A conformal mapping-based non-uniform layered space magnetic guide modeling method

Abstract

The application discloses a non-uniform layered space magnetic guide modeling method based on conformal mapping, relates to the technical field of permanent magnet motors, and is based on a motor topology model to obtain a generalized polygon calculation domain used for calculating a magnetic guide; the generalized polygon calculation domain is analyzed through multiple conformal mappings to obtain a relative magnetic guide function in a motor air gap; and a layered method is adopted based on the relative magnetic guide function to obtain a non-uniform space magnetic guide model based on conformal mapping. The method realizes complex air gap magnetic guide representation through the construction of conformal mapping, first introduces the concept that there are multiple layers of magnetic guide distribution in the air gap space, effectively corrects the model error caused by the adoption of the air gap center line magnetic guide or other single-layer magnetic guide in the traditional method, and simultaneously, the conformal mapping analysis of the motor magnetic guide only needs the motor tooth slot topology, does not need complex geometric subdivision, significantly improves the precision of the permanent magnet vernier motor magnetic field analysis model and reduces the modeling complexity, and provides a theoretical basis for the accurate modeling and performance optimization of the motor.

Description

Technical Field

[0001] This invention relates to the field of permanent magnet motor technology, and in particular to a non-uniform layered spatial magnetic permeability modeling method based on conformal mapping. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in industrial drives, robotics, and home appliances due to their high efficiency, excellent dynamic performance, and small size. With the continuous advancement of motor technology, researchers are increasingly focusing on improving the performance of PMSMs, particularly in accurately predicting their magnetic field distribution and torque characteristics during motor design. However, traditional PMSM modeling methods have certain limitations, especially when considering the non-uniformity of magnetic permeability space; existing models often struggle to effectively capture the complexity of the air gap permeability and magnetic field distribution within the motor.

[0003] Existing motor modeling methods are mostly based on finite element analysis (FEA) or analytical methods. While these methods can provide relatively accurate magnetic field distribution and torque calculation results, they have high computational complexity and still suffer from accuracy and efficiency issues when dealing with the dynamic characteristics of changes in the internal space of the motor. Especially when considering the nonlinear changes in permeability and torque distribution in permanent magnet motors, traditional modeling methods often require a large amount of computational resources, and the models have poor universality, making it difficult to adapt to optimization needs under different design conditions. Summary of the Invention

[0004] The technical problem this invention aims to solve is to overcome the shortcomings of existing technologies and provide a non-uniform hierarchical spatial permeability modeling method based on conformal mapping. This invention improves the accuracy of motor permeability modeling without requiring additional motor parameter information. By introducing higher-dimensional features through conformal mapping, it obtains the spatial permeability distribution of the air gap, ensuring that the permeability at all points in the air gap is described, rather than describing the entire air gap with a single permeability. Reasonable spatial hierarchical modeling of the air gap enhances the reliability of motor modeling and analysis. Furthermore, the analysis process no longer relies on motor mesh generation, simplifying the modeling and analysis process, improving modeling accuracy while reducing modeling difficulty.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] A non-uniform hierarchical spatial magnetic permeability modeling method based on conformal mapping proposed in this invention includes:

[0007] First, based on the motor topology model, a generalized polygonal computational domain is obtained for calculating magnetic permeability;

[0008] Secondly, the relative permeability function in the air gap of the motor is obtained by analyzing the computational domain of the generalized polygon through multiple conformal mappings.

[0009] Finally, based on the relative magnetic permeability function, a hierarchical method is used to obtain a non-uniform spatial magnetic permeability model based on conformal mapping.

[0010] As a further optimization of the non-uniform hierarchical spatial magnetic permeability modeling method based on conformal mapping described in this invention, a generalized polygonal computational domain for calculating magnetic permeability is obtained based on the motor topology model; specifically as follows:

[0011] The air gap space corresponding to a single tooth slot of the 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 including the stator inner arc segment Γ_s, the rotor outer arc segment Γ_r, and the infinite boundary of the two arc segments extending tangentially is constructed.

