CN110346719B - Rotor eccentricity analysis method based on accurate angle-preserving mapping - Google Patents

Rotor eccentricity analysis method based on accurate angle-preserving mapping Download PDF

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CN110346719B
CN110346719B CN201910554843.9A CN201910554843A CN110346719B CN 110346719 B CN110346719 B CN 110346719B CN 201910554843 A CN201910554843 A CN 201910554843A CN 110346719 B CN110346719 B CN 110346719B
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黄晓艳
李赵凯
朱加贝
吴立建
方攸同
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Zhejiang University ZJU
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Abstract

The invention discloses a rotor eccentricity analysis method based on accurate angle-preserving mapping, which comprises the following steps: the method comprises the steps of mapping an air gap region of a surface-mounted motor with eccentric rotor to an annular air gap region with a tooth slot concentric with a stator and a rotor through bilinear mapping, mapping the annular air gap region with the tooth slot to a polygonal air gap region through logarithmic mapping, mapping the polygonal air gap region to a rectangular air gap region through Schwarz-Christoffel mapping, and finally mapping the rectangular air gap region to two air gap regions without concentric slots through exponential mapping. Calculating the electromagnetic field distribution of the air gap region of the non-groove concentric circle according to a black lattice equation; the air gap magnetic field distribution of the motor under the original rotor eccentricity can be obtained through the inverse transformation of the 4 conformal mappings, and then the induction voltage and the electromagnetic torque of the rotor eccentric motor are obtained. The time for calculating the dynamic eccentricity and the static eccentricity is shorter than the finite element calculation time, and the prediction result is completely consistent with the finite element result.

Description

Rotor eccentricity analysis method based on accurate angle-preserving mapping
Technical Field
The invention relates to the field of motor electromagnetic field analysis, in particular to a rotor eccentricity analysis method based on accurate angle-preserving mapping.
Background
The analysis of the air gap field of the permanent magnet motor is the basis of motor design and performance calculation, and is generally solved by adopting a numerical method or an analytical method. The finite element has the characteristics of high calculation precision and wide application range, but the calculation time is long, so that the calculation period is prolonged. The analytic method has the characteristic of high calculation speed, but the precision is relatively low compared with a finite element. In the initial optimization design process of the motor, modeling optimization is generally performed by using an analytical method. After a proper basic structure of the motor stator and the motor rotor is obtained, a finite element method is adopted for further optimization design and result verification.
The method for solving the motor electromagnetic field by using the analytic method mainly comprises a subfield method (typically shown in patent CN105005692A), a magnetic circuit method (typically shown in patent CN103823926A) and the like, but the methods cannot consider the rotor eccentricity of the motor. Due to inevitable influence factors in actual production and manufacturing, such as manufacturing process, structural deformation, bearing errors and the like, the rotor of the motor is often eccentric to the stator, so that the performance of the motor is changed. In order to accurately predict the electromagnetic performance of the rotor eccentric motor, the invention provides a rotor eccentricity analysis method based on accurate angle-preserving mapping, which can convert an eccentric motor into a concentric motor through angle-preserving mapping, thereby considering the influence of the rotor eccentricity and having the characteristics of high precision and high calculation speed.
Disclosure of Invention
The invention mainly solves the technical problem that the analytic analysis of the electromagnetic field under the eccentricity of the motor rotor is difficult.
In order to solve the technical problems, the invention adopts a technical scheme that:
a rotor eccentricity analysis method based on accurate conformal mapping comprises the following steps:
1. mapping an air gap area of the rotor eccentric motor to an annular air gap area with tooth grooves, which is concentric with the stator and the rotor, through bilinear mapping;
2. mapping the concentric annular air gap areas with tooth grooves of the stator and the rotor to polygonal air gap areas through logarithmic mapping;
3. mapping the polygonal airgap region to the rectangular airgap region by Schwarz-Christoffel mapping;
4. mapping the rectangular air gap area to two non-groove concentric circle air gap areas through exponential mapping, and calculating the magnetic induction intensity distribution of the non-groove concentric circle air gap areas according to a black grid equation;
5. and (4) obtaining the air gap magnetic field distribution of the rotor eccentric motor by utilizing the magnetic induction intensity distribution of the air gap area of the non-groove concentric circle obtained in the step (4) through the inverse transformation of the angle keeping mapping in the step (1-4), and further obtaining the induced voltage and the electromagnetic torque of the rotor eccentric motor.
