CN115102313A - Analytic modeling method for double-layer alternating pole semi-insertion type permanent magnet motor - Google Patents

Analytic modeling method for double-layer alternating pole semi-insertion type permanent magnet motor Download PDF

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CN115102313A
CN115102313A CN202210492614.0A CN202210492614A CN115102313A CN 115102313 A CN115102313 A CN 115102313A CN 202210492614 A CN202210492614 A CN 202210492614A CN 115102313 A CN115102313 A CN 115102313A
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CN115102313B (en
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倪有源
张亮
邱志伟
肖本贤
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Hefei University of Technology
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K1/00—Details of the magnetic circuit
    • H02K1/06—Details of the magnetic circuit characterised by the shape, form or construction
    • H02K1/22—Rotating parts of the magnetic circuit
    • H02K1/27—Rotor cores with permanent magnets
    • H02K1/2706—Inner rotors
    • H02K1/272—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis
    • H02K1/274—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis the rotor consisting of two or more circumferentially positioned magnets
    • H02K1/2753—Inner rotors the magnetisation axis of the magnets being perpendicular to the rotor axis the rotor consisting of two or more circumferentially positioned magnets the rotor consisting of magnets or groups of magnets arranged with alternating polarity
    • H02K1/278—Surface mounted magnets; Inset magnets
    • H02K1/2781—Magnets shaped to vary the mechanical air gap between the magnets and the stator
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K1/00—Details of the magnetic circuit
    • H02K1/06—Details of the magnetic circuit characterised by the shape, form or construction
    • H02K1/12—Stationary parts of the magnetic circuit
    • H02K1/16—Stator cores with slots for windings
    • H02K1/165—Shape, form or location of the slots
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K1/00—Details of the magnetic circuit
    • H02K1/06—Details of the magnetic circuit characterised by the shape, form or construction
    • H02K1/22—Rotating parts of the magnetic circuit
    • H02K1/26—Rotor cores with slots for windings
    • H02K1/265—Shape, form or location of the slots
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K3/00—Details of windings
    • H02K3/04—Windings characterised by the conductor shape, form or construction, e.g. with bar conductors
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K2201/00—Specific aspects not provided for in the other groups of this subclass relating to the magnetic circuits
    • H02K2201/03—Machines characterised by aspects of the air-gap between rotor and stator
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K2213/00—Specific aspects, not otherwise provided for and not covered by codes H02K2201/00 - H02K2211/00
    • H02K2213/03—Machines characterised by numerical values, ranges, mathematical expressions or similar information

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Abstract

The invention discloses an analytic modeling method of a double-layer alternating-pole semi-insertion permanent magnet motor, which is characterized in that a layer of surface-mounted permanent magnet is added on a magnetic pole of the alternating-pole motor to improve the electromagnetic performance of the motor. And the accurate sub-domain model is used for modeling and analyzing the motor, so that the calculation precision is improved, and the air gap flux density and the electromagnetic torque of the motor are obtained. Compared with the traditional surface insertion type permanent magnet motor, the motor structure provided by the invention has the advantage that the average electromagnetic torque is obviously improved when the usage amount of the permanent magnet is the same.

Description

Analytic modeling method for double-layer alternating pole semi-insertion type permanent magnet motor
Technical Field
The invention relates to the technical field of permanent magnet motors, in particular to an analytic modeling method of a double-layer alternating-pole half-insertion type permanent magnet motor.
Background
Because of simple structure and high operation efficiency, the surface-mounted permanent magnet motor is widely applied in various fields. Compared with a surface-mounted permanent magnet motor, the surface-embedded permanent magnet motor has the characteristics of firm structure, high torque density, wide weak magnetic area and the like. In order to reduce the cost and the amount of permanent magnets used, alternating-pole permanent magnet motors have been studied. Because the other pole of the permanent magnet is replaced by the rotor convex iron, when the air gap flux density THD of the alternating pole motor is larger, the fluctuation of the electromagnetic torque is larger. The prior art shows that the double-layer magnetic pole permanent magnet motor has better electromagnetic performance compared with a single-layer magnetic pole motor. There have been many studies on surface mount motors, plug-in motors and alternating pole motors in the related papers, and there is no mention of a double-layer alternating pole half-plug-in motor.
Disclosure of Invention
The invention aims to make up for the defects of the prior art and provides an analytic modeling method of a double-layer alternating pole semi-inserted permanent magnet motor.
The invention is realized by the following technical scheme:
a double-layer alternate pole semi-insertion type permanent magnet motor analytical modeling method is characterized in that a layer of surface-mounted magnetic poles is newly added on the basis of alternate poles, a double-layer semi-insertion type magnetic pole structure and a fan-shaped stator slot structure with equal-thickness tooth tips are adopted, wherein the lower layer of a permanent magnet is an alternate pole structure with the same polarity and rotor convex iron which are alternately arranged, the upper layer of the permanent magnet is a layer of N, S magnetic poles which are alternately arranged and are surface-mounted magnetic poles, and the magnetization modes of the upper and lower magnetic poles are radial magnetization. The method adopts a precise sub-domain model method to divide the motor solution domain into: a stator slot sub-region, a slot sub-region, an air gap sub-region, an outer ring magnetic pole sub-region and a rotor slot sub-region. And a general solution expression of the vector magnetic potential A of each sub-domain under the two-dimensional plane is obtained by the ampere loop law and the Gauss law, a matrix equation is established by using boundary conditions among the sub-domains to solve each harmonic component coefficient in the vector magnetic potential equation of each sub-domain, and the radial/tangential component and the electromagnetic torque of the air gap flux density are obtained.
The following is a specific calculation procedure:
the central line of the rotor slot coincides with the central line of the sector slot when defining the initial position, namely the first rotor slot andfirst stator slot, θ 0 The serial numbers of the rotor slots are sequentially arranged into the s-th rotor slot … … of the 2 nd rotor slot with the counterclockwise direction as the positive direction as the position angle of the rotor relative to the position; the numbers of the stator slots are sequentially arranged into the 2 nd stator slot … … i-th stator slot with the counterclockwise direction as the positive direction. Theta.theta. i Is an included angle between the center line of the ith stator slot and the center line of the 1 st stator slot; theta s Is the angle between the center line of the s-th rotor slot and the center line of the 1 st rotor slot. Obviously, for one there is N r A rotor slot and N s Electric machines with individual stator slots, theta i And theta s Are respectively as follows:
Figure BDA0003631662890000021
the general solution expression of the vector magnetic potential A of each sub-domain under the two-dimensional plane is obtained by the ampere loop law and the Gauss law, a partial differential equation is established for the component of the vector magnetic potential A in the z direction under each sub-domain, and the radial direction B of the magnetic field density in the two-dimensional polar coordinate system ρ With a tangential component B θ The relationship to vector magnetic bit A can be expressed as:
Figure BDA0003631662890000022
wherein rho and theta are respectively the calculated radius and the position angle under a polar coordinate system in a two-dimensional plane.
