Method for analyzing stray magnetic field of high-precision outer rotor permanent magnet motor

Abstract

A high-precision external rotor permanent magnet motor stray magnetic field analysis method relates to the field of external rotor permanent magnet motor magnetic field calculation, and comprises the following steps: S1, dividing an external rotor permanent magnet motor into 5 + n subdomains according to a structural equivalence principle under a two-dimensional polar coordinate system; s2, establishing a Laplacian equation or a Poisson equation of each sub-domain, and solving a vector magnetic potential of each sub-domain through a variable separation method and boundary conditions of an interface of a ferromagnetic material and a non-ferromagnetic material; s3, listing a constraint equation according to boundary conditions, and solving a harmonic coefficient of each vector magnetic potential; s4, collaboratively integrating the sub-domain model and an iterative convergence algorithm, and automatically iteratively updating the rotor unsaturated permeability at different radiuses of the iron core of the motor under various working conditions; and S5, solving partial derivatives of the vector magnetic potential of the external air domain to obtain radial and tangential components of the stray magnetic field. The method can accurately predict the unsaturated permeability of the iron core at different working conditions and different radiuses of the iron core of the motor, and improves the calculation precision of the stray magnetic field.

Description

Technical Field

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

[0002] With the widespread application of permanent magnet motors, the method of diagnosing motor faults by data-driven using stray magnetic fields as signals is becoming increasingly widely used. Obtaining a large amount of accurate stray magnetic field data in a short period of time is the basis for motor fault diagnosis, such as ZL202110367107.X, CN113985282A, etc. Compared with numerical methods, analytical methods have the advantages of fast calculation speed and high efficiency, such as CN116244847A, CN116629167A and CN117574636A, etc. The analytical calculation of stray magnetic fields can be divided into the attenuation coefficient method and the subdomain method. Among them, the subdomain method can take into account the cogging effect and harmonic distribution, such as the patent documents with the publication or announcement numbers: CN116629167A, CN114006559A and CN108875168B, and the accuracy is relatively high.

[0003] However, for the outer rotor permanent magnet motor, the prior art only considers the saturated magnetic permeability of the core, and specifies the magnetic permeability of the rotor core when it is unsaturated as a constant, ignoring the influence of the unsaturated magnetic permeability on the stray magnetic field, such as the patent documents disclosed or announced with the numbers: CN108563912A, CN113343171A, CN108875168B and CN 112949146 A. When the motor is in different operating conditions, the current and the residual magnetism of the permanent magnet will jointly affect the magnetic flux density at the same position of the rotor core, thereby affecting the magnetic permeability. Even under fixed conditions, the magnetic flux density at different radii of the rotor is different, which will cause the magnetic permeability to change with the radius. The magnetic flux density will affect the magnetic permeability of the core, and the magnetic permeability determines the magnetic resistance of the core, which will affect the attenuation of the magnetic field in the radial direction, that is, the magnetic field and the magnetic permeability affect each other. Therefore, even if the various operating conditions of the motor are not considered, 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 stray magnetic field analysis calculation.

[0005] The technical solution of the present invention is as follows:

[0006] A high-precision outer rotor permanent magnet motor stray magnetic field analysis method, 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 of each subdomain, and solve the vector magnetic potential of each subdomain by separation of variables method and boundary conditions of the interface between ferromagnetic material and non-ferromagnetic material;

[0009] S3: Write constraint equations according to 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 radii of the core under various motor working conditions;

[0011] S5: Calculate the partial derivative of the vector magnetic potential in the external air domain, thereby calculating 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, take the center point of the cross section as the coordinate origin, and establish a two-dimensional polar coordinate system.

[0014] S1.2 According to the equivalent principle, the external air, rotor core, permanent magnet and air gap of the motor 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, such as Figure 2 shown.

[0015] Among them, the permanent magnet motor is an outer rotor and inner stator structure, and the magnetic field propagates radially through the permanent magnet and the rotor core. The difference in unsaturated magnetic permeability at different radii of the rotor core has a non-negligible effect on the attenuation of the magnetic field. 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 the calculation accuracy and calculation efficiency, n can usually be 3-6. In the present invention, n is 3.

[0016] After the rotor core is subdivided into n annular domains, the names of the subdomains are uniformly defined. The subdomains are arranged from the outside to the inside: 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.

[0017] The S2 comprises the following steps:

[0018] S2.1 According to the Ampere circuit law of Maxwell equations and the constitutive relation 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. μ 0 , μ r represent the magnetic permeability of vacuum and the magnetic permeability of other subdomain materials respectively;

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

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

[0023]

[0024]

[0025] In the formula, A 1 , A 2 , 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 strength of the outer air domain, the tangential magnetic field strength of the rotor core domain 1, the vector magnetic potential of the rotor core domain n, the tangential magnetic field strength of the rotor core domain n, the vector magnetic potential of the permanent magnet domain, and the tangential magnetic field strength of the permanent magnet domain. When n takes different values, they can be substituted into the above general formula. ro , R r They respectively 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.

[0026] Since the stator structure is located inside the permanent magnet, the magnetic field does not pass through the stator when it leaks radially to the outside of the motor. To simplify the calculation process, the junction of the stator slot, slot opening and stator core still adopts homogeneous boundary conditions:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] In the formula, A (n+4)i , A (n+5)i , B (n+5)iθ Respectively 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. 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 the present invention, N s Take 51. 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 The vector magnetic potential of each subdomain is obtained by separation of variables method and combining the interface conditions between the stator and rotor core and the non-ferromagnetic material:

[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] In the formula, k is the harmonic order, ω r is the angular velocity of the motor rotor, t is the time, α 0 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, μ 0 is the vacuum magnetic permeability. r , R m , R a A is 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 , B 1k , 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] In the formula, 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 Fourier series period of the vth solution domain. γ v0 is the initial phase angle of the vth solution domain.

[0044] The S3 comprises the following steps:

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

[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 intensity and magnetic permeability of the rotor core is given:

[0049] H θ|rotor =μ Fe B θ|rotor

[0050] In the formula,

[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] From the above formula, it can be seen that the boundary condition of "equal tangential magnetic field intensity" at the interface between the rotor core and the non-ferromagnetic material is related to the unsaturated magnetic permeability of the core. Accurately predicting the magnetic permeability values ​​at different radii of the core is the key to improving the calculation accuracy of the boundary conditions. The method of predicting the magnetic permeability at different radii of the core is described in detail in 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 equal 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] In the formula,

[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 domain; 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] In the formula,

[0066]

[0067]

[0068] in,

[0069]

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

[0071]

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

[0073] From the above formula, we can derive the constraint equation:

[0074]

[0075]

[0076] Similarly, the constraint equations are derived through the boundary conditions of each adjacent subdomain described in step S3.1, "equal vector magnetic potential and equal tangential magnetic field strength", and all constraint equations are solved simultaneously to obtain the harmonic coefficients of each subdomain. Substituting the harmonic coefficients into the vector magnetic potential in step S2.3, and then taking the partial derivative of the vector magnetic potential, the magnetic flux density of each subdomain can be obtained.

[0077] The S4 step is as follows:

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

[0079] S4.2 solves 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] In the formula, B 2r , B 3r , B (n+1)r Respectively 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. 2 、r 3 、r n+1 Represent the solution radius of subdomains 2, 3, and n+1 respectively. ro1 , R r1They 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. 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 are the harmonic coefficients 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 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 algorithm is used to iterate the calculation again until the magnetic permeability meets the error.

[0085] S4.3.1 Based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, ..., subdomain n+1 is 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 Feb Whether the error requirements are met, the error formula is as follows:

[0088]

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

[0090] If the error requirement is not met, the subdomain method and iterative algorithm are combined to iterate. In addition, the magnetic permeabilities at different radii of the rotor will affect each other, so the three magnetic permeabilities in the iterative procedure must meet the error requirements at the same time.

