A hybrid analytical calculation method for electromagnetic field of permanent magnet motor based on magnetic network method and subdomain method

Abstract

This invention discloses a hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method. First, the air gap region and the permanent magnet region are divided into subdomains. Then, governing differential equations are established for both the air gap and permanent magnet regions. The air gap region satisfies the Laplace equation, and the permanent magnet region satisfies the Poisson equation, yielding solutions to the governing differential equations for both regions. Next, based on the principle of scalar magnetic potential continuity, boundary conditions of the subdomain model are applied to the solutions of the governing differential equations for both the air gap and permanent magnet regions. Then, based on Kirchhoff's laws, a magnetic network model of the stator region is established, and the magnetic permeability of each node is calculated to obtain the solution equation for the magnetopotential matrix. Next, coupled boundary conditions are applied between the subdomain model region and the magnetic network region to obtain the coupled solution matrix equation, solving for the magnetic field results for each part. Finally, considering the saturation effect of ferromagnetic materials, an update process for saturated permeability is constructed. This invention combines the advantages of the magnetic network method and the subdomain method, resulting in fast calculation speed and high accuracy.

Description

Technical Field

[0001] This invention belongs to the field of analytical calculation technology of electromagnetic fields of motors, and specifically relates to a hybrid analytical calculation method for electromagnetic fields of permanent magnet motors based on the magnetic network method and the subdomain method. Background Technology

[0002] Compared to induction motors, surface-mounted permanent magnet synchronous motors (PMSMs) offer higher power density, higher efficiency, and better control performance, making them widely used in various aspects of industry and daily life, such as automotive drives, marine auxiliary machinery, and wind power generation. The electromagnetic characteristics of a PMSM are determined by the distribution of the magnetic field within the motor; therefore, understanding these characteristics requires accurately determining the distribution of the magnetic field within the motor.

[0003] The electromagnetic field calculation methods for surface-mounted permanent magnet motors are mainly divided into two types: the finite element method and the analytical method. The finite element method is a commonly used numerical method for calculating the electromagnetic field of motors. By discretizing the solution domain into a finite number of small elements, it solves Maxwell's equations, adapts to complex geometries, handles nonlinear material properties (such as the saturation effect of ferromagnetic materials), flexibly sets boundary conditions, and is suitable for multiphysics coupling analysis. It can accurately simulate the magnetic field distribution and electromagnetic characteristics within the motor. However, it involves a large computational load and is highly dependent on mesh quality. Under high-precision requirements, when the mesh is finely divided, the computation time is even longer, which is not conducive to motor characteristic evaluation and optimization.

[0004] Analytical methods generally include the magnetic network method (magnetic circuit method) and the subdomain method. The magnetic network method offers high flexibility, allowing adjustment of the reluctance element division and size according to different regions of the motor. Essentially, it simplifies the solution of the motor's magnetic field to the solution of a magnetic circuit. However, the accuracy of the magnetic network method heavily depends on the number of nodes; if too many nodes are used, the method loses its advantage of rapid computation. The subdomain method divides the solution domain into multiple subdomains, uses analytical methods within each subdomain, and then obtains the overall solution through boundary condition coupling. This method offers high accuracy and speed in solving the magnetic field, but it is only suitable for regions with simple, regular shapes and struggles to handle the saturation effects of ferromagnetic materials.

[0005] For the stator section of the motor, since the permeability of the silicon steel sheet differs from that of the air gap by thousands of times, simplifying the two-dimensional magnetic field into a one-dimensional magnetic circuit yields more accurate results. For the air gap and permanent magnet sections, the permeability is relatively uniform due to the absence of magnetically conductive material. Furthermore, the permanent magnets and air gap sections have simpler shapes and, being non-ferromagnetic materials, do not exhibit saturation. Therefore, using the subdomain method to solve for the permanent magnet and air gap regions of the motor yields even more accurate results.

[0006] In summary, how to balance computational efficiency and accuracy to quickly solve the electromagnetic characteristics of surface-mounted permanent magnet motors is a major problem at present. Summary of the Invention

[0007] The purpose of this invention is to provide a hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method. The method combines the magnetic network method and the subdomain method for analytical calculation, which is simple to solve, fast to calculate, and has high accuracy.

