An analytical calculation method for no-load stray magnetic field of outer rotor permanent magnet motor

Abstract

The application discloses a kind of outer rotor wheel hub permanent magnet motor no-load stray magnetic field analytical calculation method, it is related to the calculation method of motor stray magnetic field, comprising the following steps;S1: in two-dimensional polar coordinate system, according to the establishment of motor equivalent model including 7 subdomains according to the magnetization mode of motor structure, and the magnetization intensity expression of permanent magnet is established;S2: with vector magnetic potential as potential function, the control equation of each subdomain is established, and the general solution of the vector magnetic potential of each subdomain is solved;S3: the constraint equation is written by the boundary condition and the interface condition of each subdomain, and the harmonic coefficient is solved;S4: the expression of vector magnetic potential in outer air domain is written, and the partial derivative is calculated to calculate the radial component and tangential component of stray magnetic field.The application can meet the requirements of high precision, fast calculation speed and low cost simultaneously.The calculation efficiency of stray magnetic field is greatly improved;It can be used in the initial design stage of motor prototype.

Description

Technical Field

[0001] This invention relates to a method for calculating stray magnetic fields of motors, and more specifically, an analytical calculation method for the no-load stray magnetic field of an external rotor hub permanent magnet motor. Background Technology

[0002] External rotor hub permanent magnet motor drives are widely used in electric bicycles due to their advantages of flexible control and high integration, and will gradually be applied to distributed drive electric vehicles.

[0003] The stray magnetic field of an external rotor hub permanent magnet motor is defined as the magnetic field generated by the magnetic flux leaking out of the motor housing. On the one hand, stray magnetic fields can interfere with precision electronic equipment, affecting the safety of the control system; on the other hand, stray magnetic fields can provide motor status information, which can be used for fault diagnosis such as bearing failure, inter-turn short circuit, eccentricity, and demagnetization. Therefore, the accurate calculation and analysis of stray magnetic fields is extremely important. Methods for obtaining stray magnetic fields can be divided into analytical methods, finite element methods, and experimental methods. A comprehensive review of existing technologies reveals the following main problems with methods for obtaining stray magnetic fields:

[0004] (1) Existing analytical models can only calculate the air gap magnetic field, but cannot accurately calculate the stray magnetic field of the external rotor hub permanent magnet motor. For example, some technologies calculate the air gap magnetic field using the subdomain method, such as CN112347627A, CN114006559A, CN113343171A, and CN111651914A; some technologies calculate the air gap magnetic field using the equivalent magnetic circuit method, such as CN114626244A, CN111327170A, CN108563912A, and CN111327170A; and some technologies calculate the air gap magnetic field using the magnetomotive force and magnetic permeability method, such as CN113868929A and CN113868929A.

[0005] (2) The finite element model is only a calculation model for a specific motor structure and is not general. It is also time-consuming and inefficient. For example, in CN113985282A and CN113094952A, in order to ensure the accuracy of stray magnetic field calculation, the outer radius of the motor's outer air domain must be set to about 20 times the motor's outer diameter. This causes the number of meshes in the finite element model to increase dramatically and greatly prolongs the calculation time.

[0006] (3) Obtaining stray magnetic fields through experimental methods requires the prior preparation of a motor prototype. Therefore, this method cannot be used for the initial design of motors without a prototype, cannot reveal the characteristic mechanism of stray magnetic fields, and the testing equipment is expensive, resulting in high testing costs. For example, CN101126799A tested the normal component of the stray magnetic field on the surface of ferromagnetic materials using a magnetic sensor.

[0007] The purpose of this invention is to solve the problem that existing technologies cannot simultaneously calculate and analyze stray magnetic fields with high precision, high efficiency, and low cost, and to provide an analytical calculation method for the no-load stray magnetic field of an external rotor hub permanent magnet motor based on the subdomain method. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide an analytical calculation method for the no-load stray magnetic field of an external rotor hub permanent magnet motor that can simultaneously meet the requirements of high accuracy, fast calculation speed, and low cost.

[0009] The technical solution adopted by this invention to overcome the shortcomings of the prior art is as follows:

[0010] An analytical calculation method for the no-load stray magnetic field of an external rotor hub permanent magnet motor includes the following steps:

[0011] S1: In a two-dimensional polar coordinate system, an equivalent model of a motor containing 7 subdomains is established according to the different motor structures, and the magnetization intensity expression of the permanent magnet is established according to the magnetization method of the permanent magnet.

[0012] S2: Using vector magnetic potential as the potential function, establish the governing equations for each subdomain and solve the general solution of vector magnetic potential for each subdomain;

[0013] S3: Write the constraint equations based on the interface and boundary conditions of each subdomain, and solve for the harmonic coefficients;

[0014] S4: Write the expression for the vector magnetic potential in the outer air domain, and calculate the radial and tangential components of the stray magnetic field by taking the partial derivative.

[0015] S1 includes the following steps:

[0016] S1.1 Establish a two-dimensional polar coordinate system with a point on the rotor axis of the motor as the origin.

[0017] S1.2 Based on the structure of the motor, it is divided into 7 sub-domains, including the external air domain, rotor core domain, air gap domain, permanent magnet domain, slot opening domain, slot domain and stator core domain.

[0018] S1.3 Based on the actual shape and size of each subdomain, establish an equivalent model of the motor in a two-dimensional polar coordinate system; wherein the external air domain, rotor core domain, air gap domain, and permanent magnet domain are equivalent to a ring model, and the slot opening domain and slot subdomain are equivalent to a sector model.

[0019] The permanent magnets of the S1.4 external rotor hub permanent magnet motor are radially magnetized, with alternating N and S poles. An expression for magnetization intensity is established based on the permanent magnet pole parameters:

[0020]

[0021]

[0022] in,

[0023]

[0024] M θk =0, k / p=1, 3, 5,...

[0025] In the formula, M r and M θ These represent the radial and tangential components of the remanent magnetization of the motor's permanent magnet, respectively; k is the harmonic order; θ is the angle; and ω is the tangential component. r Let ω be the angular velocity of the motor rotor, t be time, α0 be the initial angle of the motor rotor, and B be the angular velocity of the rotor. r α is the remanence of the permanent magnet in the motor. p denoted as the pole arc coefficient of the motor, p as the number of pole pairs of the motor, and μ0 as the permeability of free space.

[0026] S2 includes the following steps:

[0027] S2.1 Based on Ampere's circuital law for Maxwell's equations in a quasi-static field and the constitutive relations in an isotropic linear medium, the governing equations for each subdomain are as follows:

[0028] Permanent magnet domain:

[0029]

[0030] Other subdomains besides the permanent magnet domain:

[0031]

[0032] In the formula, A zy The component representing the vector magnetic potential in the z-direction of the y-subdomain is denoted by y, where y represents the subdomain number, and f and θ represent the radius and angle, respectively. The subdomains are numbered as follows: outer air domain 1, rotor core domain 2, air gap domain 3, permanent magnet domain 4, slot opening domain 5i, slot subdomain 6i, and stator core domain 7.

[0033] S2.2 uses the method of separation of variables to obtain the vector magnetic potential general solution for each subdomain:

[0034] outer air domain

[0035]

[0036] In the formula, A 1k B1k 、 C 1k and D 1k R is the harmonic coefficient to be solved for the vector magnetic potential in the outer air domain. a and Ro These represent the outer and inner diameters of the outer air domain, respectively, and k represents the harmonic order.

[0037] Rotor core region

[0038]

[0039] In the formula, A 2k B 2k C 2k and D 2k R is the harmonic coefficient to be solved for the vector magnetic potential of the rotor core domain. o and R r These represent the outer and inner diameters of the rotor core, respectively.

[0040] Permanent magnet domain

[0041]

[0042] In the formula,

[0043]

[0044] A 3k B 3k C 3k and D 3k R is the harmonic coefficient to be solved for the vector magnetic potential of the permanent magnet domain. r and R m These represent the outer and inner diameters of the permanent magnet domain, respectively.

[0045] air gap domain

[0046]

[0047] In the formula, A 4k B 4k C 4k and D 4k R represents the harmonic coefficients to be solved for the vector magnetic potential in the air gap domain. m and R s These represent the outer diameter and inner diameter of the air gap region, respectively.

[0048] slot opening area

[0049] Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero:

[0050]

[0051]

[0052] In the formula, β oa θ represents the width of the slot opening.i This represents the position angle of the i-th slot opening region.

[0053] Based on the above boundary conditions, the vector magnetic potential flux solution for the i-th slot opening domain can be obtained as follows:

[0054]

[0055] In the formula,

[0056]

[0057] B 5i0 A 5im and B 5im R is the harmonic coefficient to be solved for the vector magnetic potential in the slot opening domain. s and R t represents the outer diameter and inner diameter of the slot opening, respectively, and m is the harmonic order of the slot opening.

[0058] Slot Domain

[0059] Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero:

[0060]

[0061]

[0062] In the formula, β sa θ represents the width of the slot subdomain. i It represents the position angle of the i-th slot subdomain, which is the same as the position angle of the i-th slot opening domain.

[0063] Meanwhile, at the interface between the bottom of the slot and the stator core, the tangential magnetic field strength of the slot subdomain is zero.

[0064]

[0065] Combining the two boundary conditions mentioned above, the vector magnetic potential of the slot subdomain can be obtained as follows:

[0066]

[0067] In the formula,

[0068]

[0069] A 6in R is the harmonic coefficient to be solved for the vector magnetic potential of the slot subdomain. t and R sb represents the outer diameter and inner diameter of the slot domain, respectively, and n is the harmonic order of the slot domain.

