A dynamic magnetic circuit calculation method for transient electromagnetic torque of a radial magnetic gear

Abstract

The present application belongs to the technical field of magnetic transmission and electromagnetic field numerical calculation, and relates to a dynamic magnetic circuit calculation method for transient electromagnetic torque of a radial magnetic gear. The steps are as follows: first, a two-dimensional grid model of the radial magnetic gear is established and physical material properties are assigned; second, grid element branch permeances are calculated and a global permeance sparse matrix is assembled; third, rotor positions are determined based on kinematic equations and equivalent magnetic flux source vectors are reconstructed; fourth, air gap magnetic flux density is solved and electromagnetic torque is calculated based on Maxwell stress tensor method. The dynamic magnetic circuit calculation method exhibits the characteristics of simplicity and convenience in practical engineering applications, and has excellent parameterization expandability and universality, and can flexibly adapt to performance evaluation of radial magnetic gears with different pole pair numbers and different topological sizes, providing a high-precision calculation tool with strong engineering universality for multi-objective optimization and parameter rapid iteration of magnetic gear transmission systems.

Description

Technical Field

[0001] This invention belongs to the field of magnetic transmission and electromagnetic field numerical calculation technology, and relates to a dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear. Background Technology

[0002] In modern industrial and transportation transmission systems, gear transmission devices are indispensable core components, widely used in wind power generation, new energy vehicles, aerospace, and ship propulsion. Traditional mechanical gears rely on the mechanical meshing between teeth to transmit torque and regulate speed. However, this direct-contact physical transmission inevitably brings a series of inherent problems, such as tooth surface friction and wear, mechanical fatigue fracture, the need for regular lubrication and maintenance, and high noise and severe vibration generated during high-speed operation. In addition, in the event of sudden overload or jamming, traditional mechanical gears are prone to tooth breakage or even damage to the entire transmission system. To overcome the bottlenecks of these mechanical transmissions, radial magnetic gears, as a new type of non-contact magnetic transmission technology, have emerged. It utilizes the principle of magnetic field modulation to achieve smooth conversion between different speeds and torques through the interaction of a high-speed inner rotor, a low-speed outer rotor, and a magnetic adjusting ring located between them. By employing an air gap magnetic field to transmit power, radial magnetic gears achieve physical isolation between transmission components. This not only completely eliminates mechanical friction and wear, but also provides significant advantages such as lubrication-free operation, low noise, and high reliability. Furthermore, it enables automatic magnetic pole slippage under system overload, offering natural overload protection and significantly reducing equipment operation and maintenance costs. However, radial magnetic gears generally face severe computational and evaluation challenges in electromagnetic design and performance optimization. First, the topology of radial magnetic gears is complex. Their air gap magnetic field is composed of a large number of high-order harmonics generated by the modulation of the permanent magnet magnetic fields of the inner and outer rotors by the tuning magnet block, resulting in a highly uneven magnetic field distribution. Second, accurate performance evaluation of radial magnetic gears, particularly in the study of transient electromagnetic torque and torque pulsation characteristics, relies heavily on finite element analysis (FEA) techniques. However, obtaining accurate transient torque curves using the finite element method typically requires extremely detailed meshing of the dual rotor and tuning ring regions during model building, and continuous handling of complex motion boundary conditions during time stepping. While this method offers high accuracy, it is not only cumbersome in modeling but also involves an exceptionally large and time-consuming computational process, severely limiting the efficiency of rapid parameter iteration and optimization in the early stages of design. It often exceeds the computational limits of conventional computers, especially during multi-objective optimization. Conversely, using traditional analytical methods or simplified finite element model parameter settings makes it difficult to accurately simulate complex physical boundaries and dynamic magnetic flux changes, thus compromising computational accuracy. Therefore, developing a dynamic magnetic circuit calculation method that can overcome the computational constraints of the finite element method, accurately reconstruct the dynamic equivalent magnetic flux source, and efficiently calculate the transient electromagnetic torque of radial magnetic gears, while ensuring extremely high computational accuracy, is crucial for shortening the development cycle of magnetic gears and advancing theoretical computational research and technological innovation in the field of magnetic transmission.

[0003] To address the numerical simulation and calculation of magnetic field in magnetic gears, Hao Xiuhong et al. from Yanshan University proposed a method combining the subdomain method and the Physical Information Neural Network (PINN) in their patent "Magnetic Gear Magnetic Field Simulation Method Based on Physical Information Neural Network and Subdomain Method" (CN117669377A). This method uses the governing equations, boundary conditions, and initial values ​​of each subdomain as loss terms for the neural network to simulate the magnetic field, aiming to achieve high-precision, meshless calculations. However, this method heavily relies on the construction and training of deep learning network architectures, making hyperparameter tuning difficult and model training extremely time-consuming. Furthermore, this method lacks a clear physically equivalent magnetic circuit topology, making it difficult to intuitively reflect the direct physical impact of changes in the dimensions of various local structures within the magnetic gear on leakage flux and magnetic saturation, thus hindering engineers from performing rapid parameterization iterations in the early design stages. In their 2024 paper "Calculation of Air Gap Magnetic Field and Torque of Three-Axis Ring Plate Permanent Magnet Gear," published in the journal *Mechanical Transmission*, Liu Dongning et al. from Dalian Jiaotong University used the scalar magnetic potential method to obtain a mathematical model of the air gap magnetic field of permanent magnet gears and established an electromagnetic torque model based on the principle of linear superposition of magnetic fields, aiming to improve calculation speed and replace part of the finite element analysis. However, this analytical model is overly simplified when dealing with the transient position linkage relationship during continuous rotor rotation, and the assumption of linear superposition makes it difficult to accurately account for the complex nonlinear magnetic flux changes inside the adjusting ring and yoke, resulting in difficulty in guaranteeing the accuracy of transient torque prediction under actual complex working conditions. Therefore, developing a dynamic magnetic circuit calculation method that takes into account the intuitiveness of physical principles, eliminates the need for cumbersome mesh generation and long-term training, and can efficiently and accurately solve the transient electromagnetic torque of radial magnetic gears is crucial for promoting engineering design and theoretical innovation in the field of magnetic gear transmission. Summary of the Invention

[0004] In order to overcome the shortcomings of the finite element analysis method in the prior art, which is extremely time-consuming and the traditional analytical model has low calculation accuracy, this invention provides a dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear. The aim is to rapidly and efficiently construct the underlying physical equivalent magnetic circuit topology based on the physical structural characteristics of radial magnetic gears by dividing a high-precision two-dimensional discrete mesh and assembling a global sparse magnetic permeability matrix using the reluctance series principle. Furthermore, when simulating the dynamic operation of a dual-rotor magnetic gear in continuous operation, the solution for the air gap magnetic field mainly relies on introducing kinematic equations to perform Thevenin-Norton equivalent source transformation on the mesh elements to reconstruct the global dynamic magnetic flux source column vector. This effectively captures and accounts for the complex nonlinear magnetic flux changes within the tuning ring and yoke, significantly improving the solution accuracy for the global dynamic magnetic scalar potential and air gap magnetic flux density distribution. Moreover, by integrating the underlying matrix numerical solution with the macroscopic Maxwell stress tensor method, this method achieves efficient analytical integration of the transient electromagnetic torques of the inner and outer rotors and the tuning ring, greatly shortening the calculation cycle of dynamic simulation while maintaining the intuitiveness of the physical principles. This provides strong support for the design optimization and rapid parameter iteration of radial magnetic gears.

[0005] The technical solution of the present invention: A dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear is proposed. Based on the physical structural characteristics and two-dimensional discrete mesh of the radial magnetic gear, it utilizes the magnetic equivalent circuit theory and the principle of sparse matrix assembly, combined with the kinematic equations of steady-state synchronous rotation of a dual-rotor system to reconstruct the dynamic equivalent magnetic flux source vector. This method solves for the dynamic magnetic scalar potential and air gap magnetic field distribution at all grid nodes. By integrating matrix numerical solutions with analytical integrals of the Maxwell stress tensor, the transient electromagnetic torque of the inner and outer rotors and the adjusting ring of the radial magnetic gear is calculated with high efficiency and high accuracy. The specific steps are as follows: The first step is to create a two-dimensional mesh model of the radial magnetic gear and assign it physical material properties; Obtain the physical structural parameters of the radial magnetic gear, including the number of pole pairs of the inner rotor. P HS Number of external rotor pole pairs P LS Adjusting the number of magnetic ring iron blocks Q M Remanence of permanent magnets B r Vacuum permeability m 0 Relative recovery permeability of permanent magnets m rec Relative permeability of silicon steel sheets m iron Axial effective stacking thickness L stack Total number of radial physical layers N rTotal number of tangential mesh divisions N t and radial boundary radius array R bounds ; The radial magnetic gear is divided into a two-dimensional discrete mesh along both the radial and tangential directions, and the radial mesh element layer index is defined as follows. i , i Values ​​range from 1 to N r Define the tangential mesh element layer index as j , j Values ​​range from 1 to N t ; Calculate the tangential mesh step size Dth : (1) Definition of the first i Inner diameter of layer mesh cell r in (i) With outer diameter r out (i) They are respectively: (2) (3) Based on the physical region where the two-dimensional discrete grid is located, the initial relative permeability matrix of each grid cell is calculated and assigned. m r (i,j) With magnetization matrix M(i,j) ; When the i When the layer grid cell is an inner rotor permanent magnet layer, its inner rotor magnetization matrix M HS (i,j) With relative permeability matrix m r-HS (i,j) They are respectively: (4) (5) When the i When the layer grid cell is an outer rotor permanent magnet layer, its outer rotor magnetization matrix M LS (i,j) With relative permeability matrix m r-LS (i,j) They are respectively: (6) (7) When the i When the layer mesh element is a tuning ring layer, its tuning ring magnetization matrix is ​​defined as follows: M Mod (i,j) The relative permeability matrix of the tuning ring is m r-Mod (i,j) If the following conditions are met: (8) Then, its relative permeability matrix is ​​assigned: (9) If not satisfied, then assign it a relative permeability matrix: (10) Furthermore, the magnetization matrix of the tuning ring layer is assigned as follows: (11) When the i When the layer grid cell is an inner rotor back iron layer or an outer rotor back iron layer, its back iron magnetization matrix is ​​defined. M Fe (i,j) Relative permeability matrix of the back iron m r-Fe (i,j) They are respectively: (12) (13) When the i When the layer grid cell is an inner air gap layer or an outer air gap layer, its magnetization matrix M Air (i,j) With relative permeability matrix m r-Air (i,j) They are respectively: (14) (15) The second step is to calculate the permeability of the grid cell branches and assemble the global permeability sparse matrix. Calculate the first i Layer, First j Radial permeability of the column grid cell itself P rad (i,j) With tangential magnetic permeability P tan (i,j) : (16) (17) Define the total number of nodes in the equivalent magnetic circuit system corresponding to a two-dimensional discrete grid. N total Its value is the total number of radial grid cell layers. N r Total number of tangential mesh cell layers N t The product of: (18) Define the one-dimensional global index of each grid cell in the global matrix operation. n， Used to index two-dimensional space (i,j) The mapping is to a one-dimensional linear sequence, and its calculation formula is: (19) Define nodes n The tangential index of the right adjacent node in the same layer is n rt The tangential index of the left adjacent node in the same layer is n lf The tangential index of the adjacent nodes on the same level is n up The tangential index of the adjacent nodes on the lower side of the same layer is n dn And satisfy the 360° periodic boundary condition; based on the principle of reluctance series connection, calculate the nodes respectively. n Equivalent tangential branch permeability to the right and left P t_rt (i,j) and P t_lf (i,j) : (20) (twenty one) When satisfied i < N r At that time, compute nodes n Upward equivalent radial branch permeability P r_up (i,j) : (twenty two) When satisfied i When >1, compute node n Downward equivalent radial branch permeability P r_dn (i,j) : (twenty three) Establish dimensions as N total × N total Global magnetic permeability sparse matrix K And initialize it to a zero matrix; for nodes n Take its diagonal elements K(n,n) The value is the sum of the magnetic permeabilities of all branches connected to this node: (twenty four) Assign the negative values ​​of the permeability of the corresponding branch to its off-diagonal elements: (25) (26) (27) (28) For global magnetic permeability sparse matrix K The first row of elements is cleared to zero, and the diagonal elements are forced to be cleared as well. (29) The third step is to determine the rotor position and reconstruct the equivalent flux source vector based on the kinematic equations. The steady-state synchronous rotational kinematic relationship between the inner and outer rotors is set based on the transmission ratio of the radial magnetic gears; when the mechanical angle of rotation of the inner rotor is... i in At that time, calculate the mechanical rotation angle of the outer rotor. i out : (30) Calculate the number of translation steps of the tangential mesh elements required for the inner rotor and outer rotor respectively. S in and S out : (31) (32) By using array cyclic shift operations based on periodic boundary conditions, the initial magnetization matrix is ​​tangentially translated to reconstruct the global dynamic magnetization matrix under the current mechanical angle. M dyn (i,j) The specific shift update rules are as follows: When the i When the layer grid cell is an inner rotor permanent magnet layer, its initial inner rotor magnetization matrix is ​​cyclically shifted along the tangential grid index j in one dimension, with a shift step size of . S in; When the i-th mesh element is the outer rotor permanent magnet layer, its initial inner rotor magnetization matrix is ​​cyclically shifted along the tangential mesh index j in one dimension, with a shift step size of . S out ; For the tuning ring layer, inner rotor back iron layer, outer rotor back iron layer, inner air gap layer, and outer air gap layer in the two-dimensional discrete grid, their magnetization matrices remain unchanged; the updated magnetization matrices of the inner rotor permanent magnet layer, outer rotor permanent magnet layer, tuning ring layer, inner rotor back iron layer, outer rotor back iron layer, inner air gap layer, and outer air gap layer together constitute the global dynamic magnetization matrix. M dyn (i,j); Subsequently, the global dynamic magnetization matrix is ​​traversed. M dyn (i,j) , for satisfying M dyn (i,j) For grid cells with a value ≠ 0, the Thevenin-Norton equivalent source transformation is performed to convert the discretized permanent magnet remanence into a magnetomotive force source injected into the corresponding grid node. F inj (i,j) The calculation formula is as follows: (33) Calculate the equivalent injected magnetic flux F inj (i,j) : (34) Establish dimensions as N total Global dynamic flux source column vector B vec Initialize it to zero; define the current equivalent injected magnetic flux. F inj (i,j) The i Layer, First j The column grid cell is the source grid, and the center node of this source grid is defined in the global dynamic flux source column vector. B vec The one-dimensional global index in is n Simultaneously, the adjacent grid cells of the source grid along the radial direction outward are defined as the first... i +1 floor, No. j The center node of the column grid cell is in the column vector of the global dynamic flux source. B vec The one-dimensional global index in is n up And satisfy nup =i×N t +j ; Based on Kirchhoff's nodal flux law, the equivalent injected flux is... F inj (i,j) Considered as radial from node n Outflow and inflow to the tangential index of the adjacent node on the same level. n up When traversing all grid cells for vector assembly, state change variables are used to avoid symbolic logic repetition. The iterative update formula for flux distribution and accumulation is expressed as follows: (35) (36) in, B(old)vec This represents the current state value of the global dynamic flux source column vector before the current flux allocation. B(new) vec This represents the updated state value of the global dynamic flux source column vector after this round of flux allocation; After completing the flux assignment for all grid cells, in order to match the Dirichlet boundary conditions at the system's outer boundary, the global dynamic flux source column vector is forced to... B vec The first row has an element value of zero: (37) Jointly constructed global magnetic permeability sparse matrix K With the assembled global dynamic flux source column vector B vec By solving a system of linear algebraic equations, a one-dimensional column vector of dynamic magnetic scalar potential containing the central nodes of all grid cells within the system is obtained. U dyn The matrix solution equation is: (38) The one-dimensional column vector of the dynamic magnetic scalar potential U dyn Reconstructed N r × N t From a two-dimensional matrix, obtain the dynamic magnetoscale potential matrix of all nodes. U(i,j) ; Step 4: Solve for the air gap magnetic flux density and calculate the electromagnetic torque based on Maxwell's stress tensor method; Obtain the inner air gap layer index idx in Index of outer air gap layer idx outThe radial thickness of the mesh elements in the inner air gap layer is calculated as follows: Δr in The average radius of the center is r m_in The radial thickness of the grid cells in the outer air gap layer is Δr out The average radius of the center is r m_out ; Based on the obtained dynamic magnetic scalar potential one-dimensional column vector U dyn Solve for the radial magnetic flux density components at the center of the inner and outer air gap layers. B r_in (j) , B r_out (j) With tangential magnetic flux density component B t_in (j)、B t_out (j) : (39) (40) (41) (42) Calculation of the electromagnetic torque of the inner rotor based on Maxwell's stress tensor method T in : (43) Calculate the electromagnetic torque of the external rotor T out : (44) Calculate the electromagnetic torque acting on the adjusting ring based on the system's static equilibrium equations. T mod : (45) At this point, the dynamic magnetic circuit calculation method for the transient electromagnetic torque of the radial magnetic gear is complete.

