A method for simulating electromagnetic characteristics of a non-planar coil based on a superconducting bundled cable

Abstract

The application discloses a non-planar coil electromagnetic characteristic simulation method based on a superconducting bundled cable and relates to the technical field of superconducting magnet electromagnetic simulation. The application firstly builds a three-dimensional model of the non-planar coil to obtain a magnetic field configuration distribution, determines a minimum boundary of a magnetic field extraction, a cross section position and performs a grid independence check, evaluates a local critical current by using a self-consistent model after coordinate migration, and takes the minimum value of the local critical current of each cross section as the global critical current. The application effectively solves the problems of high simulation freedom and poor convergence of the coil, can accurately simulate electromagnetic characteristics such as the critical current, and provides technical support for the design optimization and safe operation of the non-planar coil based on the superconducting bundled cable.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic simulation technology for superconducting magnets, and more specifically to a method for simulating the electromagnetic characteristics of nonplanar coils based on superconducting bundled cables. Background Technology

[0002] Superconducting nonplanar coils are coil components made by winding superconducting materials into non-flat, non-two-dimensional planar structures. They often exist in three-dimensional forms such as saddle shapes and spirals. Leveraging the zero resistance and strong current-carrying capacity of superconducting materials, combined with their unique three-dimensional structure, they possess irreplaceable advantages in complex magnetic field construction scenarios. To further enhance current-carrying capacity and overcome mechanical performance limitations, multiple superconducting tapes can be wound into compact and stable high-temperature superconducting multilayer bundled cables with a reasonable geometric structure. Compared to superconducting tapes, superconducting bundled cables have higher current-carrying capacity and lower AC losses, which can further improve the performance of nonplanar coils.

[0003] The curved shape of nonplanar coils inherently possesses complex structures and multi-field characteristics. Furthermore, the use of superconducting bundled cables, which combine multiple flat high-temperature superconducting tapes through stacking, twisting, and transposition, further exacerbates the system's complexity. This dual complexity leads to an exponential increase in the difficulty of analyzing the electromagnetic characteristics of nonplanar coils wound with superconducting bundled cables. The critical current is a crucial parameter for the stable operation of high-temperature superconducting coils. The three-dimensional geometry of the nonplanar magnet and the helical configuration of the superconducting bundled cable cause the angle between the magnetic field and the high-temperature superconducting tape to continuously change along the coil's circumference. Since the magnetic properties of the superconducting tape exhibit significant anisotropy, its critical current and other key parameters are strongly dependent on the direction of the applied external magnetic field. This results in a spatially non-uniform distribution of the critical current in superconducting nonplanar coils, easily leading to a "bottleneck effect" in the coil's global stability. The minimum local critical current directly determines the overall current margin of the coil. The spatial curved surface structure of the non-planar coil of the superconducting bundled cable completely breaks the magnetic field symmetry of the planar coil, bringing core challenges such as multi-scale structural analysis and irregular geometric modeling to its critical current assessment.

[0004] Numerical simulation, as a core method for accurately evaluating the magnetic field distribution and critical current of superconducting coils, is directly related to the design optimization, operational safety, and lifespan assurance of superconducting equipment, and has irreplaceable engineering value. However, existing simulation methods are difficult to perform electromagnetic characteristic simulation analysis of non-planar coils wound with superconducting bundled cables.

[0005] Therefore, it is necessary to develop new numerical calculation methods and models that take into account geometric adaptability, multi-field coupling accuracy and computational efficiency, so as to provide reliable technical support for the design, optimization and safe operation of non-planar coils in superconducting bundled cables. Summary of the Invention

[0006] In view of this, the present invention provides a simulation method for the electromagnetic characteristics of nonplanar coils based on superconducting bundled cables. This simulation method solves the problems of high degree of freedom and poor convergence in the simulation of the electromagnetic characteristics of this type of nonplanar superconducting coils, and can simulate the electromagnetic characteristics such as the critical current of nonplanar coils wound with superconducting bundled cables.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A simulation method for the electromagnetic characteristics of nonplanar coils based on superconducting bundled cables includes:

[0009] Step 1: Obtain the magnetic field configuration distribution of the non-planar coils of the superconducting bundled cable;

