A 2D meshless method for analyzing surface-mounted permanent magnet synchronous motors

Abstract

The present invention provides a method for analyzing a surface-mounted permanent magnet synchronous motor and its 2D mesh-free method, including discretizing the magnetic field region of the motor to be solved in a point-wise manner; based on Taylor expansion and the weighted least squares principle, the derivative values ​​of the vector magnetic potential of each discrete node can be approximated as a linear combination of the vector magnetic potential values ​​of each node in the support region; the partial differential equation is converted into an algebraic equation; the algebraic equation is solved to obtain the vector magnetic potential of each node, and the direction of the magnetic lines of force and the distribution of the magnetic flux density can be obtained. According to the electromagnetic calculation constraints of the motor, the back electromotive force of the motor winding, the electromagnetic torque and other parameters can be obtained; and finally, the results are compared with the finite element method results. The present invention is the first to perform mesh-free modeling analysis on a surface-mounted permanent magnet synchronous motor, and the provided solution can provide a reference study for this type of surface-mounted permanent magnet motor.

Description

Technical Field

[0001] The present invention relates to an analysis method of a surface-mounted permanent magnet synchronous motor and a 2D gridless method thereof, and belongs to the field of electromagnetic field calculation. Background Art

[0002] Permanent magnet motors (PMMs) are gaining increasing attention due to their high reliability, high efficiency, and high power density. Consequently, they are widely used in high-end applications such as electric vehicles and aerospace. Furthermore, the structure is widely used in surface-mounted PMMs due to its low torque ripple and relatively sinusoidal back EMF. Therefore, appropriate analysis methods are crucial, as they directly impact design efficiency and operational performance.

[0003] Mesh-based numerical methods, such as the finite difference method, the finite element method, and the boundary element method, are quite mature and have successfully solved many engineering problems. However, these methods are not perfect in all cases, especially when dealing with flows and deformations. In these cases, the mesh may be severely deformed, significantly affecting the accuracy of the solution. In this case, the mesh needs to be rebuilt, but reconstruction is difficult in complex areas. To address these problems, meshless methods have gradually developed. Unlike mesh-based methods, meshless methods only focus on node information. There are no connections between nodes, which facilitates the treatment of moving parts. Therefore, meshless methods have great potential and advantages in solving the transient magnetic field of rotor motors. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for analyzing the electromagnetic characteristics of a surface-mounted permanent magnet synchronous motor, which mainly includes discretizing the motor area to be solved in a point distribution manner, constructing the support area, and converting the partial differential equations satisfied by each area into algebraic equations for solution.

[0005] To achieve the above object, the present invention adopts a technical solution: a gridless method for analyzing a surface-mounted permanent magnet synchronous motor, comprising the following steps:

[0006] Step 1: Arrange points in various areas of the motor to be solved;

[0007] Step 2: Take any node as the center node and search for a certain number of nodes closest to the node to form a support area;

[0008] Step 3: Based on Taylor expansion and weighted least squares method, the residual function is constructed by approximating the derivative values ​​of each node as a linear combination of the function values ​​of each node in the support region.

[0009] Step 4, convert the partial differential equation satisfied by each node in the region into an algebraic equation;

[0010] Step 5: Process the nodes at the junction and the nodes at the boundary separately. The nodes at the junction need to meet the continuity conditions, while the nodes at the boundary need to meet the corresponding boundary conditions.

[0011] Step 6: According to steps 4 and 5, an algebraic equation can be obtained for each node, and the vector magnetic potential of each discrete node can be obtained by solving the algebraic equation group.

[0012] In step 7, based on the vector magnetic potential of each node solved in step 6, the direction of the magnetic lines of force and the magnetic density distribution inside the click can be obtained; according to the electromagnetic calculation constraints of the motor, the electromagnetic parameters such as the motor winding back electromotive force and electromagnetic torque can be calculated.

