Permanent magnet motor electromagnetic field hybrid analytical calculation method based on magnetic network method and subdomain method

Abstract

The invention discloses a permanent magnet motor electromagnetic field hybrid analytical calculation method based on a magnetic network method and a sub-domain method. The method comprises the following steps: firstly, dividing sub-domains for an air gap region and a permanent magnet region; an air gap and permanent magnet area control differential equation is established, the air gap area meets the Laplace equation, the permanent magnet area meets the Poisson equation, and a control differential equation solution of the air gap and permanent magnet area is obtained; according to a scalar magnetic potential continuity principle, applying a sub-domain model boundary condition to a control differential equation solution of an air gap and a permanent magnet region; according to the Kirchhoff's law, establishing a stator region magnetic network model, calculating the magnetic conductance of each node, and obtaining a magnetic potential matrix solving equation; then applying coupling boundary conditions of the sub-domain model region and the magnetic network region to obtain a coupling solution matrix equation, and solving magnetic field results of all parts; according to the method, the advantages of the magnetic network method and the sub-domain method are combined, the calculation speed is high, and the precision is high.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic field analysis and calculation of motors, and particularly relates to a hybrid analytical calculation method for the electromagnetic field of a permanent magnet motor based on the magnetic network method and the subdomain method. Background Art

[0002] Compared with induction motors, surface-mounted permanent magnet synchronous motors have high power density, high efficiency, and good control performance, and are widely used in various aspects of industry and life, such as automotive drive, marine auxiliary machinery, and wind power generation. The electromagnetic characteristics of permanent magnet motors are determined by the magnetic field distribution inside the motors. To explore the electromagnetic characteristics of motors, accurate solutions for the magnetic field distribution inside the motors are indispensable.

[0003] The electromagnetic field calculation methods for surface-mounted permanent magnet motors are mainly divided into two types, including 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 region into a finite number of small elements and solving Maxwell's equations, it can adapt to complex geometric shapes, handle nonlinear material characteristics (such as the saturation effect of ferromagnetic materials), and flexibly set boundary conditions. It is suitable for multi-physics field coupling analysis and can accurately simulate the magnetic field distribution and electromagnetic characteristics inside the motor. However, the calculation amount is large, and it is highly dependent on the mesh quality. Under high-precision requirements, when the mesh is divided finely, the calculation time is longer, which is not conducive to the evaluation and optimization of motor characteristics.

[0004] The analytical method generally includes the magnetic network method (magnetic circuit method) and the subdomain method. The magnetic network method has strong flexibility and can adjust the division and size of magnetic resistance units according to different regions of the motor. Its essence is to simplify the solution of the motor magnetic field into the solution of a magnetic circuit. However, when using the magnetic network method to solve, its accuracy will seriously depend on the number of divided nodes. If too many nodes are divided, the magnetic network method will lose its advantage of fast calculation. The subdomain method is a method of dividing the solution region into multiple subdomains, solving each subdomain using the analytical method, and then coupling through boundary conditions to obtain the overall solution. It has high accuracy and fast speed in solving the magnetic field, but is only applicable to regions with simple and regular shapes and is difficult to handle the saturation effect of ferromagnetic materials.

[0005] For the stator part of the motor, since the permeability of silicon steel sheets and the air gap differ by thousands of times, a relatively accurate result can be obtained by simplifying the two-dimensional magnetic field into a one-dimensional magnetic circuit. For the air gap and permanent magnet parts, since there is no permeable material, the magnetic permeability of the entire region is relatively uniform. For the permanent magnet and air gap parts of the motor, the shape is relatively simple, and since it is not a ferromagnetic material, no saturation phenomenon will occur. Therefore, using the subdomain method to solve the permanent magnet and air gap regions of the motor can obtain more accurate results.

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

[0007] The object of the present 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, which combines the magnetic network method and the subdomain method for analytical calculation, has simple solution, fast calculation speed and high accuracy.

[0008] To achieve the above object, the present 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, including the following steps: S1. Divide the motor from the inside to the outside into rotor, permanent magnet, air gap and stator regions, and divide subdomains for the air gap region and the permanent magnet region; S2. Establish the control 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, and obtain the solutions of the control differential equations for the air gap and permanent magnet regions; S3. According to the principle of scalar magnetic potential continuity, apply the subdomain model boundary conditions to the solutions of the control differential equations for the air gap and permanent magnet regions; S4. According to Kirchhoff's law, establish a magnetic network model for the stator region, calculate the magnetic conductance of each node, and obtain the magnetic potential matrix solution equation; S5. Apply the coupling 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 of each part; S6. Considering the saturation effect of ferromagnetic materials, construct an update process for the saturation magnetic permeability and perform iterative calculation of the magnetic permeability.

