Analytical method of air gap magnetic field of permanent magnet motor based on magnetomotive force field subdomain

Abstract

The present invention discloses a method for analyzing the air gap magnetic field of a permanent magnet motor based on a magnetomotive force field subdomain, and relates to the field of analytical calculation of the electromagnetic field of a permanent magnet motor. The method is summarized as follows: the subdomains are divided according to the motor structure, the permanent magnet domain is equivalent to a spatial magnetic charge source, and the armature winding is equivalent to the stator magnetomotive force boundary; based on the continuity of the magnetomotive force at the interface and the consistency of the magnetic induction intensity field perpendicular to the interface, the boundary equation and the interface contact equation group between the subdomains are established; the boundary equation and the interface contact equation are combined to form a linear equation group with undetermined coefficients as variables, and the linear equation group is solved. After obtaining the undetermined coefficients, the magnetomotive force equation, the magnetic induction field intensity equation and the magnetic field intensity equation of each subdomain can be obtained. The method of the present invention performs analytical calculation of the magnetic field based on the scalar potential field. The negative number of the magnetomotive force field gradient is the magnetic field intensity vector. The method simplifies the calculation process of the subdomain, the process is easier to understand, and the calculation results are verified to be accurate.

Description

Technical Field

[0001] The invention relates to the field of motor electromagnetic field calculation, and in particular to a method for analyzing the air gap magnetic field of a permanent magnet motor based on a magnetomotive force field subdomain. Background Art

[0002] Surface-mounted permanent magnet synchronous motors are widely used in vehicle drive motors, servo motors, and wind turbines due to their high efficiency, high power density, and high torque density. In order to efficiently and accurately calculate the characteristics of permanent magnet motors, it is essential to obtain accurate motor magnetic field distribution.

[0003] At present, the calculation methods of motor magnetic field are mainly divided into magnetic circuit method, finite element method and subdomain method. Among them, the magnetic circuit method is a one-dimensional analytical model, which is used in the preliminary design stage of the motor to evaluate the basic performance of the motor, as well as the qualitative evaluation of harmonics and vibration. However, the magnetic circuit method cannot obtain accurate air gap magnetic field, especially high-order harmonic magnetic field, which limits its application in the accurate calculation of motor characteristics.

[0004] The finite element method is a numerical calculation method that discretizes the entire motor into micro-structure units for calculation. The finite element method obtains rich field components and can be used to evaluate the performance of various motors. The accuracy of the finite element calculation results is highly related to the number of units. The more units there are, the higher the calculation accuracy, but at the same time the calculation efficiency decreases. Therefore, the finite element method is often used as the last step in solving the motor problem and is not suitable for motor structure optimization calculations and joint simulations with control algorithms.

[0005] The subdomain method is a two-dimensional analytical model with high computational efficiency and accurate calculation results, which is very suitable for the structural optimization of motors. The subdomain method is suitable for motor structures modeled in polar coordinates, and can accurately predict the air gap magnetic field and slot magnetic field of the motor. The current subdomain method model is based on the magnetic vector potential field, which requires the divergence of the magnetic induction field to be zero. The magnetic vector potential field is a vector field. The differential of the vector potential field in the normal direction is used to represent the tangential component of the magnetic induction field, and the negative of the differential of the vector potential field in the tangential direction is used to represent the normal component of the magnetic induction field. The representation is not intuitive enough and is not conducive to the further promotion of the subdomain method. At the same time, the boundary of the silicon steel sheet in the traditional subdomain method is equivalent to the second-class differential boundary condition, which hinders the further application of the subdomain method in other motor configurations.

