CN107612256A - A kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor - Google Patents

A kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor Download PDF

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CN107612256A
CN107612256A CN201710380841.3A CN201710380841A CN107612256A CN 107612256 A CN107612256 A CN 107612256A CN 201710380841 A CN201710380841 A CN 201710380841A CN 107612256 A CN107612256 A CN 107612256A
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mrow
msub
mfrac
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杨思雨
陈拯民
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Tongling hard core motor technology Co., Ltd.
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Hefei Hardcore Technology Co Ltd
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Abstract

A kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor, comprises the following steps：Step 1: Fourier decomposition calculating is carried out to the remanent magnetization in segmentation magnetic pole permanent magnet domain：Step 2: establishing the durface mounted permanent magnet synchronous motor magnetic field solving model for including multiple solution domains, each interregional magnetic field boundaries condition is determined：Step 3: solving model and boundary condition that step 2 is established, carry out Analytical Solution to magnetic field in each solution domain, obtain magnetic field analytical Calculation result, establish motor electromagnetic performance analysis models：Step 4: the solving model established according to step 3, chooses optimized variable and optimization aim, establishes magnetic pole sectional type durface mounted permanent magnet synchronous motor Model for Multi-Objective Optimization, magnetic pole segmentation parameter is optimized.The present invention can improve computational accuracy and reduce the electromagnetism calculating time compared with conventional method, can be segmented position by changing magnetic pole to improve the combination property of motor.

Description

A kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor
Technical field
The present invention relates to permagnetic synchronous motor technical field, specifically a kind of magnetic pole sectional type durface mounted permanent magnet synchronous motor Optimization Design.
Background technology
Although Conventional permanent magnet synchronous motor by rational pole design can air gap flux density distribution tend to be sinusoidal, It is unloaded and fully loaded turn because air-gap field unavoidably has harmonic wave, stator teeth notching causes the influences of the factors such as non-uniform air-gap Square pulsation is relatively large, is restricted in the application that positioning and control accuracy requirement high field are closed.It is segmented by magnetic pole, can be with The magnetic field of surface-mounted permanent magnet machine is adjusted, improves motor performance.Compared with other magnetic pole profile optimization methods, magnetic pole point Section do not change the profile of permanent magnet, only by by it is every extremely under permanent magnet be divided into some sections, for the processing technology of permanent magnet It is less demanding.Therefore, magnetic pole sectional type permagnetic synchronous motor has good development prospect.
The emi analysis method of magneto mainly has two classes, magnetic equivalent circuit method and numerical method at present.
(1) magnetic equivalent circuit method
In permagnetic synchronous motor engineering design process, the processing in magnetic field is analogous to electric field, introduces the concept of magnetic circuit, shape Into magnetic equivalent circuit method, this method is analogized to magnetomotive force electronic based on the Ohm's law and Kirchhoff's law in magnetic circuit Gesture, resistance is analogized to by magnetic resistance, and magnetic flux is analogized into electric current, is a kind of method of more traditional analysis Electromagnetic Field.Magnetic The purpose that road calculates is to determine the relation of magnetomotive force, magnetic flux and magnetic structure.The advantages of magnetic equivalent circuit method is simple and fast, analysis Convenient, result of calculation is directly perceived.Weak point is to be difficult to accurately portray electric machine structure, and computational solution precision is relatively low, and it is generally required By other method or empirical coefficient amendment, it is unfavorable for handling magnetic circuit saturation or the situation of permanent magnet characteristic nonlinearity.And should The estimation magnetic that method is typically only capable to obtain at air gap is close, and can not accurately obtain the magnetic field intensity of everywhere in motor.For Magnetic pole sectional type durface mounted permanent magnet synchronous motor, the situation that magnetic field is not segmented relative to magnetic pole in motor is increasingly complex, therefore etc. Effect Magnetic Circuit Method can not obtain effective result of calculation, it is difficult to effectively applied, be not easy to realize the optimization of motor performance.
