Computer program, simulation method, and simulation device

Abstract

To provide a computer program capable of efficiently solving a nonlinear equation and simulating the behaviors of an electromagnetic component.SOLUTION: A computer program which causes a computer to simulate the behaviors of an electromagnetic component including a coil and a movable element at each of a plurality of time points on the basis of an analysis model of the electromagnetic component is configured to create a lookup table correlating the position of the movable element with the magnetic flux linkage in the coil, the inductance of the coil, and current flowing through the coil, all obtained through magnetic field analysis based on the analysis mode, and simulate the behaviors of the electromagnetic component by referencing the lookup table using the current flowing through the coil calculated in the previous simulation step, the position of the movable element in the current simulation step, and the position of the movable element in the previous simulation step.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a computer program, a simulation method, and a simulation device for simulating the behavior of an electromagnetic component. [Background technology]

[0002] Simulation devices that simulate motor behavior are used in the development of motors and drive circuits. To simulate motor behavior in detail and accurately, a motor behavior simulator that simulates motor behavior using characteristics obtained by magnetic field analysis is coupled with a drive circuit simulator that simulates the operation of the motor's drive circuit. In this coupled simulator, the drive circuit simulator calls the motor behavior simulator for each simulation step corresponding to each point in time in the time series to simulate the motor's behavior in detail, and then uses the simulation results to simulate the behavior of the drive circuit.

[0003] Motor behavior simulators simulate motor behavior through magnetic field analysis using the finite element method of an analytical model that represents the motor's geometry and electromagnetic characteristics. To streamline calculations, a lookup table (LUT) is created in advance to represent characteristics such as magnetic flux linkage according to the motor's driving state, and the LUT is referenced to simulate motor behavior.

[0004] Motor behavior simulators solve voltage equations to calculate the current flowing through each coil of a motor and simulate the behavior of the motor. Furthermore, magnetic field analysis using the finite element method calculates the magnetic field and other parameters at each part of the motor by solving the governing equations of the magnetic field, and simulates the behavior of the motor. The voltage equation and the governing equations of the magnetic field are nonlinear equations, and solving them generally requires repeated calculations such as the Newton-Laplason method (see, for example, Non-Patent Document 1). The above repeated calculations also pose a problem when simulating the behavior of electromagnetic parts other than motors. [Prior art documents] [Non-patent literature]

[0005] [Non-Patent Document 1] Hiroyuki Kaimori, Kan Akatsu, Behavior Modeling of Permanent Magnet Synchronous Motors Using Flux Linkages for Coupling with Circuit Simulation, IEEJ Journal of Industry Applications, Japan, Institute of Electrical Engineers of Japan, Vol. 7, No. 1, 2018, pp. 56-63 Summary of the Invention [Problem to be solved by the invention]

[0006] As described above, the conventional method has the problem that it requires iterative calculations to solve the nonlinear equations.

[0007] An object of the present disclosure is to provide a computer program, a simulation method, and a simulation device that can efficiently solve nonlinear equations to simulate the behavior of electromagnetic components. [Means for solving the problem]

[0008] A computer program according to one aspect of the present disclosure is (1) a computer program for causing a computer to execute a process of simulating the behavior of an electromagnetic component having a coil and a mover at each of multiple points in time based on an analytical model of the electromagnetic component, the computer program creating a lookup table correlating the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil obtained by magnetic field analysis based on the analytical model, or creating a learning model by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil, and causing the computer to execute a process of simulating the behavior of the electromagnetic component by referring to the lookup table or the learning model using the current flowing in the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step.

[0009] The computer program (1) above preferably causes the computer to execute (2) a process of calculating a current based on the following formula (1):

[0010]

number

[0011] The computer program of (1) or (2) above preferably causes the computer to execute the following process: (3) calculating a current predictor based on the following formula (2-1), and calculating a current based on the following formula (2-2) using the calculated current predictor.

[0012]

number

[0013] The computer program described in any one of (1) to (3) above preferably causes the computer to execute a process of calculating a current in the k-th simulation step by (4) using the current calculated based on the above formula as the current in the (k-1)-th simulation step and repeating the process of calculating a current based on the above formula.

[0014] The computer program described in any one of (1) to (4) above preferably causes the computer to execute a process of (5) calculating the current based on the above formula by using the current and voltage obtained by weighting and adding the current and voltage of the (k-1)th simulation step and the current and voltage of the kth simulation step using a weighting coefficient τ (0<τ<1) as the current and voltage of the kth simulation step.

[0015] The computer program according to any one of (1) to (5) above is preferably configured to cause the computer to execute the following process: (6) the electromagnetic component is a motor having a plurality of the coils; the lookup table is a table correlating the flux linkages in the plurality of coils, the inductances of the plurality of coils, the currents flowing in the plurality of coils, and the position of the mover; the computer acquires the voltages applied to each coil of the motor and the position of the mover from an external drive circuit simulator; reads out the inductances of the plurality of coils and the flux linkages in each coil by referring to the lookup table using the currents flowing in each coil calculated in the previous simulation step and the position of the mover; reads out the flux linkages in the plurality of coils by referring to the lookup table using the currents flowing in each coil calculated in the previous simulation step and the position of the mover; and calculates the current in each coil in the current simulation step based on the inductances and flux linkages of each coil and the acquired voltages and position of the mover.

[0016] A computer program according to one aspect of the present disclosure is (7) a computer program for causing a computer to execute a process of simulating the behavior of an electromagnetic component at each of a plurality of time points using a finite element method based on an analytical model of the electromagnetic component, and causes the computer to execute a process of calculating a magnetic vector potential in the k-th simulation step based on the following formula (3) using the magnetic vector potential calculated in the (k-1)-th simulation step:

[0017]

number

[0018] The computer program of (7) above preferably causes the computer to execute a process of (8) calculating a magnetic vector potential predictor based on the following formula (4-1), and calculating a magnetic vector potential based on the following formula (4-2) using the calculated magnetic vector potential predictor.

[0019]

number

[0020] The computer program of (7) or (8) above preferably causes the computer to execute a process of calculating a magnetic vector potential in the k-th simulation step by repeating the process of calculating a magnetic vector potential based on the above formula (9), using the magnetic vector potential calculated based on the above formula as the magnetic vector potential in the (k-1)-th simulation step.

[0021] A simulation method according to one aspect of the present disclosure is (10) a simulation method in which a computer executes a process to simulate the behavior of an electromagnetic component having a coil and a mover at each of a plurality of points in time based on an analytical model of the electromagnetic component, wherein the computer creates a lookup table correlating the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil obtained by magnetic field analysis based on the analytical model, or creates a learning model by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil, and executes a process to simulate the behavior of the electromagnetic component by referring to the lookup table or the learning model using the current flowing in the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step.