[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 tangential magnetic field strength calculation 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 the outer rotor permanent magnet motor provided by the present invention takes into account the unsaturated magnetic permeability distribution of the rotor 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. Through the collaborative integration of the subdomain model and the iterative convergence algorithm, the automatic update of the unsaturated magnetic permeability at different rotor radii under different current magnitudes and permanent magnet operating points is realized.

[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) Under the premise of meeting the calculation accuracy requirements, the calculation speed of the analytical model is 15.2 times that of the finite element method, and the computer memory occupied is only 4% of the finite element method. The calculation accuracy of the analytical model proposed in the present invention is similar to that of the finite element method, but the calculation 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] Aiming at the problem that the unsaturated magnetic permeability of the motor changes with the rotor radius and working conditions, the present invention divides the rotor core into multiple solution domains and integrates the subdomain method and the iterative convergence algorithm in a coordinated manner. It can accurately predict the unsaturated magnetic permeability of the core at different working conditions and different radii of the motor, and based on the predicted magnetic permeability, improve the solution accuracy of the boundary conditions at the junction of the rotor core and non-ferromagnetic materials, thereby realizing the accurate calculation of the stray magnetic field. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0104] Figure 3 It 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) curve graph;

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

[0107] Figure 6 is a schematic diagram of a 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 embodiments provide detailed implementation methods and specific operating steps based on the technical methods 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 As shown, the specific steps are as follows:

[0110] S1: The outer rotor permanent magnet motor is represented 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. Among them, n represents the number of subdomains that the rotor core is evenly divided into along the radial direction, and takes a positive integer. The larger n is, the more accurately the unsaturated magnetic permeability at different radii of the rotor core is considered, but the longer the calculation time is. Considering the calculation accuracy and calculation time, n can usually be 3-6; in this embodiment, n is 3.

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

[0112] S1.2 According to the equivalent principle, the external air, rotor core, permanent magnet and air gap of the motor 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. Considering the complexity and accuracy of the calculation, n is 3 in the present invention, that is, the rotor core is divided into 3 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 as follows 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 of the interface between ferromagnetic materials and non-ferromagnetic materials.

[0115] S2.1 According to the Ampere circuit law of Maxwell 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 the two are Poisson's equations, and the governing equations for the remaining subdomains are Laplace equations.

[0118] The governing equation for the permanent magnet domain is:

[0119]

[0120] In the formula,

[0121]

[0122]

[0123] in,

[0124]

[0125] M θk =0

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

[0127]

[0128] In the formula,

[0129]

[0130] J i1 , J i2 They represent the upper and lower currents in the slot subdomain, 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 the 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 of the interface between the rotor core and non-ferromagnetic material, and the interface between the rotor core and non-ferromagnetic material. The prototype is an outer rotor permanent magnet motor. When the magnetic field decays from the permanent magnet to the outside, it passes through the rotor but not the stator. Considering the accuracy of the calculation and reducing the difficulty of derivation as much as possible, complex boundary conditions are used for the interface between the rotor and non-ferromagnetic material, and 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] In the formula, R ro =110.6mm; R m =100.8mm.

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

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] In the formula, 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 separation of variables method combined with the interface conditions between the stator and rotor core and the non-ferromagnetic material.

[0148] The vector magnetic potential of subdomain 1:

[0149]

[0150] In the formula, R a =2212mm, R ro =110.6mm.

[0151] The vector magnetic potential of subdomain 2:

[0152]

[0153] In the formula, R ro1 =107.9mm.

[0154] The vector magnetic potential of subdomain 3:

[0155]

[0156] In the formula, R r1 =105.2mm.

[0157] The vector magnetic potential of subdomain 4:

[0158]

[0159] In the formula, R r =102.5mm.

[0160] The vector magnetic potential of subdomain 5:

[0161]

[0162] In the formula, Rm =100.8mm.