(2) numerical method
Numerical method is by computer-assisted analysis, is solved based on variation principle and subdivision interpolation computing method inclined The problem of complex electromagnetic fields that the differential equation solves, obtain the discrete solution with degree of precision.Numerical methods in electromagnetic fields has very The high degree of accuracy.Finite element algorithm is disadvantageous in that arithmetic speed is slower, requires higher to computer hardware so that it should It is not directly perceived by a definite limitation, operation result expression with scope, it is difficult to embody influence of the motor each several part parameter to motor performance. For magnetic pole sectional type durface mounted permanent magnet synchronous motor, because optimized variable is more, a large amount of computing resources, therefore numerical value can be expended Application of the calculating method in magnetic pole sectional type durface mounted permanent magnet synchronous motor optimization design has larger limitation.
The content of the invention
It is an object of the invention to provide a kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor, this hair Bright to establish a kind of novel magnetic pole sectional type durface mounted permanent magnet synchronous motor electromagnetic field analytical calculation model, it improves magnetic pole segmentation Type durface mounted permanent magnet synchronous motor electromagnetism computational accuracy, improve motor electromagnetic performance, accurately calculating in motor regional On the basis of electromagnetic parameter, magnetic pole subsection optimization parameter is determined, proposes a kind of synchronous electricity of novel magnetic pole sectional type durface mounted permanent magnet The Optimization Design of machine.
The purpose of the present invention is achieved through the following technical solutions：A kind of magnetic pole sectional type durface mounted permanent magnet synchronous motor Optimization Design, comprise the following steps：
Step 1: Fourier decomposition calculating is carried out to the remanent magnetization in segmentation magnetic pole permanent magnet domain：
In magnetic pole sectional type permagnetic synchronous motor analytic modell analytical model, θ is definedm(i)For corresponding to the symmetry axis of i-th piece of segmentation Mechanical angle；lm(i)For the mechanical angle corresponding to the length of i-th piece of segmentation；P is number of pole-pairs；θ is corresponding to rotor-position Mechanical angle；MrFor intensity of magnetization radial component；MθFor intensity of magnetization tangential component, Fourier decomposition is carried out to it；
Step 2: establishing the durface mounted permanent magnet synchronous motor magnetic field solving model for including multiple solution domains, each region is determined Between magnetic field boundaries condition：
Durface mounted permanent magnet synchronous motor is divided into four regions under two-dimentional polar coordinate system：Region 1 is permanent magnet domain；Region 2 be air gap domain；Region 3i is i-th of groove body domain；Region 4i is i-th of notch domain, in order to solve the magnetic field in each subdomain of motor Distribution, magnetic potential equation is established to each subdomain；
Step 3: solving model and boundary condition that step 2 is established, Analytical Solution is carried out to magnetic field in each solution domain, Magnetic field analytical Calculation result is obtained, establishes motor electromagnetic performance analysis models：
Step 4: the solving model established according to step 3, chooses optimized variable and optimization aim, magnetic pole segmentation is established Type durface mounted permanent magnet synchronous motor Model for Multi-Objective Optimization, is optimized to magnetic pole segmentation parameter：
On the basis of step 3 accurately calculates motor-field distribution and electromagnetic performance, the optimization for choosing magnetic pole segmentation becomes Amount, establishes Model for Multi-Objective Optimization, comprises the following steps that：
(1) choose magnetic pole fragment bit and be set to optimized variable, according to the symmetry of segmentation position, optimized variable is carried out about Beam, reduce optimized variable number；
(2) according to every section of permanent magnet relevant position, existing restriction relation between each optimized variable is determined；
(3) to reduce motor cogging torque and back-emf harmonic content as optimization aim, appropriate weighted factor is introduced Multi-objective optimization question is converted into single-object problem, magnetic pole sectional type durface mounted permanent magnet synchronous motor is optimized, Improve the combination property of motor.
Beneficial effect
1st, the present invention proposes new surface-mount type permagnetic synchronous motor Electromagnetic Calculation method, can compared with conventional method The time is calculated to improve computational accuracy and reduce electromagnetism.
2nd, the present invention proposes new surface-mount type permagnetic synchronous motor optimization method, can be segmented position by changing magnetic pole To improve the combination property of motor.
Brief description of the drawings
Fig. 1 is magnetic pole sectional type durface mounted permanent magnet synchronous motor analytic modell analytical model of the present invention；
Fig. 2 is the present invention per pole permanent magnet distribution schematic diagram；
Fig. 3 is the 4 pole 24 of the invention close comparing result of groove motor gas-gap magnetic；
Fig. 4 is the 8 pole 12 of the invention close comparing result of groove motor gas-gap magnetic；
Fig. 5 is cogging torque comparing result of the present invention；
Fig. 6 is back-emf comparing result of the present invention；
Fig. 7 is magnetic pole optimized variable schematic diagram of the present invention.
Embodiment
The present invention is applied to the durface mounted permanent magnet synchronous motor of the cooperation of various pole grooves and magnetic pole division number, with reference to The present invention is described in further detail for specific example and accompanying drawing.
A kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor, comprises the following steps：
Step 1: Fourier decomposition calculating is carried out to the remanent magnetization in segmentation magnetic pole permanent magnet domain：
In magnetic pole sectional type permagnetic synchronous motor analytic modell analytical model, θ is definedm(i)For corresponding to the symmetry axis of i-th piece of segmentation Mechanical angle；lm(i)For the mechanical angle corresponding to the length of i-th piece of segmentation；P is number of pole-pairs, considers radial magnetizing and puts down The capable situation that magnetizes, then extremely descend the intensity of magnetization to be distributed as in each pair：
Work as θm(i)-lm(i)/ 2 ＜ θ ＜ θm(i)+lm(i)When/2,
For radial magnetizing
For parallel magnetization
As 2 π/p- (θm(i)+lm((i)/ 2) 2 π of ＜ θ ＜/p- (θm(i)-lm(i)/ 2) when,
For radial magnetizing
For parallel magnetization
When θ is other values,
For two kinds of modes that magnetize
In formula, θ is the mechanical angle corresponding to rotor-position；MrFor intensity of magnetization radial component；MθIt is tangential for the intensity of magnetization Component, Fourier decomposition is carried out to it, can be obtained：
For radial magnetizing
Mθk=0 (7)
For parallel magnetization
In formula
As k ≠ 1,
As k=1,
The then intensity of magnetization radial component M in the range of whole rotor permanent magnetrWith intensity of magnetization tangential component MθIt is represented by
In formula, ωrFor rotor velocity；T is the time；θ0For initial position of rotor；K is each overtone order；
Step 2: establishing the durface mounted permanent magnet synchronous motor magnetic field solving model for including multiple solution domains, each region is determined Between magnetic field boundaries condition：
Durface mounted permanent magnet synchronous motor is divided into four regions under two-dimentional polar coordinate system：Region 1 is permanent magnet domain；Region 2 be air gap domain；Region 3i is i-th of groove body domain；Region 4i is i-th of notch domain.In order to solve the magnetic field in each subdomain of motor Distribution, magnetic potential equation is established to each subdomain, following relation between vector magnetic potential A and magnetic density B be present：
▽2A=- ▽ × B (15)
In zone 1, magnetic density B is represented by
B=μ0μrH+μ0M (16)
In region 2,3i, 4i, magnetic density B is represented by
B=μ0H (17)
In formula, μ0For space permeability；μrFor permanent magnet relative permeability；H is magnetic field intensity；M is permanent magnet remanent magnetization Intensity.Under two-dimentional polar coordinate system, vector magnetic potential A only includes z-axis component, and each interregional magnetic field boundaries condition is；
In formula, B1r~B4irFor the radial component of each regional magnetic field density, A1r~A4irFor the z of each region vector magnetic potential Axis component；
Step 3: solving model and boundary condition that step 2 is established, Analytical Solution is carried out to magnetic field in each solution domain, Magnetic field analytical Calculation result is obtained, establishes motor electromagnetic performance analysis models：
Therefore in each region, vector magnetic potential general solution is represented by：
In permanent magnet domain 1
During k ≠ 1
During k=1
In air gap domain 2
In i-th of groove body region 3i
In i-th of notched areas 4i
In formula, k, n, m are each overtone order in corresponding region；A1k~D1k、A2k~D2k、C3in、D3in、C4im、D4imTo be undetermined Coefficient, each undetermined coefficient can be obtained by each interregional Boundary Condition for Solving determined by formula (24)；The close distribution of magnetic in each region Radial component BrWith tangential component BθIt can be solved by vector magnetic potential using formula (25) with formula (26)
The close distribution result of calculation of magnetic in obtained air gap region is solved according to formula (25) and formula (26), using Maxwell Power Tensor Method can try to achieve motor cogging torque TcogExpression formula is：
In formula, laFor motor axial length；
Each phase winding magnetic linkage ψ can be obtained using the close distribution result of calculation of magneticxWith back-emf ExExpression formula is：
In formula, x represents three-phase stator winding A phases, B phases and C phases respectively；NcIt is the every groove conductor number of stator；A is stator winding Parallel branch number；
