Eddy current analysis method, eddy current analysis device, and program

Abstract

To provide an eddy current analysis method which can numerically determine the eddy current that is generated when an electrical conductor is placed in a magnetic field that changes over time.SOLUTION: An analysis object defined in an analytic space is divided into a plurality of volume elements, and a magnetic moment is added to each of the plurality of volume elements. An external magnetic field to be applied to the analysis object from the outside is defined. A step of calculating a vector potential at each position of the plurality of volume elements on the basis of the magnetic moment of each of the plurality of volume elements and the external magnetic field, a step of calculating an eddy current on the basis of the vector potential at each position of the plurality of volume elements, a step of calculating an induced magnetic field at each positions of the plurality of volume elements, and a step of updating the magnetic moment of each of the plurality of volume elements on the basis of the induced magnetic field are repeated. In the step for calculating the vector potential, a binding condition that a normal direction component of the eddy current on the surface of the analysis object becomes zero is imposed.SELECTED DRAWING: Figure 5

Description

Technical Field

[0001] The present invention relates to an eddy current analysis method, an eddy current analysis apparatus, and a program.

Background Art

[0002] There is a known analysis apparatus that numerically calculates an induced magnetic field by representing an analysis object with a plurality of particles and using the induced magnetization induced in each particle due to the time variation of an external magnetic field and the magnetic field obtained by the interaction between the magnetic moments based on the induced magnetization (Patent Document 1).

Prior Art Documents

Patent Documents

[0003]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0004] When a time-varying magnetic field is applied to an electric conductor (conductor), eddy currents are generated by electromagnetic induction. When analyzing the operation of an electromagnetic actuator, there may be a case where eddy currents are desired to be obtained. An object of the present invention is to provide an eddy current analysis method, an eddy current analysis apparatus, and a program capable of numerically obtaining eddy currents generated when an electric conductor is placed in a time-varying magnetic field.

Means for Solving the Problems

[0005] According to one aspect of the present invention, an analysis object defined in an analysis space is divided into a plurality of volume elements, a magnetic moment is assigned to each of the plurality of volume elements, an external magnetic field applied externally to the analysis object is defined, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and Repeatedly, A method for analyzing eddy currents is provided in which, in the procedure for calculating the vector potential, a constraint condition is imposed such that the normal component of the eddy currents on the surface of the object being analyzed becomes zero.

[0006] According to another aspect of the present invention, An input device into which the analysis conditions are entered, A processing unit that performs eddy current analysis based on the analysis conditions input to the input device, Equipped with, The aforementioned processing apparatus is Based on the analysis conditions input to the input device, the object to be analyzed defined in the analysis space is divided into multiple volume elements. A magnetic moment is applied to each of the aforementioned multiple volume elements, An external magnetic field to be applied to the object to be analyzed is defined, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and Repeatedly, An eddy current analyzer is provided that, in the procedure for calculating the vector potential, imposes a constraint condition that the normal component of the eddy current on the surface of the object being analyzed becomes zero.

[0007] According to yet another aspect of the present invention, A function to divide the object to be analyzed defined within the analysis space into multiple volume elements, The function of imparting a magnetic moment to each of the aforementioned multiple volume elements, The function defines an external magnetic field to be applied to the object being analyzed from the outside. A program that enables a computer to implement this, moreover, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and A function to repeat the process, In the procedure for calculating the vector potential, a constraint is imposed such that the normal component of the eddy current on the surface of the object being analyzed is zero. A program is provided to achieve this. [Effects of the Invention]

[0008] By imposing a constraint condition that makes the normal component of eddy currents on the surface of the object being analyzed zero, the accuracy of eddy current analysis can be improved. [Brief explanation of the drawing]

[0009] [Figure 1] Figure 1A is a perspective view of the cylindrical object being analyzed, and Figure 1B is a schematic diagram showing the results of the eddy current analysis. [Figure 2]Figure 2A is a perspective view of the rectangular prism-shaped object being analyzed, and Figure 2B is a schematic diagram showing the results of the eddy current analysis. [Figure 3] Figure 3A is a schematic diagram showing the distribution of multiple beads in a cross-section perpendicular to the height direction (z-direction) of a rectangular prism-shaped object being analyzed, and Figure 3B is a schematic diagram showing some of the beads and hypothetical particles on the surface of the object being analyzed. [Figure 4] Figure 4 is a block diagram of an eddy current analyzer according to an embodiment. [Figure 5] Figure 5 is a flowchart showing the procedure for the eddy current analysis method according to the example. [Figure 6] Figure 6 is a schematic diagram showing the distribution of eddy currents obtained from the analysis. [Modes for carrying out the invention]

[0010] Before describing the eddy current analysis method using the examples, we will explain the eddy current analysis method using reference examples, referring to Figures 1A to 2B.