[0163] The vector magnetic potential of subdomain 6:

[0164]

[0165] In the formula, R s =100.0mm.

[0166] The vector magnetic potential of subdomain 7:

[0167]

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

[0169] The vector magnetic potential of subdomain 8:

[0170]

[0171] In the formula, R sb =86.7mm.

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

[0173] S3.1 Write constraint equations through boundary conditions and solve the harmonic coefficients of vector magnetic potential. 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] In the formula, 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] In the formula,

[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 the two subdomains are different, it is necessary to unify the Fourier series index variables. Take the equal vector magnetic potential at the interface between the slot subdomain and the slot opening domain 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] In the formula,

[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, we can derive the constraint equation:

[0197]

[0198]

[0199] Similarly, through Figure 3 The constraint equations are derived from the other boundary conditions given in step S3.1. All constraint equations are solved simultaneously to obtain the harmonic coefficients. The harmonic coefficients are substituted into the vector magnetic potential equation in step S2.3, and the partial derivative is calculated to obtain the magnetic flux density of each subdomain.

[0200] S4: μ is obtained by interpolation based on the inherent B(H) curve of the ferromagnetic material used in the rotor.Fe (B) Curve, the magnetic flux density of the rotor core is solved according to the subdomain model established by S1-3. The subdomain model established by S1-S3 is combined with the convergence algorithm to automatically iterate and update the rotor unsaturated magnetic permeability at different radii of the core under different working conditions of the motor.

[0201] S4.1 Look up 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 coefficient 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) The corresponding magnetic permeability μ is calculated by the curve Fe . Determine the initial assigned magnetic permeability and the 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 algorithm is used to iterate the calculation again until the magnetic permeability meets the error.

[0207] S4.3.1 First, based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, and subdomain 4 are Feb Assign an initial value of 2000μ 0 , 10000μ 0 , 20000μ 0 .

[0208] S4.3.2 Solve the magnetic flux density of subdomain 2, subdomain 3, and subdomain 4 through 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), (0.4492T, 0.67T). Combined with the magnetic flux density value range and the μ of the rotor ferromagnetic material Fe (B) curve, the range of the three magnetic permeabilities is calculated as (2303μ 0 ,4256μ 0 ), (6700μ 0 ,24032μ 0 ), (10260μ 0 ,24032μ 0 Finally, the initial values ​​of the permeability of subdomains 2, 3, and 4 are set to 2303 μ 0 , 6700μ 0 , 10329μ 0 , according to the empirical value, the permeability update interval is 0.5μ 0 .

[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 , determine whether the difference between the two meets the error requirement. Considering the calculation efficiency and accuracy, the error is set to no more than 0.1:

[0210]

[0211] In the formula, 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 meet the error requirements at the same time.

[0212] S4.3.4 Under the condition of input current 7A and permanent magnet remanence 0.35T, the final permeability of subdomains 2, 3, and 4 is 2307μ 0 , 24032μ 0 , 10265μ 0 .

[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 load stray magnetic field of the prototype was experimentally measured. The experimental input current is the same as that of the analytical model and the finite element model, both of which are 7A sinusoidal three-phase alternating current. The stray magnetic field is measured by a Hall probe, and the stray magnetic field data is collected using a Gaussmeter. In addition, to ensure the measurement accuracy of the stray magnetic field, the angle of the Hall probe needs to be adjusted to ensure that the magnetic field enters the Hall probe vertically. The comparison between the stray magnetic field calculation results of the method proposed in this paper and the finite element results and experimental measurement results is shown in Figure 2. 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 jth 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 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 of the two methods are 0.021 and 0.02 respectively. Under the premise of satisfying the calculation accuracy, the calculation speed of the analytical model is 15.2 times that of the finite element method, and the computer memory occupied is only 4% of that of the finite element method. It can be seen that the improved analytical model proposed in this paper has achieved a good balance between calculation efficiency and calculation accuracy.