[0008] To achieve the above objectives, this invention provides a hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method, comprising the following steps: S1. Divide the motor from the inside out into rotor, permanent magnet, air gap and stator regions, and further divide the air gap region and permanent magnet region into sub-domains. S2. Establish the governing differential equations for the air gap and permanent magnet regions. The air gap region satisfies the Laplace equation, and the permanent magnet region satisfies the Poisson equation. Obtain the solutions to the governing differential equations for the air gap and permanent magnet regions. S3. Based on the principle of scalar magnetic potential continuity, apply the subdomain model boundary conditions to the solution of the governing differential equations for the air gap and permanent magnet regions; S4. Based on Kirchhoff's laws, establish a magnetic network model of the stator region, calculate the magnetic permeability of each node, and obtain the magnetic potential matrix to solve the equation. S5. Apply coupled boundary conditions between the subdomain model region and the magnetic network region to obtain the coupled solution matrix equation and solve the magnetic field results for each part. S6. Considering the saturation effect of ferromagnetic materials, construct an update process for saturation permeability and perform iterative calculation of permeability.

[0009] Furthermore, in step S2, establishing the control differential equations for the air gap and permanent magnet regions specifically includes: The relationship between magnetic field strength and scalar magnetic potential satisfies:

[0010] in, The magnetic field strength; is the scalar magnetic potential; grad is the gradient, representing the rate and direction of change of the scalar magnetic potential in space; For a magnetic field, the divergence of magnetic flux density satisfies:

[0011] in, ρ represents magnetic flux density, and div represents divergence; In the air gap region, the relationship between magnetic field strength and magnetic flux density is as follows:

[0012] in, Permeability of free space; Substituting the relationship between magnetic field strength and magnetic flux density into the formula for magnetic field divergence, we obtain that the divergence of magnetic field strength is always 0. Substituting the zero divergence of magnetic field strength into the relationship between magnetic field strength and scalar magnetic potential, we derive the governing differential equation for the air gap region as follows:

[0013] Where Δ represents the Laplace operator; The air gap region satisfies the Laplace equation, and its solution consists only of the general solution term. The general solution formula is as follows:

[0014] in, This represents the number of pole pairs of the motor. The coefficients are undetermined and determined by the boundary conditions; r is the radius. In the permanent magnet region, the magnetic flux density is expressed as:

[0015] in, Residual magnetization The relative permeability of a permanent magnet; By analogy with the derivation process of the governing differential equation for the air gap region, the magnetic flux density expression for the air gap region is replaced with the magnetic flux density expression for the permanent magnet region to derive the governing differential equation for the permanent magnet region as follows:

[0016] Where div represents divergence. Residual magnetization The relative permeability of a permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization method is radial magnetization, the governing differential equation of the permanent magnet region is:

[0017] in, The angle between the center of the N pole of the permanent magnet and the polar axis. The relative permeability of the permanent magnet; coefficient The expression is:

[0018] in, Remanence of permanent magnets. The polar arc coefficient of the permanent magnet; Let the particular solution of the Poisson equation be:

[0019] in, These are coefficients to be determined; Substituting the particular solution into the governing differential equation of the permanent magnet region, we get:

[0020] The solution to the governing differential equation for the permanent magnet region is: .

[0021] Furthermore, in step S3, for the permanent magnet region, let the radius be... and The circumference of the circle is its boundary, and the radius is... The boundary conditions at the location are:

[0022] in, This represents the tangential component of the magnetic field intensity in the permanent magnet region.

[0023] At a radius of The boundary conditions at the location are:

[0024] in, For the scalar magnetic potential of the permanent magnet region. The scalar magnetic potential of the air gap region. The radial magnetic flux density of the permanent magnet region. The radial magnetic flux density in the air gap region; With radius , Substitute the boundary conditions at the point into the radius r of the solution of the governing differential equation for the air gap and permanent magnet regions.

[0025] Furthermore, in step S4, the magnetic network method is used to calculate the motor magnetic field. The problem of solving the magnetic field is analogized to solving the circuit problem. By simplifying the stator into individual magnetic permeability units and the current into a magnetomotive force source, the following set of equations is obtained by writing Kirchhoff's laws for each node:

[0026] in, The magnetic flux matrix of each node is formed by the magnetic potential source flowing into it. Here is the permeability matrix. Given the magnetic potential matrix, the magnetic potential matrix is ​​obtained. To obtain the scalar magnetic potential of each node of the stator; the matrix expressions are as follows:

[0027]

[0028]

[0029] Among them, matrix The expressions for each element are:

[0030] in, In order to be with the first A magnetic potential source adjacent to each node, The magnetic reluctance of the branch where the magnetomotive force source is located. The calculation formula is:

[0031] in, For the first The total current flowing out of the motor cross-section of each slot; The calculation formula is:

[0032] in, The length of the magnetoresistive unit. The cross-sectional area of ​​the magnetoresistive unit is... The permeability of the reluctance unit; Elements of the permeability matrix Represents a node and The magnetic permeability between them is calculated using the following formula:

[0033] After obtaining the magnetomotive force of each node in the magnetic network, the magnetic flux density between any two nodes is... The following formula can be used to obtain: .

[0034] Furthermore, in step S5, the coupling boundary between the subdomain model and the magnetic network model is a radius of... To directly couple and solve the subdomain model and the magnetic network model, the radius of the stator magnetic network model is [missing information]. A series of nodes are arranged at the circumference; the radius is The boundary conditions are:

[0035] in, For the scalar magnetic potential of the stator region, For the radial magnetic flux of the stator region; Substituting the radii and angles corresponding to the nodes at the magnetic network coupling boundary into the solution of the air gap region control differential equation in the subdomain model, i.e., the air gap scalar magnetic potential expression, we obtain:

[0036] in, , , , For undetermined coefficients, Let the angles of each coupling boundary point of the magnetic network and subdomain model be represented. The above formula can be written in matrix form as follows:

[0037] in, This refers to the scalar magnetic potential at the same location as each node at the boundary of the magnetic network coupling. This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the air gap scalar magnetic potential expression. The constant term after substitution. The undetermined coefficients in the expression for the scalar magnetic potential of the air gap , , , ; The relationship between the magnetic flux density and scalar magnetic potential in the air gap region is as follows:

[0038] in, Permeability of free space; Combining the above equation with the expression for the scalar magnetic potential of the air gap, the expression for the radial magnetic flux density of the air gap region is obtained as follows:

[0039] Similarly, substituting the radii and angles corresponding to each node at the magnetic network coupling boundary into the air gap radial magnetic flux expression in the subdomain model, and multiplying by the cross-sectional area, yields the radial magnetic flux, which can be written in matrix form as follows:

[0040] in, For the radial magnetic flux at the same position of each node at the boundary of the magnetic network, This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the expression for the radial magnetic flux of the air gap. The constant term after substitution. The undetermined coefficients in the expression for the radial magnetic flux in the air gap , , , ; Substituting the obtained scalar magnetic potential matrix and radial flux matrix into the magnetic network model, we get:

[0041] in, For the scalar magnetic potentials of nodes other than those at the coupling boundary in the stator magnetic network model, For magnetic flux to flow into nodes other than the coupling boundary, To obtain the corresponding magnetic permeability matrix, the above equation can be transformed to get:

[0042] The above equation is combined with the solution of the control differential equation of the air gap and permanent magnet region with boundary conditions substituted in step S3 to obtain the coefficients to be determined in the subdomain model and the scalar magnetic potential of each node in the magnetic network model, and then the magnetic flux density of each region of the motor is obtained.

[0043] Furthermore, in step S6, the process for updating the saturation permeability and performing iterative calculations of the permeability includes the following steps: S6.1 Considering stator saturation, first give the initial permeability and the maximum allowable error value; S6.2. Based on the given magnetic permeability, solve for the magnetic field through steps S1-S5 to obtain the magnetic flux density result; S6.3. The new permeability of the stator teeth is obtained by interpolation based on the BH curve of the stator material; S6.4 Calculate whether the error between the two permeabilities is less than or equal to the maximum permissible error value. If yes, the magnetic flux density obtained in step S5 is the final value. If no, substitute the new permeability into steps S1-S5 to solve the magnetic field again, and repeat steps S6.2-S6.4 until the error between the two permeabilities is less than or equal to the maximum permissible error value, and obtain the final magnetic flux density.

[0044] Furthermore, in step S6, the maximum permissible error value is 0.001.

[0045] Furthermore, after step S6, the accuracy of the final magnetic flux density is verified by measuring the no-load back EMF of the motor using the finite element method and / or experimental measurement.

[0046] Furthermore, verifying the final magnetic flux density using the finite element method includes: establishing a finite element simulation model of the motor, calculating the magnetic flux density using the finite element method, and comparing it with the magnetic flux density calculated in steps S1-S6 to see if they are consistent.

[0047] Furthermore, the final magnetic flux density is verified by measuring the no-load back EMF of the motor through experiments. This includes substituting the final air gap magnetic flux density into the back EMF calculation formula to calculate the back EMF, and comparing whether the calculated back EMF is consistent with the back EMF measured by the test motor. The test motor operates at a speed of 300 rpm, and the back EMF calculation formula is as follows:

[0048] in, The winding flux linkage is calculated using the following formula:

[0049] in, The number of turns in the winding. Where is the air gap radius, For the length of the motor, The coil pitch angle, It is the air gap magnetic flux density.

[0050] After adopting the above solution, the beneficial effects of the present invention are as follows: 1. This invention only divides the air gap and permanent magnet regions into subdomains and constructs subdomains. There is no need to refine the magnetic network nodes in the air gap and permanent magnet regions, resulting in fast calculation speed and avoiding the construction of subdomains for irregular stator slots.

[0051] 2. This invention utilizes the large difference in permeability between the stator and the air gap to construct a magnetic network model in the stator region, which simplifies the solution steps and shortens the calculation time.

[0052] 3. This invention establishes a coupling boundary at the interface between the air gap and the stator, which is simple to solve and has high accuracy. It can be combined with the optimization algorithm of the motor electromagnetic structure for rapid iterative analysis.

[0053] 4. This invention combines the subdomain method and the magnetic network method for analytical calculation, taking into account both computational efficiency and accuracy. It can quickly solve the electromagnetic characteristics of surface-mounted permanent magnet motors, which is a major breakthrough in solving the electromagnetic characteristics of motors. Attached Figure Description

[0054] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the surface-mount permanent magnet motor of the present invention; Figure 3 This is a boundary diagram of the permanent magnet model region of the present invention; Figure 4 This is a model diagram of the stator magnetic network of the present invention; Figure 5 This is the coupling boundary diagram of the subdomain model and the magnetic network model of this invention; Figure 6 This is an enlarged view of the coupling boundary between the subdomain model and the magnetic network model of this invention; Figure 7 The iterative flowchart for considering saturation permeability in this invention; Figure 8 This is a finite element model diagram of the motor of the present invention; Figure 9 This is a comparison diagram of the spatial distribution of air gap magnetic flux density of the motor under no-load conditions and the finite element method. Figure 10 This is a comparison diagram of the spatial distribution of air gap magnetic flux density of the motor under load conditions and the finite element method. Figure 11 This is a comparison chart of the analysis and experiment of the no-load back EMF mixing at 300 rpm according to the present invention; Figure 12 This is a BH curve graph.

[0055] Label Explanation: 1. Rotor; 2. Permanent magnet; 3. Air gap; 4. Stator; 5. Coupling boundary. Detailed Implementation

[0056] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0057] This invention provides a hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method, such as... Figure 1 As shown, the specific steps include: S1, such as Figure 2 As shown, the motor is divided into rotor 1, permanent magnet 2, air gap 3 and stator 4 regions from the inside out. Among them, the permeability of air gap 3 and permanent magnet 2 regions is relatively uniform, and the subdomain method can be used to solve the problem. The air gap 3 region and permanent magnet 2 region are divided into subdomains to establish subdomain models, and the stator 4 region is solved using the magnetic network method.

[0058] S2. Establish the governing differential equations for the air gap and permanent magnet regions. The air gap region satisfies the Laplace equation, and the permanent magnet region satisfies the Poisson equation. Obtain the solutions to the governing differential equations for the air gap and permanent magnet regions, including the following steps: The relationship between magnetic field strength and scalar magnetic potential satisfies:

[0059] in, The magnetic field strength; is the scalar magnetic potential; grad is the gradient, representing the rate and direction of change of the scalar magnetic potential in space; For a magnetic field, the divergence of magnetic flux density satisfies:

[0060] in, ρ represents magnetic flux density, and div represents divergence; In the air gap region, the relationship between magnetic field strength and magnetic flux density is as follows:

[0061] in, Permeability of free space; Substituting the relationship between magnetic field strength and magnetic flux density into the formula for magnetic field divergence, we obtain that the divergence of magnetic field strength is always 0. Substituting the zero divergence of magnetic field strength into the relationship between magnetic field strength and scalar magnetic potential, we can derive the governing differential equation for the air gap region as follows:

[0062] Where Δ represents the Laplace operator; The air gap region satisfies the Laplace equation, and its solution consists only of the general solution term, which can be written as:

[0063] in, This represents the number of pole pairs of the motor. The coefficients are undetermined and determined by the boundary conditions; r is the radius. In the permanent magnet region, the magnetic flux density can be expressed as:

[0064] in, Residual magnetization The relative permeability of a permanent magnet; By analogy with the derivation of the governing differential equation for the air gap region, and replacing the magnetic flux density expression for the air gap region with that for the permanent magnet region, the governing differential equation for the permanent magnet region can be derived as follows:

[0065] Where div represents divergence. Residual magnetization The relative permeability of a permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization method is radial magnetization, the governing differential equation of the permanent magnet region is:

[0066] in, The angle between the center of the N pole of the permanent magnet and the polar axis. The relative permeability of the permanent magnet; coefficient The expression is:

[0067] in: Remanence of permanent magnets. The polar arc coefficient of the permanent magnet; Let the particular solution of the Poisson equation be:

[0068] in, These are coefficients to be determined; Substituting the particular solution into the governing differential equation of the permanent magnet region, we get:

[0069] The solution to the governing differential equation for the permanent magnet region can be written as: .

[0070] S3. Based on the principle of scalar magnetic potential continuity, apply subdomain model boundary conditions to the solutions of the governing differential equations for the air gap and permanent magnet regions, including the following steps: For the permanent magnet region, let the radius be... and The circumference of the circle is its boundary, such as Figure 3 As shown, radius The corresponding circumference is the outer circumference of the rotor, which is also the inner circumference of the permanent magnet, with a radius of... The corresponding circumference is the outer circumference of the permanent magnet, and therefore the radius is and The circumference is the boundary of the permanent magnet region; At a radius of The boundary conditions at the location are:

[0071] in, This represents the tangential component of the magnetic field strength in the permanent magnet region. At a radius of The boundary conditions at the location are:

[0072] in, For the scalar magnetic potential of the permanent magnet region. The scalar magnetic potential of the air gap region. The radial magnetic flux density of the permanent magnet region. The radial magnetic flux density in the air gap region; With radius , Substituting the boundary conditions at the given location into the radius *r* of the solution to the governing differential equations for the air gap and permanent magnet regions, 6n equations can be formed. Any 6n undetermined coefficients in the solutions to the governing differential equations for the air gap and permanent magnet regions can be eliminated. The solutions to the governing differential equations for the air gap and permanent magnet regions each have 4n undetermined coefficients, which are respectively... After eliminating 6n undetermined coefficients, 2n undetermined coefficients remain.

[0073] S4. Based on Kirchhoff's laws, establish a magnetic network model for the stator region, calculate the magnetic permeability of each node, and obtain the magnetic potential matrix. Solve the equations, including the following steps: The magnetic network method is used to calculate the magnetic field of a motor. Solving the magnetic field problem is analogous to solving a circuit problem. The stator is simplified into individual magnetic permeability units, and the current is simplified into a magnetomotive force source. The stator magnetic network model is as follows: Figure 4 As shown; By writing Kirchhoff's laws equations for each node, we can obtain the following system of equations:

[0074] in, The magnetic flux matrix of each node is formed by the magnetic potential source flowing into it. Here is the permeability matrix. Given the magnetic potential matrix, the magnetic potential matrix is ​​obtained. To obtain the scalar magnetic potential of each node of the stator; the matrix expressions are as follows:

[0075]

[0076]

[0077] Among them, matrix The expressions for each element are:

[0078] in, In order to be with the first A magnetic potential source adjacent to each node, The magnetic reluctance of the branch where the magnetomotive force source is located. The calculation formula is:

[0079] in, For the first The total current flowing out of the motor cross-section of each slot; The calculation formula is:

[0080] in, The length of the magnetoresistive unit. The cross-sectional area of ​​the magnetoresistive unit is... The permeability of the reluctance unit; Elements of the permeability matrix Represents a node and The magnetic permeability between them is calculated using the following formula:

[0081] After obtaining the magnetomotive force of each node in the magnetic network, the magnetic flux density between any two nodes is... The following formula can be used to obtain: .

[0082] S5. Apply the coupling boundary condition 5 between the subdomain model region and the magnetic network region to obtain the coupling solution matrix equation, and solve for the magnetic field results of each part, including the following steps: like Figure 5-6As shown, the coupling boundary 5 of the subdomain model and the magnetic network model has a radius of... To directly couple and solve the subdomain model and the magnetic network model, the radius of the stator magnetic network model is [missing information]. A series of nodes are arranged around the circumference; radius is The boundary conditions are:

[0083] in, For the scalar magnetic potential of the stator region, For the radial magnetic flux of the stator region; Substituting the radius and angle corresponding to the node at point 5 of the magnetic network coupling boundary into the solution of the air gap region control differential equation in the subdomain model, i.e., the air gap scalar magnetic potential expression, we obtain:

[0084] in, , , , For undetermined coefficients, Let the angles of each coupling boundary point of the magnetic network and subdomain model be represented. The above formula can be written in matrix form as follows:

[0085] in, This refers to the scalar magnetic potential at the same location as each node at the boundary of the magnetic network coupling. This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the air gap scalar magnetic potential expression. The constant term after substitution. The undetermined coefficients in the expression for the scalar magnetic potential of the air gap , , , ; The relationship between the magnetic flux density and scalar magnetic potential in the air gap region is as follows:

[0086] in, Permeability of free space; Combining the above equation with the expression for the scalar magnetic potential of the air gap, the expression for the radial magnetic flux density of the air gap region is obtained as follows:

[0087] Similarly, substituting the radii and angles corresponding to each node at the magnetic network coupling boundary into the air gap radial magnetic flux expression in the subdomain model, and multiplying by the cross-sectional area, yields the radial magnetic flux, which can be written in matrix form as follows:

[0088] in, For the radial magnetic flux at the same position of each node at the boundary of the magnetic network, This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the expression for the radial magnetic flux of the air gap. The constant term after substitution. The undetermined coefficients in the expression for the radial magnetic flux in the air gap , , , ; Substituting the obtained scalar magnetic potential matrix and radial flux matrix into the magnetic network model, we get:

[0089] in, For the scalar magnetic potentials of nodes other than those at the coupling boundary in the stator magnetic network model, For magnetic flux to flow into nodes other than the coupling boundary, To obtain the corresponding magnetic permeability matrix, the above equation can be transformed to get:

[0090] By combining the above equation with the solutions to the governing differential equations for the air gap and permanent magnet regions obtained in step S3, which incorporate the boundary conditions, we can solve for the coefficients to be determined in the subdomain model. Scalar magnetic potential of each node in the magnetic network model , and through Furthermore, the magnetic flux density of each region of the motor (permanent magnet and air gap region) can be obtained.

[0091] S6. Since magnetic permeability is not a constant value, to make the calculation more accurate, the saturation effect of ferromagnetic materials is considered. An update process for saturation permeability is constructed, and iterative calculations of magnetic permeability are performed to reduce errors. For example... Figure 7 As shown, it includes the following steps: S6.1 Considering stator saturation, first give the initial permeability and the maximum permissible error value. ; S6.2. Based on the given magnetic permeability, solve for the magnetic field through steps S1-S5 to obtain the magnetic flux density result. ; S6.3. The new permeability of the stator teeth is obtained by interpolation based on the BH curve of the stator material. ,like Figure 12 As shown, the BH curve is a fixed curve determined by the inherent properties of the stator material, which can be silicon steel sheet. Specifically, the magnetic field strength H can be obtained from the magnetic flux density result and the BH curve. Then, the new permeability can be obtained according to the relationship between magnetic field strength and magnetic flux density. ; S6.4 Calculate whether the error between the two permeabilities is less than or equal to the maximum permissible error value. The calculation formula is as follows:

[0092] If yes, the calculated permeability is close to the true value, and the magnetic flux density obtained in step S5 is the final value; if no, the new permeability is substituted into steps S1-S5 to solve the magnetic field again, and steps S6.2-S6.4 are repeated until the error between the two permeabilities is less than or equal to the maximum allowable error value, and the final magnetic flux density is obtained.

[0093] In addition, the maximum permissible error value The smaller the value, the closer the final permeability is to the true value. In this embodiment, The preferred value is 0.001.

[0094] Furthermore, after step S6, a verification can be performed to check whether the final magnetic flux density is accurate. Specifically, this can be done through the finite element method and / or by experimentally measuring the no-load back EMF of the motor.

[0095] The final magnetic flux density is verified using the finite element method, which includes establishing a finite element simulation model of the motor, as shown in the figure below. Figure 8 As shown, the air gap magnetic flux density of the motor under no-load and load conditions can be calculated using the finite element method. Comparison with the radial magnetic surface calculated in steps S1-S6 is then performed to determine if they are consistent. Figure 9 and Figure 10 As shown, it can be found that the radial magnetic flux density calculated using the hybrid analytical method of the present invention is basically consistent with the result calculated using the finite element method. Through verification by the finite element method, it can be proved that the hybrid analytical model established by the present invention has high accuracy in solving the motor magnetic field under no-load and loaded conditions.

[0096] The verification of the final magnetic flux density by experimentally measuring the no-load back EMF of the motor includes: substituting the final air gap magnetic flux density into the back EMF calculation formula to calculate the back EMF, and comparing whether the calculated back EMF is consistent with the back EMF measured by the test motor; wherein, the test motor speed is 300 rpm, and the no-load back EMF is an important indicator in motor design, which can be calculated from the no-load magnetic field of the motor. The formula for calculating the back EMF is:

[0097] in, The winding flux linkage is calculated using the following formula:

[0098] in, The number of turns in the winding. Where is the air gap radius, For the length of the motor, The coil pitch angle, It is the air gap magnetic flux density.

[0099] The no-load back EMF is determined by the motor's magnetic field, and the magnetic field can be reflected through the back EMF, such as... Figure 11 As shown, the waveform and amplitude of the no-load back EMF measured by the experiment are basically consistent with the results calculated by the hybrid analytical method of the present invention. This proves that the hybrid analytical model established by the present invention has high accuracy in solving the motor magnetic field under no-load conditions.

[0100] The above description is only a preferred embodiment of the present invention and is not intended to limit the design of this case. All equivalent changes made based on the key design features of this case shall fall within the protection scope of this case.