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

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

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

[0012] Wherein, is the vacuum magnetic permeability; Substitute the relational expressions of magnetic field intensity and magnetic flux density into the magnetic field divergence formula, and it is obtained that the divergence of the magnetic field intensity is constantly 0. Substitute the fact that the divergence of the magnetic field intensity is constantly zero into the relational expression between the magnetic field intensity and the scalar magnetic potential, and the controlled differential equation in the air-gap region is derived 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 terms. The general solution formula is:

[0014] where, is the number of pole pairs of the motor, is the undetermined coefficient, which is determined by the boundary conditions, and r is the radius; In the permanent magnet region, the magnetic flux density is expressed as:

[0015] where, is the remanence magnetization, is the relative magnetic permeability of the permanent magnet; By analogy with the derivation process of the controlled differential equation in the air-gap region, replace the magnetic flux density expression in the air-gap region with the magnetic flux density expression in the permanent magnet region to derive the controlled differential equation in the permanent magnet region as:

[0016] where, div represents divergence, is the remanence magnetization, is the relative magnetic permeability of the permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization mode is radial magnetization, the controlled differential equation in the permanent magnet region is:

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

[0018] where, is the remanence of the permanent magnet, is the pole arc coefficient of the permanent magnet; The particular solution corresponding to the Poisson equation is set as:

[0019] where, is the undetermined coefficient; Substituting the particular solution into the control differential equation in the permanent magnet region gives:

[0020] Then the solution of the control differential equation in the permanent magnet region is: .

[0021] Furthermore, in step S3, for the permanent magnet region, let the circumferences with radii and be its boundaries. The boundary conditions at a radius of are:

[0022] where is the tangential component of the magnetic field intensity in the permanent magnet region.

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

[0024] where is the scalar magnetic potential in the permanent magnet region, is the scalar magnetic potential in the air gap region, is the radial magnetic flux density in the permanent magnet region, is the radial magnetic flux density in the air gap region; Substitute the boundary conditions at radii , into the radius r of the solution of the control differential equation in the air gap and permanent magnet regions.

[0025] AndFurthermore, in step S4, the magnetic network method is used to calculate the magnetic field of the motor. The problem of solving the magnetic field is analogized to the problem of solving an electric circuit. By simplifying the stator into individual magnetic conductance units and the current into magnetic potential sources, and writing Kirchhoff's law equations for each node, the following system of equations is obtained:

[0026] where is the magnetic flux matrix flowing into each node by the magnetic potential source, [[ID=6)], is the magnetic conductance matrix, is the magnetic potential matrix. By solving for the magnetic potential matrix

[0027]

[0028]

[0029] where the matrix The expressions for each element are as follows:

[0030] Among them, is the magnetomotive force source adjacent to the th node, is the magnetic resistance of the branch where the magnetomotive force source is located, The calculation formula for

[0031] Among them, is the total current of the motor cross-section flowing out of the th slot; The calculation formula for

[0032] Among them, is the length of the magnetic resistance unit, is the cross-sectional area of the magnetic resistance unit, is the magnetic permeability of the magnetic resistance unit; Each element of the permeance matrix represents the permeance between node [[ID=4②]] and , and its calculation formula is as follows:

[0033] After obtaining the magnetomotive forces of each node in the magnetic network, the magnetic flux density between any two nodes is obtained according to the following formula: .

[0034] Furthermore, in step S5, the coupling boundary of the subdomain model and the magnetic network model is a circle with a radius of . To directly couple and solve the subdomain model and the magnetic network model, a series of nodes are arranged at the circle with a radius of in the stator magnetic network model; the boundary condition at a radius of is:

[0035] Among them, is the scalar magnetic potential in the stator region, is the radial magnetic flux in the stator region; Substituting the radius and angle corresponding to the nodes at the magnetic network coupling boundary into the solution of the control differential equation in the air-gap region of the subdomain model, that is, the air-gap scalar magnetic potential expression, we get:

[0036] Among them, , , , are undetermined coefficients, is the angle of each coupling boundary point of the magnetic network and sub-domain model. Writing the above formula in matrix form gives:

[0037] where, is the scalar magnetic potential at the same position as each node at the magnetic network coupling boundary, is the coefficient matrix formed by substituting the radius and angle of each node at the coupling boundary into the air-gap scalar magnetic potential expression, is the constant term after substitution, is the undetermined coefficient of the air-gap scalar magnetic potential expression , , , ; The relationship between the magnetic flux density and scalar magnetic potential in the air-gap region is:

[0038] where, is the vacuum permeability; Combining the above formula with the air-gap scalar magnetic potential expression gives the radial magnetic flux density expression in the air-gap region as:

[0039] Similarly, substituting the radius and angle corresponding to each node at the magnetic network coupling boundary into the air-gap radial magnetic flux density expression in the sub-domain model, multiplying by the cross-sectional area to obtain the radial magnetic flux, and writing it in matrix form gives:

[0040] where, is the radial magnetic flux at the same position as each node at the magnetic network coupling boundary, is the coefficient matrix formed by substituting the radius and angle of each node at the coupling boundary into the air-gap radial magnetic flux expression, is the constant term after substitution, is the undetermined coefficient of the air-gap radial magnetic flux expression , , , ; Substituting the obtained scalar magnetic potential matrix and radial magnetic flux matrix into the magnetic network model gives:

[0041] where, is the scalar magnetic potential of other nodes in the stator magnetic network model except at the coupling boundary, It is the magnetic flux flowing into other nodes except the coupling boundary. For the corresponding permeance matrix, transform the above formula to get:

[0042] Solve the above formula together with the solutions of the control differential equations of the air gap and permanent magnet regions substituted with boundary conditions in step S3 to obtain the coefficients to be solved in the subdomain model and the scalar magnetic potentials of each node in the magnetic network model, and then obtain the magnetic flux density in each region of the motor.

[0043] Furthermore, in step S6, construct an update process for the saturated permeability and perform iterative calculation of the permeability, including the following steps: S6.1. Considering stator saturation, first give the initial permeability and the maximum allowable error value; S6.2. Solve the magnetic field according to the given permeability through steps S1 - S5 to obtain the magnetic flux density result; S6.3. Interpolate the new permeability of the stator teeth according to the B - H curve of the stator material; S6.4. Calculate whether the error between the two permeabilities before and after is less than or equal to the maximum allowable error value. If so, the magnetic flux density obtained in step S5 is the final value; if not, substitute the new permeability into steps S1 - S5 to re - solve the magnetic field, and repeat steps S6.2 - S6.4 until the error between the two permeabilities before and after is less than or equal to the maximum allowable error value to obtain the final magnetic flux density.

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

[0045] Furthermore, after step S6, verify whether the finally obtained magnetic flux density is accurate by the finite element method and / or experimental measurement of the no - load back - electromotive force of the motor.

[0046] Furthermore, verifying the finally obtained magnetic flux density by the finite element method includes: establishing a finite element simulation model of the motor, calculating the magnetic flux density by the finite element method, and comparing whether it is consistent with the magnetic flux density calculated in steps S1 - S6.

[0047] Furthermore, verifying the finally obtained magnetic flux density by experimental measurement of the no - load back - electromotive force of the motor includes: substituting the finally obtained air - gap magnetic flux density into the back - electromotive force calculation formula to calculate the back - electromotive force, and comparing whether the calculated back - electromotive force is consistent with the back - electromotive force measured by the test motor; where the rotational speed of the test motor is 300 rpm, and the back - electromotive force calculation formula is:

[0048] Where, is the winding magnetic flux linkage, and the calculation formula of the winding magnetic flux linkage is as follows:

[0049] Among them, is the number of turns of the winding, is the air-gap radius, is the length of the motor, is the coil pitch angle, is the air-gap magnetic flux density.

[0050] After adopting the above scheme, the beneficial effects of the present invention are as follows: 1. The present invention only divides and constructs sub-domains for the air-gap and permanent magnet regions. There is no need to refine the magnetic network nodes in the air-gap and permanent magnet regions, the calculation speed is fast, and the construction of sub-domains for irregular stator slots is avoided.

[0051] 2. The present invention utilizes the characteristic that the magnetic permeability difference between the stator and the air-gap is large to construct a magnetic network model in the stator region, simplifies the solution steps, and shortens the calculation time.

[0052] 3. The present invention establishes a coupling boundary at the interface between the air-gap and the stator, the solution is simple, the accuracy is high, and it can be combined with the optimization algorithm of the electromagnetic structure of the motor for rapid iterative analysis.

