000μ0, 10000μ0, and 20000μ0.

[0208] S4.3.2 The magnetic flux density of subdomain 2, subdomain 3, and subdomain 4 is solved by the subdomain model. In this embodiment, the maximum magnetic flux density of the three subdomains is 0.039T, 0.1826T, and 0.5615T respectively. In order to improve the fault tolerance of the model, the magnetic flux density value ranges of the three solution domains are calculated based on 20% of the maximum magnetic flux density of each subdomain: (0.0312T, 0.0468T), (0.1466T, 0.2191T), and (0.4492T, 0.67T). Combined with the magnetic flux density value range and the μ of the rotor ferromagnetic material, Fe (B) Curve. The calculated permeabilities range for the three parameters are (2303μ0, 4256μ0), (6700μ0, 24032μ0), and (10260μ0, 24032μ0). Ultimately, the initial permeabilities for subdomains 2, 3, and 4 are set to 2303μ0, 6700μ0, and 10329μ0, respectively. Based on empirical values, the permeabilities are updated every 0.5μ0.

[0209] S4.3.3 μ Feb After the value is assigned, it is inserted into the subdomain model to obtain the magnetic flux density of the rotor core, and then the magnetic flux density of the rotor core is substituted into the function curve μ Fe The magnetic permeability μ is calculated in (B) Fetb , to determine whether the difference between the two meets the error requirement. Taking into account the calculation efficiency and accuracy, the error is set to no more than 0.1:

[0210]

[0211] Where μ Feb is the magnetic permeability in the subdomain model, which is automatically assigned by the loop program; μ Fetb is the magnetic permeability obtained by the B(H) curve. To ensure the accuracy of the calculation, the magnetic permeabilities of the three rotor core domains must simultaneously meet the error requirements.

[0212] S4.3.4 Under the working conditions of input current 7A and permanent magnet remanence 0.35T, the final magnetic permeabilities of subdomains 2, 3, and 4 are 2307μ0, 24032μ0, and 10265μ0 respectively.

[0213] S5: Calculate the partial derivative of the vector magnetic potential in the external air domain to calculate the radial and tangential components of the stray magnetic field. The calculation results are as follows Figure 6 shown.

[0214]

[0215]

[0216] The stray magnetic field of the prototype load was experimentally measured. The experimental input current was the same as that of the analytical model and the finite element model, both of which were 7A sinusoidal three-phase alternating current. The stray magnetic field was measured using a Hall probe, and the stray magnetic field data was collected using a Gaussmeter. In addition, to ensure the measurement accuracy of the stray magnetic field, the angle of the Hall probe must be adjusted to ensure that the magnetic field enters the Hall probe vertically. The stray magnetic field calculation results of the proposed method are compared with the finite element results and the experimental measurement results. Figure 7 As shown in the figure, the analytical results of stray magnetic field and electromagnetic torque are in good agreement with the experimental results. The root mean square error of the analytical and experimental results is selected to evaluate the solution accuracy of the model proposed in this paper. The root mean square error of the stray magnetic field is defined as:

[0217]

[0218] Where W p is the number of sampling points evenly distributed along the circumference of the cross section at 1 mm outside the motor, j represents the number of the sampling point, and The calculation results and experimental measurement results of the j-th sampling point of the stray magnetic field are shown in Table 1. The workstation used for calculation is Dell Precision 7920, which has a 36-core CPU and 512GB of memory. The calculation accuracy and efficiency of the analytical model and the finite element model are shown in Table 1. Under the same calculation case and computing hardware conditions, the B 1r and B 1t The root mean square errors of the finite element model are 0.036 and 0.029 respectively. 1r and B 1t The root mean square errors (RMS) are 0.021 and 0.02, respectively. While maintaining computational accuracy, the analytical model is 15.2 times faster than the finite element method and occupies only 4% of the computer memory. This demonstrates that the improved analytical model proposed in this paper strikes a good balance between computational efficiency and accuracy.

[0219] Table 1 Accuracy and efficiency verification of analytical model and finite element model

[0220]

Claims

1. A high-precision outer rotor permanent magnet motor stray magnetic field analysis method, characterized in that The following steps are involved: S1: In a two-dimensional polar coordinate system, according to the principle of structural equivalence, the outer rotor permanent magnet motor is divided into 5+n subdomains; S2: Establish the Laplace equation or Poisson equation for each subdomain, and solve the vector magnetic potential of each subdomain by separation of variables method and boundary conditions at the interface between ferromagnetic material and non-ferromagnetic material; S3: Write constraint equations based on boundary conditions and solve the harmonic coefficients of each vector magnetic potential; S4: The subdomain model is integrated with the iterative convergence algorithm to automatically iteratively update the rotor unsaturated permeability at different core radii under various motor operating conditions. S5: Calculate the partial derivative of the vector magnetic potential in the external air domain to calculate the radial and tangential components of the stray magnetic field; The S3 comprises the following steps: S3.1 Write the constraint equations according to the boundary conditions and solve the harmonic coefficients of the vector magnetic potential. The boundary conditions between each subdomain are: the vector magnetic potential is equal, the tangential magnetic field strength is equal, A x | r＝R =A y | r＝R H xθ | r＝R =H yθ | r＝R The relationship between the magnetic field strength and magnetic permeability of the rotor core: H θ|rotor =μ Fe B θ|rotor Where, x and y represent the numbers of two adjacent subdomains, respectively, and R represents the radius of the interface between two adjacent subdomains; B θ|rotor 、H θ|rotor 、A rotor 、μ Fe They represent the tangential flux density, tangential magnetic field intensity, vector magnetic potential and magnetic permeability of the rotor core respectively; S3.2 When the Fourier series periods of two adjacent subdomains are different, the Fourier series index variables of the two subdomains are unified by the Fourier expansion method; Substituting the harmonic coefficient into the vector magnetic potential in step S2 and then taking the partial derivative of the vector magnetic potential, the magnetic flux density of each subdomain can be obtained. The S4 step is as follows: S4.1 Interpolate the B(H) curve inherent to the rotor ferromagnetic material and estimate the function between magnetic permeability and magnetic flux density to obtain μ Fe (B) Curve; S4.2 solves the partial derivatives of the magnetic potential vector of n rotor core domains to obtain the magnetic flux density at different radii of the rotor core. Where B 2r 、B 3r 、B (n+1)r They represent the radial magnetic flux density of subdomain 2, the radial magnetic flux density of subdomain 3, and the radial magnetic flux density of subdomain n+1, r2, r3, and r n+1 Represent the solution radius of subdomains 2, 3, and n+1 respectively, R ro represents the radius of the interface between the external air domain and the rotor core domain 1, R ro1 、R r1 They represent the radius of the interface between subdomain 2 and subdomain 3, and the radius of the interface between subdomain 3 and subdomain n+1, respectively. 2k 、B 2k 、C 2k 、D 2k is the harmonic coefficient of the vector magnetic potential of subdomain 2, A 3k 、B 3k 、C 3k 、D 3k is the harmonic coefficient of the vector magnetic potential of subdomain 3, A (n+1)k 、B (n+1)k 、C (n+1)k 、D (n+1)k is the harmonic coefficient of the vector magnetic potential of subdomain n+1; S4.3 combines the subdomain method in steps S1-S3 with the iterative convergence algorithm to automatically iteratively update the values ​​of the magnetic permeability at different radii of the rotor core under various working conditions of the motor; wherein, combining the subdomain method with the iterative convergence algorithm means first assigning initial values ​​to the magnetic permeability at different radii of the rotor core through empirical values, and then solving the magnetic flux density at different radii of the rotor core through the subdomain method on this basis, finding the B(H) curve of the material used for the rotor core, and obtaining μ by interpolation calculation of the B(H) curve. Fe (B) Curve, using the rotor core magnetic flux B calculated by the subdomain method as the independent variable, through μ Fe (B) The corresponding magnetic permeability μ is calculated from the curve Fe , determine the initial assigned magnetic permeability and the μ Fe (B) Whether the difference in magnetic permeability calculated by the curve is within the allowable error range. If it does not meet the error requirement, the magnetic permeability is updated according to the value range of the magnetic permeability, and the iterative calculation is performed again through the iterative algorithm until the magnetic permeability meets the error requirement.

2. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 1 is characterized in that The S1 comprises the following steps: S1.1 Take a circular section of the motor and establish a two-dimensional polar coordinate system with the center point of the section as the coordinate origin; S1.2 According to the equivalence principle, the motor's external air, rotor core, permanent magnets, and air gap are equivalent to an annular domain in the polar coordinate system, and the slot opening and stator slot are equivalent to a sector domain in the polar coordinate system; In which, the permanent magnet motor has an outer rotor and inner stator structure, and the magnetic field propagates radially through the permanent magnets and the rotor core; the rotor core is evenly divided into n annular domains along the radial direction; after the rotor core is subdivided into n annular domains, the names of the subdomains are uniformly defined: the outer air domain is subdomain 1, the rotor core domain 1 is subdomain 2, the rotor core domain 2 is subdomain 3, ..., the rotor core domain n is subdomain n+1, the permanent magnet domain is subdomain n+2, the air gap domain is subdomain n+3, the slot opening domain is subdomain n+4, and the slot subdomain is subdomain n+5.

3. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 2 is characterized in that The n is 3-6.

4. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 2 is characterized in that The S2 comprises the following steps: S2.1 Based on the Ampere circuit law of Maxwell's equations and the constitutive relation of isotropic linear media, the governing equation of the subdomain can be obtained: Where A is the vector magnetic potential, M r 、M θ They are the radial and tangential components of the residual magnetization intensity of the permanent magnet, r and θ represent the radial radius and tangential angle of the polar coordinate system, J is the current density, μ0, μ r represent the vacuum permeability and the permeability of other subdomain materials respectively; S2.2 Determine the boundary conditions for the interface between the rotor core and non-ferromagnetic material, and the interface between the stator core and non-ferromagnetic material: Complex boundary conditions are used at the interface between the rotor core, permanent magnets and external air: Where A1, A2, H 1θ 、H 2θ 、A n+1 、H (n+1)θ 、A n+2 、H (n+2)θ They represent the vector magnetic potential of the outer air domain, the vector magnetic potential of the rotor core domain 1, the tangential magnetic field intensity of the outer air domain, the tangential magnetic field intensity of the rotor core domain 1, the vector magnetic potential of the rotor core domain n, the tangential magnetic field intensity of the rotor core domain n, the vector magnetic potential of the permanent magnet domain, and the tangential magnetic field intensity of the permanent magnet domain. When n takes different values, they can be substituted into the above general formula; R ro 、R r They represent the radius of the interface between the external air domain and the rotor core domain 1, and the radius of the interface between the rotor core domain n and the permanent magnet domain respectively; Homogeneous boundary conditions are used at the intersection of the stator slots, slot openings, and the stator core: Where A (n+4)i 、A (n+5)i 、B (n+5)iθ Represent the vector magnetic potential of the i-th slot opening domain, the vector magnetic potential of the i-th slot subdomain, and the tangential magnetic flux density of the i-th slot subdomain respectively; θ i Refers to the center position angle of the i-th slot; i represents the number of the slot subdomain and slot opening domain, i ranges from 1 to N s , N s Refers to the total number of slot subdomains or slot opening domains of the motor; β sa , β oa Represent the width of the slot subdomain and slot opening domain, R sb represents the groove bottom radius of the groove subdomain; S2.3 Use the separation of variables method and the interface conditions between the stator and rotor cores and the non-ferromagnetic material to calculate the vector magnetic potential of each subdomain: The vector magnetic potentials of the permanent magnet domain and the external air domain are: in, Where k is the harmonic order, ω r is the angular velocity of the motor rotor, t is the time, α0 is the initial angle of the motor rotor, B r is the residual magnetic strength of the motor permanent magnet, α p is the pole arc coefficient of the motor, p is the number of motor pole pairs, μ0 is the vacuum permeability; R r 、R m 、R a are the radius of the interface between the rotor core and the permanent magnet, the radius of the interface between the permanent magnet and the air gap, and the outer radius of the external air; A (n+2)k 、B (n+2)k 、C (n+2)k 、D (n+2)k is the desired harmonic coefficient of the vector magnetic potential of the permanent magnet domain; A 1k 、B 1k 、C 1k 、D 1k is the desired harmonic coefficient of the vector magnetic potential in the outer air domain; The vector magnetic potentials of other subdomains all satisfy the following formula, which can be further simplified by boundary conditions: Where A v0 、B v0 、A vx 、B vx 、C vx 、D vx Represents the harmonic coefficient of the vector magnetic potential of the vth solution domain, R vou 、R vin Represent the outer radius and inner radius of the vth solution domain, x represents the harmonic order of the vth subdomain, r represents the solution radius, γ v is the Fourier series period of the vth solution domain, γ v0 is the initial phase angle of the vth solution domain.

5. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 2 is characterized in that The S3 comprises the following steps: Take the case where the vector magnetic potential at the interface between the slot opening domain and the slot subdomain is equal as an example: A (n+4)i 、A (n+5)i are the vector magnetic potentials of the slot opening domain and the slot subdomain respectively; R t is the radius of the interface between the slot opening domain and the slot subdomain; The two expand into: Where, R s is the radius of the interface between the slot opening domain and the air gap domain; m and f are the harmonic coefficients of the slot opening domain and the slot subdomain respectively; B (n+4)i0 、A (n+4)im 、B (n+4)im is the desired harmonic coefficient of the vector magnetic potential in the slot opening region; B (n+5)if is the desired harmonic coefficient of the vector magnetic potential in the slot subdomain; In order to unify the Fourier series index variables of the two subdomains, the vector magnetic potential of the slot subdomain is Fourier expanded: Where, in, The boundary conditions after unifying the Fourier series indicator variables can be expressed as follows: Where A (n+5)Fou. is the vector magnetic potential of the slot subdomain after Fourier expansion; From the above formula, the constraint equation can be derived: Similarly, the constraint equations are derived using the boundary conditions of "equal vector magnetic potential and equal tangential magnetic field strength" for each adjacent subdomain described in step S3.1, and all constraint equations are solved simultaneously to obtain the harmonic coefficients of each subdomain.

6. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 2 is characterized in that The S4 step is as follows: S4.3.1 Based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, ..., and subdomain n+1 are respectively Feb Assign an initial value; b takes 2, 3, ..., n+1, representing the unsaturated permeability μ of subdomains 2, 3, and subdomain n+1 respectively Fe2 、μ Fe3 、μ Fen+1 ; S4.3.2 Substitute the initial value of magnetic permeability into the subdomain model. The rotor flux density formula in S4.2 can be obtained by the subdomain method to obtain the flux density at different rotor radii. In order to improve the fault tolerance of the model, the magnetic density value range of the three solution domains is calculated based on 20% of the maximum magnetic density of each subdomain, and the magnetic density value is used as the function curve μ Fe (B) The independent variable can be used to find the range of magnetic permeability; S4.3.3 Take the magnetic flux density at different rotor radii as the independent variable and use the function curve μ Fe (B) The corresponding μ can be obtained Fetb , judge μ Fetb and μ Feb Whether the error requirements are met, the error formula is as follows: Where μ Feb is the magnetic permeability in the subdomain model, which is automatically assigned through the iterative loop program; μ Fetb is the magnetic permeability obtained by the B(H) curve; If the error requirements are not met, the subdomain method and iterative algorithm are combined to iterate. In addition, the magnetic permeabilities at different rotor radii will affect each other, so the three magnetic permeabilities in the iterative procedure must meet the error requirements at the same time. The accurately predicted magnetic permeability is updated to the boundary condition "Equal tangential magnetic field strength" in step S3.1 to improve the accuracy of the tangential magnetic field strength calculation, thereby improving the accuracy of the stray magnetic field calculation.

7. The high-precision outer rotor permanent magnet motor stray magnetic field analysis method according to claim 2 is characterized in that The S5 step is as follows: S5.1 Take the partial derivative of the vector magnetic potential in the external air domain to obtain the radial and tangential components of the stray magnetic field:

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