CN116306007A - Analytical calculation method for no-load stray magnetic field of outer rotor hub permanent magnet motor - Google Patents

Analytical calculation method for no-load stray magnetic field of outer rotor hub permanent magnet motor Download PDF

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CN116306007A
CN116306007A CN202310349789.0A CN202310349789A CN116306007A CN 116306007 A CN116306007 A CN 116306007A CN 202310349789 A CN202310349789 A CN 202310349789A CN 116306007 A CN116306007 A CN 116306007A
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permanent magnet
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马琮淦
位昱慧
周生森
张星星
安原圣
赵亮
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Weihai Taiyi Motor Technology Co ltd
Harbin Institute of Technology Weihai
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Weihai Taiyi Motor Technology Co ltd
Harbin Institute of Technology Weihai
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Abstract

An external rotor hub permanent magnet motor no-load stray field analysis and calculation method relates to a motor stray field calculation method, which comprises the following steps of; s1: under a two-dimensional polar coordinate system, a motor equivalent model comprising 7 subdomains is established according to a motor structure, and a magnetization intensity expression of the permanent magnet is established according to a magnetization mode of the permanent magnet; s2: using the vector magnetic bits as bit functions, establishing a control equation of each subdomain, and solving a vector magnetic bit general solution of each subdomain; s3: the constraint equation is written through interface conditions and boundary conditions of all subfields, and the harmonic coefficient is solved; s4: and writing an outer air domain vector magnetic potential expression, and calculating the radial and tangential components of the stray magnetic field by solving the partial derivative. The invention can simultaneously meet the requirements of high precision, high calculation speed and low cost. The calculation efficiency of the stray magnetic field is greatly improved; can be used for the initial design stage of the motor-free prototype.

Description

Analytical calculation method for no-load stray magnetic field of outer rotor hub permanent magnet motor
Technical Field
The invention relates to a calculation method of a motor stray field, in particular to an analysis calculation method of an idle stray field of an outer rotor hub permanent magnet motor.
Background
The outer rotor hub permanent magnet motor has the advantages of flexible control, high integration level and the like, and is widely applied to electric bicycles and gradually applied to distributed driving electric automobiles.
The stray field of an outer rotor in-hub permanent magnet motor is defined as the magnetic field created by the magnetic flux leaking out of the motor housing. On the one hand, stray magnetic fields can interfere with precise electronic equipment to influence the safety of a control system; on the other hand, stray magnetic fields can provide state information of the motor, and the information can be used for fault diagnosis of bearing faults, turn-to-turn short circuits, eccentricity, demagnetization and the like. Thus, accurate calculation and analysis of stray magnetic fields is extremely important. The method for obtaining the stray magnetic field can be classified into an analytical method, a finite element method and an experimental method. Comprehensive investigation of the prior art, the stray magnetic field acquisition method mainly has the following problems:
(1) The existing analytical model can only calculate the air gap magnetic field, and cannot accurately analyze and calculate the stray magnetic field of the outer rotor hub permanent magnet motor. For example, there are techniques for calculating the air gap field by the subfield method, such as CN112347627A, CN114006559A, CN113343171A, CN111651914a; some technologies calculate an air gap magnetic field through an equivalent magnetic circuit method, such as CN114626244A, CN111327170A, CN108563912A, CN111327170A; some techniques calculate the air gap field by magnetic potential flux guiding methods, such as CN113868929A, CN113868929a.
(2) The finite element model is only a calculation model of a specific motor structure, has no generality, and is long in calculation time and low in calculation efficiency. For example, in CN113985282a and CN113094952a, in order to ensure accuracy of stray magnetic field calculation, the outer radius of the outer air domain of the motor needs to be set to be about 20 times of the outer diameter of the motor, which increases the number of meshes of the finite element model, and greatly prolongs the calculation time.
(3) The method for acquiring the stray magnetic field through the experimental method needs to prepare a motor prototype in advance, so that the method cannot be used for initial design of a motor without the prototype, cannot reveal the characteristic mechanism of the stray magnetic field, and has the advantages of high test equipment cost and high test cost. For example, CN101126799a tests the normal component of the stray magnetic field on the surface of ferromagnetic materials by means of magnetic sensors.
The invention aims to solve the problem that the prior art cannot calculate and analyze the stray magnetic field at high precision, high efficiency and low cost, and provides an external rotor hub permanent magnet motor no-load stray magnetic field analysis and calculation method based on a subdomain method.
Disclosure of Invention
The invention aims to solve the defects of the prior art, and provides an analysis and calculation method for the no-load stray magnetic field of the outer rotor hub permanent magnet motor, which can simultaneously meet the requirements of high precision, high calculation speed and low cost.
The invention solves the defects of the prior art, adopts the following technical proposal:
an external rotor hub permanent magnet motor no-load stray magnetic field analysis and calculation method comprises the following steps:
s1: under a two-dimensional polar coordinate system, according to different motor structures, establishing a motor equivalent model comprising 7 subdomains, and according to a magnetizing mode of a permanent magnet, establishing a magnetization intensity expression of the permanent magnet;
s2: using the vector magnetic bits as bit functions, establishing a control equation of each subdomain, and solving a vector magnetic bit general solution of each subdomain;
s3: the constraint equation is written through interface conditions and boundary conditions of all subfields, and the harmonic coefficient is solved;
s4: and writing an outer air domain vector magnetic potential expression, and calculating the radial and tangential components of the stray magnetic field by solving the partial derivative.
The step S1 comprises the following steps:
s1.1, a two-dimensional polar coordinate system is established by taking a point on the axis of the motor rotor as a coordinate origin.
S1.2, dividing the motor into 7 subdomains including an outer air domain, a rotor core domain, an air gap domain, a permanent magnet domain, a slot opening domain, a slot subdomain and a stator core domain according to the structure of the motor.
S1.3, establishing a motor equivalent model in a two-dimensional polar coordinate system according to the actual shape and size of each subdomain; wherein the outer air domain, the rotor core domain, the air gap domain and the permanent magnet domain are equivalent to form an annular model, and the slot opening domain and the slot subdomain are equivalent to form a fan-shaped model.
S1.4, permanent magnets of the outer rotor hub permanent magnet motor adopt a radial magnetizing mode, N poles and S poles are alternately distributed, and an expression of magnetization intensity is established according to magnetic pole parameters of the permanent magnets:
Figure BDA0004161150040000021
Figure BDA0004161150040000022
wherein,,
Figure BDA0004161150040000023
M θk ＝0，k/p＝1，3，5，...
wherein M is r And M θ Respectively representing radial and tangential components of the residual magnetization intensity of the motor permanent magnet, wherein k is harmonic frequency, theta is angle and omega r Is the angular velocity of the rotation of the motor rotor, t is time, alpha 0 For the initial angle of the motor rotor, B r Is the residual magnetic intensity of the permanent magnet of the motor, alpha p Is the pole arc coefficient of the motor, p is the pole pair number of the motor, mu 0 Is vacuum magnetic permeability.
The step S2 comprises the following steps:
s2.1, according to ampere loop law of Maxwell equation set in quasi-static field and constitutive relation in isotropic linear medium, each subdomain control equation is as follows:
permanent magnet domain:
Figure BDA0004161150040000024
other subfields than permanent magnets:
Figure BDA0004161150040000025
wherein A is zy Representing the component of the y-subfield vector in the z direction, y representing the subfield number, f and θ representing the radius and angle, respectively. The number of each subdomain is: outer air field number 1, rotor core field number 2, air gap field number 3, permanent magnet field number 4, slot opening field number 5i, slot subfield number 6i, stator core field number 7.
S2.2, vector magnetic flux potential general solution of each subdomain is obtained through a separation variable method:
outside air area
Figure BDA0004161150040000026
Wherein A is 1k 、B 1 k 、 C 1k And D 1k To-be-solved harmonic coefficient for outer air domain vector magnetic potential, R a And R is o Represents the outer and inner diameters of the outer air field, respectively, and k represents the harmonic order.
Rotor core field
Figure BDA0004161150040000027
Wherein A is 2k 、B 2k 、C 2k And D 2k To-be-solved harmonic coefficient for rotor core domain vector magnetic potential, R o And R is r Representing the outer and inner diameters of the rotor core field, respectively.
Permanent magnet field
Figure BDA0004161150040000031
In the method, in the process of the invention,
Figure BDA0004161150040000032
A 3k 、B 3k 、C 3k and D 3k To-be-solved harmonic coefficient for vector magnetic position of permanent magnet domain, R r And R is m Representing the outer and inner diameters of the permanent magnet fields, respectively.
Air gap domain
Figure BDA0004161150040000033
Wherein A is 4k 、B 4k 、C 4k And D 4k To-be-solved harmonic coefficient for air gap domain vector magnetic potential, R m And R is s Representing the outer and inner diameters of the air gap region, respectively.
Groove opening area
Since the magnetic permeability of the stator core is infinite, in
Figure BDA0004161150040000034
And->
Figure BDA0004161150040000035
Where the radial magnetic density of the slot opening domain is zero:
Figure BDA0004161150040000036
Figure BDA0004161150040000037
wherein beta is oa Represents the width of the opening area of the groove, theta i Indicating the ith slot openingAngle of position of the mouth region.
Combining the boundary conditions, the vector magnetic flux potential general solution of the ith slot opening domain can be obtained as follows:
Figure BDA0004161150040000038
in the method, in the process of the invention,
Figure BDA0004161150040000039
B 5i0 、A 5im and B 5im To-be-solved harmonic coefficient for slot opening domain vector magnetic potential, R s And R is t Represents the outer diameter and the inner diameter of the slot opening area respectively, and m is the harmonic frequency of the slot opening area.
Groove subdomain
Since the magnetic permeability of the stator core is infinite, in
Figure BDA00041611500400000310
And->
Figure BDA00041611500400000311
Where the radial magnetic density of the slot opening domain is zero:
Figure BDA00041611500400000312
Figure BDA00041611500400000313
wherein beta is sa Represents the slot subdomain width, θ i The position angle of the ith slot sub-field is shown, which is the same as the position angle of the ith slot opening field.
Meanwhile, at the interface between the bottom of the groove and the stator core, the tangential magnetic field strength of the groove subdomain is zero:
Figure BDA0004161150040000041
combining the two boundary conditions, the vector magnetic potential of the slot subdomain can be obtained as follows:
Figure BDA0004161150040000042
in the method, in the process of the invention,
Figure BDA0004161150040000043
A 6in to-be-solved harmonic coefficient for slot subdomain vector magnetic bits, R t And R is sb Respectively representing the outer diameter and the inner diameter of the groove subdomain, and n is the harmonic frequency of the groove subdomain.
The step S3 comprises the following steps:
s3.1, the vector magnetic potential of the interface between the permanent magnet domain and the air gap domain is equal, and the tangential magnetic field strength is equal:
Figure BDA0004161150040000044
Figure BDA0004161150040000045
s3.2, the interface vector magnetic potential of the air gap domain and the slot opening domain are equal, and the tangential magnetic densities are equal:
Figure BDA0004161150040000046
Figure BDA0004161150040000047
s3.3, establishing that the interface vector magnetic potential of the slot opening domain and the slot subdomain are equal, and the tangential magnetic densities are equal:
Figure BDA0004161150040000048
Figure BDA0004161150040000049
s3.4, the vector magnetic potential of the interface between the permanent magnet domain and the rotor core domain is equal, and the tangential magnetic field strength is equal:
Figure BDA00041611500400000410
Figure BDA00041611500400000411
s3.5, the vector magnetic potential of the interface between the rotor core domain and the outer air domain is equal, and the tangential magnetic field strength is equal:
Figure BDA00041611500400000412
Figure BDA00041611500400000413
s3.6 vector magnetic potential at outer radius of outer air domain is 0:
Figure BDA00041611500400000414
s3.7, establishing a matrix equation and solving harmonic coefficients
The constraint equations of the 6 interfaces are combined and are arranged into a matrix form, and the harmonic coefficient A is obtained by solving the matrix equation 1k 、B 1k 、C 1k 、D 1k 。
The step S4 comprises the following steps:
s4.1, carrying out bias conduction on vector magnetic potential of an external air domain to obtain expressions of radial magnetic flux density and tangential magnetic flux density of a stray magnetic field:
Figure BDA0004161150040000051
Figure BDA0004161150040000052
s4.2 harmonic coefficient A of the vector magnetic potential of the outer air domain 1k 、B 1k 、C 1k 、D 1k Substituting the magnetic field to solve the stray magnetic field at any radius of the outer air domain.
The motor is an outer rotor hub permanent magnet motor.
The subdomain includes: outer air field, rotor core field, air gap field, permanent magnet field, slot opening field, slot sub-field and stator core field.
Compared with the prior art, the invention has the following advantages:
1) The method can consider the slotting effect of the external rotor hub motor and the attenuation effect of the rotor core and the shell on the stray magnetic field, and compared with experimental test results, the precision is higher than 96%;
2) Compared with a finite element method, the method has the advantages that the calculation speed is 5.5 times faster, and the calculation efficiency of the stray magnetic field is greatly improved;
3) Compared with an experimental method, the method has extremely low cost and can be used for the initial design stage of a motor-free model machine.
The invention can simultaneously meet the requirements of high precision, high calculation speed and low cost. The calculation efficiency of the stray magnetic field is greatly improved; can be used for the initial design stage of the motor-free prototype.
Drawings
Fig. 1 is a motor sub-domain division schematic diagram.
Fig. 2 is a schematic diagram of motor boundary conditions and interface conditions.
FIG. 3 is a three-dimensional spatiotemporal distribution plot of the radial component of the stray field at 1mm from the outer surface of the outer rotor hub permanent magnet motor shown in Table 1 calculated using the method of the present invention.
FIG. 4 is a three-dimensional spatiotemporal distribution plot of the tangential component of the stray field at 1mm from the outer surface of an outer rotor hub permanent magnet motor as shown in Table 1 calculated using the method of the present invention.
Fig. 5 is a schematic diagram of the spatial distribution result of the radial component of the stray magnetic field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated by using the method of the present invention.
Fig. 6 is a schematic diagram of the spatial order of the radial component of the stray field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated using the method of the present invention.
Fig. 7 is a schematic diagram of the spatial distribution result of tangential components of stray magnetic fields at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated by using the method of the present invention.
Fig. 8 is a schematic diagram of the spatial order of the tangential component of the stray field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated using the method of the present invention.
FIG. 9 is a graph showing the time course of the radial component of the stray field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in Table 1 calculated using the method of the present invention.
Fig. 10 is a graph showing the result of the spectral characteristics of the radial component of the stray field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated using the method of the present invention.
FIG. 11 is a graph showing the time course of the tangential component of the stray field at 1mm from the outer surface of the outer rotor hub permanent magnet motor shown in Table 1 calculated using the method of the present invention.
Fig. 12 is a schematic diagram showing the result of the spectrum characteristic of the tangential component of the stray magnetic field at 1mm of the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated by using the method of the present invention.
Fig. 13 is a schematic diagram of an experimental apparatus for measuring an outer rotor hub permanent magnet motor as shown in table 1 using a direct test experiment.
Fig. 14 is a schematic diagram of the construction of the outer rotor hub permanent magnet motor stray field simulation model shown in table 1 using the finite element method.
Fig. 15 is a graph showing the comparison between the spatial distribution result of the radial component of the stray field and the experimental result at the position of 1mm on the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated by using the method of the present invention and the finite element method.
Fig. 16 is a graph showing the comparison between the spatial distribution result of tangential components of stray magnetic fields and the experimental result at the position of 1mm on the outer surface of the outer rotor hub permanent magnet motor shown in table 1 calculated by using the method and the finite element method of the present invention.
Detailed Description
The invention will now be described in detail with reference to the drawings and specific examples.
The embodiment takes an outer rotor hub permanent magnet motor with 46 poles and 51 grooves as an example. The main parameters of the motor are shown in table 1:
table 1 main parameters of motor
Figure BDA0004161150040000061
S1, dividing an internal solving domain of a motor:
s1.1, according to the structural characteristics of the outer rotor hub permanent magnet motor, a two-dimensional polar coordinate system is established by taking the center point of the outer surface of the motor end cover as the origin of coordinates.
S1.2, dividing the motor into 7 solving domains according to the difference of the internal structure and magnetic permeability of the motor, as shown in FIG. 1, wherein a subdomain 1 is an outer air domain, a subdomain 2 is a rotor core domain, a subdomain 3 is a permanent magnet domain, a subdomain 4 is an air gap domain, a subdomain 5i is an ith slot opening domain, a subdomain 6i is an ith slot subdomain, and a subdomain 7 is a stator core domain.
S1.3, the outer air domain, the rotor core domain, the air gap domain and the permanent magnet domain are equivalent to form an annular model, and the slot opening domain and the slot subdomain are equivalent to form a fan-shaped model. At the same time, to ensure the accuracy of stray magnetic field calculation, the outer radius R of the outer air domain a Set to 20 times the motor outer diameter.
S1.4 permanent magnet adoption of external rotor hub permanent magnet motorIn a radial magnetizing mode, N poles and S poles are alternately distributed. Residual magnetic strength B of permanent magnet r 0.35T, and vacuum permeability of 4pi×10 -7 Wb/(A.m) establishes an expression of magnetization according to the permanent magnet pole parameters:
Figure BDA0004161150040000071
Figure BDA0004161150040000072
wherein,,
Figure BDA0004161150040000073
M θk ＝0，k/p＝1，3，5，...
s2: and (3) taking the vector magnetic bits as bit functions, establishing a control equation of each subdomain, and solving a vector magnetic bit general solution of each subdomain:
s2.1, according to ampere loop law of Maxwell equation set in quasi-static field and constitutive relation in isotropic linear medium, each subfield control equation can be expressed:
permanent magnet domain:
Figure BDA0004161150040000074
other subfields than permanent magnets:
Figure BDA0004161150040000075
s2.2, vector magnetic flux position general solution of each subdomain can be obtained by combining part of interface conditions through a separation variable method, and the boundary conditions and the interface conditions of the motor are shown in figure 2.
S2.2.1 outside air region
Figure BDA0004161150040000076
S2.2.2 rotor core domain
Figure BDA0004161150040000077
S2.2.3 permanent magnet domains
Figure BDA0004161150040000078
In the method, in the process of the invention,
Figure BDA0004161150040000079
s2.2.4 air gap region
Figure BDA00041611500400000710
S2.2.5 slot opening area
Since the magnetic permeability of the stator core is infinite, in
Figure BDA0004161150040000081
And->
Figure BDA0004161150040000082
Where the radial magnetic density of the slot opening domain is zero:
Figure BDA0004161150040000083
Figure BDA0004161150040000084
combining the boundary conditions, the vector magnetic flux potential general solution of the ith slot opening domain can be obtained as follows:
Figure BDA0004161150040000085
in the method, in the process of the invention,
Figure BDA0004161150040000086
s2.2.6 groove subdomain
Since the magnetic permeability of the stator core is infinite, in
Figure BDA0004161150040000087
And->
Figure BDA0004161150040000088
Where the radial magnetic density of the slot opening domain is zero:
Figure BDA0004161150040000089
Figure BDA00041611500400000810
meanwhile, at the interface between the bottom of the groove and the stator core, the tangential magnetic field strength of the groove subdomain is zero:
Figure BDA00041611500400000811
combining the two boundary conditions, the vector magnetic potential of the slot subdomain can be obtained as follows:
Figure BDA00041611500400000812
in the method, in the process of the invention,
Figure BDA00041611500400000813
s3: and solving harmonic coefficients by writing constraint equations through interface conditions and boundary conditions of all subfields:
s3.1 establishing an interface condition constraint equation of the permanent magnet domain and the air gap domain
S3.1.1 the vector flux at the interface of the permanent magnet domain and the air gap domain is equal:
Figure BDA00041611500400000814
the interface conditions were expanded:
Figure BDA0004161150040000091
Figure BDA0004161150040000092
the sine term coefficient and cosine term coefficient of the two equations are respectively equal:
Figure BDA0004161150040000093
Figure BDA0004161150040000094
the process is carried out by the steps of,
Figure BDA0004161150040000095
Z 11 ＝diag(Z 11_k )Z 21 ＝diag(Z 21_k )
Z 12 ＝eye(K)Z 22 ＝eye(K)
Z 13 ＝-eye(K)Z 23 ＝-eye(K)
Figure BDA0004161150040000096
Z 14 ＝diag(Z 14_k )Z 24 ＝diag(Z 24_k )
f 1s_k ＝kR m M k sin(kω r t+kα 0 )f 1c_k ＝-kR m M k cos(kω r t+kα 0 )
f 1s ＝f 1s_k f 1c ＝f 1c_k
in the formula, diag is a diagonal matrix, eye is a unit matrix, and f' 1s_k Is f 1s_k Transposed matrix of f' 1c_k Is f 1c_k Is the transposed matrix of K, K is the maximum harmonic order of K, Z 11 、Z 12 、Z 13 、Z 14 、Z 21 、Z 22 、Z 23 、Z 24 Coefficient matrix as constraint equation, f 1s And f 1c Is a constant quantity matrix of constraint equations.
From this, constraint equations can be established:
Z 11 A 3k +Z 12 B 3k +Z 13 A 4k +Z 14 B 4k ＝f 1s
Z 21 C 3k +Z 22 D 3k +Z 23 C 4k +Z 24 D 4k ＝f 1c
s3.1.2 the tangential magnetic field strength of the permanent magnet domain and air gap domain interface is equal:
Figure BDA0004161150040000097
the interface conditions were expanded:
Figure BDA0004161150040000101
Figure BDA0004161150040000102
wherein mu is r Is the relative permeability of the permanent magnet.
From this, constraint equations can be established:
Z 31 A 4k +Z 32 B 3k +Z 33 A 3k +Z 34 B 4k ＝f 2s
Z 41 C 4k +Z 42 D 3k +Z 43 C 3k +Z 44 D 4k ＝f 2c
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 31 、Z 32 、Z 33 、Z 34 、Z 41 、Z 42 、Z 43 、Z 44 Coefficient matrix as constraint equation, f 2s And f 2c Is a constant quantity matrix of constraint equations.
S3.2 establishing an interface condition constraint equation of the air gap domain and the groove opening domain
S3.2.1 the vector flux at the interface of the air gap domain and slot opening domain is equal:
Figure BDA0004161150040000103
the interface conditions were expanded:
Figure BDA0004161150040000104
Figure BDA0004161150040000105
in order to unify the Fourier index variable of the air gap domain and the slot opening domain vector magnetic bits, the vector magnetic bits of the air gap domain are spread out in the slot opening domain:
Figure BDA0004161150040000106
wherein,,
Figure BDA0004161150040000107
Figure BDA0004161150040000111
Figure BDA0004161150040000112
Figure BDA0004161150040000113
Figure BDA0004161150040000114
Figure BDA0004161150040000115
from this, constraint equations can be established:
Z 51 A 4k +Z 52 B 4k +Z 53 C 4k +Z 54 D 4k +Z 55 B 5i0 ＝0
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
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 51 、Z 52 、Z 53 、Z 54 、Z 55 、Z 61 、Z 62 、Z 63 、Z 64 、Z 65 、Z 66 For constraint equationsIs a coefficient matrix of (a).
S3.2.2 the tangential magnetic densities of the air gap domain and slot opening domain interfaces are equal:
Figure BDA0004161150040000116
the interface conditions were expanded:
Figure BDA0004161150040000117
Figure BDA0004161150040000118
in order to unify the Fourier index variable of the tangential magnetic densities of the air gap domain and the slot opening domain, the tangential magnetic density of the slot opening domain is spread out in the air gap domain:
Figure BDA0004161150040000119
wherein,,
Figure BDA00041611500400001110
Figure BDA00041611500400001111
Figure BDA0004161150040000121
Figure BDA0004161150040000122
Figure BDA0004161150040000123
from this, constraint equations can be established:
Z 71 A 4k +Z 72 B 4k +Z 73 B 5i0 +Z 74 A 5im +Z 75 B 5im ＝0
Z 81 C 4k +Z 82 D 4k +Z 83 B 5i0 +Z 84 A 5im +Z 85 B 5im ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 71 、Z 72 、Z 73 、Z 74 、Z 75 、Z 81 、Z 82 、Z 83 、Z 84 、Z 85 Is a coefficient matrix of constraint equations.
S3.3 establishing an interface condition constraint equation of the slot opening domain and the slot subdomain
S3.3.1 the vector magnetic potential of the slot opening domain and slot subdomain interface is equal to:
Figure BDA0004161150040000124
the interface conditions were expanded:
Figure BDA0004161150040000125
Figure BDA0004161150040000126
in order to unify the Fourier series index variable of the vector magnetic positions of the slot subdomains and the slot opening domains, the vector magnetic positions of the slot subdomains are Fourier unfolded in the slot opening domains:
Figure BDA0004161150040000127
in the method, in the process of the invention,
Figure BDA0004161150040000128
Figure BDA0004161150040000129
Figure BDA00041611500400001210
from this constraint equation:
Z 61 B 5i0 +Z 92 A 6in ＝0
Z 101 A 5im +Z 102 B 5im +Z 103 A 6in ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 91 、Z 92 、Z 101 、Z 102 、Z 103 Is a coefficient matrix of constraint equations.
S3.3.2 the tangential magnetic density of the slot opening domain and slot subdomain interface is equal:
Figure BDA0004161150040000131
the interface conditions were expanded:
Figure BDA0004161150040000132
Figure BDA0004161150040000133
in order to unify the Fourier series index variables of the tangential magnetic densities of the slot subdomains and the slot opening domains, the tangential magnetic densities of the slot opening domains are Fourier unfolded in the slot subdomains:
Figure BDA0004161150040000134
/>
in the method, in the process of the invention,
Figure BDA0004161150040000135
Figure BDA0004161150040000136
from this constraint equation:
Z 111 B 5i0 +Z 112 A 5im +Z 113 B 5im +Z 114 A 6in ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 111 、Z 112 、Z 113 、Z 114 Is a coefficient matrix of constraint equations.
S3.4, establishing an interface condition constraint equation of the permanent magnet domain and the rotor core domain
S3.4.1 the vector magnetic potential of the interface between the permanent magnet domain and the rotor core domain is equal to that of the interface:
Figure BDA0004161150040000137
the interface conditions were expanded:
Figure BDA0004161150040000138
Figure BDA0004161150040000139
from this constraint equation:
Z 121 A 2k +Z 122 B 2k +Z 123 A 3k +Z 124 B 3k ＝f 3s
Z 131 C 2k +Z 132 D 2k +Z 133 C 3k +Z 134 D 3k ＝f 3c
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 121 、Z 122 、Z 123 、Z 124 、Z 131 、Z 132 、Z 133 、Z 134 Coefficient matrix as constraint equation, f 3s And f 3c Is a constant quantity matrix of constraint equations.
S3.4.2 the tangential magnetic field strength of the interfaces of the permanent magnet domain and the rotor core domain is equal:
Figure BDA0004161150040000141
the interface conditions were expanded:
Figure BDA0004161150040000142
/>
Figure BDA0004161150040000143
wherein mu is ro Is the relative permeability of the rotor core.
From this constraint equation:
Z 141 A 2k +Z 142 B 2k +Z 143 A 3k +Z 144 B 3k ＝f 4s
Z 151 C 2k +Z 152 D 2k +Z 153 C 3k +Z 154 D 3k ＝f 4c
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 141 、Z 142 、Z 143 、Z 144 、Z 151 、Z 152 、Z 153 、Z 154 Coefficients for constraint equationsMatrix f 4s And f 4c Is a constant quantity matrix of constraint equations.
S3.5, establishing an interface condition constraint equation of a rotor core domain and an outer air domain
S3.5.1 rotor core domain and outer air domain interface vector magnetic potential are equal
Figure BDA0004161150040000144
The interface conditions were expanded:
Figure BDA0004161150040000145
Figure BDA0004161150040000146
from this constraint equation:
Z 161 A 1k +Z 162 B 1k +Z 163 A 2k +Z 164 B 2k ＝0
Z 171 C 1k +Z 172 D 1k +Z 173 C 2k +Z 174 D 2k ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 161 、Z 162 、Z 163 、Z 164 、Z 171 、Z 172 、Z 173 、Z 174 Is a coefficient matrix of constraint equations.
Tangential magnetic field strength of interface between S3.5.2 rotor core domain and external air domain is equal
Figure BDA0004161150040000147
The interface conditions were expanded:
Figure BDA0004161150040000151
Figure BDA0004161150040000152
from this constraint equation:
Z 181 A 1k +Z 182 B 1k +Z 183 A 2k +Z 184 B 2k ＝0
Z 191 C 1k +Z 192 D 1k +Z 193 C 2k +Z 194 D 2k ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 181 、Z 182 、Z 183 、Z 184 、Z 191 、Z 192 、Z 193 、Z 194 Is a coefficient matrix of constraint equations.
S3.6 establishing a boundary surface condition constraint equation of the outer air domain
At the outside radius of the air, a boundary condition is set that the vector magnetic potential is 0:
Figure BDA0004161150040000153
expanding the boundary surface condition:
Figure BDA0004161150040000154
from this constraint equation:
Z 201 A 1k +Z 202 B 1k ＝0
Z 211 C 1k +Z 212 D 1k ＝0
the process of establishing constraint equations from interface conditions may be referred to as S3.1.1.Z is Z 201 、Z 202 、Z 211 、Z 212 Is a coefficient matrix of constraint equations.
S3.7, establishing a matrix equation and solving harmonic coefficients
Solving equations by combining harmonic coefficients of the 6 interfaces, and arranging the equations into a matrix form:
Figure BDA0004161150040000155
by solving the matrix equation, the harmonic coefficient A can be obtained 1k 、B 1k 、C 1k 、D 1k 。
S4: and (3) writing an outer air domain vector magnetic level expression, solving bias guide and establishing a stray magnetic field radial and tangential component analysis model:
s4.1 the harmonic coefficient A 1k 、B 1k 、C 1k 、D 1k Substituting the expression of the vector magnetic potential of the external air domain into the expression of the vector magnetic potential of the external air domain to obtain the expression of the radial magnetic flux density and the tangential magnetic flux density of the stray magnetic field:
Figure BDA0004161150040000161
/>
Figure BDA0004161150040000162
s4.2, solving the space-time distribution of the radial and tangential magnetic flux density of the idle stray magnetic field at a position 1mm away from the outer surface of the motor according to a mathematical expression, wherein the result is shown in fig. 3 and 4. As can be seen from the figure, the amplitude of the no-load stray magnetic field of the external rotor hub permanent magnet motor is smaller, about 1.0 multiplied by 10 -4 The order of T is slightly larger than the geomagnetic field and is far smaller than the residual magnetism of the permanent magnet. The time-space distribution of the method shows obvious periodicity. For this purpose, spatial order analysis and amplitude-frequency characteristic analysis are further performed thereon.
The spatial distribution and spatial order of the radial components of the stray magnetic field are shown in fig. 5 and 6, respectively, and the spatial distribution and spatial order of the tangential components are shown in fig. 7 and 8, respectively. As can be seen from the four figures, for thisThe motor has a spatial distribution of stray magnetic fields with 23 cycles in one mechanical cycle. The primary spatial order of the stray magnetic field is (2 n-1) p+ -uQ s N and u are nonnegative integers, p is the pole pair number, Q s For example, 23, 28, etc.
The time history and spectral characteristics of the radial component of the stray magnetic field are shown in fig. 9 and 10, respectively, and the time history and spectral characteristics of the tangential component are shown in fig. 11 and 12, respectively. As can be seen from the four figures, the time course of the stray field has 23 cycles in one mechanical cycle for the motor. The main frequency component of the stray magnetic field is (2 k-1) f 0 Wherein k is a positive integer, f 0 For example, 168.67Hz, 506.00Hz, 843.35Hz, etc.
The invention has the beneficial effects that:
in order to verify the accuracy of the analytic calculation method provided by the invention, a direct test experiment of the stray magnetic field of the outer rotor hub permanent magnet motor is carried out. The experimental device is shown in fig. 13, and comprises a rotary workbench, a tested motor, a Hall probe, a probe support and a Tesla gauge, wherein the rotary workbench is used for fixing the motor and driving the motor to rotate, the Hall probe is used for measuring a stray magnetic field of the motor, the probe support is used for adjusting and fixing the position of the Hall probe, and the Tesla gauge is used for displaying the numerical value of the measured magnetic field in real time. In order to realize accurate measurement of the stray magnetic field, the magnetic field needs to be ensured to vertically enter the Hall probe.
In order to compare the calculation efficiency and occupied resources of the analytic calculation method provided by the invention, a finite element model of the no-load stray magnetic field of the motor is established, as shown in fig. 14. To ensure accuracy of finite element calculations as much as possible, the outer diameter of the outer air field is set to 20 times the outer diameter of the motor while applying a boundary condition that the vector magnetic potential is 0.
The radial and tangential components of the no-load stray magnetic field of the motor are obtained by respectively adopting the analytic calculation method, the finite element calculation method and the experimental measurement method, and the obtained data results at the position of 1mm on the surface of the motor shell are respectively shown in fig. 15 and 16.
The analysis and calculation result provided by the invention is well matched with the experimental result.
By R 2 Measuring the calculation accuracy of the model:
Figure BDA0004161150040000163
the actual test is carried out according to the method of the embodiment, and the error, efficiency and occupied resource pair of the analytic calculation method and the finite element method are shown in the table 2.
The calculation accuracy of the analytical model is not lower than 96%, and the time is about 20 minutes, and the memory of 42.5MB is occupied. The calculation accuracy of the finite element model is not lower than 97%, the time is about 110 minutes, and 6245MB of memory is occupied. Under the same conditions of the same case and the same calculation hardware, the calculation time of the method is only 18% of the finite element, and the calculation time occupies 7% of the finite element with insufficient memory. Therefore, the method provided by the embodiment has the advantages of high efficiency, small memory occupation and the like, and can achieve good balance between calculation precision and efficiency.
The motors involved in this embodiment are all external rotor hub permanent magnet motors with 46-pole 51 slots, the main parameters of which are shown in table 1.
Table 2 comparison of errors, efficiency and occupancy resources of the analytical method and the finite element method
Figure BDA0004161150040000171
/>

Claims (5)

1. The method for analyzing and calculating the no-load stray magnetic field of the outer rotor hub permanent magnet motor is characterized by comprising the following steps of:
s1: under a two-dimensional polar coordinate system, according to different motor structures, establishing a motor equivalent model comprising 7 subdomains, and according to a magnetizing mode of a permanent magnet, establishing a magnetization intensity expression of the permanent magnet;
s2: using the vector magnetic bits as bit functions, establishing a control equation of each subdomain, and solving a vector magnetic bit general solution of each subdomain;
s3: the constraint equation is written through interface conditions and boundary conditions of all subfields, and the harmonic coefficient is solved;
s4: and writing an outer air domain vector magnetic potential expression, and calculating the radial and tangential components of the stray magnetic field by solving the partial derivative.
2. The method for analyzing and calculating the no-load stray field of the outer rotor hub permanent magnet motor according to claim 1, wherein the step S1 comprises the following steps:
s1.1, establishing a two-dimensional polar coordinate system by taking a point on the axis of a motor rotor as a coordinate origin;
s1.2, dividing the motor into 7 subdomains including an outer air domain, a rotor core domain, an air gap domain, a permanent magnet domain, a slot opening domain, a slot subdomain and a stator core domain according to the structure of the motor;
s1.3, establishing a motor equivalent model in a two-dimensional polar coordinate system according to the actual shape and size of each subdomain;
s1.4, permanent magnets of the outer rotor hub permanent magnet motor adopt a radial magnetizing mode, N poles and S poles are alternately distributed, and an expression of magnetization intensity is established according to magnetic pole parameters of the permanent magnets:
Figure FDA0004161150020000011
Figure FDA0004161150020000012
wherein,,
Figure FDA0004161150020000013
M θk ＝0,k/p＝1,3,5,...
wherein M is r And M θ Represents the radial and tangential components of the residual magnetization of the permanent magnet of the motor, k is the harmonic frequency, theta is the angle, r is the angular velocity of the rotation of the motor rotor, t is time, alpha 0 For the initial angle of the motor rotor, B r Is the residual magnetic intensity of the permanent magnet of the motor, alpha p Is the pole arc coefficient of the motor, p is the pole pair number of the motor, 0 is vacuum magnetic permeability.
3. The method for calculating the no-load stray field analysis of the outer rotor hub permanent magnet motor according to claim 2, wherein the step S2 comprises the following steps:
s2.1, according to ampere loop law of Maxwell equation set in quasi-static field and constitutive relation in isotropic linear medium, each subdomain control equation is as follows:
permanent magnet domain:
Figure FDA0004161150020000014
other subfields than permanent magnets:
Figure FDA0004161150020000015
wherein A is zy Representing the component of the vector magnetic bit z direction of the y subdomain, y represents the subdomain number, and r and theta represent the radius and angle respectively; the number of each subdomain is: outer air domain number 1, rotor core domain number 2, air gap domain number 3, permanent magnet domain number 4, slot opening domain number 5i, slot subdomain number 6i, stator core domain number 7;
s2.2, vector magnetic flux potential general solution of each subdomain is obtained through a separation variable method:
outside air area
Figure FDA0004161150020000021
Wherein A is 1k 、B 1k 、C 1k And D 1k To-be-solved harmonic coefficient for outer air domain vector magnetic potential, R a And R is o Respectively representing the outer diameter and the inner diameter of the outer air domain, and k represents the harmonic frequency;
rotor core field
Figure FDA0004161150020000022
Wherein A is 2k 、B 2k 、C 2k And D 2k To-be-solved harmonic coefficient for rotor core domain vector magnetic potential, R o And R is r Representing the outer diameter and the inner diameter of the rotor core field, respectively;
permanent magnet field
Figure FDA0004161150020000023
In the method, in the process of the invention,
Figure FDA0004161150020000024
A 3k 、B 3k 、C 3k and D 3k To-be-solved harmonic coefficient for vector magnetic position of permanent magnet domain, R r And R is m Representing the outer and inner diameters of the permanent magnet domains, respectively;
air gap domain
Figure FDA0004161150020000025
Wherein A is 4k 、B 4k 、C 4k And D 4k To-be-solved harmonic coefficient for air gap domain vector magnetic potential, R m And R is s Representing the outer and inner diameters of the air gap region, respectively;
groove opening area
Since the magnetic permeability of the stator core is infinite, in
Figure FDA0004161150020000026
And->
Figure FDA0004161150020000027
Where the radial magnetic density of the slot opening domain is zero:
Figure FDA0004161150020000028
Figure FDA0004161150020000029
wherein beta is oa Represents the width of the opening area of the groove, theta i Representing the position angle of the i-th slot opening domain;
combining the boundary conditions, the vector magnetic flux potential general solution of the ith slot opening domain can be obtained as follows:
Figure FDA00041611500200000210
in the method, in the process of the invention,
Figure FDA00041611500200000211
β 5i0 、A 5im and B 5im To-be-solved harmonic coefficient for slot opening domain vector magnetic potential, R s And R is t Respectively representing the outer diameter and the inner diameter of the slot opening area, wherein m is the harmonic frequency of the slot opening area;
groove subdomain
Since the magnetic permeability of the stator core is infinite, in
Figure FDA0004161150020000031
And->
Figure FDA0004161150020000032
Where the radial magnetic density of the slot opening domain is zero:
Figure FDA0004161150020000033
Figure FDA0004161150020000034
wherein beta is sa Represents the slot subdomain width, θ i Representing the position angle of the ith slot sub-field, which is the same as the position angle of the ith slot opening field;
meanwhile, at the interface between the bottom of the groove and the stator core, the tangential magnetic field strength of the groove subdomain is zero:
Figure FDA0004161150020000035
combining the two boundary conditions, the vector magnetic potential of the slot subdomain can be obtained as follows:
Figure FDA0004161150020000036
in the method, in the process of the invention,
Figure FDA0004161150020000037
A 6in to-be-solved harmonic coefficient for slot subdomain vector magnetic bits, R t And R is sb Respectively representing the outer diameter and the inner diameter of the groove subdomain, and n is the harmonic frequency of the groove subdomain.
4. The method for calculating the no-load stray field analysis of the outer rotor hub permanent magnet motor according to claim 3, wherein the step S3 comprises the following steps:
s3.1, the vector magnetic potential of the interface between the permanent magnet domain and the air gap domain is equal, and the tangential magnetic field strength is equal:
Figure FDA0004161150020000038
Figure FDA0004161150020000039
s3.2, the interface vector magnetic potential of the air gap domain and the slot opening domain are equal, and the tangential magnetic densities are equal:
Figure FDA00041611500200000310
Figure FDA00041611500200000311
s3.3, establishing that the interface vector magnetic potential of the slot opening domain and the slot subdomain are equal, and the tangential magnetic densities are equal:
Figure FDA00041611500200000312
Figure FDA00041611500200000313
s3.4, the vector magnetic potential of the interface between the permanent magnet domain and the rotor core domain is equal, and the tangential magnetic field strength is equal:
Figure FDA00041611500200000314
Figure FDA0004161150020000041
s3.5, the vector magnetic potential of the interface between the rotor core domain and the outer air domain is equal, and the tangential magnetic field strength is equal:
Figure FDA0004161150020000042
Figure FDA0004161150020000043
s3.6 vector magnetic potential at outer radius of outer air domain is 0:
Figure FDA0004161150020000044
s3.7, establishing a matrix equation, and solving harmonic coefficients:
the constraint equations of the 6 interfaces are combined and are arranged into a matrix form, and the harmonic coefficient A is obtained by solving the matrix equation 1k 、B 1k 、C 1k 、D 1k 。
5. The method for calculating the no-load stray field analysis of the outer rotor hub permanent magnet motor according to claim 4, wherein S4 comprises the following steps:
s4.1, carrying out bias conduction on vector magnetic potential of an external air domain to obtain expressions of radial magnetic flux density and tangential magnetic flux density of a stray magnetic field:
Figure FDA0004161150020000045
Figure FDA0004161150020000046
s4.2 harmonic coefficient A of the vector magnetic potential of the outer air domain 1k 、B 1k 、C 1k 、D 1k Substituting the magnetic field to solve the stray magnetic field at any radius of the outer air domain.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117749010A (en) * 2023-10-30 2024-03-22 天津大学 Magnetic field calculation method of parallel magnetic circuit axial flux permanent magnet motor

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