[0022] A simulation method according to one aspect of the present disclosure is (11) a computer program that causes a computer to execute a process of simulating the behavior of an electromagnetic component at each of a plurality of time points using a finite element method based on an analytical model of the electromagnetic component, wherein the computer executes a process of calculating the magnetic vector potential in the k-th simulation step based on the following equation (5) using the magnetic vector potential calculated in the (k-1)-th simulation step.

[0023]

number

[0024] A simulation device according to one aspect of the present disclosure is (12) a simulation device including a processing unit that simulates the behavior of an electromagnetic component having a coil and a mover at each of multiple points in time based on an analytical model of the electromagnetic component, wherein the processing unit creates a lookup table that associates the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil, which are obtained by magnetic field analysis based on the analytical model, or creates a learning model by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing in the coil, and simulates the behavior of the electromagnetic component by referring to the lookup table or the learning model using the current flowing in the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step.

[0025] A simulation device according to one aspect of the present disclosure (13) is a simulation device including a processing unit that simulates the behavior of an electromagnetic component at each of a plurality of time points using a finite element method based on an analytical model of the electromagnetic component, and the processing unit calculates the magnetic vector potential in the k-th simulation step based on the following equation (6) using the magnetic vector potential calculated in the (k-1)-th simulation step.

[0026]

number

[0027] According to the present disclosure, nonlinear equations can be efficiently solved to simulate the behavior of electromagnetic components. [Brief explanation of the drawings]

[0028] [Figure 1] 1 is a block diagram showing the configuration of a simulation device according to a first embodiment. [Figure 2] FIG. 2 is a schematic diagram showing the motor as viewed from the direction of the rotation axis. [Figure 3] FIG. 1 is a conceptual diagram illustrating an outline of a coupled analysis performed by a simulation device. [Figure 4] 10 is a flowchart showing the processing procedure of a processing unit related to creation of an LUT. [Figure 5] 10 is a flowchart showing a processing procedure of a processing unit related to coupled analysis. [Figure 6] 5 is a flowchart showing a processing procedure of a processing unit related to the motor behavior simulation according to the first embodiment. [Figure 7] 10 is a table showing the calculation time required for simulation according to the first embodiment and the modified example. [Figure 8] 10 is a graph showing simulation results of d-axis current according to the first embodiment and the modified example. [Figure 9] 6 is a graph showing simulation results of q-axis current according to the first embodiment and the modified example. [Figure 10] 6 is a graph showing torque simulation results according to the first embodiment and the modified example. [Figure 11] 10 is a flowchart showing the processing procedure of a processing unit related to the creation of a learning model in the second embodiment. [Figure 12] 10 is a flowchart showing the processing procedure of a processing unit related to a motor behavior simulation in the second embodiment. [Figure 13] 10 is a table showing the calculation time required for simulation according to the third embodiment and the modified example. [Figure 14] 10 is a graph showing simulation results of a u-phase current according to the third embodiment and a modification thereof. [Figure 15] 10 is a graph showing torque simulation results according to the third embodiment and a modification thereof. DETAILED DESCRIPTION OF THE INVENTION

[0029] The present invention will be described in detail below with reference to the drawings showing embodiments thereof. (Embodiment 1) Fig. 1 is a block diagram showing the configuration of a simulation device 1 according to a first embodiment, and Fig. 2 is a schematic diagram showing a motor 4 as viewed from the direction of the rotation axis. The simulation device 1 according to the first embodiment is a device that simulates the behavior of electromagnetic parts of the motor 4. The simulation device 1 includes a processing unit 11, a storage unit 12, an input device 13, an output device 14, and a communication unit 15.

[0030] The storage unit 12 is an external storage device having a nonvolatile memory such as a hard disk or a flash memory. The storage unit 12 stores a simulator program 21. The simulator program 21 includes, for example, a motor behavior simulator program 21a, a drive circuit simulator program 21b, a magnetic field analysis simulator program 21c, etc. The simulator program 21 or the motor behavior simulator program 21a corresponds to the computer program (program product) according to the first embodiment.

[0031] The storage unit 12 also stores an analytical model 12a of the motor 4, a drive circuit model that drives the motor 4, etc. The analytical model 12a is data that represents the two-dimensional or three-dimensional shapes and electromagnetic characteristics of the stator 41, the multiple coils 42, and the rotor 43. The motor 4 to be simulated is, for example, a three-phase permanent magnet synchronous motor. The motor 4 shown in FIG. 2 is an 8-pole, 48-slot, concentrated winding, interior permanent magnet synchronous motor (PMSM). The motor 4 includes a cylindrical stator 41 in which U-phase coils 42u, V-phase coils 42v, and W-phase coils 42w are evenly arranged in the circumferential direction to generate field flux, and a rotor 43 arranged concentrically on the inner diameter side of the stator 41. Each coil 42 is, for example, star-connected or delta-connected. The rotor 43 is cylindrical and includes multiple pairs of permanent magnets 43a. Note that the number of poles, slots, and coils 42 is not limited to these. The analysis model 12a includes, for example, a three-dimensional geometric model, such as three-dimensional CAD data, representing the shapes of the multiple coils 42, stator 41, and rotor 43 that constitute the motor 4, and material properties of each component that constitutes the three-dimensional geometric model. The material properties include magnetization properties, electrical properties, mechanical properties, thermal properties, iron loss properties, etc. The electrical properties include conductivity, dielectric constant, etc. The drive circuit to be simulated is composed of, for example, a driver and an inverter. The storage unit 12 stores a drive circuit model that represents a plurality of circuit elements that compose the driver and inverter, as well as the connection state and characteristics of each circuit element.

[0032] Furthermore, the storage unit 12 stores an LUT 12b and a torque LUT 12c as characteristic databases for simulating the behavior of the motor 4. Each characteristic database is created prior to simulating the behavior of the motor 4. The LUT 12b and the torque LUT 12c will be described in detail later.

[0033] The processing unit 11 is a processor that includes one or more arithmetic circuits such as a CPU (Central Processing Unit), a GPU (Graphics Processing Unit), a GPGPU (General-purpose computing on graphics processing units), a TPU (Tensor Processing Unit), etc., internal storage devices such as a ROM (Read Only Memory) and a RAM (Random Access Memory), input / output terminals, a timer, etc. By executing the simulator program 21, the processing unit 11 functions as a motor behavior simulator that simulates the behavior of the motor 4 at each of multiple points in time, a drive circuit simulator that simulates the behavior of a drive circuit that drives the motor 4, and a magnetic field analysis simulator that performs magnetic field analysis of the behavior of the motor 4 using magnetic field analysis such as the finite element method or the boundary element method. Note that the functions of the simulation device 1 may be realized by software, or some or all of them may be realized by hardware using an FPGA, ASIC, or the like.