To facilitate the solution of the sub-domain vector magnetic potential and harmonic coefficients, a function P (x, y, z) and a function E (x, y, z) are defined, which scales the harmonic component coefficients for an expression of the general solution of the above partial differential equation:
Figure BDA0003631662890000023
and analyzing the magnetic field of each sub-domain to obtain a vector magnetic potential A and performing coefficient scaling to obtain a final form, wherein the final form is as follows:
(1) stator slot domain magnetic field analysis
Under two-dimensional polar coordinates i A 1 (ρ, θ) is the z-component equation for vector magnetic bit a in the slot sub-domain in the ith slot, where i ═ 1,2,3 s . Then in the slot sub-domain with respect to the ith slot when the excitation current in the coil is zero i A 1 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000031
wherein R is 5 Is the outer diameter of the stator tooth tip, R 6 Is the outer diameter of the stator slot, xi 1 The radian of the fan-shaped stator slot corresponding to the central angle is adopted. The differential equation can be solved by using a separation variable method, and the original form of the solution of the differential equation can be finally obtained as follows:
Figure BDA0003631662890000032
to facilitate subsequent calculations, the coefficients are scaled
Figure BDA0003631662890000033
And
Figure BDA0003631662890000034
and scaling the coefficient, wherein the scaling of the coefficient does not influence the final result according to the property of the solution of the differential equation. The solution of the coefficient scaled differential equation is:
Figure BDA0003631662890000035
in the formula: i W 1 ， i X 1 ，
Figure BDA0003631662890000036
and
Figure BDA0003631662890000037
respectively are a direct current component coefficient and a harmonic component coefficient of the stator slot domain vector magnetic potential equation, m is a harmonic order of the stator slot domain vector magnetic potential equation, and tau is m The expression of (a) is:
Figure BDA0003631662890000038
(2) notch subfield magnetic field analysis
Under two-dimensional polar coordinates j A 2 (ρ, θ) is the z-component equation for the vector magnetic bit A in the notch sub-domain in the jth notch, where j is 1,2,3 s Then, when the exciting current in the coil is zero, the slot subfield with respect to the j-th stator slot j A 2 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000039
in the formula: r is 4 Is the stator inner diameter, R 5 Is the outer diameter of the tooth tip of the stator, xi 2 The radian of the notch of the fan-shaped stator slot corresponding to the central angle is adopted. The form and the domain range of the differential equation are similar to those of a tank sub-domain without excitation current, a separation variable method can be used for solving, all coefficients are scaled simultaneously, and the solution of the differential equation after processing is as follows:
Figure BDA0003631662890000041
wherein: j W 2 ， j X 2 ，
Figure BDA0003631662890000042
and
Figure BDA0003631662890000043
respectively are DC component coefficient and harmonic component coefficient of stator slot mouth domain vector magnetic potential equation, n is stator slotHarmonic order of the vector magnetic potential equation in the mouth region, θ j Is the position of the center line of the jth notch, theta j And τ n Are respectively:
Figure BDA0003631662890000044
Figure BDA0003631662890000045
(3) air gap sub-domain magnetic field analysis
The solution domain is located at the outer ring magnetic pole outer diameter R 3 And the inner diameter R of the stator 4 The annular region in between. Let A under two-dimensional polar coordinates 3 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 3 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000046
the differential equation is also solved using the discrete variational method, the original form of the solution being:
Figure BDA0003631662890000047
in the formula: a is 3 ，b 3 ， k W 3 ， k X 3 ， k Y 3 And k Z 3 respectively are a direct current component coefficient and a harmonic component coefficient of the air gap sub-domain vector magnetic potential equation, and k is a harmonic order of the air gap sub-domain vector magnetic potential equation;
unlike the slot and notch sub-regions, the air gap sub-region is a connected annular region, so that according to ampere-loop law, the integral of the magnetic field strength at a circumference of any radius ρ within the air gap region is equal to the total amount of current passing through that circumference, while the current through that region is always zero, so that the solution to the differential equation should not contain a dc component. The solution of the differential equation is simplified and scaled by coefficients, the final form of the solution being:
Figure BDA0003631662890000048
(4) magnetic field analysis in outer ring magnetic pole subdomain
The solution domain is located at the outer diameter R of the alternating pole magnetic pole 2 Outer diameter R of outer ring magnetic pole 3 The annular magnetic pole region in between. Let A under two-dimensional polar coordinates 4 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 4 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000051
in the formula: mu.s 0 Is the magnetic permeability of a vacuum, M ρ And M θ The radial and tangential components of the magnetization of the permanent magnet, respectively, for a radial magnetization polar arc coefficient of 1, the radial and tangential components of the magnetization have the expressions:
Figure BDA0003631662890000052
in the formula: m ρu And M θu Respectively as follows:
Figure BDA0003631662890000053
in the formula B r And p is the remanence and the pole pair number of the permanent magnet, respectively.
Using the discrete variational method, taking into account the nature of the solution of the inhomogeneous partial differential equation and the ampere-loop law, the original form of its solution can be calculated:
Figure BDA0003631662890000054
scaling the solution by coefficients can obtain the final form of the solution as follows:
Figure BDA0003631662890000055
in the formula: u W 4 ， u X 4 ， u Y 4 and u Z 4 the vector magnetic potential equation comprises a direct current component coefficient and a harmonic component coefficient of the annular magnetic pole subdomain vector magnetic potential equation respectively, and u is the harmonic order of the outer annular magnetic pole subdomain vector magnetic potential equation. Function F u The expression of (ρ) is:
Figure BDA0003631662890000061
(5) rotor slot domain magnetic field analysis
Under two-dimensional polar coordinates s A 5 (ρ, θ) is the z-component equation of the vector magnetic bit a in the slot sub-domain in the s-th slot, where s is 1,2,3 r . Then with respect to the slot sub-domain when the excitation current in the coil is 0 s A 5 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000062
in the formula: r 1 Is the inner diameter of alternating pole magnetic pole, xi 3 The radian of the rotor slot corresponding to the central angle, s M ρ and s M θ the radial and tangential components of the magnetization of the permanent magnet are respectively, the polar arc coefficient of the radial magnetization is 1, and the expressions of the radial and tangential components of the magnetization of the permanent magnet in the rotor slot at the position are as follows:
Figure BDA0003631662890000063
in the formula: c, M ρv And M θv Respectively as follows:
Figure RE-GDA0003806255740000064
Figure BDA0003631662890000065
and (3) calculating a final form of the rotor slot domain vector magnetic potential solution by using a separation variable method and considering the property of the solution of the non-homogeneous partial differential equation and coefficient scaling:
Figure BDA0003631662890000071
in the formula:
Figure BDA0003631662890000072
and
Figure BDA0003631662890000073
the direct current component coefficient and the harmonic component coefficient of the rotor slot domain vector magnetic potential equation are respectively shown, and v is the harmonic order of the rotor slot domain vector magnetic potential equation. Tau is v And equation F v (ρ) are:
Figure BDA0003631662890000074
Figure BDA0003631662890000075
in summary, the domain vector magnetic potential equations containing the coefficients to be solved in the five regions are established. Wherein i W 1 ， i X 1 ，
Figure BDA0003631662890000076
j W 2 ， j X 2 ，
Figure BDA0003631662890000077
k W 3 ， k X 3 ， k Y 3 ， k Z 3 ， u W 4 ， u X 4 ， u Y 4 ， u Z 4 ，
Figure BDA0003631662890000078
And
Figure BDA0003631662890000079
the total of 20 sets of coefficients (including the dc component coefficient and the harmonic component coefficient) will be determined by the boundary conditions between the regions. Using p ═ R 1 ，R 2 ，R 3 ，R 4 ，R 5 ，R 6 And establishing a matrix equation to solve each harmonic component coefficient in each sub-domain vector magnetic potential equation under the boundary conditions at six positions to obtain the radial/tangential component and the electromagnetic torque of the air gap flux density. Consider that at ρ ═ R 1 And ρ ═ R 6 The part of the molecular domain vector magnetic potential equation can be partially simplified at the position, so that the two positions are analyzed first, and then the other positions are analyzed in sequence. The method comprises the following specific steps:
(1) at ρ ═ R 1 Analysis of boundary conditions of
At rho ═ R 1 The position is the junction of the rotor slot subdomain and the rotor iron at the bottom of the rotor slot. The surface of the rotor iron is provided with:
Figure BDA00036316628900000710
by substituting formula (25) for formula (28), the following can be solved:
Figure BDA00036316628900000711
Figure BDA0003631662890000081
and (3) taking the equation (30) as an equation to be solved subsequently, substituting the equation (29) into the equation (25), and simplifying the vector magnetic potential equation of the rotor slot domain into:
Figure BDA0003631662890000082
in the following, expressions regarding the rotor slot regions are all calculated using equation (31).
(2) At rho ═ R 6 Analysis of boundary conditions of
At rho ═ R 6 The position is the junction of the stator iron of the stator slot subdomain and the bottom of the stator slot. The stator iron surface has:
Figure BDA0003631662890000083
by substituting formula (6) for formula (32), we can understand:
i X 1 ＝0 (33)
Figure BDA0003631662890000084
by substituting equations (33) and (34) for equation (6), the simplified domain vector magnetic potential equation for the slot domain can be obtained as follows:
Figure BDA0003631662890000085
in the following, expressions regarding the stator slot regions are all operated using equation (35).
(3) At rho ═ R 5 Analysis of boundary conditions of
At rho ═ R 5 Is the intersection of the stator slot region and the stator slot opening. The radial air gap flux density is continuous according to the boundary condition, which can be expressed as:
Figure BDA0003631662890000086
obviously, the stator slots have a one-to-one correspondence with the stator slots, so that i ═ j is always present in the calculation process. According to the nature of the Fourier series and equation (36) taking into account the boundary conditions, the DC component coefficients and harmonic component coefficients for the notch sub-regions are:
Figure BDA0003631662890000091
Figure BDA0003631662890000092
in the formula: function F 1 (m) and function F 2 (m, n) are respectively:
Figure BDA0003631662890000093
Figure BDA0003631662890000094
at rho ═ R 5 Another boundary condition at (b) is that the tangential magnetic field strength is continuous, which can be expressed as:
Figure BDA0003631662890000095
similarly, let ρ be R 5 Substituting, considering the boundary condition expression (41), the direct current coefficient and harmonic component coefficient of the Fourier series are characterized by:
j X 3 ＝0 (42)
Figure BDA0003631662890000096
at rho ═ R 5 The process of substituting the boundary conditions is completed, and equation (37), equation (38), equation (42), and equation (43) are equations of the obtained harmonic component coefficients. The subsequent substitution process of each boundary condition is similar, in order to avoid repetition, in the subsequent boundary condition analysis process, only the boundary condition and the obtained harmonic component coefficient equation are given, and the calculation process is not repeated.
(4) At ρ ═ R 4 Analysis of boundary conditions of
ρ＝R 4 The stator slot and air gap interface. The two boundary conditions are:
Figure BDA0003631662890000101
the vector magnetic potential function is also subjected to Fourier decomposition, and the equation of the coefficient of the direct current component and the harmonic component is obtained as follows:
Figure BDA0003631662890000102
Figure BDA0003631662890000103
Figure BDA0003631662890000104
Figure BDA0003631662890000105
in the formula: function F 3 ～F 6 Respectively as follows:
Figure BDA0003631662890000106
Figure BDA0003631662890000107
Figure BDA0003631662890000108
Figure BDA0003631662890000109
(5) at ρ ═ R 3 Analysis of boundary conditions of
ρ＝R 3 Is the interface of the air gap region and the ring pole region. Because both the two regions are annular regions and the vector magnetic potential expressions of the two regions have similar forms, when the boundary condition is substituted, Fourier operation is not needed, and only the harmonic component coefficients of each order corresponding to the forms are repeatedly substituted into the boundary condition for operation. Therefore, in this section, k is always u. The boundary conditions here are:
Figure BDA0003631662890000111
substituting the vector magnetic potential function can obtain:
Figure BDA0003631662890000112
Figure BDA0003631662890000113
Figure BDA0003631662890000114
Figure BDA0003631662890000115
(6) at rho ═ R 2 Analysis of boundary conditions of
ρ＝R 2 Is the interface of the annular pole region and the rotor slot. Its boundary condition and substitution process and rho ═ R 4 Similarly. The boundary conditions are as follows:
Figure BDA0003631662890000116
the direct current component and harmonic component coefficients of the vector magnetic potential function obtained by carrying out Fourier decomposition on the vector magnetic potential function are as follows:
Figure BDA0003631662890000121
Figure BDA0003631662890000122
Figure BDA0003631662890000123
Figure BDA0003631662890000124
the united equation (29), the equation (30), the equation (33), the equation (34), the equation (37), the equation (38), the equation (42), the equation (43), the equation (45), the equation (46), the equation (47), the equation (48), the equation (54), the equation (55), the equation (56), the equation (57), the equation (59), the equation (60), the equation (61) and the equation (62) are 20 equations in total, that is, the direct current component coefficient and the harmonic component coefficient of each region vector magnetic potential equation can be solved. Then the air gap region vector magnetic potential equation A 3 And (rho, theta) is substituted for the formula (2), and the air gap flux density of the motor is obtained through solving.
For any rotor position, the flux linkage linked by a coil can be calculated by the difference of the average vector flux positions of the upper layer side and the lower layer side, and the no-load induced electromotive force can be solved by the differential of the coil flux linkage. Its flux linkage can be expressed as:
ψ x ＝ψ x+ -ψ x- (63)
Figure BDA0003631662890000125
in the formula: l is the axial length of the motor, N c S is the sectional area of the upper layer edge or the lower layer edge of the coil, and the expression is as follows:
Figure BDA0003631662890000131
for any phase winding there is N λ The coils are connected in series, so that the total flux linkage of the phase is as follows:
Figure BDA0003631662890000132
the no-load induced electromotive force of this phase is:
Figure BDA0003631662890000133
in the formula: and omega is the rotating speed of the motor. The electromagnetic torque expression of the three-phase motor is as follows:
Figure BDA0003631662890000134
in the formula: e A ，E B ，E C Is the induced electromotive force of each phase; i.e. i A ，i B And i C Is a three-phase symmetric armature current.
The invention has the advantages that: the invention adopts a double-layer half-insertion type magnetic pole structure, a layer of surface-mounted magnetic poles are added on the basis of alternate poles, the magnetization modes of the permanent magnets are radial magnetization, and compared with the traditional insertion type permanent magnet motor, the permanent magnet motor has larger air gap flux density fundamental wave amplitude and higher average electromagnetic torque under the condition that the usage amount of the permanent magnets passes through the same permanent magnet motor firstly.
Drawings
Fig. 1 is a schematic diagram of a double-layer alternating pole half-inserted magnetic pole structure of the present invention.
Fig. 2 is a schematic structural diagram of a double-layer alternating-pole half-insertion permanent magnet motor.
Fig. 3 is a schematic structural diagram of a surface-insertion permanent magnet motor.
FIG. 4 is radial air gap flux density analysis and finite element comparison verification.
FIG. 5 is tangential air gap flux density analysis and finite element comparison verification.