Further, the formula of the bilinear mapping in step 1 is as follows:
Figure BDA0002106599320000021
Figure BDA0002106599320000022
Figure BDA0002106599320000023
d＝c0ejβ
Figure BDA0002106599320000024
Figure BDA0002106599320000025
wherein S represents the air gap region of the eccentric motor, T represents the annular air gap region, and the stator inner diameter of the motor is RsThe outer diameter of the rotor is Rr，ρdRepresents OsOrDistance, beta, represents OsOrAngle to the horizontal axis.
Further, the formula of the logarithmic mapping in step 2 is Z ═ log (t), where Z represents a polygonal air gap region.
Further, the formula of the Schwarz-christofel mapping in step 3 is as follows:
Figure BDA0002106599320000026
where W represents the rectangular air gap area,Z0,Z1,w0,w,wk,αjis a parameter of the Schwarz-christofel mapping, derived from the values of the points of the polygon on the polygonal airgap region Z.
Further, the formula of the exponential mapping in step 4 is as follows:
Figure BDA0002106599320000027
where Ψ represents the circular air gap region, Δ x and Δ y are the length and width of the rectangular air gap region W, and j is an imaginary unit;
the black lattice equation is as follows:
Figure BDA0002106599320000028
Figure BDA0002106599320000029
Figure BDA0002106599320000031
Figure BDA0002106599320000032
wherein B isrψkIs the internal polar coordinate (r) of the air gap region psi of the circular ringψ,αψ) Distribution of radial magnetic induction in position, BtψkIs a tangential magnetic induction intensity distribution on polar coordinates in a psi annular air gap regionrψAnd RsψIs the inner and outer diameters, mu, of the annular air gap region psi0Is a vacuum permeability, IkThe magnitude of the current in polar coordinates in the circular air gap region Ψ.
Further, the step 5 specifically comprises the following steps:
the inverse of the conformal mapping is to map the radial direction at the annular air gap region ΨMagnetic induction intensity distribution BrψkAnd tangential magnetic induction distribution BtψkAir-gap field distribution B mapped to air-gap region S of eccentric motorskI.e. by
Bsk(r,α)＝Bsrk(r,α)+jBstk(r,α)＝λ*(Brψk(r,α)+jBtψk(r,α))
Figure BDA0002106599320000033
Wherein,
Figure BDA0002106599320000034
respectively are partial derivative functions of the four conformal mapping functions;
according to the air-gap magnetic field distribution B of the rotor eccentric motorskObtaining the induced voltage E of the rotor eccentric motorpAnd an electromagnetic torque T, the calculation formula being as follows:
Figure BDA0002106599320000035
Figure BDA0002106599320000036
Figure BDA0002106599320000037
wherein N iscFor the number of turns per slot of the motor, lefIs the effective length of the motor, alphaiIs the initial angle of the coil winding, τ is the winding span, and r is the radius of any circle within the air gap region S of the eccentric machine.
The invention has the beneficial effects that:
1. the electromagnetic performance of the eccentric motor is directly calculated by an analytical method, and the calculation time under the condition of calculating the dynamic eccentricity and the static eccentricity is shorter than that of a finite element;
2. the predicted result is completely consistent with the result calculated by the finite element.
Drawings
FIG. 1 is a diagram of a surface-mounted permanent magnet machine with rotor eccentricity in the S domain;
FIG. 2 is a diagram of the concentric air gap region of the stator and rotor in the T field;
FIG. 3 is a diagram of a polygonal air gap region in the Z domain;
FIG. 4 is a diagram of a rectangular air gap region under the W domain;
FIG. 5 is a diagram of a slotless concentric circular air gap region under the Ψ -domain;
fig. 6 is an induced voltage waveform of the rotor eccentric motor;
fig. 7 is an electromagnetic torque waveform of the rotor eccentric motor.
Detailed Description
The following detailed description of the preferred embodiments of the present invention is provided to enable those skilled in the art to more readily understand the advantages and features of the present invention, and to clearly and unequivocally define the scope of the present invention.
The embodiment of the invention is as follows:
step one, for the eccentric motor shown in figure 1, the center of a circle of a rotor is OrThe center of the stator is OsThe center of the rotating shaft is OaThe stator of the motor has an inner diameter of RsThe outer diameter of the rotor is Rr. When O is presentaAnd OrCoincident with but not bound to OsWhen the motor is overlapped, the motor is in static eccentricity; when O is presentaAnd OsCoincident with but not bound to OrWhen the motor is overlapped, the motor is in dynamic eccentricity; when O is presentaIs not reacted with OrCoincident with O and notsWhen the two phases are coincident, the motor is in mixed eccentricity. Mapping a surface-mounted motor air gap region S with eccentric rotor to an annular air gap region T with a tooth slot and concentric with a stator and a rotor through bilinear mapping, as shown in FIG. 2;
Figure BDA0002106599320000041
Figure BDA0002106599320000042
Figure BDA0002106599320000043
d＝c0ejβ
Figure BDA0002106599320000044
Figure BDA0002106599320000045
wherein S represents the air gap region of the eccentric motor, T represents the annular air gap region, and the stator inner diameter of the motor is RsThe outer diameter of the rotor is Rr，ρdRepresents OsOrDistance, beta, represents OsOrAngle to the horizontal axis.
Step two, utilizing the annular air gap area T with the tooth grooves, which is obtained in the step one, of the stator and the rotor, and mapping the annular air gap area T with the tooth grooves, which is obtained in the step one, of the stator and the rotor to a polygonal air gap area Z through logarithmic mapping, as shown in FIG. 3;
Z＝log(T)
step three, mapping the polygonal air gap region Z obtained in the step two to a rectangular air gap region W through Schwarz-Christoffel mapping by utilizing the polygonal air gap region Z, as shown in FIG. 4;
Figure BDA0002106599320000051
wherein: z0,Z1,w0,w,wk,αjThe parameters are parameters mapped by Schwarz-Christoffel, and are obtained according to the values of the points of the polygon on the complex domain;
step four, using the rectangular air gap region W obtained in step three, mapping the rectangular air gap region W to two concentric circular air gap regions psi without grooves by exponential mapping, as shown in fig. 5,
Figure BDA0002106599320000052
wherein: Δ x and Δ y are the length and width of the rectangular air gap region W, and j is an imaginary unit;
the electromagnetic field distribution of the air gap region psi of concentric circles without grooves can be calculated according to the equation of black grids:
Figure BDA0002106599320000053
Figure BDA0002106599320000054
Figure BDA0002106599320000055
Figure BDA0002106599320000056
wherein B isrψkIs the internal polar coordinate (r) of the air gap region psi of the circular ringψ,αψ) Distribution of radial magnetic induction in position, BtψkIs a tangential magnetic induction intensity distribution on polar coordinates in a psi annular air gap regionrψAnd RsψIs the inner and outer diameters, mu, of the annular air gap region psi0Is a vacuum permeability, IkThe magnitude of the current in polar coordinates in the circular air gap region Ψ.
Step five, utilizing the electromagnetic field distribution of the non-groove concentric circle region psi obtained in the step four, and obtaining the air gap magnetic field distribution of the motor under the original rotor eccentricity through inverse transformation of 4 conformal mappings from the step one to the step four, wherein the inverse transformation of the conformal mapping is to obtain the radial magnetic induction intensity distribution B under the annular air gap region psirψkAnd tangential magnetic induction distribution BtψkAir-gap field distribution B mapped to air-gap region S of eccentric motorskI.e. by
Bsk(r,α)＝Bsrk(r,α)+jBstk(r,α)＝λ*(Brψk(r,α)+jBtψk(r,α))
Figure BDA0002106599320000061
Wherein,
Figure BDA0002106599320000062
respectively are partial derivative functions of the four conformal mapping functions;
according to the air-gap magnetic field distribution B of the rotor eccentric motorskObtaining the induced voltage E of the rotor eccentric motorpAnd an electromagnetic torque T, the calculation formula being as follows:
Figure BDA0002106599320000063
Figure BDA0002106599320000064
Figure BDA0002106599320000065
wherein N iscFor the number of turns per slot of the motor, lefIs the effective length of the motor, alphaiIs the initial angle of the coil winding, τ is the winding span, and r is the radius of any circle within the air gap region S of the eccentric machine.
As shown in fig. 6-7, fig. 6 is the waveform of the induced voltage of the rotor eccentric motor, and fig. 7 is the waveform of the electromagnetic torque of the rotor eccentric motor, it can be seen that the induced voltage and the electromagnetic torque calculated by the method are very consistent with the calculation result of the finite element, and the calculation time by the method is shorter than that by the finite element method.
