A High-Precision Calculation Method for Transient Electromagnetic Field-Circuit Coupling Based on the A-α Formula

Abstract

This application relates to an A- α The formula-based high-precision calculation method for transient electromagnetic field-circuit coupling divides the mesh data of a 1 / 8-cycle model of a three-dimensional motor into several polyhedral smooth regions centered on the corresponding edges, and calculates the smooth vector shape functions of the edges and the smooth shape functions of the nodes in the polyhedral smooth regions; based on A- α The transient field-circuit coupling equations are constructed using formulas, transient eddy currents, and Coulomb's rules. These equations are then rewritten based on the smooth vector shape functions of each edge and node, as well as the Galerkin method, yielding the system equations for transient field-circuit coupling. These system equations are further rewritten as Newton's iterative linear equations. The Newton iterative method is then used to iteratively solve these linear equations until convergence, yielding the magnetic vector potential in the field-circuit coupling. This achieves a high-precision numerical solution to the transient field-circuit coupling problem in complex motor systems.

Description

Technical Field

[0001] This application relates to the field of engineering electromagnetic calculation technology, and in particular to a method based on A- α A high-precision calculation method for transient electromagnetic field-circuit coupling formulas. Background Technology

[0002] The coupling problem between transient electromagnetic fields and external circuits is widespread in engineering applications such as motors, power equipment, and power electronic systems. Numerical calculations typically rely on the finite element method. The current mainstream method is the scalar potential in the magnetic vector potential A-circuit. α (abbreviated as A-) α The transient field-path coupling control equations were established, and edge element and nodal element methods were used to analyze A and α Spatial discretization is performed, and combined with improved node element techniques in circuits, to achieve strongly coupled calculations of transient electromagnetic fields and circuits. However, existing A- α In multi-connected conductor structures, coils, and thick conductor regions, the transient field-circuit coupling method is prone to non-physical solutions or numerical non-convergence due to the insufficient uniqueness of the scalar potential inside the conductor, thus affecting the reliability of the calculation results. Meanwhile, the traditional A-... α Field-circuit coupling methods require simultaneous coupling of nodal finite element methods, edge finite element methods, and improved nodal analysis algorithms in the circuit. Different algorithms differ in variable types, degree-of-freedom definitions, and discretization forms, leading to complex coupling implementation processes and a lack of a unified, universal coupling mechanism. Furthermore, in the numerical modeling of complex electromagnetic equipment (such as motor systems), the computational domain typically includes multiple structural components such as stators, rotors, windings, and housings, exhibiting complex geometry and uneven material distribution. Therefore, unstructured meshes are usually required for discretization. Under these conditions, traditional nodal and edge element methods still exhibit limitations in numerical accuracy and stability when performing transient field-circuit coupling calculations. Summary of the Invention

[0003] For the existing A- α The transient field-circuit coupling method has shortcomings in terms of conductor region uniqueness, algorithm coupling compatibility, and computational accuracy in unstructured meshes. Therefore, it is necessary to provide a numerical computation method for transient field-circuit coupling to achieve high-precision numerical solutions to transient field-circuit coupling problems in complex motor systems. Specifically, this involves a method based on A- α The formula provides a high-precision calculation method for transient electromagnetic field-circuit coupling, including:

[0004] S1: Import mesh data of the 3D motor 1 / 8 cycle model;

[0005] S2: Divide the mesh data into several polyhedral smooth regions centered on the corresponding edges according to each edge. The polyhedral smooth region is formed by the nodes, body centers, and face centers of the tetrahedral elements in the mesh data surrounding the corresponding edges. Calculate the smooth vector shape function of the corresponding edge based on the vector shape function of each tetrahedral element in the polyhedral smooth region corresponding to the edge. Calculate the smooth shape function of the corresponding node based on the shape function of all nodes in the tetrahedral element corresponding to the node.