[0011] [Reference example] Figure 1A is a perspective view of the cylindrical object 10 to be analyzed. The object 10 is placed in the analysis space. The object 10 is made of an electrically conductive magnetic material. An xyz Cartesian coordinate system is defined with the height direction of the object 10 as the z direction.

[0012] In the reference examples and embodiments, the object to be analyzed is divided into multiple volume elements (hereinafter referred to as beads), and the magnetic field state within the object to be analyzed is determined by calculating the magnetic interaction between the beads. The number of beads constituting the object to be analyzed 10 is N, and the multiple beads are numbered sequentially from 0. Vector potential A on the i-th bead i It can be expressed by the following formula.

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[0013] Here, A extHere, is the externally applied vector potential, μ is the permeability, σ is the conductivity, A is the vector potential on the bead, r is the position vector, and v is the volume. The subscripts i and j are the sequential numbers assigned to the beads, respectively. In equation (1), r' is the position vector of a point inside the j-th bead, and the integration range is the internal region of the j-th bead.

[0014] The eddy current j is expressed by the following equation.

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[0015] Eddy current analysis of the cylindrical object 10 (Figure 1A) was performed using equation (1). Conductivity σ was set to 10 6 The S / m value, the cylinder radius was set to 10 mm, and the height to 100 mm. An external magnetic flux density in the z direction with amplitude 1T and frequency 50 Hz is applied to the entire analysis space where the object 10 is located. External magnetic flux density B ext and external vector potential A ext The following relationships exist:

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[0016] Figure 1B is a schematic diagram showing the results of the eddy current analysis. Figure 1B shows the distribution of eddy currents in a cross-section perpendicular to the z-axis of the object being analyzed 10. The arrows shown in Figure 1B indicate the direction of the eddy currents. It can be seen that eddy currents are generated in an almost circumferential direction. The magnitude of the eddy currents on the surface of the object being analyzed 10 was 2.91 MA. The difference from the analysis results using a general finite element method was 0.5%.

[0017] Figure 2A is a perspective view of the rectangular prism-shaped object 10 to be analyzed. An xyz Cartesian coordinate system is defined where the height direction of the object 10 is the z direction and the normal direction of one side is the x direction.

[0018] Figure 2B is a schematic diagram showing the results of the eddy current analysis. Figure 2B shows the distribution of eddy currents in a cross-section perpendicular to the z-axis of the object being analyzed 10. The arrows shown in Figure 2B indicate the direction of the eddy currents. It can be seen that the eddy currents have a normal component at the surface of the object being analyzed 10. The results of the eddy current analysis of the rectangular prism-shaped object being analyzed 10 using the analysis method in the reference example cannot be said to represent actual physical phenomena. In contrast, the results of the analysis using the general finite element method showed that the normal component of the eddy currents at the surface of the object being analyzed 10 was almost zero.

[0019] If we denote the normal vector of the conductor surface as n, the current j flowing through the conductor surface must satisfy the following equation. That is, the normal component of the current flowing through the conductor surface must be zero.

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[0020] [Constraints under which the normal component of surface eddy currents is zero] In the examples described below, the constraint condition of equation (4) is imposed and the analysis is performed. Next, the method for imposing the constraint condition of equation (4) will be explained with reference to Figures 3A and 3B.

[0021] Figure 3A is a schematic diagram showing the distribution of multiple beads 20 in a cross-section perpendicular to the z-direction of a rectangular prism-shaped object 10 for analysis. Multiple virtual particles 21 are arranged outside the beads 20 located on the surface of the object 10 for analysis, so as to follow the surface. For example, the surface of the object 10 for analysis is covered with multiple virtual particles 21 in one layer.

[0022] Figure 3B is a schematic diagram showing a part of the beads 20 and the virtual particles 21 on the surface of the analysis object 10. When the eddy current flowing through the i-th bead 20 is marked as j i the boundary condition of Equation (4) is represented by the following Equation (5).

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[0023] For the beads 20 on the surface of the analysis object 10, solve the simultaneous equations of Equation (5) and the following Equation (6) to calculate the eddy current j b flowing through the virtual particles 21.

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[0024] Here, k is the serial number sequentially assigned from 0 to the virtual particle 21, and N b is the number of virtual particles 21. r b ’ in Equation (6) is the position vector of a point inside the k-th virtual particle 21, and the integration range of the third term on the right side of Equation (6) is the region inside the k-th virtual particle 21. The first and second terms on the right side of Equation (6) are the same as the first and second terms on the right side of Equation (1).

[0025] For the solution method of the simultaneous equations of Equation (5) and Equation (6), for example, an iterative method, a Gauss elimination method, etc. can be used. When the eddy current j b flowing through the virtual particle 21 is obtained by solving the simultaneous equations of Equation (5) and Equation (6), from the obtained j b and Equation (6), for the beads 20 inside the analysis object 10, the vector potential A i is also calculated. Thereby, the vector potential when the boundary condition that the normal component of the surface eddy current becomes zero is imposed is obtained.

[0026] Next, an example of the iterative calculation for solving the simultaneous equations of Equation (5) and Equation (6) will be described. The dot of the second term on the right side of Equation (6) for A j is calculated by the following equation.

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[0027] Equation (6) for performing iterative calculations can be expressed as follows by introducing the iterative calculation step s+1:

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[0028] Next, with reference to Figure 4, an eddy current analysis apparatus according to an embodiment will be described. Figure 4 is a block diagram of the eddy current analysis apparatus 30 according to this embodiment. The eddy current analysis apparatus 30 according to this embodiment includes a processing unit 31, an input device 32, an output device 33, and a storage device 34. The processing unit 31 includes an analysis information acquisition unit 311, a magnetization application unit 312, a calculation unit 313, and an output control unit 314.

[0029] Each block shown in Figure 4 can be implemented in hardware terms using components and mechanical devices, including the central processing unit (CPU) of a computer, and in software terms using computer programs, etc. Figure 4 shows functional blocks that are realized through the cooperation of hardware and software. Therefore, these functional blocks can be realized in various ways by combining hardware and software.

[0030] The processing unit 31 is connected to an input device 32 and an output device 33. The input device 32 receives commands and data from the user related to the processing performed by the processing unit 31. The input device 32 can be, for example, a keyboard or mouse that the user operates to input data, a communication device that receives input via a network such as the Internet, or a reader that receives input from removable media such as CDs or DVDs.

[0031] The analysis information acquisition unit 311 acquires information necessary for eddy current analysis via the input device 32. The information necessary for eddy current analysis includes, for example, the shape of the object to be analyzed 10 (Figures 1A and 2A) defined in the virtual space, information for dividing the object to be analyzed 10 into multiple beads 20 (Figures 3A and 3B), and the physical properties of the object to be analyzed 10.

[0032] The magnetization application unit 312 divides the object to be analyzed 10 in the analysis space into a plurality of beads 20 based on the information acquired by the analysis information acquisition unit 311, and applies magnetization to each of the beads 20. Each of the plurality of beads 20 is assigned a bead identifier (sequential number) to identify the bead 20. The magnetization applied to the beads 20 may be applied as an initial condition, or the magnetization generated in each bead 20 may be calculated based on the external magnetic field obtained by the calculation unit 313 described later, and a magnetic moment based on the calculated magnetization may be applied. Since the magnetization is the value per unit volume of the vector sum of the total magnetic moments in the bead 20, the magnetic moment to be applied to the bead 20 can be determined from the magnetization obtained for each bead 20. The magnetization application unit 312 stores the information of the applied magnetic moment in the storage device 34, associating it with the bead identifier.

[0033] The calculation unit 313 calculates the magnetic field and eddy currents at multiple observation points in the analysis space based on the magnetic moment applied to the beads 20 and the external magnetic field supplied from the outside, and stores the calculation results in the storage device 34. The observation points are, for example, placed at the center of each bead 20. The calculations performed by the calculation unit 313 will be described in detail later.

[0034] The output control unit 314 outputs the analysis results, such as the calculation results of the magnetic field and eddy currents for each observation point stored in the memory device 34, to the output device 33. The output device 33 can be, for example, a display, a removable media writing device, a communication device, etc.

