A method and system for obtaining the mutual inductance of two-bit coils placed arbitrarily in space

Abstract

The present invention discloses a method and system for obtaining the mutual inductance of two Bitter coils placed arbitrarily in space. The method comprises: using a spatial rotation and translation method to obtain an expression for two Bitter rings at any spatial position; performing loop integration on vector length infinitesimals on the two Bitter rings to obtain an expression for the mutual inductance of the two Bitter rings; and then combining the expression for the mutual inductance of the two Bitter coils at any spatial position with a numerical discretization expression of the two Bitter coils at any spatial position to obtain an expression for the mutual inductance of the two Bitter coils at any spatial position. The present invention solves the problem of calculating the mutual inductance of two Bitter coils placed at any spatial position.

Description

Technical Field

[0001] The present invention relates to the technical field of water-cooled magnets, and in particular to a method and system for obtaining the mutual inductance of two-bit coils arbitrarily placed in space. Background Art

[0002] Water-cooled magnets are basic devices for generating steady-state strong magnetic fields. For example, the WM6 water-cooled magnet is composed of six Bitter coils ABCDEF, as detailed in Reference 1. Multiple Bitter coils are connected in series or nested to form a hybrid magnet. Due to installation errors and coil processing and assembly errors, it is impossible for the axes of the several Bitter coils that make up the magnet to completely coincide with the midplane. There are three basic positional relationships between Bitter coils, as shown in Figures 1(a) to 1(c): Figure 1(a) shows the axis Z of the two coils A and B. A 、Z B coincident, midplane O A , O B There is a certain distance dz. Figure 1 (b) shows the midplane O of the two coils A and B. A , O B Coincident, axis Z A 、Z B There is a certain distance dr. Figure 1 (c) shows that the center points of the two coils coincide with each other, and the axis Z A 、Z B There is a certain angle da. The positional relationship between the two Bitter coils is usually a combination of three basic positional relationships. This combination of positional relationships inevitably affects the calculation of the mutual inductance between the water-cooled magnet coils, and thus the total inductance and energy storage calculation of the water-cooled magnet. The magnitude of the induced voltage between the coils during excitation is also directly proportional to the mutual inductance, making the mutual inductance between Bitter coils a very important physical quantity.

[0003] References 2 and 3, for example, focus on calculating the mutual inductance of two Bitter coils in the first basic positional relationship shown in Figure 1(a), omitting the second and third cases and the three combinations corresponding to Figures 1(b) and 1(c). Reference 4 considers the mutual inductance of two Bitter coils in the combination of the first and second cases, but ignores the third case and the general case of the three combinations.

[0004] Among them, the above-mentioned references 1-4 are specifically:

[0005] Reference document 1 is: Z. Fang, J. Li, XX Qian, ZJ Wang, J. Su, Y. Zhou, Y. Zhang, SY Chen, and GL Kuang, "Design of a 42 T Resistive Magnet at the CHMFL," IEEE Trans. Appl. Supercond., vol. 34, no. 5, Aug. 2024, Art. no. 4300504.

[0006] Reference document 2 is: Slobodan Babic, Cevdet Akyel, “Calculation of mutualinductance and magnetic force between two thick coaxial Bitter coils ofrectangular cross section” IET Electr. Power Appl., 2017, Vol. 11, Iss. 3, pp. 441–446.

[0007] Reference 3: Ren, Y., Kuang, G., Chen, W.: 'Inductance of bitter coilwith rectangular cross-section', J. Supercond. Novel Magn., 2013, 6, (6), pp.2159–2163.

[0008] Reference document 4 is: Yue Yu, Yao Luo, "Inductance calculations for non-coaxialBitter coils with rectangular cross-section using inverse Mellin transform" IET Electr. Power Appl., 2019, Vol. 13 Iss. 1, pp. 119-125. Summary of the Invention

[0009] The technical problem to be solved by the present invention is to solve the problem of calculating the mutual inductance of two Bitter coils placed at arbitrary positions in space.

[0010] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0011] A method for obtaining the mutual inductance of two bit coils arbitrarily placed in space, comprising:

[0012] Using the spatial rotation and translation method, we can obtain the expression of two Bitter rings at any position in space.

[0013] The vector length infinitesimals on the two Bitter rings are loop-integrated to obtain the mutual inductance expression of the two Bitter rings. Then, combined with the numerical discretization expression of the two Bitter coils at any position in space, the mutual inductance expression of the two Bitter coils at any position in space is obtained.

[0014] In this embodiment, the mutual inductance expression of two Bitter coils at any position in space is based on the inversely proportional relationship between the current density distribution of the Bitter coils and their radius, and a numerical integration method is used to calculate the mutual inductance of the two Bitter coils. The three-dimensional electromagnetic force of the two Bitter coils in space is calculated by using second-order differences instead of differentials based on the mutual inductance obtained by the numerical integration method.

[0015] In this embodiment, the expression for obtaining two Bitter rings at any position in space includes:

[0016] Place any point on the Bitter ring The coordinates of the circle are expressed using the central angle The equation representation of ;

[0017] Assume fixed point is any point on the circle with the coordinate origin of Bitter circle 1 as the center, and at the same time uses the central angle The equation representation of ;

[0018] Assume that the point on Bitter Ring 2 for point After the corresponding points are rotated and translated, the points are obtained. with dot relational expressions;

[0019] According to the point with dot The relational expression of with dot The coordinate relationship of

[0020] Then according to the point Use the central angle The equation shows that the junction point with dot The coordinate relationship of is used to obtain the expression of two Bitter rings at any position in space.

[0021] In this embodiment, the expression of the second Bitter ring at any position in space is as follows:

[0022] ;

[0023] ;

[0024] ;

[0025] Where, for point The coordinate value in the three-dimensional coordinate system, is the radius of Bitter Ring 2, are the coordinates of the center of Bitter Ring 2, Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the figure, the plane normal of Bitter Ring 2 is Planar projection and The angle between the axes, is the normal of the plane where the second Bitter ring is located and C The angle between the axes.

[0026] In this embodiment, the expression for the mutual inductance of two Bitter rings at any position in space is obtained, including:

[0027] Get a point on the Bitter Ring Go to the second point of Bitter Circle The vector length differential expression of ;

[0028] Combined with the vector length differential expression, the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2 is obtained;

[0029] Get the Powered Bitter Ring One on Point The resulting gravitational vector potential expression;

[0030] Substitute the spherical vector potential expression into the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2 to simplify it and obtain the Newman expression that only includes the mutual inductance of Bitter ring 2;

[0031] Substitute the expression of two Bitter rings at any position in space into the vector length differential expression, and simplify it again by combining it with the Newman expression that only contains the mutual inductance of the two Bitter rings to obtain the central angle of the two Bitter rings. is the integrand of the parameter, which is the expression of the mutual inductance of two Bitter rings at any position in space.

[0032] In this embodiment, the mutual inductance of two Bitter rings at any position in space is expressed as follows:

[0033] ;

[0034] in, ;

[0035] ;

[0036] ;

[0037] Where, is the mutual inductance of two Bitter rings at any position in space, is the magnetic permeability, for point In the coordinate system x Axis and y The coordinate values of the axis, is the radius of Bitter ring 1, for point The distance from the projection point to the origin in the xoy plane, are the z-axis coordinate values of Bitter Ring 1 and Bitter Ring 2 respectively. Contains k Parameterized elliptic integrals of the first and second kind, Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the figure, the plane normal of Bitter Ring 2 is Planar projection and The angle between the axes, is the normal of the plane where the second Bitter ring is located and C The angle between the axes, is the radius of Bitter Ring 2.

[0038] In this embodiment, obtaining the three-dimensional electromagnetic force of two Bitter coils in space includes:

[0039] Place the Bitter coil on an axial cross section of a two-dimensional plane A 1 Along the two axes of the two-dimensional plane, they are divided into M Share N parts, forming multiple grid rings; similarly, place the Bitter coil two on its own axial section A 2 Along the two axes of the two-dimensional plane, they are divided into Share parts, forming multiple grid rings;

[0040] According to the mutual inductance expression of two Bitter coils at any position in space, the mutual inductance expression of the corresponding axial section of the two Bitter coils is obtained;

[0041] Obtain the current density distribution on the corresponding axial cross-sections of Bitter coil 1 and Bitter coil 2, and then combine the mutual inductance expressions of the corresponding axial cross-sections of the two Bitter coils to obtain the numerical expression of the mutual inductance of the two Bitter coils;

[0042] According to the numerical expression of the mutual inductance of the two Bitter coils, after calculating the mutual inductance, the three-dimensional electromagnetic force between the two Bitter coils in space is obtained.

[0043] In this embodiment, the three-dimensional electromagnetic force of the two Bitter coils is obtained by the following formula:

[0044] ;

[0045] ;

[0046] ;

[0047] Where, 、 、 The two Bitter coils in the space after mutual inductance are x, y, z The electromagnetic force in the axial direction, I 1. I 2 are the currents flowing through the two Bitter coils, is the numerical expression of the mutual inductance of two Bitter coils.

[0048] In this embodiment, the numerical expression of the mutual inductance of the two Bitter coils is as follows:

[0049] ;

[0050] Where, is the numerical expression of the mutual inductance of the two Bitter coils, are the turns of Bitter coil 1 and Bitter coil 2 respectively, is the width of Bitter coil 1, is the width of Bitter coil 2, is the ratio of the outer ring to the inner ring of Bitter coil 1, is the ratio of the outer ring to the inner ring of Bitter coil 2, Bitter coil Mesh Ring and Bitter Coil II Mutual inductance of mesh rings.

[0051] The present invention further provides a system for obtaining the mutual inductance of two-bit coils placed arbitrarily in space, which uses the above-mentioned method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space, comprising:

[0052] Position module, used to obtain the expression of two Bitter rings at any position in space using the spatial rotation and translation method;

[0053] The mutual inductance module is used to perform loop integration on the vector length elements on the two Bitter rings to obtain the mutual inductance expression of the two Bitter rings. This is then combined with the numerical discretization expression of the two Bitter coils at any position in space to obtain the mutual inductance expression of the two Bitter coils at any position in space.

[0054] Compared with the prior art, the present invention has the following advantages: it solves the problem of calculating the mutual inductance of two Bitter coils placed at arbitrary positions in space. It first uses the spatial rotation and translation method to obtain the expression of two circular rings at arbitrary positions in space, then applies the Newman formula for the mutual inductance of any two closed loops in space to the calculation of the mutual inductance of two circular rings at arbitrary positions, introduces the relationship that the current density distribution of the two Bitter coils is inversely proportional to the radius into the mutual inductance expression, and uses the grid division summation method to replace the double cross-sectional area integral to calculate the mutual inductance of the two circular rings. The localized angular integral is replaced by a 20-node Gaussian integral, resulting in a method for calculating the mutual inductance of two bitter coils at arbitrary locations in space. Specifically, the two bitter coils are divided into multiple circular rings with varying cross-sections. A loop integral is then performed on each of the two bitter coils using vector length elements according to the Newman formula. The calculated mutual inductances for all the rings are summed to obtain an expression for the mutual inductance between the bitter coil and the superconducting coil.

[0055] By comparing the mutual inductance calculation of two Bitter coils with the position relationship in references 2 and 4, the correctness and high accuracy of this method are verified.

[0056] The present invention solves the problem of calculating the mutual inductance and electromagnetic force of two Bitter coils at any position in space. Its correctness and high precision are confirmed by comparing with the calculation results of references 2 and 4, and the calculation error can reach 1.0E-4. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] FIG1( a ) is a schematic diagram of the first basic position relationship of two Bitter coils according to an embodiment of the present invention.

[0058] FIG1( b ) is a schematic diagram of the second basic position relationship of two Bitter coils according to an embodiment of the present invention.

[0059] FIG1( c ) is a schematic diagram of the third basic position relationship of two Bitter coils according to an embodiment of the present invention.

[0060] Figure 2 A flow chart of a method for obtaining the mutual inductance of two-bit coils arbitrarily placed in space according to an embodiment of the present invention.

[0061] Figure 3 Schematic diagram of the spatial position relationship of two Bitter coils according to an embodiment of the present invention.

[0062] Figure 4 Schematic diagram of any two closed loops in space according to an embodiment of the present invention.

[0063] Figure 5 The positional relationship and cross-sectional mesh division diagram of any two Bitter coils in space according to an embodiment of the present invention.

[0064] Figure 6 Parameter diagram of the position relationship between two coaxial Bitter coils according to an embodiment of the present invention.

[0065] Figure 7 Axial cross-sectional position relationship diagram of two Bitter coils in space according to an embodiment of the present invention.

[0066] Figure 8 Block diagram of a system for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to an embodiment of the present invention. DETAILED DESCRIPTION

[0067] To facilitate those skilled in the art to understand the technical solution of the present invention, the technical solution of the present invention is further described with reference to the accompanying drawings.

[0068] The terms "first" and "second" are used for descriptive purposes only and should not be understood to indicate or imply relative importance or implicitly specify the number of the technical features indicated. Therefore, a feature specified as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of this application, "plurality" means two or more, unless otherwise specifically defined.

[0069] See also Figure 2 As shown, this embodiment provides a method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space, including:

[0070] S10, using the spatial rotation and translation method, obtain the expression of two Bitter rings at any position in space.

[0071] In one embodiment of the present invention, in the expression method of two Bitter rings at arbitrary positions in space, for the convenience of expression, Figure 3 Assume that the coordinates of the center of the Bitter ring are And parallel to the xoy plane, the radius is Assuming that Bitter Ring 2 is a ring at any position in space, then Bitter Ring 2 needs to know three elements:

[0072] The coordinates of the center of Bitter Ring 2 in the three-dimensional coordinate system xyz .

[0073] The radius of Bitter Ring 2 .

[0074] Bitter ring two normals Azimuth of direction .

[0075] See Figure 3 As shown, parallel to the x-axis, parallel to the y-axis, Parallel to the z-axis. Normal On the plane Projection on. is the two normals of the Bitter ring exist Planar projection and The angle between the axes ( ), is the normal of the plane where Bitter ring 2 is located The angle with the z-axis (i.e. ). Then Bitter Ring 2 can always be solved by Ring Rotate around the y-axis , rotate around the z axis , then pan to the point The following is the specific derivation process of the expression of any position in the two-ring space.

[0076] See Figure 2 As shown, in one embodiment of the present invention, obtaining an expression for two Bitter rings at any position in space includes:

[0077] S11, move the Bitter ring to any point The coordinates of the circle are expressed using the central angle The equation represents .

[0078] In this embodiment, any point on the Bitter ring The coordinates can be expressed using angle parameters The equations (1), (2), and (3) are expressed as follows, where the angle parameter Also the central angle.

[0079] ,(1);

[0080] , (2);

[0081] , (3);

[0082] Where, for point The coordinate value in the three-dimensional coordinate system, is the radius of Bitter ring 1, sin is the sine trigonometric function, and cos is the cosine trigonometric function.

[0083] S12, assuming fixed point is a ring with the coordinate origin of Bitter ring 1 as its center Any point on the circle, and use the central angle The equation represents .

[0084] In this embodiment, it is assumed that is the radius r 2 rings Any point on It can be expressed by formulas (4)-(6).

[0085] , (4);

[0086] , (5);

[0087] , (6);

[0088] Where, is the radius of Bitter Ring 2.

[0089] S13, assuming that the point on Bitter ring 2 for point After the corresponding points are rotated and translated, the points are obtained. with dot A relational expression.

[0090] In this embodiment, it is assumed that point 2 on the Bitter ring is for point Corresponding points after rotation and translation. Rotation around the y-axis The unit rotation matrix of , see formula (7) for details, Rotation around the z axis The unit rotation matrix of , see formula (8), then the point with dot The relational expression of can be expressed by formula (9).

[0091] , (7);

[0092] , (8);

[0093] , (9);

[0094] In this embodiment, for point The coordinate value in the three-dimensional coordinate system, are the coordinates of the center of Bitter Ring 2, Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the figure, the plane normal of Bitter Ring 2 is Planar projection and The angle between the axes, is the normal of the plane where the second Bitter ring is located and C The angle between the axes.

[0095] S14, according to the point with dot The relational expression of with dot coordinate relationship.

[0096] In this embodiment, it can be deduced from formula (9) that with dot The specific coordinate relationship equations are (10)-(12).

[0097] , (10);

[0098] , (11);

[0099] , (12);

[0100] S15, then according to point Use the central angle The equation shows that the junction point with dot The coordinate relationship of is used to obtain the expression of two Bitter rings at any position in space.

[0101] In this embodiment, substituting formulas (4)-(6) into formulas (10)-(12) yields specific expressions (13)-(15) of the spatial Bitter ring II.

[0102] , (13);

[0103] , (14);

[0104] , (15);

[0105] At this point, the specific expression of two Bitter rings at two arbitrary positions in space is completed.

[0106] S20, loop integration is performed on the vector length infinitesimals on the two Bitter rings to obtain the mutual inductance expression of the two Bitter rings, and then combined with the numerical discretization expression of the two Bitter coils at any position in space to obtain the mutual inductance expression of the two Bitter coils at any position in space.

[0107] In one embodiment of the present invention, Figure 4 The two closed loops in the space shown are Bitter ring 1 and Bitter ring 2, and the currents are respectively and , r is a point on the Bitter ring With Bitter Ring 2 on point The distance between is the magnetic permeability in vacuum or air, and The vector length elements on Bitter ring 1 and Bitter ring 2 are respectively. By performing full loop integration on the length elements of the two rings, we can obtain the Newman formula (18) for calculating the mutual inductance of the two Bitter rings. The resulting equation (19) for the spherical vector potential and the first and second elliptic integral equations (22)-(23) can be used to easily solve equation (18). The following is the specific formula reasoning process, which includes:

[0108] S21, obtain the data from Bitter Ring 1 Go to the second point of Bitter Circle The vector length differential expression of .

[0109] In this embodiment, it is assumed that and Respectively represent a point from the Bitter ring and Bitter Ring Two Points The starting vector length is infinitesimal, then we can get formula (16) and formula (17). 、 、 are the unit vectors of the x, y, and z axes respectively.

[0110] , (16);

[0111] , (17);

[0112] Where, is the differential symbol.

[0113] S22, combining the vector length infinitesimal expression, obtains the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2.

[0114] In this embodiment, the Newman formula for calculating the mutual inductance between Bitter Ring 1 and Bitter Ring 2 is shown in formula (18):

[0115] , (18);

[0116] Where, is the mutual inductance of two Bitter rings at any position in space.

[0117] S23, get the energized Bitter ring at point The resulting gravitational vector potential expression.

[0118] In this embodiment, the Bitter ring is powered at point The resulting spherical vector potential is calculated as shown in formula (19):

[0119] , (19);

[0120] , (20);

[0121] ,(twenty one);

[0122] ,(twenty two);

[0123] ,(twenty three);

[0124] ,(twenty four);

[0125] Where, Bitter Ring for Powering on One at Point The resulting spherical vector potential, for point In the coordinate system x Axis and y The coordinate values of the axis, for point The distance from the projection point to the origin in the xoy plane is The distance from the point to the origin, are the z-axis coordinate values of Bitter Ring 1 and Bitter Ring 2 respectively. Contains k Parameterized elliptic integrals of the first and second kind, is an integral variable, and the integral range is , for point The circular unit vector in the xyz coordinate system is represented by the point The angular coordinate unit vector in cylindrical coordinates.

[0126] S24, substitute the spherical vector potential expression into the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2 to simplify, and obtain the Newman expression that only includes the mutual inductance of Bitter ring 2.

[0127] In this embodiment, formula (19) is simplified to obtain formula (25):

[0128] , (25);

[0129] Substituting formula (25) into formula (18) yields formula (26):

[0130] , (26);

[0131] S25, substitute the expression of two Bitter rings at any position in space into the vector length infinitesimal expression, and simplify it again by combining it with the Newman expression containing only the mutual inductance of Bitter ring two, and obtain the central angle of Bitter ring two is the integrand of the parameter, which is the expression of the mutual inductance of two Bitter rings at any position in space.

[0132] In this embodiment, formulas (13), (14), and (15) are substituted into formula (17) to obtain formulas (27)-(29).

[0133] , (27);

[0134] , (28);

[0135] , (29);

[0136] make:

[0137] , (30);

[0138] , (31);

[0139] , (32);

[0140] Substituting formulas (30)-(32) into formulas (27)-(29) yields formulas (33)-(35):

[0141] , (33);

[0142] , (34);

[0143] , (35);

[0144] Formula (17) can be transformed into formula (36):

[0145] , (36);

[0146] Substituting formula (36) and formula (24) into formula (26) and simplifying it, we can obtain formula (37):

[0147] , (37);

[0148] Formula (37) is Figure 4 The mutual inductance expression of two Bitter rings at arbitrary positions in the space shown in the figure can be obtained by analysis. The integrand of the entire formula (37) is the central angle of the two Bitter rings. As a function of the parameters, the analytical solution of formula (37) can be replaced by the high-precision Gaussian numerical integration result.

[0149] See also Figure 5 As shown, this embodiment also includes: S30, introducing the relationship that the current density distribution of the Bitter coil is inversely proportional to the radius into the mutual inductance expression of the two Bitter coils at any position in space, and using a numerical integration method to obtain the calculation result of the mutual inductance of the two Bitter coils. The mutual inductance of the two Bitter coils obtained by the numerical integration method is replaced by a second-order difference to obtain the three-dimensional electromagnetic force of the two Bitter coils in space.

[0150] In this embodiment, obtaining the three-dimensional electromagnetic force of two Bitter coils in the space behind the mutual inductance includes:

[0151] S31, place the Bitter coil on an axial cross section of a two-dimensional plane. A 1 Along the two axes of the two-dimensional plane, they are divided into M Share N parts, forming multiple grid rings; similarly, place the Bitter coil two on its own axial section A2 Along the two axes of the two-dimensional plane, they are divided into Share parts, forming multiple grid rings.

[0152] In this embodiment, the Bitter coil 1 and the Bitter coil 2 at any spatial position are as follows: Figure 5 As shown, the Bitter coil is placed on the axial section of the xOz plane. A 1 is divided equally along the x direction M parts, divided equally along the z direction N Parts, so that the Bitter coil is discretized into NM Each ring is divided into two grid rings. ... Coordinate pair identification Similarly, put the Bitter coil 2 in Axial section on the surface A 2 along Direction is divided into along Direction is divided into parts, so that the Bitter coil 2 is discretized into grid rings, each ring uses Coordinate pairs to identify.

[0153] S32: Obtain the mutual inductance expression of the two Bitter coils corresponding to the axial cross-section according to the mutual inductance expression of the two Bitter coils at any position in space.

[0154] In this embodiment, Indicates the Bitter coil Mesh Ring and Bitter Coil II Mutual inductance of mesh rings, It can be calculated using formula (37) for the two axial sections. A 1 and A 2 The mutual inductance of the two magnet coils can be obtained by integrating them separately Expressed as formula (38).

[0155] , (38);

[0156] , (39);

[0157] , (40);

[0158] , (41);

[0159] , (42);

[0160] , (43);

[0161] Where, J 1. J 2 are the current density distributions on the corresponding axial sections of Bitter coil 1 and Bitter coil 2, I 1. I 2 are the currents of Bitter coil 1 and Bitter coil 2 respectively, N 1 is the number of turns of Bitter coil 1, N 2 is the number of turns of Bitter coil 2, is the height of Bitter coil 1, Bitter coil height, is the ratio of the outer ring to the inner ring of Bitter coil 1, is the ratio of the outer ring to the inner ring of Bitter coil 2, is the width of Bitter coil 1, is the width of Bitter coil 2.

[0162] S33, obtaining the current density distribution on the corresponding axial cross-sections of the Bitter coil 1 and the Bitter coil 2, and then combining the mutual inductance expressions of the corresponding axial cross-sections of the two Bitter coils to obtain a numerical expression of the mutual inductance of the two Bitter coils.

[0163] In this embodiment, substituting formula (39) into formula (38) yields formula (44):

[0164] , (44);

[0165] , (45);

[0166] S34, calculating the mutual inductance according to the numerical expression of the mutual inductance of the two Bitter coils, and obtaining the three-dimensional electromagnetic force of the two Bitter coils in space.

[0167] In this embodiment, formula (45) is substituted into formula (44) and simplified to obtain formula (46):

[0168] , (46);

[0169] Where, is the numerical expression of the mutual inductance of two Bitter coils.

[0170] The calculation formulas for the three-dimensional electromagnetic force between the two coils in space after mutual inductance are shown in (47)-(49):

[0171] , (47);

[0172] , (48);

[0173] , (49);

[0174] Where, 、 、 The two Bitter coils in the space after mutual inductance are x, y, z Electromagnetic force in the axial direction.

[0175] See also Figure 6 As shown, in one embodiment of the present invention, the correctness and high precision of this embodiment are verified by comparing the mutual inductance calculation of the two Bitter coils with the first and second position relationships in references 2 and 5.

[0176] (1) Mutual inductance calculation verification when the central axes of two Bitter coils coincide, such as Figure 6 shown.

[0177] Table 1 Parameters of the position relationship between the two Bitter coils in Reference 2

[0178]

[0179] