Permanent magnet synchronous motor heterogeneous grid adaptive high-precision magnetic network modeling method

Abstract

The invention relates to a permanent magnet synchronous motor heterogeneous grid adaptive high-precision magnetic network modeling method, and belongs to the technical field of permanent magnet synchronous motor electromagnetic analysis. The method comprises the following steps: extracting a single-slot or single-pole minimum periodic unit of a motor structure, carrying out symmetric fan-shaped grid division, accurately representing the boundary of a heterogeneous material by utilizing a polar coordinate function, and calculating magnetic conductance based on a geometric overlapping principle; the permanent magnet and the stator winding magnetomotive force are fused through unified matrix modeling, and the stability of nonlinear solution is improved in combination with a Newton-Raphson iterative algorithm. According to the method, the modeling process is simplified, the preprocessing time and the calculation time are greatly shortened compared with those of a finite element method, and the efficiency of multi-working-condition iterative optimization is remarkably improved on the premise that high precision is guaranteed.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic analysis technology for permanent magnet synchronous motors, specifically involving a high-precision magnetic network modeling method for permanent magnet synchronous motors with heterogeneous mesh adaptive modeling. This method balances modeling accuracy and efficiency and is suitable for electromagnetic performance prediction and multi-condition iterative analysis scenarios of permanent magnet synchronous motors. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs), with their comprehensive advantages of high torque density, high operating efficiency, low noise, and long service life, have become core components in high-performance applications such as electric vehicle drives, industrial servo systems, and wind power generation. In these fields, accurate calculation of the motor's internal electromagnetic field is crucial for evaluating its efficiency, torque ripple, vibration noise, and verifying the rationality of the design scheme. Currently, the mainstream electromagnetic analysis methods mainly include three categories: analytical methods, finite element analysis methods, and equivalent magnetic network methods. Each method has its applicable scenarios and characteristics, collectively forming a complete toolbox for motor design and analysis.

[0003] However, magnetic field analysis of permanent magnet synchronous motors (especially those with complex rotor topologies such as embedded ones) faces a series of challenges. The geometric asymmetry of the rotor structure brings significant difficulties to the mathematical description of boundary conditions, limiting the direct application of classical analytical models. Although the finite element method can simulate multiphysics coupling problems with high accuracy through fine mesh generation, its computational cost is very high. Especially in the early stages of design, which requires iterative optimization under multiple parameters and operating conditions, the sensitivity to complex laminated structures and nonlinear material properties leads to long computation times, thus significantly extending the development cycle.

[0004] The equivalent magnetic network method treats the internal magnetic circuit of a motor as a topological network composed of magnetic permeability and magnetomotive force sources, and solves for the magnetic flux distribution by establishing nodal magnetic potential equations. Its advantages are that it does not require dense meshing like the finite element method, resulting in faster calculation speed. At the same time, it can consider nonlinear factors such as core magnetic saturation effect, complex leakage magnetic paths, and rotor dynamic processes in greater detail than traditional analytical magnetic circuit methods.

[0005] However, the equivalent magnetic network method may fall short of the accuracy of the finite element method. Currently, the main approach is to change the mesh shape, but accurate calculation requires a finer mesh, which significantly increases the solution time. How to enable the equivalent magnetic network to achieve the accuracy of the finite element method in a shorter time is an urgent technical problem to be solved. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing electromagnetic modeling methods for permanent magnet synchronous motors (PMSMs) and provide a high-precision magnetic network modeling method for PMSMs that is adaptive to heterogeneous meshes, specifically addressing the following technical problems: 1. Simplify the modeling process, reduce preprocessing workload, and avoid the tedious operation of manually defining nodes; 2. Improve the accuracy of magnetic permeability calculation in heterogeneous material regions without relying on mesh refinement; 3. Effectively integrates rotor skew effect, improving the prediction accuracy of parameters such as torque and air gap magnetic flux density; 4. Optimize the iterative solution algorithm to enhance the numerical stability in the magnetic saturation region and avoid solution divergence; 5. While maintaining accuracy comparable to the finite element analysis method, it significantly reduces computational costs and adapts to the needs of iterative optimization under multiple operating conditions.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A high-precision magnetic network modeling method for heterogeneous mesh adaptation in permanent magnet synchronous motors, the modeling method comprising the following: The minimum periodic unit of the stator (single stator slot) and the minimum periodic unit of the rotor (single pole) are determined. The minimum periodic units are then divided using a sector-shaped grid structure to form a structured grid. Each sector-shaped grid unit is equivalent to two perpendicular magnetic permeabilities, defined as tangential magnetic permeability. G t and radial permeability G r ; Adaptive magnetic permeability calculation of heterogeneous material region: First, input the basic position information of the smallest periodic unit of the research object, construct the geometric boundary mathematical description in polar coordinates, define the fan-shaped grid traversed by the heterogeneous contour of the smallest periodic unit as heterogeneous grid, and the rest of the grids are uniform grids. All heterogeneous grids constitute heterogeneous material region. The heterogeneous material region contains two materials: silicon steel and non-silicon steel. The radial and tangential magnetic permeabilities are calculated first within each heterogeneous mesh, and then the overall radial and tangential magnetic permeabilities of the entire sector mesh cell are calculated. The radial and tangential magnetic permeabilities of the air domain are subtracted from the overall radial and tangential magnetic permeabilities to obtain the radial and tangential magnetic permeabilities of the silicon steel material within the heterogeneous mesh. By comparing the relative magnitudes of the radial and tangential magnetic permeability of silicon steel materials within the heterogeneous mesh with those of the air domain, mesh optimization and partitioning are automatically completed based on the principle of "merging materials with small magnetic permeability and retaining materials with large magnetic permeability," and a cell model is established. Finally, the Matlab program automatically generates a global model based on the element model through rotation and copying, and applies periodic boundary conditions and Dirichlet boundary conditions to ensure the continuity and consistency of the physical field; the Matlab program is used to calculate the magnetic permeability of the heterogeneous material region of the motor. The adaptive magnetic permeability calculation results are used to perform an iterative solution using Newton-Raphson.

[0008] Furthermore, the principle of "merging small magnetic permeability materials and retaining large magnetic permeability materials" specifically refers to: grouping the silicon steel material and the air domain with relatively small magnetic permeability into the adjacent fan-shaped grid area of ​​the same material, while retaining the other relatively large magnetic permeability material in the original heterogeneous grid, thereby achieving the homogenization of materials within the heterogeneous grid.

[0009] In this invention, the non-silicon steel material is used for the stator as an air-domain material and the rotor as a permanent magnet material.

[0010] Furthermore, the magnetic permeability of adjacent materials is classified as follows: specifically, the composite tangential magnetic permeability viewed tangentially. G Yt This represents the parallel configuration of the two tangential magnetic permeabilities before synthesis; from a radial perspective, the synthesized radial magnetic permeability... G Yr This is a series configuration of two radial magnetic permeators before synthesis.

[0011] Furthermore, the specific process of iteratively solving using Newton-Raphson methods based on the adaptive magnetic permeability calculation results is as follows: The Newton-Raphson method is used for iterative solution, and a variable Newton damping factor α is introduced. k To adaptively adjust the iteration step size, α k The pattern of change is as follows: (15) in, α base It is the basic step size. B s It is the inflection point magnetic flux density of the BH curve of silicon steel material. B sat It is the adjustment coefficient; In each iteration step k In evaluating nonlinear residuals F ( x k ) and Jacobi matrix J ( x k The solution is updated, and if entry into the saturation region is detected, the variable damping factor is adaptively adjusted. α k To ensure numerical stability, iteration continues until the solution satisfies the convergence criterion. x k+1 - x k |< ε ,in ε It is a user-defined tolerance.

[0012] Furthermore, by introducing a rotor skewed pole structure into the motor design, and simulating the electromagnetic torque by segmenting along the axial direction and applying a phase shift, the integral equation is: (18) In the formula, m Represents the number of oblique pole segments, φ m It is the polar angle of each axial section. K Indicates the number of skew poles. B t,m and B r,m The tangential and radial magnetic flux density under each skew pole segment are given. The magnetic flux density is obtained by iteratively solving the equivalent magnetic network method based on the adaptive magnetic permeability calculation results. B t,m ( t )and B r,m ( t (This is time) t The function; R outer and R inner These represent the outer radius and inner radius of the sector grid cell, respectively; θ Indicates the central angle of a sector grid cell; μ 0 It is the permeability of vacuum; L d It is the effective axial length; T It is electromagnetic torque.

[0013] Compared with the prior art, the beneficial effects of the present invention are: 1. This invention extracts the smallest periodic elements of the motor stator and rotor separately for individual modeling, and then utilizes the periodic symmetry of the motor structure to quickly generate a global model through rotational replication. This method avoids repetitive modeling of the global model, significantly reducing preprocessing time. Regarding mesh generation, this method borrows advanced mesh generation techniques, automatically generating meshes that better match the magnetic field distribution in key areas. This significantly reduces the number of mesh nodes compared to traditional finite element methods while maintaining accuracy, resulting in a substantial improvement in modeling efficiency.

[0014] 2. The equivalent magnetic network method discretizes the continuous magnetic field problem into a circuit network problem composed of magnetic permeability. The scale of the unknown magnetic potential to be solved is much smaller than the nodal degrees of freedom in the finite element method. Furthermore, the resulting system matrix is ​​a sparse matrix, which is highly beneficial for efficient solution, significantly reducing computation time. In addition, this method typically exhibits good convergence when dealing with nonlinear magnetic materials, avoiding potential divergence problems in the magnetic saturation region, and demonstrating good numerical stability.

[0015] 3. The method of this invention can accurately handle complex structures such as heterogeneous material boundaries and rotor skewed poles. For rotor skewed poles, they can be simulated by segmenting along the axial direction and applying appropriate phase offsets, without the need for complex three-dimensional global modeling. Furthermore, this method is specifically optimized for the deep magnetic saturation region that may occur in permanent magnet motors, ensuring robustness and stability during iterative optimization under various operating conditions.

[0016] 4. In terms of data, when using the same processor, the number of meshes in the finite element method is 30799, while the equivalent magnetic network method in this invention has only 5400. The equivalent magnetic network method proposed in this invention has an error of no more than 5% in terms of calculation accuracy compared with the finite element method, and the calculation time is only 30% of that of the finite element method. While improving accuracy, it greatly shortens the time of electromagnetic simulation of motors. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of the structure of the built-in permanent magnet synchronous motor used in the embodiments of the present invention; Figure 2 This is a structural diagram of a sector mesh element in the equivalent magnetic network method; Figure 3 The node connection diagram for the smallest periodic unit; Figure 4 This is a model diagram of the stator before its equivalent operation. Figure 5 This is a diagram of the heterogeneous boundary model before equivalence. Figure 6 This is an equivalent treatment diagram for non-homogeneous media in the air domain; Figure 7 This is a diagram of the equivalent heterogeneous boundary model; Figure 8 This is the node connection diagram for the half-motor model; Figure 9 Node connection diagram of the motion boundary of the semi-electric model Figure 10 This is a diagram showing the stator winding connection of a permanent magnet synchronous motor. Figure 11 This is a flowchart of the iterative process of the equivalent magnetic network method. Figure 12 Comparison of simulation waveforms of unloaded air gap magnetic flux density; Figure 13 Comparison of simulated waveforms of air gap magnetic flux density under load; Figure 14 Comparison of simulated waveforms without considering rotor skewed pole electromagnetic torque; Figure 15 Comparison of simulated electromagnetic torque waveforms considering rotor skew poles. Detailed Implementation

[0018] The present invention will be further explained below with reference to the embodiments and accompanying drawings, but this is not intended to limit the scope of protection of this application.

[0019] This invention focuses on the processing effect of heterogeneous boundaries. It performs sector-shaped meshing of the smallest periodic unit of the motor research object. Processing the heterogeneous mesh only requires handling one smallest periodic unit; the rest can be completed through rotational replication. Furthermore, considering the influence of rotor skew, which inevitably leads to accuracy issues when processing motors with skewed rotors, this invention incorporates rotor skew into the equivalent magnetic network method. It also makes the Newtonian damping factor vary according to a set function, which significantly improves convergence and fully considers the nonlinearity of silicon steel sheets.

[0020] Existing equivalent magnetic network modeling methods are not applicable when dealing with irregular boundaries of stator slots, and their applicability is limited. However, the heterogeneous boundary processing of this invention can be applied to all boundary conditions, and the stator and rotor generate their own meshes. Except for boundary processing, the fan-shaped mesh can be automatically generated. The advantage of this method is that it assumes that the meshes at all locations are consistent, and then processes the heterogeneous parts separately. The heterogeneous parts only need to be automatically calculated by inputting the position information in polar coordinates according to the formula, which simplifies the preprocessing process and greatly reduces the mesh preprocessing time.

[0021] This invention employs heterogeneous boundary adaptive processing, such as heterogeneous processing of the stator slot area, enabling the equivalent magnetic network to achieve the accuracy of the finite element method in a shorter time. It provides an excellent balance between computational efficiency and accuracy, improving the accuracy of the equivalent magnetic network method and significantly shortening the solution time compared to the finite element method. This allows researchers to quickly adjust motor dimensions and predict magnetic flux distribution with minimal computational error, greatly reducing the time required for electromagnetic analysis. It is particularly suitable for extensive early-stage optimization iterations and system-level simulations.

[0022] Example 1 This embodiment presents a high-precision magnetic network modeling method for heterogeneous mesh adaptive permanent magnet synchronous motors, comprising the following steps: Step 1: Generation of symmetric sector mesh and generation of minimum periodic element mesh The smallest periodic unit of the stator and the smallest periodic unit of the rotor are identified, and then divided using a sector-shaped mesh structure to form a structured mesh. When using a sector-shaped mesh structure, each sector-shaped mesh unit is equivalent to two perpendicular magnetic permeabilities, defined as tangential magnetic permeability. G t and radial permeability G r .

[0023] Step 2: Adaptive magnetic permeability calculation of heterogeneous material regions The Matlab program was used to calculate the magnetic permeability of heterogeneous material regions such as stator slots and rotor poles in motors. The process is automated and only requires basic position information. Taking the stator as an example, the region containing the stator slots is the smallest periodic unit. First, the basic position information of the stator slots is input, and a mathematical description of the geometric boundary in polar coordinates is constructed. The sector-shaped mesh traversed by the heterogeneous contour of the smallest periodic unit is defined as the heterogeneous mesh, while the remaining meshes are uniform meshes. All heterogeneous meshes constitute the heterogeneous material region. Within each heterogeneous grid, the permeability decomposition and series / parallel synthesis of irregular shapes are autonomously performed to calculate the permeability in the air domain.