[0012] As a further optimization of the non-uniform layered spatial permeability modeling method based on conformal mapping described in this invention, the relative permeability function in the air gap of the motor is obtained by analyzing the generalized polygon computational domain through multiple conformal mappings; specifically as follows:

[0013] Step A: Through logarithmic transformation in the conformal mapping of the computational domain complex plane, the generalized polygon computational domain is transformed into a polygonal domain with a vertex set {V_1,V_2,…,V_n}, where each vertex is located at the intersection of the stator yoke, the surface of the rotor permanent magnet 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;

[0014] Step B: Define the boundary magnetic potential constraint conditions. The boundary magnetic potential constraint conditions include: the equivalent surface magnetic potential Φ_m1 of the rotor-side permanent magnet, the magnetic potential Φ_m2 of the stator-side iron core, the set of vertices of the extended boundary on both sides of the air gap {V_+∞,V_-∞}, and the equivalent air gap distance σ between the stator and rotor; where V_+∞ is the node at positive infinity and V_-∞ is the node at negative infinity.

[0015] Step C: Based on the Schwarz-Christophe map, construct a conformal mapping between the polygonal domain and another upper half-complex plane domain. Then, through a conformal mapping, map the other upper half-complex plane domain into a regular rectangular domain. According to the boundary magnetic potential constraint conditions, obtain the analytical solution of the magnetic potential at any spatial location within the regular rectangular domain. According to Maxwell's equations, the expression for the magnetic field strength inside the air gap is:

[0016]

[0017] Where H is the air gap magnetic field strength, For gradient operators, The magnetic potential difference between the stator and rotor is given by f(x,y), which is the analytical result of the relative magnetic permeability function and is used to characterize the change in the magnetic field distribution in the air gap before and after slotting.

[0018] Under the boundary conditions of the same air gap length and stator-rotor magnetic potential difference when the motor is not slotted:

[0019]

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

[0021] As a further optimization of the non-uniform layered spatial magnetic permeability modeling method based on conformal mapping described in this invention, 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 spatial magnetic permeability modeling method based on conformal mapping described in this invention, 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. Based on this analytical domain, the air gap region within the slot uniformly sandwiched by the two slots is taken as the analytical result of the magnetic permeability function.

[0023] As a further optimization of the non-uniform layered spatial magnetic permeability modeling method based on conformal mapping described in this invention, a non-uniform spatial magnetic permeability model based on conformal mapping is obtained by combining the relative permeability function with the layering function; specifically as follows:

[0024] The air gap is divided into multiple layers, with 'l' representing the layer number, for a total of 1. max layer;

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

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

[0027] Where, r min The minimum stratification scale is denoted by 'a', and 'a' is the stratification coefficient.

[0028] The permeability function is obtained by multiplying the relative permeability function by the average permeability of the air gap:

[0029] λ(θ,l)=f(x,y)×λ mean

[0030] Where λ(θ,l) is the air gap spatial permeability, λ mean This represents the average air gap permeability.

[0031] Fourier decomposition of the permeability function yields a permeability result represented as a cosine function in the following form:

[0032]

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

[0034] As a further optimization scheme of the non-uniform layered spatial magnetic permeability modeling method based on conformal mapping described in this invention, a is [0.1 0.5], which represents that as the position moves away from the slot opening, the magnetic permeability layering gradually expands with the spatial region being represented, where the size of a is proportional to the air gap length.

[0035] As a further optimization scheme of the non-uniform layered spatial magnetic permeability modeling method based on conformal mapping described in this invention, the Schwarz-Christophe map precondition is to complete the mapping transformation from the generalized polygonal domain to the polygonal domain of the annular region enclosed by the inner arc segment of the stator and the outer arc segment of the rotor through logarithmic mapping; the completed polygonal domain includes real vertex information and vertex information at two infinity locations.

[0036] As a further optimization of the non-uniform hierarchical spatial magnetic permeability modeling method based on conformal mapping described in this invention, the conformal mapping of the upper half-complex plane domain to a regular rectangular domain takes the following two forms:

[0037] dw=Re jφ dz

[0038]

[0039] Where R is the coordinate radius in the z-plane, e is the natural base, j is the imaginary number, w, z, and t are the three planar transformation domains, S and K are two constants used to determine the size and location of the polygon domain, φ is the z-plane angle, a0, b0, and c0 are the complex coordinates of the three vertices in the polygon 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 and location of the polygon domain.