[0035] Next, we will explain the calculation methods for various physical quantities calculated in the eddy current analysis method according to the examples. Note that the following calculation methods are described in Japanese Patent Publication No. 2022-112214, so we will only briefly explain them here.

[0036] [Calculation of magnetic flux density, magnetic field vector, magnetization vector, and magnetic moment in magnetic materials] This section explains how to calculate the magnetic flux density, magnetic field vector, and time derivative of the magnetic flux density within bead 20. iThe magnetic flux density generated can be expressed by the following formula.

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[0037] Position r within a magnetic body i The magnetic field vector at this point is expressed by the following equation.

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[0038] Total external magnetic field vector H o (r i ) is expressed by the following formula.

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[0039] The magnetization vector M(H) in equations (9) and (10) is expressed by the following equations when the magnetic material is a linear material.

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[0040] To find the magnetic flux density in the i-th bead and the time derivative of the magnetic flux density, use the position vector r in equations (9) and (11). i You can substitute the center position of the bead into the formula. If the object being analyzed 10 is a non-magnetic material, the magnetization vector M is zero and the magnetic susceptibility χ is also zero.

[0041] If the magnetic material is a linear magnetic material, the total external magnetic field vector H in equation (12) o (r i Once ) is determined, the position r can be found from equation (13). i The magnetization vector M can be determined. If the magnetic material is a nonlinear magnetic material, the magnetization vector M can be determined using equations (10) and (14) and the total external magnetic field vector H o (r i From the expression written as a function of ), the position r i The magnetization vector M can be determined at position r. i From the magnetization vector M, the magnetic moment m to be imparted to the i-th bead isi It is possible to find this.

[0042] [Calculation of external magnetic field due to magnetic moment within a magnetic material] Next, we will explain how to calculate the external magnetic field due to the magnetic moment within the magnetic material. External magnetic field H due to the magnetic moment within the magnetic material m The calculation method is described in detail in Japanese Patent Publication No. 6249912. Here, the external magnetic field H due to the magnetic moment within the magnetic material is described. m I will briefly explain how to calculate it.

[0043] The magnetic moment attached to the j-th bead in the virtual space is m j This is how it is denoted. Magnetic moment m j It can be expressed using the respective components of the xyz Cartesian coordinate system as follows:

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[0044] position r i External magnetic field vector H m (r i ) is expressed by the following formula.

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[0045] [Calculation of induced magnetic field] Next, we will explain how to calculate the induced magnetic field within the object 10 being analyzed. The center of gravity position r of the i-th bead i Induced magnetic field vector H ind (r i ) can be calculated using the following formula.

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[0046] Next, with reference to Figure 5, the procedure for the eddy current analysis method according to this embodiment will be described. Figure 5 is a flowchart showing the procedure for the eddy current analysis method according to this embodiment. Each step shown in Figure 5 is performed, for example, by the processing unit 31 of the eddy current analysis device 30 (Figure 4).

[0047] First, the processing unit 31 acquires the analysis conditions (step S1). Next, based on the analysis conditions, the object to be analyzed 10 (Figures 1A and 2A) is divided into multiple beads, and an initial value for the magnetic moment is assigned to each of the multiple beads (step S2). The initial value for the magnetic moment is, for example, zero.

[0048] Next, the total external magnetic field H at each position of the multiple beads. o (r i )(Equation (12)), and the vector potential A i Calculate equation (6) (step S3). Vector potential A i Once this is determined, the eddy current j at each position of the multiple beads can be calculated. i Calculate (Equation (2)).

[0049] Next, the induced magnetic field H at each position of the multiple beads.ind (r i )(Calculate equation (17) (step S5). Update the magnetic moment applied to each bead from the magnetization vector M (equations (13) and (14)) at the position of each of the plurality of beads (step S6). Using the updated magnetic moment of each of the plurality of beams, at each position of the plurality of beads, the magnetic field H due to the magnetic moment m (r i )(Equation (16)) is calculated (step S7).

[0050] Repeat the procedure from step S3 to step S7 until the analysis end condition is satisfied (step S8). When the analysis end condition is satisfied, output the analysis result, for example, the distribution of eddy currents, to the output device 33 (Figure 4) (step S9).

[0051] Next, the excellent effects of this embodiment will be described. In this embodiment, since the calculation is performed by imposing the boundary condition (Equation (5)) that the normal component of the eddy current on the surface of the analysis object 10 becomes zero, the analysis result satisfies the condition that the normal component of the current on the surface of the conductor becomes zero. Therefore, more accurate analysis of eddy currents can be performed.