[0006] S3; Based on A- α The transient field-path coupling equations are constructed using formulas, transient eddy currents, and Coulomb's specification. The transient field-path coupling equations are then rewritten based on the smooth vector shape functions of each edge, the smooth shape functions of each node, and the Galerkin method, resulting in the system equations for transient field-path coupling. The system equations for transient field-path coupling are further rewritten as Newton's iterative linear equations.

[0007] S4: The Newton iteration method is used to iteratively solve the Newton iteration linear equation until it converges, and the magnetic vector potential in the field-circuit coupling is obtained.

[0008] Preferably, the formula for calculating the smooth vector shape function of the edge is:

[0009] ;

[0010] in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Indicates edge e vector shape function, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n The edges of a tetrahedral unit e vector shape function, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

[0011] Preferably, the formula for calculating the smooth shape function of a node is:

[0012] ;

[0013] in, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Represents a node i Shape function at the location, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n Nodes in a tetrahedral element i Shape function at the location, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

[0014] Preferably, the smoothing function is the reciprocal of the volume of the smooth region of the corresponding polyhedron, and the smoothing function satisfies the following condition:

[0015] ;

[0016] in, Indicates the first k A polyhedral smooth region Indicates the first k Smoothing functions for polyhedral smooth regions This indicates differentiation over the integration domain.

[0017] Preferably, the expression for the transient field-path coupling equations is:

[0018] ;

[0019] in, Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. This indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. Represents the integration field. This indicates differentiation over the integration domain.

[0020] Preferably, the process of obtaining the system equations for transient field-path coupling includes:

[0021] Based on the smooth vector shape function of each edge, the smooth shape function of each node, and the Galerkin method, the system equations of transient field-path coupling are rewritten to obtain the weak form of the system equations of transient field-path coupling.

[0022] Based on the smooth vector shape function of each edge, the smooth shape function of each node, and the difference theorem, expand the magnetic vector potential and the virtual scalar potential;

[0023] By using the weak form of the coupled transient field-circuit coupling equations, the expansion of the magnetic vector potential, and the expansion of the virtual scalar potential, the system equations for the coupled transient field-circuit are obtained.

[0024] Preferably, the weak form of the transient field-path coupling equations is expressed as:

[0025] ;

[0026] in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates the first k A polyhedral smooth region This represents a smooth region representing the coarse conductor region in the periodic model. Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. This indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. This indicates differentiation over the integration domain.

[0027] Preferably, the expansions of the magnetic vector potential and the virtual scalar potential are as follows:

[0028] ;

[0029] ;

[0030] in, Indicates magnetic vector potential. Represents a virtual scalar potential. Representing a smooth region of a polyhedron The number of middle edges, Representing a smooth region of a polyhedron The number of nodes in the middle, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates edge e Tangential magnetic vector value on, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Represents a node i The virtual scalar potential value at that location.

[0031] Preferably, the Newton iterative linear equation is expressed as:

[0032] ;

[0033] ;

[0034] ;

[0035] in, Represents the Jacobian matrix of the system; Represents the residual of the magnetic vector potential; Represents the residuals of the system equations; Indicates the first h +1 iteration step magnetic vector position; Indicates the first h The magnetic vector position of the iteration step; Represents the system stiffness matrix; This represents a known external force vector.

[0036] Preferably, the iterative solution process of the Newton iteration method includes:

[0037] Step 1: Initialize the iteration steps;

[0038] Step 2: Solve the Newton-based linear equation for the current iteration step to obtain the magnetic vector potential for the current iteration step;

[0039] Step 3: Add the magnetic vector bit of the current iteration step to the residual of the magnetic vector bit to obtain the magnetic vector bit of the next iteration step;

[0040] Step 4: Update the iteration step;

[0041] Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the field-circuit coupling is determined.