[0024] By analyzing the series and parallel magnetic circuits, the equivalent magnetic permeability of silicon steel material can be automatically derived from the overall magnetic permeability.

[0025] Subsequently, the magnetic permeability values ​​of the two materials within each heterogeneous grid are compared. Based on the principle of "merging materials with small magnetic permeability and retaining materials with large magnetic permeability", the grid optimization and partitioning are automatically completed, which can significantly improve the magnetic permeability calculation accuracy of the region without the need for grid refinement.

[0026] Finally, the Matlab program automatically generates a global model based on the element model through rotation and replication, and applies periodic boundary conditions and Dirichlet boundary conditions to ensure the continuity and consistency of the physical field.

[0027] The aforementioned adaptive magnetic permeability calculation process embeds complex modeling, calculation, and processing decision-making processes into the algorithm, achieving fully automated and high-precision solutions from geometric input to magnetic field analysis.

[0028] Step 3: Unified matrix modeling and optimization iterative solution algorithm Based on Kirchhoff's laws, mathematical relationships are established among the magnetic permeability flux matrix, nodal magnetomotive force matrix, and magnetic flux matrix of the magnetic network nodes. The total magnetomotive force is formed by the vector superposition of the permanent magnet magnetomotive force and the stator winding current excitation magnetomotive force. The Newton-Raphson method is used for iterative solution, and a variable damping factor is introduced to adaptively adjust the iteration step size, enhancing the numerical stability in the magnetic saturation region.

[0029] Step 4: Integration of rotor skew effect and torque calculation The rotor global model obtained from steps 2-3 is segmented along the axial direction, and a phase offset angle is applied to each segment. The air gap magnetic flux density after pole skew is calculated using the magnetic flux superposition method. The electromagnetic torque is solved using the Maxwell stress tensor method, and the pole skew magnetic flux is substituted into the electromagnetic torque integral equation to improve the torque prediction accuracy.

[0030] Example 2 This embodiment is combined with the appendix Figure 1 The 18-slot 8-pole V-type built-in permanent magnet synchronous motor in the invention provides a detailed explanation of the specific implementation process. Figure 1 The motor described herein consists of a stator 1, windings 2 wound around the stator, and a rotor 3 coaxially arranged with the stator. The rotor has a built-in permanent magnet 4, which together constitute the design area.

[0031] The specific process of the high-precision magnetic network modeling method for heterogeneous mesh adaptation of permanent magnet synchronous motor in this embodiment is as follows: Step 1: Sector mesh generation and motor minimum cycle unit division A sector-shaped grid structure is adopted, where each sector grid cell is equivalent to two perpendicular magnetic permeabilities, defined as tangential magnetic permeabilities. G t and radial permeability G r ,like Figure 2 As shown, the tangential magnetic permeability in the sector grid cell G t and radial permeability G r The calculation formula is formula (1): (1) in, μ 0 It is the permeability of vacuum; μ r It is the relative permeability; L d It is the effective axial length; R It is the radius of the inner diameter of the sector element; h It is the radial length. θ It is the central angle opposite to the sector.

[0032] Based on the inherent radial symmetry of the stator and rotor of the permanent magnet synchronous motor, the minimum periodic elements of the stator and rotor are selected separately for modeling, forming their respective minimum modeling units. The combination of the minimum periodic elements of the stator and rotor can reflect the basic electromagnetic characteristics of the motor and significantly reduce the computational scale. Subsequently, the two minimum modeling units are divided into sector meshes. The sector mesh lines are distributed along the circumferential and radial directions, which naturally fits the radial geometric contour and magnetic field path of the motor. The generated mesh nodes naturally form the basis of the element model.

[0033] Based on this, the number of rows in the sector grid is explicitly defined ( i This typically indicates the number of radial layers (e.g., the number of layers from the rotor's inner diameter to the stator's outer diameter) and the number of columns ( b This typically represents the tangential segmentation number, which is the number of mesh segments in the circumferential direction of a minimum modeling unit, thus forming a structured mesh. i × b A mesh array, where the intersections of the radial and circumferential directions are called nodes, such as... Figure 3 As shown.

[0034] Each grid can be further equivalent to a series or parallel combination of tangential and radial magnetic permeability, providing a discretization basis for subsequent magnetic network analysis. Through precise control... i and b The value can effectively balance computational accuracy and efficiency while ensuring the grid density in critical areas.

[0035] Step 2: Adaptive magnetic permeability calculation of heterogeneous material regions Taking the stator region as an example, the boundaries of different material regions within the smallest periodic unit of the stator are first accurately described by mathematical functions in polar coordinates. Equation (2) is the radial coordinate of the stator slot boundary with respect to the polar angle. θ and radius r These analytical expressions can accurately characterize the complex contours of stator slots, such as pear-shaped slots or parallel tooth slots.