[0034] The input device 13 is an interface that accepts operations such as data input. The input device 13 includes, for example, a keyboard, a mouse, a touch panel device with a built-in display, etc. The input device 13 sends a control signal to the processing unit 11 according to the content of the accepted operation.

[0035] The output device 14 includes, for example, a display device such as a liquid crystal display, an organic EL (Electro Luminescence) display, or a CRT display, a printing device, etc. The output device 14 outputs information such as simulation results according to instructions from the processing unit 11.

[0036] The communication unit 15 includes a communication circuit connected to the external server 3 etc. via a network (not shown). The communication unit 15 sends and receives various information in accordance with instructions from the processing unit 11.

[0037] The simulation device 1 may be a stand-alone computer or a server device connected to a network. The simulation device 1 may be an on-premise device or a cloud system. The simulation device 1 may be configured with multiple computers that perform distributed processing, or may be configured with multiple virtual machines.

[0038] The simulator program 21 or the motor behavior simulator program 21a including the computer program (program product) according to this embodiment may be provided by a non-transitory recording medium 2 on which the computer program is readably recorded. The storage unit 12 stores the simulator program 21 or the motor behavior simulator program 21a read from the recording medium 2 by a reading device (not shown). The recording medium 2 is, for example, a magnetic disk, an optical disk, or a semiconductor memory. The simulation device 1 may also download the simulator program 21 or the motor behavior simulator program 21a from an external server 3 connected to a communication network and store it in the storage unit 12. The simulator program 21 or the motor behavior simulator program 21a may be a single computer program or may be composed of multiple computer programs, and may be executed on a single computer or multiple computers interconnected by a communication network.

[0039] <Conceptual diagram of coupled analysis> FIG. 3 is a conceptual diagram illustrating an outline of the coupled analysis performed by the simulation device 1. Before simulating the behavior of the motor 4, the simulation device 1 first calculates various characteristics of the motor 4 through magnetic field analysis based on an analytical model 12a, such as a finite element method model. For example, the processing unit 11 generates, as the characteristics of the motor 4, a LUT (Φ, L, I, θ) 12b that associates the total magnetic flux Φ (hereinafter simply referred to as "flux linkage"), which is the sum of the flux linkage components associated with the permanent magnets 43a of the rotor 43 and the phase currents, with the inductance L of each coil 42, the current I flowing through each coil 42, and the mechanical angle θ of the rotor 43. The inductance L includes the self-inductance of each coil 42 and the mutual inductance between the coils 42. Hereinafter, the self-inductance and mutual inductance will be simply referred to as "inductance." Furthermore, the processing unit 11 creates a torque LUT 12c that associates the torque T generated in the rotor 43, the current I flowing through each coil 42, and the mechanical angle θ of the rotor 43.

[0040] Although an example using the mechanical angle θ will be described, it goes without saying that the simulation device 1 may create a lookup table using an electrical angle and simulate the behavior of the motor 4.

[0041] The simulation device 1 then couples the motor behavior simulator with the drive circuit simulator to simulate the behavior of the motor 4. The drive circuit simulator passes the voltage [V] = [Vu, Vv, Vw] applied to the terminals of each coil 42 of the motor 4 and the mechanical angle of the rotor 43 to the motor behavior simulator. The motor behavior simulator uses the voltage [V] and the mechanical angle of the rotor 43, etc., to refer to the LUT 12b and torque LUT 12c to determine the current [I] = [Iu, Iv, Iw] of each coil 42 and the torque of the motor 4, and returns the simulation results to the drive circuit simulator. By repeatedly performing similar processing, the behavior of the motor 4 can be simulated.

[0042] <Simulation method> As a simulation method according to this embodiment, a procedure for creating the LUT 12b and a procedure for simulating the behavior of the motor 4 will be described below in order.

[0043] 4 is a flowchart showing the processing procedure of the processing unit 11 for creating the LUT 12b. The processing unit 11 of the simulation device 1 executes the following processing in accordance with the motor behavior simulator program 21a stored in the storage unit 12. First, the processing unit 11 receives the selection of the analysis model 12a and drive circuit model of the motor 4 to be simulated, as well as various other settings, via the input device 13 (step S11).

[0044] Next, the processing unit 11 performs magnetic field analysis using the finite element method (FEM) while varying parameters indicating the drive state, i.e., the current flowing through each coil 42 and the mechanical angle of the rotor 43 (step S12). When varying the current flowing through each coil 42, the processing unit 11 sets the current flowing through each coil 42 so as to satisfy the current conservation principle. In the FEM, the processing unit 11 divides the three-dimensional geometric model of the motor 4 into multiple elements. For example, the processing unit 11 divides the three-dimensional geometric model of the motor 4 into multiple tetrahedral elements, hexahedral elements, square pyramidal elements, triangular prism elements, etc. The processing unit 11 calculates the magnetic vector potential of each element by numerically calculating a multi-dimensional simultaneous linear equation obtained from Maxwell's equations under specific boundary conditions, such as Dirichlet boundary conditions and Neumann boundary conditions. The magnetic field or magnetic flux density of each part of the motor 4 is obtained from the magnetic vector potential. The magnetic field or magnetic flux density is basic information for calculating current, torque, etc.

[0045] Next, based on the magnetic field analysis results of step S12, the processing unit 11 calculates the current of each coil 42 and the interlinkage magnetic flux in each coil 42 according to the mechanical angle of the rotor 43 (step S13). The processing unit 11 also calculates the inductance of each coil 42 (step S14). The inductance calculated here is differential inductance, and is expressed by the following formula (7). Hereinafter, it will be simply referred to as inductance.

[0046]

number

[0047] Next, based on the magnetic field analysis result of step S12, the processing unit 11 calculates the electromagnetic force acting on the rotor 43 according to the current of each coil 42 and the mechanical angle of the rotor 43, and calculates the torque acting on the rotor 43 (step S15). The processing unit 11 calculates the electromagnetic force acting on the rotor 43 using a method such as the nodal force method.

[0048] Next, based on the magnetic field analysis results of step S12, the processing unit 11 creates an LUT 12b that stores the linkage magnetic flux in each coil, the inductance of each coil 42, the current flowing through each coil 42, and the mechanical angle of the rotor 43 in correspondence with each other (step S16).

[0049] Next, the processing unit 11 creates a torque LUT 12c that stores the torque calculated in step S15, the current flowing through each coil 42, and the mechanical angle of the rotor 43 in association with each other (step S17), and ends the process.

[0050] Although the LUT 12b and the torque LUT 12c have been described as separate tables, these tables may be configured as a single table.

[0051] 5 is a flowchart showing the processing procedure of the coupled analysis performed by the processing unit 11. The processing unit 11 sets initial values ​​for the voltage and current applied to the coil 42, the mechanical angle of the rotor 43, etc. (step S31).

[0052] Next, the processing unit 11 calculates the voltage to be applied to the coil 42 and the mechanical angle of the rotor 43 in the next simulation step based on the current in the coil 42 and the torque acting on the rotor 43 calculated in the previous simulation step (step S32). The processing of step S32 is executed by a drive circuit simulator (see FIG. 3), and the drive circuit simulator provides the simulation results, that is, the voltage [V]=[Vu, Vv, Vw] and the mechanical angle of the rotor 43, to the motor behavior simulator.

[0053] Next, the processing unit 11 simulates the behavior of the motor 4 based on the voltage applied to the motor 4, the current of each coil 42 calculated in the previous or previous simulation step, the flux linkage in each coil 42, the mechanical angle of the rotor 43, etc., and calculates the current flowing through each coil 42 and the torque generated in the rotor 43 (step S33). The processing of step S33 is executed by a motor behavior simulator (see FIG. 3), and the simulation results, i.e., the current [I]=[Iu, Iv, Iw] of the coils 42 and the torque generated in the rotor 43, are passed to a drive circuit simulator. The processing of step S33 will be described in detail later.

[0054] Next, the processing unit 11 determines whether or not the simulation termination condition is satisfied (step S34). For example, when a predetermined number of simulation steps corresponding to a predetermined real time have been executed, the processing unit 11 terminates the simulation. When it is determined that the simulation termination condition is not satisfied (step S34: NO), the processing unit 11 returns the processing to step S32 and repeatedly executes the processing of steps S32 and S33. When it is determined that the simulation termination condition is satisfied (step S34: YES), the processing unit 11 terminates the processing.

[0055] Before describing the details of the processing procedure of the processing unit 11 in step S33, the theory of the method for calculating the current flowing through each coil 42 based on the voltage applied to each coil 42 will be described below.

[0056] The voltage equation of each coil 42 is expressed by the following equation (8), and when the time differential of the interlinkage magnetic flux is replaced with the time difference Δt, it is expressed as the following equation (9).

[0057]

number

[0058]

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[0059] In the case of a star-connected motor, the voltage V of the coil 42 in the above equation (9) is the phase voltage. In other words, the voltage V of the coil 42 is the voltage applied to the terminals of the coil 42 with respect to the neutral point potential. When the neutral point is grounded, the voltage applied to the terminals of the coil 42 is the voltage V of the coil 42 in the above equation (9). In the case of a delta-connected motor, the voltage V of the coil 42 in the above equation (9) is the line voltage.

[0060] Here, the flux linkage Φ at the kth simulation step depends on the unknown current I at the kth simulation step, but we consider expressing this using the current I at the (k-1)th simulation step, which is explicitly known. The flux linkage Φ at the kth simulation step can be approximately expressed as in equation (10) below.

[0061]

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[0062] Here, by substituting the above formula (10) into the above formula (9), the voltage equation of the above formula (9) can be expressed as the following formula (11).

[0063]

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[0064] In the above equation (11), the inductance and flux linkage can be read by referencing the LUT 12b using the current I and the mechanical angle θ of the rotor 43 as keys. According to the above equation (11), the current flowing through each coil 42 in the current simulation step (the kth step) can be calculated without iterative calculations by using information from the previous simulation step (the (k-1)th step). In a star-connected motor, if the neutral point is ungrounded, the potential at the neutral point can be calculated by considering the law of conservation of current, which states that the sum of the currents flowing through each coil is zero, along with the voltage equations for each phase. Specifically, the processing unit 11 can calculate four unknowns, namely, the U-phase current, the V-phase current, the W-phase current, and the neutral point potential, based on four equations, namely, the voltage equations for the U-phase coil 42u, the V-phase coil 42v, and the W-phase coil 42w, and the current conservation equation.

[0065] 6 is a flowchart showing the processing procedure of the processing unit 11 for the motor behavior simulation according to the first embodiment. The processing content of step S33 in the kth step will be described below. k is an integer equal to or greater than 2. The processing unit 11 acquires the voltages applied to each coil 42 and the mechanical angle of the rotor 43 from the drive circuit simulator (step S51). For example, if the drive circuit simulator is configured to output the simulation results as a file, the processing unit 11 reads the applied voltages to each coil 42 and the mechanical angle of the rotor 43 from the file.

[0066] The processing unit 11 refers to the LUT 12b using the current of each coil 42 in the (k-1)th simulation step and the mechanical angle of the rotor 43 in the kth simulation step as keys, and reads out the inductance and flux linkage of each coil 42 (step S52).

[0067] Next, the processing unit 11 refers to the LUT 12b using the current of each coil 42 in the (k-1)th simulation step and the mechanical angle of the rotor 43 in the (k-1)th simulation step as keys, and reads out the interlinkage magnetic flux in each coil 42 (step S53).

[0068] For convenience of explanation, the steps have been explained in the order of steps S52 to S53, but the order of the processes is not particularly limited.

[0069] The processing unit 11 calculates the current flowing through each coil 42 in the k-th simulation step using the above equation (11) based on the voltage applied to each coil 42, the acquired mechanical angle of the rotor 43, and the inductance and flux linkage read out in steps S52 and S53 (step S54). In a star-connected motor, if the neutral point is not grounded, the potential of the neutral point is calculated taking into account the law of conservation of current.

[0070] Next, the processing unit 11 reads out the torque acting on the rotor 43 from the torque LUT 12c using the current of each coil 42 calculated in step S54 and the mechanical angle of the rotor 43 as keys (step S55).

[0071] Then, the processing unit 11 outputs the current of each coil 42 calculated in step S54 and the torque read out in step S55 to the drive circuit simulator (step S56), and ends the process.

[0072] As described above, the simulation device 1, the simulation method, and the simulator program 21 according to the first embodiment use the above formula (11) to efficiently solve the nonlinear equation and simulate the behavior of the motor 4. In other words, the current of the coil 42 can be calculated without performing iterative calculations to solve the nonlinear equation.

[0073] Next, a modified example of the simulation device 1 etc. according to the first embodiment will be described.

[0074] (Variation 1) The processing unit 11 may calculate the current using the current predictor in step S54. Specifically, the processing unit 11 calculates the current predictor by the following formula (12) based on the voltage applied to each coil 42, the acquired mechanical angle of the rotor 43, and the inductance and flux linkage read in steps S52 and S53.

[0075]

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[0076] Next, the processing unit 11 calculates the current of the coil 42 in the k-th simulation step based on the following equation (13) using the calculated current predictor.

[0077]

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[0078] (Variation 2) In step S54, the processing unit 11 may calculate the current of the coils 42 in the k-th simulation step by repeating the calculation using the above formula (11). Specifically, the processing unit 11 calculates the first iterative current by the following formula (14) based on the voltage applied to each coil 42, the acquired mechanical angle of the rotor 43, and the inductance and flux linkage read out in steps S52 and S53.

[0079]

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[0080] Next, the processing unit 11 calculates the current of the coil 42 in the k-th simulation step based on the following equation (15) using the calculated first iterative current.

[0081]

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[0082] Here, as shown in the above formulas (14) and (15), an example has been described in which the current of the coil 42 is calculated by repeating the above formula (11) twice. However, if the calculation speed is faster than that of a conventional iterative method such as the Newton-Laplason method, the current may be calculated by repeating the above formula (11) two or more times. For example, considering the actual simulation results described later, it can be seen that the processing unit 11 may calculate the current by repeating the above formula (11) two to five times. Note that two to five times is an example of the iterative calculation. The number of iterations required to obtain the effect of the simulation method according to the second modification varies depending on the analysis model 12a and the calculation conditions.

[0083] (Variation 3) In step S54, the processing unit 11 may apply a so-called theta method to calculate the current of the coil 42 in the k-th simulation step. Specifically, the processing unit 11 calculates the current of the coil 42 in the k-th simulation step by the following equation (16) based on the voltage applied to each coil 42, the acquired mechanical angle of the rotor 43, and the inductance and flux linkage read out in steps S52 and S53.

[0084]

number

[0085] More specifically, when the theta method is applied to the above formula (11), the processing unit 11 may calculate the current of the coil 42 in the k-th simulation step by the following formula (17).

[0086]

number

[0087] The current of the coil 42 may be calculated by applying the theta method to the above equations (12) and (13) of Modification 1. The current of the coil 42 may also be calculated by applying the theta method to the above equations (14) and (15) of Modification 2.

[0088] Furthermore, the processing unit 11 may calculate the current of the coil 42 in the k-th simulation step using the following equation (18) to which the theta method is applied:

[0089]

number

[0090] When calculating the current using the above formula (18), the processing unit 11 refers to the LUT 12b using the current of the coil 42 in the (k-1)th simulation step and the mechanical angle of the rotor 43 in the kth simulation step as keys, and reads out the inductance and flux linkage of the coil 42. The processing unit 11 also refers to the LUT 12b using the current of the coil 42 in the (k-2)th simulation step and the mechanical angle of the rotor 43 in the (k-1)th simulation step as keys, and reads out the inductance and flux linkage of the coil 42.

[0091] Then, the processing unit 11 calculates the current flowing through each coil 42 in the k-th simulation step using the above equation (18) based on the voltage applied to each coil 42, the mechanical angle of the rotor 43 acquired from the drive circuit simulator, and the inductance and flux linkage read from the LUT 12b as described above. The subsequent processing is the same as in the above embodiment 1.

[0092] <Actual simulation results> Next, the effects of the above-described embodiment 1 and modifications 1 to 3 will be described using the results of an actual simulation. The actual simulation to demonstrate the effects was performed using an analytical model 12a of a 4-pole, 12-slot, concentrated winding, interior permanent magnet synchronous motor (PMSM).

[0093] FIG. 7 is a diagram showing the calculation time required for the simulation according to the first embodiment and the modified example, and FIGS. 8 to 10 are graphs showing the simulation results of the d-axis current, q-axis current, and torque according to the first embodiment and the modified example.

[0094] 7 shows the calculation time required to calculate the behavior of the motor 4 over 10 electrical cycles using the conventional Newton-Laplason method and the simulation methods of Embodiment 1, Modification 1, and Modification 2. The number of simulation steps for 10 electrical cycles is 18,000.

[0095] As shown in the diagram of FIG. 7, the calculation time according to the Newton-Laplason method is 2.15 seconds. The time required for calculation using the calculation formula (11) of the first embodiment is 0.557 seconds. The times required for calculation using the calculation formula (13) of the first modification and the calculation formula (15) of the second modification are 0.796 seconds and 0.790 seconds, respectively. In the first modification, since it is necessary to solve the calculation formulas (13) and (14), the calculation time is longer than that of the method using the calculation formula (11). For the same reason, the calculation time is also longer in the second modification than that of the method using the calculation formula (11).

[0096] In Figures 8, 9, and 10, the black solid line graphs represent the calculation results obtained by simulating the behavior of the motor 4 using the Newton-Laplason method, which is a conventional technique, and the gray solid line graphs represent the simulation results according to the first embodiment and the first and second modifications. The black solid line graphs in Figures 8A, 9A, and 10A represent the calculation results obtained by simulating the behavior of the motor 4 using the calculation method based on equation (11) according to the first embodiment. The gray solid line graphs in Figures 8B, 9B, and 10B represent the calculation results obtained by simulating the behavior of the motor 4 using the calculation method based on equation (13) according to the first modification. The gray solid line graphs in Figures 8C, 9C, and 10C represent the calculation results obtained by simulating the behavior of the motor 4 using the calculation method based on equation (15) according to the second modification. The horizontal axis represents the electrical angle of the rotor 43. The vertical axes of the graphs in Figures 8 to 10 represent the d-axis current, the q-axis current, and the torque, respectively.

[0097] Regarding the calculation accuracy, the d-axis current, the q-axis current, and the torque are all reproduced with high accuracy, as shown in Figures 8 to 10. The calculation accuracy of Modifications 1 and 2 is slightly higher than that of the calculation method using Equation (11) of Embodiment 1.

[0098] The calculation time required for the simulation according to the third modification is approximately the same as that for the calculation formula (11). The calculation accuracy depends on the weighting coefficient τ.

[0099] As described above, the simulation device 1, the simulation method, and the simulator program 21 according to the first embodiment and the first to third modifications can efficiently solve nonlinear equations and simulate the behavior of the motor 4.

[0100] In the present embodiment and the modified examples, the motor 4 has been described as a rotating machine in which the mover rotates, but the present invention can also be applied to a motor 4 as a direct-acting motor in which the mover moves linearly, thereby simulating the behavior of the motor 4. The behavior of a direct-acting motor can be simulated using the same processing procedure, with only the shape of the analysis model 12a being different.

[0101] Furthermore, although the motor 4 in which the mover moves linearly or rotationally has been described as the subject of analysis, the manner in which the mover moves is not particularly limited, and the present invention can also be applied to a motor 4 in which the mover vibrates, a solenoid actuator, etc. The present invention can also be applied to an induction machine.

[0102] Furthermore, the application of the present invention is not limited to simulating a motor 4 having a mover, but can also be applied to any electromagnetic component having multiple coils. For example, the present invention can be applied to simulating the behavior of a stationary device such as a transformer or a contactless charger. Regarding the behavior of an electromagnetic component that does not have a mover, the current flowing through the coil can be found based on the following equation (19).

[0103]

number

[0104] Furthermore, in the present embodiment and its modifications, an example has been described in which the voltage and the mechanical angle of the rotor 43 are passed from the drive circuit simulator to the motor behavior simulator, and the current and torque are returned from the motor behavior simulator to the drive simulator, but the physical quantities exchanged between the simulators are not limited to this and may be selected as appropriate. Also, a configuration may be adopted in which physical constants representing the state of the motor 4 or the generator are exchanged. For example, the drive circuit simulator may be configured to transfer current to the motor behavior simulator, and return voltage to the drive simulator. In this case, the motor behavior simulator solves the above equation (11) with the current as a known quantity and the voltage of each coil 42 as an unknown quantity, calculates the voltage generated in each coil 42, and returns the calculated voltage to the drive circuit simulator.

[0105] Furthermore, in the present embodiment and the modified examples, the LUT 12b has been described as information that associates the current flowing through each coil 42, the mechanical angle of the rotor 43, the flux linkage in each coil 42, and the inductance of each coil 42. However, it is also possible to use a first LUT that outputs the inductance of each coil 42 when the current of each coil and the mechanical angle of the rotor 43 are input, and a second LUT that outputs the flux linkage in each coil 42 when the current of each coil and the mechanical angle of the rotor 43 are input.

[0106] (Embodiment 2) The simulation device 1, the simulation method, and the simulator program 21 according to the second embodiment are different from those according to the first embodiment in that a machine learning model is used to calculate the inductance, the flux linkage, and the torque of the coil 42, instead of the LUT 12b and the torque LUT 12c. The following mainly describes the above differences.

[0107] 11 is a flowchart showing the processing procedure of the processing unit 11 for creating a learning model in embodiment 2. The processing unit 11 according to embodiment 2 performs magnetic field analysis by the finite element method while varying the values ​​of the current flowing through the coil 42 and the mechanical angle of the rotor 43, by the same processing as in steps S11 to S15 of embodiment 1, and calculates the flux linkage, inductance, and torque (steps S211 to S215).

[0108] Next, the processing unit 11 creates a magnetic characteristic learning model that outputs the flux linkage and inductance of the coil 42 when the current of the coil 42 and the mechanical angle of the rotor 43 are input (step S216).

[0109] The magnetic property learning model is a model including a neural network (NN) such as a deep neural network (DNN). The magnetic property learning model is a neural network having an input layer to which the current of the coil 42 and the mechanical angle of the rotor 43 are input, an intermediate layer that extracts feature quantities of the magnetic properties of the motor 4, and an output layer that outputs the flux linkage and inductance of the coil 42 calculated based on the extracted feature quantities.

[0110] The magnetic characteristic learning model may output the flux linkage and inductance of the coil 42 when the analytical model 12a, the current of the coil 42, and the mechanical angle of the rotor 43 are input.

[0111] Furthermore, in the second embodiment, an example has been described in which the magnetic property learning model is a general neural network, but the model may be configured as another neural network using a transformer such as a large language model (LLM), a support vector machine (SVM), a Bayesian network, or a regression tree.

[0112] Next, the processing unit 11 creates a torque learning model that outputs the torque generated in the rotor 43 when the current of the coil 42 and the mechanical angle of the rotor 43 are input (step S217).

[0113] The torque learning model is a model including a neural network such as a deep neural network. The torque learning model is a neural network having an input layer to which the current of the coil 42 and the mechanical angle of the rotor 43 are input, an intermediate layer to extract feature quantities related to the torque generated in the motor 4, and an output layer to output the torque generated in the coil 42 calculated based on the extracted feature quantities.

[0114] The torque learning model may be one that outputs the flux linkage and inductance of the coil 42 when the analytical model 12a, the current of the coil 42, and the mechanical angle of the rotor 43 are input.

[0115] Furthermore, in the second embodiment, an example has been described in which the torque learning model is a general neural network, but the torque learning model may be a model having a configuration such as another neural network such as a transformer, an SVM (Support Vector Machine), a Bayesian network, or a regression tree.

[0116] The processing unit 11 may create a learning model that outputs the flux linkage, inductance, and torque of the coil 42 when the analytical model 12a, the current of the coil 42, and the mechanical angle of the rotor 43 are input.

[0117] 12 is a flowchart showing the processing procedure of the processing unit 11 for the motor behavior simulation in embodiment 2. The processing unit 11 acquires the voltages applied to the coils 42 and the mechanical angle of the rotor 43 from the drive circuit simulator (step S251).

[0118] The processing unit 11 calculates the inductance and flux linkage of each coil 42 by inputting the current of each coil 42 in the (k-1)th simulation step and the mechanical angle of the rotor 43 in the kth simulation step into the magnetic characteristic learning model (step S252).

[0119] Next, the processing unit 11 calculates the interlinkage magnetic flux in each coil 42 by inputting the current in each coil 42 in the (k-1)th simulation step and the mechanical angle of the rotor 43 in the (k-1)th simulation step into the magnetic characteristic learning model (step S253).

[0120] The processing unit 11 calculates the current flowing through each coil 42 in the kth simulation step using the above equation (11) etc. based on the voltage applied to each coil 42, the acquired mechanical angle of the rotor 43, and the inductance and interlinkage magnetic flux calculated in steps S252 to S253 (step S254).

[0121] Next, the processing unit 11 inputs the current of each coil 42 calculated in step S254 and the mechanical angle of the rotor 43 into the torque learning model, thereby calculating the torque acting on the rotor 43 (step S255).

[0122] Then, the processing unit 11 outputs the current of each coil 42 calculated in step S254 and the torque calculated in step S255 to the drive circuit simulator (step S256), and ends the process.

[0123] As described above, the simulation device 1, simulation method, and simulator program 21 according to the second embodiment can efficiently solve nonlinear equations and simulate the behavior of the motor 4, similar to the first embodiment. In particular, in the second embodiment, by using the magnetic property learning model and the torque learning model, it is possible to flexibly estimate the relationship between the current, inductance, flux linkage, and torque of the coil 42, and to widely simulate the behavior of the motor 4. In other words, the magnetic property learning model and the torque learning model make it possible to estimate the flux linkage and inductance for combinations of the current of the coil 42 and the mechanical angle of the rotor 43 that are not included in the learning data of the learning model.

[0124] (Embodiment 3) The simulation device 1, simulation method, and simulator program 21 according to the third embodiment are techniques for simulating the behavior of an electromagnetic component such as a motor 4 at each of a plurality of points in time by magnetic field analysis using the finite element method, based on an analysis model 12a of the electromagnetic component. The simulator program 21 or the magnetic field analysis simulator program 21c corresponds to a computer program (program product) according to the third embodiment.

[0125] The processing unit 11 of the simulation device 1 according to the third embodiment efficiently solves the governing equations (nonlinear equations) related to the magnetic field analysis by the finite element method, and calculates the magnetic vector potential related to the behavior of the motor 4 at each of a plurality of points in time.

[0126] Specifically, the processing unit 11 uses the magnetic vector potential calculated in the (k-1)th simulation step, which is the previous simulation step, to calculate the magnetic vector potential in the kth simulation step based on the following formula (20). Specifically, the processing unit 11 calculates the magnetic vector potential in the kth simulation step based on the following formula (20).

[0127]

number

[0128] The processing unit 11 can calculate the magnetic flux density or magnetic field in each element of the analysis model 12a of the motor 4 using the calculated magnetic vector potential. If the magnetic flux density is known, the current density in each element can be calculated, for example, using the above formula (8). Thereafter, by finding the magnetic vector potential in each simulation step using the above formula (20), it is possible to simulate the behavior of the electromagnetic parts at each of multiple time points.

[0129] For example, the processing unit 11 may be configured not to create the LUT 12b and the torque LUT 12c in the first embodiment, but to obtain the magnetic vector potential by magnetic field analysis using the finite element method and the above formula (20) in step S54, and to calculate the current and torque flowing in each coil 42.

[0130] The simulation device 1, simulation method, and simulator program 21 according to the third embodiment can efficiently perform magnetic field analysis using the finite element method, calculate magnetic vector potential, and simulate the behavior of electromagnetic parts such as the motor 4.

[0131] The processing unit 11 may calculate the flux linkage and inductance based on the magnetic vector potential calculated by the above formula (20), and calculate the LUT 12b and the torque LUT 12c according to the first embodiment.

[0132] Next, a modified example of the simulation device 1 etc. according to the third embodiment will be described.

[0133] (Variation 1) The processing unit 11 may calculate the magnetic vector potential using the magnetic vector potential predictor. Specifically, the processing unit 11 calculates the magnetic vector potential predictor using the following equation (21).

[0134]

number

[0135] Next, the processing unit 11 calculates the magnetic vector potential in the k-th simulation step based on the following equation (22) using the calculated magnetic vector potential predictor.

[0136]

number

[0137] (Variation 2) The processing unit 11 may calculate the magnetic vector potential in the k-th simulation step by repeating the calculation using the above formula (20). Specifically, the processing unit 11 calculates the first iteration magnetic vector potential using the following formula (23).

[0138]

number

[0139] Next, the processing unit 11 calculates the magnetic vector potential in the k-th simulation step based on the following formula (24) using the calculated first iterative magnetic vector potential.

[0140]

number

[0141] Here, an example has been described in which the magnetic vector potential is calculated by repeating the above formula (20) twice, but the magnetic vector potential may be calculated by repeating the above formula (20) two or more times if the calculation speed is faster than that of conventional iterative methods such as the Newton-Laplason method.

[0142] Here, as shown in the above formulas (23) and (24), an example has been described in which the magnetic vector potential is calculated by repeating the above formula (20) twice. However, if the calculation speed is faster than conventional iterative methods such as the Newton-Laplason method, the current may be calculated by repeating the above formula (20) two or more times. For example, considering the actual simulation results described below, it can be seen that the processing unit 11 may calculate the current by repeating the above formula (20) two to six times. Note that two to six times is an example of the iterative calculation. The number of iterations required to obtain the effect of the simulation method according to the second modification varies depending on the analysis model 12a and the calculation conditions.

[0143] <Actual simulation results> Next, the effects of the above-described embodiment 3 and modification 2 will be described using the results of an actual simulation. The analytical model 12a used in the actual simulation to demonstrate the effects is the same as that of embodiment 1, and the behavior of the motor 4 was simulated by magnetic field analysis using the finite element method.

[0144] FIG. 13 is a diagram showing the calculation time required for the simulation according to the third embodiment and the modified example, and FIGS. 14 and 15 are graphs showing the simulation results of the u-phase current and torque according to the third embodiment and the modified example.

[0145] As shown in the chart of Fig. 13, the calculation time according to the Newton-Laplason method is 6.8 seconds. The time required for calculation using the calculation formula (20) of the third embodiment is 1.2 seconds. The time required for calculation using the calculation formula (24) of the second modification is 2.2 seconds. In the second modification, since it is necessary to solve the calculation formulas (23) and (24), the calculation time is longer than that of the method using the calculation formula (20).

[0146] In Figures 14 and 15, the black solid line graphs represent the calculation results obtained by simulating the behavior of motor 4 using the Newton-Laplason method, which is a conventional technique, and the gray solid line graphs represent the simulation results according to embodiment 3 and modification 2. The black solid line graphs shown in Figures 14A and 15A represent the calculation results obtained by simulating the behavior of motor 4 using the calculation method that uses equation (20) according to embodiment 3. The gray solid line graphs shown in Figures 14B and 15B represent the calculation results obtained by simulating the behavior of motor 4 using the calculation method that uses equation (24) according to modification 2. The horizontal axis represents the electrical angle of rotor 43. The vertical axes of the graphs shown in Figures 14 and 15 represent the u-phase current and torque, respectively.

[0147] Regarding the calculation accuracy, both the u-phase current and torque are reproduced with high accuracy, as shown in Figures 14 and 15. The calculation accuracy of Modification 2 is higher than that of the calculation method using Equation (20) of Embodiment 3.

[0148] Although there are no actual simulation results for the simulation method according to the first modification, it is estimated that the same level of calculation speed and calculation accuracy as those of the first modification can be obtained.

[0149] As described above, the simulation device 1, the simulation method, and the simulator program 21 according to the third embodiment and the modified examples can efficiently perform magnetic field analysis by the finite element method, calculate the magnetic vector potential, and simulate the behavior of electromagnetic parts such as the motor 4.

[0150] The embodiments disclosed herein are illustrative in all respects and should not be considered limiting. The technical features described in each embodiment can be combined with each other, and the scope of the present invention is intended to include all modifications within the scope of the claims and the scope equivalent to the claims. The sequences shown in each embodiment are not limited, and the order of each process may be changed within a range consistent with the present invention, and multiple processes may be executed in parallel. The entity that performs each process is not limited, and the process of each device may be executed by another device within a range consistent with the present invention.

[0151] The matters described in each embodiment can be combined with each other. In addition, the independent claims and dependent claims described in the claims can be combined with each other in any combination, regardless of the reference format. Furthermore, although the claims use a format in which a claim references two or more other claims (multiple claim format), this is not limited to this format. A multiple claim (multi-multi claim) that references at least one other multiple claim may also be used. [Explanation of symbols]

[0152] 1 Simulation device 2. Recording media 3. Server Computer 4 motors 11 Processing section 12 Storage section 12a Analysis Model 12b LUT 12c Torque LUT 13 Input Devices 14 Output Devices 15 Communication Interface 21 Simulator Program 21a Motor behavior simulator program 21b Drive circuit simulator program 21c Magnetic Field Analysis Simulator Program 41 Stator 42 Coil 42u U-phase coil 42v V phase coil 42w W phase coil 43 Rotor

Claims

1. A computer program for causing a computer to execute a process of simulating behavior of an electromagnetic component having a coil and a mover at each of a plurality of points in time based on an analytical model of the electromagnetic component, the computer program comprising: a look-up table associating the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil, which are obtained by magnetic field analysis based on the analysis model, or a learning model is created by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil; The behavior of the electromagnetic component is simulated by referring to the lookup table or the learning model using the current flowing through the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step. A computer program that causes the computer to execute a process.

2. The current is calculated based on the following formula (1): The computer program product according to claim 1 , which causes the computer to execute a process. [Equation 1]

3. A current predictor is calculated based on the following formula (2-1): The current is calculated based on the following formula (2-2) using the calculated current predictor. The computer program product according to claim 1 , which causes the computer to execute a process. [Equation 2]

4. The current calculated based on the above formula (1) is used as the current in the (k-1)th simulation step, and the process of calculating the current based on the above formula (1) is repeated again to calculate the current in the kth simulation step. The computer program according to claim 2, which causes the computer to execute a process.

5. The current and voltage obtained by weighting and adding the current and voltage of the (k-1)th simulation step and the current and voltage of the kth simulation step with a weighting coefficient τ (0<τ<1) are defined as the current and voltage of the kth simulation step, and the current is calculated based on the above formula.

5. The computer program according to claim 2, which causes the computer to execute a process.

6. the electromagnetic component is a motor having a plurality of the coils, The lookup table is a table associating interlinkage magnetic fluxes in the plurality of coils, inductances of the plurality of coils, currents flowing through the plurality of coils, and positions of the mover; The voltage applied to each coil of the motor and the position of the mover are acquired from an external drive circuit simulator; referring to the lookup table using the current flowing through each coil calculated in the previous simulation step and the position of the mover in the current simulation step, to read out the inductance of the plurality of coils and the flux linkage in each coil; reading out the flux linkages in the plurality of coils by referring to the look-up table using the currents flowing through the coils and the position of the mover calculated in the previous simulation step; Calculate the current of each coil in the current simulation step based on the inductance and interlinkage magnetic flux of each coil that have been read out, the acquired voltage, and the position of the mover.

5. A computer program according to claim 1, for causing a computer to execute a process.

7. A computer program for causing a computer to execute a process of simulating behavior of an electromagnetic component at each of a plurality of points in time using a finite element method based on an analytical model of the electromagnetic component, Using the magnetic vector potential calculated in the (k-1)th simulation step, the magnetic vector potential in the kth simulation step is calculated based on the following formula (3): A computer program that causes the computer to execute a process. [Equation 3]

8. A magnetic vector potential predictor is calculated based on the following formula (4-1): The magnetic vector potential is calculated based on the following formula (4-2) using the calculated magnetic vector potential predictor. The computer program product according to claim 7, which causes the computer to execute a process. [Equation 4]

9. The magnetic vector potential calculated based on the above formula (3) is used as the magnetic vector potential in the (k-1)th simulation step, and the process of calculating the magnetic vector potential based on the above formula (3) is repeated to calculate the magnetic vector potential in the kth simulation step. The computer program product according to claim 7, which causes the computer to execute a process.

10. A simulation method in which a computer executes a process of simulating behavior of an electromagnetic component having a coil and a mover at each of a plurality of points in time based on an analytical model of the electromagnetic component, the method comprising: The computer a look-up table associating the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil, which are obtained by magnetic field analysis based on the analysis model, or a learning model is created by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil; The behavior of the electromagnetic component is simulated by referring to the lookup table or the learning model using the current flowing through the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step. A simulation method for performing processing.

11. A computer program that causes a computer to execute a process of simulating behavior of an electromagnetic component at each of a plurality of points in time using a finite element method based on an analytical model of the electromagnetic component, The computer Using the magnetic vector potential calculated in the (k-1)th simulation step, the magnetic vector potential in the kth simulation step is calculated based on the following formula (5): A simulation method for performing processing. [Equation 5]

12. A simulation device including a processing unit that simulates behavior of an electromagnetic component having a coil and a mover at each of a plurality of points in time based on an analytical model of the electromagnetic component, The processing unit a look-up table associating the position of the mover with the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil, which are obtained by magnetic field analysis based on the analysis model, or a learning model is created by machine learning the relationship between the position of the mover, the flux linkage in the coil, the inductance of the coil, and the current flowing through the coil; The behavior of the electromagnetic component is simulated by referring to the lookup table or the learning model using the current flowing through the coil calculated in the previous simulation step, the position of the mover in the current simulation step, and the position of the mover in the previous simulation step. Simulation device.

13. A simulation device including a processing unit that simulates behavior of an electromagnetic component at each of a plurality of points in time using a finite element method based on an analytical model of the electromagnetic component, The processing unit Using the magnetic vector potential calculated in the (k-1)th simulation step, the magnetic vector potential in the kth simulation step is calculated based on the following formula (6): Simulation device. [Equation 6]

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