Claims

1. A hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method, characterized in that, Includes the following steps: S1. Divide the motor from the inside out into rotor, permanent magnet, air gap and stator regions, and further divide the air gap region and permanent magnet region into sub-domains. S2. Establish the governing differential equations for the air gap and permanent magnet regions. The air gap region satisfies the Laplace equation, and the permanent magnet region satisfies the Poisson equation. Obtain the solutions to the governing differential equations for the air gap and permanent magnet regions. S3. Based on the principle of scalar magnetic potential continuity, apply the subdomain model boundary conditions to the solution of the governing differential equations for the air gap and permanent magnet regions; S4. Based on Kirchhoff's laws, establish a magnetic network model of the stator region, calculate the magnetic permeability of each node, and obtain the magnetic potential matrix to solve the equation. S5. Apply coupling boundary conditions to the subdomain model region and the magnetic network region to obtain the coupling solution matrix equations and solve for the magnetic field results of each part; the coupling boundary between the subdomain model and the magnetic network model has a radius of... To directly couple and solve the subdomain model and the magnetic network model, the radius of the stator magnetic network model is [missing information]. A series of nodes are arranged at the circumference; the radius is The boundary conditions are: in, The scalar magnetic potential of the air gap region. The radial magnetic flux density in the air gap region; For the scalar magnetic potential of the stator region, For the radial magnetic flux of the stator region; Substituting the radii and angles corresponding to the nodes at the magnetic network coupling boundary into the solution of the air gap region control differential equation in the subdomain model, i.e., the air gap scalar magnetic potential expression, we obtain: Where n is the order of the spatial harmonics. This represents the number of pole pairs of the motor. , , , For undetermined coefficients, Let be the angle of each coupling boundary point in the magnetic network and subdomain model, and m be the coupling boundary node number. The above formula can be written in matrix form as follows: in, This refers to the scalar magnetic potential at the same location as each node at the boundary of the magnetic network coupling. This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the air gap scalar magnetic potential expression. The constant term after substitution. The undetermined coefficients in the expression for the scalar magnetic potential of the air gap , , , ; The relationship between the magnetic flux density and scalar magnetic potential in the air gap region is as follows: in, Permeability of free space; Combining the above equation with the expression for the scalar magnetic potential of the air gap, the expression for the radial magnetic flux density of the air gap region is obtained as follows: Similarly, substituting the radii and angles corresponding to each node at the magnetic network coupling boundary into the air gap radial magnetic flux expression in the subdomain model, and multiplying by the cross-sectional area, yields the radial magnetic flux, which can be written in matrix form as follows: in, For the radial magnetic flux at the same position of each node at the boundary of the magnetic network, This is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the expression for the radial magnetic flux of the air gap. The constant term after substitution. The undetermined coefficients in the expression for the radial magnetic flux in the air gap , , , ; Substituting the obtained scalar magnetic potential matrix and radial flux matrix into the magnetic network model, we get: in, For the scalar magnetic potentials of nodes other than those at the coupling boundary in the stator magnetic network model, For magnetic flux to flow into nodes other than the coupling boundary, To obtain the corresponding magnetic permeability matrix, the above equation can be transformed to get: The above equation is combined with the solution of the control differential equation for the air gap and permanent magnet region in step S3, which incorporates the boundary conditions, to obtain the coefficients to be determined in the subdomain model and the scalar magnetic potential of each node in the magnetic network model, thereby obtaining the magnetic flux density of each region of the motor; S6. Considering the saturation effect of ferromagnetic materials, construct an update process for saturation permeability and perform iterative calculation of permeability.

2. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 1, characterized in that: In step S2, establishing the control differential equations for the air gap and permanent magnet regions specifically includes: The relationship between magnetic field strength and scalar magnetic potential satisfies: in, The magnetic field strength; is the scalar magnetic potential; grad is the gradient, representing the rate and direction of change of the scalar magnetic potential in space; For a magnetic field, the divergence of magnetic flux density satisfies: in, ρ represents magnetic flux density, and div represents divergence; In the air gap region, the relationship between magnetic field strength and magnetic flux density is as follows: in, Permeability of free space; Substituting the relationship between magnetic field strength and magnetic flux density into the formula for magnetic field divergence, we obtain that the divergence of magnetic field strength is always 0. Substituting the zero divergence of magnetic field strength into the relationship between magnetic field strength and scalar magnetic potential, we derive the governing differential equation for the air gap region as follows: Where Δ represents the Laplace operator; The air gap region satisfies the Laplace equation, and its solution consists only of the general solution term. The general solution formula is as follows: in, This represents the number of pole pairs of the motor. The coefficients are undetermined and determined by the boundary conditions; r is the radius. In the permanent magnet region, the magnetic flux density is expressed as: in, Residual magnetization The relative permeability of a permanent magnet; By analogy with the derivation process of the governing differential equation for the air gap region, the magnetic flux density expression for the air gap region is replaced with the magnetic flux density expression for the permanent magnet region to derive the governing differential equation for the permanent magnet region as follows: Where div represents divergence. Residual magnetization The relative permeability of a permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization method is radial magnetization, the governing differential equation of the permanent magnet region is: in, The angle between the center of the N pole of the permanent magnet and the polar axis. The relative permeability of the permanent magnet; coefficient The expression is: in, Remanence of permanent magnets. The polar arc coefficient of the permanent magnet; Let the particular solution of the Poisson equation be: in, These are coefficients to be determined; Substituting the particular solution into the governing differential equation of the permanent magnet region, we get: The solution to the governing differential equation for the permanent magnet region is: 。 3. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 2, characterized in that: In step S3, for the permanent magnet region, let the radius be... and The circumference of the circle is its boundary, and the radius is... The boundary conditions at the location are: in, This represents the tangential component of the magnetic field strength in the permanent magnet region. At a radius of The boundary conditions at the location are: in, For the scalar magnetic potential of the permanent magnet region. The scalar magnetic potential of the air gap region. The radial magnetic flux density of the permanent magnet region. The radial magnetic flux density in the air gap region; With radius , Substitute the boundary conditions at the point into the radius r of the solution of the governing differential equation for the air gap and permanent magnet regions.

4. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 3, characterized in that: In step S4, the magnetic network method is used to calculate the motor magnetic field. The problem of solving the magnetic field is analogous to solving the circuit problem. By simplifying the stator into individual magnetic permeability units and the current into a magnetomotive force source, the following set of equations is obtained by writing Kirchhoff's laws for each node: in, The magnetic flux matrix of each node is formed by the magnetic potential source flowing into it. Here is the permeability matrix. Given the magnetic potential matrix, the magnetic potential matrix is ​​obtained. To obtain the scalar magnetic potential of each node of the stator; the matrix expressions are as follows: in, The total number of nodes in the magnetic network, matrix The expressions for each element are: in, In order to be with the first A magnetic potential source adjacent to each node, The magnetic reluctance of the branch where the magnetomotive force source is located. The calculation formula is: in, For the first The total current flowing out of the motor cross-section of each slot; The calculation formula is: in, The length of the magnetoresistive unit. The cross-sectional area of ​​the magnetoresistive unit is... The permeability of the reluctance unit; Elements of the permeability matrix Represents a node and The magnetic permeability between them is calculated using the following formula: In the formula, In order to be with the first All nodes connected to The total number; After obtaining the magnetomotive force of each node in the magnetic network, the magnetic flux density between any two nodes is... The following formula can be used to obtain: 。 5. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 4, characterized in that: In step S6, the process for updating the saturation permeability and performing iterative calculations of the permeability includes the following steps: S6.1 Considering stator saturation, first give the initial permeability and the maximum allowable error value; S6.

2. Based on the given magnetic permeability, solve for the magnetic field through steps S1-S5 to obtain the magnetic flux density result; S6.

3. The new permeability of the stator teeth is obtained by interpolation based on the BH curve of the stator material; S6.4 Calculate whether the error between the two permeabilities is less than or equal to the maximum permissible error value. If yes, the magnetic flux density obtained in step S5 is the final value. If no, substitute the new permeability into steps S1-S5 to solve the magnetic field again, and repeat steps S6.2-S6.4 until the error between the two permeabilities is less than or equal to the maximum permissible error value, and obtain the final magnetic flux density.

6. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 5, characterized in that: In step S6, the maximum permissible error value is 0.

001.

7. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 6, characterized in that: After step S6, the accuracy of the final magnetic flux density is verified by measuring the no-load back EMF of the motor using the finite element method and / or by experimental measurement.

8. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 7, characterized in that: The verification of the final magnetic flux density using the finite element method includes: establishing a finite element simulation model of the motor, calculating the magnetic flux density using the finite element method, and comparing it with the magnetic flux density calculated in steps S1-S6 to see if they are consistent.

9. The analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method as described in claim 7, characterized in that: The final magnetic flux density is verified by measuring the no-load back EMF of the motor through experiments. This includes substituting the final air gap magnetic flux density into the back EMF calculation formula to calculate the back EMF, and comparing whether the calculated back EMF is consistent with the back EMF measured by the test motor. The test motor operates at a speed of 300 rpm, and the back EMF calculation formula is as follows: in, The winding flux linkage is calculated using the following formula: in, The number of turns in the winding. Where is the air gap radius, For the length of the motor, The coil pitch angle, It is the air gap magnetic flux density.

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