[0053] 4. The present invention combines the sub-domain method and the magnetic network method for analytical calculation, taking into account both the calculation efficiency and the calculation accuracy, and can quickly solve the electromagnetic characteristics of the surface-mounted permanent magnet motor, which is a major breakthrough in solving the electromagnetic characteristics of the motor. Description of the Drawings

[0054] Figure 1 is the method flow chart of the present invention; Figure 2 is the structural schematic diagram of the surface-mounted permanent magnet motor of the present invention; Figure 3 is the boundary diagram of the permanent magnet model region of the present invention; Figure 4 is the stator magnetic network model diagram of the present invention; Figure 5 is the coupling boundary diagram of the sub-domain model and the magnetic network model of the present invention; Figure 6 is the enlarged view of the coupling boundary of the sub-domain model and the magnetic network model of the present invention; Figure 7 is the iterative flow chart of the present invention considering the saturation magnetic permeability; Figure 8 is the finite element model diagram of the motor of the present invention; Figure 9 is the comparison diagram of the mixed analysis and finite element of the air-gap magnetic flux density spatial distribution of the motor under no-load condition of the present invention; Figure 10 is the comparison diagram of the mixed analysis and finite element of the air-gap magnetic flux density spatial distribution of the motor under load condition of the present invention; Figure 11 This is the no-load back electromotive force hybrid analysis and test comparison diagram of the present invention at 300 rpm; Figure 12 This is the B-H curve diagram.

[0055] Label description: 1. Rotor; 2. Permanent magnet; 3. Air gap; 4. Stator; 5. Coupling boundary. Specific implementation manner

[0056] The following is a detailed description of the present invention in conjunction with the accompanying drawings and specific embodiments.

[0057] The present 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, as Figure 1 shown, specifically including the following steps: S1. As Figure 2 shown, the motor is divided into the rotor 1, permanent magnet 2, air gap 3, and stator 4 regions from the inside to the outside. Among them, the magnetic permeability of the air gap 3 and the permanent magnet 2 regions is relatively uniform, and the subdomain method can be used for solution. Subdomains are divided for the air gap 3 region and the permanent magnet 2 region to establish a subdomain model, and the magnetic network method is used for solution of the stator 4 region.

[0058] S2. Establish the control 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 of the control differential equations for the air gap and permanent magnet regions, including the following steps: The relationship between the magnetic field intensity and the scalar magnetic potential satisfies:

[0059] Among them, is the magnetic field intensity; is the scalar magnetic potential; grad is the gradient, indicating the change rate and direction of the scalar magnetic potential in space; For the magnetic field, the divergence of the magnetic flux density satisfies:

[0060] Among them, is the magnetic flux density, and div represents the divergence; In the air gap region, the relationship between the magnetic field intensity and the magnetic flux density is:

[0061] Among them, is the vacuum permeability; Substitute the relational expressions of the magnetic field intensity and the magnetic flux density into the magnetic field divergence formula to obtain that the divergence of the magnetic field intensity is always 0. Substitute the fact that the divergence of the magnetic field intensity is always zero into the relational expression between the magnetic field intensity and the scalar magnetic potential, and the control differential equation for the air gap region can be deduced as:

[0062] Among them, Δ represents the Laplace operator; The air-gap region satisfies the Laplace equation, and its solution consists only of the general solution terms. The general solution formula can be written as:

[0063] Among them, is the number of pole pairs of the motor, is the undetermined coefficient, which is determined by the boundary conditions, and r is the radius; In the permanent magnet region, the magnetic flux density can be expressed as:

[0064] Among them, is the remanence, is the relative permeability of the permanent magnet; By analogy with the derivation process of the control differential equation in the air-gap region, replacing the magnetic flux density expression in the air-gap region with the magnetic flux density expression in the permanent magnet region, the control differential equation in the permanent magnet region can be derived as:

[0065] Among them, div represents the divergence, is the remanence, is the relative permeability of the permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization mode is radial magnetization, the control differential equation in the permanent magnet region is:

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

[0067] Among them: is the remanence of the permanent magnet, is the pole arc coefficient of the permanent magnet; The particular solution corresponding to the Poisson equation is set as:

[0068] Among them, is the undetermined coefficient; Substituting the particular solution into the control differential equation in the permanent magnet region gives:

[0069] Then the solution of the control differential equation in the permanent magnet region can be written as: 。

[0070] S3. According to the principle of scalar magnetic potential continuity, applying the sub-domain model boundary conditions to the solutions of the control differential equations in the air-gap and permanent magnet regions includes the following steps: For the permanent magnet region, let the circumferences with radii and be its boundaries. As shown in Figure 3 , the circumference corresponding to the radius is the outer circumference of the rotor, which is also the inner circumference of the permanent magnet. The circumference corresponding to the radius is the outer circumference of the permanent magnet. Thus, the circumferences with radii and are the boundaries of the permanent magnet region; The boundary condition at the radius of is:

[0071] where is the tangential component of the magnetic field intensity in the permanent magnet region; The boundary condition at the radius of is:

[0072] where is the scalar magnetic potential in the permanent magnet region, is the scalar magnetic potential in the air-gap region, is the radial magnetic flux density in the permanent magnet region, is the radial magnetic flux density in the air-gap region; Substituting the boundary conditions at the radii of , into the radius r of the solutions of the control differential equations in the air-gap and permanent magnet regions, 6n equations can be formed, and any 6n undetermined coefficients in the solutions of the control differential equations in the air-gap and permanent magnet regions can be eliminated. There are 4n undetermined coefficients in the solutions of the control differential equations in the air-gap and permanent magnet regions respectively, which are . After eliminating 6n undetermined coefficients, 2n undetermined coefficients remain.

[0073] S4. According to Kirchhoff's law, establish the magnetic network model of the stator region, calculate the permeance of each node, and obtain the magnetic potential matrix solution equation, including the following steps: Using the magnetic network method to calculate the magnetic field of the motor, the problem of solving the magnetic field is analogized to the problem of solving an electric circuit. By simplifying the stator into magnetic permeance units and the current into magnetic potential sources, the stator magnetic network model is as shown in Figure 4 ; By writing Kirchhoff's law equations for each node, the following system of equations can be obtained:

[0074] Among them, is the magnetic potential source inflow flux matrix for each node, is the magnetic conductance matrix, is the magnetic potential matrix, and the magnetic potential matrix is obtained to get the scalar magnetic potential of each stator node; the expressions of each matrix are as follows:

[0075]

[0076]

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

[0078] Among them, is the magnetic potential source adjacent to the th node, is the magnetic resistance of the branch where the magnetic potential source is located, The calculation formula of

[0079] Among them, is the total current flowing out of the motor cross-section from the th slot; [[ID=4&]] The calculation formula of e

[0080] Among them, is the length of the magnetic resistance unit, is the cross-sectional area of the magnetic resistance unit, is the magnetic permeability of the magnetic resistance unit; Each element of the magnetic conductance matrix represents the magnetic conductance between node and , and its calculation formula is as follows:

[0081] After obtaining the magnetic potential of each node of the magnetic network, the magnetic flux density between any two nodes is obtained according to the following formula: .

[0082] S5. Apply the coupling boundary 5 conditions of the subdomain model area and the magnetic network area to obtain the coupled solution matrix equation and solve the magnetic field results of each part, including the following steps: As Figures 5-6As shown, the coupling boundary 5 of the sub-domain model and the magnetic network model is a circle with a radius of To directly couple and solve the sub-domain model and the magnetic network model, a series of nodes are arranged at the circle with a radius of in the stator magnetic network model; The boundary condition at the radius of is:

[0083] where is the scalar magnetic potential in the stator region, and is the radial magnetic flux in the stator region; Substitute the radius and angle corresponding to the nodes at the magnetic network coupling boundary 5 into the control differential equation solution in the air-gap region of the sub-domain model, that is, the air-gap scalar magnetic potential expression, to obtain:

[0084] where , , , are undetermined coefficients, and is the angle of each coupling boundary point of the magnetic network and the sub-domain model. Write the above formula in matrix form as:

[0085] where is the scalar magnetic potential at the same position as each node at the magnetic network coupling boundary, is the coefficient matrix formed after substituting the radius and angle of each node at the coupling boundary into the air-gap scalar magnetic potential expression, is the constant term after substitution, and is the undetermined coefficient of the air-gap scalar magnetic potential expression , , , ; The relationship between the magnetic density and the scalar magnetic potential in the air-gap region is:

[0086] where is the vacuum magnetic permeability; Combining the above formula with the air-gap scalar magnetic potential expression, the radial magnetic density expression in the air-gap region is obtained as:

[0087] Similarly, substitute the radius and angle corresponding to each node at the magnetic network coupling boundary into the air-gap radial magnetic density expression in the sub-domain model, multiply by the cross-sectional area to obtain the radial magnetic flux, and write it in matrix form as:

[0088] Among them, is the radial magnetic flux at the same position of each node at the magnetic network coupling boundary, is the coefficient matrix formed by substituting the radii and angles of each node at the coupling boundary into the air-gap radial magnetic flux expression, is the constant term after substitution, is the undetermined coefficient of the air-gap radial magnetic flux expression , , , ; Substitute the obtained scalar magnetic potential matrix and radial magnetic flux matrix into the magnetic network model to obtain:

[0089] Among them, is the scalar magnetic potential of other nodes in the stator magnetic network model except at the coupling boundary, is the incoming magnetic flux of other nodes except at the coupling boundary, is the corresponding permeance matrix. Transforming the above formula gives:

[0090] Solve the above formula together with the solutions of the control differential equations of the air-gap and permanent magnet regions substituted with boundary conditions in step S3 to obtain the coefficients to be solved in the sub-domain model and the scalar magnetic potential of each node in the magnetic network model , and through the magnetic flux density in each region of the motor (permanent magnet and air-gap regions) can be further obtained.

[0091] S6. Since the magnetic permeability is not a constant value, in order to make the calculation more accurate, considering the saturation effect of ferromagnetic materials, a renewal process of the saturation magnetic permeability is constructed to perform iterative calculation of the magnetic permeability to reduce errors. As Figure 7 shown, it includes the following steps: S6.1. Considering stator saturation, first give the initial magnetic permeability and the maximum allowable error value ; S6.2. According to the given magnetic permeability, perform magnetic field solution through steps S1 - S5 to obtain the magnetic flux density result ; S6.3. Interpolate the B-H curve of the stator material to obtain the new magnetic permeability of the stator teeth. As Figure 12 shown, the B-H curve is determined by the inherent properties of the stator material and is a fixed curve. The stator material can be silicon steel sheet. Specifically, the magnetic field strength H can be obtained through the magnetic flux density result and the B-H curve, and then the new magnetic permeability can be obtained according to the relationship formula between the magnetic field strength and the magnetic flux density ; S6.4. Calculate whether the error between the two permeabilities before and after is less than or equal to the maximum allowable error value. The calculation formula is:

[0092] If it is, the calculated permeability is close to the true value, and the magnetic flux density obtained in step S5 is the final value; if not, substitute the new permeability into steps S1 - S5 to re - solve the magnetic field, and repeat steps S6.2 - S6.4 until the error between the two permeabilities before and after is less than or equal to the maximum allowable error value to obtain the final magnetic flux density.

[0093] In addition, the maximum allowable error value The smaller it is, the closer the finally obtained permeability is to the true value. In this embodiment, it is preferably 0.001.

[0094] Furthermore, after step S6, verification can also be carried out to check whether the finally obtained magnetic flux density is accurate. Specifically, it can be verified by the finite element method and / or experimental measurement of the no - load back - electromotive force of the motor.

[0095] Verifying the finally obtained magnetic flux density by the finite element method includes: establishing a finite element simulation model of the motor. The finite element simulation model diagram is as Figure 8 shown. Using the finite element method, the air - gap magnetic flux density under no - load and load conditions of the motor can be calculated, and compare whether it is consistent with the radial magnetic surface calculated in steps S1 - S6. As Figure 9 and Figure 10 shown, it can be found that the radial magnetic flux density calculated by using the hybrid analytical method of the present invention is basically consistent with the calculation result by 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 magnetic field of the motor under no - load and load states.

[0096] Verifying the finally obtained magnetic flux density by experimental measurement of the no - load back - electromotive force of the motor includes: substituting the finally obtained air - gap magnetic flux density into the back - electromotive force calculation formula to calculate the back - electromotive force, and comparing whether the calculated back - electromotive force is consistent with the back - electromotive force measured by the test motor; among them, the rotational speed of the test motor is 300 rpm. The no - load back - electromotive force is an important index in motor design and can be calculated from the no - load magnetic field of the motor. The calculation formula of the back - electromotive force is:

[0097] Among them, is the winding magnetic chain, and the calculation formula of the winding magnetic chain is as follows:

[0098] Among them, is the number of winding turns, is the air-gap radius, is the length of the motor, is the coil pitch angle, is the air-gap magnetic flux density.

