Non-uniform layered space magnetic conductance modeling method based on conformal mapping

Abstract

The invention discloses a non-uniform layered space magnetic conductance modeling method based on conformal mapping, and relates to the technical field of permanent magnet motors, and the method comprises the steps: obtaining a generalized polygon calculation domain for calculating magnetic conductance based on a motor topology model; analyzing the generalized polygon computational domain through multiple conformal mapping to obtain a relative permeability function in the air gap of the motor; and obtaining a non-uniform space magnetic conductance model based on conformal mapping by adopting a layering method based on the relative specific magnetic conductance function. According to the method, complex air gap magnetic conductance characterization is achieved by constructing conformal mapping, the concept that multi-layer magnetic conductance distribution exists in an air gap space is introduced for the first time, and model errors caused by the fact that linear magnetic conductance or other single-layer magnetic conductance is adopted in an air gap in a traditional method are effectively corrected; and meanwhile, the conformal mapping analysis of the magnetic conductance of the motor only needs the tooth space topology of the motor and does not need complex geometric subdivision, so that the precision of a magnetic field analysis model of the permanent magnet vernier motor is remarkably improved, the modeling complexity is reduced, and a theoretical basis is provided for accurate modeling and performance optimization of the motor.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of permanent magnet motors, in particular to a non-uniform layered space magnetic guide modeling method based on conformal mapping. BACKGROUND

[0002] Permanent magnet synchronous motors (PMSM) are widely used in industrial drives, robots, household appliances and other fields due to their high efficiency, good dynamic performance and small size. With the continuous progress of motor technology, researchers have gradually focused on how to improve the performance of permanent magnet vernier motors, especially in accurately predicting the magnetic field distribution and torque characteristics during motor design. However, the traditional permanent magnet vernier motor modeling method has certain limitations, especially when considering the non-uniformity of the magnetic guide space, the existing model often has difficulty in effectively capturing the complexity of the air gap magnetic guide and magnetic field distribution inside the motor.

[0003] Existing motor modeling methods are mostly based on finite element analysis (FEA) or analytical methods. Although these methods can provide relatively accurate magnetic field distribution and torque calculation results, the calculation complexity is high, and there are still certain accuracy and efficiency problems when dealing with the dynamic characteristics of the internal space of the motor. Especially when considering the nonlinear changes of the magnetic guide and torque distribution in the permanent magnet motor, the traditional modeling method often requires a large amount of computing resources, and the universality of the model is poor, making it difficult to adapt to the optimization needs under different design conditions. SUMMARY

[0004] The technical problem to be solved by the present application is to overcome the shortcomings of the prior art and provide a non-uniform layered space magnetic guide modeling method based on conformal mapping. The present application improves the accuracy of motor magnetic guide modeling without additional motor parameter information, introduces higher dimensional features through conformal mapping to obtain air gap space magnetic guide distribution, so that the magnetic guide at each point in the air gap is described, and the air gap is no longer described by a single magnetic guide. Reasonable space layered modeling of the air gap makes the motor modeling analysis more reliable. At the same time, the modeling and analysis process no longer depends on the motor grid division, making the modeling and analysis process more simple, improving the modeling accuracy while reducing the modeling difficulty.

[0005] The application adopts the following technical solutions to solve the above technical problems:

[0006] According to the non-uniform layered space magnetic guide modeling method based on conformal mapping provided by the present application, the method comprises the following steps:

[0007] Firstly, based on the motor topology model, a generalized polygon calculation domain for calculating the magnetic guide is obtained;

[0008] Secondly, the generalized polygon calculation domain is analyzed through multiple conformal mapping to obtain a relative permeance function in a motor air gap;

[0009] Finally, a non-uniform spatial permeance model based on conformal mapping is obtained by using a layered method based on the relative permeance function.

[0010] As a further optimization scheme of the non-uniform layered spatial permeance modeling method based on conformal mapping, a generalized polygon calculation domain for calculating the permeance is obtained based on a motor topology model; the specific implementation is as follows:

[0011] The air gap space corresponding to a single tooth slot of the permanent magnet Vernier motor is separated into an independent analysis unit, the annular air gap topology structure from the stator tooth top to the surface of the rotor permanent magnet in the independent analysis unit is extracted, and the generalized polygon calculation domain including the stator inner arc segment Γ_s, the rotor outer arc segment Γ_r, and the infinite boundary extending in the tangent direction of the two arc segments is constructed according to the annular air gap topology structure.