[0070] S3 includes the following steps:

[0071] S3.1 The vector magnetic potential and tangential magnetic field strength at the interface between the permanent magnet domain and the air gap domain are equal:

[0072]

[0073]

[0074] The vector magnetic potential and tangential magnetic flux density at the interface between the air gap region and the slot opening region are equal in S3.2.

[0075]

[0076]

[0077] S3.3 Establish that the interface vector magnetic potential and tangential magnetic flux density are equal between the slot opening domain and the slot subdomain:

[0078]

[0079]

[0080] S3.4 The interface vector magnetic potential and tangential magnetic field strength are equal at the junction of the permanent magnet domain and the rotor core domain:

[0081]

[0082]

[0083] The vector magnetic potential and tangential magnetic field strength at the interface between the rotor core region and the external air region of S3.5 are equal.

[0084]

[0085]

[0086] The vector magnetic potential at the outer radius of the S3.6 outer air domain is 0:

[0087]

[0088] S3.7 Establish the matrix equation and solve for the harmonic coefficients.

[0089] By simultaneously solving the constraint equations of the above six interfaces and rearranging them into matrix form, the harmonic coefficient A can be obtained by solving the matrix equations. 1k B 1k C 1k D 1k .

[0090] S4 includes the following steps:

[0091] S4.1 Taking the partial derivative of the vector magnetic potential in the outer air domain, we obtain the expressions for the radial and tangential magnetic flux densities of the stray magnetic field:

[0092]

[0093]

[0094] S4.2 The harmonic coefficient A of the vector magnetic potential in the outer air domain 1k B 1k C 1k D 1k By substituting the values, the stray magnetic field at any radius in the outer air domain can be solved.

[0095] The motor described in this invention is an external rotor hub permanent magnet motor.

[0096] The subdomains include: external air domain, rotor core domain, air gap domain, permanent magnet domain, slot opening domain, slot subdomain, and stator core domain.

[0097] Compared with the prior art, the advantages of the present invention are as follows:

[0098] 1) This method can take into account the slotting effect of the external rotor hub motor and the attenuation effect of the rotor core and housing on the stray magnetic field. Compared with the experimental test results, the accuracy is higher than 96%.

[0099] 2) Compared with the finite element method, this method is 5.5 times faster, which greatly improves the calculation efficiency of stray magnetic fields;

[0100] 3) Compared to the experimental method, this method is extremely low in cost and can be used in the initial design stage of motorless prototypes.

[0101] This invention simultaneously meets the requirements of high accuracy, fast calculation speed, and low cost. It greatly improves the calculation efficiency of stray magnetic fields and can be used in the initial design stage of motorless prototypes. Attached Figure Description

[0102] Figure 1 This is a schematic diagram of the motor subdomain partitioning.

[0103] Figure 2 This is a schematic diagram of the motor boundary conditions and interface conditions.

[0104] Figure 3 The three-dimensional spatiotemporal distribution diagram of the radial component of the stray magnetic field at 1 mm on the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, is calculated using the method of the present invention.

[0105] Figure 4The three-dimensional spatiotemporal distribution of the stray magnetic field tangential component at 1 mm on the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, is calculated using the method of the present invention.

[0106] Figure 5 This is a schematic diagram showing the spatial distribution of the radial component of the stray magnetic field at 1 mm on the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0107] Figure 6 This is a schematic diagram of the spatial order of the radial component of the stray magnetic field at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0108] Figure 7 This is a schematic diagram showing the spatial distribution of the stray magnetic field tangential component at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0109] Figure 8 This is a schematic diagram of the spatial order of the stray magnetic field tangential component at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0110] Figure 9 This is a schematic diagram showing the time history of the radial component of the stray magnetic field at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0111] Figure 10 This is a schematic diagram showing the spectral characteristics of the radial component of the stray magnetic field at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0112] Figure 11 This is a schematic diagram showing the time history of the stray magnetic field tangential component at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0113] Figure 12 This is a schematic diagram showing the spectral characteristics of the stray magnetic field tangential component at 1 mm from the outer surface of the external rotor hub permanent magnet motor, as shown in Table 1, calculated using the method of the present invention.

[0114] Figure 13 This is a schematic diagram of the experimental setup for measuring the external rotor hub permanent magnet motor shown in Table 1 using direct testing experiments.

[0115] Figure 14 This is a schematic diagram of the stray magnetic field simulation model of the external rotor hub permanent magnet motor established using the finite element method, as shown in Table 1.

[0116] Figure 15This is a comparison chart of the spatial distribution of the radial component of the stray magnetic field at 1 mm on the outer surface of the external rotor hub permanent magnet motor calculated using the method of the present invention and the finite element method, and the experimental results.

[0117] Figure 16 This is a comparison chart of the spatial distribution of the stray magnetic field tangential component at 1 mm on the outer surface of the external rotor hub permanent magnet motor calculated using the method of the present invention and the finite element method, and the experimental results. Detailed Implementation

[0118] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0119] This embodiment uses a 46-pole, 51-slot external rotor hub permanent magnet motor as an example. The main parameters of the motor are shown in Table 1:

[0120] Table 1 Main parameters of the motor

[0121]

[0122] Division of the solution domain within motor S1:

[0123] S1.1 Based on the structural characteristics of the external rotor hub permanent magnet motor, a two-dimensional polar coordinate system is established with the center point of the outer surface of the motor end cover as the origin.