[0036] (2) In the formula, L(r,θ) and C(r,θ) Represent the polar coordinate representation of the stator slot boundary under straight lines and circular regions, respectively; r k , θ k ) represent the node radius and angle, respectively. k It can take the value 1 or 2. r c Indicates the radius of the circular region, ( r o ,θ o () represents the center of the circle. θ t This indicates the angle corresponding to the endpoint of the semicircular region.

[0037] The angular relationship of boundary functions in polar coordinates can be precisely described by formula (3): (3) In the formula θ C This represents the angle of the circular region of the stator slot in polar coordinates. θ L This represents the angle of the straight region of the stator slot in polar coordinates. They respectively correspond to... Figure 4 The semicircular region centered at point 4, and the straight-line region consisting of points 1, 2, and 3.

[0038] exist Figure 4 The sector-shaped grids traversed by the stator slot profile are all heterogeneous grids, while the remaining grids are all uniform grids. Figure 5 That is Figure 4A diagram illustrating a heterogeneous boundary model where a heterogeneous mesh connects to an adjacent homogeneous mesh within a straight-line region. G rair and G tair Let be the radial and tangential magnetic permeability of the uniform grid within the stator slot air domain, respectively, which can be expressed as: (4) In the formula △ r Indicates the length of the sector's side. r This indicates the radius at that location.

[0039] Two different materials can exist within a heterogeneous mesh, therefore, it is necessary to define the radial and tangential magnetic permeability of each material within this region. G rh Represented as radial magnetic permeability of the air domain within a heterogeneous mesh, G th Represented as the tangential magnetic permeability of the air domain within a heterogeneous mesh, such as Figure 5 As shown. Since its actual shape is irregular, a piecewise solution is used, with the intersection of the lower boundary of the sector mesh and the straight region of the stator slot as the starting point. θ' The dividing point is used to decompose it into two parts for solution. θ 0 The angle corresponding to the right boundary of the sector grid is... θ 0 and θ' The tangential permeability of the region between is defined as G th1 . θ* The angle represents the intersection point of the upper boundary of the sector grid and the straight area of ​​the stator slot. θ' and θ* The tangential permeability of the region between is defined as G th2 Specifically, as follows Figure 6 As shown, due to the different paths along the sector-shaped magnetic permeability, the tangential magnetic permeability at the same location is different. G th Consider it as two tangential magnetic permeabilities G th1 and G th2 The series configuration, while for radial permeability G rh It is a parallel configuration with tangential magnetic permeability. G th and radial permeability G rh The final form is formula (5): (5) Within the heterogeneous grid, silicon steel material also exists, therefore... Grr Defined as the radial magnetic permeability of silicon steel material within a heterogeneous boundary grid and G tr Defined as the tangential magnetic permeability of silicon steel material within a heterogeneous boundary grid. In solving for this magnetic permeability, a method based on magnetic circuit analysis was adopted: first, the overall magnetic permeability of the sector grid cells was calculated, and then the magnetic permeability of the corresponding air domain within the heterogeneous grid, as described by formula (5), was subtracted, thus indirectly obtaining the magnetic permeability contribution of the silicon steel material itself. This process essentially decomposes the magnetic permeability of the composite material based on the series and parallel relationship of magnetic circuits, as shown in formula (6): (6) The magnetic permeability of different materials within a heterogeneous mesh can be obtained using formulas (5) and (6). Figure 5 Taking the interior of a heterogeneous mesh as an example, since relative permeability is not considered in the comparison stage, the permeability comparison in this process can be simplified to an area comparison. Figure 5 The area of ​​the air domain within the heterogeneous mesh is significantly smaller than that of the silicon steel material. Therefore, using formula (7), the magnetic permeability of the air domain within the heterogeneous mesh is incorporated into the adjacent sector region of the same material, while the magnetic permeability of the silicon steel material remains within the original heterogeneous mesh, thus achieving material homogenization within the mesh. After incorporating the magnetic permeability of the air domain into the magnetic permeability of the adjacent same material, the magnetic flux path is lengthened from the tangential perspective, resulting in a decrease in magnetic permeability. The synthesized tangential magnetic permeability is... G Yt This indicates that the magnetic circuit behavior is similar to a parallel configuration. Conversely, from a radial perspective, the effective cross-sectional area of ​​the magnetic flux increases, leading to an increase in permeability. The resulting radial permeability is... G Yr This is equivalent to a series configuration. The equivalent internal mesh structure is as follows: Figure 7 As shown. G Yt , G Yr It can be obtained through formula (7): (7) Electromagnetic modeling must strictly adhere to the periodic symmetry characteristics of the motor. The 18-slot, 8-pole motor structure studied in this embodiment has a periodicity of 2; therefore, the smallest model capable of fully characterizing the motor's electromagnetic behavior is a half-motor model. A physically complete half-motor model structure is constructed by rotating and replicating the smallest periodic unit. This complete half-motor model mainly includes the unit model and the symmetric model generated through rotation, together forming a structured model. i × j Grid array, such as Figure 8As shown. Because the obtained model is a half-motor model rather than a full-motor model, specific boundary conditions need to be applied to ensure its physical consistency. The boundary conditions used include: Dirichlet boundary conditions applied to the inner and outer radial boundaries, and periodic boundary conditions used to characterize the geometric periodic symmetry. Under the periodic boundary conditions, the first column of the mesh and the... j The columns are connected to form a closed grid structure, such as... Figure 9 As shown.

[0040] Step 3: Unified matrix modeling and optimization iterative solution algorithm Formula (8) in the equivalent magnetic network method is defined with reference to Kirchhoff's voltage law and Kirchhoff's current law, which together define the magnetic permeability matrix ( G n ), nodal magnetomotive force matrix ( F n ) and flux matrix ( Φ n The strict mathematical relationship between them.

[0041] (8) Where the magnetic permeability matrix G n This is based on the discussion above. Figure 8 The matrix form of the connection form defined in the table is as follows: (9) In the formula n This represents the total number of nodes in the sector mesh in the equivalent magnetic network method. G n If the subscripts are the same, it indicates self-permeability; if the subscripts are different, it indicates mutual permeability.

[0042] The constraints inside the magnetic permeability matrix are: (10) Total magnetomotive force F n As the core input for solving the magnetic field in the equivalent magnetic network modeling, its essence is the comprehensive magnetomotive force formed by the vector superposition of the inherent magnetomotive force of the permanent magnet and the magnetomotive force generated by the current excitation of the stator winding. Among them, the permanent magnet, as the constant magnetic source of the motor, provides a magnetomotive force component based on its own material properties and structural parameters, which can be accurately calculated by formula (11). This formula can objectively reflect the magnetic energy output characteristics of the permanent magnet in the magnetic network.

[0043] (11) In the formula B r It is the remanence of a permanent magnet.h pm The height of the permanent magnet. μ pm It is the relative permeability of the permanent magnet, which is usually set to 1.05 for rare earth materials.

[0044] exist Figure 10 In the three-phase winding connection shown, the three-phase current excitation will generate a corresponding magnetomotive force (MTF) in the motor stator. The spatial distribution amplitude and direction of this MMF depend on the excitation method of the three-phase current. To accurately analyze its synthesis effect, this invention adopts a phase-by-phase calculation and resynthesis approach: First, the MMF generated by the A, B, and C phase windings in the stator tooth region is calculated separately, where each phase MMF can be regarded as a pulsating MMF with a fixed spatial position and an amplitude that varies with time according to the current law; then, the three pulsating MMFs that satisfy a specific phase relationship in both space and time are vector-superimposed to obtain the synthesized winding MMF of the three-phase windings. Its comprehensive expression can be expressed in the following form: (12) in, H and dl These are the magnetic field strength and the path of the integration loop, respectively. N It is the number of turns in the winding. I i It is the current injected into the phase windings of A, B, and C.

[0045] Therefore, in equation (8) F n It can be generated by the magnetomotive force of a permanent magnet. F pm and winding magnetomotive force F s It is formed by the superposition of , To represent the superimposed magnetomotive force, then F n Represented as: (13) The iterative method employs the Newton-Raphson method. To ensure numerical stability and avoid divergence in highly nonlinear regions, a variable Newton damping factor is introduced to adaptively adjust the iteration step size. This method helps maintain convergence and enhances the algorithm's stability. The update equation is defined as follows: (14) in, x k It is the first k The magnetic network state variables in the next iteration; x k+1 It is the first k+ The state variables of the magnetic network in one iteration; F( x k ) is the nonlinear equation corresponding to the permeability solution; J ( x k ) is a Jacobian matrix, α k It is a variable damping factor.

[0046] α k The pattern of change is shown in formula (15): (15) In the formula, α base It is the basic step size. B s It is the inflection point magnetic flux density of the BH curve of silicon steel material. B sat It is the adjustment coefficient, and N is the magnetic flux density.

[0047] The following describes the implementation steps of the Newton-Raphson method. This process begins with initializing the magnetic state variables. x 0 Initial damping factor α base and the corresponding magnetic permeability matrix G ( x 0 Begin with this step. In each iteration... k In evaluating nonlinear residuals F ( x k ) and Jacobi matrix J ( x k The solution is updated accordingly. If entry into the saturation region is detected, the variable damping factor is adaptively adjusted. α k To ensure numerical stability, iteration continues until the solution satisfies the convergence criterion. x k+1 - x k |< ε ,in ε This is a user-defined tolerance, typically set to 10. -6 This iterative approach can reliably and efficiently solve the magnetic network equations at different saturation levels.

[0048] Figure 11The overall flowchart of processing the electromagnetic model using the proposed magnetic network method is shown. First, the minimum modeling unit is created, and sector meshes are applied to both the stator and rotor. Then, the boundary locations are heterogeneously processed to unify them into a uniform mesh. The motor model is generated through rotation and replication, obtaining the magnetomotive force matrix and magnetic permeability matrix of the motor model, and then proceeding to the iterative section. The iterative section starts with initializing the magnetic state variable x. 0 Initial damping factor α base and the corresponding magnetic permeability matrix G ( x 0 Begin. In each iteration step... k In evaluating nonlinear residuals F ( x k ) and Jacobi matrix J ( x k The solution is updated accordingly. If entry into the saturation region is detected, the variable damping factor is adaptively adjusted. α k To ensure numerical stability, the iterative process will continue until the solution satisfies the convergence criterion. x k +1 - x k | < ε ,in ε The user-defined tolerance is typically set to 10. -6 After the current iteration completes its solution, the electromagnetic parameters for the next time step are solved by rotating the nodes using node transformations, until the set position is reached and the solution is complete. If the maximum number of iterations is exceeded, it indicates that no correct solution has been found, and therefore, the model needs to be rebuilt.

[0049] Step 4: Integration of rotor skew effect and torque calculation Electromagnetic torque is commonly calculated using two classical methods: the virtual work method and the Maxwell stress tensor method. The Maxwell stress tensor method is particularly prevalent in equivalent magnetic network modeling. In scenarios where rotor skewness is not considered, the electromagnetic torque can be accurately solved using the Maxwell stress tensor method, and its expression is: (16) in, R outer and R inner These represent the outer radius and inner radius of the sector grid cell, respectively; θ Indicates the central angle of a sector grid cell. B t and B r These represent the tangential and radial magnetic flux density, respectively.

[0050] When a skewed rotor structure is introduced into motor design, the rotor's permanent magnets are axially segmented, leading to a spatial phase shift in the magnetic field. This causes asynchronous magnetic field changes and phase differences between the segments. To analyze the total rotor flux, it should be obtained by superimposing the fluxes within partitioned grid cells at the same axial position. These cells accurately represent local magnetic fields, and their flux superposition yields the precise total rotor flux, expressed mathematically as follows: (17) in m Represents the number of oblique pole segments, φ m It is the polar angle of each axial section. K Indicates the number of skew poles. B t,m and B r,m The tangential and radial magnetic flux density under each skew pole segment.

[0051] After considering the skewed poles, substituting the skewed pole flux into the torque integral equation yields: (18) In the formula B t,m ( t )and B r,m ( t (This is time) t The function.

[0052] Most existing equivalent magnetic network modeling studies neglect rotor skewness, which limits their application in motor designs that suppress cogging torque and torque ripple through rotor skewness. By considering rotor skewness through axial segmentation and phase-shifted flux synthesis, accurate modeling of motors with arbitrary skewness angles becomes possible, and generalization ability is improved.

[0053] Step 5: Compare the accuracy of the finite element analysis method and the equivalent magnetic network method. To fully verify the accuracy of the modeling method of this invention, Figures 12-15 This is a comparison of the calculation errors between the equivalent magnetic network method and the finite element analysis method.

[0054] Figure 12 and Figure 13The results of the air gap radial magnetic flux density distribution obtained by the equivalent magnetic network method and the finite element analysis method are presented intuitively. From the overall distribution trend, the calculation results of the two methods show a high degree of consistency; the peak position, variation law, and overall distribution pattern of the magnetic flux density are basically consistent, fully verifying the reliability of the equivalent magnetic network method proposed in this invention in magnetic field characterization. This high degree of consistency not only proves the effectiveness of the equivalent magnetic network method in solving the air gap magnetic flux density, but also provides solid support for its application in the prediction and design optimization of the electromagnetic performance of permanent magnet synchronous motors.

[0055] Figure 14 and Figure 15 The electromagnetic torque waveforms are presented with and without considering the rotor skew effect. Analysis of the two figures clearly demonstrates that the Equivalent Magnetic Network (EMN) method proposed in this invention can stably adapt to the torque calculation requirements of motors with different rotor designs. Whether in scenarios with or without skew, the error between the calculation results of the EMN method and the finite element method (FEA) is controlled within an acceptable engineering range, fully verifying the method's versatility and reliability. More importantly, comparing the two sets of waveforms calculated using the same method reveals that without incorporating the post-processing mechanism for rotor skew into the EMN modeling, the amplitude characteristics, fluctuation frequency, and overall variation of the torque waveform show significant differences compared to the results considering skew. Without considering skew, the torque pulsation amplitude is larger, and harmonic interference is more pronounced, failing to truly reflect the optimizing effect of the skew structure on motor torque performance. Therefore, incorporating the rotor skew effect into the post-processing stage of the EMN modeling is crucial for ensuring the accuracy of torque calculation and aligning with actual motor design and operating characteristics; its importance is self-evident.

[0056] This invention is disclosed with reference to the preferred embodiments described above, but it should not be construed as being limited to the details of these embodiments. For those skilled in the art, any obvious modifications, refinements, or technical solutions derived from the inventive concept that are made to this invention without departing from its spirit and essence, as long as they contain all the technical features of the modified claims and the added technical features are described in the original claims, fall within the protection scope of this invention.

[0057] Any aspects not covered in this invention are applicable to existing technologies.

Claims

1. A high-precision magnetic network modeling method for heterogeneous mesh adaptive permanent magnet synchronous motor, characterized in that, The modeling method includes the following: The minimum periodic unit of the stator and the minimum periodic unit of the rotor are determined, and the minimum periodic unit is divided using a sector-shaped grid structure to form a structured grid; each sector-shaped grid unit is equivalent to two perpendicular magnetic permeabilities, which are defined as tangential magnetic permeabilities. G t and radial permeability G r ; Adaptive magnetic permeability calculation of heterogeneous material region: First, input the basic position information of the smallest periodic unit of the research object, construct the geometric boundary mathematical description in polar coordinates, define the fan-shaped grid traversed by the heterogeneous contour of the smallest periodic unit as heterogeneous grid, and the rest of the grids are uniform grids. All heterogeneous grids constitute heterogeneous material region. The heterogeneous material region contains two materials: silicon steel and non-silicon steel. The radial and tangential magnetic permeabilities are calculated first within each heterogeneous mesh, and then the overall radial and tangential magnetic permeabilities of the entire sector mesh cell are calculated. The radial and tangential magnetic permeabilities of the air domain are subtracted from the overall radial and tangential magnetic permeabilities to obtain the radial and tangential magnetic permeabilities of the silicon steel material within the heterogeneous mesh. By comparing the relative magnitudes of the radial and tangential magnetic permeability of silicon steel materials within a heterogeneous mesh with those of the air domain, mesh optimization and partitioning are automatically completed based on the principle of "merging materials with small magnetic permeability and retaining materials with large magnetic permeability," and a cell model is established. Finally, the Matlab program automatically generates a global model based on the element model through rotation and copying, and applies periodic boundary conditions and Dirichlet boundary conditions to ensure the continuity and consistency of the physical field; the Matlab program is used to calculate the magnetic permeability of the heterogeneous material region of the motor. The adaptive magnetic permeability calculation results are used to perform an iterative solution using Newton-Raphson.

2. The method according to claim 1, characterized in that, The principle of "merging materials with small magnetic permeability and retaining materials with large magnetic permeability" specifically refers to: grouping the silicon steel material and the air domain material with relatively small magnetic permeability into the adjacent fan-shaped grid area of ​​the same material, while retaining the other material with relatively large magnetic permeability in the original heterogeneous grid, thereby achieving the homogenization of materials within the heterogeneous grid.

3. The method according to claim 2, characterized in that, Classifying the magnetic permeability into that of adjacent materials of the same type, specifically: the composite tangential magnetic permeability viewed tangentially. G Yt This represents the parallel configuration of the two tangential magnetic permeabilities before synthesis; from a radial perspective, the synthesized radial magnetic permeability... G Yr This is a series configuration of two radial magnetic permeators before synthesis.

4. The method according to claim 1, characterized in that, The specific process of iterative solution using Newton-Raphson algorithm based on adaptive magnetic permeability calculation results is as follows: The Newton-Raphson method is used for iterative solution, and a variable Newton damping factor α is introduced. k To adaptively adjust the iteration step size, α k The pattern of change is as follows: (15) in, α base It is the basic step size. B s It is the inflection point magnetic flux density of the BH curve of silicon steel material. B sat It is the adjustment coefficient; In each iteration step k In evaluating nonlinear residuals F ( x k ) and Jacobi matrix J ( x k The solution is updated, and if entry into the saturation region is detected, the variable damping factor is adaptively adjusted. α k To ensure numerical stability, iteration continues until the solution satisfies the convergence criterion. x k+1 - x k |< ε ,in ε It is a user-defined tolerance.

5. The method according to claim 1, characterized in that, After introducing a rotor skewed pole structure into the motor design, and simulating it by segmenting along the axial direction and applying a phase shift, the integral equation of electromagnetic torque is: (18) In the formula, m Represents the number of oblique pole segments, φ m It is the polar angle of each axial section. K Indicates the number of skew poles. B t,m and B r,m The tangential and radial magnetic flux density under each skew pole segment are given. The magnetic flux density is obtained by iteratively solving the equivalent magnetic network method based on the adaptive magnetic permeability calculation results. B t,m ( t )and B r,m ( t (This is time) t The function; R outer and R inner These represent the outer radius and inner radius of the sector grid cell, respectively; θ Indicates the central angle of a sector grid cell; μ 0 It is the permeability of vacuum; L d It is the effective axial length; T It is electromagnetic torque.

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