[0013] Furthermore, in step 1, points are arranged inside each region of the motor magnetic field to be solved, at the junction and at the boundary between two regions; the sub-regions to be solved include the stator, slots, air gap, and permanent magnets; since the rotor core of the surface-mounted permanent magnet synchronous motor is generally unsaturated, the second type of boundary conditions can be given directly on the outer surface of the rotor, thereby eliminating the need to solve the rotor core, thereby improving computational efficiency.

[0014] Furthermore, in step 2, it is necessary to use any node in the solution sub-region as the central node and find a certain number of adjacent nodes closest to the node to form a support region; all nodes constituting the support region must be in the same sub-region.

[0015] Furthermore, the specific process of step 3 is as follows: within the support region, a second-order Taylor expansion is performed at the central node for all nodes except the central node, and then the expression of the remainder is obtained and multiplied by the corresponding weight function to construct the corresponding residual function, and finally the corresponding algebraic equation is obtained according to the extreme value principle; solving the algebraic equation can express the derivative value of each node as a linear combination of the node function values ​​in the support region.

[0016] Furthermore, in step 4, the partial differential equations satisfied by each sub-region are converted into algebraic equations. The permanent magnet region and the air region satisfy the Laplace equation, the slot region satisfies the Poisson equation, and the stator core satisfies a two-dimensional nonlinear partial differential equation.

[0017] Furthermore, in step 5, for the nodes distributed at the junction of the two sub-areas, it is necessary to use the node as the central node to construct support areas in the two areas respectively, and then obtain the corresponding equations based on the magnetic field continuity conditions; for the nodes distributed on the rotor boundary, the second type of boundary conditions are met; and the nodes on the outer surface of the stator meet the first type of boundary conditions.

[0018] Furthermore, in step 6, all the obtained algebraic equations are combined to construct a set of algebraic equations; wherein the coefficient matrix G of the algebraic equations depends on the magnetic permeability, the node coordinates, and the weight function; and the source matrix S of the algebraic equations depends on the current density in the winding and the magnetization intensity of the magnet;

[0019] Furthermore, in step 7, the direction of the magnetic lines of force and the magnetic flux density of each node can be further solved based on the vector magnetic potential of each node obtained by solving the algebraic equation group; according to the electromagnetic calculation constraints of the motor, the magnetic flux flowing through the motor stator teeth at a moment in the electrical angle cycle can be obtained, and the next rotor position is recalculated to obtain the magnetic flux of each tooth in one electrical angle cycle, thereby obtaining the electromagnetic parameters such as the three-phase magnetic flux and induced electromotive force of the motor, which can be used to calculate the output torque of the motor if it is loaded.

[0020] The surface-mounted permanent magnet synchronous motor of the present invention is a three-phase motor with 12 slots and 10 poles, and is divided into four parts: a stator, an air gap, a rotor, and a rotating shaft. The stator includes a stator yoke, stator teeth, stator slots, and an armature winding. The armature slots are flat-bottomed slots, and the armature winding is wound in a centralized manner with a span of one stator tooth. The rotor is cylindrical, and a permanent magnet is attached to its surface. The permanent magnet material is neodymium iron boron with a grade of N42UH. The permanent magnet has a fan-shaped cross-section and is eccentrically processed and evenly distributed in the circumferential direction of the rotor. The material of the stator core and the rotor core is both silicon steel sheet DW310_35. The air gap is between the stator and the rotor, and the air gap thickness is 1.5 mm. The motor shaft is made of non-magnetic material, is solid cylindrical, and is coaxially connected to the rotor.

[0021] The present invention has the following beneficial effects:

[0022] 1. In modeling, the present invention does not require meshing like traditional numerical methods, but only requires point distribution; it can greatly simplify pre-processing work and has good adaptability to complex motor geometric structures;

[0023] 2. In the present invention, the coefficient matrix forming the algebraic equation system is sparse, which is beneficial to speeding up the CPU calculation speed and saving memory resources;

[0024] 3. The point density can be freely adjusted to balance calculation efficiency and accuracy. In areas with drastic magnetic field changes, the point density can be increased to improve accuracy, while in areas with gentle magnetic field changes, the point density can be reduced to improve calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 is a 2D structural diagram of the motor used in the present invention;

[0026] Figure 2 It is a schematic diagram of the solution area of ​​the present invention;

[0027] Figure 3 It is a schematic diagram of the interface of the present invention;

[0028] Figure 4 It is the magnetic field line distribution diagram given by the finite element software;

[0029] Figure 5 It is the magnetic force line distribution diagram provided by the present invention;

[0030] Figure 6 This is a schematic diagram comparing the single-phase no-load back EMF modeled by finite element software and mesh-free method.

[0031] Figure 7 Flowchart of the modeling method of the present invention. DETAILED DESCRIPTION

[0032] The technical solutions in the embodiments of the present invention will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present invention.

[0033] In order to more simply and clearly illustrate the beneficial effects of the present invention, a detailed description is given below in conjunction with a specific surface-mounted permanent magnet synchronous motor: Figure 1 The topological structure diagram of the motor is shown in FIG1 , in which 1-1 is the stator part of the motor, 1-2 is the rotor part, 1-3 is the stator slot, 1-4 is the permanent magnet, and 1-5 is the air gap. The embodiment of the present invention is a three-phase motor with 12 slots and 10 poles, which is divided into four parts: stator, air gap, rotor and rotating shaft. The stator includes a stator yoke, stator teeth, stator slots and armature windings. The armature slots are flat-bottomed slots. The armature windings are wound in a centralized manner with a span of one stator tooth. The rotor is cylindrical, with parallel magnetized permanent magnets attached to its surface. The permanent magnet material is neodymium iron boron, brand N42UH. The permanent magnets have a sector-shaped cross-section and are eccentrically processed, evenly distributed in the circumferential direction of the rotor. The stator core and rotor core are both made of silicon steel sheet DW310_35. The air gap between the stator and rotor has a thickness of 1.5mm. The motor shaft is made of non-magnetic material, is solid cylindrical, and is coaxially connected to the rotor.

[0034] like Figure 7 The flowchart shown is divided into the following steps:

[0035] Step 1: Arrange points in various areas of the motor to be solved.

[0036] Points are placed inside each region of the motor magnetic field to be solved, at the intersection and boundary between two regions; the sub-regions to be solved include the stator, slots, air gap, and permanent magnets; because the rotor core of a surface-mounted permanent magnet synchronous motor is generally unsaturated, the second type of boundary conditions can be given directly on the outer surface of the rotor, eliminating the need to solve the rotor core, thereby improving computational efficiency.

[0037] First, it is necessary to arrange points in the motor geometric area to be solved; Figure 1 As shown, the points need to be arranged in the 1-1 stator area, the 1-3 slot area, the 1-4 permanent magnet area, the 1-5 air gap area, and the boundaries of these areas; since the rotor of the surface-mounted permanent magnet motor is generally not saturated, it is not used as a solution area, and only boundary conditions need to be given on the outer surface of the rotor.

[0038] Step 2: Take any node as the center node and search for a certain number of nodes closest to the node to form a support area. All nodes forming the support area must be in the same sub-area.

[0039] Figure 2 Schematic diagram of the solution region; points are placed inside the region 1 and on the boundary 2. Any node in the solution region is selected as the central node 2-1, and then a certain number of surrounding nodes 2-2 closest to the node are searched to form a support region 2-3.

[0040] Step 3: Construct the residual function based on Taylor expansion and weighted least squares method, by approximating the derivative values ​​of each node as a linear combination of the function values ​​of each node in the support region.

[0041] The specific process is as follows: within the support region, a second-order Taylor expansion is performed at the central node for all nodes except the central node, and then the expression of the remainder is obtained and multiplied by the corresponding weight function to construct the corresponding residual function. Finally, the corresponding algebraic equation is obtained according to the extreme value principle. Solving the algebraic equation can express the derivative value of each node as a linear combination of the node function values ​​in the support region.