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

[0220]

Claims

1. A high-precision stray magnetic field analysis method for an outer rotor permanent magnet motor, 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 of each subdomain, and solve the vector magnetic potential of each subdomain by separation of variables method and boundary conditions of the interface between ferromagnetic material and non-ferromagnetic material; S3: Write constraint equations according to 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 radii of the core under various motor working conditions; S5: Take 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.

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, take the center point of the section as the origin, and establish a two-dimensional polar coordinate system; S1.2 According to the equivalent principle, the external air, rotor core, permanent magnet and air gap of the motor 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. Among them, the permanent magnet motor is an outer rotor and inner stator structure, and the magnetic field propagates radially through the permanent magnet 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 According to the Ampere circuit law of Maxwell 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 are the radial and tangential components of the residual magnetization 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 magnetic permeability of vacuum and the magnetic permeability of other subdomain materials respectively; S2.2 Determine the boundary conditions of the interface between the rotor core and non-ferromagnetic material, and the interface between the stator core and non-ferromagnetic material respectively: Complex boundary conditions are used at the interface between the rotor core, permanent magnets and external air: In the formula, 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: In the formula, A (n+4)i , A (n+5)i , B (n+5)iθ Respectively 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; 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 sub-domains 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 The vector magnetic potential of each subdomain is obtained by separation of variables method and combining the interface conditions between the stator and rotor core and the non-ferromagnetic material: The vector magnetic potentials of the permanent magnet domain and the external air domain are: in, In the formula, 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: In the formula, 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: S3.1 Write constraint equations according to boundary conditions and solve the harmonic coefficients of vector magnetic potential. The boundary conditions between subdomains are: equal vector magnetic potential and equal tangential magnetic field strength. 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 In the formula, 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 are unified by the Fourier expansion method; taking the vector magnetic potential at the interface between the slot opening domain and the slot subdomain 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: In the formula, 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 domain; 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: In the formula, in, The boundary conditions after unifying the Fourier series indicator variables can be expressed as follows: In the formula, A (n+5)Fou. is the vector magnetic potential of the slot subdomain after Fourier expansion; From the above formula, we can derive the constraint equation: Similarly, the constraint equations are derived through the boundary conditions of each adjacent subdomain described in step S3.1, "equal vector magnetic potentials and equal tangential magnetic field intensities", and all constraint equations are solved simultaneously to obtain the harmonic coefficients of each subdomain; the harmonic coefficients are substituted into the vector magnetic potential in step S2.3, and the partial derivative of the vector magnetic potential is then taken to obtain the magnetic flux density 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: S4.1 Interpolate the B(H) curve inherent to the rotor ferromagnetic material to estimate the function between magnetic permeability and magnetic flux density, and obtain μ Fe (B) Curve; S4.2 solves 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. In the formula, 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, r2, r3, and r n+1 Represent the solution radius of subdomains 2, 3, and n+1 respectively, 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 iterate and 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 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 on this basis, finding the B(H) curve of the material used for the rotor core, and obtaining μ through 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; S4.3.1 Based on empirical values, the magnetic permeability μ of subdomain 2, subdomain 3, ..., subdomain n+1 is 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, 20% of the maximum value of the flux density in each subdomain is used as the basis to calculate the flux density value range of the three solution domains, and the flux density value is used as the function curve μ Fe (B) is the independent variable, and the range of magnetic permeability can be obtained; 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: In the formula, μ Feb is the magnetic permeability in the subdomain model, which is automatically assigned a value through the iterative loop program; μ Fetb is the magnetic permeability obtained by the B(H) curve. If the error requirement is not met, the subdomain method and iterative algorithm are combined to iterate; in addition, the magnetic permeabilities at different radii of the rotor 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 "tangential magnetic field intensity is equal" in step S3.1 to improve the accuracy of the tangential magnetic field intensity calculation, thereby improving the calculation accuracy of the stray magnetic field.

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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