[0010] Step 2: Based on the configuration distribution results, set the magnetic field parameter transfer and boundary conditions to obtain the magnetic field boundary parameters and specific locations;

[0011] Step 3: Perform an independence check on the extracted magnetic field boundary parameters and the mesh density to obtain the checked magnetic field boundary data;

[0012] Step 4: Extract the magnetic field based on the verified magnetic field boundary data, and transfer the original three-dimensional coordinates of the magnetic field data to the local coordinate system of the two-dimensional cross-sectional model of the superconducting bundled cable;

[0013] Step 5: Based on the superconducting EJ characteristics, establish a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, calculate the critical current, and take the minimum local critical current of the cross-section as the global critical current.

[0014] Preferably, step 1 specifically includes:

[0015] Ignoring the helical distribution characteristics of the superconducting tape on the superconducting bundled cable, the superconducting bundled cable is equivalent to a cylindrical structure;

[0016] A three-dimensional geometric model of a non-planar coil is constructed, and a current excitation is applied to a superconducting bundled cable.

[0017] The three-dimensional geometric model is solved using the magnetic vector potential A method to obtain the configuration distribution of the target magnetic field, as shown in the following formula:

[0018]

[0019]

[0020] In the formula, B is the magnetic induction intensity, E is the electric field intensity, J is the current density, A is the magnetic vector potential, H is the magnetic field intensity, Ωsc is the superconducting domain, and Ωair is the air domain.

[0021] Preferably, the magnetic field parameter transfer and magnetic field boundary conditions in step 2 include the selection range of the magnetic field boundary and the specific location of the magnetic field boundary.

[0022] Preferably, the selection range of the magnetic field boundary is obtained by establishing a two-dimensional cross-sectional model of a superconducting bundled cable of a planar coil, resulting in a minimum boundary magnetic field extraction ring, specifically including:

[0023] The discrete sampling points of the magnetic field are preprocessed. Specifically, the rectangular coordinates of the discrete points are converted into polar coordinates and sorted according to the angular dimension.

[0024] The sorted discrete magnetic field signals are resampled by isoangular interpolation to obtain uniformly distributed periodic signals;

[0025] The periodic signal is processed using Fast Fourier Transform to extract its Fourier coefficients and calculate the amplitude of each harmonic.

[0026] A magnetic field roundness distortion index is defined, which is quantitatively characterized by harmonic distortion (THD). The calculation logic of THD is as follows: by comparing the total energy of all harmonic components in the magnetic field signal with the energy ratio of the fundamental frequency component, the degree of distortion of the magnetic field distribution deviating from the ideal concentric circle shape is assessed. The formula is as follows:

[0027]

[0028] Where H1 represents the energy of the fundamental frequency component, H2, H3, ..., H n THD represents the energy of each harmonic component and is expressed as a percentage.

[0029] When the harmonic distortion (THD) calculated from the selected circular ring is less than the distortion criterion, the size of the circular magnetic field boundary is the minimum boundary magnetic field extraction ring.

[0030] Preferably, the specific location of the magnetic field boundary is determined by a cross section parallel to the fundamental plane of the Cartesian coordinate system.

[0031] Preferably, step 3 specifically includes:

[0032] Baseline model construction: Based on the three-dimensional geometric model of the non-planar coil, the circular ring extracted by the minimum boundary magnetic field is used as the target boundary. The grid is divided using the default grid parameter grid density level, and the magnetic field parameters of the target boundary are extracted as the baseline data set.

[0033] Variable mesh model construction: Keeping the geometric parameters, magnetic field excitation conditions, and boundary selection position and size of the three-dimensional geometric model of the non-planar coil unchanged, only changing the size and number of boundary mesh divisions, multiple sets of variable mesh models are constructed to form mesh models with different density gradients;

[0034] Magnetic field parameter extraction: For each set of variable grid models, the same magnetic field extraction method as the baseline data set is used to extract the magnetic field parameters of the target boundary, forming multiple sets of comparative data;

[0035] Mesh independence analysis: Calculate the deviation values ​​between each set of comparative data and the baseline data set, set an error threshold, and determine that the independence requirement is met when the deviations of the magnetic field parameters of all variable mesh models are within the threshold range; if there are deviations exceeding the threshold, the mesh division strategy needs to be optimized and the verification needs to be performed again until the requirement is met, and the verified magnetic field boundary data is obtained.