[0012] As a further optimization scheme of the non-uniform layered spatial permeance modeling method based on conformal mapping, the generalized polygon calculation domain is analyzed through multiple conformal mapping to obtain a relative permeance function in a motor air gap; the specific implementation is as follows:

[0013] Step A, the generalized polygon calculation domain is converted into a polygon domain with a vertex set {V_1, V_2, …, V_n} through a logarithmic transformation in the conformal mapping of the calculation domain complex plane, wherein each vertex is located at the stator yoke, the surface of the rotor permanent magnet, and the air gap extension junction, V_i is the i th vertex, n≥i≥1 and i is an integer, and n is the total number of vertices;

[0014] Step B, a boundary magnetic potential constraint condition is defined, and the boundary magnetic potential constraint condition includes: a rotor side permanent magnet equivalent surface magnetic potential Φ_m1, a stator side iron core magnetic potential Φ_m2, an air gap two-side extension boundary vertex set {V_+∞, V_-∞}, and an equivalent air gap distance σ between the stator and the rotor; wherein V_+∞ is a positive infinite node, and V_-∞ is a negative infinite node;

[0015] Step C, based on the Schwarz-Christoffel mapping, a conformal mapping of the polygon domain and another upper half complex plane domain is constructed, and then the other upper half complex plane domain is mapped into a regular rectangular domain through a conformal mapping; according to the boundary magnetic potential constraint condition, an analytical solution of the magnetic potential at any spatial position in the regular rectangular domain is obtained According to Maxwell's equation, the expression of the air gap magnetic field intensity is:

[0016]

[0017] wherein H is the air gap magnetic field intensity, For gradient operator, For the magnetic potential difference of stator and rotor, f(x, y) is the analytical result of relative permeance function, which is used to represent the change of magnetic field distribution in the air gap before and after slotting;

[0018] When the motor is not slotted, under the boundary conditions of the same air gap length and the magnetic potential difference of stator and rotor:

[0019]

[0020] Wherein, x, y are the transverse coordinates of the air gap position and the longitudinal coordinates of the air gap position respectively.

[0021] As a further optimization scheme of the non-uniform layered space permeance modeling method based on conformal mapping, f(x, y) is consistent with the modulation function in the air gap magnetic field modulation theory.

[0022] As a further optimization scheme of the non-uniform layered space permeance modeling method based on conformal mapping, when the equivalent air gap distance σ between the stator and the rotor is less than 2% of the air gap arc length corresponding to a single slot, two slot positions are selected to form the starting solution domain when constructing the annular air gap topology structure, and based on the analytical domain, the analytical result of the relative permeance function is taken in the air gap region within the slot range sandwiched by the two slots.

[0023] As a further optimization scheme of the non-uniform layered space permeance modeling method based on conformal mapping, based on the relative permeance function, the layered function is combined to obtain the non-uniform space permeance model based on conformal mapping; Specifically as follows:

[0024] The air gap is divided into multiple layers, l is the layer number, and there are l max layers in total;

[0025] The layering method is non-uniform layering, and the width of each layer is r l The specific acquisition is as follows:

[0026] r l = r min (1+a) l

[0027] Wherein, r min is the minimum layering scale, and a is the layering coefficient;

[0028] The relative permeance function is multiplied by the average air gap permeance to obtain the permeance function:

[0029] λ(θ, l) = f(x, y) x λ mean

[0030] Wherein, λ(θ, l) is the air gap space permeance, and λ mean is the average air gap permeance;

[0031] The magnetic conductance function is Fourier decomposed, and the magnetic conductance result is characterized as a cosine function and form:

[0032]

[0033] Wherein, λ0(l) is the DC component of the magnetic conductance at the position of l, λ m (m) is the amplitude of the magnetic conductance at the position of l, N s is the number of stator slots, and θ is the rotor angle.

[0034] As a further optimization scheme of the non-uniform layered space magnetic conductance modeling method based on conformal mapping, a is [0.1 0.5], and the magnetic conductance layer is gradually expanded along with the space region represented by a, wherein the size of a is proportional to the length of the air gap.

[0035] As a further optimization scheme of the non-uniform layered space magnetic conductance modeling method based on conformal mapping, the pre-basis of the Schwarz-Christoffel mapping is to complete the mapping transformation of the annular region surrounded by the inner arc segment of the stator and the outer arc segment of the rotor from the generalized polygon domain to the polygon domain through the logarithmic mapping; the polygon domain completed by the mapping includes real vertex information and vertex information at two infinite places.

[0036] As a further optimization scheme of the non-uniform layered space magnetic conductance modeling method based on conformal mapping, the conformal mapping of the upper half complex plane domain to a regular rectangular domain is as follows:

[0037] dw=Re jφ dz

[0038]

[0039] Wherein, R is the z-plane lower coordinate radius, e is the natural base, j is the imaginary number, w, z, t are three plane transformation domains, S, K are two constants used to determine the size range and position of the polygon domain, φ is the z-plane angle, a0, b0, c0 are the complex coordinates of the three vertices in the polygon domain, respectively, α, β, γ are the angles of the three vertices and the origin, S, K are constants, used to determine the size range and position of the polygon domain.

[0040] As a further optimization scheme of the non-uniform layered space magnetic conductance modeling method based on conformal mapping, the relative permeance function is obtained using all the magnetic conductance information in the air gap, and the relative permeance function form is three-dimensional data including motor radial, motor tangential and magnetic conductance amplitude; when the air gap is divided into x max layers, at least the magnetic potential function is divided into (x max +4) layers.

[0041] Compared with the prior art, the present application has the following technical effects:

[0042] (1) The present application breaks the conventional motor air gap magnetic field modeling method, first introduces the air gap permeance spatial distribution, and adopts the non-uniform non-uniform layering method, improves the analysis dimension and modeling accuracy of the motor air gap permeance, makes the motor air gap modeling more reasonable and accurate, and avoids direct derivation of the permeance function, avoids grid division and iterative calculation of numerical methods.

[0043] (2) The present application uses conformal mapping to transform complex boundary into uniform condition through function transformation, reduces the difficulty of equation solving, and for motor design, only the topological model of the solution domain needs to be changed to conveniently obtain the permeance parameters of the motor after correction, without the need for time-consuming finite element software, which is efficient and friendly to the early design of the motor.

[0044] (3) The spatial permeance given by the present application overcomes the drawbacks of traditional conformal mapping which only uses the air gap center line electromagnetic parameters, by starting from a global point of view, fully using various parameters in the analytical result, without increasing the additional mathematical difficulty, the accuracy of the permeance calculation is improved. BRIEF DESCRIPTION OF DRAWINGS

[0045] Figure 1 Flowchart of the method of the present application;

[0046] Figure 2 For the selected motor tooth slot topology and polygon domain diagram, each parameter for calculating the spatial permeance using the value transformation is included;

[0047] Figure 3 For the selected motor tooth slot structure, the spatial distribution diagram of the motor air gap magnetic potential is shown.

[0048] Figure 4 For the selected motor tooth slot structure, the spatial distribution diagram of the air gap permeance when the air gap is divided into 40 layers under the non-uniform layering structure. DETAILED DESCRIPTION

[0049] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be described in detail below with reference to the drawings and specific embodiments.

[0050] In recent years, the modeling method based on conformal transformation has gradually attracted the attention of researchers. As a mathematical tool, conformal transformation can transform complex geometric and physical problems into a more concise and manageable form, while ensuring the accuracy and computational efficiency of the model. Therefore, a non-uniform layered space magnetic guide modeling method based on conformal mapping is proposed, which can effectively describe the spatial magnetic guide distribution in the motor air gap and provide more accurate calculation results. This method not only reduces the computational complexity, but also better adapts to the needs of different motor designs, with strong universality and practicality. Figure 1 As shown in the figure, the steps of the present application are as follows:

[0051] Step S1: Separate the air gap space corresponding to a single tooth slot of the permanent magnet vernier motor into independent analysis units, extract the annular air gap topology from the stator tooth top to the surface of the rotor permanent magnet, and construct a generalized polygon calculation domain containing the inner arc segment Γ_s of the stator, the outer arc segment Γ_r of the rotor, and the extended boundaries on both sides;

[0052] Step S2: Convert the generalized polygon calculation domain into a polygon domain with a vertex set {V_1, V_2, …, V_n} through logarithmic transformation in the complex plane conformal mapping of the calculation domain, wherein each vertex is located at the stator yoke, the surface of the rotor permanent magnet, and the air gap extension junction, V_i is the i-th vertex, n≥i≥1 and i is an integer, and n is the total number of vertices.

[0053] Step S3: Define boundary magnetic potential constraint conditions, including:

[0054] a. Rotor side permanent magnet equivalent surface magnetic potential Φ_m1

[0055] b. Stator side iron core magnetic potential Φ_m2

[0056] c. Air gap extension boundary vertex set {V_+∞, V_-∞}

[0057] d. Equivalent air gap distance σ between stator and rotor;

[0058] V_+∞ is a positive infinity node, and V_-∞ is a negative infinity node;

[0059] The overall topological boundary and the label are shown in Figure 2 .

[0060] Step S4: Based on the mapping theory of Schwarz-Christoffel mapping, construct the conformal correspondence relationship between the polygon domain and the upper half complex plane, and obtain the analytical solution of the magnetic potential in the regular rectangular domain according to the constraint conditions, as shown in Figure 3 .

[0061] Step S5: Perform the inverse conformal mapping operation to map the regular domain magnetic field decoupling result back to the original air gap topology, combine the Maxwell method to derive the spatial magnetic field intensity distribution, and establish the relative spatial permeance model before and after slotting by comparing the magnetic field intensity change before and after slotting.

[0062] The pre-requisite of Schwarz-Christoffel mapping theory is to complete the mapping from the annular region surrounded by the stator inner arc segment and the rotor outer arc segment to the polygonal domain through the logarithmic mapping.

[0063] The above-mentioned polygonal domain should contain real vertex information and vertex information at two infinite places.

[0064] The conformal mapping for processing the annular domain to the polygonal domain is as follows:

[0065] dw=Re jφ dz

[0066]

[0067] wherein R is the coordinate radius under the z plane, e is the natural base, j is the imaginary number, w, z, t are three plane transformation domains, S, K are two constants for determining the size range and position of the polygonal domain, φ is the angle of the z plane, a0, b0, c0 are the complex coordinates of the three vertices in the w plane, respectively, α, β, γ are the angles of the three vertices and the origin, S, K are constants for determining the size range and position of the polygonal domain.

[0068] The result finally obtained by the two conformal mappings is the magnetic potential function at any position in the analytical domain According to the Maxwell equation, the expression of the air gap magnetic field intensity is:

[0069]

[0070] When the motor is not slotted, under the same boundary conditions:

[0071]

[0072] wherein H is the air gap magnetic field intensity, is the gradient operator, is the stator-rotor magnetic potential difference, σ is the equivalent air gap distance between the stator and the rotor, f(x, y) is the distribution function, f(x, y) represents the change of the magnetic field distribution in the air gap before and after slotting, f(x, y) is consistent with the modulation function in the air gap magnetic field modulation theory, and f(x, y) is the relative permeance function;

[0073] When the motor is not slotted, under the same boundary conditions of air gap length and stator-rotor magnetic potential difference:

[0074]

[0075] where, is the magnetic potential function.

[0076] The air gap is divided into multiple layers, x is the layer number, and the total number of layers is x max layers, and the results are shown in Figure 4

[0077] The layering method is non-uniform layering, and the width of each layer is r l The specific process is as follows:

[0078] r l = r min (1 + a) l

[0079] where, r min is the minimum layering scale, a is the layering coefficient, and is usually [0.1 0.5], representing the gradual expansion of the magnetic permeability layering with the characteristic spatial region away from the slot position. The size of a is proportional to the air gap length.