Distribution of Magnetic Field in each region of motor model and motor cogging torque, opposite can accurately be calculated by above step The electromagnetic performances such as potential；
Step 4: the solving model established according to step 3, chooses optimized variable and optimization aim, magnetic pole segmentation is established Type durface mounted permanent magnet synchronous motor Model for Multi-Objective Optimization, is optimized to magnetic pole segmentation parameter：
On the basis of step 3 accurately calculates motor-field distribution and electromagnetic performance, the optimization for choosing magnetic pole segmentation becomes Amount, establishes Model for Multi-Objective Optimization, comprises the following steps that：
(1) choose magnetic pole fragment bit and be set to optimized variable, according to the symmetry of segmentation position, optimized variable is carried out about Beam, reduce optimized variable number；
(2) according to every section of permanent magnet relevant position, existing restriction relation between each optimized variable is determined；
(3) to reduce motor cogging torque and back-emf harmonic content as optimization aim, appropriate weighted factor is introduced Multi-objective optimization question is converted into single-object problem, magnetic pole sectional type durface mounted permanent magnet synchronous motor is optimized, Improve the combination property of motor.
Now tested using the magnetic pole sectional type durface mounted permanent magnet synchronous motor of the groove of 4 pole 24 and the groove of 8 pole 12, specific step Suddenly it is：
(1) as shown in figure 1, establishing durface mounted permanent magnet synchronous motor analytic modell analytical model, R in figurerFor rotor core outer radius；Rm For permanent magnet outer radius；RsFor stator core inside radius；RtFor stator slot top radius；RsbFor stator slot bottom radius；boaFor Central angle is corresponded at notch；bsaCorresponding central angle is in for groove；θiFor i-th of groove center line mechanical angle, in each region Vector magnetic potential meets such as formula (15)~(18) with magnetic field boundaries condition.
(2) per the distribution of pole permanent magnet as shown in Fig. 2 θm(i)For the mechanical angle corresponding to the symmetry axis of i-th piece of segmentation； lm(i)For the mechanical angle corresponding to the length of i-th piece of segmentation；P is number of pole-pairs, to ensure the symmetry of magnetic pole, θm(i)With lm(i) Meet following condition：
When segments n is odd number
When segments n is even number
The intensity of magnetization radial component M in the range of whole rotor permanent magnet is derived according to formula (6)~(14)rWith the intensity of magnetization Tangential component Mθ。
(3) each undetermined coefficient in formula (19)~(24) is solved, and it is strong using magnetic field in each region of calculating of formula (25)~(26) Degree, further calculate the property such as magnetic pole sectional type surface-mounted permanent magnet machine cogging torque and back-emf using formula (27), formula (29) Energy.
(4) under the conditions of identical parameters using electromagnetic finite element algorithm with to air-gap field, cogging torque with it is opposite electric Kinetic potential is calculated, and the result of calculation of computation model proposed by the invention is verified, its result such as Fig. 3~Fig. 6 It is shown.It can draw what the result of calculation obtained by the analytic modell analytical model established of the present invention obtained with finite element method from contrast Result of calculation is respectively provided with the higher goodness of fit.
(5) magnetic pole optimized variable is determined, as shown in Figure 7.In figure, lm(1)、lm(2)、lm(3)Respectively the 1st, 2,3 section of permanent magnet Corresponding central angle radian；θm(1)、θm(2)Respectively the 1st, 2 section of permanent magnet center corresponding angle.Due to it is every extremely under each section of permanent magnetism Body distribution has symmetry, therefore five variables can determine every section of permanent magnet relevant position more than, and between each variable Restriction relation forth below be present：
(6) to suppress motor cogging torque and back-emf harmonic wave as optimization aim, the synchronization of magnetic pole segment permanent magnet is established Motor Optimized model, introduce appropriate weighted factor by multi-objective optimization question be converted to following formula expression single object optimization ask Topic：
In formula, f is the fitness value of each scheme, and its value is smaller to think that motor is more preferable；λ1、λ2To correspond to the weight of target, And λ1+λ2=1；Tcog_maxWith THDmaxThe maximum that respectively motor cogging torque and back-emf harmonic content allow.