[0036] Preferably, step 5, based on the superconducting EJ characteristics, establishes a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, and calculates the critical current by including:

[0037] The expression for the superconducting EJ property is as follows:

[0038]

[0039] In the formula, E is the electric field strength, Ec is the superconducting field quench criterion, J is the current density, JcB is the critical current density of superconductivity affected by the magnetic field, and n is the coefficient for the degree of transition from the superconducting state to the resistive state. By slightly rewriting the above formula, we obtain...

[0040]

[0041] Define the self-consistency coefficient P;

[0042]

[0043] When P = 1 and E = Ec, the electric field reaches the quench criterion. At this point, the input current is obtained as the critical current of the superconducting bundled cable, as shown in the following formula:

[0044]

[0045] The input current is calculated by extracting the cross-sectional area of ​​the coil containing the loop based on the current density and the minimum boundary magnetic field.

[0046] Preferably, the strip distribution on the superconducting bundled cable is rotated to obtain the cable critical current of different strip distributions on the superconducting bundled cable under the same magnetic field. The minimum value is taken as the critical current of the cross section, and the minimum value of the local critical current of all cross sections is taken as the global critical current of the entire non-planar coil to obtain the electromagnetic characteristic simulation results.

[0047] As can be seen from the above technical solution, compared with the prior art, this invention discloses a simulation method for the electromagnetic characteristics of non-planar coils based on superconducting bundled cables. This method, based on the principle of dimensional superposition, considers a three-dimensional superconducting coil as composed of countless two-dimensional planes, which can be discretized into countless two-dimensional planes. The three-dimensional magnetic field of the superconducting non-planar superconducting coil is decomposed into the superposition of as many finite number of two-dimensional cross-sectional magnetic fields as possible. Combining the axial continuity condition of the magnetic field, the calculation of the three-dimensional overall magnetic field is decoupled from the calculation of the two-dimensional local electromagnetic characteristics. Based on the boundary value problem solution logic in electromagnetic field theory, the transitivity of the magnetic field boundary conditions—that is, the boundary magnetic fields extracted from the three-dimensional model (such as tangential and normal components)—can be used as the external boundary conditions of the two-dimensional model to calculate the two-dimensional planar electromagnetic characteristics, thus completing the solution of the electromagnetic characteristics from the local to the global aspects of the superconducting coil. This simulation method solves the problems of high degree of freedom and poor convergence in the simulation of the electromagnetic characteristics of this type of non-planar superconducting coil, and can simulate the electromagnetic characteristics such as the critical current of non-planar coils wound with superconducting bundled cables. Attached Figure Description

[0048] To more clearly illustrate the technical solutions in the embodiments of the present invention 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 embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0049] Figure 1 The diagram shows the configuration distribution of the target magnetic field provided by this invention.

[0050] Figure 2 A cross-sectional diagram showing the specific location of the magnetic field boundary provided by this invention.

[0051] Figure 3 The coordinate migration diagram provided by this invention.

[0052] Figure 4 The method flowchart provided by the present invention. Detailed Implementation

[0053] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0054] like Figure 4 As shown, this invention discloses a method for simulating the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable, comprising:

[0055] Step 1: Obtain the magnetic field configuration distribution of the non-planar coils of the superconducting bundled cable;

[0056] Step 2: Based on the configuration distribution results, set the magnetic field parameter transfer and boundary conditions to obtain the magnetic field boundary parameters and specific locations;

[0057] Step 3: Perform an independence check on the extracted magnetic field boundary parameters and the mesh density to obtain the checked magnetic field boundary data;

[0058] Step 4: Extract the magnetic field based on the verified magnetic field boundary data, and transfer the original three-dimensional coordinates of the magnetic field data to the local coordinate system of the two-dimensional cross-sectional model of the superconducting bundled cable;

[0059] Step 5: Based on the superconducting EJ characteristics, establish a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, calculate the critical current, and take the minimum local critical current of the cross-section as the global critical current.