[0042] Beneficial effects: First, based on the edges of the mesh data of the 1 / 8 cycle model of the 3D motor, the mesh data is divided into several polyhedral smooth regions centered on the corresponding edges, and the smooth vector shape functions of the edges and the smooth shape functions of the nodes in the polyhedral smooth regions are calculated; second, based on A- α The transient field-circuit coupling equations are constructed using formulas, transient eddy currents, and Coulomb's rules. These equations are then rewritten based on the smooth vector shape functions of each edge and node, as well as the Galerkin method, yielding the system equations for transient field-circuit coupling. These system equations are further rewritten as Newton's iterative linear equations. Finally, the Newton iterative method is used to iteratively solve these linear equations until convergence, obtaining the magnetic vector potential in the field-circuit coupling. This achieves a high-precision numerical solution to the transient field-circuit coupling problem in complex motor systems. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0044] Figure 1 In the embodiments of this application, based on A- α A flowchart of the high-precision calculation method for transient electromagnetic field-circuit coupling formula.

[0045] Figure 2 This is a schematic diagram of the construction of the smooth polyhedral region in an embodiment of this application.

[0046] Figure 3a This is a thermogram of magnetic induction intensity obtained by the method in this embodiment of the application.

[0047] Figure 3b This is a thermal map of magnetic induction intensity obtained by the benchmarking software in the embodiments of this application. Detailed Implementation

[0048] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the specific embodiments of this application are described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of this application. However, this application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.

[0049] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0050] like Figure 1 As shown, this embodiment provides a method based on A- α The formula provides a high-precision calculation method for transient electromagnetic field-circuit coupling, including:

[0051] S1: Import the mesh data of the 3D motor 1 / 8 cycle model.

[0052] S2: Divide the mesh data into several polyhedral smooth regions centered on the corresponding edges according to each edge. The polyhedral smooth region is formed by the nodes, body centers (midpoints of tetrahedrons), and face centers (centers of face points) of the tetrahedral elements in the mesh data surrounding the corresponding edges. Calculate the smooth vector shape function of the corresponding edge based on the vector shape function of each tetrahedral element in the polyhedral smooth region corresponding to the edge. Calculate the smooth shape function of the corresponding node based on the shape function of all nodes in the tetrahedral element corresponding to the node.

[0053] For transient field-circuit coupling, edge-based smoothing techniques can more accurately simulate the skin effect and proximity effect of eddy currents inside solid conductors by ensuring smoothness at the edges of the elements and reducing numerical oscillations. This avoids deviations in eddy current loss calculations caused by unsmooth field quantities. Simultaneously, the induced eddy current terms obtained from the field quantity solution can be accurately fed back to the circuit section, making the entire field-circuit coupling solution and iteration more stable.

[0054] Specifically, the strategy for constructing the smooth domain based on edge-based smooth finite element method is as follows: the entire computational domain is discretized into... N Each polyhedral smooth region is a non-overlapping polyhedral smooth region, and each polyhedral smooth region uniquely corresponds to a central edge. e The mathematical expression is: , ,in This indicates the number of smooth regions on the polyhedron. Indicates the first A polyhedral smooth region Indicates the first A polyhedral smooth region This represents emptiness, thus ensuring the mutual exclusivity between smooth polyhedral regions. For three-dimensional problems, the geometric boundary of this polyhedral volume partition is defined by the area surrounding the edges. e It is formed by connecting the nodes, body centers, and face centers of adjacent tetrahedral elements, such as Figure 2 As shown.

[0055] In this embodiment, the core concept of edge-smoothing finite element technology is to smooth the gradient of the element shape function. The formula for calculating the smooth vector shape function of the edge is:

[0056] ;

[0057] in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Indicates edge e vector shape function, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n The edges of a tetrahedral unit e vector shape function, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

[0058] The formula for calculating the smooth shape function of a node is:

[0059] ;

[0060] in, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Represents a node i Shape function at the location, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n Nodes in a tetrahedral element i Shape function at the location, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

[0061] In this embodiment, the smoothing function is the reciprocal of the volume of the smooth region of the corresponding polyhedron, and the smoothing function satisfies the following condition:

[0062] ;

[0063] in, Indicates the first k A polyhedral smooth region Indicates the first k Smoothing functions for polyhedral smooth regions This indicates differentiation over the integration domain.

