All-condition analysis method of non-electric insulation superconducting magnet system based on field-circuit coupling

Abstract

The invention discloses a field-circuit coupling-based all-working-condition analysis method for a non-electrically-insulated superconducting magnet system. The method specifically comprises the following steps of: defining geometry, material attributes, boundary condition parameters and model discrete unit settings of the non-electrically-insulated superconducting magnet system; based on a circuit equivalence principle, establishing an equivalent circuit model of the superconducting magnet system to calculate current and loss distribution of the magnet system; current and loss are used as input, a magnetic field model and a thermal circuit model of the magnet system are constructed through an analysis or numerical calculation method, and the magnetic field and temperature distribution of the magnet system are calculated respectively; calculating a vector magnetic potential of a discrete unit of the magnet system so as to obtain an equivalent global voltage of the magnet system and simplify circuit model calculation; and calculating strong nonlinear resistivity on which magnetic field and temperature parameters of the superconducting magnet system depend, and realizing mutual coupling of each field model and each path model. According to the method, the complex discrete electrical parameter calculation cost is avoided, and the method has the advantages of high calculation rate and high calculation precision.

Description

Technical Field

[0001] The invention belongs to the technical field of superconductivity, and in particular relates to a full-operating-condition analysis method of a non-electrically-insulated superconducting magnet system based on field-circuit coupling. Background Art

[0002] Superconductors are new electrical materials with high current carrying capacity and low conductive loss. Superconducting magnets made of superconductors have advantages that conventional magnets cannot match, such as compact structure and high energy density. They have broad application prospects in the fields of clean energy, maglev transportation, national defense engineering, high-end medical care, and large scientific facilities. Typical applications include controlled nuclear fusion, superconducting magnetic energy storage, high-speed maglev trains, electromagnetic launch, magnetic resonance imaging, particle accelerators, etc. However, for superconducting magnets, their magnetothermal stability is an important parameter to ensure the safe operation of superconducting equipment. Therefore, the non-electrically insulated magnet technology has become one of the main technical approaches for the development of superconducting magnets due to its good self-protection ability and thermal stability.

[0003] The technology of non-electrically insulated magnets removes the electrical insulation layer between the magnet turns, or replaces the electrical insulation layer with a good conductor. The introduction of the inter-turn electrical path provides a redistribution path for the overcurrent, improves the self-protection ability of the magnet during local overheating, and enhances the thermal stability of the magnet. However, the current transmission path of non-electrically insulated superconducting magnets is not unique. During dynamic operation, the changing electromagnetic field causes the transmission current, induced current, and shielding current in the magnet to be transmitted in multiple directions, which also makes the traditional finite element method (such as the magnetic field intensity H method, etc.) unable to be used for the calculation and analysis of non-electrically insulated superconducting magnets.

[0004] In order to calculate and analyze the characteristics of non-electrically insulated superconducting magnets, most existing studies have been implemented through the current vector potential-vector magnetic potential TA method or the vector magnetic potential A method combined with the distributed parameter circuit network method, or by setting the rotational resistivity property for the superconducting material and combining it with the magnetic field intensity H method. However, on the one hand, the circuit network model needs to solve the parameters such as self-inductance, mutual inductance, and resistance of discrete units, which has high calculation cost. At the same time, in dynamic scenarios, the inductance parameters of superconducting magnets change with the changes of transmission current and external magnetic field. Therefore, the calculation method of simple numerical solution of inductance parameters without considering the change of current in the magnet is flawed, which makes the distributed parameter circuit network model difficult to use for the accurate calculation of the multi-physics field model of the magnet under dynamic conditions. On the other hand, the modeling strategies of other methods are complex, and there is a problem that the superconducting magnet working conditions that can be analyzed are single. Summary of the invention

[0005] In order to overcome the above technical problems, the present invention provides a full operating condition analysis method for a non-electrically insulated superconducting magnet system based on field-circuit coupling.

[0006] A full-operating-condition analysis method of a non-electrically-insulated superconducting magnet system based on field-circuit coupling of the present invention comprises the following steps:

[0007] Step 1: Define the geometry, material properties, boundary condition parameters and model discrete element settings of the non-electrically insulated superconducting magnet system.

[0008] Step 2: Based on the circuit equivalence principle, an equivalent circuit model of the superconducting magnet system is established to calculate the current and loss distribution of the magnet system.

[0009] Step 3: Using current and loss as input, construct the magnetic field and thermal circuit model of the magnet system through analytical or numerical calculation methods to calculate the magnetic field and temperature distribution of the magnet system respectively.

[0010] Step 4: Calculate the vector magnetic potential of the discrete units of the magnet system to obtain the equivalent global voltage of the magnet system and simplify the circuit model calculation.

[0011] Step 5: Calculate the strong nonlinear resistivity of the superconducting magnet system that depends on the magnetic field and temperature parameters to achieve mutual coupling of various field and path models.

[0012] Furthermore, the equivalent circuit model in step 2 includes a distributed parameter circuit network model, a single-turn or multi-turn equivalent circuit model, a coil equivalent model, and a coil group equivalent model.

[0013] Furthermore, the calculation method of the magnetic field model in step 3 includes: (1) multiple integral solution method based on Biotschaff's law: finite difference and mathematical analysis method; (2) finite element method: H method, TA method, Law.

[0014] Furthermore, the calculation method of the thermal circuit model in step 3 includes: finite difference method, finite volume method, and finite element method.

[0015] Furthermore, the calculation method of the vector magnetic potential in step 4 includes: (1) a multiple integral solution method based on Biotschaff's law: a finite difference method and a mathematical analytical method; (2) indirect calculation through a finite element magnetic field model.

[0016] Furthermore, the global voltage in step 4 is obtained based on a formula derived from Faraday's law of electromagnetic induction, and the calculation formula is as follows:

[0017]

[0018] Where A, B, E are vector magnetic potential, magnetic induction intensity and electric field intensity respectively; l refers to the integral path for calculating the resistive voltage; is the Hamilton operator, used in curl, gradient and divergence operations; V HTS , Vr , V i They are global voltage, resistive voltage and inductive voltage respectively; the inductive voltage is obtained by the linear integration of the time derivative of the vector magnetic potential, while the resistive voltage is obtained by the nonlinear resistivity of the magnetic system;

[0019] The nonlinear resistivity calculation methods include: power exponential calculation formula, discrete data interpolation method, and discrete data fitting formula.

[0020] The beneficial technical effects of the present invention are:

[0021] 1. Compared with the traditional method, the present invention uses the law of electromagnetic induction to quickly solve the global voltage of the discrete units of the superconducting domain of the non-electrically insulated superconducting magnet, so as to cancel the solution of the inductance and resistance parameters of the discrete units of the superconducting domain. This method will not reduce the calculation accuracy of the model and can be objectively and efficiently used for the simulation analysis of the non-electrically insulated superconducting magnet.

[0022] 2. The present invention can be directly used in combination with other conventional finite element methods and has good applicability.

[0023] 3. The present invention can perform dimension-up or dimension-down processing on the model according to the feature detail level and calculation rate to be solved, and can be applied to various working conditions such as optimization design of non-electrically insulated superconducting magnets, characteristic estimation, AC loss calculation, quench state analysis, etc., and has excellent scalability. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 This is a schematic diagram of the multi-physics field modeling process based on field-circuit coupling of the present invention.

[0025] Figure 2 This is a schematic diagram of a single-turn equivalent circuit model of the present invention.

[0026] Figure 3 This is a schematic diagram of the TA method modeling strategy of the present invention.

[0027] Figure 4 The calculation results of electromagnetic parameters such as AC loss of the uninsulated superconducting magnet in the case of the present invention.

[0028] Figure 5 Schematic diagram of the modeling strategy for the racetrack-type non-insulated superconducting magnet extended by the present invention. DETAILED DESCRIPTION

[0029] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0030] A full-operating-condition analysis method of a non-electrically-insulated superconducting magnet system based on field-circuit coupling of the present invention comprises the following steps:

[0031] Step 1: Define the geometry, material properties, boundary condition parameters and model discrete element settings of the non-electrically insulated superconducting magnet system.

[0032] The material property definition mainly includes: air domain definition and superconducting domain definition, where the conductivity of the air domain is set to zero. For the superconducting domain properties, they are mainly calculated through strong nonlinear constitutive relationship formulas that depend on physical parameters such as magnetic field and temperature, where non-uniform critical current properties can be added to the superconducting domain through position functions.

[0033] The boundary condition settings mainly include magnetic field boundary settings, heat flow boundary settings, and temperature boundary settings. The magnetic field boundary can not only be directly superimposed through calculation formulas or interpolation functions, but can also be applied through magnetic field boundary conditions of finite element software such as COMSOL Multiphysic. For heat flow boundary and temperature boundary, it mainly depends on the simulation conditions, which can simulate local hot spots of magnets, coil bath cooling, coil insulation, etc.

[0034] Step 2: Based on the circuit equivalence principle, an equivalent circuit model of the superconducting magnet system is established to calculate the current and loss distribution of the magnet system.

[0035] The equivalent circuit model can use a variety of modeling strategies to calculate different coil characteristic parameters, including: distributed parameter circuit network model, single-turn or multi-turn equivalent circuit model, coil equivalent model and coil group equivalent model, and the calculation rate increases step by step.

[0036] Step 3: Using current and loss as input, construct the magnetic field and thermal circuit model of the magnet system through analytical or numerical calculation methods to calculate the magnetic field and temperature distribution of the magnet system respectively.

[0037] The calculation methods of the magnetic field model include: (1) multiple integral solution methods based on Biotschaff's law: finite difference and mathematical analysis methods; (2) finite element methods: H method, TA method, Law.

[0038] The calculation methods of thermal circuit model include: finite difference method, finite volume method and finite element method.

[0039] The multi-physics field models such as circuit, magnetic field, thermal circuit, etc. can be preferably set as Cartesian rectangular coordinate system, cylindrical coordinate system or curvilinear coordinate system according to the shape of the superconducting magnet.

[0040] Step 4: Calculate the vector magnetic potential of the discrete units of the magnet system to obtain the equivalent global voltage of the magnet system and simplify the circuit model calculation.

[0041] The calculation methods of vector magnetic potential include: (1) multiple integral solution method based on Biotschaff's law: finite difference and mathematical analysis method; (2) indirect calculation through finite element magnetic field model.

[0042] Among them, the mathematical analytical method for solving the multiple integrals of Biotsalvar's law has the fastest calculation rate, that is, it is solved by the substitution integral method and the partial integration method. The one-dimensional calculation formula of the vector magnetic potential distribution of the discrete arc unit is:

[0043]

[0044]

[0045] θ i =|α i |

[0046] γ=z i -z

[0047]

[0048] a 2 =γ 2 +(r i +r) 2

[0049]

[0050]

[0051] θ i =π-|α i |

[0052]

[0053] The source point coordinates are The field point coordinates are and is the upper and lower limits of the angle of the discrete arc unit boundary. i ,k),E(θ i ,k), snμ i cnμ i and dnμ i They are the incomplete elliptic integral function and the Jacobi elliptic integral function respectively. The sgn function is used to obtain the positive and negative signs of the parameters. The corresponding field point is located inside the superconducting domain, and vice versa, it is located outside the superconducting domain. In the formula, the parameters μ0 and I are the vacuum permeability and the source point line current, respectively.

[0054] Correspondingly, the one-dimensional calculation formula of the magnetic field distribution of discrete arc units is:

[0055]

[0056] For discrete linear units with current along the z-axis, the one-dimensional calculation formula for the vector magnetic potential distribution is:

[0057]

[0058] The source point coordinates are (x2, y2, z i ), the field point coordinates are (x, y, z), z1 and z2 are the upper and lower limits of the discrete linear unit boundary. Accordingly, the one-dimensional calculation formula for the magnetic field distribution is:

[0059]

[0060] The core parameter of the present invention, the global voltage of the magnet system, is obtained based on the formula derived from Faraday's law of electromagnetic induction, and the calculation formula is as follows:

[0061]

[0062] Where V HTS , V r , V i They are global voltage, resistive voltage and inductive voltage. The inductive voltage is obtained by linear integration of the time derivative of the vector magnetic potential, while the resistive voltage is obtained by the nonlinear resistivity of the magnetic system.

[0063] Step 5: Calculate the strong nonlinear resistivity of the superconducting magnet system that depends on the magnetic field and temperature parameters to achieve mutual coupling of various field and path models.

[0064] The nonlinear resistivity calculation methods include: power exponential calculation formula, discrete data interpolation method, discrete data fitting formula and other calculation methods that can be used to characterize the nonlinear characteristics of superconductors. The critical current parameter used to describe and calculate the resistivity of the superconducting domain can be obtained using the Bean model formula, Kim model and its modified formula and experimental data interpolation fitting formula method.

[0065] Bean Model J c (T) = J c0 (T)

[0066] Kim Model

[0067] Kim's modified model

[0068] Among them J c0 (T) is the critical current density of the superconducting material under zero magnetic field and temperature T, B ∥ and B ⊥ are the parallel field and perpendicular field relative to the surface of the superconducting material, k, B c, β are fitting parameters related to superconducting materials.

[0069] Example:

[0070] This example describes the modeling strategy for analyzing the loss characteristics of a circular uninsulated coil under an alternating magnetic field. The multi-physics modeling process is as follows: Figure 1 As shown, a two-dimensional axisymmetric model is selected to demonstrate the extensibility of the present invention. First, the geometric parameters, turn-to-turn resistivity distribution, material properties, boundary condition parameters and model discrete unit settings, current and external magnetic field parameters of the non-insulated superconducting magnet system are defined, wherein the turn-to-turn resistivity can be set to a non-uniform distribution. In addition, the definition of material properties mainly includes: air domain definition and superconducting domain definition, wherein the conductivity of the air domain is set to zero. For the superconducting domain properties, they are mainly calculated through strong nonlinear constitutive relationship formulas that depend on physical parameters such as magnetic field and temperature, wherein non-uniform critical current properties can be added to the superconducting domain through position functions.

[0071] Then, a single-turn equivalent circuit model of the magnetic system is established, mainly through the global voltage (V HTS,k ) and turn-to-turn resistivity (R ct,k ) parallel circuit represents the electrical circuit of each turn of the magnet, and its schematic diagram is as follows Figure 2 As shown. From this, the circumferential current and radial current of the magnet system can be calculated;

[0072]

[0073] I HTS,k =I op -I ct,k

[0074] Among them I op Input current for the power supply.

[0075] The second is the magnetic field model. The TA method is used here. The vector magnetic potential A method mainly calculates the magnetic field distribution in all domains, and the current vector potential T method only calculates the current density in the superconductor. The modeling idea (such as Figure 3 As shown in Figure 1, each superconductor is approximated as a single thin strip, thereby reducing the calculation freedom of the dependent variable. That is, in this two-dimensional model, the superconductor is simplified to a line unit, and then the control equation is derived based on Ampere's law and the law of electromagnetic induction:

[0076]

[0077]

[0078] The current density in method A is J zIt is calculated by the T method, and the magnetic field in the T method is provided by the A method. In addition, the external alternating magnetic field is applied through the Dirichlet boundary, and the current application of the T method is also realized through the Dirichlet boundary. The calculation formula is:

[0079]

[0080] The current I HTS,k is the circumferential current per turn of superconductor calculated by the circuit model.

[0081] For the thermal circuit model, since this example only calculates the loss, the model is only set to fixed temperature boundary conditions. However, if necessary, heat flow boundaries and temperature boundaries can also be set, mainly depending on the simulation conditions.

[0082] Next, we calculate the core parameters of the model, which is the global voltage of a single turn of the magnet:

[0083]

[0084] Finally, the strong nonlinear resistivity ρ, on which the magnetic field, temperature and other physical parameters of the magnet system depend, is calculated to achieve mutual coupling of various field and path models.

[0085]

[0086] Using the above modeling method, the AC loss of the uninsulated magnet can be calculated, such as Figure 4 shown.

[0087] Similarly, for non-circular coils, a three-dimensional model can be used. The modeling strategy is similar to the above two-dimensional model, but it is no longer simplified to line element units, but simplified to thin strips, such as Figure 5 The example shown is a racetrack coil. The remaining constraints can also be expanded according to the reference 2D method.

Claims

1. A full-operating condition analysis method for a non-electrically insulated superconducting magnet system based on field-circuit coupling, characterized in that: The following steps are involved: Step 1: Define the geometry, material properties, boundary condition parameters and model discrete unit settings of the non-electrically insulated superconducting magnet system; Step 2: Based on the circuit equivalent principle, an equivalent circuit model of the superconducting magnet system is established to calculate the current and loss distribution of the magnet system; Step 3: Using current and loss as input, construct the magnetic field and thermal circuit model of the magnet system through analytical or numerical calculation methods to calculate the magnetic field and temperature distribution of the magnet system respectively; Step 4: Calculate the vector magnetic potential of the discrete units of the magnet system to obtain the equivalent global voltage of the magnet system and simplify the circuit model calculation; Step 5: Calculate the strong nonlinear resistivity of the superconducting magnet system that depends on the magnetic field and temperature parameters to achieve mutual coupling of various field and path models.

2. The full-operating condition analysis method of a non-electrically insulated superconducting magnet system based on field-circuit coupling according to claim 1 is characterized in that: The equivalent circuit model in step 2 includes a distributed parameter circuit network model, a single-turn or multi-turn equivalent circuit model, a coil equivalent model, and a coil group equivalent model.

3. The full-operating condition analysis method of a non-electrically insulated superconducting magnet system based on field-circuit coupling according to claim 1 is characterized in that: The calculation method of the magnetic field model in step 3 includes: (1) a multiple integral solution method based on Biotschaff's law: finite difference and mathematical analysis method; (2) a finite element method: H method, TA method, Law.

4. The full-operating condition analysis method of a non-electrically insulated superconducting magnet system based on field-circuit coupling according to claim 1 is characterized in that: The calculation method of the thermal circuit model in step 3 includes: finite difference method, finite volume method, and finite element method.

5. The full-operating condition analysis method of a non-electrically insulated superconducting magnet system based on field-circuit coupling according to claim 1 is characterized in that: The calculation method of the vector magnetic potential in step 4 includes: (1) a multiple integral solution method based on Biotschaffer's law: a finite difference method and a mathematical analysis method; (2) indirect calculation through a finite element magnetic field model.

6. The full-operating condition analysis method of a non-electrically insulated superconducting magnet system based on field-circuit coupling according to claim 1 is characterized in that: The global voltage in step 4 is obtained based on the formula derived from Faraday's law of electromagnetic induction, and the calculation formula is as follows: Where A, B, E are vector magnetic potential, magnetic induction intensity and electric field intensity respectively; l refers to the integral path for calculating the resistive voltage; is the Hamiltonian operator, V HTS , V r , V i They are global voltage, resistive voltage and inductive voltage respectively; the inductive voltage is obtained by the linear integration of the time derivative of the vector magnetic potential, while the resistive voltage is obtained by the nonlinear resistivity of the magnetic system; The nonlinear resistivity calculation method includes: a power exponential calculation formula, a discrete data interpolation method, and a discrete data fitting formula.

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