METHOD FOR EVALUATING ELECTROMAGNETIC PERFORMANCE OF ELECTRIC MACHINES IN PARTICULAR OF PERMANENT-MAGNET MACHINES

Abstract

Method for evaluating electromagnetic performance of a permanent-magnet machine, wherein the method is implemented in a computer processer or in a controller of a permanent-magnet machine, calculating the magnetic field distribution by a sub-domain method; using current density of the equivalent current sheet to represent the boundary condition, iterating by the sub-do-main method and by the magnetic circuit method until a convergent result is obtained, calculating the electro-magnetic performance parameters on basis of the convergent result, so as to evaluate the electromagnetic performance.

Claims

1. Method for evaluating electromagnetic performance of a permanent-magnet machine, wherein the method is implemented in a computer processer or in a controller of a permanent-magnet machine, building an electromagnetic analytical model for an electric machine, the solution domain of the analytical model is at least divided into a permanent magnet sub-domain (SD-1), an air gap sub-domain (SD-2), and a stator slot sub-domain (SD-3), and calculating the current magnetic field distributions of the air gap sub-domain (SD-2), stator slot sub-domain (SD-3) and permanent magnet sub-domain (SD-1) through a Poisson equation or a Laplace equation; characterized in adding an equivalent current sheet (A) to the boundary of the target sub-domain and to an interface between the target sub-domain and an adjacent sub-domain respectively, using current density of the equivalent current sheet (A) to represent the boundary condition and interface condition, and calculating the magnetic field distribution on basis of the sub-domains; calculating the magnetic potential difference of each part of the core by a magnetic circuit method according to the current magnetic field distribution, and converting the magnetic potential difference to a current density of the equivalent current sheet (A); iterating the magnetic field distribution obtained on basis of the sub-domains and the current density obtained by the magnetic circuit method until a convergent current density is obtained, solving and obtaining the magnetic field distribution of the target sub-domain by using the convergent current density; on basis of the magnetic field distribution, calculating the electromagnetic performance parameters, wherein standard electromagnetic performance parameters are presto red in the processer or the controller, the processer or the controller comparing the calculated electromagnetic performance parameters with the standard electromagnetic performance parameters, and obtaining a evaluating result of the electromagnetic performance.

2. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the analytical model divides the solution domain on the 2D plane when the two-dimensional magnetic circuit is used as the target magnetic field of the analytical model.

3. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 2, wherein when the magnetic circuit method is used to calculate the magnetic potential difference of each stator tooth and stator yoke, a node magnetic potential difference matrix equation is constructed according to Kirchhoff's law, and then a convergent solution of the node magnetic potential difference matrix is obtained using Newton-Raphson's law.

4. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 2, wherein when iterating the magnetic field distribution obtained on basis of the sub-domains and the current density obtained by magnetic circuit method starts, the current density of the equivalent current sheet (A) is preset to 0, and the sub-domain method is used to calculate the magnetic field distribution for the first time, to obtain the magnetic flux into the stator; then the magnetic potential difference of each tooth portion and yoke portion of the stator is calculated by the magnetic circuit method, and the current density value of corresponding equivalent current sheet (A) is obtained by the magnetic potential difference; then current density value is used as a boundary condition of the sub-domain method to calculate again; the above iterative process is repeated until the difference between the current density obtained in the previous iteration and the current density obtained in this iteration is within the preset threshold range.

5. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 3, wherein the mutual iteration between the magnetic circuit method and the sub-domain method is: the current density obtained by the convergent solution of the node magnetic potential difference matrix is taken into the sub-domain method as a boundary condition, to obtain the current magnetic field distribution by the sub-domain method, then calculate the status core magnetic field distribution to judge if the current density of this time and the current density of the previous time are converged; and if not, the current magnetic field distribution is taken into the magnetic circuit method, to calculate the convergent solution of the node magnetic potential difference matrix again, and repeat the iterations until the current density of this time and the current density of the previous time are converged.

6. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein when the rotor is a consequent-pole rotor, the range of the consequent-pole machine permanent magnet sub-domain is: α.sub.im−α.sub.mag/2≤α≤α.sub.im+α.sub.mag/2, where α.sub.mag is the width angle of the permanent magnet, α.sub.im is the center line of the i-th permanent magnet, the consequent-pole rotor permanent magnet sub-domain is added with a boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density: $B_{rmi}{|}_{\alpha ={\alpha }_{im}}\pm {\alpha }_{\mathrm{mag}}=0,$ B.sub.rmi represents the radial magnetic flux density in the permanent magnet; or, when the rotor is a non-consequent-pole rotor, the range of rotor permanent magnet sub-domain is 0-2π, the rotator permanent magnet sub-domain in the sub-domain method is added with a boundary condition at the interface between the permanent magnet and the rotor yoke, and the boundary condition is represented by a current density of the equivalent current sheet (A).

7. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a permanent magnet machine whose stator slot is an open slot, wherein the stator slot sub-domain (SD-3) boundary conditions of the analytical model comprise the boundary conditions of two slot sides and the boundary condition of the slot bottom, the boundary condition is represented by: H.sub.3ri|.sub.α=α.sub.i.sub.+b.sub.sa.sub./2=J.sub.i1; H.sub.3ri|.sub.α=α.sub.i.sub.−b.sub.sa.sub./2=−J.sub.i2; H.sub.3αi|.sub.r=R.sub.sb=−J.sub.i3; where, b.sub.oa is the slot width, J.sub.i1, is the current density of the equivalent current sheet (A) on the first slot side of the stator slot, J.sub.i2 is the current density of the equivalent current sheet (A) on the second slot side of the stator slot, J.sub.i3 is the current density of the equivalent current sheet (A) at the slot bottom of the stator slot.

8. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a permanent magnet machine whose stator slot has a slot opening, wherein for a semi-closed slot: the stator slot sub-domain (SD-3) comprises the boundary condition of two slot sides, the boundary condition of the slot bottom and the boundary condition of two sides of the slot opening, the boundary condition is represented by: H.sub.3ri|.sub.α=α.sub.i.sub.+b.sub.sa.sub./2=J.sub.i1; H.sub.3ri|.sub.α=α.sub.i.sub.−b.sub.sa.sub./2=J.sub.i2; H.sub.3αi|.sub.r=R.sub.sb=−J.sub.i3; H.sub.4ri|α=.sub.α.sub.i.sub.+b.sub.oa.sub./2=J.sub.i4; H.sub.4ri|.sub.α=α.sub.i.sub.−b.sub.oa.sub./2=−J.sub.i5; where, b.sub.oa is the width of slot opening, J.sub.i1, is the current density of the equivalent current sheet (A) on the first slot side of the stator slot, J.sub.i2 is the current density of the equivalent current sheet (A) on the second slot side of the stator slot, J.sub.i3 is the current density of the equivalent current sheet (A) at the slot bottom of the stator slot; J.sub.i4, is the current density of the equivalent current sheet (A) on the first slot side of the stator slot opening, Ja is the current density of the equivalent current sheet (A) on the second slot side of the stator slot opening.

9. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a dual-stator surface-mounted permanent magnet machine, wherein equivalent current sheets (A) are added in the internal current slot (SD-7), the internal terminal slot opening (SD-5), the external terminal slot (SD-6), and the external terminal slot opening (SD-4), to simulate nonlinear effects in the internal and external stator cores; boundary conditions of the stator slot sub-domain (SD-3) comprise: the equivalent current density of two slot sides and one slot bottom of the internal stator slot (SD-7), the equivalent current density of two slot sides of the internal stator slot opening (SD-5), the equivalent current density of two slot sides and one slot bottom of the external stator slot (SD-6), and the equivalent current density of two slot sides of the external stator slot opening (SD-4).

10. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a dual-stator consequent-pole permanent magnet machine, wherein boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the internal stator slot (SD-7), the equivalent current density of two slot sides of the internal stator slot opening (SD-5), the equivalent current density of two slot sides and one slot bottom of the external stator slot (SD-6), and the equivalent current density of two slot sides of the external stator slot opening (SD-4), and the consequent-pole rotor permanent magnet sub-domain is added with a boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density: $B_{rmi}{|}_{\alpha ={\alpha }_{im}\pm {\alpha }_{mag}}=0,$ B.sub.rmi, represents the radial magnetic flux density in the permanent magnet.

11. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a surface plug-in consequent-pole permanent magnet machine, wherein boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot (SD-3), the equivalent current density of two slot sides of the stator slot opening (SD-4), and the consequent-pole rotor permanent magnet sub-domain is added with a boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density: $B_{rmi}{|}_{\alpha ={\alpha }_{im}\pm {\alpha }_{mag}}=0,$ B.sub.rmi represents the radial magnetic flux density in the permanent magnet.

12. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a V-shaped built-in internal rotor permanent magnet machine, wherein boundary conditions of the sub-domains comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain (SD-2), the equivalent current density of the two slot sides of the slot opening sub-domain (SD-3), and the equivalent current density of the interface between the rotor and the air gap sub-domain (SD-1).

13. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a I-shaped built-in internal rotor permanent magnet machine, wherein boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain (SD-2), the equivalent current density of the two slot sides of the slot opening sub-domain (SD-3), and the equivalent current density of the interface between the rotor and the air gap sub-domain (SD-1).

14. Method for evaluating electromagnetic performance of a permanent-magnet machine according to claim 1, wherein the method is applied to a split-tooth permanent magnet vernier machine, wherein boundary conditions of the sub-domains comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain (SD-3), and the equivalent current density of two slot sides of the slot opening; and the equivalent current density of all slot sides and slot bottoms in split teeth (SD-4).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0044] FIG. 1 is a schematic diagram of a model of a nonlinear surface-mounted permanent magnet machine of the present invention.

[0045] FIG. 2 is a nonlinear composite model in which an equivalent current sheet is added as a boundary condition.

[0046] FIG. 3 is a flow chart of a solving method of the present invention when used in electromagnetic design no-load condition of a surface-mounted permanent magnet machine.

[0047] FIG. 4 shows a magnetic circuit of variable magnetic reluctance numbers used by the magnetic circuit method in the embodiment of the present invention. The more severity the saturation condition, the more magnetic reluctance numbers selected.

[0048] FIG. 5 shows a correspondence relationship between the equivalent current sheet current density and the magnetic potential difference of the magnetic circuit used under load conditions according to the embodiment of the present invention.

[0049] FIG. 6 is a schematic diagram for comparing the predicted radial magnetic flux density in the air gap under no-load conditions between the present invention and other three methods.

[0050] FIG. 7 is a schematic diagram for comparing the predicted tangential magnetic flux density in the air gap under no-load conditions between the present invention and other three methods.

[0051] FIG. 8 is a schematic diagram comparing the predicted A-phase flux linkage under no-load conditions between the present invention and other three methods.

[0052] FIG. 9 is a schematic diagram for comparing the predicted A-phase counter electromotive force under no-load conditions between the present invention and other three methods.

[0053] FIG. 10 is a schematic diagram for comparing the predicted cogging torque under no-load conditions between the present invention and other three methods.

[0054] FIG. 11 is a schematic diagram for comparing the predicted radial magnetic flux density in the air gap under load conditions between the present invention and other three methods.

[0055] FIG. 12 is a schematic diagram for comparing the predicted tangential magnetic flux density in the air gap under load conditions between the present invention and other three methods.

[0056] FIG. 13 is a schematic diagram for comparing the predicted A-phase flux linkage under load conditions between the present invention and other three methods.

[0057] FIG. 14 is a schematic diagram for comparing the predicted A-phase induced electromotive force under load conditions between the present invention and other three methods.

[0058] FIG. 15 is a schematic diagram of the predicted total torque and the prototype experimental results under load conditions between the present invention and other three methods.

[0059] FIG. 16 is a schematic diagram of a three-dimensional magnetic circuit in which sub-sheets are axially connected by taking two sub-sheets as an example.

[0060] FIG. 17 is a nonlinear composite model suitable for use when a surface-mounted permanent magnet machine has tooth tips.

[0061] FIG. 18 is a composite model of an outer rotor surface-mounted permanent magnet machine suitable for consideration of the rotor nonlinear effects.

[0062] FIG. 19 is a nonlinear composite model suitable for a dual-stator surface-mounted permanent magnet machine.

[0063] FIG. 20 is a nonlinear composite model suitable for a dual-stator consequent-pole permanent magnet machine.

[0064] FIG. 21 is a nonlinear composite model suitable for a surface plug-in consequent-pole permanent magnet machine.

[0065] FIG. 22 is a nonlinear composite model suitable for a V-shaped built-in internal rotor permanent magnet machine.

[0066] FIG. 23 is a nonlinear composite model suitable for an I-shaped built-in internal rotor permanent magnet machine.

[0067] FIG. 24 is a magnetic circuit model used in FIG. 22 and FIG. 23.

[0068] FIG. 25 is a nonlinear composite model suitable for a split-tooth permanent magnet vernier machine.

[0069] FIG. 26 is a three-dimensional rotor model of permanent magnet block and oblique pole, suitable for the three-dimensional magnetic field analytical model proposed by the present invention.

[0070] FIG. 27 is a three-dimensional stator model including an axial ventilation slot, suitable for the three-dimensional magnetic field analytical model proposed by the present invention.

[0071] FIG. 28 is a schematic diagram of co-simulation between the magnetic field distribution solved by the present invention and the control algorithm provided by MATLAB.

[0072] The foregoing embodiments describe the technical solutions and beneficial effects of the present invention in detail. It should be understood that the above embodiments are only specific embodiments of the present invention and are not intended to limit the present invention.

DETAILED DESCRIPTION

[0073] The present invention will be further described in detail with reference to the accompanying drawings and embodiments. It should be noted that the following embodiments are intended to facilitate the understanding of the invention but not to limit the invention.

[0074] A method for evaluating electromagnetic performance of a PM machine, establishing an electromagnetic analytical model of the machine, the solution domain of the analytical model being at least divided into a permanent magnet sub-domain, an air gap sub-domain, and a stator slot sub-domain, and calculating the current magnetic field distributions of the air gap sub-domain, stator slot sub-domain and/or permanent magnet sub-domain based on a sub-domain method by the analytical model through a Poisson equation or a Laplace equation; adding an equivalent current sheet to the boundary of the target sub-domain and interface between the target sub-domain and an adjacent sub-domain respectively, representing the boundary condition and interface condition by the current density of the equivalent current sheet, and solving the magnetic field distribution by a sub-domain method; solving the magnetic potential difference of each part of the core by a magnetic circuit method according to the current magnetic field distribution, and converting the magnetic potential difference to a current density of the equivalent current sheet; iterating the magnetic field distribution obtained by the sub-domain method and the current density obtained by the magnetic circuit method until a convergent current density is obtained, solving the magnetic field distribution of the target sub-domain with the convergent current density; calculating the electromagnetic performance parameters based on the magnetic field distribution and evaluating the electromagnetic performance of the machine based on the electromagnetic performance parameters.

[0075] The two-dimensional magnetic circuit is used as the target magnetic field of the analytical model, and the solution domain is divided on a 2D plane of the analytical model.

[0076] According to the boundary condition and interface condition in each sub-domain, the undetermined coefficients in each sub-domain expression are solved by simultaneous equations, and the equation set is written as a matrix form. There is a matrix inversion process for solving the undetermined coefficients; when the undetermined coefficients are determined, the magnetic field distribution in each sub-domain is obtained.

[0077] when solving with the magnetic circuit method, the silicon steel of stator core tooth portion and yoke portion is equivalent to different magnetic reluctance to construct a magnetic network, and the obtained current magnetic field distribution is converted into the magnetic flux into the stator as a magnetic flux source in the magnetic circuit method, to obtain the magnetic potential difference of each stator tooth and stator yoke by solving, the magnetic potential difference of the stator tooth and stator yoke is equivalent to the current density of equivalent current sheets on the slot side and slot bottom. When the magnetic circuit method is used to solve the magnetic potential difference of each stator tooth and stator yoke, a node magnetic potential difference matrix equation is constructed according to kirchhoff s law, and then a convergent solution of the node magnetic potential difference matrix is obtained using Newton-Raphson's law. Where, the node magnetic potential difference matrix equation is:

V=(AΛA.sup.T).sup.−1Φ  (22)

[0078] where, A is the associated matrix, Λ is the branch magnetic reluctance matrix, V is the node magnetic potential difference matrix, Φ is the node magnetic flux matrix, and the magnetic flux into the stator is obtained by integrating the magnetic flux density and the area in the stator slot and the air gap.

[0079] when iteration of the magnetic field distribution obtained by sub-domain method and the current density obtained by magnetic circuit method is performed, the current density of the equivalent current sheet is preset to 0, and the sub-domain method is used to solve the magnetic field distribution for the first time, to obtain the magnetic flux into the stator; then the magnetic potential difference of each tooth portion and yoke portion of the stator is solved by the magnetic circuit method, and the current density value of corresponding equivalent current sheet is obtained by the magnetic potential difference; then current density value is used as a boundary condition of the sub-domain method to resolve; the above iterative process is repeated until the difference between the current density obtained in the previous iteration and the current density obtained in this iteration is within the preset threshold range.

[0080] In some embodiments, when considering the saturation of the rotor yoke, a magnetic network is established at the rotor yoke to calculate the current density of the equivalent current sheet between the rotor yoke and the rotor permanent magnet, and the current density is taken as the boundary condition at the interface between the rotor yoke and the rotor permanent magnet.

[0081] In some embodiments, the stator slot sub-domain includes a boundary condition of two slot sides and a boundary condition of the slot bottom, and the boundary condition is expressed as:

H.sub.3ri|.sub.α=α.sub.i.sub.+b.sub.sa.sub./2=J.sub.i1  (1)

H.sub.3ri|.sub.α=α.sub.i.sub.−b.sub.sa.sub./2=J.sub.i2  (2)

H.sub.3αi|.sub.r=R.sub.sb=−J.sub.i3  (3)

[0082] where, J.sub.i1 is the current density of the equivalent current sheet on the first slot side of the stator slot, J.sub.i2 is the current density of the equivalent current sheet on the second slot side of the stator slot, J.sub.i3 is the current density of the equivalent current sheet at the slot bottom of the stator slot.