[0064] S3; Based on A- α The transient field-path coupling equations are constructed using formulas, transient eddy currents, and Coulomb's specification. The transient field-path coupling equations are then rewritten based on the smooth vector shape functions of each edge, the smooth shape functions of each node, and the Galerkin method, resulting in the system equations for transient field-path coupling. The system equations for transient field-path coupling are further rewritten as Newton's iterative linear equations.

[0065] Specifically, considering the transient induced eddy current term, for the coarse conductor region in the periodic model, based on the novel A- α The transient eddy current field coupled voltage excitation equation is as follows:

[0066] ;

[0067] Based on the novel A- α The transient eddy current coupling current excitation equation for the formula is as follows:

[0068] ;

[0069] ;

[0070] The current I and voltage V mentioned above are placed as excitation sources in the vector on the right. If an external circuit is connected at this point, the excitation sources I and V are not directly given, and a circuit equation needs to be introduced for coupling and solution. For the non-conductor region, to ensure unique solution, a virtual scalar potential needs to be introduced for Coulomb regulation. Therefore, the expression of the transient field-circuit coupling equation set is:

[0071] ;

[0072] in, Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. This indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. Represents the integration field. This indicates differentiation over the integration domain. In this embodiment, the external resistor, external inductor, and voltage across the winding in the external circuit are discretized using an improved nodal method.

[0073] Introducing source potential This is used to represent the voltage degree of freedom across the terminals. For a terminal in the domain after finite element discretization... =0, another terminal =1; at this point, combined with the terminal voltage difference , This can be used to represent directional source electric fields. This quantity transforms the terminal voltage, which is essentially a globally constrained quantity, into a local volume source term. This avoids the traditional A- The method suffers from problems such as non-unique electric scalar potential, difficulty in applying boundary conditions, and numerical ill-conditioning.

[0074] Assuming the source current is a closed current field, this is numerically correct. However, when calculating the divergence on both sides of the transient eddy current field equations above, it can be found that... It satisfies the Laplace equation. And on the boundary surface of the entire domain... By forcibly applying the homogeneous Dirichlet boundary conditions, it can be easily known that... It is zero in the non-conductive region. Therefore, the Coulomb gauge can be introduced here, namely the novel A- α The formula is based on the transient field-path coupled eddy current field Coulomb specification.

[0075] At this time, based on the new A- α The formula for the induced eddy current field equations of current excitation and voltage excitation, along with the introduction of external circuits and their circuit equations, thus realizes the coupling of electromagnetic field and circuit, that is, the establishment of the transient field-circuit coupling equation set.

[0076] Furthermore, the process of obtaining the system equations for transient field-path coupling includes:

[0077] Based on the smooth vector shape function of each edge, the smooth shape function of each node, and the Galerkin method, the system equations of transient field-path coupling are rewritten to obtain the weak form of the system equations of transient field-path coupling.

[0078] By using the Galerkin method and edge-based smooth finite element technique, A- α By spatially discretizing edge elements and node elements, and further discretizing the components in the external circuit using improved node elements, the weak form of the transient field-circuit coupling equations is expressed as follows:

[0079] ;

[0080] in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates the first k A polyhedral smooth region This represents a smooth region representing the coarse conductor region in the periodic model. Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. This indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. This indicates differentiation over the integration domain.

[0081] Based on the smooth vector shape functions of each edge, the smooth vector shape function of each node, and the difference theorem, expand the magnetic vector potential and the virtual scalar potential; the expansions of the magnetic vector potential and the virtual scalar potential are as follows:

[0082] ;

[0083] ;

[0084] in, Indicates magnetic vector potential. Represents a virtual scalar potential. Representing a smooth region of a polyhedron The number of middle edges, Representing a smooth region of a polyhedron The number of nodes in the middle, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates edge e Tangential magnetic vector value on, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Represents a node i The virtual scalar potential value at that location.

[0085] By using the weak form of the coupled transient field-circuit equations, the expansion of the magnetic vector potential, and the expansion of the virtual scalar potential, we obtain the system equations for the coupled transient field-circuit. The expression for the system equations is as follows:

[0086] ;

[0087] in, Represents the system stiffness matrix; Indicates magnetic vector potential; This represents a known external force vector.

[0088] At this point, the equation takes the form of a directly solvable linear equation. However, when there is a nonlinear material in the transient solution domain, i.e., the magnetic permeability is nonlinear, the above system equation becomes nonlinear. To handle the nonlinear transient field-circuit coupling problem, the Newton-Raphson iteration method is a good choice due to its quadratic convergence speed. Specifically, the system equation is further rewritten as a Newton-Raphson iterative linear equation, which is expressed as:

[0089] ;

[0090] ;

[0091] ;

[0092] in, Represents the Jacobian matrix of the system; Represents the residual of the magnetic vector potential; Represents the residuals of the system equations; Indicates the first h +1 iteration step magnetic vector position; Indicates the first h The magnetic vector position of the iteration step; Represents the system stiffness matrix; This represents a known external force vector.

[0093] In this embodiment, the residual of the magnetic vector potential and the residuals of the system equations The following conditions must be met:

[0094] ;

[0095] ;

[0096] in, Indicates the first preset number. This represents the second preset number, and in this embodiment, it will be... , Set them to numbers with very small absolute values.

[0097] S4: The Newton iteration method is used to iteratively solve the Newton iteration linear equation until it converges, and the magnetic vector potential in the field-circuit coupling is obtained.

[0098] Furthermore, the iterative solution process of Newton's iterative method includes:

[0099] Step 1: Initialize the iteration steps;

[0100] Step 2: Solve the Newton-based linear equation for the current iteration step to obtain the magnetic vector potential for the current iteration step;

[0101] Step 3: Add the magnetic vector bit of the current iteration step to the residual of the magnetic vector bit to obtain the magnetic vector bit of the next iteration step;

[0102] Step 4: Update the iteration step;

[0103] Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the field-circuit coupling is determined.

[0104] In this embodiment, the method further includes calculating the magnetic induction intensity and magnetic field intensity based on the magnetic vector potential, and the calculation formulas are as follows:

[0105] ;

[0106] ;

[0107] in, Indicates magnetic flux density. Indicates curl, Indicates magnetic vector potential. Indicates magnetic field strength. The magnetic permeability of the model is represented by the magnetic induction intensity and the magnetic field intensity, which are used to provide quantitative indicators for transient field-circuit coupling optimization.

[0108] To better illustrate the field-circuit coupling calculation formula, a supporting case is provided: A typical 3D motor 1 / 8 cycle model is used. For both the conductor and non-conductor regions within the model, different excitation formulas are reflected in the calculations described above. All numerical simulation time discretizations employ the backward Euler algorithm, the magnetic vector potential A uses edge-smooth nodal finite element method, and ESP uses linear nodal finite element method. Finally, the magnetic flux density thermograms calculated by this method and the benchmark software are shown below. Figure 3a , Figure 3b As shown.

[0109] This embodiment provides an A-based... α The formula-based high-precision calculation method for transient electromagnetic field-circuit coupling has the following advantages:

[0110] 1. This application proposes a synergistic coupling solution method among the nodal element method, the edge element method, and the improved nodal analysis method for circuits. In the finite element discretization process of the electromagnetic field, edge elements are used to spatially discretize the magnetic vector potential A to ensure numerical consistency in the calculation of the magnetic field curl. Simultaneously, nodal elements are used to discretize the scalar potential variables to establish a correspondence with the potential variables at circuit nodes. In the circuit system, the improved nodal analysis method is used to establish the circuit equations. By constructing a unified system of discrete equations, the nodal element variables, edge element variables, and circuit node variables are uniformly coupled, thereby achieving the synergistic solution of the electromagnetic field equations and the circuit equations. This method solves the problems of complex coupling and insufficient compatibility between different numerical methods in traditional field-circuit coupling calculations.

[0111] 2. To address the issue that complex motor systems often require discretization using unstructured meshes, this application introduces an edge-based smoothing finite element technique based on nodal elements, edge elements, and improved nodal elements. This technique smooths the gradient calculation process in traditional finite element units by constructing smooth regions on the edges of the finite element mesh, thereby reducing the impact of mesh distortion on numerical results and improving the stability and accuracy of the calculation results.

[0112] 3. The proposed numerical method is well-suited for low-order unstructured meshes and effectively addresses the instability issues in transient field-circuit coupling. Therefore, this method is applicable to complex motor systems containing various structural components such as stators, rotors, windings, and housings. By uniformly modeling different material regions, conductor structures, and external circuits within the motor, and combining finite element discretization with circuit equation solving, the transient coupling calculation between the electromagnetic field and external circuits in complex motor systems can be achieved. This method can effectively describe the electromagnetic field changes during motor startup, load variations, and electromagnetic transient processes.

[0113] 4. In the traditional method based on magnetic vector potential A and source potential α In traditional transient field-circuit coupling models, insufficient potential constraints in the conductor region often lead to non-uniqueness of solutions and are prone to non-physical solutions in multi-connected regions, coil structures, and thick conductor regions. This application introduces virtual scalar potential variables in the conductor region during field-circuit coupling modeling. By applying reasonable constraints to the electromagnetic field variables inside the conductor, a more complete potential description system is established, effectively eliminating the non-uniqueness problem in traditional models and improving the numerical stability of multi-conductor systems.

[0114] 5. A unified computational framework for transient electromagnetic field-circuit coupling: This application proposes a unified computational framework for transient electromagnetic field-circuit coupling in complex motor systems. This framework couples nodal elements, edge elements, and an improved nodal method, and uses edge-based smooth finite element technology to improve the overall consistency and computational accuracy of field-circuit coupling calculation.

[0115] In summary, by introducing a virtual scalar potential into the field-circuit coupling model, the electromagnetic field variables in the conductor region are reasonably constrained, effectively avoiding the problems of traditional A-type electromagnetic field coupling. α The field-circuit coupling method addresses the non-uniqueness of solutions in the conductor region, improving the stability and convergence of numerical calculations. Simultaneously, it achieves synergistic coupling between the nodal element method, edge element method, and improved circuit nodal analysis method within a unified mathematical framework, resolving the insufficient compatibility of multiple numerical methods in existing technologies, thus enhancing the algorithm's versatility and scalability. Furthermore, by introducing edge-based smooth finite element technology, the traditional finite element discretization method is improved, effectively reducing numerical errors and increasing the accuracy of transient electromagnetic field calculations under unstructured mesh conditions. This allows the method to be stably applied to transient coupling calculations of motor systems with complex structures such as stators, rotors, windings, and housings, providing a more accurate and stable numerical calculation method for electromagnetic transient analysis of complex motors and power equipment, demonstrating significant engineering application value.

[0116] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0117] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method based on A- α The formula-based high-precision calculation method for transient electromagnetic field-circuit coupling is characterized by, include: S1: Import mesh data of the 3D motor 1 / 8 cycle model; S2: Divide the mesh data into several polyhedral smooth regions centered on the corresponding edges according to each edge. The polyhedral smooth region is formed by the nodes, body centers, and face centers of the tetrahedral elements in the mesh data surrounding the corresponding edges. Calculate the smooth vector shape function of the corresponding edge based on the vector shape function of each tetrahedral element in the polyhedral smooth region corresponding to the edge. Calculate the smooth shape function of the corresponding node based on the shape function of all nodes in the tetrahedral element corresponding to the node. S3； Based on A- α The transient field-path coupling equations are constructed using formulas, transient eddy currents, and Coulomb's specification. These equations are then rewritten based on the smooth vector shape functions of each edge and node, as well as the Galerkin method, yielding the system equations for transient field-path coupling. These system equations are further rewritten as Newton's iterative linear equations. The expression for the transient field-path coupling equations is as follows: ； in, Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. Indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. Represents the integration field. This indicates differentiation over the integration domain; S4: The Newton iteration method is used to iteratively solve the Newton iteration linear equation until it converges, and the magnetic vector potential in the field-circuit coupling is obtained.

2. The method according to claim 1, characterized in that, The formula for calculating the smooth vector shape function of the edge is: ； in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Indicates edge e vector shape function, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n The edges of a tetrahedral unit e vector shape function, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

3. The method according to claim 1, characterized in that, The formula for calculating the smooth shape function of a node is: ； in, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates partial derivative, express r direction, r Directions include direction, direction, direction, Indicates the first k A polyhedral smooth region Represents a node i Shape function at the location, Indicates the first k Smoothing functions for polyhedral smooth regions Indicates the first k The number of tetrahedral elements in a polyhedral smooth domain. Indicates the first n Nodes in a tetrahedral element i Shape function at the location, Indicates the first n The volume of a tetrahedral unit cell This indicates differentiation over the integration domain.

4. The method according to claim 2 or 3, characterized in that, The smoothing function is the reciprocal of the volume of the smooth region of the corresponding polyhedron, and the smoothing function satisfies the following conditions: ； in, Indicates the first k A polyhedral smooth region Indicates the first k Smoothing functions for polyhedral smooth regions This indicates differentiation over the integration domain.

5. The method according to claim 1, characterized in that, The process of obtaining the system equations for transient field-path coupling includes: Based on the smooth vector shape function of each edge, the smooth shape function of each node, and the Galerkin method, the system equations of transient field-path coupling are rewritten to obtain the weak form of the system equations of transient field-path coupling. Based on the smooth vector shape function of each edge, the smooth shape function of each node, and the difference theorem, expand the magnetic vector potential and the virtual scalar potential; By using the weak form of the coupled transient field-circuit coupling equations, the expansion of the magnetic vector potential, and the expansion of the virtual scalar potential, the system equations for the coupled transient field-circuit are obtained.

6. The method according to claim 5, characterized in that, The weak form of the transient field-path coupling equations is as follows: ； in, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Indicates the first k A polyhedral smooth region This represents a smooth region representing the coarse conductor region in the periodic model. Indicates curl, Represents magnetoresistance. Indicates magnetic vector potential. Represents a virtual scalar potential. This represents the transient induced eddy current term. Indicates time, Indicates partial derivative, Indicates the terminal voltage difference. Indicates the source potential. Represents current. Indicates the voltage applied by the external circuit. Indicates external resistance. Indicates external inductance. This indicates differentiation over the integration domain.

7. The method according to claim 5, characterized in that, The expansions of the magnetic vector potential and the virtual scalar potential are as follows: ； ； in, Indicates magnetic vector potential. Represents a virtual scalar potential. Representing a smooth region of a polyhedron The number of middle edges, Representing a smooth region of a polyhedron The number of nodes in the middle, Representing a smooth region of a polyhedron k Middle edge e Smooth vector shape function, Indicates edge e Tangential magnetic vector value on, Representing a smooth region of a polyhedron k Middle node i The smooth shape function at the location, Represents a node i The virtual scalar potential value at that location.

8. The method according to claim 1, characterized in that, Newton's iterative linear equation is expressed as: ； ； ； in, Represents the Jacobian matrix of the system; Represents the residual of the magnetic vector potential; Represents the residuals of the system equations; Indicates the first h +1 iteration step magnetic vector position; Indicates the first h The magnetic vector position of the iteration step; Represents the system stiffness matrix; This represents a known external force vector.

9. The method according to claim 8, characterized in that, The iterative solution process of Newton's iterative method includes: Step 1: Initialize the iteration steps; Step 2: Solve the Newton-based linear equation for the current iteration step to obtain the magnetic vector potential for the current iteration step; Step 3: Add the magnetic vector bit of the current iteration step to the residual of the magnetic vector bit to obtain the magnetic vector bit of the next iteration step; Step 4: Update iteration step; Step 5: Repeat steps 2-4 until the Newton iterative linear equation converges and the magnetic vector potential in the field-circuit coupling is determined.

Images