[0006] The beneficial effects of this invention are as follows: This invention proposes a dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear. This method breaks away from the heavy reliance of traditional finite element analysis (FEA) on massive computing power and extremely long simulation time when dealing with complex dual-rotor motion boundaries. By dividing the data into two-dimensional discrete meshes and assembling a global magnetic permeability sparse matrix, it achieves rapid construction and solution of the underlying physically equivalent magnetic circuit topology, significantly improving the calculation efficiency of the dynamic air gap magnetic field and transient torque of the radial magnetic gear, and greatly shortening the product development cycle. Furthermore, this invention introduces the kinematic equations of steady-state synchronous rotation of the dual rotors to further refine the mesh elements. The Thevenin-Norton equivalent source transformation is used to reconstruct the global dynamic flux source column vector, which can accurately capture the complex nonlinear flux changes and spatial harmonic effects inside the tuning ring and yoke. This overcomes the accuracy limitations of traditional analytical methods caused by oversimplification of linkage relationships. While ensuring extremely high computational speed, it achieves high-precision numerical calculations comparable to extremely fine mesh finite element simulations. Furthermore, this calculation method maintains the intuitiveness of clear physical principles; its underlying reluctance series and node network model directly reflects the direct physical impact of local structural dimensions and material properties variations of the magnetic gear on overall leakage flux and electromagnetic performance. In practical engineering applications, this method demonstrates simplicity and speed, and possesses excellent parametric scalability and versatility. It can flexibly adapt to the performance evaluation of radial magnetic gears with different pole pairs and topological dimensions, providing a highly accurate calculation tool with strong engineering applicability for multi-objective optimization and rapid parameter iteration of magnetic gear transmission systems. Attached Figure Description

[0007] Figure 1 This is a flowchart of a dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear.

[0008] Figure 2 This is a schematic diagram of the physical structure and two-dimensional mesh division of the radial magnetic gear in an embodiment of the present invention.

[0009] Figure 3 This is a schematic diagram of the equivalent magnetic circuit model of the magnetic permeability of the grid cell branch in an embodiment of the present invention.

[0010] Figure 4 This is a waveform diagram of the steady-state synchronous rotation electromagnetic torque of the radial magnetic gear calculated according to an embodiment of the present invention.

[0011] In the diagram: 1-Inner rotor core, 2-Inner rotor permanent magnet, 3-Inner air gap, 4-Adjusting magnetic ring block, 5-Adjusting magnetic ring air groove, 6-Outer air gap, 7-Outer rotor permanent magnet, 8-Outer rotor core. Detailed Implementation

[0012] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0013] Example A radial magnetic gear was selected, and its transient electromagnetic torque was dynamically calculated using magnetic circuit calculations. The number of inner rotor pole pairs of this radial magnetic gear was determined. P HS =2, number of external rotor pole pairs P LS =3, adjust the number of iron blocks in the magnetic ring. Q M =5, Remanence of permanent magnet B r =1.2T, vacuum permeability m 0 = 4π × 10 -7 H / m, relative restoring permeability of permanent magnet m rec =1.05, relative permeability of silicon steel sheet m iron =3000, effective axial thickness L stack =0.1m, total number of radial physical layers N r =71, Total number of tangential mesh divisions N t =720, Radial boundary radius array R bounds =[0.030,0.040,0.050,0.052,0.062,0.064,0.074,0.084]m.

[0014] A dynamic magnetic circuit calculation method for transient electromagnetic torque of a radial magnetic gear, such as... Figure 1 As shown, the specific steps are as follows: The first step is to create a two-dimensional mesh model of the radial magnetic gear and assign it physical material properties; First, the physical structural parameters of the radial magnetic gear are obtained. The radial magnetic gear is then divided into a two-dimensional discrete mesh along both the radial and tangential directions. The radial mesh element layer index is defined as follows: i , i Values ​​range from 1 to N r =71; Define the tangential mesh element layer index as j , j Values ​​range from 1 to N t =720, the tangential mesh step size is calculated using equation (1). Dth =0.5°. Taking the 14th layer of mesh cells as an example, the inner and outer diameters of the mesh cells in this layer are defined and calculated. The radial boundary inner diameter of the mesh in this layer is calculated by equations (2) and (3) respectively. r in (14) =0.040m and outer diameter r out(14) =0.0408m, based on the physical region where the two-dimensional discrete grid is located, calculate and assign an initial relative permeability to each grid cell. m r (i,j) With magnetization matrix M(i,j) When the 14th layer and the 1st column of the grid cells are inner rotor permanent magnet layers, their magnetization intensity can be calculated using equations (4) and (5). M HS 14,1) =9.55×10 5 A / m, relative permeability m r-HS (14,1)= 1.05; When the 46th layer and the 1st column of the grid cells are the outer rotor permanent magnet layer, the magnetization intensity is calculated by equations (6) and (7). M LS 14,1) =9.55×10 5 A / m, relative permeability m r-LS (14,1)= 1.05; When the 30th layer of the mesh is a tuning ring layer, its tuning ring magnetization matrix is ​​defined as follows: M Mod (i,j) The relative permeability matrix of the tuning ring is m r-Mod (i,j) Taking the second grid cell in the same layer as an example, it is determined by equation (8) that it meets the condition, and then by equation (9) it is assigned the relative permeability matrix. m r-Mod (30,2) =3000; Taking the first grid cell in the same layer as an example, if this condition is not met, then the relative permeability matrix is ​​assigned by equation (10). m r-Mod (30,1) =1; and by equation (11), the magnetization matrix of the tuning ring layer is assigned as M Mod (30,j) =0; When the first layer and the first column of grid cells are the inner rotor back iron layer (or the outer rotor back iron layer), the back iron magnetization intensity matrix is ​​given by equation (12). M Fe (1,1) =0; Equation (13) gives it the relative permeability matrix of the back iron. m r-Fe (1,1) =3000; When the 27th layer and the 1st column grid cell are inner air gap layers (or outer air gap layers), the magnetization matrix is ​​assigned by equation (14). M Air(27, 1) =0; its relative permeability matrix is ​​given by equation (15). m r-Air (27,1) =1.

[0015] The second step is to calculate the permeability of the grid cell branches and assemble the global permeability sparse matrix. First, calculate the radial and tangential magnetic permeability of the 14th layer and the 1st column grid cell: the radial magnetic permeability is calculated by equation (16). P rad (14,1) =6.04×10 -8 H is calculated from equation (17) to obtain the tangential magnetic permeability. P tan (14,1) =2.88×10 -7 H; The total number of nodes in the equivalent magnetic circuit system corresponding to the two-dimensional discrete grid can be calculated from equation (18). N total =51120, defines the one-dimensional global index of each grid cell in global matrix operations. n The 14th layer and the 1st column of the grid cells are mapped and calculated using equation (19). n =9361, definition n Tangential index of the right adjacent node at the same level =9361 n rt =2, tangential index of the left adjacent node in the same layer. n lf =720, Tangential index of adjacent nodes on the same level. n up =8641, Tangential index of adjacent nodes on the lower side of the same layer n dn =10081; Based on the principle of reluctance series connection, such as Figure 3 As shown, the nodes are calculated using equation (20). n Equivalent tangential branch permeability to the right P t_rt (i,j)(14,1) =2.88×10 -7 H is calculated from equation (21) as the equivalent tangential branch permeability to the left. P t_lf (14,1) =2.88×10 -7 H is calculated from equation (22) as the equivalent radial branch permeability upwards from the node. P r_up (14,1) =6.10×10 -8 H is calculated using equation (23) for the equivalent radial branch permeability downwards from the node. P r_dn(14,1) =1.21×10 -7 H, with dimensions defined as N total × N total Global magnetic permeability sparse matrix K And initialize it to a zero matrix; for nodes n =9361, from equation (24) its diagonal elements K(9361,9361) The value is assigned to the sum of the magnetic permeabilities of all branches connected to this node, i.e., 7.58 × 10⁻⁶. -7 H; assign the negative values ​​of the corresponding branch permeability to its off-diagonal elements respectively: calculate and assign the values ​​of the elements on the right side by equations (25), (26), (27), and (28) respectively. K(9361,9362) =7.58×10 -7 H, left-side element K(9361,9360) =-2.88×10 -7 H, Top element K(9361,10081) =- 6.10×10 -8 H and the elements below K(9361,8641) =- 1.21×10 -7 H, for the global magnetic permeability sparse matrix K All elements in the first row are cleared to zero, and the diagonal elements are forced to be zeroed by equation (29). K(1,1) =1.

[0016] The third step is to determine the rotor position and reconstruct the equivalent flux source vector based on the kinematic equations. The steady-state synchronous rotational kinematic relationship between the inner and outer rotors is set based on the transmission ratio of the radial magnetic gears; taking the simulation step as an example, when the mechanical angle of rotation of the inner rotor is... i in When the angle is 1.5°, the mechanical rotation angle of the outer rotor can be calculated using equation (30). i out =-1.0°; The translation steps of the tangential mesh elements required for the inner and outer rotors are calculated using equations (31) and (32) respectively. S in =83 and S out =-2; Using array cyclic shift operations based on periodic boundary conditions, the initial magnetization matrix is ​​tangentially shifted to reconstruct the global dynamic magnetization matrix under the current mechanical angle. M dyn (i,j) (e.g., to obtain a specific mesh) M dyn (14,1) =-9.55×10 5 A / m); then iterate through the global dynamic magnetization matrix to find the values ​​that satisfy...M dyn (i,j) For grid cells ≠0, the Thevenin-Norton equivalent source transformation is performed, and the equivalent transformation into a magnetomotive force source injected into the corresponding grid node is calculated by equation (33). F inj (14,1) =-699.58A; the equivalent injected magnetic flux is calculated from equation (34). F inj (14,1) =-4.23×10 - 5 Wb; establish dimensions as N total Global dynamic flux source column vector B vec And initialize to zero; define the source mesh center node at B vec The one-dimensional global index in is n =9361; Defines the center node of adjacent mesh cells radially outward. B vec The one-dimensional global index in is n up =10081; Based on Kirchhoff's nodal flux law, the iterative update of flux distribution and accumulation is calculated by equations (35) and (36) respectively to obtain the updated state value: B(old)vec(9361) =4.23×10 -5 Wb, and B(old)vec (10081) =8.064.23×10 -7 Wb; After completing the flux distribution of all grid cells, the global dynamic flux source column vector is forced by equation (37). B vec The first row element has a value of zero, that is... B vec (1)= 0; Jointly constructed global magnetic permeability sparse matrix K With the assembled global dynamic flux source column vector B vec The one-dimensional column vector of the dynamic magnetic scalar potential is obtained by solving the matrix equation of equation (38). U dyn For example, it can be concluded that... U dyn (9361) = -4427.47 A; The one-dimensional column vector of the dynamic magnetic scalar potential... U dyn Reconstructed N r × N t From a two-dimensional matrix, obtain the dynamic magnetoscale potential matrix of all nodes. U(i,j) The specific node of the internal air gap is obtained. i The magnetoscale potential at 0° over five consecutive rotational steps U 1 =-4427.47A、 U 2 =-4551.74A、 U 3 =-4674.36A、 U 4 =-4793.90A、 U 5 =-4909.07A.

[0017] Step 4: Solve for the air gap magnetic flux density and calculate the electromagnetic torque based on Maxwell's stress tensor method; Obtain the inner air gap layer index idx in Index of outer air gap layer idx out The radial thickness of the mesh elements in the inner air gap layer is calculated as follows: Δr in =0.000667m, the average radius of the center is r m_in =0.0510m; the radial thickness of the grid cells in the outer air gap layer is Δr out =0.000667m, the average radius of the center is r m_out =0.0630m; based on the obtained one-dimensional column vector of dynamic magnetic scalar potential U dyn Solve for the magnetic flux density components at the center of the inner and outer air gap layers: The radial magnetic flux density component at the center of the inner air gap layer is calculated by equation (39). B r_in1 =-0.401T、 B r_in2 =-0.377T、 B r_in3 =-0.353T、 B r_in4 =-0.331T、 B r_in5 =-0.309T, the tangential magnetic flux density component of the inner air gap layer is calculated by equation (40). B t_in1 =0.252T B t_in2 =0.259T B t_in3 =0.265T B t_in4 =0.268T Bt_in5 =0.270T, the radial magnetic flux density component of the outer air gap layer is calculated by equation (41). B r_out1 =0.167T B r_out2 =0.200T B r_out3 =0.221T B r_out4 =-0.331T、 B r_out5 =0.244T, the tangential magnetic flux density component of the outer air gap layer is calculated by equation (42). B t_out1 =0.740T B t_out2 =0.668T B t_out3 =0.604T B t_out4 =0.545T B t_out5 =0.493T; Based on Maxwell's stress tensor method, the electromagnetic torque of the inner rotor is calculated by equation (43). T in1 =-53.73 N·m T in2 =-51.44 N·m T in3 =-50.66 N·m T in4 =-51.15 N·m T in5 =-52.69 N·m; The electromagnetic torque of the outer rotor is calculated using equation (44). T out1 =81.71 N·m T out2 =80.67 N·m T out3 =80.83 N·m T out4 =82.52 N·m T out5 =85.77 N·m; Based on the system static equilibrium equation, the electromagnetic torque acting on the adjusting ring is calculated by equation (45), and the instantaneous bearing torque is obtained. T mod1 =135.44 N·m T mod2 =132.11 N·m T mod3 =131.50 N·m T mod4 =133.67 N·mT mod5 =138.46 N·m, such as Figure 4 As shown.