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

[0100] The above are only the preferred embodiments of the present invention, and do not limit the design of this case. All equivalent changes made according to the key design of this case 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, It includes the following steps: S1. Divide the motor from the inside out into rotor, permanent magnet, air gap, and stator regions, and divide sub-domains for the air gap region and the permanent magnet region; S2. Establish the control 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, and obtain the solutions of the control differential equations for the air gap and permanent magnet regions; S3. According to the principle of scalar magnetic potential continuity, apply the sub-domain model boundary conditions to the solutions of the control differential equations for the air gap and permanent magnet regions; S4. According to Kirchhoff's law, establish a magnetic network model for the stator region, calculate the magnetic permeance of each node, and obtain the magnetic potential matrix solution equation; S5. Apply the coupling boundary conditions between the sub-domain model region and the magnetic network region to obtain the coupled solution matrix equation, and solve the magnetic field results of each part; S6. Considering the saturation effect of ferromagnetic materials, construct an update process for the saturation magnetic permeability and perform iterative calculations of the magnetic permeability.

2. 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: In step S2, establishing the control differential equations for the air gap and permanent magnet regions specifically includes: The relationship between magnetic field intensity and scalar magnetic potential satisfies: Among them, is the magnetic field strength; is the scalar magnetic potential; grad is the gradient, indicating the rate and direction of change of the scalar magnetic potential in space; For the magnetic field, the divergence of magnetic flux density satisfies: Among them, is the magnetic flux density, and div represents divergence; In the air gap region, the relationship between magnetic field intensity and magnetic flux density is: Among them, is the vacuum permeability; Substitute the relationship expressions of magnetic field intensity and magnetic flux density into the magnetic field divergence formula, and it is obtained that the divergence of magnetic field intensity is always 0. Substitute the fact that the divergence of magnetic field intensity is always zero into the relationship expression between magnetic field intensity and scalar magnetic potential, and the control differential equation for the air gap region is derived as: where Δ represents the Laplace operator; The air gap region satisfies the Laplace equation, and its solution consists only of the general solution terms. Its general solution formula is: Among them, is the number of pole pairs of the motor, is a coefficient to be determined, which is determined by the boundary conditions, and r is the radius; In the permanent magnet region, the magnetic flux density is expressed as: Among them, is the residual magnetization intensity, is the relative permeability of the permanent magnet; By analogy with the derivation process of the control differential equation for the air gap region, replace the magnetic flux density expression in the air gap region with the magnetic flux density expression in the permanent magnet region to derive the control differential equation for the permanent magnet region as: where div represents divergence, is the remanent magnetization, is the relative permeability of the permanent magnet; The permanent magnet region satisfies the Poisson equation. When the magnetization mode is radial magnetization, the control differential equation for the permanent magnet region is: Wherein, is the included angle between the center of the N pole of the permanent magnet and the polar axis, is the relative permeability of the permanent magnet; the coefficient has the following expression: Among them, is the remanence of the permanent magnet, is the pole arc coefficient of the permanent magnet; The particular solution corresponding to the Poisson equation is set as: Among them, is an undetermined coefficient; Substitute the particular solution into the control differential equation for the permanent magnet region to get: Then the solution of the control differential equation for the permanent magnet region is: 。 3. 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: In step S3, for the permanent magnet region, let the circumference with a radius of and be its boundary. The boundary condition at a radius of is: Among them, is the tangential component of the magnetic field strength in the permanent magnet region; The boundary condition at a radius of is as follows: Among them, is the scalar magnetic potential in the permanent magnet region, is the scalar magnetic potential in the air gap region, is the radial magnetic flux density in the permanent magnet region, is the radial magnetic flux density in the air gap region; Substitute the boundary conditions at a radius of and into the radius r of the solution of the governing differential equations in the air gap and permanent magnet regions.