[0042] like Figure 2 As shown, in the support region 2-3, the Taylor expansion of the vector magnetic potential A of the N surrounding nodes 2-2 at the central node 2-1 is:

[0043]

[0044] Where A o Represents the central node vector magnetic potential value, A n The vector magnetic potential value of the nth node around the central node, Δx n =x n -x0 represents the difference between the horizontal coordinates of the surrounding nth node and the central node, Δy n =y n -y0 represents the difference between the vertical coordinates of the surrounding nth node and the central node, o(p 3 ) represents a high-order infinitesimal quantity, p=(Δx n 2 +Δy n2 ) 1 / 2 Represents the distance between two nodes. Then the residual function can be defined as:

[0045]

[0046] Where A o Represents the central node vector magnetic potential value, A n表示 The vector magnetic potential value of the nth node around the central node, Δx n =x n -x0 represents the difference between the horizontal coordinates of the surrounding nth node and the central node, Δy n =y n -y0 represents the difference d between the vertical coordinates of the surrounding nth node and the central node n =(Δx n 2 +Δy n 2 ) 1 / 2 represents the distance between two nodes, and w is the weight function; the weight function is a function of distance. If the surrounding nodes are farther away from the central node, the weight function value will be lower. Obviously, the residual function value should be as small as possible, so according to the extreme value principle:

[0047]

[0048] Where R represents the residual function defined by formula (2); this gives a set of algebraic equations:

[0049] FD=C (4)

[0050] Where:

[0051]

[0052] In the formula

[0053]

[0054] U=(A0 A1 A2……A N ) T

[0055] By solving equation (4), the central node (x o ,y o ) is expressed as a linear combination of the function values ​​of each node in the support region:

[0056] D 5×1 =(F 5×5 -1 E 5×(N+1) )U (N+1)×1 =K5×(N+1) U (N+1)×1 (5)

[0057] Step 4: Convert the partial differential equations satisfied by each node in the region into algebraic equations.

[0058] The partial differential equations satisfied by each sub-region are converted into algebraic equations, where the permanent magnet region and the breath region satisfy the Laplace equation, the slot region satisfies the Poisson equation, and the stator core satisfies a two-dimensional nonlinear partial differential equation.

[0059] Figure 1 The vector magnetic potential of the permanent magnet region 1-4 and the air gap region 1-5 satisfies the two-dimensional Laplace equation:

[0060]

[0061] Where A represents the vector magnetic potential function, x and y represent coordinate variables respectively; the discrete format of formula (6) can be written as:

[0062]

[0063] Where A i Indicates the vector magnetic potential value of the i-th node in the support area, the coefficient k 3,i+1 It represents the element in the 3rd row and the i+1th column of the matrix K in step 3 formula (5), the coefficient k 4,i+1 It represents the element in the 3rd row and the i+1th column of the matrix K in formula (5). The area 1-3 in the slot satisfies the Poisson equation:

[0064]

[0065] Where A represents the vector magnetic potential function, x and y represent coordinate variables, μ0 is the magnetic permeability in vacuum, and J is the current density in the slot. Similarly, its discrete format can be written as:

[0066]

[0067] Where A i Indicates the vector magnetic potential value of the i-th node in the support area, the coefficient k 3,i+1 It represents the element in the 3rd row and the i+1th column of the matrix K in step 3 formula (5), the coefficient k 4,i+1 It represents the element in the 3rd row and the i+1th column of the matrix K in formula (5), μ0 is the magnetic permeability in vacuum, and J is the current density in the slot. The stator core region satisfies the second-order nonlinear partial differential equation:

[0068]

[0069] Where A represents the vector magnetic potential function, x and y represent coordinate variables, and v is the magnetic permeability of the iron core. Similarly, its discrete format can be written as:

[0070]

[0071] Where A i represents the vector magnetic potential value of the i-th node in the support area, v i represents the permeability value of the i-th node in the support area, v0 represents the permeability value of the central node, and the coefficient k 1,i+1 It represents the element in the 1st row and the i+1th column of the matrix K in step 3 formula (5), and the coefficient k 2,i+1 It represents the element in the 2nd row and the i+1th column of the matrix K in formula (5), and the coefficient k 3,i+1 It represents the element in the 3rd row and the i+1th column of the matrix K in step 3 formula (5), the coefficient k 4,i+1 It represents the element in the 4th row and the i+1th column of the matrix K in formula (5).