[0124] S1.2 Based on the different internal structures and permeability of the motor, the motor is divided into 7 solution domains, such as... Figure 1 As shown, subdomain 1 is the external air domain, subdomain 2 is the rotor core domain, subdomain 3 is the permanent magnet domain, subdomain 4 is the air gap domain, subdomain 5i is the i-th slot opening domain, subdomain 6i is the i-th slot subdomain, and subdomain 7 is the stator core domain.

[0125] S1.3 The outer air domain, rotor core domain, air gap domain, and permanent magnet domain are equivalent to a ring model, and the slot opening domain and slot sub-domain are equivalent to a sector model. Simultaneously, to ensure the accuracy of stray magnetic field calculations, the outer radius R of the outer air domain is... a Set to 20 times the outer diameter of the motor.

[0126] The permanent magnets of the S1.4 external rotor hub permanent magnet motor are radially magnetized, with alternating N and S poles. The remanence of the permanent magnets is B. r Its T is 0.35, and its free permeability is 4π×10⁻⁶. -7 Wb / (Am) is an expression for magnetization based on the magnetic pole parameters of a permanent magnet:

[0127]

[0128]

[0129] in,

[0130]

[0131] M θk =0, k / p=1, 3, 5,...

[0132] S2: Using vector magnetic potential as the potential function, establish the governing equations for each subdomain and solve the general solution of vector magnetic potential for each subdomain:

[0133] S2.1 Based on Ampere's circuital law for Maxwell's equations in a quasi-static field and the constitutive relations in an isotropic linear medium, the governing equations for each subdomain can be expressed as follows:

[0134] Permanent magnet domain:

[0135]

[0136] Other subdomains besides the permanent magnet domain:

[0137]

[0138] S2.2 By using the method of separation of variables and combining partial interface conditions, the vector magnetic potential general solution for each subdomain can be obtained. The motor boundary conditions and interface conditions are as follows: Figure 2 As shown.

[0139] S2.2.1 Outer air domain

[0140]

[0141] S2.2.2 Rotor core region

[0142]

[0143] S2.2.3 Permanent Magnet Domain

[0144]

[0145] In the formula,

[0146]

[0147] S2.2.4 Air Gap Domain

[0148]

[0149] S2.2.5 slot opening area

[0150] Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero:

[0151]

[0152]

[0153] Based on the above boundary conditions, the vector magnetic potential flux solution for the i-th slot opening domain can be obtained as follows:

[0154]

[0155] In the formula,

[0156]

[0157] S2.2.6 Slot Domain

[0158] Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero:

[0159]

[0160]

[0161] Meanwhile, at the interface between the bottom of the slot and the stator core, the tangential magnetic field strength of the slot subdomain is zero.

[0162]

[0163] Combining the two boundary conditions mentioned above, the vector magnetic potential of the slot subdomain can be obtained as follows:

[0164]

[0165] In the formula,

[0166]

[0167] S3: Write the constraint equations based on the interface and boundary conditions of each subdomain, and solve for the harmonic coefficients:

[0168] S3.1 Establish the interface constraint equations between the permanent magnet domain and the air gap domain.

[0169] S3.1.1 The vector magnetic potential at the interface between the permanent magnet domain and the air gap domain is equal:

[0170]

[0171] Expand the interface conditions:

[0172]

[0173]

[0174] The coefficients of the sine and cosine terms in both expressions are equal:

[0175]

[0176]

[0177] make,

[0178]

[0179] Z 11 =diag(Z) 11_k )Z 21 =diag(Z) 21_k )

[0180] Z 12 =eye(K)Z 22 =eye(K)

[0181] Z 13 =-eye(K)Z 23 =-eye(K)

[0182]

[0183] Z 14 =diag(Z) 14_k )Z 24 =diag(Z) 24_k )

[0184] f 1s_k =kR m M k sin(kω r t+kα0)f 1c_k =-kR m M k cos(kω r t+kα0)

[0185] f 1s =f 1s_k f 1c =f 1c_k

[0186] In the formula, diag is a diagonal matrix, eye is a unit matrix, and f′ 1s_k f 1s_k The transpose of f′ 1c_k f 1c_k The transpose of Z, where K is the maximum harmonic order of k, and Z 11 Z 12 Z 13 Z 14 Z 21Z 22 Z 23 Z 24 Let f be the coefficient matrix of the constraint equation. 1s and f 1c Let be the constant matrix of the constraint equations.

[0187] From this, constraint equations can be established:

[0188] Z 11 A 3k +Z 12 B 3k +Z 13 A 4k +Z 14 B 4k =f 1s

[0189] Z 21 C 3k +Z 22 D 3k +Z 23 C 4k +Z 24 D 4k =f 1c

[0190] S3.1.2 The tangential magnetic field strength at the interface between the permanent magnet domain and the air gap domain is equal:

[0191]

[0192] Expand the interface conditions:

[0193]

[0194]

[0195] In the formula, μ r is the relative permeability of the permanent magnet.

[0196] From this, constraint equations can be established:

[0197] Z 31 A 4k +Z 32 B 3k +Z 33 A 3k +Z 34 B 4k =f 2s

[0198] Z 41 C 4k +Z 42 D 3k +Z 43 C3k +Z 44 D 4k =f 2c

[0199] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 31 Z 32 Z 33 Z 34 Z 41 Z 42 Z 43 Z 44 Let f be the coefficient matrix of the constraint equation. 2s and f 2c Let be the constant matrix of the constraint equations.

[0200] S3.2 Establish the interface constraint equations for the air gap region and the slot opening region.

[0201] S3.2.1 The vector magnetic potential at the interface between the air gap domain and the slot opening domain is equal:

[0202]

[0203] Expand the interface conditions:

[0204]

[0205]

[0206] To unify the Fourier series index variables of the vector magnetic potential in the air gap domain and the slot opening domain, the vector magnetic potential in the air gap domain is expanded within the slot opening domain:

[0207]

[0208] in,

[0209]

[0210]

[0211]

[0212]

[0213]

[0214]

[0215] From this, constraint equations can be established:

[0216] Z 51 A 4k +Z52 B 4k +Z 53 C 4k +Z 54 D 4k +Z 55 B 5i0 =0

[0217] Z 61 A 4k +Z 62 B 4k +Z 63 C 4k +Z 64 D 4k +Z 65 A 5im +Z 66 B 5im =0

[0218] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 51 Z 52 Z 53 Z 54 Z 55 Z 61 Z 62 Z 63 Z 64 Z 65 Z 66 Let be the coefficient matrix of the constraint equation.

[0219] S3.2.2 The tangential magnetic flux density at the interface between the air gap region and the slot opening region is equal:

[0220]

[0221] Expand the interface conditions:

[0222]

[0223]

[0224] To unify the Fourier series index variables of the tangential magnetic flux density in the air gap domain and the slot opening domain, the tangential magnetic flux density in the slot opening domain is expanded within the air gap domain:

[0225]

[0226] in,

[0227]

[0228]

[0229]

[0230]

[0231]

[0232] From this, constraint equations can be established:

[0233] Z 71 A 4k +Z 72 B 4k +Z 73 B 5i0 +Z 74 A 5im +Z 75 B 5im =0

[0234] Z 81 C 4k +Z 82 D 4k +Z 83 B 5i0 +Z 84 A 5im +Z 85 B 5im =0

[0235] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 71 Z 72 Z 73 Z 74 Z 75 Z 81 Z 82 Z 83 Z 84 Z 85 Let be the coefficient matrix of the constraint equation.

[0236] S3.3 Establish the interface constraint equations for the slot opening region and the slot sub-region.

[0237] S3.3.1 The vector magnetic potential at the interface between the slot opening region and the slot sub-region is equal:

[0238]

[0239] Expand the interface conditions:

[0240]

[0241]

[0242] To unify the Fourier series index variables of the vector magnetic potential in the slot subdomain and the slot opening domain, the vector magnetic potential in the slot subdomain is Fourier expanded within the slot opening domain:

[0243]

[0244] In the formula,

[0245]

[0246]

[0247]

[0248] Therefore, the constraint equations can be obtained:

[0249] Z 61 B 5i0 +Z 92 A 6in =0

[0250] Z 101 A 5im +Z 102 B 5im +Z 103 A 6in =0

[0251] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 91 Z 92 Z 101 Z 102 Z 103 Let be the coefficient matrix of the constraint equation.

[0252] S3.3.2 The tangential magnetic flux density at the interface between the slot opening region and the slot sub-region is equal:

[0253]

[0254] Expand the interface conditions:

[0255]

[0256]

[0257] To unify the Fourier series index variables of the tangential magnetic flux density in the slot subdomain and the slot opening domain, the tangential magnetic flux density in the slot opening domain is Fourier expanded within the slot subdomain:

[0258]

[0259] In the formula,

[0260]

[0261]

[0262] Therefore, the constraint equations can be obtained:

[0263] Z 111 B 5i0 +Z 112 A 5im +Z 113 B 5im +Z 114 A 6in =0

[0264] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 111 Z 112 Z 113 Z 114 Let be the coefficient matrix of the constraint equation.

[0265] S3.4 Establish the interface constraint equations for the permanent magnet domain and the rotor core domain.

[0266] S3.4.1 The vector magnetic potential at the interface between the permanent magnet domain and the rotor core domain is equal:

[0267]

[0268] Expand the interface conditions:

[0269]

[0270]

[0271] Therefore, the constraint equations can be obtained:

[0272] Z 121 A 2k +Z 122 B 2k +Z 123 A 3k +Z 124 B 3k =f 3s

[0273] Z 131 C 2k +Z 132 D 2k +Z 133 C 3k +Z 134 D 3k =f 3c

[0274] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 121 Z 122 Z 123 Z 124 Z 131Z 132 Z 133 Z 134 Let f be the coefficient matrix of the constraint equation. 3s and f 3c Let be the constant matrix of the constraint equations.

[0275] S3.4.2 The tangential magnetic field strength at the interface between the permanent magnet domain and the rotor core domain is equal:

[0276]

[0277] Expand the interface conditions:

[0278]

[0279]

[0280] In the formula, μ ro denoted as ρ, where ρ is the relative permeability of the rotor core.

[0281] Therefore, the constraint equations can be obtained:

[0282] Z 141 A 2k +Z 142 B 2k +Z 143 A 3k +Z 144 B 3k =f 4s

[0283] Z 151 C 2k +Z 152 D 2k +Z 153 C 3k +Z 154 D 3k =f 4c

[0284] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 141 Z 142 Z 143 Z 144 Z 151 Z 152 Z 153 Z 154 Let f be the coefficient matrix of the constraint equation. 4s and f 4c Let be the constant matrix of the constraint equations.

[0285] S3.5 Establish the interface constraint equations between the rotor core region and the external air region.

[0286] S3.5.1 The vector magnetic phase at the interface between the rotor core region and the external air region is equal.

[0287]

[0288] Expand the interface conditions:

[0289]

[0290]

[0291] Therefore, the constraint equations can be obtained:

[0292] Z 161 A 1k +Z 162 B 1k +Z 163 A 2k +Z 164 B 2k =0

[0293] Z 171 C 1k +Z 172 D 1k +Z 173 C 2k +Z 174 D 2k =0

[0294] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 161 Z 162 Z 163 Z 164 Z 171 Z 172 Z 173 Z 174 Let be the coefficient matrix of the constraint equation.

[0295] S3.5.2 The tangential magnetic field strength at the interface between the rotor core region and the outer air region is equal.

[0296]

[0297] Expand the interface conditions:

[0298]

[0299]

[0300] Therefore, the constraint equations can be obtained:

[0301] Z 181 A 1k +Z 182B 1k +Z 183 A 2k +Z 184 B 2k =0

[0302] Z 191 C 1k +Z 192 D 1k +Z 193 C 2k +Z 194 D 2k =0

[0303] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 181 Z 182 Z 183 Z 184 Z 191 Z 192 Z 193 Z 194 Let be the coefficient matrix of the constraint equation.

[0304] S3.6 Establish the boundary surface constraint equations for the outer air domain.

[0305] At the outer radius of the air domain, set the boundary condition with a vector magnetic potential of 0:

[0306]

[0307] Expand the boundary surface conditions:

[0308]

[0309] Therefore, the constraint equations can be obtained:

[0310] Z 201 A 1k +Z 202 B 1k =0

[0311] Z 211 C 1k +Z 212 D 1k =0

[0312] The process of establishing constraint equations based on interface conditions can be found in S3.1.1. 201 Z 202 Z 211 Z 212 Let be the coefficient matrix of the constraint equation.

[0313] S3.7 Establish the matrix equation and solve for the harmonic coefficients.

[0314] Solve the equations for the harmonic coefficients of the above six interfaces simultaneously, and then rearrange them into matrix form:

[0315]

[0316] By solving the above matrix equations, the harmonic coefficient A can be obtained. 1k B 1k C 1k D 1k .

[0317] S4: Write the expression for the vector magnetic potential in the outer air domain, and take the partial derivatives to establish an analytical model of the radial and tangential components of the stray magnetic field:

[0318] S4.1 will reduce the harmonic coefficient A 1k B 1k C 1k D 1k Substituting into the vector magnetic potential expression for the outer air domain, and taking the partial derivative of the vector magnetic potential for the outer air domain, we obtain the expressions for the radial and tangential magnetic flux densities of the stray magnetic field:

[0319]

[0320]

[0321] S4.2 Based on the mathematical expression, the spatiotemporal distribution of the radial and tangential magnetic flux density of the unloaded stray magnetic field at a position 1 mm from the outer surface of the motor is solved. The results are as follows: Figure 3 and Figure 4 As shown in the figure, the amplitude of the stray magnetic field of the external rotor hub permanent magnet motor under no-load conditions is relatively small, approximately 1.0 × 10⁻⁶. -4 Its magnitude is on the order of T, slightly larger than the Earth's magnetic field, but much smaller than the remanence of a permanent magnet. Its spatiotemporal distribution exhibits a clear periodicity. Therefore, further spatial order analysis and amplitude-frequency characteristic analysis were conducted.

[0322] The spatial distribution and spatial order of the radial component of the stray magnetic field are as follows: Figure 5 and Figure 6 As shown, the spatial distribution and spatial order of the tangential components are respectively as follows: Figure 7 and Figure 8 As shown in the four figures above, for this motor, the spatial distribution of the stray magnetic field has 23 cycles within one mechanical cycle. The main spatial order of the stray magnetic field is (2n-1)p±uQ. s n and u are non-negative integers, p is the pole pair, and Q is the polar pair. s The number of slots, for example, 23, 28, etc.

[0323] The time history and spectral characteristics of the radial component of the stray magnetic field are as follows: Figure 9 and Figure 10As shown, the time history and spectral characteristics of the tangential component are respectively as follows: Figure 11 and Figure 12 As shown in the four figures above, for this motor, the time history of the stray magnetic field has 23 cycles within one mechanical cycle. The main frequency component of the stray magnetic field is (2k-1)f0, where k is a positive integer and f0 is the fundamental frequency of the motor, such as 168.67Hz, 506.00Hz, 843.35Hz, etc.

[0324] The beneficial effects of this invention are:

[0325] To verify the accuracy of the analytical calculation method provided by this invention, a direct test experiment of the stray magnetic field of an external rotor hub permanent magnet motor was conducted. The experimental setup is as follows: Figure 13 As shown, the system includes a rotary table, a motor under test, a Hall probe, a probe holder, and a Tesla meter. The rotary table is used to fix the motor and rotate it. The Hall probe is used to measure the stray magnetic field of the motor. The probe holder is used to adjust and fix the position of the Hall probe. The Tesla meter is used to display the measured magnetic field value in real time. To achieve accurate measurement of the stray magnetic field, it is necessary to ensure that the magnetic field enters the Hall probe perpendicularly.

[0326] To compare the computational efficiency and resource consumption of the analytical calculation method provided in this invention, a finite element model of the stray magnetic field of the motor under no-load is established, such as... Figure 14 As shown. To ensure the accuracy of the finite element calculation as much as possible, the outer diameter of the outer air domain is set to 20 times the outer diameter of the motor, and a boundary condition with a vector magnetic potential of 0 is applied.

[0327] The radial and tangential components of the stray magnetic field of the motor under no-load were obtained using the analytical calculation method, finite element method, and experimental measurement method proposed in this invention, respectively. The data results obtained at 1 mm on the surface of the motor casing are as follows: Figure 15 and Figure 16 As shown.

[0328] The analytical calculation results proposed in this invention agree well with the experimental results.

[0329] Use R 2 Measuring the computational accuracy of the model:

[0330]

[0331] Table 2 shows a comparison of the errors, efficiency, and resource consumption of the analytical calculation method and the finite element method of the present invention, obtained through actual testing based on the method of this embodiment.

[0332] The analytical model achieves a computational accuracy of at least 96%, takes approximately 20 minutes, and occupies 42.5 MB of memory. The finite element model achieves a computational accuracy of at least 97%, takes approximately 110 minutes, and occupies 6245 MB of memory. Under the same case and the same computing hardware conditions, the proposed method's computation time is only 18% of that of the finite element model, and its memory usage is less than 7% of that of the finite element model. Therefore, the method proposed in this example has advantages such as high efficiency and low memory consumption, achieving a good balance between computational accuracy and efficiency.

[0333] The motors involved in this embodiment are all 46-pole, 51-slot external rotor hub permanent magnet motors with the main parameters shown in Table 1.

[0334] Table 2 Comparison of error, efficiency, and resource consumption between analytical methods and finite element methods.

[0335]

Claims

1. A method for analytical calculation of the no-load stray magnetic field of an external rotor hub permanent magnet motor, characterized in that, Includes the following steps: S1: In a two-dimensional polar coordinate system, an equivalent model of a motor containing 7 subdomains is established according to the different motor structures, and the magnetization intensity expression of the permanent magnet is established according to the magnetization method of the permanent magnet. S2: Using vector magnetic potential as the potential function, establish the governing equations for each subdomain and solve the general solution of vector magnetic potential for each subdomain; S3: Write the constraint equations based on the interface and boundary conditions of each subdomain, and solve for the harmonic coefficients; S4: Write the expression for the vector magnetic potential in the outer air domain, and calculate the radial and tangential components of the stray magnetic field by taking the partial derivatives; S1 includes the following steps: S1.1 Establish a two-dimensional polar coordinate system with a point on the rotor axis of the motor as the origin; S1.2 Based on the structure of the motor, it is divided into 7 sub-domains, including the external air domain, rotor core domain, air gap domain, permanent magnet domain, slot opening domain, slot domain and stator core domain; S1.3 Based on the actual shape and size of each subdomain, establish an equivalent model of the motor in a two-dimensional polar coordinate system; The permanent magnets of the S1.4 external rotor hub permanent magnet motor are radially magnetized, with alternating N and S poles. An expression for magnetization intensity is established based on the permanent magnet pole parameters: in, M θk =0,k / p=1,3,5,... In the formula, M r and M θ These represent the radial and tangential components of the remanent magnetization of the motor's permanent magnet, respectively; k is the harmonic order; θ is the angle; and ω is the tangential component. r Let ω be the angular velocity of the motor rotor, t be time, α0 be the initial angle of the motor rotor, and B be the angular velocity of the rotor. r α is the remanence of the permanent magnet in the motor. p denoted as the pole arc coefficient of the motor, p as the number of pole pairs of the motor, and μ0 as the permeability of free space. S2 includes the following steps: S2.1 Based on Ampere's circuital law for Maxwell's equations in a quasi-static field and the constitutive relations in an isotropic linear medium, the governing equations for each subdomain are as follows: Permanent magnet domain: Other subdomains besides the permanent magnet domain: In the formula, A zy The component of the vector magnetic potential in the z direction of the y subdomain is represented by y, where y represents the subdomain number, and r and θ represent the radius and angle, respectively. The subdomains are numbered as follows: outer air domain number 1, rotor core domain number 2, permanent magnet domain number 3, air gap domain number 4, slot opening domain number 5i, slot subdomain number 6i, and stator core domain number 7. S2.2 uses the method of separation of variables to obtain the vector magnetic potential general solution for each subdomain: outer air domain In the formula, A 1k B 1k C 1k and D 1k R is the harmonic coefficient to be solved for the vector magnetic potential in the outer air domain. a and R o These represent the outer and inner diameters of the outer air domain, respectively, and k represents the harmonic order. Rotor core region In the formula, A 2k B 2k C 2k and D 2k R is the harmonic coefficient to be solved for the vector magnetic potential of the rotor core domain. o and R r These represent the outer and inner diameters of the rotor core region, respectively. Permanent magnet domain In the formula, A 3k B 3k C 3k and D 3k R is the harmonic coefficient to be solved for the vector magnetic potential of the permanent magnet domain. r and R m These represent the outer and inner diameters of the permanent magnet domain, respectively. air gap domain In the formula, A 4k B 4k C 4k and D 4k R represents the harmonic coefficients to be solved for the vector magnetic potential in the air gap domain. m and R s These represent the outer and inner diameters of the air gap region, respectively. slot opening area Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero: In the formula, β oa θ represents the width of the slot opening. i Indicates the position angle of the i-th slot opening region; Based on the above boundary conditions, the vector magnetic potential flux solution for the i-th slot opening domain can be obtained as follows: In the formula, B 5i0 A 5im and B 5im R is the harmonic coefficient to be solved for the vector magnetic potential in the slot opening domain. s and R t These represent the outer and inner diameters of the slot opening, respectively, and m is the harmonic order of the slot opening. Slot Domain Because the permeability of the stator core is infinite, therefore... and At this location, the radial magnetic flux density of the slot opening region is zero: In the formula, β sa θ represents the width of the slot subdomain. i This represents the position angle of the i-th slot subdomain, which is the same as the position angle of the i-th slot opening domain; Meanwhile, at the interface between the bottom of the slot and the stator core, the tangential magnetic field strength of the slot subdomain is zero. Combining the two boundary conditions mentioned above, the vector magnetic potential of the slot subdomain can be obtained as follows: In the formula, A 6in R is the harmonic coefficient to be solved for the vector magnetic potential of the slot subdomain. t and R sb These represent the outer and inner diameters of the slot sub-domain, respectively, and n is the harmonic order of the slot sub-domain. S3 includes the following steps: S3.1 The vector magnetic potential and tangential magnetic field strength at the interface between the permanent magnet domain and the air gap domain are equal: The vector magnetic potential and tangential magnetic flux density at the interface between the air gap region and the slot opening region are equal in S3.

2. S3.3 Establish that the interface vector magnetic potential and tangential magnetic flux density are equal between the slot opening domain and the slot subdomain: S3.4 The interface vector magnetic potential and tangential magnetic field strength are equal at the junction of the permanent magnet domain and the rotor core domain: The vector magnetic potential and tangential magnetic field strength at the interface between the external air domain and the rotor core domain are equal in S3.

5. The vector magnetic potential at the outer radius of the S3.6 outer air domain is 0: S3.7 Establish the matrix equation and solve for the harmonic coefficients: By simultaneously solving the constraint equations of the above six interfaces and rearranging them into matrix form, the harmonic coefficient A can be obtained by solving the matrix equations. 1k B 1k C 1k D 1k .

2. The analytical calculation method for the no-load stray magnetic field of an external rotor hub permanent magnet motor according to claim 1, characterized in that, S4 includes the following steps: S4.1 Taking the partial derivative of the vector magnetic potential in the outer air domain, we obtain the expressions for the radial and tangential magnetic flux densities of the stray magnetic field: S4.2 The harmonic coefficient A of the vector magnetic potential in the outer air domain 1k B 1k C 1k D 1k By substituting the values, the stray magnetic field at any radius in the outer air domain can be solved.

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