4. 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: In step S4, use the magnetic network method to calculate the motor magnetic field. Analogy the problem of solving the magnetic field to the problem of solving an electric circuit. By simplifying the stator into individual magnetic permeance units and simplifying the current into a magnetic potential source, and writing Kirchhoff's law equations for each node, the following system of equations is obtained: Among them, is the magnetic potential source inflowing into the magnetic flux matrix of each node, is the magnetic conductance matrix, is the magnetic potential matrix, and the magnetic potential matrix is obtained to get the scalar magnetic potential of each stator node; the expressions of each matrix are as follows: Among them, the matrix The expression of each element is: Among them, is the magnetomotive force source adjacent to the th node, is the magnetic resistance of the branch where the magnetomotive force source is located, The calculation formula of is: Among them, is the total current of the motor cross-section flowing out from the th slot; The calculation formula is as follows: Among them, is the length of the magnetoresistive element, is the cross-sectional area of the magnetoresistive element, is the magnetic permeability of the magnetoresistive element; Elements of the magnetic conductance matrix representing the node and the magnetic conductance therebetween, and its calculation formula is as follows: After obtaining the magnetic potential of each node in the magnetic network, the magnetic flux density between any two nodes is obtained according to the following formula: 。 5. 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: In step S5, the coupling boundary between the sub-domain model and the magnetic network model is a circle with a radius of . To directly couple and solve the sub-domain model and the magnetic network model, a series of nodes are arranged at the circle with a radius of in the stator magnetic network model; the boundary condition at the radius of is as follows: Among them, is the scalar magnetic potential of the stator region, is the radial magnetic flux of the stator region; Substitute the radius and angle corresponding to the nodes at the magnetic network coupling boundary into the solution of the control differential equation for the air gap region in the sub-domain model, that is, the air gap scalar magnetic potential expression, to get: Among them, , , , are undetermined coefficients, is the angle of each coupling boundary point of the magnetic network and sub-domain model. The above formula can be written in matrix form as: wherein, is the scalar magnetic potential at the same positions of each node at the coupling boundary of the magnetic network, is the coefficient matrix formed after substituting the radii and angles of each node at the coupling boundary into the expression of the scalar magnetic potential in the air gap, is the constant term after substitution, is the undetermined coefficient in the expression of the scalar magnetic potential in the air gap , , , ; The relationship between the magnetic flux density and the scalar magnetic potential in the air gap region is: Among them, is the vacuum permeability; Combine the above formula with the air gap scalar magnetic potential expression to obtain the radial magnetic flux density expression for the air gap region as: Similarly, substitute the radius and angle corresponding to each node at the magnetic network coupling boundary into the radial magnetic flux density expression for the air gap in the sub-domain model, multiply by the cross-sectional area to obtain the radial magnetic flux, and write it in matrix form as: Among them, is the radial magnetic flux at the same position of each node at the coupling boundary of the magnetic network, is the coefficient matrix formed after substituting the radius and angle of each node at the coupling boundary into the expression of the radial magnetic flux in the air gap, is the constant term after substitution, is the undetermined coefficient of the expression of the radial magnetic flux in the air gap , , , ; Substitute the obtained scalar magnetic potential matrix and radial magnetic flux matrix into the magnetic network model to get: Among them, is the scalar magnetic potential of other nodes of the stator magnetic network model except at the coupling boundary, is the incoming magnetic flux of other nodes except the coupling boundary, is the corresponding magnetic conductance matrix, and the above formula is deformed to obtain: Solve the above equation by combining it with the solutions of the controlled differential equations for the air gap and permanent magnet regions with boundary conditions substituted in step S3, to obtain the coefficients to be determined in the sub-domain model and the scalar magnetic potentials at each node in the magnetic network model, and then obtain the magnetic flux densities in each region of the motor.

6. 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: In step S6, construct an update process for the saturation permeability and perform iterative calculations of the permeability, including the following steps: S6.

1. Considering stator saturation, first specify the initial permeability and the maximum allowable error value. S6.

2. Solve the magnetic field through steps S1 - S5 according to the specified permeability to obtain the magnetic flux density result. S6.

3. Interpolate the new permeability of the stator teeth based on the B - H curve of the stator material. S6.

4. Calculate whether the error between the two permeabilities before and after is less than or equal to the maximum allowable error value. If so, the magnetic flux density obtained in step S5 is the final value; if not, substitute the new permeability into steps S1 - S5 to re - solve the magnetic field, and repeat steps S6.2 - S6.4 until the error between the two permeabilities before and after is less than or equal to the maximum allowable error value to obtain the final magnetic flux density.

7. 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: In step S6, the maximum allowable error value is 0.

001.

8. 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: After step S6, verify whether the finally obtained magnetic flux density is accurate by using the finite element method and / or experimental measurement of the no - load back - electromotive force of the motor.

9. 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: Verifying the finally obtained magnetic flux density by using the finite element method includes: establishing a finite element simulation model of the motor, calculating the magnetic flux density by the finite element method, and comparing it with the magnetic flux density calculated in steps S1 - S6 to check if they are consistent.

10. 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: Verifying the finally obtained magnetic flux density by experimental measurement of the no - load back - electromotive force of the motor includes: substituting the finally obtained air - gap magnetic flux density into the back - electromotive force calculation formula to calculate the back - electromotive force, and comparing the calculated back - electromotive force with the back - electromotive force measured by the test motor; where the rotational speed of the test motor is 300 rpm, and the back - electromotive force calculation formula is: Among them, is the winding magnetic flux, and the calculation formula of the winding magnetic flux is as follows: Among them, is the number of winding turns, is the air-gap radius, is the motor length, is the coil pitch angle, is the air-gap magnetic flux density.

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