[0060] Specifically, step 1 includes:

[0061] Ignoring the helical distribution characteristics of the superconducting tape on the superconducting bundled cable, the superconducting bundled cable is equivalent to a cylindrical structure;

[0062] A three-dimensional geometric model of a non-planar coil is constructed, and a current excitation is applied to a superconducting bundled cable.

[0063] The three-dimensional geometric model is solved using the magnetic vector potential A method to obtain the configuration distribution of the target magnetic field, as shown in the following formula:

[0064]

[0065]

[0066] In the formula, B is the magnetic flux density, E is the electric field strength, J is the current density, A is the magnetic vector potential, and H is the magnetic field strength (Ω). sc For the superconducting domain, Ω air For air domain, It is the nabla operator, which is the vector differential operator and is a common mathematical operator.

[0067] In a specific embodiment of the present invention, according to Ampere's circuital law, in a steady magnetic field, the line integral of the magnetic induction intensity B along any closed path is equal to the product of the algebraic sum of all currents enclosed by that closed path and the permeability. To simplify the magnetic field calculation model of the superconducting bundled cable and coil, and to focus on the core magnetic field distribution, the helical distribution characteristics of the superconducting tape on the superconducting bundled cable are ignored, and the superconducting bundled cable is equivalent to a cylindrical structure. A three-dimensional geometric model of the nonplanar coil is constructed, and current excitation is applied to the coil. The superconducting characteristics of the superconducting coil are temporarily disregarded, i.e., the inherent nonlinear EJ characteristics of the superconducting material are ignored, and it is equivalent to a conventional metallic material to simplify the current field distribution. This three-dimensional model is solved using the magnetic vector potential A method, and the final configuration distribution result of the target magnetic field is obtained, as shown below. Figure 1 As shown.

[0068] In step 1, to simplify the simulation calculation, it is assumed that the superconducting bundled cable has a cylindrical structure. The magnetic field generated by this ideal structure has perfect central annular symmetry, and its magnetic field contour lines are distributed in a series of concentric circles. The magnetic field strength decreases uniformly from the center outwards, with the strongest magnetic field in the central region and gradually decreasing in the peripheral region, without local distortion or magnetic field concentration. However, in actual applications, superconducting bundled cables are formed by winding superconducting tapes onto a central skeleton through processes such as stacking, twisting, and transposition. Gaps inevitably exist between the tapes. Due to the influence of this structure, the magnetic field contour lines on the surface of the bundled cable are not ideal concentric circles. Magnetic field distortion occurs in the gap regions, manifested as locally dense contour lines and irregular shapes.

[0069] As described above, there is a significant difference between the magnetic field distribution of the ideal 3D coil model and the actual structure in step 1, resulting in a lack of unified technical standards for the transfer of magnetic field parameters and the matching of magnetic field boundary conditions between the 3D and 2D models at different scales. If the selection and setting of the magnetic field boundary are unreasonable, it is easy to cause difficulties in coordinating the accurate description of the magnetic field at the 2D scale and the efficient solution of the simulation at the 3D scale. This will not only lead to a significant increase in model complexity and computational cost, but may also affect the accuracy of the simulation results. To solve the above technical problems, this embodiment constructs a standardized system for the transfer of magnetic field parameters and the setting of boundary conditions through step 2: step 2 is used to determine the selection range of the magnetic field boundary and the specific location of the magnetic field boundary, and step 3 is used to determine the magnetic field boundary mesh, ensuring that the transfer of magnetic field parameters from the 3D magnetic field model to the 2D cross-sectional model has sufficient data support, and ensuring that the 2D cross-sectional model can accurately reproduce the magnetic field distribution characteristics of the corresponding region.

[0070] Specifically, the magnetic field parameter transfer and magnetic field boundary conditions in step 2 include the selection range of the magnetic field boundary and the specific location of the magnetic field boundary.

[0071] Specifically, the selection range of the magnetic field boundary is obtained by establishing a two-dimensional cross-sectional model of a superconducting bundled cable of a planar coil, resulting in a minimum boundary magnetic field extraction ring, specifically including:

[0072] The discrete sampling points of the magnetic field are preprocessed. Specifically, the rectangular coordinates of the discrete points are converted into polar coordinates and sorted according to the angular dimension.