[0018] Thus, a dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear is completed.

[0019] This method characterizes the internal spatial magnetic field distribution of a magnetic gear by constructing a high-precision two-dimensional discrete mesh and an equivalent magnetic circuit model. It obtains the magnetic flux density of the inner and outer air gaps and the transient electromagnetic torque of the rotor and the adjusting ring under continuous dynamic rotation conditions. At the same time, it comprehensively considers the complex nonlinear magnetic flux changes and motion linkage effects inside the radial magnetic gear, and makes high-precision predictions of the dynamic electromagnetic transmission performance of the magnetic gear. This provides a solid theoretical reference for the electromagnetic design and multi-objective optimization of actual products. It is applicable to radial magnetic gear structures with different pole pairs and different topological dimensions. The calculation is efficient and fast, and it is a rapid calculation method with strong engineering universality and practical value.

Claims

1. A method for calculating the dynamic magnetic circuit of transient electromagnetic torque of a radial magnetic gear, characterized in that, The steps are as follows: The first step is to create a two-dimensional mesh model of the radial magnetic gear and assign it physical material properties; The second step is to calculate the permeability of the grid cell branches and assemble the global permeability sparse matrix. The third step is to determine the rotor position and reconstruct the equivalent flux source vector based on the kinematic equations. The fourth step is to solve for the air gap magnetic flux density and calculate the electromagnetic torque based on Maxwell's stress tensor method.