(7) using the Modified particle swarm optimization algorithm given by following formula, the optimization problem in step (5) and (6) is carried out Optimization.
In formula, ωkInertia weight during flight secondary for particle kth；ωmaxWith ωminFor inertia weight maximum with most Small value；kmaxFor maximum iteration, obtained optimization result of calculation is as shown in table 1 and table 2：
The position of magnetic pole optimum results of table 1
Table 2 optimizes front and rear motor performance comparing result
Permagnetic synchronous motor after optimizing before and after the optimization provided in table 2 it can be seen from motor performance comparing result leads to The distribution for changing magnetic pole segmentation position is crossed so that the sine degree of motor gas-gap Distribution of Magnetic Field improves, and then effectively reduces motor teeth groove Torque peak and back-emf harmonic content.For the groove motor of 4 pole 24, under the conditions of radial magnetizing and parallel magnetization, teeth groove turns Square peak value reduces 71.69% and 21.81% respectively, and back-emf harmonic content reduces 33.35% and 55.81% respectively；It is right In the groove motor of 8 pole 12, under the conditions of two kinds magnetize, cogging torque peak value reduces 98.59% and 86.14%, back-emf respectively Harmonic content reduces 94.20% and 63.81% respectively.Illustrate to count magnetic pole segmentation position using particle swarm optimization algorithm Calculate, the combination property of magnetic pole sectional type durface mounted permanent magnet synchronous motor can be effectively improved.
Although above in conjunction with figure, invention has been described, and the invention is not limited in above-mentioned specific embodiment party Formula, above-mentioned embodiment is only schematical, rather than restricted, and one of ordinary skill in the art is in this hair Under bright enlightenment, without deviating from the spirit of the invention, many variations can also be made, these belong to the guarantor of the present invention Within shield.

Claims (6)

1. a kind of Optimization Design of magnetic pole sectional type durface mounted permanent magnet synchronous motor, it is characterised in that comprise the following steps：
Step 1: Fourier decomposition calculating is carried out to the remanent magnetization in segmentation magnetic pole permanent magnet domain：
Step 2: establishing the durface mounted permanent magnet synchronous motor magnetic field solving model for including multiple solution domains, each interregional magnetic is determined Field boundary condition：
Step 3: solving model and boundary condition that step 2 is established, carry out Analytical Solution to magnetic field in each solution domain, obtain Magnetic field analytical Calculation result, establishes motor electromagnetic performance analysis models：
Step 4: the solving model established according to step 3, chooses optimized variable and optimization aim, establishes magnetic pole sectional type table Mounted permagnetic synchronous motor Model for Multi-Objective Optimization, is optimized to magnetic pole segmentation parameter.
2. according to the method for claim 1, it is characterised in that the step 1, in magnetic pole sectional type permagnetic synchronous motor In analytic modell analytical model, θ is definedm(i)For the mechanical angle corresponding to the symmetry axis of i-th piece of segmentation；lm(i)For the length institute of i-th piece of segmentation Corresponding mechanical angle；P is number of pole-pairs；θ is the mechanical angle corresponding to rotor-position；MrFor intensity of magnetization radial component；MθFor Intensity of magnetization tangential component, Fourier decomposition is carried out to it, can be obtained：
For radial magnetizing
<mrow> <msub> <mi>M</mi> <mrow> <mi>r</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>M</mi> <mi>p</mi> </mrow> <mrow> <mi>k</mi> <mi>&amp;pi;</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>k&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mfrac> <msub> <mi>l</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
Mθk=0 (7)
For parallel magnetization
<mrow> <msub> <mi>M</mi> <mrow> <mi>r</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <mi>p</mi> </mrow> <mi>&amp;pi;</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mi>M</mi> <mi>p</mi> </mrow> <mi>&amp;pi;</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
In formula
<mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mfrac> <msub> <mi>l</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mn>2</mn> </mfrac> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>k&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
As k ≠ 1,
<mrow> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <mn>2</mn> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>k</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mfrac> <msub> <mi>l</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mn>2</mn> </mfrac> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>k&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
As k=1,
<mrow> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>l</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>sin&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
The then intensity of magnetization radial component M in the range of whole rotor permanent magnetrWith intensity of magnetization tangential component MθIt is represented by：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <msub> <mi>M</mi> <mrow> <mi>r</mi> <mi>k</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>k&amp;omega;</mi> <mi>r</mi> </msub> <mi>t</mi> <mo>-</mo> <msub> <mi>k&amp;theta;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <msub> <mi>M</mi> <mrow> <mi>r</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>M</mi> <mrow> <mi>r</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>k</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>k&amp;omega;</mi> <mi>r</mi> </msub> <mi>t</mi> <mo>-</mo> <msub> <mi>k&amp;theta;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
In formula, ωrFor rotor velocity；T is the time；θ0For initial position of rotor；K is each overtone order.
3. according to the method for claim 1, it is characterised in that the step 2, by surface-mount type under two-dimentional polar coordinate system Permagnetic synchronous motor is divided into four regions：Region 1 is permanent magnet domain；Region 2 is air gap domain；Region 3i is i-th of groove body domain；Area Domain 4i is i-th of notch domain, and in order to solve the Distribution of Magnetic Field in each subdomain of motor, magnetic potential equation, vector magnetic are established to each subdomain Following relation be present between the A and magnetic density B of position：
<mrow> <msup> <mo>&amp;dtri;</mo> <mn>2</mn> </msup> <mi>A</mi> <mo>=</mo> <mo>-</mo> <mo>&amp;dtri;</mo> <mo>&amp;times;</mo> <mi>B</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
In zone 1, magnetic density B is represented by
B=μ0μrH+μ0M (16)
In region 2,3i, 4i, magnetic density B is represented by
B=μ0H (17)
In formula, μ0For space permeability；μrFor permanent magnet relative permeability；H is magnetic field intensity；M is that permanent magnet remanent magnetization is strong Degree, under two-dimentional polar coordinate system, vector magnetic potential A only includes z-axis component, and each interregional magnetic field boundaries condition is：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mrow> <mn>3</mn> <mi>i</mi> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>r</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mn>3</mn> <mi>i</mi> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>z</mi> </mrow> </msub> <msub> <mo>|</mo> <mrow> <mi>r</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
In formula, B1r~B4irFor the radial component of each regional magnetic field density, A1r~A4irFor the z-axis point of each region vector magnetic potential Amount.
4. according to the method for claim 1, it is characterised in that the step 3, in each region, vector magnetic potential general solution can It is expressed as：
In permanent magnet domain 1
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mi>z</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <mo>&amp;lsqb;</mo> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>m</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>k</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>r</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <mo>&amp;lsqb;</mo> <msub> <mi>C</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>m</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>k</mi> </msup> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>r</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mo>&amp;rsqb;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>A</mi> <mi>p</mi> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
During k ≠ 1
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> <mi>r</mi> <munder> <mo>&amp;Sigma;</mo> <mi>k</mi> </munder> <mfrac> <mn>1</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>kM</mi> <mrow> <mi>r</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>k</mi> <mi>p</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>kM</mi> <mrow> <mi>r</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>p</mi> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
During k=1
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>p</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> <mi>ln</mi> <mi> </mi> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>kM</mi> <mrow> <mi>r</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow> <mi>&amp;theta;</mi> <mi>s</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>kM</mi> <mrow> <mi>r</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mi>&amp;theta;</mi> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>
In air gap domain 2
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mi>z</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>A</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>s</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>k</mi> </msup> <mo>+</mo> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>m</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <munder> <mi>&amp;Sigma;</mi> <mi>k</mi> </munder> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>s</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mi>k</mi> </msup> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>m</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
In i-th of groove body region 3i
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mi>z</mi> <mn>3</mn> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>n</mi> </munder> <mo>&amp;lsqb;</mo> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mi>b</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mfrac> <mrow> <mi>n</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mfrac> </msup> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mn>3</mn> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>t</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>n</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>cos</mi> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mfrac> <mrow> <mi>n</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>&amp;theta;</mi> <mo>+</mo> <mfrac> <msub> <mi>b</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>
In i-th of notched areas 4i