[0073] The sorted discrete magnetic field signals are resampled by isoangular interpolation to obtain uniformly distributed periodic signals;

[0074] The periodic signal is processed using Fast Fourier Transform to extract its Fourier coefficients and calculate the amplitude of each harmonic.

[0075] A magnetic field roundness distortion index is defined, which is quantitatively characterized by harmonic distortion (THD). The calculation logic of THD is as follows: by comparing the total energy of all harmonic components in the magnetic field signal with the energy ratio of the fundamental frequency component, the degree of distortion of the magnetic field distribution deviating from the ideal concentric circle shape is assessed. The formula is as follows:

[0076]

[0077] Where H1 represents the energy of the fundamental frequency component, H2, H3, ..., H n THD represents the energy of each harmonic component and is expressed as a percentage.

[0078] When the harmonic distortion (THD) calculated from the selected circular ring is less than the distortion criterion, the size of the circular magnetic field boundary is the minimum boundary magnetic field extraction ring.

[0079] In a specific embodiment of the present invention, addressing the technical bottleneck of magnetic field coordination matching between models of different scales, this embodiment creatively proposes the technical concept of a minimum boundary magnetic field extraction ring. Specifically, in the near-field region of the superconducting bundled cable (especially near the strip gap), the magnetic field contour lines are distorted due to local disturbances; as they extend outward from the cable to a certain ring region, the magnetic field contour lines gradually recover to a uniform concentric circle arrangement. The "minimum boundary magnetic field extraction ring" refers to the smallest ring region from this irregular magnetic field distribution that completely restores the magnetic field contour lines to a concentric circle shape—its core is to select the smallest range ring that satisfies the "internal magnetic field distribution returning to a regular concentric circle characteristic" by avoiding local magnetic field disturbances caused by the strip gap, thereby achieving coordinated consistency between three-dimensional magnetic field calculation and the application of the boundary magnetic field of the two-dimensional model, effectively solving the problem of magnetic field parameter transfer and boundary matching between models of different scales.

[0080] The specific selection method for the "minimum boundary magnetic field extraction ring" is as follows:

[0081] After extracting the boundary magnetic field distribution of the magnetic field to be analyzed from the ring, it is represented in the form of a mathematical function. The magnetic field function can be decomposed into a series of superposition combinations of sine and cosine functions through Fourier transform. By analyzing the coefficient characteristics of each decomposed function, key information about the magnetic field distribution can be obtained. Among them, the smaller the amplitude of the higher harmonic coefficient, the closer the magnetic field distribution of the ring boundary is to the magnetic field distribution law of a conventional conductor. The three-dimensional magnetic field calculation and the application of the boundary magnetic field of the two-dimensional model are highly consistent.

[0082] Preprocessing of discrete magnetic field sampling points: First, convert the rectangular coordinates of the discrete points to polar coordinates (including angle and radius parameters), and sort them according to the angle dimension;

[0083] The sorted discrete magnetic field signals are resampled by isoangular interpolation to obtain uniformly distributed periodic signals;

[0084] The above periodic signal is processed by fast Fourier transform, its Fourier coefficients are extracted, and the amplitude of each harmonic is calculated.

[0085] A "magnetic field circularity distortion" index is defined, which is quantitatively characterized by harmonic distortion (THD). The calculation logic of the harmonic distortion is as follows: by comparing the total energy of all harmonic components in the magnetic field signal with the energy ratio of the fundamental frequency component, the degree of distortion in the magnetic field distribution that deviates from the ideal concentric circle distribution is assessed.

[0086]

[0087] Where H1 represents the energy of the fundamental frequency component, H2, H3, ..., H n The energy of each harmonic component is usually expressed as a percentage, representing the percentage of harmonic components in the output signal, and also representing the degree of non-uniformity of the circular boundary magnetic field.

[0088] When the harmonic distortion (THD) calculated from the selected circular ring is less than the distortion criterion, the size of the circular magnetic field boundary is the minimum boundary magnetic field extraction ring.