[0052] Next, referring to FIG. 6, the results of actual analysis using the eddy current analysis method according to this embodiment will be described. As the analysis object 10, a regular square prism shown in FIG. 2A was adopted. An external magnetic field H ext (Equation (12)) parallel to the z-axis direction whose magnitude changes with time was generated in the analysis space where the analysis object 10 was arranged.

[0053] Figure 6 is a schematic diagram showing the distribution of eddy currents obtained from the analysis. The direction of the arrows indicates the direction of the eddy currents. It can be seen that the normal component of the eddy currents is almost zero on the surface of the object being analyzed 10. The difference between the peak value of the eddy currents on the surface of the object being analyzed 10 and the peak value of the eddy currents obtained from the analysis using a general finite element method is 1.4%, confirming that the two are in close agreement. In the analysis performed without imposing the constraint condition of equation (5) (Figure 2B), the normal component of the eddy currents on the surface of the object being analyzed 10 is not zero. It was confirmed that a more accurate analysis of eddy currents can be performed by imposing the constraint condition of equation (5).

[0054] The embodiments described above are illustrative, and the present invention is not limited to the embodiments described above. For example, it will be obvious to those skilled in the art that various modifications, improvements, and combinations are possible. [Explanation of symbols]

[0055] 10. Objects to be analyzed 20 beads (volume element) 21. Virtual Particles 30 Eddy current analysis device 31 Processing Unit 32 Input devices 33 Output device 34 Storage device 311 Analysis information acquisition unit 312 Magnetization application section 313 Arithmetic section 314 Output Control Unit

Claims

1. The object to be analyzed, defined within the analysis space, is divided into multiple volume elements. A magnetic moment is applied to each of the aforementioned multiple volume elements, An external magnetic field to be applied to the object to be analyzed is defined, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and Repeatedly, An eddy current analysis method in which, in the procedure for calculating the vector potential, a constraint condition is imposed such that the component of the eddy current in the normal direction on the surface of the object to be analyzed is zero.

2. In the procedure for calculating the aforementioned vector potential, The eddy current analysis method according to claim 1, which involves arranging a plurality of virtual particles along the surface of the object to be analyzed and applying an electric current to the plurality of virtual particles to calculate a vector potential that satisfies the constraint conditions.

3. An input device into which the analysis conditions are entered, A processing unit that performs eddy current analysis based on the analysis conditions input to the input device, Equipped with, The aforementioned processing apparatus is Based on the analysis conditions input to the input device, the object to be analyzed defined in the analysis space is divided into multiple volume elements. A magnetic moment is applied to each of the aforementioned multiple volume elements, An external magnetic field to be applied to the object to be analyzed is defined, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and Repeatedly, An eddy current analyzer that, in the procedure for calculating the vector potential, imposes a constraint condition such that the normal component of the eddy current on the surface of the object to be analyzed becomes zero.

4. In the procedure for calculating the vector potential, the processing apparatus, The eddy current analysis apparatus according to claim 3, which calculates a vector potential that satisfies the constraint conditions by arranging a plurality of virtual particles along the surface of the object to be analyzed and applying an electric current to the plurality of virtual particles.

5. A function to divide the object to be analyzed defined within the analysis space into multiple volume elements, The function of imparting a magnetic moment to each of the aforementioned multiple volume elements, The function defines an external magnetic field to be applied to the object being analyzed from the outside. A program that enables a computer to implement this, moreover, A procedure for calculating the vector potential at each position of the plurality of volume elements based on the magnetic moment of each of the plurality of volume elements and the external magnetic field, A procedure for calculating eddy currents based on the vector potential at each of the aforementioned multiple volume elements, A procedure for calculating the induced magnetic field at each of the aforementioned multiple volume elements, A procedure for updating the magnetic moment of each of the multiple volume elements based on the induced magnetic field, and A function to repeat the process, In the procedure for calculating the vector potential, a constraint is imposed such that the normal component of the eddy current on the surface of the object being analyzed is zero. A program that makes this possible.

6. In the procedure for calculating the aforementioned vector potential, The program according to claim 5, which calculates a vector potential that satisfies the constraint conditions by arranging a plurality of virtual particles along the surface of the object to be analyzed and applying an electric current to the plurality of virtual particles.

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