2. The dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear according to claim 1, characterized in that, The specific implementation process of the first step is as follows: Obtain the physical structural parameters of the radial magnetic gear, including the number of pole pairs of the inner rotor. P HS Number of external rotor pole pairs P LS Adjust the number of magnetic ring iron blocks Q M Remanence of permanent magnets B r Vacuum permeability μ 0 Relative recovery permeability of permanent magnets μ rec Relative permeability of silicon steel sheets μ iron Axial effective stacking thickness L stack Total number of radial physical layers N r Total number of tangential mesh divisions N t and radial boundary radius array R bounds Radial boundary radius array R bounds Include N r +1 element, recording the interface radius of each radial physical layer from the inside out; The radial magnetic gear is divided into a two-dimensional discrete mesh along both the radial and tangential directions, and the radial mesh element layer index is defined as follows. i , i Values ​​range from 1 to N r Define the tangential mesh element layer index as j , j Values ​​range from 1 to N t ; Calculate the tangential mesh step size Δθ : Definition of the first i Inner diameter of layer mesh cell r in (i) With outer diameter r(i) They are respectively: Based on the physical region where the two-dimensional discrete grid is located, the initial relative permeability matrix of each grid cell is calculated and assigned. μ r (i, j) With magnetization matrix M(i,j) ; When the i When the layer grid cell is an inner rotor permanent magnet layer, its inner rotor magnetization matrix M HS (i,j) With relative permeability matrix μ r-HS (i,j) They are respectively: When the i When the layer grid cell is an outer rotor permanent magnet layer, its outer rotor magnetization matrix M LS (i,j) With relative permeability matrix μ r-LS (i,j) They are respectively: When the i When the layer mesh element is a tuning ring layer, its tuning ring magnetization matrix is ​​defined as follows: M Mod (i,j) The relative permeability matrix of the tuning ring is μ r-Mod (i,j) If the following conditions are met: Then, its relative permeability matrix is ​​assigned: If not satisfied, then assign it a relative permeability matrix: Furthermore, the magnetization matrix of the tuning ring layer is assigned as follows: When the i When the layer grid cell is an inner rotor back iron layer or an outer rotor back iron layer, its back iron magnetization matrix is ​​defined. M Fe (i, j) Relative permeability matrix of the back iron μ r-Fe (i,j) They are respectively: When the i When the layer grid cell is an inner air gap layer or an outer air gap layer, its magnetization matrix M Air (i,j) With relative permeability matrix μ r-Air (i,j) They are respectively: 。 3. The dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear according to claim 2, characterized in that, The specific implementation process of the second step is as follows: Calculate the first i Layer, First j Radial permeability of the grid cell itself P rad (i,j) With tangential magnetic permeability P tan (i,j) : Define the total number of nodes in the equivalent magnetic circuit system corresponding to a two-dimensional discrete grid. N total Its value is the total number of radial grid cell layers. N r Total number of tangential mesh cell layers N t The product of: Define the one-dimensional global index of each grid cell in the global matrix operation. n， Used to index two-dimensional space (i,j) The mapping is to a one-dimensional linear sequence, and its calculation formula is: Define nodes n The tangential index of the right-side adjacent node in the same layer is n rt The tangential index of the left adjacent node in the same layer is n lf The tangential index of the adjacent nodes on the same level is n up The tangential index of the adjacent nodes on the lower side of the same layer is n dn And satisfy the 360° periodic boundary condition; Based on the principle of reluctance series connection, calculate the nodes respectively. n Equivalent tangential branch permeability to the right and left P t_rt (i,j) and P t_lf (i, j) : When satisfied i < N r At that time, compute nodes n Upward equivalent radial branch permeability P r_up (i,j) : When satisfied i When >1, compute node n Downward equivalent radial branch permeability P r_dn (i,j) : Establish dimensions as N total × N total Global magnetic permeability sparse matrix K And initialize it to a zero matrix; for nodes n Take its diagonal elements K(n,n) The value is the sum of the magnetic permeabilities of all branches connected to this node: Assign the negative values ​​of the permeability of the corresponding branch to its off-diagonal elements: For global magnetic permeability sparse matrix K The first row of elements is cleared to zero, and the diagonal elements are forced to be cleared as well. 。 4. The dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear according to claim 3, characterized in that, The specific implementation process of the third step is as follows: The steady-state synchronous rotational kinematic relationship between the inner and outer rotors is set based on the transmission ratio of the radial magnetic gears; when the mechanical angle of rotation of the inner rotor is... θ in At that time, calculate the mechanical rotation angle of the outer rotor. θ out : Calculate the number of translation steps of the tangential mesh elements required for the inner rotor and outer rotor respectively. S in and S out : By using array cyclic shift operations based on periodic boundary conditions, the initial magnetization matrix is ​​tangentially translated to reconstruct the global dynamic magnetization matrix under the current mechanical angle. M dyn (i,j) The specific shift update rules are as follows: When the i When the layer grid cell is an inner rotor permanent magnet layer, its initial inner rotor magnetization matrix is ​​cyclically shifted along the tangential grid index j in one dimension, with a shift step size of . S in ; When the i-th mesh element is the outer rotor permanent magnet layer, its initial inner rotor magnetization matrix is ​​cyclically shifted along the tangential mesh index j in one dimension, with a shift step size of . S out ; For the tuning ring layer, inner rotor back iron layer, outer rotor back iron layer, inner air gap layer, and outer air gap layer in the two-dimensional discrete grid, their magnetization matrices remain unchanged; the updated magnetization matrices of the inner rotor permanent magnet layer, outer rotor permanent magnet layer, tuning ring layer, inner rotor back iron layer, outer rotor back iron layer, inner air gap layer, and outer air gap layer together constitute the global dynamic magnetization matrix. M dyn (i,j); Subsequently, the global dynamic magnetization matrix is ​​traversed. M dyn (i,j) , for satisfying M dyn (i,j) For grid cells with a value ≠ 0, the Thevenin-Norton equivalent source transformation is performed to convert the discretized permanent magnet remanence into a magnetomotive force source injected into the corresponding grid node. F inj (i,j) The calculation formula is as follows: Calculate the equivalent injected magnetic flux Φ inj (i,j) : Establish dimensions as N total Global dynamic flux source column vector B vec Initialize it to zero; define the current equivalent injected magnetic flux. Φ inj (i,j) The i Layer, First j The column grid cell is the source grid, and the center node of this source grid is defined in the global dynamic flux source column vector. B vec The one-dimensional global index in is n Simultaneously, the adjacent grid cells of the source grid along the radial direction outward are defined as the first... i +1 floor, No. j The center node of the column grid cell is in the column vector of the global dynamic flux source. B vec The one-dimensional global index in is n up And satisfy n up =i×N t +j ; Based on Kirchhoff's nodal flux law, the equivalent injected flux is... Φ inj (i,j) Considered as radial from node n Outflow and inflow to the tangential index of the adjacent node on the same level. n up When traversing all grid cells for vector assembly, state change variables are used to avoid symbolic logic repetition. The iterative update formula for flux distribution and accumulation is expressed as follows: in, B(old)vec This represents the current state value of the global dynamic flux source column vector before the current flux allocation. B(new)vec This represents the updated state value of the global dynamic flux source column vector after the current flux allocation. After completing the flux assignment for all grid cells, in order to match the Dirichlet boundary conditions at the system's outer boundary, the global dynamic flux source column vector is forced to... B vec The first row has an element value of zero: Jointly constructed global magnetic permeability sparse matrix K With the assembled global dynamic flux source column vector B vec By solving a system of linear algebraic equations, a one-dimensional column vector of dynamic magnetic scalar potential containing the central nodes of all grid cells within the system is obtained. U dyn The matrix solution equation is: The one-dimensional column vector of the dynamic magnetic scalar potential U dyn Reconstructed N r × N t A two-dimensional matrix is ​​used to obtain the dynamic magnetoscale potential matrix of all nodes in the domain. U(i,j) .

5. The dynamic magnetic circuit calculation method for the transient electromagnetic torque of a radial magnetic gear according to claim 4, characterized in that, The specific implementation process of the fourth step is as follows: Obtain the inner air gap layer index idx in Index of outer air gap layer idx out ; The radial thickness of the mesh element in the inner air gap layer is calculated as follows: Δr in The average radius of the center is r m_in The radial thickness of the grid cells in the outer air gap layer is Δr out The average radius of the center is r m_out ; Based on the obtained dynamic magnetic scalar potential one-dimensional column vector U dyn Solve for the radial magnetic flux density components at the center of the inner and outer air gap layers. B r_in (j) , B r_out (j) With tangential magnetic flux density component B t_in (j), B t_out (j) : Calculation of the electromagnetic torque of the inner rotor based on Maxwell's stress tensor method T in : Calculate the electromagnetic torque of the external rotor T out : Calculate the electromagnetic torque acting on the adjusting ring based on the system's static equilibrium equations. T mod : 。

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