[0089] Specifically, the exact location of the magnetic field boundary is determined by a cross-section parallel to the fundamental plane of the Cartesian coordinate system.

[0090] In a specific embodiment of the present invention, such as Figure 2As shown, non-planar coils typically possess large torsional angles and curvatures, resulting in a highly asymmetrical overall geometry. Consequently, the magnetic field they generate exhibits an asymmetrical and complex spiral shape. If magnetic field data is obtained by extracting a cross-section perpendicular to the magnetic field direction at each selected point, the normal directions of the two-dimensional cross-sections will be inconsistent due to the continuous change in magnetic field direction with the coil's three-dimensional geometry. This not only leads to a chaotic morphology of the two-dimensional cross-sections, making subsequent data transfer between multiple cross-sections lack a unified benchmark, but also significantly increases the difficulty and error risk of data integration. In contrast, the magnetic field generated by a superconducting coil exhibits continuity in both intensity and direction, without any instantaneous abrupt changes. Based on this magnetic field continuity, a magnetic field extraction cross-section parallel to the fundamental planes of the Cartesian coordinate system (XY, XZ, or YZ planes) is selected. Although not strictly perpendicular to the magnetic field direction, the magnetic field data within this cross-section still reflects the magnetic field distribution in the corresponding region, avoiding data distortion issues that could affect the simulation.

[0091] Therefore, while ensuring the accuracy of magnetic field extraction, by selecting a fixed cross-section parallel to the three basic planes of the Cartesian coordinate system, XY, XZ, or YZ, a unified migration reference can be established. Even if the coil geometry is asymmetrical, a regular magnetic field cross-section dataset can be formed, which greatly reduces the complexity of subsequent data processing.

[0092] Specifically, step 3 includes:

[0093] Baseline model construction: Based on the three-dimensional geometric model of the non-planar coil, the circular ring extracted by the minimum boundary magnetic field is used as the target boundary. The grid is divided using the default grid parameter grid density level, and the magnetic field parameters of the target boundary are extracted as the baseline data set.

[0094] Variable mesh model construction: Keeping the geometric parameters, magnetic field excitation conditions, and boundary selection position and size of the three-dimensional geometric model of the non-planar coil unchanged, only changing the size and number of boundary mesh divisions, multiple sets of variable mesh models are constructed to form mesh models with different density gradients;

[0095] Magnetic field parameter extraction: For each set of variable grid models, the same magnetic field extraction method as the baseline data set is used to extract the magnetic field parameters of the target boundary, forming multiple sets of comparative data;

[0096] Mesh independence analysis: Calculate the deviation values ​​(relative errors of magnetic field components) between each set of comparative data and the baseline data, set an error threshold (relative error ≤ 1%), and if the deviations of magnetic field parameters of all variable mesh models are within the threshold range, it is determined that the independence requirement is met; if there are deviations exceeding the threshold, the mesh generation strategy needs to be optimized and the verification needs to be performed again until the requirement is met, and the verified magnetic field boundary data is obtained.

[0097] In a specific embodiment of the present invention, the boundary mesh extraction irrelevance verification refers to a specific verification process conducted during the simulation of the electromagnetic characteristics of a non-planar coil to verify that the extracted magnetic field boundary parameters are independent of the boundary mesh density. Its core purpose is to ensure the stability and reliability of the magnetic field boundary data, avoid distortion in magnetic field parameter transmission due to differences in mesh partitioning, and thus guarantee the accuracy and consistency of subsequent co-simulation of two-dimensional / three-dimensional models.

[0098] In the simulation of magnetic fields of nonplanar coils wound with superconducting bundled cables, the boundary mesh directly affects the extraction results of magnetic field data: if the mesh density is too low, local magnetic field features may be missed; if the mesh density is too high, computational redundancy will increase. If the magnetic field boundary parameters are related to the mesh division, the data extracted from the same boundary under different mesh settings will be inconsistent, leading to deviations in the application of boundary conditions to the two-dimensional model, ultimately affecting the reliability of the simulation results. Therefore, independence verification can eliminate the interference of mesh division on boundary magnetic field data, providing stable data support for subsequent magnetic field parameter transfer and model coordination.