[0006] In the process of calculating various fields, the scalar potential field is more widely used and easier to understand, such as temperature field and electric potential field. The potential field gradient is the lower-level vector field, the tangential differential of the potential field is the tangential component, and the normal differential of the potential field is the normal component. In addition, by treating the armature current as the boundary, the computational complexity of modeling can be reduced. At present, the methods involving motor subdomain analysis have not yet involved solutions based on scalar potential fields. Therefore, how to introduce scalar potential fields in the analytical calculation of motor electromagnetic fields to promote the further development of subdomain methods is a technical difficulty in motor electromagnetic field calculations. Summary of the invention

[0007] The present invention aims to solve the problem that the calculation of the motor electromagnetic field subdomain method is limited to the magnetic vector potential field, and proposes a method for calculating the air gap magnetic field of a permanent magnet motor based on the magnetomotive force field subdomain. The present invention uses the magnetomotive force field of the scalar potential as the basic field to perform analytical calculations of the motor electromagnetic field. This method can establish a complete magnetomotive force field that includes the entire motor domain, and calculate the magnetic induction field distribution of the motor based on the magnetomotive force field.

[0008] The present invention provides a method for analyzing the air gap magnetic field of a permanent magnet motor based on a magnetomotive force field subdomain, which comprises the following steps:

[0009] 1) Decompose the permanent magnet motor into motor subdomains and select the subdomain to be analyzed;

[0010] 2) Equivalence is performed on the permanent magnet domain and the armature winding so that the curl of the magnetic field intensity in each subdomain is zero;

[0011] 3) Establish the differential equations of each subdomain according to the shape and excitation conditions of each subdomain; solve the differential equations of each subdomain to obtain the general solution expression and the special solution expression with undetermined coefficients;

[0012] 4) Process the boundary conditions of each subdomain and establish the boundary equations of undetermined coefficients corresponding to the general solution of the subdomain;

[0013] 5) According to the continuity characteristics of magnetomotive force and normal magnetic induction intensity, the contact interface equations between subdomains are established to form a group of interface contact equations with undetermined coefficients for general solutions of each subdomain;

[0014] 6) Combine the boundary equations of step 4) and the interface contact equations of step 5) to form a linear equation system with the undetermined coefficients as independent variables, solve the equation system to obtain the solution of the undetermined coefficients, assign the value of the undetermined coefficients to the general solution expression of each subdomain, and combine the special solution expression to obtain the magnetomotive potential field of each subdomain of the motor;

[0015] 7) The gradient of the magnetomotive force field of the subdomain is solved, and the reverse of the gradient is the magnetic field intensity. The product of the magnetic field intensity and the magnetic permeability of the material is the magnetic induction field intensity of the subdomain.

[0016] Compared with the prior art, the present invention has the following beneficial effects:

[0017] The analytical method based on the magnetomotive force field adopted in the present invention is more intuitive than the magnetic vector potential field, thereby simplifying the complex traditional subdomain method and making the subdomain method easy to understand and apply.

[0018] The present invention equates the armature current in the slot to the boundary value, so that the differential equation in the slot domain is equivalent to the Laplace equation from the Poisson equation, eliminating the derivation and calculation of the special solution in the slot domain and simplifying the calculation process of the subdomain model.

[0019] The subdomain method based on magnetomotive force can obtain the complete magnetomotive force expression of the motor. Combined with the current motor magnetic circuit analysis method, it can be used to perform various analyses and optimizations of the motor. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 is a flow chart of the method of the present invention;

[0021] Figure 2 is a schematic structural diagram of a surface-mounted permanent magnet motor used in an embodiment of the present invention;

[0022] Figure 3 yes Figure 2 The enlarged structural diagram of the surface-mounted permanent magnet motor part A;

[0023] Figure 4 It is a schematic diagram of the magnetomotive force distribution of the armature winding at a certain moment;

[0024] Figure 5 It is the equivalent schematic diagram of the magnetomotive force boundary of the motor slot;

[0025] Figure 6 is the comparison result of radial magnetic field intensity in the middle of the air gap under no-load condition;

[0026] Figure 7 It is the comparison result of radial magnetic field intensity in the middle of the air gap under load conditions. DETAILED DESCRIPTION

[0027] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings in the embodiments of the present invention. The motor air gap magnetic field analytical calculation method based on the magnetomotive force field subdomain described in the present invention is applicable to surface-mounted permanent magnet synchronous motors including brushless DC motors. This embodiment uses a 48-slot 52-pole outer rotor surface-mounted permanent magnet motor for illustration.

[0028] Obviously, the described embodiments are only part of the embodiments of the invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative work are within the scope of protection of the present invention.

[0029] The method of the present invention is based on the following assumptions:

[0030] a. Ignore the end effect of the motor;

[0031] b. The magnetic permeability of silicon steel sheet is considered infinite;

[0032] c. The armature current is evenly distributed in the slot area.

[0033] The above assumptions are reasonable equivalences based on motor design practice, so these assumptions are valid and the calculation results of the present invention have high accuracy.

[0034] like Figure 1 As shown, the method for analyzing the magnetic field subdomain of a permanent magnet motor based on the magnetomotive force field proposed in this embodiment is specifically as follows:

[0035] Step A: Decompose the motor into subdomains based on the motor’s geometric structure and material properties, and select the subdomain to be analyzed;

[0036] like Figure 2 and Figure 3 As shown, the motor can be divided into a permanent magnet domain 4, an air gap domain 3, a slot domain 5, an in-slot domain 1, a rotor core 6 and a stator core 2. The division of each sub-domain is shown in Figure 3 Before solving the overall domain of the motor, each subdomain needs to be modeled.

[0037] Since the rotor core 6 and the stator core 2 are made of silicon steel sheets, their magnetic permeability is approximately infinite compared to air or permanent magnets, and they can be regarded as equipotential bodies with no magnetic pressure drop inside. The magnetomotive force of the rotor core 6 is defined as zero, and the boundary of the stator core 2 is determined by the distribution state of the armature current. Therefore, in the specific modeling and solution process of this embodiment, the rotor core 6 and the stator core 2 are not considered. This embodiment only selects the permanent magnet domain 4, the air gap domain 3, the slot domain 5 and the slot domain 1 as the subdomains to be analyzed, and establishes a subdomain model, and solves the corresponding general solution and special solution according to the boundary conditions.

[0038] Step B: Equivalence is performed on the permanent magnet domain and the armature winding so that the curl of the magnetic field intensity in each subdomain is zero.

[0039] Step B1: The permanent magnet domain is equivalent to a charge source, and the charges are distributed at both ends of the permanent magnet magnetization. After the equivalent, the residual magnetic field intensity of the permanent magnet domain is considered to be the polarization intensity of the charge source. The polarization intensity of the charge source and the magnetic field intensity vector jointly determine the magnetic induction field intensity of the permanent magnet domain.

[0040] Step B2: The armature current is equivalent to the magnetomotive force of the contact interface between the stator core and the slot area, the slot area, and the air gap area. In this example, assuming that the motor control adopts the d-axis current control method, the magnetomotive force distribution of the armature at a certain moment is as follows: Figure 4 As shown, the magnetomotive force transition in the slot domain is temporarily transitioned by oblique lines.

[0041] The magnetomotive force distribution of the slot area and the slot area is as follows: Figure 5 As shown, the slot and the boundaries in the slot are expanded in the order of abcdefgh and projected on the x-axis. Assuming that the current in the motor slot is uniformly distributed and the magnetic voltage drop of the stator core is zero, the magnetomotive force is linearly distributed on the left and right boundaries of the slot domain, and the other boundaries are constant distribution.

[0042] Step C: Establish the differential equations of each subdomain according to the shape and excitation conditions of each subdomain; further combine the known radial or circumferential boundary conditions on both sides of each subdomain to solve the differential equations of each subdomain and obtain the general solution and special solution expressions with undetermined coefficients;

[0043] Step C1: Establish the expression of the magnetomotive force field of the permanent magnet domain:

[0044] For the permanent magnet domain, the differential equation is expressed as: , when the permanent magnet is radially excited, , the expanded form is:

[0045]

[0046] in, represents the magnetomotive force of the subdomain, represents the radial coordinate, represents the circumferential coordinate, represents the relative magnetic permeability, Indicates the order, Represents a permanent magnet The radial remanent magnetization intensity is Indicates the mechanical speed of the motor. represents the initial position of the rotor, Indicates time.

[0047] Radial component of residual magnetization The expression is:

[0048]

[0049] Indicates The radial residual magnetization intensity is expressed as:

[0050]

[0051] The expression of the permanent magnet domain solution is as follows:

[0052]

[0053] in , used to simplify expressions, , , , express The unknown coefficient sequence of order, , is a single undetermined coefficient. It represents the special solution of the subdomain, which is determined by the magnetization state of the permanent magnet and is expressed by Fourier decomposition. The expression is:

[0054]

[0055] in, represents the component of the particular solution on the cosine, represents the component of a particular solution on the sine.

[0056] Step C2: Establish the expression of magnetomotive force field in the air gap domain:

[0057] For the air gap domain, since it is passive and irrotational, the magnetomotive potential field can be used to describe the entire air gap subdomain. The differential expression of the air gap domain is: , the equation is the Laplace equation, and the expanded form is:

[0058]

[0059] The expression of the solution is:

[0060]

[0061] in, , , , express The unknown coefficient sequence of order, , is a single undetermined coefficient.

[0062] Step C3: Establish the magnetomotive force field expression of the slot domain:

[0063] For the slot domain, since it is passive and irrotational, the magnetomotive potential field can be used to describe the entire air gap subdomain. The differential equation is the Laplace equation, and the initial expression of the general solution is:

[0064]

[0065] in, ,in represents the circumferential order of the notch domain, represents the angular span of the slot area in the circumferential direction, ,in represents the radial order of the notch domain, represents the span of the slot domain in the radial direction, Indicates The radial starting coordinates of the slot domain, , , , Respectively represent Slot Field Stage, The unknown coefficient sequence of order, , , , For the A single undetermined coefficient for the slot domain;

[0066] The unknown parameters of the circumferential sine expansion part of the expression are determined by the circumferential boundary, the unknown parameters of the radial sine expansion part of the expression are determined by the radial boundary, and the remaining parameters are determined by the four corner point values ​​of the subdomain. For the slot domain, the magnetomotive force of the radial boundary is known and the boundary value is constant. The general solution equation of the slot domain can be simplified to:

[0067]

[0068] in: , , , are the magnetomotive force values ​​of the radial starting boundary and radial ending boundary of the slot domain, , They are the circumferential coordinate values ​​of the radial starting boundary and radial ending boundary of the slot domain respectively.

[0069] Step C4: Establish the expression of magnetomotive force field in the slot:

[0070] For the slot domain, according to the magnetomotive force analysis method, after the armature current is equivalent to the magnetomotive force boundary, the slot domain is sourceless and irrotational, and the magnetomotive force field can be used to describe the entire air gap subdomain. The differential equation is the Laplace equation, and the initial expression of the general solution is:

[0071]

[0072] in, ,in represents the circumferential order of the slot domain, It represents the angular span of the groove area in the circumferential direction. ,in represents the radial order of the slot domain, It represents the radial span of the slot area. Indicates The radial starting coordinate of the domain in the slot, , , , Respectively represent In-slot domain Stage, The unknown coefficient sequence of order, , , , For the A single undetermined coefficient for the slot domain;

[0073] For the inner domain of the slot, the radial boundary is known and can be expressed by the following boundary expression:

[0074]

[0075] in, , are the expressions of the magnetomotive force at the radial starting boundary and radial ending boundary of the slot domain, respectively. , are the values ​​of the magnetomotive force at the initial position of the radial boundary, , are the slopes of the respective magnetomotive force at the radial boundaries.

[0076] Substituting the magnetomotive force expression of the radial boundaries on both sides into the initial expression of the general solution of the slot domain, the general solution expression of the slot domain can be simplified to:

[0077]

[0078] in, is a known expression, from The radial boundary states on both sides of the slot are determined by the internal domain, and the specific expression is:

[0079]

[0080] Step D: Process the boundary conditions of each subdomain and establish the boundary equations of undetermined coefficients of the corresponding subdomain general solution; the boundary conditions and processing methods of each subdomain are as follows:

[0081] Step D1: The interface between the permanent magnet domain and the rotor core is used as the boundary of the permanent magnet domain, and the magnetomotive force value of the boundary is zero. Establish the equation of undetermined coefficients, and keep the sine, cosine and constant terms at the boundary after the general solution and the special solution are superimposed to zero, that is:

[0082]

[0083]

[0084]

[0085] in, represents the radial coordinate of the outer boundary of the permanent magnet domain, express The logarithm of .

[0086] Step D2: The radial boundary of the notch domain has been shown in the form of a solution, and there are no other boundaries.

[0087] Step D3: The radial boundary of the slot domain has been presented in the form of a solution. The slot bottom of the slot domain contacts the stator core and is defined as the boundary. The magnetomotive force of the boundary is a constant value, which is determined by the state of the armature winding and is expressed as The equation of undetermined coefficients is established to keep the consistency of the coefficients of the sine component at the bottom boundary of the slot:

[0088]

[0089] in, Indicates The coefficient of the bottom boundary magnetomotive force of the slot domain is transformed into the sinusoidal part of the slot domain, and the expression is:

[0090]

[0091] Indicates that when converting the coefficients of the sine part in the slot domain, the The solved part of the domain in the slot The impact on the conversion results is specifically expressed as follows:

[0092]

[0093] Step E: Establish the contact interface equation based on the continuity principle of the magnetomotive potential field and the consistency principle of the normal magnetic induction intensity; ensure the consistency of the corresponding coefficients of the contact interface equation, and form an interface contact equation group with undetermined coefficients of the general solution of each subdomain;

[0094] The boundaries between subdomains in the present invention should meet the following two conditions:

[0095] 1. The magnetomotive force is continuous at the interface.

[0096] 2. The magnetic induction field strength perpendicular to the interface is consistent.

[0097] Based on the above conditions, step E of the present invention specifically includes:

[0098] Step E1: Establish the continuous interface equation of magnetomotive force between the permanent magnet domain and the air gap domain, and keep the consistency of sine coefficient, cosine coefficient and constant coefficient. The equations are as follows:

[0099]

[0100]

[0101]

[0102] in, shows the radial coordinates of the inner boundary of the permanent magnet domain, express The logarithm of Represents the logarithm of the radial coordinate of the outer boundary of the air gap domain.

[0103] Step E2: Establish the continuous interface equation of the normal magnetic induction field intensity between the permanent magnet domain and the air gap domain, including the sum of the differential part of the general solution, the differential part of the special solution and the residual magnetization intensity part, and keep the consistency of the sine coefficients, cosine coefficients and constant coefficients on both sides of the boundary. The equations of the undetermined coefficients are:

[0104]

[0105]

[0106]

[0107] in, and Represent the magnetic permeability correlation coefficients of the permanent magnet domain and the air gap domain, respectively.

[0108] Step E3: Establishing the continuity equation of the magnetomotive force field at the contact interface between the air gap domain and the notch domain;

[0109] The boundary between the air gap domain and the stator is divided into two cases. One part is in contact with the slot domain ( Figure 3 3-2) belongs to the interface; the other part is in direct contact with the stator punching ( Figure 3 3-1) in the figure belongs to the boundary. The two cases are processed separately and then the results of the two parts are superimposed.