[0072] Step 5: Process the nodes at the junction and the nodes at the boundary separately. The nodes at the junction need to meet the continuity conditions, while the nodes at the boundary need to meet the corresponding boundary conditions.

[0073] For nodes distributed at the junction of two sub-areas, it is necessary to use this node as the central node to construct support areas in the two areas respectively, and then obtain the corresponding equations based on the magnetic field continuity conditions; for nodes distributed on the rotor boundary, the second type of boundary conditions are met; and the nodes on the outer surface of the stator meet the first type of boundary conditions.

[0074] Since the derivative of the vector magnetic potential is discontinuous at the junction of the two regions, the nodes distributed there need to be processed according to the continuity condition. At the junction, the vector magnetic potential and the tangential magnetic field strength are continuous. Since the nodes distributed at the junction belong to both regions at the same time, the vector magnetic potential continuity is automatically satisfied, and only the tangential magnetic field continuity condition needs to be considered. First, as Figure 3 As shown, for the node 3 at the junction, it is necessary to construct support area 3-1 and support area 3-2 in the two areas respectively. Then, according to step 4, the derivative values ​​of the node at the junction in the two areas are expressed as linear combinations of the function values ​​of each point in the two support areas. Finally, the expression of the continuity condition is discretized.

[0075] The vector magnetic potential at the junction of two permanent magnets satisfies the following relationship:

[0076]

[0077] Where n x and n yare the x-direction component and y-direction component of the unit tangent vector at the junction, A I represents the vector magnetic potential function in the first permanent magnet, A II represents the vector magnetic potential function in the second permanent magnet, μ is the magnetic permeability of the permanent magnet, H cx and H cy are the x-direction and y-direction components of the permanent magnet's coercive force, respectively. According to step 4, the derivative of (12) is expressed as a linear combination of the function values ​​in the support region, so the discrete format can be expressed as:

[0078]

[0079] Where A i I A represents the magnetic potential value of the i-th node in the first permanent magnet. i II represents the vector magnetic potential value of the i-th node in the second permanent magnet, μ is the magnetic permeability of the permanent magnet, H cx and H cy are the x-direction and y-direction components of the permanent magnet’s coercive force, represents the element in the 1st row and the i+1th column of the matrix K determined by formula (5) in the first permanent magnet region, represents the element in the 2nd row and the i+1th column of the matrix K determined by formula (5) in the first permanent magnet region, represents the element in the 1st row and the i+1th column of the matrix K determined by formula (5) in the second permanent magnet region, represents the element in the 2nd row and the i+1th column of the matrix K determined by formula (5) in the second permanent magnet region. At the junction of the permanent magnet and the air gap region, the vector magnetic potential satisfies the following relationship:

[0080]

[0081] Where A PM represents the vector magnetic potential function in the permanent magnet, A air represents the vector magnetic potential function in the air gap, μ PM is the magnetic permeability of the permanent magnet, μ0 is the magnetic permeability of air, H cx and H cy are the x-direction and y-direction components of the permanent magnet's coercive force, respectively. Its discrete format can be written as:

[0082]

[0083] A i PM A represents the magnetic potential value of the i-th node in the permanent magnet. i air Represents the vector magnetic potential value of the i-th node in the air gap, μPM is the magnetic permeability of the permanent magnet, μ0 is the magnetic permeability of air, H cx and H cy are the x-direction and y-direction components of the permanent magnet’s coercive force, It represents the element in the 1st row and the i+1th column of the matrix K determined by formula 5 in the permanent magnet region. It represents the element in the 2nd row and i+1th column of the matrix K determined by formula 5 in the permanent magnet region. It represents the element in the 1st row and the i+1th column of the matrix K determined by formula 5 in the air gap area. It represents the element in the 2nd row and i+1th column of the matrix K in the air gap region determined by formula 5. At other intersections, the vector magnetic potential satisfies the following relationship:

[0084]

[0085] Among them A I and A II are the vector magnetic functions in regions I and II, n x and n y are the unit tangent vectors at the junction, μ1 and μ2 are the magnetic permeabilities of the two solution areas. The discrete format of Equation (16) can be expressed as:

[0086]

[0087] Where n x and n y are the x-direction component and y-direction component of the unit tangent vector at the junction, A i I is the vector magnetic potential value of the i-th node in region I, A i II is the vector magnetic potential value of the ith node in region II, is the element in row 1 and column i+1 of the matrix K determined by formula (5) in region I, k2 I ,i+1 is the element in the 2nd row and the i+1th column of the matrix K determined by formula (5) in region I, is the element in row 1 and column i+1 of the matrix K determined by formula (5) in region II, is the element in the 2nd row and the i+1th column of the matrix K determined by formula (5) in region II; the boundary conditions can be directly given for the nodes at the boundary. The first type of boundary condition is given on the outer surface of stator 1-1:

[0088] A stator =0 (18)

[0089] Where A statorRepresents the vector magnetic potential value on the outer surface of the stator. For the nodes distributed on the outer surface of rotor 1-2, the second type of boundary conditions can be given:

[0090]

[0091] where n is the boundary unit tangent vector, μ pm It represents the magnetic permeability of permanent magnets, A rotor Represents the vector magnetic potential function of the rotor outer surface, H c Its discrete format can be expressed as:

[0092]

[0093] Where n x and n y are the x- and y-direction components of the unit tangent vector of the rotor outer surface, μ pm Indicates the magnetic permeability of the permanent magnet, A rotor,i Represents the magnetic potential value of the node vector on the outer surface of the rotor, H cx and H cy are the x-direction and y-direction components of the permanent magnet’s coercive force, k 1,i+1 is the element in the 1st row and the i+1th column of the matrix K determined by formula (5), k 2,i+1 is the element in the 2nd row and the i+1th column in the matrix K determined by formula (5).

[0094] Step 6: According to steps 4 and 5, an algebraic equation can be obtained for each node, and the vector magnetic potential of each discrete node can be obtained by solving the algebraic equation group.

[0095] For all the obtained algebraic equations, a set of algebraic equations is constructed by combining them; the coefficient matrix G of the algebraic equations depends on the magnetic permeability, node coordinates, and weight function; the source matrix S of the algebraic equations depends on the current density in the winding and the magnetization intensity of the magnet.

[0096] According to steps 4 and 5, each node can get an algebraic equation. Combining them can get a matrix equation system:

[0097] GA=S (21)

[0098] Where G is the coefficient matrix, which depends on the node coordinates, weight function and magnetic permeability; A is the column vector of the vector magnetic potential variables to be solved for each node, and S is the source matrix, which depends on the current density and the magnetization intensity of the permanent magnet. Since the magnetic permeability in the iron core is not constant, the equation group (21) is a nonlinear algebraic equation. The successive linearization method can be used to solve equation (21), and its iterative format is:

[0099]

[0100] Where A k is the vector magnetic potential array vector obtained by the k-th calculation, A k-1 is the vector magnetic potential vector obtained by the k-1th calculation, f1 and f2 are relaxation factors used to control the convergence speed, v k is the permeability value used in the kth calculation, v k-1 is the permeability value used in the k-1 calculation, v k-1 (B k-1 ) is the magnetic density B obtained from the k-1th calculation k-1 The magnetic permeability value obtained from the BH curve.

[0101] In step 7, based on the vector magnetic potential of each node solved in step 6, the direction of the magnetic lines of force and the magnetic density distribution inside the click can be obtained; according to the electromagnetic calculation constraints of the motor, the electromagnetic parameters such as the motor winding back electromotive force and electromagnetic torque can be calculated.

[0102] Through step 6, the vector magnetic potential of each node can be obtained, and the magnetic flux density of each node can be further calculated through the vector magnetic potential; the direction of the magnetic lines of force can be obtained by drawing the contour lines of the vector magnetic potential; through the node vector magnetic potential difference, the magnetic flux flowing through the motor stator teeth at a moment in the electrical angle cycle can be obtained, and the next rotor position is recalculated to obtain the magnetic flux linkage of each tooth in one electrical angle cycle, and the electromagnetic parameters such as the three-phase magnetic flux and induced electromotive force of the motor can be obtained based on this. If it is under load, it can be used to calculate the output torque of the motor. In order to verify that the 2D gridless analysis method for analyzing surface-mounted permanent magnet synchronous motors proposed in this invention is accurate and reliable, the results obtained using finite element commercial software are compared and verified.

[0103] Figure 4 The magnetic field line distribution diagram given by finite element; Figure 5 The magnetic field line distribution given by the meshless method is very close. Figure 6 The results of the back EMF were compared, where A1 is the waveform obtained by the finite element software and A2 is the waveform obtained by the meshless modeling method. It can be found that the results given by this method are basically consistent with those of the commercial finite element software.

[0104] In summary, the present invention provides a 2D gridless method for an analytical surface-mounted permanent magnet synchronous motor, which includes discretizing the magnetic field region to be solved for the motor in the form of points; for each discrete node in the region, finding several points closest to the node to form a support region, and based on Taylor expansion and weighted least squares principle, the derivative values ​​of the vector magnetic potential of each discrete node can be approximated as a linear combination of the vector magnetic potential values ​​of each node in the support region; in this way, the partial differential equation satisfied by the vector magnetic potential can be converted into an algebraic equation; and for the nodes at the junction of the two regions, since the vector magnetic potential derivative at the junction is discontinuous, additional processing is required according to the continuity condition; the nodes at the boundary will satisfy the corresponding boundary conditions; performing such operations on the nodes can obtain a set of algebraic equations, and the number of algebraic equations is equal to the number of discrete nodes; solving the algebraic equations can obtain the vector magnetic potential of each node, and then the direction of the magnetic lines and the distribution of the magnetic flux can be obtained, and according to the electromagnetic calculation constraints of the motor, the parameters such as the back electromotive force and electromagnetic torque of the motor winding can be obtained, and finally compared with the finite element results. This invention is the first to conduct meshless modeling analysis on a surface-mounted permanent magnet synchronous motor, and the provided solution can provide a reference study for this type of surface-mounted permanent magnet motor.

[0105] Although the present invention has been disclosed above with reference to preferred embodiments, the embodiments are not intended to limit the present invention. Any equivalent changes or modifications made without departing from the spirit and scope of the present invention shall fall within the scope of protection defined by the appended claims of this application.

Claims

1. A 2D meshless method for analyzing surface-mounted permanent magnet synchronous motors, characterized in that: The following steps are involved: Step 1: Arrange points in various areas of the motor to be solved; Step 2: Take any node as the center node and search for a certain number of nodes closest to the node to form a support area; Step 3: Based on Taylor expansion and weighted least squares method, the residual function is constructed by approximating the derivative values ​​of each node as a linear combination of the function values ​​of each node in the support region. Step 4, convert the partial differential equation satisfied by each node in the region into an algebraic equation; Step 5: Process the nodes at the junction and the nodes at the boundary separately. The nodes at the junction need to meet the continuity conditions, while the nodes at the boundary need to meet the corresponding boundary conditions. In step 5, for the nodes distributed at the junction of the two sub-regions, it is necessary to use the node as the central node to construct support regions in the two regions respectively, and then obtain the corresponding equations based on the magnetic field continuity condition; for the nodes distributed on the rotor boundary, the second type of boundary conditions are met; and the nodes on the outer surface of the stator meet the first type of boundary conditions; Step 6: According to Step 4 and Step 5, each node obtains an algebraic equation, and the vector magnetic potential of each discrete node is obtained by solving the algebraic equations. In step 6, all the obtained algebraic equations are combined to construct a set of algebraic equations; wherein the coefficient matrix G of the algebraic equations depends on the magnetic permeability, the node coordinates, and the weight function; and the source matrix S of the algebraic equations depends on the current density in the winding and the magnetization intensity of the magnet; Step 7: Based on the vector magnetic potential of each node solved in step 6, the direction of the magnetic lines of force and the magnetic density distribution inside the click are obtained; according to the electromagnetic calculation constraints of the motor, the electromagnetic parameters such as the back electromotive force and electromagnetic torque of the motor winding are calculated; In step 7, the direction of the magnetic lines of force and the magnetic flux density of each node are further solved based on the vector magnetic potential of each node obtained by solving the algebraic equation group; according to the electromagnetic calculation constraints of the motor, the magnetic flux flowing through the motor stator teeth at a moment in the electrical angle cycle is obtained, and the next rotor position is recalculated to obtain the magnetic flux Φ of each tooth in one electrical angle cycle. Based on this, the electromagnetic parameters such as the three-phase magnetic flux and induced electromotive force of the motor are obtained, which are used to calculate the output torque of the motor if it is loaded.

2. A 2D meshless method for analyzing a surface-mounted permanent magnet synchronous motor according to claim 1, characterized in that: In step 1, points are arranged inside each region of the motor magnetic field to be solved, at the intersection and boundary between two regions; the sub-regions to be solved include the stator, slots, air gap, and permanent magnets; since the rotor core of the surface-mounted permanent magnet synchronous motor is not saturated, it is only necessary to give boundary conditions on the outer surface of the rotor, so there is no need to solve the rotor core, so as to improve calculation efficiency.

3. A 2D meshless method for analyzing a surface-mounted permanent magnet synchronous motor according to claim 1, characterized in that: In step 2, it is necessary to use any node in the solution sub-region as the central node and find a certain number of adjacent nodes closest to the node to form a support region. All nodes constituting the support region must be in the same sub-region.

4. A 2D meshless method for analyzing a surface-mounted permanent magnet synchronous motor according to claim 1, characterized in that: The specific process of step 3 is as follows: within the support region, a second-order Taylor expansion is performed at the central node for all nodes except the central node, and then the expression of the remainder is obtained and multiplied by the corresponding weight function to construct the corresponding residual function. Finally, the corresponding algebraic equation group is obtained according to the extreme value principle, and the algebraic equation is solved to express the derivative value of each node as a linear combination of the node function values ​​in the support region.

5. The 2D meshless method for analyzing a surface-mounted permanent magnet synchronous motor according to claim 1, characterized in that: In step 4, the partial differential equations satisfied by each sub-region are converted into algebraic equations, wherein the permanent magnet region and the air region satisfy the Laplace equation, the slot region satisfies the Poisson equation, and the stator core satisfies the two-dimensional nonlinear partial differential equation.

6. A surface-mounted permanent magnet synchronous motor implemented according to the method of claim 1, characterized in that: The surface-mounted permanent magnet synchronous motor is a three-phase motor with 12 slots and 10 poles, divided into four parts: stator, air gap, rotor and rotating shaft; the stator includes a stator yoke, stator teeth, stator slots and armature windings, the armature slots are flat-bottomed, and the armature windings are wound in the stator slots in a centralized manner with a span of one stator tooth; the rotor is cylindrical, with parallel magnetized permanent magnets attached to its surface, the permanent magnet material is neodymium iron boron with a grade of N42UH, the cross-section of the permanent magnet is fan-shaped and eccentrically processed, and is evenly distributed in the circumferential direction of the rotor; the material of the stator core and the rotor core is both silicon steel sheet DW310_35; the air gap is between the stator and the rotor, and the air gap thickness is 1.5 mm; the motor shaft is made of non-magnetic material, is solid cylindrical, and is coaxially connected to the rotor.

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