[0040] As a further optimization of the non-uniform layered spatial magnetic permeability modeling method based on conformal mapping described in this invention, a relative permeability function is obtained using all magnetic permeability information within the air gap. The relative permeability function is in the form of three-dimensional data including the motor radial direction, the motor tangential direction, and the magnetic permeability amplitude. The air gap is divided into x... max When layering, at least ensure that the magnetic potential function is divided into (x max +4) layers.

[0041] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0042] (1) This invention breaks with the conventional modeling method of air gap magnetic field of motor. It introduces the spatial distribution of air gap magnetic permeability for the first time and adopts a non-uniform layering method, which improves the analysis dimension and modeling accuracy of motor air gap magnetic permeability, making motor air gap modeling more reasonable and accurate. At the same time, it avoids the direct derivation of magnetic permeability function and avoids the mesh division and iterative calculation of numerical methods.

[0043] (2) This invention uses conformal mapping to transform complex boundaries into uniform conditions through function transformation, reducing the difficulty of solving equations. At the same time, for motor design, only the topology model of the solution domain needs to be modified to conveniently obtain the corrected magnetic permeability parameters of the motor, without the need for time-consuming finite element software, which is efficient and friendly to the early design of motors.

[0044] (3) The spatial magnetic permeability provided by this invention overcomes the drawback of traditional conformal mapping which only uses the electromagnetic parameters of the air gap centerline. By taking a global perspective and making full use of the parameters in the analytical results, the accuracy of magnetic permeability is improved without increasing the additional mathematical difficulty. Attached Figure Description

[0045] Figure 1 Flowchart of the method of this invention;

[0046] Figure 2 The diagram shows the selected motor cogging topology and polygonal domain, including the parameters for calculating spatial permeability using valence transformation;

[0047] Figure 3 This is a spatial distribution diagram of the air gap magnetic potential of the selected motor cog structure.

[0048] Figure 4 This is a spatial distribution diagram of the air gap magnetic permeability when the air gap is divided into 40 layers under the selected motor toothed structure and non-uniform layered structure. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0050] In recent years, conformal mapping-based modeling methods have gradually attracted researchers' attention. As a mathematical tool, conformal mapping can transform complex geometric and physical problems into simpler, more manageable forms while ensuring model accuracy and computational efficiency. Therefore, this paper proposes a non-uniform hierarchical spatial permeability modeling method based on conformal mapping. This method can effectively describe the spatial permeability distribution within the air gap of a motor and provide more accurate calculation results. This method not only reduces computational complexity but also better adapts to the needs of different motor designs, exhibiting strong versatility and practicality. Figure 1 As shown, the steps of this invention are as follows:

[0051] Step S1: Separate the air gap space corresponding to a single tooth slot of the permanent magnet vernier motor into an independent analysis unit, extract the annular air gap topology from the stator tooth tip to the surface of the rotor permanent magnet, and construct a generalized polygonal computational domain including 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: Through logarithmic transformation in the conformal mapping of the computational domain complex plane, the generalized polygon computational domain is transformed into a polygonal domain with vertex set {V_1,V_2,…,V_n}, where each vertex is located at the intersection of the stator yoke, the surface of the rotor permanent magnet 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.

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

[0054] a. Magnetic potential Φ_m1 of the equivalent surface of the rotor-side permanent magnet

[0055] b. Stator side core magnetic displacement Φ_m2

[0056] c. Set of vertices of the extended boundary on both sides of the air gap: {V_+∞,V_-∞}

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

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

[0059] Overall topology boundaries and annotations as follows Figure 2 As shown.

[0060] Step S4: Based on the Schwarz-Christophe map theory, construct the conformal correspondence between the polygonal domain and the upper half-complex plane. According to the constraints, obtain the analytical solution of the magnetic potential within the regular rectangular domain. The result is as follows: Figure 3 As shown;

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

[0062] The Schwarz-Christopheel mapping theory is based on the mapping of the annular region enclosed by the inner arc segment of the stator and the outer arc segment of the rotor to a polygonal domain through logarithmic mapping.

[0063] The aforementioned polygonal domain should contain information about real vertices as well as information about vertices at two points at infinity.

[0064] The conformal mapping for processing toroidal domains to polygonal domains takes the following two forms:

[0065] dw=Re jφ dz

[0066]

[0067] Where R is the coordinate radius in the z-plane, e is the natural base, j is the imaginary number, w, z, and t are the three planar transformation domains, S and K are two constants used to determine the size and location of the polygon domain, φ is the z-plane angle, a0, b0, and c0 are the complex coordinates of the three vertices in the polygon 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 and location of the polygon domain.

[0068] The final analytical result obtained through two conformal mappings is the magnetic potential function at any position in the analytic domain. According to Maxwell's equations, the expression for the magnetic field strength within the air gap is:

[0069]

[0070] Under the same boundary conditions, when the motor is not slotted:

[0071]

[0072] Where H is the air gap magnetic field strength, For gradient operators, σ is the magnetic potential difference between the stator and rotor, σ is the equivalent air gap distance between the stator and rotor, f(x,y) is the distribution function, f(x,y) characterizes the change in 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 permeability function.

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

[0074]

[0075] in, This is the magnetic potential function.

[0076] The air gap is divided into multiple layers, with x being the layer number, for a total of x layers. max The air gap is divided into 40 layers, and the result is as follows: Figure 4 As shown;

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

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

[0079] Where, r min The minimum stratification scale is denoted by 'a', which is the stratification coefficient, typically [0.1 0.5]. This coefficient represents the gradual expansion of magnetic permeability stratification as the distance from the slot opening increases with the spatial region being represented. The value of 'a' is proportional to the air gap length.

[0080] The permeability function is obtained by multiplying the relative permeability function by the average permeability of the air gap:

[0081] λ(θ,l)=f(x,y)×λ mean

[0082] Where λ(θ,l) is the air gap spatial permeability, λ mean This represents the average air gap permeability.

[0083] Fourier decomposition of the permeability function yields a permeability result represented as a cosine function in the following form:

[0084]

[0085] Where λ0(l) is the DC component of the magnetic permeability at position l, λ m (l) represents the m-order magnetic permeability amplitude at position l, N s θ represents the number of stator slots and θ represents 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 magnetic permeability distortion is higher, and the magnetic field energy is richer than that at the far end. Using geometrically uniform layering would dilute some information, so a suitable layering method is needed.

[0087] The relative permeability function should be in the form of three-dimensional data including the radial direction of the motor, the tangential direction of the motor, and the permeability amplitude.

[0088] In addition, it should be noted that, to prevent discontinuities in the permeability due to local abrupt changes in the solution data near the stator and rotor walls when solving the air gap permeability function, the air gap is divided into x...max When layering, at least ensure that the magnetic potential function to be obtained is divided into (x max +4) layer. Furthermore, when the equivalent air gap distance σ between the stator and 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 choosing the annular air gap topology. Correspondingly, the analytical result of the relative permeability function should be taken from the air gap region within the slot uniformly sandwiched by the two slots, in order to reduce the influence of the existence of the set of vertices of the extended boundary on both sides of the air gap {V_+∞,V_-∞} on the solution result.

[0089] The spatial permeability modeling method based on conformal mapping designed in this invention adopts a method of motor tooth cogging topology partitioning, boundary information assignment, multiple conformal mappings, and full utilization of electromagnetic information at all points in the analytical domain. Without increasing the analytical difficulty, it analytically obtains a higher-dimensional spatial air gap permeability model. Furthermore, it spatially layers the air gap permeability through an exponential model, which not only improves the accuracy of air gap permeability modeling but also effectively achieves the rapid combination of motor topology and air gap permeability analytical results. To a certain extent, it overcomes the drawbacks of traditional permeability calculation methods that require mesh partitioning of the analytical domain or strong dependence on global motor parameters, thereby improving the accuracy of motor modeling and reducing the modeling difficulty.

[0090] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for modeling non-uniform hierarchical spatial magnetic permeability based on conformal mapping, characterized in that, include: First, based on the motor topology model, a generalized polygonal computational domain is obtained for calculating magnetic permeability; Secondly, the relative permeability function in the air gap of the motor is obtained by analyzing the computational domain of the generalized polygon through multiple conformal mappings. Finally, based on the relative permeability function, a hierarchical method is used to obtain a non-uniform spatial permeability model based on conformal mapping; the details are as follows: The air gap is divided into multiple layers. Numbered in layers, a total of l max layer; The stratification method is non-uniform stratification, and the width of each layer is... The specific details are as follows: ； in, For the smallest stratification scale, Stratification coefficient; Relative permeability Multiplying by the average air gap permeability yields the permeability function: ； in, For air gap space magnetic permeability, This represents the average air gap permeability. Fourier decomposition of the permeability function yields a permeability result represented as a cosine function in the following form: ； in, for DC component of magnetic permeability at position for The m-th order permeability amplitude at position, The number of stator slots This represents the rotor angle.

2. The non-uniform layered spatial magnetic permeability modeling method based on conformal mapping according to claim 1, characterized in that, Based on the motor topology model, a generalized polygonal computational domain for calculating magnetic permeability is obtained; specifically as follows: The air gap space corresponding to a single tooth slot of the 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 including the stator inner arc segment Γ_s, the rotor outer arc segment Γ_r, and the infinite boundary of the two arc segments extending tangentially is constructed.

3. The non-uniform layered spatial magnetic permeability modeling method based on conformal mapping according to claim 1, characterized in that, The relative permeability function in the air gap of the motor is obtained by analyzing the computational domain of the generalized polygon through multiple conformal mappings; the details are as follows: Step A: Through logarithmic transformation in the conformal mapping of the computational domain complex plane, the generalized polygon computational domain is transformed into a polygonal domain with a vertex set {V_1,V_2,…,V_n}, where each vertex is located at the intersection of the stator yoke, the surface of the rotor permanent magnet 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: Define the boundary magnetic potential constraint conditions. The boundary magnetic potential constraint conditions include: the equivalent surface magnetic potential Φ_m1 of the rotor-side permanent magnet, the magnetic potential Φ_m2 of the stator-side iron core, the set of vertices of the extended boundary on both sides of the air gap {V_+∞,V_-∞}, and the equivalent air gap distance σ between the stator and rotor; where V_+∞ is the node at positive infinity and V_-∞ is the node at negative infinity. Step C: Based on the Schwarz-Christophe map, construct a conformal mapping between the polygonal domain and another upper half-complex plane domain. Then, through a conformal mapping, map the other upper half-complex plane domain into a regular rectangular domain. According to the boundary magnetic potential constraint conditions, obtain the analytical solution of the magnetic potential at any spatial location within the regular rectangular domain. According to Maxwell's equations, the expression for the magnetic field strength inside the air gap is: ； in, The air gap magnetic field strength, For gradient operators, For the magnetic potential difference between the stator and rotor, The analytical results of the relative permeability function are used to characterize the change in the magnetic field distribution in the air gap before and after slotting; Under the boundary conditions of the same air gap length and stator-rotor magnetic potential difference when the motor is not slotted: ； in, These are the horizontal coordinates and vertical coordinates of the air gap position, respectively.

4. The non-uniform layered spatial magnetic permeability modeling method based on conformal mapping according to claim 1, characterized in that, The is [0.1 0.5], where, The size is proportional to the air gap length.

5. The non-uniform hierarchical spatial magnetic permeability modeling method based on conformal mapping according to claim 3, characterized in that, The Schwarz-Christopheel mapping prerequisite is to complete the mapping transformation from the generalized polygonal domain to the polygonal domain of the annular region enclosed by the inner arc segment of the stator and the outer arc segment of the rotor through logarithmic mapping. The mapped polygonal domain includes information on real vertices as well as information on vertices at two points infinity.

6. The non-uniform layered spatial magnetic permeability modeling method based on conformal mapping according to claim 3, characterized in that, The conformal mapping of the upper half-plane domain to a regular rectangular domain takes the following two forms: ； ； Where R is the coordinate radius in the z-plane, e is the natural base, j is the imaginary number, w, z, and t are the three planar transformation domains, and S and K are two constants used to determine the size and location of the polygonal domain. Let the angle be in the z-plane. , , These are the complex coordinates of the three vertices in the w-plane corresponding to the three vertices of the polygon domain. , , Let be the angle between the three vertices and the origin.

7. The non-uniform layered spatial magnetic permeability modeling method based on conformal mapping according to claim 1, characterized in that, The relative permeability function is obtained by using all magnetic permeability information within the air gap. The relative permeability function is in the form of three-dimensional data including motor radial, motor tangential, and permeability amplitude data. The air gap is divided into x... max When layering, at least ensure that the magnetic potential function is divided into (x max +4) layers.

Images