The above description is only an embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes performed by the present specification and drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (5)

1. A rotor eccentricity analysis method based on accurate conformal mapping is characterized by comprising the following steps:
1) mapping an air gap area of the rotor eccentric motor to an annular air gap area with tooth grooves, which is concentric with the stator and the rotor, through bilinear mapping; step 1) the bilinear mapping formula is as follows:
Figure FDA0002690025070000011
Figure FDA0002690025070000012
Figure FDA0002690025070000013
d＝c0ejβ
Figure FDA0002690025070000014
Figure FDA0002690025070000015
wherein S represents the air gap region of the eccentric motor, T represents the annular air gap region, and the stator inner diameter of the motor is RsThe outer diameter of the rotor is Rr，ρdRepresents OsOrDistance, beta, represents OsOrThe included angle between the horizontal shaft and the horizontal shaft;
2) mapping the concentric annular air gap areas with tooth grooves of the stator and the rotor to polygonal air gap areas through logarithmic mapping;
3) mapping the polygonal airgap region to the rectangular airgap region by Schwarz-Christoffel mapping;
4) mapping the rectangular air gap region to a circular air gap region through exponential mapping, and calculating the magnetic induction intensity distribution of the circular air gap region according to a black grid equation;
5) and (3) obtaining the air gap magnetic field distribution of the rotor eccentric motor by utilizing the magnetic induction intensity distribution of the annular air gap area obtained in the step 4) through inverse transformation of the angle-preserving mapping from the step 1) to the step 4), and further obtaining the induction voltage and the electromagnetic torque of the rotor eccentric motor.
2. The method for analyzing the eccentricity of the rotor based on the precise conformal mapping of claim 1, wherein the logarithmic mapping of step 2) has a formula of Z ═ log (T), wherein Z represents a polygonal air gap region.
3. The method for analyzing the rotor eccentricity based on the precise conformal mapping of claim 1, wherein the formula of the Schwarz-Christoffel mapping in step 3) is as follows:
Figure FDA0002690025070000016
wherein W represents a rectangular air gap region, Z0,Z1,w0,w,wk,αjIs a parameter of the Schwarz-christofel mapping, derived from the values of the points of the polygon on the polygonal airgap region Z.
4. The method for analyzing the eccentricity of a rotor based on a precise conformal mapping according to claim 1, wherein the formula of the exponential mapping of step 4) is as follows:
Figure FDA0002690025070000021
where Ψ represents the circular air gap region, Δ x and Δ y are the length and width of the rectangular air gap region W, and j is an imaginary unit;
the black lattice equation is as follows:
Figure FDA0002690025070000022
Figure FDA0002690025070000023
Figure FDA0002690025070000024
Figure FDA0002690025070000025
Figure FDA0002690025070000026
wherein B isrψkIs the internal polar coordinate (r) of the air gap region psi of the circular ringψ,αψ) Distribution of radial magnetic induction in position, BtψkIs a tangential magnetic induction intensity distribution on polar coordinates in a psi annular air gap regionrψAnd RsψIs the inner and outer diameters, mu, of the annular air gap region psi0Is a vacuum permeability, IkThe magnitude of the current in polar coordinates in the circular air gap region Ψ.
5. The method for analyzing the rotor eccentricity based on the precise conformal mapping according to claim 1, wherein the step 5) is as follows:
the inverse of said conformal mapping isThe radial magnetic induction intensity distribution B under the annular air gap region psi is distributedrψkAnd tangential magnetic induction distribution BtψkAir-gap field distribution B mapped to air-gap region S of eccentric motorskI.e. by
Bsk(r,α)＝Bsrk(r,α)+jBstk(r,α)
＝λ*(Brψk(r,α)+jBtψk(r,α))
Figure FDA0002690025070000031
Wherein,
Figure FDA0002690025070000032
respectively are partial derivative functions of the four conformal mapping functions;
according to the air-gap magnetic field distribution B of the rotor eccentric motorskObtaining the induced voltage E of the rotor eccentric motorpAnd an electromagnetic torque T, the calculation formula being as follows:
Figure FDA0002690025070000033
Figure FDA0002690025070000034
Figure FDA0002690025070000035
wherein N iscFor the number of turns per slot of the motor, lefIs the effective length of the motor, alphaiIs the initial angle of the coil winding, τ is the winding span, R is the radius of any circle in the air gap region S of the eccentric machine, RsIs the stator inner diameter of the motor, mu0Is a vacuum magnetic permeability.
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