[0110] When calculating the direct contact part between the air gap and the stator punching, the magnetomotive force of the direct contact part is determined by the stator armature winding, and the magnetomotive force of the interface in contact with the slot is defined as zero, thus forming the boundary magnetomotive force equation , Fourier decomposition is used to transform the boundary magnetomotive force equation into the general solution form in the air gap domain, and the corresponding sine and cosine term coefficients are:

[0111]

[0112]

[0113] Fourier decomposition is used to convert boundary data from the slot domain to the air gap domain, and the boundary function of the air gap domain and the contact part of the stator punching are superimposed to jointly maintain the consistency of the sine coefficient, cosine coefficient and constant coefficient. The cosine coefficient equation is as follows:

[0114]

[0115] in, represents the logarithm of the radial coordinate of the boundary within the air gap domain, Indicates The notch domain is opposite to the air gap domain. The contribution of the coefficient of the order cosine component is expressed as:

[0116]

[0117] in, Indicates The logarithm of the radial coordinates of the outer boundary of the slot domain, Indicates The starting circumferential coordinates of the slot domain, Indicates The ending circumferential coordinate of the slot domain.

[0118] The sin coefficient equation is as follows:

[0119]

[0120] in, Indicates The notch domain is opposite to the air gap domain. The contribution of the coefficient of the order sine component is expressed as:

[0121]

[0122] The constant coefficient equation is as follows:

[0123]

[0124] in, Indicates The contribution of the slot domain to the average value of the air gap domain is expressed as:

[0125]

[0126] Step E4: Establish a continuity equation for the radial magnetic induction field strength in the air gap domain and the slot domain. The interface data is converted from the air gap domain to the slot domain to maintain the consistency of the sinusoidal function coefficients. The equation is as follows:

[0127]

[0128] Step E5: Establish the continuity equation of the magnetomotive force field in the slot domain and the slot domain;

[0129] A portion of the slot area close to the slot side boundary is connected to the stator core ( Figure 31-1, 1-2 in the figure), the remaining part is in contact with the slot domain, the two parts are calculated independently, and the results are superimposed. The left and right magnetomotive forces of the part connected to the stator core are constant, but the values ​​are different. Temporarily set the magnetomotive force of the part in contact with the slot domain to zero, and the boundary function close to the boundary on one side of the slot is expressed as , the sinusoidal component coefficients after the boundary function is transformed into the general solution form are as follows:

[0130]

[0131] The interface data is converted from the slot domain to the slot domain boundary, and the contact part between the slot interface and the stator core is superimposed ( Figure 3 The magnetomotive force of 1-1 and 1-2) in the above equations is as follows:

[0132]

[0133] in, The coefficient transformation from the sinusoidal part of the slot domain to the sinusoidal part of the slot domain is expressed as:

[0134]

[0135] Indicates the solved portion of the slot domain Coefficient transformation to the sinusoidal component in the slot domain and the solved part in the slot domain The impact on coefficient transformation is expressed as:

[0136]

[0137] Step E6: Establish the continuity equation of the magnetic induction intensity field in the slot domain and the slot domain. The interface data is converted from the slot domain to the slot domain boundary to maintain the consistency of the sinusoidal function coefficients. The equation is as follows:

[0138]

[0139] in, The coefficient transformation from the sinusoidal part in the slot to the sinusoidal component in the slot mouth is expressed as:

[0140]

[0141] The coefficient transformation from the known part in the slot domain to the sinusoidal component in the slot domain is expressed as:

[0142]

[0143] in, Indicates Solved part of slot domain On the outer boundary The partial differential equation of is expressed as follows:

[0144]

[0145] Step F: Combine the boundary equations of step D and the interface contact equations of step E to form a linear equation group with unknown coefficients as independent variables, solve the linear equation group, assign the unknown coefficient values ​​to the general solution expression of each subdomain, and combine the specific solution expression to obtain the magnetomotive force field of each subdomain of the motor, thereby obtaining the magnetomotive force field of the entire motor.

[0146] Step G: The gradient of the magnetomotive force field of the subdomain is solved. The reverse of the gradient is the magnetic field intensity. The product of the magnetic field intensity and the magnetic permeability of the material is the magnetic induction field intensity of the subdomain.

[0147] Using the above steps, the calculation results of the embodiment are compared with the finite element simulation results. The radial magnetic field intensity in the middle of the air gap domain under no load is as follows Figure 6 As shown in the figure, the red curve is the finite element simulation result, and the blue curve is the result of the analytical method calculation, which is completely consistent with the radial magnetic field strength in the middle of the air gap domain under load. Figure 7 As shown in the figure, the light blue curve is the calculation result of this analytical method, and the orange curve is the finite element simulation result. The two curves are almost completely overlapped. This calculation method can obtain accurate calculation results under both no-load and loaded conditions, which verifies the feasibility of this method.

[0148] The above-mentioned embodiments only express several implementation methods of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the attached claims.

Claims

1. A method for analyzing the air gap magnetic field of a permanent magnet motor based on a magnetomotive force field subdomain, characterized in that: The steps include: 1) Decompose the permanent magnet motor into motor subdomains and select the subdomain to be analyzed; 2) Equivalence is performed on the permanent magnet domain and the armature winding so that the curl of the magnetic field intensity in each subdomain is zero; 3) Establish the differential equations of each subdomain according to the shape and excitation conditions of each subdomain; solve the differential equations of each subdomain to obtain the general solution expression and the special solution expression with undetermined coefficients; In step 3), the differential equations of each subdomain are: The differential equation in the permanent magnet domain is as follows: ； in, represents the magnetomotive force of the subdomain, represents the radial coordinate, represents the circumferential coordinate, represents the relative magnetic permeability, Indicates the order, Represents a permanent magnet The radial remanent magnetization intensity is Indicates the mechanical speed of the motor. represents the initial position of the rotor, Indicates time; The air gap domain and the slot domain are passive irrotational fields. After the armature current is equivalent to the magnetomotive force boundary, the slot domain is equivalent to a passive irrotational field. The passive irrotational field satisfies the condition that the magnetic field intensity curl is zero. The differential equations of the air gap domain, the slot domain, and the slot domain are all Laplace equations. 4) Process the boundary conditions of each subdomain and establish the boundary equations of undetermined coefficients corresponding to the general solution of the subdomain; 5) According to the continuity characteristics of magnetomotive force and normal magnetic induction intensity, the contact interface equations between subdomains are established to form a group of interface contact equations with undetermined coefficients for general solutions of each subdomain; The continuity characteristics of the magnetomotive force and the normal magnetic induction intensity in step 5) are: the magnetomotive force is continuous at the interface and the magnetic induction field intensity perpendicular to the interface is consistent between the subdomains; Step 5) specifically includes: Step 5.1) Establish the continuous interface equation of magnetomotive force and the continuous interface equation of normal magnetic induction intensity between the permanent magnet domain and the air gap domain, and keep the consistency of sine coefficients, cosine coefficients and constant coefficients on both sides of the contact interface; Step 5.2) Establish the continuity equation of the magnetomotive force field at the contact interface between the air gap domain and the slot domain, use Fourier decomposition to convert data from the slot domain to the air gap domain, superimpose the boundary magnetomotive force of the contact part between the air gap domain and the stator core, and maintain the consistency of the sine coefficient, cosine coefficient and constant coefficient on both sides of the boundary; Step 5.3) Establish the continuity equation of the normal magnetic induction intensity of the air gap domain and the slot domain, perform Fourier transformation on the interface data from the air gap domain to the slot domain, and keep the consistency of the sine coefficients on both sides of the boundary; Step 5.4) Establish the continuity equation of the magnetomotive force field in the slot domain and the slot domain, perform Fourier transformation of the interface data from the slot domain to the slot domain, and superimpose the magnetomotive force of the contact part between the slot domain interface and the stator core to maintain the consistency of the boundary sine coefficient; Step 5.5) Establish the continuity equation of the magnetic induction intensity field in the slot inner domain and the slot outer domain, and transform the interface data from the slot inner domain to the slot outer domain, keeping the consistency of the sinusoidal coefficients on both sides of the interface; 6) Combine the boundary equations of step 4) and the interface contact equations of step 5) to form a linear equation system with the undetermined coefficients as independent variables, solve the equation system to obtain the solution of the undetermined coefficients, assign the value of the undetermined coefficients to the general solution expression of each subdomain, and combine the special solution expression to obtain the magnetomotive potential field of each subdomain of the motor; 7) The gradient of the magnetomotive force field of the subdomain is solved, and the reverse of the gradient is the magnetic field intensity. The product of the magnetic field intensity and the magnetic permeability of the material is the magnetic induction field intensity of the subdomain.

2. The method according to claim 1, characterized in that The permanent magnet motor is a surface mounted permanent magnet synchronous motor.

3. The method according to claim 1, characterized in that The step 1) is: decomposing the motor into subdomains according to the geometric structure and material properties of the permanent magnet motor, and selecting the permanent magnet domain, the air gap domain, the slot domain and the slot domain as the subdomains to be analyzed.

4. The method according to claim 1, characterized in that: The step 2) is: the permanent magnet domain is equivalent to a magnetic charge source, the magnetic charge is distributed at both ends of the permanent magnet magnetization, and the armature winding is equivalent to the magnetomotive force boundary of the contact interface between the stator core and the slot domain, the slot domain, and the air gap domain.

5. The method according to claim 1, characterized in that In step 3), the differential equations of each subdomain are solved to obtain general solution expressions and special solution expressions with undetermined coefficients, specifically: General solution for permanent magnet domain The expression is as follows: ； in, , , , express The unknown coefficient sequence of order, , is a single undetermined coefficient, represents the logarithm of the radial coordinate; Special solution for permanent magnet domain The expression is: ； in, represents the component of the particular solution on the cosine, represents the component of the particular solution on the sine; There is no special solution in the air gap domain, and the general solution The expression is as follows: ； in, , , , express The unknown coefficient sequence of order, , is a single undetermined coefficient; There is no special solution in the slot domain. Combined with the known radial boundary magnetomotive force on both sides, the general solution is The expression is as follows: ； in, , represents the circumferential order of the notch domain, Indicates The angular span of the slot area in the circumferential direction, Indicates The starting angle of the slot area, , Indicates Slot Field The unknown coefficient sequence of order, , is a known coefficient; There is no special solution in the slot, and the general solution is given by combining the known radial boundary magnetomotive force on both sides. The expression is as follows: ； in, ,in represents the circumferential order of the slot domain, Indicates The angular span of the inner domain of each slot in the circumferential direction, Indicates The starting angle of the slot domain, , Indicates In-slot domain The coefficient sequence of the order to be determined, is a known expression, from The left and right boundary conditions of the domain within a slot are determined.

6. The method according to claim 5, characterized in that The step 4) includes: Step 4.1) The interface between the permanent magnet domain and the rotor core is equivalent to the boundary of the permanent magnet domain. The boundary magnetomotive force value is defined as zero. The established boundary equation of undetermined coefficients keeps the sine, residual and constant terms at the boundary after the general solution and the special solution are superimposed to zero; Step 4.2) There are no independent boundary equations for the notch domain and the air gap domain; Step 4.3) The magnetomotive force at the bottom of the slot is determined by the armature current. The boundary magnetomotive force value is converted to the general solution form of the slot domain. The boundary equation of the unknown coefficient is established based on the slot bottom boundary to maintain the consistency of the sine coefficient.

7. The method according to claim 6, characterized in that The boundary equation of the undetermined coefficients established in step 4.1) is: ； ； ； in, Represents the logarithmic value of the radial coordinate of the outer boundary of the permanent magnet domain; The boundary equation of the undetermined coefficient established by the groove bottom boundary in step 4.3) is: ； in, Indicates The coefficient of the bottom boundary magnetomotive force of the slot domain is transformed into the sinusoidal part of the slot domain, and the expression is: ； Indicates that when the coefficients of the sine part of the slot are converted, the The non-sinusoidal part of the general solution in the slot The impact on the conversion result is expressed as: 。

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