Fig. 6 is an analytic method and finite element comparison verification of electromagnetic torque of a double-layer alternating pole semi-inserted permanent magnet motor.
Fig. 7 is a comparison of electromagnetic torque for a double-layer alternating pole half-plug-in permanent magnet machine and a conventional surface-plug-in permanent magnet machine.
Detailed Description
An analytic modeling method for a double-layer alternating pole semi-insertion permanent magnet motor adopts a double-layer semi-insertion magnetic pole structure and a fan-shaped stator slot structure with equal-thickness tooth tips, carries out analytic modeling on the double-layer semi-insertion magnetic pole structure and the fan-shaped stator slot structure by using a sub-domain model method, and divides a motor solution domain into: a stator slot sub-region, a slot sub-region, an air gap sub-region, an outer ring magnetic pole sub-region and a rotor slot sub-region. A general solution expression of the vector magnetic potential A of each sub-domain under the two-dimensional plane is obtained by the ampere loop law and the Gauss law, a matrix equation is established by using boundary conditions among the sub-domains to solve each harmonic component coefficient in the vector magnetic potential equation of each sub-domain, and the radial/tangential component and the electromagnetic torque of the air gap flux density are obtained analytically.
Defining the initial position, the central line of the rotor slot is coincident with that of the sector slot, i.e. the first rotor slot and the first stator slot, theta 0 The serial numbers of the rotor slots are sequentially arranged into the s-th rotor slot of the 2 nd rotor slot … … by taking the anticlockwise direction as the positive direction; the numbers of the stator slots are sequentially arranged into the 2 nd stator slot … … i-th stator slot with the counterclockwise direction as the positive direction. Theta.theta. i Is the ith stator slot center line and the 1 stThe included angle between the central lines of the stator slots; theta.theta. s Is the angle between the center line of the s-th rotor slot and the center line of the 1 st rotor slot. Obviously, for one there is N r A rotor slot and N s Electric machines with individual stator slots, theta i And theta s Are respectively:
Figure BDA0003631662890000141
the general solution expression of the vector magnetic potential A of each sub-domain under the two-dimensional plane is obtained by the ampere loop law and the Gauss law, a partial differential equation is established for the component of the vector magnetic potential A in the z direction under each sub-domain, and the radial direction B of the magnetic field density in the two-dimensional polar coordinate system ρ With a tangential component B θ The relationship to vector magnetic bit A can be expressed as:
Figure BDA0003631662890000142
wherein rho and theta are respectively the calculated radius and the position angle under a polar coordinate system in a two-dimensional plane.
To facilitate the solution of the sub-domain vector magnetic potential and harmonic coefficients, a function P (x, y, z) and a function E (x, y, z) are defined, which scales the harmonic component coefficients for an expression of the general solution of the above partial differential equation:
Figure BDA0003631662890000143
and analyzing the magnetic field of each sub-domain to obtain a vector magnetic potential A and performing coefficient scaling to obtain a final form, wherein the final form is as follows:
(1) stator slot domain magnetic field analysis
Under two-dimensional polar coordinates i A 1 (ρ, θ) is the z-component equation for vector magnetic bit a in the slot sub-domain in the ith slot, where i ═ 1,2,3 s . Then in the slot sub-domain with respect to the ith slot when the excitation current in the coil is zero i A 1 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000151
wherein R is 5 Is the outer diameter of the stator tooth tip, R 6 Is the outer diameter of the stator slot, xi 1 The radian of the fan-shaped stator slot corresponding to the central angle is adopted. The differential equation can be solved by using a separation variable method, and the original form of the solution of the differential equation can be finally obtained as follows:
Figure BDA0003631662890000152
to facilitate subsequent calculations, the coefficients are calculated
Figure BDA0003631662890000153
And
Figure BDA0003631662890000154
and scaling the coefficient, wherein the scaling of the coefficient does not influence the final result according to the property of the solution of the differential equation. The solution of the coefficient scaled differential equation is:
Figure BDA0003631662890000155
in the formula: i W 1 ， i X 1 ，
Figure BDA0003631662890000156
and
Figure BDA0003631662890000157
respectively are a direct current component coefficient and a harmonic component coefficient of the stator slot domain vector magnetic potential equation, m is a harmonic order of the stator slot domain vector magnetic potential equation, and tau m The expression of (c) is:
Figure BDA0003631662890000158
(2) notch subfield magnetic field analysis
Under two-dimensional polar coordinates j A 2 (ρ, θ) is the z-component equation for the vector magnetic bit A in the slot sub-domain in the jth slot, where j is 1,2,3 s Then, when the exciting current in the coil is zero, the slot subfield with respect to the j-th stator slot j A 2 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000159
in the formula: r is 4 Is the stator inner diameter, R 5 Is the outer diameter of the tooth tip of the stator, xi 2 The radian of the notch of the fan-shaped stator slot corresponding to the central angle is adopted. The form and the domain range of the differential equation are similar to those of a tank subdomain without excitation current, a separation variable method can be used for solving, all coefficients are scaled at the same time, and the solution of the processed differential equation is as follows:
Figure BDA0003631662890000161
wherein: j W 2 ， j X 2 ，
Figure BDA0003631662890000162
and
Figure BDA0003631662890000163
respectively are the direct current component coefficient and harmonic component coefficient of the stator slot domain vector magnetic potential equation, n is the harmonic order of the stator slot domain vector magnetic potential equation, and theta j Is the position of the center line of the jth notch, theta j And τ n Are respectively:
Figure BDA0003631662890000164
Figure BDA0003631662890000165
(3) air gap sub-domain magnetic field analysis
The solution domain is located at the outer ring magnetic pole outer diameter R 3 And the inner diameter R of the stator 4 The annular region in between. Let A under two-dimensional polar coordinates 3 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 3 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000166
the differential equation is also solved using the discrete variational method, the original form of the solution being:
Figure BDA0003631662890000167
in the formula: a is 3 ，b 3 ， k W 3 ， k X 3 ， k Y 3 And k Z 3 respectively are a direct current component coefficient and a harmonic component coefficient of the air gap sub-domain vector magnetic potential equation, and k is a harmonic order of the air gap sub-domain vector magnetic potential equation;
unlike the slot and notch sub-regions, the air gap sub-region is a connected annular region, so that according to ampere-loop law, the integral of the magnetic field strength at a circumference of any radius ρ within the air gap region is equal to the total amount of current passing through that circumference, while the current through that region is always zero, so that the solution to the differential equation should not contain a dc component. The solution of the differential equation is simplified and scaled by coefficients, the final form of the solution being:
Figure BDA0003631662890000171
(4) magnetic field analysis in outer ring magnetic pole subdomain
The solution domain is located at the outer diameter R of the alternating pole magnetic pole 2 Outer diameter R of outer ring magnetic pole 3 The annular magnetic pole region in between. Let A under two-dimensional polar coordinates 4 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 4 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000172
in the formula: mu.s 0 Magnetic permeability of vacuum, M ρ And M θ The radial and tangential components of the magnetization of the permanent magnet, respectively, for a radial magnetization polar arc coefficient of 1, the radial and tangential components of the magnetization have the expressions:
Figure BDA0003631662890000173
in the formula: m ρu And M θu Respectively as follows:
Figure BDA0003631662890000174
in the formula B r And p is the remanence and the pole pair number of the permanent magnet, respectively.
Using the discrete variable method, the original form of its solution can be calculated, taking into account the nature of the solution of the non-homogeneous partial differential equation and the ampere-loop law:
Figure BDA0003631662890000175
the final form of the solution obtained by scaling the coefficients of the solution is:
Figure BDA0003631662890000181
in the formula: u W 4 ， u X 4 ， u Y 4 and u Z 4 the direct current component coefficient and the harmonic component coefficient of the vector magnetic potential equation of the annular magnetic pole subdomain are respectively, and u is the harmonic order of the vector magnetic potential equation of the outer ring magnetic pole subdomain. Function F u (ρ) is expressed by:
Figure BDA0003631662890000182
(5) rotor slot domain magnetic field analysis
Under two-dimensional polar coordinates s A 5 (ρ, θ) is the z-component equation of the vector magnetic bit a in the slot sub-domain in the s-th slot, where s is 1,2,3 r . Then with respect to the slot sub-domain when the excitation current in the coil is 0 s A 5 The partial differential equation and domain range of (ρ, θ) can be expressed as:
Figure BDA0003631662890000183
in the formula: r 1 Is the inner diameter of alternating pole magnetic pole, xi 3 The radian of the rotor slot corresponding to the central angle, s M ρ and s M θ the radial and tangential components of the magnetization of the permanent magnet are respectively, the polar arc coefficient of the radial magnetization is 1, and the expressions of the radial and tangential components of the magnetization of the permanent magnet in the rotor slot at the position are as follows:
Figure BDA0003631662890000184
in the formula: c, M ρv And M θv Respectively as follows:
Figure RE-GDA0003806255740000191
Figure BDA0003631662890000192
and (3) calculating a final form of the rotor slot domain vector magnetic potential solution by using a separation variable method and considering the property of the solution of the non-homogeneous partial differential equation and coefficient scaling:
Figure BDA0003631662890000193
in the formula:
Figure BDA0003631662890000194
and
Figure BDA0003631662890000195
the direct current component coefficient and the harmonic component coefficient of the rotor slot domain vector magnetic potential equation are respectively shown, and v is the harmonic order of the rotor slot domain vector magnetic potential equation. Tau is v And equation F v (ρ) are:
Figure BDA0003631662890000196
Figure BDA0003631662890000197
in summary, the vector magnetic potential equations containing the coefficients to be solved in the five regions are established. Wherein i W 1 ， i X 1 ，
Figure BDA0003631662890000198
j W 2 ， j X 2 ，
Figure BDA0003631662890000199
k W 3 ， k X 3 ， k Y 3 ， k Z 3 ， u W 4 ， u X 4 ， u Y 4 ， u Z 4 ，
Figure BDA00036316628900001910
And
Figure BDA00036316628900001911
the total of 20 sets of coefficients (including the dc component coefficient and the harmonic component coefficient) will be determined by the boundary conditions between the regions. Using p ═ R 1 ，R 2 ，R 3 ，R 4 ，R 5 ，R 6 And establishing a matrix equation to solve each harmonic component coefficient in each sub-domain vector magnetic potential equation under the boundary conditions at six positions to obtain the radial/tangential component and the electromagnetic torque of the air gap flux density. Consider that in ρ ═ R 1 And rho ═ R 6 The part of the molecular domain vector magnetic potential equation can be partially simplified at the position, so that the two positions are analyzed first, and then the other positions are analyzed in sequence. The method comprises the following specific steps:
(1) at ρ ═ R 1 Analysis of boundary conditions of
At rho ═ R 1 The position is the junction of the rotor slot subdomain and the rotor iron at the bottom of the rotor slot. The surface of the rotor iron is provided with:
Figure BDA0003631662890000201
by substituting formula (25) for formula (28), we can understand that:
Figure BDA0003631662890000202
Figure BDA0003631662890000203
and (3) taking the equation (30) as an equation to be solved subsequently, substituting the equation (29) into the equation (25), and simplifying the vector magnetic potential equation of the rotor slot domain into:
Figure BDA0003631662890000204
in the following, expressions regarding the rotor slot regions are all calculated using equation (31).
(2) At rho ═ R 6 Analysis of boundary conditions of
At ρ ═ R 6 The position is the junction of the stator iron of the stator slot subdomain and the stator slot bottom. The surface of the stator iron is provided with:
Figure BDA0003631662890000205
by substituting formula (6) for formula (32), we can solve:
i X 1 ＝0 (33)
Figure BDA0003631662890000206
by substituting equations (33) and (34) for equation (6), the simplified vector magnetic potential equation for the slot domain can be obtained as follows:
Figure BDA0003631662890000207
in the following, expressions regarding the stator slot regions are all calculated using equation (35).
(3) At ρ ═ R 5 Analysis of boundary conditions of
At rho ═ R 5 Is the intersection of the stator slot region and the stator slot opening. The radial air gap flux density is continuous according to the boundary condition, which can be expressed as:
Figure BDA0003631662890000211
obviously, the stator slots have a one-to-one correspondence with the stator slots, so that i ═ j is always present in the calculation process. According to the nature of the Fourier series and equation (36) taking into account the boundary conditions, the DC component coefficients and harmonic component coefficients for the notch sub-regions are:
Figure BDA0003631662890000212
Figure BDA0003631662890000213
in the formula: function F 1 (m) and function F 2 (m, n) are respectively:
Figure BDA0003631662890000214
Figure BDA0003631662890000215
at rho ═ R 5 Another boundary condition at (b) is that the tangential magnetic field strength is continuous, which can be expressed as:
Figure BDA0003631662890000216
similarly, let ρ be R 5 Substituting, considering the boundary condition expression (41), according to the properties of the direct current coefficient and harmonic component coefficient of the Fourier series, obtaining:
j X 3 ＝0 (42)
Figure BDA0003631662890000221
at rho ═ R 5 The process of substituting the boundary conditions is completed, and equation (37), equation (38), equation (42), and equation (43) are equations of the obtained harmonic component coefficients. The substitution process of each subsequent boundary condition is similar, and in order to avoid repetition, only the boundary condition is given and obtained in the analysis process of the subsequent boundary conditionThe calculation process of the harmonic component coefficient equation will not be described in detail.
(4) At ρ ═ R 4 Analysis of boundary conditions of
ρ＝R 4 The stator slot and air gap interface. The two boundary conditions are:
Figure BDA0003631662890000222
similarly, the fourier decomposition is performed on the vector magnetic potential function, and the equation of the coefficient of the direct current component and the harmonic component of the vector magnetic potential function is obtained as follows:
Figure BDA0003631662890000223
Figure BDA0003631662890000224
Figure BDA0003631662890000225
Figure BDA0003631662890000226
in the formula: function F 3 ～F 6 Respectively as follows:
Figure BDA0003631662890000231
Figure BDA0003631662890000232
Figure BDA0003631662890000233
Figure BDA0003631662890000234
(5) at ρ ═ R 3 Analysis of boundary conditions of
ρ＝R 3 Is the interface of the air gap region and the ring pole region. Because the two areas are both annular areas and the vector magnetic potential expressions have similar forms, when the boundary condition is substituted, Fourier operation is not needed, and only the harmonic component coefficients of each order corresponding to the forms are repeatedly substituted into the boundary condition for operation. Therefore, in this section, k is always u. The boundary conditions here are:
Figure BDA0003631662890000235
substituting the vector magnetic potential function can obtain:
Figure BDA0003631662890000236
Figure BDA0003631662890000237
Figure BDA0003631662890000238
Figure BDA0003631662890000241
(6) at rho ═ R 2 Analysis of boundary conditions of
ρ＝R 2 Is the interface of the annular pole region and the rotor slot. Its boundary condition and substitution process and rho ═ R 4 Similarly. The boundary conditions are as follows:
Figure BDA0003631662890000242
the direct current component and harmonic component coefficients of the vector magnetic potential function obtained by carrying out Fourier decomposition on the vector magnetic potential function are as follows:
Figure BDA0003631662890000243
Figure BDA0003631662890000244
Figure BDA0003631662890000245
Figure BDA0003631662890000246
the united equation (29), the equation (30), the equation (33), the equation (34), the equation (37), the equation (38), the equation (42), the equation (43), the equation (45), the equation (46), the equation (47), the equation (48), the equation (54), the equation (55), the equation (56), the equation (57), the equation (59), the equation (60), the equation (61) and the equation (62) are 20 equations in total, that is, the direct current component coefficient and the harmonic component coefficient of each region vector magnetic potential equation can be solved. Then the air gap region vector magnetic potential equation A 3 And (rho, theta) is substituted for the formula (2), and the air gap flux density of the motor is obtained through solving.
For any rotor position, the flux linkage linked by a coil can be calculated by the difference of the average vector flux positions of the upper layer side and the lower layer side, and the no-load induced electromotive force can be solved by the differential of the coil flux linkage. Its flux linkage can be expressed as:
ψ x ＝ψ x+ -ψ x- (63)
Figure BDA0003631662890000251
in the formula: l is the axial length of the motor, N c S is the sectional area of the upper layer edge or the lower layer edge of the coil, and the expression is as follows:
Figure BDA0003631662890000252
for any phase winding there is N λ The coils are connected in series, so that the total flux linkage of the phase is as follows:
Figure BDA0003631662890000253
the no-load induced electromotive force of this phase is:
Figure BDA0003631662890000254
in the formula: and omega is the rotating speed of the motor. The electromagnetic torque expression of the three-phase motor is as follows:
Figure BDA0003631662890000255
in the formula: e A ，E B ，E C Is the induced electromotive force of each phase; i.e. i A ，i B And i C Is a three-phase symmetric armature current.
Fig. 1 is a schematic diagram of a double-layer alternating pole half-inserted magnetic pole structure of the present invention. The magnetic poles of the motor are divided into two layers, and S poles 1.1 and N poles 1.2 on the upper layer are alternately arranged to form a surface-mounted structure. The magnetic poles in the lower rotor groove are all N poles 1.2, the pole arc coefficient is 1, and the convex iron 1.3 between the permanent magnets is magnetized into the other pole to form an alternate pole structure in which the permanent magnets and the magnetized rotor convex iron are alternately combined; the magnetization modes of the upper and lower magnetic poles are radial magnetization.
Fig. 2 and 3 are schematic structural diagrams of a double-layer alternating-pole half-insertion permanent magnet motor and a conventional surface-insertion permanent magnet motor according to the present invention, respectively. Both are 8-pole 9-slot structures, and the rated rotating speed is 750 r/min. The stator iron core and the rotor iron core are both made of 50W470 silicon steel sheets, and the permanent magnet is made of NdFeB N35H. The main structural parameters of the motor in the embodiment are as follows: the stator outer radius is 85mm, the stator slot bottom radius is 76mm, the stator slot opening outer radius is 55mm, the stator slot opening inner radius is 53mm, the stator slot span angle is 26 degrees, the stator slot span angle is 4 degrees, the motor axial length is 50mm, the permanent magnet relative permeability is 1.05, the permanent magnet remanence is 1.2T, and the number of serial turns of each phase of winding is 330. The outer radius of the outer ring permanent magnet of the double-layer alternating pole semi-insertion permanent magnet motor is 50mm, the inner radius of the outer ring permanent magnet is 45mm, the radius of the bottom of the rotor groove is 40mm, and the span angle of the rotor groove is 45 degrees. The outer radius of the traditional surface-inserted permanent magnet motor permanent magnet is 50mm, the thickness of the traditional surface-inserted permanent magnet motor permanent magnet is 10mm, the span angle of the N pole 2.1 of the permanent magnet and the S pole 2.2 of the permanent magnet is 34.4 degrees, and the span angle of the rotor nose iron is 10.6 degrees.
Fig. 4 and 5 are the analytic results and the finite element results of the radial air gap flux density and the tangential air gap flux density of the double-layer alternating pole half-plug-in permanent magnet motor in one circle at the middle position of the air gap respectively. As can be seen from the figure, the result calculated by the accurate subdomain model method is basically consistent with the finite element result within the error tolerance range, and the correctness of the discussed analytic modeling is verified.
Fig. 6 is a comparison of the electromagnetic torque analysis result and the finite element result of the double-layer alternating pole half-inserted permanent magnet motor, and the two results are basically consistent within the error range. The error mainly comes from the deficiency of the harmonic order and the defect of the method for calculating the torque.
Fig. 7 is a comparison of the electromagnetic torque of a double-layer alternating-pole half-inserted permanent magnet motor and a conventional surface-inserted permanent magnet motor, where the torque ripple is not increased and the average electromagnetic torque is improved significantly compared to the conventional surface-inserted permanent magnet motor when the usage amount of the permanent magnets is the same.

Claims (3)

1. An analytic modeling method of a double-layer alternating pole semi-insertion permanent magnet motor is characterized in that: adding a layer of surface-mounted magnetic poles on the basis of the alternating poles of the permanent magnet motor, and adopting a double-layer half-inserted magnetic pole structure and a fan-shaped stator slot structure with equal-thickness tooth tips, wherein the lower layer of the permanent magnet is of an alternating pole structure in which permanent magnets with the same polarity and rotor convex iron are alternately arranged, the upper layer of the permanent magnet is a layer of N, S magnetic poles which are alternately arranged to form the surface-mounted magnetic poles, and the magnetization modes of the upper and lower magnetic poles are radial magnetization;
and dividing the solution domain of the permanent magnet motor into: a stator slot sub-region, a slot sub-region, an air gap sub-region, an outer ring magnetic pole sub-region, and a rotor slot sub-region; the central line of the rotor slot coincides with the central line of the sector slot when defining the initial position, i.e. the first rotor slot and the first stator slot, theta 0 The serial numbers of the rotor slots are sequentially arranged into the s-th rotor slot … … of the 2 nd rotor slot with the counterclockwise direction as the positive direction as the position angle of the rotor relative to the position; the serial numbers of the stator slots are sequentially arranged into the 2 nd stator slot … … ith stator slot by taking the anticlockwise direction as the positive direction; theta i Is an included angle between the center line of the ith stator slot and the center line of the 1 st stator slot; theta s Is the included angle between the center line of the s-th rotor slot and the center line of the 1 st rotor slot; for one has N r A rotor slot and N s Permanent magnet machine with individual stator slots, theta i And theta s Are respectively:
Figure RE-FDA0003806255730000011
the general solution expression of the vector magnetic potential A of each sub-domain under the two-dimensional plane is obtained by the ampere loop law and the Gauss law, a partial differential equation is established for the component of the vector magnetic potential A in the z direction under each sub-domain, and the radial direction B of the magnetic field density in the two-dimensional polar coordinate system ρ With a tangential component B θ The relationship to the vector magnetic bit A is expressed as:
Figure RE-FDA0003806255730000012
rho and theta are respectively the calculated radius and the position angle under a polar coordinate system in a two-dimensional plane;
to facilitate the solution of the sub-domain vector magnetic potential and harmonic coefficients, a function P (x, y, z) and a function E (x, y, z) are defined that scales the harmonic component coefficients for an expression of the general solution of the above partial differential equation:
Figure RE-FDA0003806255730000021
through analyzing the magnetic field of each sub-domain, a vector magnetic potential A is obtained and the coefficient is scaled, and the final form is obtained, which is as follows:
(1) stator slot domain magnetic field analysis
Under two-dimensional polar coordinates i A 1 (ρ, θ) is the z-component equation for vector magnetic bit a in the slot sub-domain in the ith slot, where i ═ 1,2,3 s (ii) a Then in the slot sub-domain with respect to the ith slot when the excitation current in the coil is zero i A 1 The partial differential equation and domain range of (ρ, θ) are expressed as:
Figure RE-FDA0003806255730000022
wherein R is 5 Is the outer diameter of the stator tooth tip, R 6 Is the outer diameter of the stator slot, xi 1 The radian of the fan-shaped stator slot corresponding to the central angle is adopted; the differential equation is solved by using a separation variable method, and finally the original form of the solution of the differential equation is obtained as follows:
Figure RE-FDA0003806255730000023
coefficient of friction
Figure RE-FDA0003806255730000024
And
Figure RE-FDA0003806255730000025
scaling the coefficient, and obtaining the scaling coefficient without influencing the final result according to the property of the differential equation solution; meridian systemThe solution to the number scaled differential equation is:
Figure RE-FDA0003806255730000026
in the formula: i W 1 ， i X 1 ，
Figure RE-FDA0003806255730000027
and
Figure RE-FDA0003806255730000028
respectively are the direct current component coefficient and harmonic component coefficient of the stator slot domain vector magnetic potential equation, m is the harmonic order of the stator slot domain vector magnetic potential equation, tau m The expression of (a) is:
Figure RE-FDA0003806255730000029
(2) notch subfield magnetic field analysis
Under two-dimensional polar coordinates j A 2 (ρ, θ) is the z-component equation for the vector magnetic bit A in the notch sub-domain in the jth notch, where j is 1,2,3 s Then, when the exciting current in the coil is zero, the slot subfield with respect to the jth stator slot j A 2 The partial differential equation and domain range of (ρ, θ) are expressed as:
Figure RE-FDA0003806255730000031
in the formula: r 4 Is the stator inner diameter, R 5 Is the outer diameter of the tooth tip of the stator, xi 2 The radian of the notch of the fan-shaped stator slot corresponding to the central angle is adopted; solving by using a separation variable method, simultaneously scaling each coefficient, and solving a processed differential equation as follows:
Figure RE-FDA0003806255730000032
wherein: j W 2 ， j X 2 ，
Figure RE-FDA0003806255730000033
and
Figure RE-FDA0003806255730000034
respectively are the direct current component coefficient and harmonic component coefficient of the stator slot domain vector magnetic potential equation, n is the harmonic order of the stator slot domain vector magnetic potential equation, and theta j Is the position of the center line of the jth notch, theta j And τ n Are respectively:
Figure RE-FDA0003806255730000035
Figure RE-FDA0003806255730000036
(3) air gap sub-domain magnetic field analysis
The solution domain is located at the outer ring magnetic pole outer diameter R 3 And the inner diameter R of the stator 4 An annular region therebetween; let A under two-dimensional polar coordinates 3 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 3 The partial differential equation and domain range of (ρ, θ) are expressed as:
Figure RE-FDA0003806255730000037
the differential equation is also solved using a discrete variate method, the original form of the solution being:
Figure RE-FDA0003806255730000038
in the formula: a is 3 、b 3 、 k W 3 、 k X 3 、 k Y 3 And k Z 3 the direct current component coefficient and the harmonic component coefficient of the air gap sub-domain vector magnetic potential equation are respectively, and k is the harmonic order of the air gap sub-domain vector magnetic potential equation;
the air gap sub-region is a connected annular region, according to ampere loop law, the integral of the magnetic field strength at the circumference of any radius rho in the air gap region is equal to the total current quantity passing through the circumference range, and the current passing through the region is always zero, so that the solution of the differential equation should not contain direct current component; the solution of the differential equation is simplified and scaled by coefficients, and the final form of the solution is:
Figure RE-FDA0003806255730000041
(4) outer ring magnetic pole subdomain magnetic field analysis
The solution domain is located at the outer diameter R of the alternating pole magnetic pole 2 Outer diameter R of outer ring magnetic pole 3 A ring-shaped magnetic pole region in between; let A under two-dimensional polar coordinates 4 (ρ, θ) is the z-component equation for the vector magnetic bit A of the air-gap sub-domain, with respect to A within the air-gap sub-domain 4 The partial differential equation and domain range of (ρ, θ) are expressed as:
Figure RE-FDA0003806255730000042
in the formula: mu.s 0 Is the magnetic permeability of a vacuum, M ρ And M θ Radial and tangential components of the permanent magnet magnetization, respectively, for a radial magnetization polar arc coefficient of 1, the radial and tangential components of the magnetization of the ring-shaped permanent magnet have the expressions:
Figure RE-FDA0003806255730000043
in the formula: m ρu And M θu Respectively as follows:
Figure RE-FDA0003806255730000044
in the formula B r And p is the remanence and the pole pair number of the permanent magnet respectively;
calculating the original form of the solution of the inhomogeneous partial differential equation by using a discrete variable method and considering the nature of the solution and ampere-loop law:
Figure RE-FDA0003806255730000051
the solution is coefficient scaled to obtain the final form of the solution:
Figure RE-FDA0003806255730000052
in the formula: u W 4 ， u X 4 ， u Y 4 and u Z 4 the vector magnetic potential is divided into a direct current component coefficient and a harmonic component coefficient of the annular magnetic pole subdomain vector magnetic potential equation, and u is the harmonic order of the outer annular magnetic pole subdomain vector magnetic potential equation; function F u The expression of (ρ) is:
Figure RE-FDA0003806255730000053
(5) rotor slot domain magnetic field analysis
Under two-dimensional polar coordinates s A 5 (ρ, θ) is the z-component equation for vector magnetic bit a in the slot sub-domain in the s-th slot, where s is 1,2,3 r (ii) a Then with respect to the slot sub-domain when the excitation current in the coil is 0 s A 5 The partial differential equation and domain range of (ρ, θ) are expressed as:
Figure RE-FDA0003806255730000054
in the formula: r is 1 Is the inner diameter of alternating pole magnetic pole, xi 3 The radian of the rotor slot corresponding to the central angle, s M ρ and s M θ the radial and tangential components of the magnetization of the permanent magnet are respectively, the polar arc coefficient of the radial magnetization is 1, and the expressions of the radial and tangential components of the magnetization of the permanent magnet in the rotor slot at the position are as follows:
Figure RE-FDA0003806255730000061
in the formula: c, M ρv And M θv Respectively as follows:
Figure RE-FDA0003806255730000062
Figure RE-FDA0003806255730000063
and calculating the final form of the rotor slot domain vector magnetic potential solution by using a separation variable method and considering the property of the solution of the inhomogeneous partial differential equation and coefficient scaling:
Figure RE-FDA0003806255730000064
in the formula:
Figure RE-FDA0003806255730000065
and
Figure RE-FDA0003806255730000066
respectively rotor slot sub-field vector magnetic potential squareThe direct current component coefficient and harmonic component coefficient of the equation, v is the harmonic order of the rotor slot domain vector magnetic potential equation; tau is v And equation F v (ρ) are:
Figure RE-FDA0003806255730000067
Figure RE-FDA0003806255730000068
in conclusion, vector magnetic potential equations containing coefficients to be solved in the five regions are established; wherein i W 1 ， i X 1 ，
Figure RE-FDA0003806255730000069
j W 2 ， j X 2 ，
Figure RE-FDA00038062557300000610
k W 3 ， k X 3 ， k Y 3 ， k Z 3 ， u W 4 ， u X 4 ， u Y 4 ， u Z 4 ，
Figure RE-FDA00038062557300000611
And
Figure RE-FDA00038062557300000612
the total of 20 sets of coefficients are determined by the boundary conditions between the regions.
2. The analytical modeling method for the double-layer alternating-pole half-inserted permanent magnet motor according to claim 1, characterized in that: using p ═ R 1 ，R 2 ，R 3 ，R 4 ，R 5 ，R 6 Boundary conditions at six positions, establishing matrix squareSolving the coefficient of each harmonic component in each sub-domain vector magnetic potential equation to obtain the radial/tangential component and the electromagnetic torque of the air gap flux density; consider that at ρ ═ R 1 And rho ═ R 6 The positions partially simplify the molecular domain vector magnetic potential equation, so that the two positions are analyzed first, and then other positions are analyzed in sequence; the method comprises the following specific steps:
(1) at rho ═ R 1 Analysis of boundary conditions of
At rho ═ R 1 The position is the junction of the rotor slot subdomain and the rotor iron at the bottom of the rotor slot; the surface of the rotor iron is provided with:
Figure FDA0003631662880000071
substituting formula (25) for formula (28) to obtain:
Figure FDA0003631662880000072
Figure FDA0003631662880000073
and (3) taking the equation (30) as an equation to be solved subsequently, substituting the equation (29) into the equation (25), and simplifying the rotor slot domain vector magnetic potential equation into:
Figure FDA0003631662880000074
the expressions for the rotor slot regions are all calculated using equation (31);
(2) at ρ ═ R 6 Analysis of boundary conditions of
At rho ═ R 6 The position is the junction of the stator iron of the stator slot subdomain and the stator slot bottom; the surface of the stator iron is provided with:
Figure FDA0003631662880000075
substituting formula (6) for formula (32) to obtain:
i X 1 ＝0 (33)
Figure FDA0003631662880000076
formula (6) is substituted by formula (33) and formula (34), and the simplified vector magnetic potential equation of the slot sub-domain is:
Figure FDA0003631662880000077
the expressions for the stator slot sub-fields are all calculated using equation (35);
(3) at rho ═ R 5 Analysis of boundary conditions of
At rho ═ R 5 The position is the junction of the stator slot sub-region and the stator slot opening; the magnetic flux density of the radial air gap is continuous according to the boundary condition, and is expressed as follows:
Figure FDA0003631662880000081
the stator slots and the stator slots are in one-to-one correspondence, so that i is equal to j in the calculation process; according to the nature of the Fourier series and equation (36) taking into account the boundary conditions, the DC component coefficients and harmonic component coefficients for the notch sub-regions are:
Figure FDA0003631662880000082
Figure FDA0003631662880000083
in the formula: function F 1 (m) and functionF 2 (m, n) are respectively:
Figure FDA0003631662880000084
Figure FDA0003631662880000085
at rho ═ R 5 Another boundary condition at (b) is that the tangential magnetic field strength is continuous, expressed as:
Figure FDA0003631662880000086
similarly, let ρ be R 5 Substituting, considering the boundary condition formula (41), according to the properties of the direct current component coefficient and harmonic component coefficient of Fourier series, obtaining:
j X 3 ＝0 (42)
Figure FDA0003631662880000091
at rho ═ R 5 The process of substituting the boundary conditions is finished, and the formula (37), the formula (38), the formula (42) and the formula (43) are the equations of the obtained harmonic component coefficients;
(4) at rho ═ R 4 Analysis of boundary conditions of
ρ＝R 4 The interface of the stator notch and the air gap; the two boundary conditions are:
Figure FDA0003631662880000092
and carrying out Fourier decomposition on the vector magnetic potential function to obtain the direct current component and harmonic component coefficients of the vector magnetic potential function according to the following equations:
Figure FDA0003631662880000093
Figure FDA0003631662880000094
Figure FDA0003631662880000095
Figure FDA0003631662880000096
in the formula: function F 3 ～F 6 Respectively as follows:
Figure FDA0003631662880000101
Figure FDA0003631662880000102
Figure FDA0003631662880000103
Figure FDA0003631662880000104
(5) at rho ═ R 3 Analysis of boundary conditions of
ρ＝R 3 Is the interface of the air gap region and the ring-shaped magnetic pole region; since both regions are annular regions, k ═ u is always present; the boundary conditions here are:
Figure FDA0003631662880000105
substituting the vector magnetic potential function to obtain:
Figure FDA0003631662880000106
Figure FDA0003631662880000107
Figure FDA0003631662880000108
Figure FDA0003631662880000109
(6) at rho ═ R 2 Analysis of boundary conditions of
ρ＝R 2 Is the interface of the annular magnetic pole area and the rotor slot; the boundary conditions are as follows:
Figure FDA0003631662880000111
the equation for obtaining the coefficient of the direct current component and the harmonic component by performing Fourier decomposition on the vector magnetic potential function is as follows:
Figure FDA0003631662880000112
Figure FDA0003631662880000113
Figure FDA0003631662880000114
Figure FDA0003631662880000115
the united equation (29), the equation (30), the equation (33), the equation (34), the equation (37), the equation (38), the equation (42), the equation (43), the equation (45), the equation (46), the equation (47), the equation (48), the equation (54), the equation (55), the equation (56), the equation (57), the equation (59), the equation (60), the equation (61) and the equation (62) are used for calculating 20 equations, namely, the direct current component coefficient and the harmonic component coefficient of each region vector magnetic potential equation are solved; the air gap area vector magnetic potential equation A 3 And (rho, theta) is substituted for the formula (2), and the air gap flux density of the permanent magnet motor is obtained through solving.
3. The analytic modeling method of a double-layer alternating-pole semi-plug-in permanent magnet motor according to claim 2, characterized in that: for any rotor position, the flux linkage of a coil is calculated by the difference of average vector flux positions of an upper layer edge and a lower layer edge, and the no-load induced electromotive force is obtained by differential solution of the coil flux linkage; the flux linkage is represented as:
ψ x ＝ψ x+ -ψ x- (63)
Figure FDA0003631662880000121
in the formula: l is the axial length of the motor, N c S is the sectional area of the upper layer edge or the lower layer edge of the coil, and the expression is as follows:
Figure FDA0003631662880000122
for any phase winding there is N λ Each coil is connected in series, and the total flux linkage of the phase is as follows:
Figure FDA0003631662880000123
the no-load induced electromotive force of this phase is:
Figure FDA0003631662880000124
in the formula: omega is the rotating speed of the motor; the electromagnetic torque expression of the three-phase motor is as follows:
Figure FDA0003631662880000125
in the formula: e A ，E B ，E C Is the induced electromotive force of each phase; i.e. i A ，i B And i C Is a three-phase symmetric armature current.
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