[0080] The relative permeance function is multiplied by the average air gap permeance to obtain the permeance function:

[0081] λ(θ, l) = f(x, y) x λ mean

[0082] where, λ(θ, l) is the air gap space permeance, and λ mean is the average air gap permeance;

[0083] The Fourier decomposition of the permeance function is performed, and the permeance result is represented as a cosine function and form:

[0084]

[0085] where, λ0(l) is the DC component of the permeance at position l, and λ m (l) is the m-order permeance amplitude at position l, N s is the number of stator slots, and θ is the rotor angle.

[0086] The reason for using exponential layering is that near the source end, which refers to the slot end that causes air gap distortion, the degree of air gap permeability distortion is higher, and the magnetic field energy is more abundant than the far end. Using geometric uniform layering will dilute part of the information, so a suitable layering method is needed.

[0087] The relative permeance function form should be a three-dimensional data containing the radial direction of the motor, the tangential direction of the motor, and the permeance amplitude.

[0088] In addition, it needs to be additionally explained that, in order to prevent the air gap permeance function from being discontinuous due to local mutations in the solving data near the stator and rotor wall, the air gap is divided into x​max When the layer is guaranteed, at least the magnetic potential function is divided into (x max +4) layers. When the equivalent air gap distance σ between the stator and the rotor is less than the air gap arc length corresponding to a single slot, two slot positions should be selected to form the initial solution domain when selecting the annular air gap topology. Correspondingly, the analytical result of the relative permeance function is taken as the air gap region within the slot range sandwiched by the two slots, so as to reduce the influence of the existence of the extended boundary vertex set {V_+∞,V_-∞} on the solution result.

[0089] The space permeance modeling method based on conformal mapping designed by the application adopts motor tooth slot topology division, boundary information assignment, multiple conformal mapping, and complete utilization of electromagnetic information at each place in the analytical domain. On the premise of not additionally increasing the analytical difficulty, a higher-dimensional space air gap permeance model is analytically obtained, and the air gap permeance is spatially layered through an exponential model. Not only is the air gap permeance modeling precision improved, but also the fast combination of the motor topology and the air gap permeance analytical result is well realized. To some extent, the drawbacks of the traditional permeance obtaining method, i.e., the need for grid division of the analytical domain or strong dependence on global parameters of the motor, are overcome, the precision of motor modeling is improved, and the difficulty of modeling is reduced.

[0090] The above is only a specific embodiment of the application, but the protection scope of the application is not limited thereto. Any person skilled in the art can easily think of changes or replacements within the technical range disclosed by the application, which should be covered within the protection scope of the application.

Claims

1. A method for modeling non-uniform layered spatial magnetic permeability based on conformal mapping, characterized in that: include: First, based on the motor topology model, a generalized polygonal computational domain for calculating the magnetic permeance is obtained. Secondly, the generalized polygonal computational domain is analyzed through multiple conformal mappings to obtain the relative permeance function in the motor air gap; Finally, a hierarchical method is adopted based on the relative contrast permeance function to obtain an inhomogeneous spatial permeance model based on conformal mapping.

2. The method for modeling non-uniform layered spatial magnetic permeability based on conformal mapping according to claim 1, characterized in that: Based on the motor topology model, the generalized polygonal computational domain for calculating the magnetic permeance is obtained; the details are as follows: The air gap space corresponding to a single tooth slot of a permanent magnet vernier motor is separated into independent analysis units. The annular air gap topology from the stator tooth tip to the rotor permanent magnet surface in the independent analysis unit is extracted. Based on the annular air gap topology, a generalized polygonal computational domain is constructed, including the stator inner arc segment Γ_s, the rotor outer arc segment Γ_r, and the infinite boundary extending along the tangential direction of the two arc segments.

3. The method for modeling non-uniform layered spatial magnetic permeability based on conformal mapping according to claim 1, characterized in that: The relative permeance function in the motor air gap is obtained by analyzing the generalized polygonal computational domain through multiple conformal mappings. The details are as follows: Step A: convert the generalized polygonal computational domain into a polygonal domain with a vertex set {V_1, V_2, …, V_n} by a logarithmic transformation in the computational domain complex plane conformal mapping, wherein each vertex is located at the junction of the stator yoke, the rotor permanent magnet surface, and the air gap extension, V_i is the i-th vertex, n ≥ i ≥ 1 and i is an integer, and n is the total number of vertices; Step B: defining boundary magnetic potential constraints, which include: the equivalent surface magnetic potential Φ_m1 of the permanent magnet on the rotor side, the magnetic potential Φ_m2 of the iron core on the stator side, the set of boundary vertices {V_+∞,V_-∞} extending on both sides of the air gap, and the equivalent air gap distance σ between the stator and rotor; where V_+∞ is a positive infinity node and V_-∞ is a negative infinity node; Step C: Based on the Schwarz-Christoffel mapping, a conformal mapping is constructed between the polygonal domain and another upper half complex plane domain, and then the other upper half complex plane domain is mapped into a regular rectangular domain through a conformal mapping; according to the boundary magnetic potential constraint condition, an analytical solution of the magnetic potential at any spatial position in the regular rectangular domain is obtained. According to Maxwell's equations, the expression of the magnetic field strength in the air gap is: Where H is the air gap magnetic field strength, is the gradient operator, is the stator-rotor magnetic potential difference, f(x,y) is the analytical result of the relative permeability function, which is used to characterize the change of the magnetic field distribution in the air gap before and after slotting; When the motor is not slotted, under the boundary conditions of the same air gap length and stator-rotor magnetic potential difference: Among them, x and y are the horizontal coordinate and the vertical coordinate of the air gap position respectively.

4. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 3, characterized in that: f(x,y) is consistent with the modulation function in the air gap magnetic field modulation theory.

5. The method for modeling non-uniform layered spatial permeability based on conformal mapping according to claim 2, characterized in that: When the equivalent air gap distance σ between the stator and rotor is less than 2% of the air gap arc length corresponding to a single slot, two slot positions are selected to form the initial solution domain when constructing the annular air gap topology structure. Based on this analytical domain, the air gap area within the range of one slot uniformly sandwiched between the two slots is taken by comparing the analytical results of the magnetic permeance function.

6. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 3, characterized in that: Based on the relative permeance function and the layered function, a non-uniform spatial permeance model based on conformal mapping is obtained; the details are as follows: The air gap is divided into multiple layers, l is the layer number, a total of l max layer; The stratification method is non-uniform stratification, and the width of each layer is r l Get the details as follows: r l ＝r min (1+a) l Among them, r min is the minimum stratification scale, a is the stratification coefficient; The relative permeance function multiplied by the average value of the air gap permeance is the permeance function: λ(θ,l)=f(x,y)×λ mean Among them, λ(θ,l) is the air gap permeability, λ mean is the mean value of air gap permeability; Perform Fourier decomposition on the permeance function, and the permeance result is represented by the cosine function and form: Where λ0(l) is the DC component of the permeance at position l, λ m (l) is the m-order permeability amplitude at position l, N s is the number of stator slots, and θ is the rotor angle.

7. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 6, characterized in that: a is [0.1 0.5], indicating that as the distance from the slot position increases, the magnetic permeability stratification gradually expands with the spatial region represented, where the size of a is proportional to the air gap length.

8. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 3, characterized in that: The pre-condition of Schwarz-Christoffel mapping is to complete the mapping transformation from the generalized polygonal domain of the annular area formed by the inner arc segment of the stator and the outer arc segment of the rotor to the polygonal domain through logarithmic mapping. The mapped polygonal domain includes real vertex information and two vertex information at infinity.

9. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 3, characterized in that: The conformal mapping of the upper half complex plane domain to a regular rectangular domain is as follows: dw=Re jφ dz Where R is the coordinate radius in the z plane, e is the natural base, j is an imaginary number, w, z, and t are three plane transformation domains, S and K are two constants used to determine the size range and position of the polygonal domain, φ is the z plane angle, a0, b0, and c0 are the complex coordinates of the three vertices in the polygonal domain corresponding to the w plane, α, β, and γ are the angles between the three vertices and the origin, and S and K are constants used to determine the size range and position of the polygonal domain.

10. The method for modeling non-uniform layered spatial permeance based on conformal mapping according to claim 6, characterized in that: The relative permeance function is obtained by using all the permeance information in the air gap. The relative permeance function is in the form of three-dimensional data including the motor radial direction, the motor tangential direction and the permeance amplitude. The air gap is divided into x max layer, at least ensure that the magnetic potential function is divided into (x max +4) layers.

Images