[0099] Furthermore, such as Figure 3 As shown, coordinate migration unifies the magnetic field data extracted from the 3D model to the same local coordinate system, facilitating comparison and analysis with the 2D model. Specifically, in the 3D model, the center of the circular cross-section is O(x, y), and the coordinate point A(x, y) is migrated to A'(x', y').

[0100] The relationship between the new coordinates and the coordinates before migration is as follows:

[0101] .

[0102] Specifically, step 5, based on the superconducting EJ characteristics, establishes a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, and calculates the critical current. The process includes:

[0103] The expression for the superconducting EJ property is as follows:

[0104]

[0105] In the formula, E is the electric field strength, Ec is the superconducting field quench criterion, J is the current density, JcB is the critical current density of superconductivity affected by the magnetic field, and n is the coefficient for the degree of transition from the superconducting state to the resistive state. By slightly rewriting the above formula, we obtain...

[0106]

[0107] Define the self-consistency coefficient P;

[0108]

[0109] When P = 1 and E = Ec, the electric field reaches the quench criterion. At this point, the input current is obtained as the critical current of the superconducting bundled cable, as shown in the following formula:

[0110]

[0111] The input current is calculated by extracting the cross-sectional area of ​​the coil containing the loop based on the current density and the minimum boundary magnetic field.

[0112] After this derivation, most of the physical quantities in the original theoretical system can be represented by P, and P does not conflict with the original system. This process of defining a physical quantity from an existing system, and then reasoning it out so that the physical quantity does not produce a paradox with the existing system, is called self-consistency. Similarly, a model built using the newly defined physical quantity is called a self-consistent model.

[0113] Specifically, by rotating the strip distribution on the superconducting bundled cable, the cable critical currents of different strip distributions on the superconducting bundled cable under the same magnetic field are obtained. The minimum value is taken as the critical current of the cross section, and the minimum value of the local critical current of all cross sections is taken as the global critical current of the entire non-planar coil, thus obtaining the electromagnetic characteristic simulation results.

[0114] In one specific embodiment of the present invention, the angular phase of the spirally twisted strip stack is crucial, but the actual phase after the non-planar coil assembly is unknown. Therefore, by rotating the strip distribution on the cable, the critical current values ​​of the cable with different strip distributions on the cable under the same magnetic field are obtained, and then the minimum value is selected as the critical current of a certain cross-section. The minimum value of the local critical current of all cross-sections is taken as the global critical current of the entire non-planar coil.

[0115] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0116] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A simulation method for the electromagnetic characteristics of nonplanar coils based on superconducting bundled cables, characterized in that, include: Step 1: Obtain the magnetic field configuration distribution of the non-planar coils of the superconducting bundled cable; Step 2: Based on the configuration distribution results, set the magnetic field parameter transfer and boundary conditions to obtain the magnetic field boundary parameters and specific locations; Step 3: Perform an independence check on the extracted magnetic field boundary parameters and the mesh density to obtain the checked magnetic field boundary data; Step 4: Extract the magnetic field based on the verified magnetic field boundary data, and transfer the original three-dimensional coordinates of the magnetic field data to the local coordinate system of the two-dimensional cross-sectional model of the superconducting bundled cable; Step 5: Based on the superconducting EJ characteristics, establish a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, calculate the critical current, and take the minimum local critical current of the cross-section as the global critical current. The magnetic field parameter transfer and magnetic field boundary conditions in step 2 include the selection range of the magnetic field boundary and the specific location of the magnetic field boundary. The selection range of the magnetic field boundary is obtained by establishing a two-dimensional cross-sectional model of a superconducting bundled cable of a planar coil, resulting in a minimum boundary magnetic field extraction ring, specifically including: The discrete sampling points of the magnetic field are preprocessed. Specifically, the rectangular coordinates of the discrete points are converted into polar coordinates and sorted according to the angular dimension. The sorted discrete magnetic field signals are resampled by isoangular interpolation to obtain uniformly distributed periodic signals; The periodic signal is processed using Fast Fourier Transform to extract its Fourier coefficients and calculate the amplitude of each harmonic. A magnetic field roundness distortion index is defined, which is quantitatively characterized by harmonic distortion (THD). The calculation logic of THD is as follows: by comparing the total energy of all harmonic components in the magnetic field signal with the energy ratio of the fundamental frequency component, the degree of distortion of the magnetic field distribution deviating from the ideal concentric circle shape is assessed. The formula is as follows: Where H1 represents the energy of the fundamental frequency component, H2, H3, ..., H n THD represents the energy of each harmonic component and is expressed as a percentage. When the harmonic distortion (THD) calculated from the selected circular ring is less than the distortion criterion, the size of the circular magnetic field boundary is the minimum boundary magnetic field extraction ring.

2. The simulation method for the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable according to claim 1, characterized in that, Step 1 specifically includes: Ignoring the helical distribution characteristics of the superconducting tape on the superconducting bundled cable, the superconducting bundled cable is equivalent to a cylindrical structure; A three-dimensional geometric model of a non-planar coil is constructed, and a current excitation is applied to a superconducting bundled cable. The three-dimensional geometric model is solved using the magnetic vector potential A method to obtain the configuration distribution of the target magnetic field, as shown in the following formula: In the formula, B is the magnetic flux density, E is the electric field strength, J is the current density, A is the magnetic vector potential, and H is the magnetic field strength (Ω). sc For the superconducting domain, Ω air This refers to the air domain.

3. The simulation method for the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable according to claim 1, characterized in that, The specific location of the magnetic field boundary is determined by a cross section parallel to the fundamental plane of the Cartesian coordinate system.

4. The simulation method for the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable according to claim 3, characterized in that, Step 3 specifically includes: Baseline model construction: Based on the three-dimensional geometric model of the non-planar coil, the circular ring extracted by the minimum boundary magnetic field is used as the target boundary. The grid is divided using the default grid parameter grid density level, and the magnetic field parameters of the target boundary are extracted as the baseline data set. Variable mesh model construction: Keeping the geometric parameters, magnetic field excitation conditions, and boundary selection position and size of the three-dimensional geometric model of the non-planar coil unchanged, only changing the size and number of boundary mesh divisions, multiple sets of variable mesh models are constructed to form mesh models with different density gradients; Magnetic field parameter extraction: For each set of variable grid models, the same magnetic field extraction method as the baseline data set is used to extract the magnetic field parameters of the target boundary, forming multiple sets of comparative data; Mesh independence analysis: Calculate the deviation values ​​between each set of comparative data and the baseline data set, set an error threshold, and determine that the independence requirement is met when the deviations of the magnetic field parameters of all variable mesh models are within the threshold range; if there are deviations exceeding the threshold, the mesh division strategy needs to be optimized and the verification needs to be performed again until the requirement is met, and the verified magnetic field boundary data is obtained.

5. The simulation method for the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable according to claim 4, characterized in that, Step 5, based on the superconducting EJ characteristics, establishes a self-consistent model to evaluate the critical current in the local coordinate system of the two-dimensional cross-section model, and calculates the critical current. The process includes: The expression for the superconducting EJ property is as follows: In the formula, E is the electric field strength, E c Let J be the superconducting field quench criterion, J be the current density, JcB be the critical current density of superconductivity under the influence of a magnetic field, and n be the coefficient for the degree of transition from the superconducting state to the resistive state. By slightly rewriting the above equation, we obtain... Define the self-consistency coefficient P; When P = 1, E = E c When the electric field reaches the quench criterion, the input current is used to obtain the critical current of the superconducting bundled cable, as shown in the following formula: The input current is calculated by extracting the cross-sectional area of ​​the coil containing the loop based on the current density and the minimum boundary magnetic field.

6. The method for simulating the electromagnetic characteristics of a nonplanar coil based on a superconducting bundled cable according to claim 5, characterized in that, By rotating the strip distribution on the superconducting bundled cable, the critical current of the cable with different strip distributions on the superconducting bundled cable under the same magnetic field is obtained. The minimum value is taken as the critical current of the cross section, and the minimum value of the local critical current of all cross sections is taken as the global critical current of the entire non-planar coil, so as to obtain the electromagnetic characteristic simulation results.

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