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mrow> <mi>z</mi> <mn>4</mn> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>m</mi> </munder> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>C</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>t</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mfrac> <mrow> <mi>m</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>o</mi> <mi>a</mi> </mrow> </msub> </mfrac> </msup> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mn>4</mn> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mfrac> <mi>r</mi> <msub> <mi>R</mi> <mi>s</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>m</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>o</mi> <mi>a</mi> </mrow> </msub> </mfrac> </mrow> </msup> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>cos</mi> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mfrac> <mrow> <mi>m</mi> <mi>&amp;pi;</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>o</mi> <mi>a</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>&amp;theta;</mi> <mo>+</mo> <mfrac> <msub> <mi>b</mi> <mrow> <mi>o</mi> <mi>a</mi> </mrow> </msub> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
In formula, k, n, m are each overtone order in corresponding region；A1k~D1k、A2k~D2k、C3in、D3in、C4im、D4imFor system undetermined Number, can obtain each undetermined coefficient by each interregional Boundary Condition for Solving determined by formula (24)；The close distribution footpath of magnetic in each region To component BrWith tangential component BθIt can be solved by vector magnetic potential using formula (25) with formula (26)
<mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>A</mi> <mi>z</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;theta;</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>B</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>A</mi> <mi>z</mi> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>r</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
The close distribution result of calculation of magnetic in obtained air gap region is solved according to formula (25) and formula (26), using Maxwell's tension force Amount method can try to achieve motor cogging torque TcogExpression formula is：
<mrow> <msub> <mi>T</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mi>a</mi> </msub> <msubsup> <mi>R</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> </mfrac> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </msubsup> <msub> <mi>B</mi> <mrow> <mi>r</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow> <mi>&amp;theta;</mi> <mn>2</mn> </mrow> </msub> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
In formula, laFor motor axial length；
Each phase winding magnetic linkage ψ can be obtained using the close distribution result of calculation of magneticxWith back-emf ExExpression formula is：
<mrow> <msub> <mi>&amp;psi;</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>N</mi> <mi>c</mi> </msub> <msub> <mi>l</mi> <mi>a</mi> </msub> <msub> <mi>R</mi> <mi>s</mi> </msub> </mrow> <mi>a</mi> </mfrac> <msub> <mo>&amp;Integral;</mo> <mi>x</mi> </msub> <msub> <mi>B</mi> <mrow> <mi>r</mi> <mn>2</mn> </mrow> </msub> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>E</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msub> <mi>d&amp;psi;</mi> <mi>x</mi> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>
In formula, x represents three-phase stator winding A phases, B phases and C phases respectively；NcIt is the every groove conductor number of stator；A is that stator winding is in parallel Circuitry number；
The Distribution of Magnetic Field and motor cogging torque, opposite potential in each region of motor model can be accurately calculated by above step Deng electromagnetic performance.
5. according to the method for claim 1, it is characterised in that the step 4, motor-field is accurately calculated in step 3 On the basis of distribution and electromagnetic performance, the optimized variable of magnetic pole segmentation is chosen, establishes Model for Multi-Objective Optimization, specific steps are such as Under：
(1) choose magnetic pole fragment bit and be set to optimized variable, according to the symmetry of segmentation position, row constraint is entered to optimized variable, drop Low optimized variable number；
(2) according to every section of permanent magnet relevant position, existing restriction relation between each optimized variable is determined；
(3) to reduce motor cogging torque and back-emf harmonic content as optimization aim, it will be more to introduce appropriate weighted factor Objective optimisation problems are converted to single-object problem, and magnetic pole sectional type durface mounted permanent magnet synchronous motor is optimized, and improve The combination property of motor.
6. according to the method for claim 2, it is characterised in that consider radial magnetizing and parallel magnetization situation, then every To extremely descending the intensity of magnetization to be distributed as：
Work as θm(i)-lm(i)/ 2 ＜ θ ＜ θm(i)+lm(i)When/2,
For radial magnetizing
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <mi>M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
For parallel magnetization
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
As 2 π/p- (θm(i)+lm((i)/ 2) 2 π of ＜ θ ＜/p- (θm(i)-lm(i)/ 2) when,
For radial magnetizing
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>M</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
For parallel magnetization
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <mi>&amp;theta;</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mrow> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
When θ is other values,
For two kinds of modes that magnetize
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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