[0083] In some embodiments, the stator slot is a semi-closed slot, the stator slot sub-domain includes boundary conditions of two slot sides and the boundary condition of both sides of the slot opening. The boundary condition is expressed as:

H.sub.3ri|.sub.α=α.sub.i.sub.+b.sub.sa.sub./2=J.sub.i1;

H.sub.3ri|.sub.α=α.sub.i.sub.−b.sub.sa.sub./2=J.sub.i2;

H.sub.3αi|.sub.r=R.sub.sb=−J.sub.i3;

H.sub.4ri|.sub.α=α.sub.i.sub.+b.sub.oa.sub./2=J.sub.i4;

H.sub.4ri|.sub.α=α.sub.i.sub.−b.sub.oa.sub./2=−J.sub.i5;

[0084] where, b.sub.oa is the width of slot opening, J.sub.i1 is the current density of the equivalent current sheet on the first slot side of the stator slot, J.sub.i2 is the current density of the equivalent current sheet on the second slot side of the stator slot, J.sub.i3 is the current density of the equivalent current sheet at the slot bottom of the stator slot; J.sub.i4 is the current density of the equivalent current sheet on the first slot side of the stator slot opening, J.sub.i5 is the current density of the equivalent current sheet on the second slot side of the stator slot opening.

[0085] In some embodiments, the permanent magnet pole of the rotor is a consequent-pole, and a soft magnetic material is added between two adjacent permanent magnets. The non-consequent-pole has a permanent magnet sub-domain range of 0-2n, and the range of the consequent-pole machine permanent magnet sub-domain is:

α.sub.im−α.sub.mag/2≤α≤α.sub.im+α.sub.mag/2

[0086] Where, α.sub.mag is the width angle of the permanent magnet, α.sub.im is the center line of the i-th permanent magnet.

[0087] Compared to the non-consequent-pole structure, the consequent-pole rotor permanent magnet sub-domain is added with the boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density:

B.sub.rmi|.sub.α=αim±αmag/2=0.

Example 1: Electromagnetic Analytical Model of a Surface-Mounted Permanent Magnet Machine

[0088] As shown in FIG. 1, the present invention is applied to a surface-mounted permanent magnet machine electromagnetic design, and a 4-pole 6-slot surface-mounted permanent magnet machine is taken as an example, but the number of pole slots of the machine is not limited to the examples of this embodiment. The stator of the surface-mounted permanent magnet machine herein is a nonlinear silicon steel, SD-1, SD-2 and SD-3 are sub-domains of permanent magnets, air gap and stator slot, ω.sub.r is the rotor rotational angular velocity, and α.sub.i is the angle of the centerline of the i-th slot, r and α are the radial and tangential positions in the polar coordinate system, b.sub.sa is the width of slot opening, R.sub.r, R.sub.m, R.sub.s and R.sub.sb are the radius of rotor yoke, the permanent magnet surface, the slot top, and slot bottom to the center of the circle respectively, α.sub.0 is the initial angle of the rotor, k and n are the number of harmonics on the circumference and the slot opening, respectively, and p is the number of poles of the permanent magnet.

[0089] In this embodiment, when the sub-domain method is used to establish the vector magnetic potential model of the sub-domain, the equivalent current sheet is added as a boundary condition on both slot sides and the slot bottom of the stator slot, so that the vector magnetic potential expression of the stator slot is associated with the current density of the equivalent current sheet, for details, refer to FIGS. 2 and 3, the equivalent current sheets of the slot side are Ji1 and Ji2, and the equivalent current sheet of the slot bottom is Ji3, where, i=1, 2, 3 . . . the nonlinearity of the stator is reflected by the current sheet, therefore, the stator silicon steel can be considered linear at this time.

[0090] Based on the result of the sub-domain method, the magnetic circuit method is used to solve the magnetic potential difference of each part of the stator, and is converted into the current density of the equivalent current sheet in the slot. Then iteration is performed by the sub-domain method and magnetic circuit method, to calculate the convergent solution of the magnetic field distribution of each sub-domain. Based on the convergence result, electromagnetic performance such as permanent magnet machine flux linkage, counter electromotive force, cogging torque, terminal voltage, electromagnetic torque, and unbalance force are calculated.

[0091] Taking the surface-mounted open slot internal rotor machine as an example, the specific calculation of the sub-domain method is as follows:

[0092] By solving the Poisson equation and Laplace equation and applying the boundary condition, the expressions of the vector magnetic potential Az1 in the permanent magnet and the vector magnetic potential Az2 in the air gap are as follows:

[00002]$\begin{array}{cc}{A}_{z1}=\underset{k}{.Math.}\left(C_{1k}{A}_1+C_{2k}{M}_{\alpha \mathrm{ck}}-C_{3k}{M}_{\mathrm{rsk}}\right)\cos \left(k\alpha \right)+\underset{k}{.Math.}\left(C_{1k}{C}_1+C_{2k}{M}_{\alpha \mathrm{sk}}+C_{3k}{M}_{\mathrm{rck}}\right)\sin \left(k\alpha \right)& \left(1\right)\\ A_{z2}=\underset{k}{.Math.}\left[{A_2\left(r/R_s\right)}^k+{B_2\left(r/R_m\right)}^{-k}\right]\cos \left(k\alpha \right)+\underset{k}{.Math.}\left[{C_2\left(r/R_s\right)}^k+{D_2\left(r/R_m\right)}^{-k}\right]\sin \left(k\alpha \right)& \left(2\right)\end{array}$

[0093] Thus, the radial magnetic flux density B.sub.1r and the tangential magnetic flux density B.sub.1a in the permanent magnet can be solved as follows:

[00003]$\begin{array}{cc}{B}_{1r}=-\underset{k}{.Math.}\left(k/r\right).Math.\left(C_{1k}{A}_1+C_{2k}{M}_{\alpha \mathrm{ck}}-C_{3k}{M}_{\mathrm{rsk}}\right)\sin \left(k\alpha \right)+\underset{k}{.Math.}\left(k/r\right).Math.\left(C_{1k}{C}_1+C_{2k}{M}_{\alpha \mathrm{sk}}-C_{3k}{M}_{\mathrm{rck}}\right)\cos \left(k\alpha \right)& \left(3\right)\\ B_{1\alpha }=-\underset{k}{.Math.}\left(1/r\right).Math.\left(C_{4k}{A}_1+C_{5k}{M}_{\alpha \mathrm{ck}}-C_{6k}{M}_{\mathrm{rsk}}\right)\cos \left(k\alpha \right)+\underset{k}{.Math.}\left(1/r\right).Math.\left(C_{4k}{C}_1+C_{5k}{M}_{\alpha \mathrm{sk}}-C_{6k}{M}_{\mathrm{rck}}\right)\sin \left(k\alpha \right)& \left(4\right)\end{array}$

[0094] The radial magnetic flux density B.sub.2r and the tangential magnetic flux density B.sub.2a in the air gap are:

[00004]$\begin{array}{cc}{B}_{2r}=-\underset{k}{.Math.}\left(k/r\right).Math.\left[{A_2\left(r/R_s\right)}^k+{B_2\left(r/R_m\right)}^{-k}\right]\sin \left(k\alpha \right)+\underset{k}{.Math.}\left(k/r\right).Math.\left[{C_2\left(r/R_s\right)}^k+{D_2\left(r/R_m\right)}^{-k}\right]\cos \left(k\alpha \right)& \left(5\right)\\ B_{2\alpha }=-\underset{k}{.Math.}\left(k/r\right).Math.\left[{A_2\left(r/R_s\right)}^k-{B_2\left(r/R_m\right)}^{-k}\right]\cos \left(k\alpha \right)-\underset{k}{.Math.}\left(k/r\right).Math.\left[{C_2\left(r/R_s\right)}^k-{D_2\left(r/R_m\right)}^{-k}\right]\sin \left(k\alpha \right)& \left(6\right)\end{array}$

[0095] Where, A.sub.1, B.sub.1, A.sub.2, B.sub.2, C.sub.2 and D.sub.2 are undetermined coefficients, M.sub.rsk, M.sub.rck, M.sub.ask, and M.sub.ack are magnetization components of the permanent magnet:

M.sub.rck=M.sub.rk cos(kω.sub.rt+kα.sub.0)  (7)

M.sub.rsk=M.sub.rk sin(kω.sub.rt+kα.sub.0)  (8)

M.sub.αck=−M.sub.αk sin(kω.sub.rt+kα.sub.0)  (9)

M.sub.αsk=M.sub.αk cos(kω.sub.rt+kα.sub.0)  (10)

[0096] For Radial Magnetization:

M.sub.rk=(4pB.sub.r/kπμ.sub.0)sin(kπα.sub.p/2p), k/p=1,3,5, . . .  (11)

M.sub.αk=0, k/p=1,3,5, . . .  (12)

[0097] For Tangential Magnetization:

M.sub.rk=(B.sub.r/μ.sub.0)α.sub.p(A.sub.1k+A.sub.2k), k/p=1,3,5, . . .  (13)

M.sub.αk=(B.sub.r/μ.sub.0)α.sub.p(A.sub.1k−A.sub.2k), k/p=1,3,5, . . .  (14)

[0098] Other Coefficients:

[00005]$\begin{array}{cc}{C}_{1k}=\left[{\left(r/R_m\right)}^k+{G_1\left(r/R_r\right)}^{-k}\right]& \left(15\right)\\ C_{2k}=\frac{{\mu }_0}{k^2-1}\left[R_r{k\left(\frac{r}{R_r}\right)}^{-k}+r\right]& \left(16\right)\\ C_{3k}=\frac{{\mu }_0}{k^2-1}\left[R_r{k\left(\frac{r}{R_r}\right)}^{-k}+\mathrm{kr}\right]& \left(17\right)\\ C_{4k}=k\left[{\left(r/R_m\right)}^k-{G_1\left(r/R_r\right)}^{-k}\right]& \left(18\right)\\ C_{5k}=\frac{{\mu }_0}{k^2-1}\left[-k^2{R_r\left(\frac{r}{R_r}\right)}^{-k}+r\right]& \left(19\right)\\ C_{6k}=\frac{{\mu }_{0}k}{k^2-1}\left[-{R_r\left(\frac{r}{R_r}\right)}^{-k}+r\right]& \left(20\right)\\ G_1={\left(R_r/R_m\right)}^k& \left(21\right)\end{array}$

[0099] The vector magnetic potential A.sub.z3 in the slot satisfies the Poisson equation. If it is no-load, the current density of the armature in the equation is set to 0, and its boundary condition can be expressed as:

H.sub.3ri|.sub.α=α.sub.i.sub.+b.sub.sa.sub./2=J.sub.i1  (22)

H.sub.3ri|.sub.α=α.sub.i.sub.−b.sub.sa.sub./2=−J.sub.i2  (23)

H.sub.3αi|.sub.r=R.sub.sb=−J.sub.i3  (24)

[0100] where, J.sub.i1, J.sub.i2 and J.sub.i3 are the current density of equivalent current sheets on three sides of the slot, the value of which is determined by the magnetic circuit method.

[0101] The expression formula for a vector magnetic potential of non-overlapping winding stator slot is:

[00006]$\begin{array}{cc}{A}_{z3i}\left(r,\alpha \right)=\underset{n}{.Math.}{A}_{z3\mathrm{in}}\cos \left[E_n\left(\alpha +b_{\mathrm{sa}}/2\text{-}{\alpha }_i\right)\right]+A_{z3i0}\mathrm{where},& \left(25\right)\\ A_{z3\mathrm{in}}=\left(B_{3n}{G}_3-\frac{E_n{W}_n{R}_{\mathrm{sb}}}{\left(1-E_n^2\right)}-\frac{2{\mu }_0{J}_{\mathrm{am}n}{R}_{\mathrm{sb}}^2}{E_n\left(E_n^2-4\right)}\right){\left(\frac{r}{R_{\mathrm{sb}}}\right)}^{E_n}+{B_{3n}\left(\frac{r}{R_s}\right)}^{-E_n}+\frac{{\mathrm{rE}}_n^2{W}_n}{1-E_n^2}+\frac{{\mu }_0{J}_{\mathrm{am}n}{r}^2}{E_n^2-4}& \left(26\right)\\ A_{z3i0}={\mu }_0{J}_{\mathrm{am}0}\left(2R_{\mathrm{sb}}^2\ln r-r^2\right)/4+C_i\ln r-{\mu }_0\left(J_{i1}+J_{i2}\right)r/b_{\mathrm{sa}}+Q_{3i}& \left(27\right)\\ E_n=n\pi /b_{\mathrm{sa}}& \left(28\right)\\ G_3={\left(R_s/R_{\mathrm{sb}}\right)}^{E_n}& \left(29\right)\\ W_n=2{\mu }_0\left\{-\left(J_{i1}+J_{i2}\right){\left(-1\right)}^n+J_{i2}\left[{\left(-1\right)}^n-1\right]\right\}/\left(E_n^2{b}_{\mathrm{sa}}\right)& \left(30\right)\\ C_i=R_{\mathrm{sb}}\left[{\mu }_0{J}_{i3}+{\mu }_0\left(J_{i1}+J_{i2}\right)/b_{\mathrm{sa}}\right]& \left(31\right)\\ J_{\mathrm{iam}0}=\left(J_{\mathrm{iam}1}+J_{\mathrm{iam}2}\right)/2& \left(32\right)\\ J_{\mathrm{iam}n}=2\left(J_{\mathrm{iam}1}-J_{\mathrm{iam}2}\right)\sin \left(n\pi /2\right)/\left(n\pi \right)& \left(33\right)\end{array}$

[0102] Where, α.sub.i is the angle of the centerline of the i-th groove, r and α are the radial and tangential positions in the polar coordinate system, b.sub.sa is the width of slot opening, R.sub.s and R.sub.sb are the radius of the slot top and the slot bottom to the center of the circle, respectively. J.sub.ami1 and f.sub.ami2 are the current density of the two coil sides in the i-th slot. If it is no-load, the current density of the coil is 0, and B.sub.3in and Q.sub.3i are undetermined coefficients.

[0103] Then the radial magnetic flux density and the tangential magnetic flux density are:

[00007]$\begin{array}{cc}{B}_{3\mathrm{ir}}=\underset{n}{.Math.}{B}_{3\mathrm{irn}}\sin \left[E_n\left(\alpha +b_{\mathrm{sa}}/2\text{-}{\alpha }_i\right)\right]& \left(34\right)\\ B_{3i\alpha }\left(r,\alpha \right)=\underset{n}{.Math.}{B}_{3i\alpha n}\cos \left[E_n\left(\alpha +b_{\mathrm{sa}}/2\text{-}{\alpha }_i\right)\right]+B_{3i\mathrm{\alpha 0}}\mathrm{where}\text{:}& \left(35\right)\\ B_{3\mathrm{irn}}=-E_n\left\{\frac{E_n^2{W}_n}{\left(1-E_n^2\right)}+\frac{{\mu }_0{J}_{\mathrm{am}n}r}{E_n^2-4}+\frac{B_{3n}}{R_s}{\left(\frac{r}{R_s}\right)}^{-E_n-1}+[\frac{B_{3n}{G}_3}{R_{\mathrm{sb}}}-\frac{E_n{W}_n}{\left(1-E_n^2\right)}-\frac{2{\mu }_0{J}_{\mathrm{am}n}{R}_{\mathrm{sb}}}{E_n\left(E_n^2-4\right)}]{\left(\frac{r}{R_{\mathrm{sb}}}\right)}^{E_n-1}\right\}& \left(36\right)\\ B_{3i\alpha n}=-\left[\left(\frac{B_{3n}{E}_n{G}_3}{R_{\mathrm{sb}}}-\frac{E_n^2{W}_n}{1-E_n^2}-\frac{2{\mu }_0{R}_{\mathrm{sb}}{J}_{\mathrm{am}n}}{E_n^2-4}\right){\left(\frac{r}{R_{\mathrm{sb}}}\right)}^{E_n-1}-\frac{B_{3n}{E}_n}{R_s}{\left(\frac{r}{R_s}\right)}^{-E_n-1}+\frac{E_n^2{W}_n}{1-E_n^2}+\frac{2{\mu }_0{J}_{\mathrm{am}n}r}{E_n^2-4}\right]& \left(37\right)\\ B_{3i\mathrm{\alpha 0}}=-\frac{{\mu }_0{J}_{\mathrm{am}0}\left(R_{\mathrm{sb}}^2/r-r\right)}{2}-\frac{C_i}{r}+\frac{{\mu }_0\left(J_{i1}+J_{i2}\right)}{b_{\mathrm{sa}}}& \left(38\right)\end{array}$

[0104] For overlapping windings, the vector magnetic potential expression for the slot bottom is:

[00008]$\begin{array}{cc}{A}_{\mathrm{zb}3i}\left(r,\alpha \right)=\underset{n}{.Math.}{A}_{\mathrm{zb}3\mathrm{in}}\cos \left[E_n\left(\alpha +b_{\mathrm{sa}}/2\text{-}{\alpha }_i\right)\right]+A_{\mathrm{zb}3i0}\mathrm{where},& \left(39\right)\\ A_{\mathrm{zb}3i0}=-{\mu }_0\left(J_{i1}+J_{i2}\right)r/b_{\mathrm{sa}}+C_i\ln r+{\mu }_0{J}_{\mathrm{iam}2}\left(2R_{\mathrm{sb}}^2\ln r-r^2\right)/4+Q_{3\mathrm{bi}}& \left(40\right)\\ A_{\mathrm{zb}3\mathrm{in}}=\left(B_{3n}{G}_3-E_n{W}_n{R}_{\mathrm{sb}}/\left(1-E_n^2\right)\right){\left(r/R_{\mathrm{sb}}\right)}^{E_n}+{B_{3n}\left(r/R_s\right)}^{-E_n}+{\mathrm{rE}}_n^2{W}_n/\left(1-E_n^2\right)& \left(41\right)\end{array}$

[0105] radial magnetic flux density of slot bottom is:

[00009]$\begin{array}{cc}{B}_{b3\mathrm{ir}}=\underset{n}{.Math.}{B}_{b3\mathrm{irn}}\sin \left[E_n\left(\alpha +b_{\mathrm{sa}}/2\text{-}{\alpha }_i\right)\right]& \left(42\right)\end{array}$

[0106] magnetic flux density of slot bottom is:

[00010]$\begin{array}{cc}{B}_{b3ia}=\underset{n}{.Math.}{B}_{b3ian}\cos \left[E_n\left(\alpha +b_{sa}\text{/}2-{\alpha }_i\right)\right]+B_{b3ia0}\mathrm{Where}& \left(43\right)\\ B_{b{3}_{\mathrm{irn}}}=-\left(E_n\text{/}r\right)\left[\left(B_{3n}{G}_3-E_n{W}_n{R}_{sb}\text{/}\left(1-E_n^2\right)\right){\left(r\text{/}{R}_{sb}\right)}^{E_n}+{B_{3n}\left(r\text{/}{R}_s\right)}^{-E_n}+r{E}_n^2{W}_n\text{/}\left(1-E_n^2\right)\right]& \left(44\right)\\ B_{b{3}_{\mathrm{ian}}}=-\left(E_n\text{/}r\right)\left[\left(B_{3n}{G}_3-E_n{W}_n{R}_{sb}\text{/}\left(1-E_n^2\right)\right){\left(r\text{/}{R}_{sb}\right)}^{E_n}-{B_{3n}\left(r\text{/}{R}_s\right)}^{-E_n}+r{E}_n{W}_n\text{/}\left(1-E_n^2\right)\right]& \left(45\right)\\ B_{b3ia0}={\mu }_0\left(J_{i1}+J_{i2}\right)\text{/}{b}_{sa}-C_i\text{/}r-{\mu }_0{J}_{iam2}\left(R_{\mathrm{sb}}^2\text{/}r-r\right)\text{/}2& \left(46\right)\end{array}$

[0107] The vector magnetic potential expression of the slot top is:

[00011]$\begin{array}{cc}{A}_{\mathrm{zt}3i}\left(r,\alpha \right)=\underset{n}{.Math.}{A}_{\mathrm{zt}3\mathrm{in}}\cos \left[E_n\left(\alpha +b_{sa}\text{/}2-{\alpha }_i\right)\right]+A_{\mathrm{zt}3i0}\mathrm{where},& \left(47\right)\\ A_{\mathrm{zt}3i0}=-{\mu }_{0}r\left(J_{i1}+J_{i2}\right)\text{/}{b}_{sa}+C_i\ln r+C_{\mathrm{ti}}\ln r-{\mu }_0{J}_{iam1}{r}^2\text{/}4+Q_{3\mathrm{ti}}& \left(48\right)\\ A_{zt3in}=\left(B_{3tn}{G}_3-E_n{W}_n{R}_{sb}\text{/}\left(1-E_n^2\right)\right){\left(r\text{/}{R}_{sb}\right)}^{E_n}+{B_{3tn}\left(r\text{/}{R}_s\right)}^{-E_n}+r{E}_n^2{W}_n\text{/}\left(1-E_n^2\right)& \left(49\right)\\ B_{3tn}=B_{3n}& \left(50\right)\\ C_{ti}={\mu }_0{J}_{iam1}{R}_{sm}^2\text{/}2+{\mu }_0{J}_{iam2}\left(R_{sb}^2-R_{sm}^2\right)\text{/}2& \left(51\right)\\ Q_{3bi}=Q_{3ti}+C_{ti}\ln R_{sm}-{\mu }_0{J}_{iam1}{R}_{sm}^2\text{/}4-{\mu }_0{J}_{iam2}\left(2R_{sb}^2\ln R_{sm}-R_{sm}^2\right)\text{/}4& \left(52\right)\end{array}$

[0108] The radial magnetic flux density and tangential magnetic flux density of the slot top are:

[00012]$\begin{array}{cc}{B}_{t3ir}=\underset{n}{.Math.}{B}_{t3irn}\sin \left[E_n\left(\alpha +b_{sa}\text{/}2-{\alpha }_i\right)\right]& \left(53\right)\\ B_{t3ia}=\underset{n}{.Math.}{B}_{t3ian}\cos \left[E_n\left(\alpha +b_{sa}\text{/}2-{\alpha }_i\right)\right]+B_{t3ia0}& \left(54\right)\\ B_{t3\mathrm{irn}}=B_{b3\mathrm{irn}},B_{t3\mathrm{ian}}=B_{b3\mathrm{ian}}& \left(55\right)\\ B_{t3\mathrm{ia}0}={\mu }_0\left(J_{i1}+J_{i2}\right)\text{/}{b}_{sa}-\left(C_i+C_{\mathrm{ti}}\right)\text{/}r+{\mu }_0{J}_{iam1}r\text{/}2& \left(56\right)\end{array}$

[0109] Using boundary condition, the undetermined coefficients in the above three domains can be solved by a matrix equation:

[00013]$\begin{array}{cc}\left[\begin{array}{ccccccc}{K}_{11}& 0& K_{13}& K_{14}& 0& 0& 0\\ 0& K_{22}& 0& 0& K_{25}& K_{26}& 0\\ K_{31}& 0& K_{33}& K_{34}& 0& 0& 0\\ 0& K_{42}& 0& 0& K_{45}& K_{46}& 0\\ 0& 0& K_{53}& K_{54}& 0& 0& K_{57}\\ 0& 0& 0& 0& K_{65}& K_{66}& K_{67}\\ 0& 0& K_{73}& K_{74}& K_{75}& K_{76}& K_{77}\end{array}\right]\left[\begin{array}{c}{A}_1\\ C_1\\ A_2\\ B_2\\ C_2\\ D_2\\ B_{3T}\end{array}\right]=\left[\begin{array}{c}{Y}_1\\ Y_1\\ Y_3\\ Y_4\\ Y_5\\ Y_6\\ Y_7\end{array}\right]& \left(57\right)\end{array}$

[0110] Till now, the magnetic field distributions inside the permanent magnet, and in the air gap and stator slot have been solved.

[0111] The specific process of the step 2): the silicon steel of stator core tooth portion and yoke portion is equivalent to different magnetic reluctance to construct a magnetic network, and the obtained current magnetic field distribution is converted into the magnetic flux into the stator as a magnetic flux source in the magnetic circuit method, to obtain the magnetic potential difference of each stator tooth and stator yoke by solving, the magnetic potential difference of the stator tooth and stator yoke is equivalent to the current density of equivalent current sheets on the slot side and slot bottom.

[0112] When the magnetic circuit method is used to solve the magnetic potential difference of each stator tooth and stator yoke, a node magnetic potential difference matrix equation is constructed according to kirchhoff's law, and then a convergent solution of the node magnetic potential difference matrix is obtained using Newton-Raphson's law. Where, the node magnetic potential difference matrix equation is:

V=(AΛA.sup.T).sup.−1Φ  (58)

[0113] where, A is the associated matrix, Λ is the branch magnetic reluctance matrix, V is the node magnetic potential difference matrix, Φ is the node magnetic flux matrix, and the magnetic flux into the stator is obtained by integrating the magnetic flux density and the area in the stator slot and the air gap.

[0114] In the step of iteration in the magnetic circuit method and the sub-domain method, the current density of the equivalent current sheet is preset to 0, and the sub-domain method is used to solve the magnetic field distribution for the first time, to obtain the magnetic flux into the stator; then the magnetic potential difference of each tooth portion and yoke portion of the stator is solved by the magnetic circuit method, and the current density value of corresponding equivalent current sheet is obtained by the magnetic potential difference; then current density value is used as a boundary condition of the sub-domain method to resolve; the above iterative process is repeated until a stable current density of the equivalent current sheet is obtained.

[0115] After obtaining a stable current density value of the equivalent current sheet, the flux linkage, counter electromotive force, the cogging torque of the permanent magnet machine can be calculated under no-load condition; and under load conditions, the terminal voltage, total torque and other electromagnetic performance of the permanent magnet machine can be calculated. The main process is as follows:

[0116] The calculation of the machine flux linkage is divided into two cases: non-overlapping winding and overlapping winding. In the non-overlapping winding, the corresponding flux linkages of the two coil sides in one slot are:

[00014]$\begin{array}{cc}{\psi }_{i1}=l_{ef}\left(N_c\text{/}{A}_c\right)\left[Z_{0}d+\underset{n}{.Math.}\left(Z_n\text{/}{E}_n\right)\sin \left(E_{n}d\right)\right]& \left(59\right)\\ {\psi }_{i2}=l_{ef}\left(N_c\text{/}{A}_c\right)\left[Z_{0}d-\underset{n}{.Math.}\left(Z_n\text{/}{E}_n\right)\sin \left(n\pi -E_{n}d\right)\right]\mathrm{Where}& \left(60\right)\\ Z_0=\left(R_{\mathrm{sb}}^2-R_s^2\right)g{Q}_{3i}\text{/}2-{\mu }_0\left(J_{i1}+J_{i2}\right)\left(R_{\mathrm{sb}}^3-R_s^3\right)\text{/}\left(3b_{sa}\right)+C_i\left[2R_{sb}^2\ln R_{sb}-2R_s^2\ln R_s+R_s^2-R_{sb}^2\right]\text{/}4& \left(61\right)\\ Z_n=\left[B_{3in}{G}_3-E_n{W}_{in}{R}_{sb}\text{/}\left(1-E_n^2\right)\right]g\left(R_{sb}^2-G_3{R}_s^2\right)\text{/}\left(E_n+2\right)+B_{3in}\left(R_{sb}^2{G}_3-R_s^2\right)\text{/}\left(2-E_n\right)+E_n^2{W}_{in}\left(R_{sb}^3-R_s^3\right)\text{/}\left[3\left(1-E_n^2\right)\right]& \left(62\right)\end{array}$

[0117] where, N.sub.c is the number of turns per coil, A.sub.c is the area of one coil side, and d is the width of the coil side.

[0118] In the overlapping winding, the corresponding magnetic fluxes of the two coil sides in one slot are:

[00015]$\begin{array}{cc}{\psi }_{i1}=l_{ef}\left(N_c\text{/}{A}_c\right)b_{sa}{Z}_{t0}& \left(63\right)\\ {\psi }_{i2}=l_{ef}\left(N_c\text{/}{A}_c\right)b_{sa}{Z}_{b0}\mathrm{Where}& \left(64\right)\\ Z_{b0}=-{\mu }_0\left(J_{i1}+J_{i2}\right)\left(R_{sb}^3-R_{sm}^3\right)\text{/}\left(3b_{sa}\right)+\left(R_{sb}^2-R_{sm}^2\right)Q_{3i}\text{/}2+C_i\left[2R_{sb}^2\ln R_{sb}-2R_{sm}^2\ln R_{sm}+R_{sm}^2-R_{sb}^2\right]\text{/}4& \left(65\right)\\ Z_{t0}=-{\mu }_0\left(J_{i1}+J_{i2}\right)\left(R_{sm}^3-R_s^3\right)\text{/}\left(3b_{sa}\right)+\left(R_{sm}^2-R_s^2\right)Q_{3i}\text{/}2+C_i\left[2R_{sm}^2\ln R_{sm}-2R_s^2\ln R_s+R_s^2-R_{sm}^2\right]\text{/}4& \left(66\right)\end{array}$

[0119] The counter electromotive force is calculated by:

[00016]$\begin{array}{cc}{E}_{ph}=-\frac{d{\psi }_{ph}}{dt}\mathrm{ph}=A,B,C.& \left(67\right)\end{array}$

[0120] where, ψ.sub.ph is the total flux linkage of a phase, which is obtained by superimposing the flux linkages of all relevant coil sides.

[0121] The torque algorithm is the Maxwell tensor method, namely:

[00017]$\begin{array}{cc}{T}_c=\left(\pi l_a{r}^2\text{/}{\mu }_0\right)\underset{k}{.Math.}\left(B_{rck}{B}_{\alpha ck}+B_{rsk}{B}_{\alpha sk}\right)& \left(68\right)\end{array}$

[0122] The algorithm for the unbalanced forces between the horizontal and vertical directions is:

F.sub.x=rl.sub.a∫.sub.0.sup.2πσ cos αdα+−rl.sub.α∫.sub.0.sup.2πτ sin αdα  (69)

F.sub.y=rl.sub.a∫.sub.0.sup.2πσ sin αdα+rl.sub.a∫.sub.0.sup.2πτ cos αdα  (70)

where:

σ=(B.sub.r.sup.2−B.sub.α.sup.2)/(2μ.sub.0)  (71)

τ=(B.sub.r.Math.B.sub.α)/μ.sub.0  (72)

[0123] In the calculation process of the present invention, for a conventional permanent magnet machine whose rotor dose not saturate, the rotor yoke is regarded as infinite permeability to simplify the model and speed up the solving; for some unconventional permanent magnet machines that need to consider the rotator saturation, similar to the stator, an equivalent current sheet is added between the permanent magnet and the rotor yoke to replace the magnetic potential difference of the rotor yoke.

[0124] In order to verify the correctness of the proposed algorithm, in this embodiment, the no-load and load conditions of one set of 8-pole 24-slot surface-mounted permanent magnet machine are calculated respectively, and its main parameters are shown in Table 1. The calculation results are compared with the results obtained by the two-dimensional finite element calculation program.

TABLE-US-00001 TABLE 1 Parameter Prototype Unit Stator outer diameter 50 mm Stator inner diameter 27.85 mm Thickens of permanent magnet 3.5 mm Rotor outer diameter 27.35 mm Armature length 50 mm Width of stator tooth 3.16 mm Height of stator yoke 3.2 mm Polar arc coefficient 1 Permanent magnet remanence 1.20 T Relative recovery permeability of 1.05 permanent magnet Magnetization direction Parallel magnetization Rated speed 400 rpm Number of pole pairs 4 Number of slots 24 Rated current (peak) 6.36 A Type of silicon steel sheet WG35WW300

[0125] Four methods are used for electromagnetic analysis and solution of the machine, respectively the present invention, the finite element nonlinear algorithm, the sub-domain method, and the finite element linear algorithm. For the magnetic circuit model of the variable magnetic reluctance number used in the present invention, refer to FIG. 4. Where, the stator silicon steel is divided into a tooth portion and a yoke portion, which are all equivalent to equivalent resistance. The number of equivalent magnetic reluctances can be determined by the saturation of the stator, and the magnetic flux tri entering from the air gap and slot is determined by the sub-domain, and the convergence result is obtained by the Kirchhoff s law and Newton-Raphson method, to obtain the node voltage of each node. FIG. 5 shows the correspondence relationship between the equivalent current sheet current density and the magnetic potential difference.

J.sub.i1=(V.sub.5Ns+i+1−V.sub.2i+1)/(R.sub.sb−R.sub.s)  (73)

J.sub.i2=(V.sub.2i−1−V.sub.5Ns+i)/(R.sub.sb−R.sub.s)  (74)

J.sub.i3=(V.sub.2i+1−V.sub.2i−1)/(R.sub.sb.Math.b.sub.sa)  (75)

The present invention improves the traditional sub-domain method and combines it with the magnetic circuit method. The improved sub-domain method is used to accurately calculate the magnetic field distribution in permanent magnets, air gaps and slots, and then integration of the magnetic flux density and its area entering the stator is performed, converted to magnetic flux. The magnetic flux is used as the input of the magnetic circuit, to analyze the saturation of the stator. According to different saturation situations, that is, the different magnetic potential difference of the tooth portion and the yoke portion, the current density of the equivalent current sheet is solved, and the current density is used as the input of the improved magnetic circuit method to solve the magnetic field distribution. Therefore, the algorithm proposed in this patent has an iterative process, and the convergent solution is a magnetic field distribution in consideration of the nonlinearity of the stator silicon steel sheet.

[0126] The no-load magnetic field analysis results obtained by the four methods are shown in FIG. 6 to FIG. 10. FIG. 6 and FIG. 7 show the radial magnetic flux density and tangential magnetic flux density in the air gap predicted by the four methods. FIG. 8, FIG. 9 and FIG. 10 are A-phase flux linkage, A-phase counter electromotive force, and cogging torque predicted by four methods.

[0127] The analysis results of the load magnetic field obtained by four methods are shown in FIG. 11-FIG. 15. FIG. 11 and FIG. 12 are the radial magnetic flux density and tangential magnetic flux density in the air gap predicted by the four methods. FIG. 13, FIG. 14 and FIG. 15 are phase A flux linkages, phase A induced electromotive force and total torque predicted by four methods.

[0128] From the figures, the prediction results of the present invention are highly consistent with the finite element nonlinear algorithm, and the sub-domain method is highly consistent with the finite element linear algorithm. However, in actual machine design, the nonlinearity of the prototype is a characteristic that must be considered. Therefore, the non-linear algorithm proposed by the present invention has higher calculation accuracy than that of the sub-domain method and the finite element linear algorithm. Compared with the finite element non-linear algorithm, the non-linear algorithm proposed by the present invention can provide suggestions for optimization because the machine performance corresponds to the size of the machine.

[0129] The above process includes 61 field calculation points in one electrical cycle. The four methods use the same computer, with CPU of i7-7700. The finite element method is based on the An sys Maxwell platform, and the present invention and the sub-domain method are based on MATLAB. The number of meshes in the finite element model is 23,750. Table 2 shows the calculation time of the four methods for the machine under no-load and load conditions. It can be observed from the table, the model proposed by the present invention can greatly save the solving time compared to the conventional finite element model.

TABLE-US-00002 TABLE 2 Finite Conven- Finite Calcu- element tional element lation The nonlinear sub-domain linear time(s) invention algorithm method algorithm No-load 13 230 0.64 197 Load 16.92 287 0.78 216

[0130] Therefore, the electromagnetic analysis of the prototypes proposed in Table 1 through these four methods has proven that the non-linear algorithm proposed by the present invention can accurately calculate the electromagnetic performance of the machine, greatly shortening the design cycle of the machine, which can provide suggestions for optimization and is very suitable for the machine design process.

[0131] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 2: Analytical Model of Surface-Mounted Permanent Magnet Machine with Tooth Tips

[0132] FIG. 17 shows an applicable non-linear composite model when the surface-mounted permanent magnet machine has tooth tips. SD-1 is a permanent magnet sub-domain, SD-2 is an air gap sub-domain, SD-3 is a slot sub-domain, and SD-4 is a slot opening sub-domain. Among them, an equivalent current sheet is added to the slot opening part to simulate the non-linear effect of the tooth tip part. The difference between this embodiment and Example 1 is that: two slot sides of the slot opening sub-domain SD-4 add the current density of the equivalent current sheet as a boundary condition of slot opening sub-domain SD-4 when boundary conditions are set for the sub-domain method. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1, and is not described again.

[0133] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 3: External Rotor Surface-Mounted Permanent Magnet Machine Considering Rotor Non-Linear Effects

[0134] FIG. 18 shows an applicable composite model of an external rotor surface-mounted permanent magnet machine considering the nonlinear effects of the rotor. The difference between this embodiment and Example 2 is that: an equivalent current sheet is added between the permanent magnet and the rotor yoke to simulate the nonlinear effect of the rotor yoke. That is, the boundary conditions of sub-domain method comprise a current density of equivalent current sheets at the interface between the rotor permanent magnet SD-1 and the rotor yoke as a boundary condition, the boundary condition of stator slot sub-domain SD-3 and the boundary condition of slot opening sub-domain SD-2. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Examples 1 and 2, and is not described again.

[0135] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

[0136] It should be noted that, if the stator slot of this permanent magnet machine is an open slot (i.e., without slot opening SD-2), an equivalent current sheet is added between the permanent magnet and the rotor yoke to simulate the nonlinear effect of the rotor yoke. That is, the boundary conditions of sub-domain method comprise a current density of equivalent current sheets at the interface between the rotor permanent magnet SD-1 and the rotor yoke, the equivalent current density of two slot sides of stator slot sub-domain SD-3 and the equivalent current density of slot bottom. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

Example 4: Electromagnetic Analytical Model of a Dual-Stator Surface-Mounted Permanent Magnet Machine

[0137] FIG. 19 shows an applicable nonlinear composite model of a dual-stator surface-mounted permanent magnet machine. The difference between this embodiment and Example 1 is that: SD-7 in internal stator slot, SD-5 in internal stator slot opening, SD-6 in external stator slot and SD-4 in external stator slot opening have been added with equivalent current sheet to simulate non-linear effects inside and outside the status core. That is, the boundary conditions of sub-domain method comprise equivalent current density of two slot sides and one slot bottom of the internal stator slot SD-7, equivalent current density of two slot sides of the internal stator slot opening SD-5, the equivalent current density of two slot sides and one slot bottom of external stator slot SD-6, and the equivalent current density of two slot sides of the external stator slot opening SD-4. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

[0138] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 5: Electromagnetic Analytical Model of a Dual-Stator Consequent-Pole Permanent Magnet Machine

[0139] FIG. 20 shows an applicable nonlinear composite model of a dual-stator consequent-pole permanent magnet machine. The range of consequent-pole machine permanent magnet sub-domain is α.sub.im−α.sub.mag/2≤α≤α.sub.im+α.sub.mag/2; where αmag is the width angle of the permanent magnet, α.sub.im is the center line of the i-th permanent magnet, the consequent-pole rotor permanent magnet sub-domain adds the boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density:

[00018]$B_{rmi}{|}_{\alpha ={\alpha }_{\mathrm{im}}\pm {\alpha }_{\mathrm{mag}}}=0,$

B.sub.rmi represents the radial magnetic flux density in the permanent magnet.

[0140] That is, the difference between this embodiment and Example 1 is: the boundary conditions of sub-domain method comprise the equivalent current density of two slot sides and one slot bottom of the internal stator slot SD-7, the equivalent current density of two slot sides of the internal stator slot opening SD-5, equivalent current density of two slot sides and one slot bottom of the external stator slot SD-6, and the equivalent current density of two slot sides of the external stator slot opening SD-4, and the consequent-pole rotor permanent magnet sub-domain is added with the boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density:

[00019]$B_{rmi}{|}_{\alpha ={\alpha }_{im}\pm {\alpha }_{mag}}=0,$

B.sub.rmi represents the radial magnetic flux density in the permanent magnet. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

[0141] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 6: Analytical Model of a Surface Plug-in Consequent-Pole Permanent Magnet Machine

[0142] FIG. 21 shows an applicable nonlinear composite model of a surface plug-in consequent-pole permanent magnet machine. The difference between this embodiment and Example 5 is that this embodiment is an internal rotor electron and has only one stator.

[0143] The boundary conditions of the sub-domain method of this embodiment comprise the equivalent current density of two slot sides and one slot bottom of the internal stator slot SD-3, the equivalent current density of two slot sides of the stator slot opening SD-4, and the consequent-pole rotor permanent magnet sub-domain is added with a boundary condition on the interface with the adjacent soft magnetic material, and the boundary condition is represented by the magnetic flux density:

[00020]$B_{rmi}{|}_{\alpha ={\alpha }_{im}\pm {\alpha }_{mag}}=0,$

B.sub.rmi represents the radial magnetic flux density in the permanent magnet. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Examples 1 and 5 and is not described again.

[0144] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 7: Analytical Model of a V-Shaped Built-in Internal Rotor Permanent Magnet Machine

[0145] FIG. 22 shows an applicable nonlinear composite model of a V-shaped built-in internal rotor permanent magnet machine. The permanent magnets of the rotor portion are arranged in a V-shape, and are provided with a magnetically isolated magnetic bridge. Permanent magnet and rotor yoke are solved by magnetic circuit method, while air gap and slot are solved by sub-domain method considering current density. Equivalent current sheet is added in stator slot and slot opening to simulate nonlinear effects in core. That is, the difference between this embodiment and Example 1 is that: boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain SD-2, the equivalent current density of two slot sides of the slot opening sub-domain SD-3, and the equivalent current density on the interface between the rotor and the air gap sub-domain SD-1. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

[0146] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 8: Electromagnetic Analytical Model of I-Type Built-in Internal Rotor Permanent Magnet Machine

[0147] FIG. 23 shows an applicable nonlinear composite model of an I-shaped built-in internal rotor permanent magnet machine. The permanent magnets of the rotor portion are arranged in a I-shape, and are provided with a magnetically isolated magnetic bridge. Permanent magnet and rotor yoke are solved by magnetic circuit method, while air gap and slot are solved by sub-domain method considering current density. Equivalent current sheet is added in stator slot and slot opening to simulate nonlinear effects in core.

[0148] That is, the difference between this embodiment and Example 1 is that: boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain SD-2, the equivalent current density of two slot sides of the slot opening sub-domain SD-3, and the equivalent current density on the interface between the rotor and the air gap sub-domain SD-1. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

[0149] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

[0150] FIG. 24 is one of the magnetic circuit models used in FIG. 22 and FIG. 23. Now, the idea of solving the nonlinear composite model of the built-in permanent magnet machine is described as follows: 1. Establish the entire machine magnetic network model and use the magnetic circuit method to solve the node magnetic pressure matrix. 2. According to the node magnetic pressure matrix, the magnetic potential difference between the nodes on the circumferential surface of the rotor can be calculated. Furthermore, the current density of the equivalent current sheet between each node on the circumferential surface can be obtained as the boundary condition of air gap sub-domain. At the same time, the current density of the equivalent current sheet at the edge of the stator core is calculated. The calculation method of the current density is the same as the foregoing method of the present invention, and the current density is used as the boundary condition. 3. According to the stator and rotor boundary conditions, the sub-domain method is used to solve the magnetic field distribution of air gaps, slots, and slot openings (excluding permanent magnets). 4. The iterative process is performed. On the one hand, the magnetic flux entering the rotor surface calculated by the sub-domain method is used as the magnetic flux source of the rotor magnetic circuit to calculate the rotor magnetic circuit, to obtain a new equivalent current density of the rotor circumferential surface. On the other hand, the magnetic flux entering the stator calculated by the sub-domain method is used as the magnetic flux source of the stator magnetic circuit, to solve the stator magnetic circuit and obtain a new equivalent current density on the stator surface. The equivalent current density on the rotor and stator is used as the boundary condition for the new round of sub-domain method calculations, until the current density error before and after the two iterations does not exceed the threshold, it is determined to be convergent, which is the final magnetic field distribution.

Example 9: Electromagnetic Analytical Model of a Split-Tooth Permanent Magnet Vernier Machine

[0151] FIG. 25 shows an applicable nonlinear composite model of a split-tooth permanent magnet vernier machine. Among them, an equivalent current sheet is added to the stator slot and the split tooth to be equivalent to the nonlinear effect in the stator core.

[0152] That is, the difference between this embodiment and Example 1 is that: boundary conditions of sub-domain method comprise: the equivalent current density of two slot sides and one slot bottom of the stator slot sub-domain SD-3, and the equivalent current density of two slot sides of the slot opening; and the equivalent current density of all slot sides and slot bottoms in the split tooth SD-4. Other calculation processes, for example, the iteration way of the magnetic circuit method, the sub-domain method and the magnetic circuit method is the same as that of Example 1 and is not described again.

[0153] When the object of this embodiment is a two-dimensional magnetic circuit, an analytical model of two-dimensional magnetic circuit is used; when the object of this embodiment is a three-dimensional magnetic circuit, an analytical model of three-dimensional magnetic circuit can be used, as long as the boundary condition of the corresponding analytical model are set to the boundary condition of this embodiment.

Example 10: A Method for Evaluating Electromagnetic Performance of a Machine Based on Three-Dimensional Magnetic Circuit

[0154] A method for evaluating electromagnetic performance of a PM machine based on three-dimensional magnetic circuit, comprising: establishing an analytical model by a three-dimensional magnetic circuit, the three-dimensional magnetic circuit including a plurality of sub-sheets divided by the non-ferromagnetic material in the axial direction, as shown in FIG. 16, the solution domain of each sub-sheet being at least divided into a permanent magnet sub-domain, an air gap sub-domain, and a stator slot sub-domain, and calculating the current magnetic field distributions of the air gap sub-domain, stator slot sub-domain and/or permanent magnet sub-domain based on a sub-domain method by the analytical model through a Poisson equation or a Laplace equation; adding an equivalent current sheet to the boundary of the target sub-domain and interface between the target sub-domain and an adjacent sub-domain respectively, representing the boundary condition and interface condition by the current density of the equivalent current sheet, and solving the magnetic field distribution by a sub-domain method; the magnetic field node of all the sub-sheet being arranged in the same position, the magnetic field node at the same position of the adjacent sub-sheet being connected, the magnetic path connecting direction of the adjacent sub-sheets being axially connected, the magnetic reluctance distribution of all the sub-sheets and the connecting magnetic reluctance between adjacent sub-sheets forming a three-dimensional magnetic circuit of the machine, and solving the magnetic potential difference convergent solution of the three-dimensional magnetic circuit method according to the magnetic circuit method, and converting magnetic potential difference convergent solution to a current density of the equivalent current sheet; iterating the magnetic field distribution obtained by the sub-domain method and the current density obtained by the magnetic circuit method until a convergent current density is obtained, solving the magnetic field distribution of the target sub-domain with the convergent current density; calculating the electromagnetic performance parameters based on the magnetic field distribution and evaluating the electromagnetic performance of the machine based on the electromagnetic performance parameters.

[0155] The solving of the sub-domain method is the same as the above two-dimensional magnetic field analytical model. A three-dimensional magnetic field is constructed by the magnetic circuit method. The solving of three-dimensional magnetic field includes the magnetic reluctance in each sub-sheet and the connected magnetic reluctance between adjacent sub-sheets, and the other solving is the same as that of the two-dimensional magnetic field analytical model. Because of different temperature distribution at the axial direction, the degree of nonlinearity is different, and the magnetic field distribution of each sub-sheet is different; therefore, the uneven distribution of the magnetic field in the axial direction can be calculated by the sub-sheets elaborately, and for complex and diverse machine structures, the oblique pole and stator core discontinuity can be calculated.

[0156] The axial sub-sheet methods of three-dimensional magnetic circuit are mainly divided into two types:

[0157] The first type: the nonlinear state of the stator core is different caused by the difference in temperature in the axial direction, that is, the BH curve of the stator core is different, so the part with similar nonlinearity (that is, the part that the same BH curve can be used) is used as a sub-sheet.

[0158] The second type: because of axially different structures, such as rotor block, oblique pole, stator slot discontinuity, etc., the part with the same axial section is used as a sub-sheet.

[0159] Regardless of the sub-sheet method, the position of the magnetic field node in all sub-sheets is the same, each sub-sheet uses the respective BH curve to solve the magnetic reluctance of the current sheet; in the three-dimensional magnetic circuit, the stator core of each sub-sheet is equivalent to a two-dimensional magnetic circuit. The BH curve of the current sub-sheet is used to solve the magnetic reluctance in the sub-sheet. The same node position of the adjacent sub-sheets is connected by the connected magnetic reluctance, and the connected magnetic reluctance meter and the magnetic flux. The solution parameters of the entire three-dimensional magnetic circuit include the magnetic potential difference between the nodes in each sub-sheet and the magnetic potential difference between the same node positions of two adjacent sub-sheets. The solving by the magnetic circuit method can obtain the magnetic potential difference convergent solution of the three-dimensional magnetic circuit. The solving way of magnetic circuit method is the same as that of the two-dimensional magnetic circuit method, but the parameters of three-dimensional magnetic circuit include the magnetic reluctance of all sub-sheets and the connected magnetic reluctance between adjacent sub-sheets.

[0160] For the first sub-sheet mode, each sub-sheet has its own BH curve because of the different temperatures. When solving the entire nonlinear three-dimensional magnetic circuit, the magnetic circuit in each sub-sheet needs to be iterated according to their respective BH curve in the internal iterative process of the magnetic circuit, but the whole three-dimensional magnetic circuit (including the magnetic reluctance of the above meter and axial magnetic flux) is solved when solving. For the second sub-sheet mode, because the magnetic reluctance structure of each sub-sheet is different, the two-dimensional magnetic circuit structure of each sub-sheet may be different, but even if the magnetic circuit structure is different, the arrangement of the magnetic field node is the same, and the three-dimensional magnetic circuit is connected by the magnetic field node at the same position; this sub-sheet mode is also a solution to the entire three-dimensional magnetic circuit.

[0161] The detailed description of the iterative process of the three-dimensional magnetic circuit method is as follows: The external magnetic flux source injected into the magnetic circuit is fixed. The magnetic flux source (or magnetic field distribution) is obtained by the sub-domain method. The iterative process is essentially to find the permeability of each magnetic reluctance, permeability μ=B/H, B is the magnetic induction strength, H is the magnetic field strength. When the magnetic circuit method is solved, B and H of each sub-sheet are determined according to the BH curve. Different abscissas H will correspond to different permeability. At the beginning of the iteration, each magnetic reluctance is given an initial permeability, and the initial permeability is a random value. You can choose a larger value first, that is, assuming that it is not saturated at the beginning; then according to the magnetic flux source (magnetic field distribution) obtained by the sub-domain method and the initial permeability, the magnetic potential difference of each node is obtained, and then the magnetic field strength H on each magnetic reluctance is calculated. H corresponds to permeability μ, and it is required to judge if the difference between the permeability μ and the permeability μ (or initial permeability μ) of the previous iteration is within the allowable error range. If it is, the permeability is determined to be converged; if not, the permeability is used as the current permeability to repeat the iterative calculation until the permeability converges. After convergent permeability is obtained, the convergent node magnetic potential difference is obtained from permeability, and the node magnetic potential difference is converted into equivalent current density. The equivalent current density is entered in the sub-domain method as the boundary condition to obtain the current magnetic field distribution; the current magnetic field distribution is entered into the magnetic circuit method to continue iteration to obtain a new converging magnetic potential difference; repeat this process until a convergent current density is obtained.

[0162] FIG. 26 is a three-dimensional rotor model of permanent magnet block and oblique pole. According to the axial length sub-sheet of the permanent magnet 1, each sub-sheet uses a sub-domain method to solve the magnetic field distribution, and the status core (not shown in the figure) is solved by the magnetic circuit method based on a three-dimensional magnetic circuit.

[0163] FIG. 27 is a three-dimensional stator model including an axial ventilation slot 3. In the axial direction, there is a ventilation slot 3 between adjacent status cores 2. When performing axial sub-sheet, the stator core 2 and ventilation slot 3 are on different sub-sheets. The magnetic field distribution is solved by the sub-domain method for each sub-sheet, and the stator core is solved by the magnetic circuit method based on the three-dimensional magnetic circuit.

[0164] FIG. 28 is a schematic diagram of co-simulation between the magnetic field distribution solved by the present invention and the control algorithm provided by MATLAB.

[0165] The foregoing embodiments describe the technical solutions and beneficial effects of the present invention in detail. It should be understood that the above embodiments are only specific embodiments of the present invention and are not intended to limit the present invention. Any modification, supplement and equivalent replacement made shall fall within the scope of protection of the present invention.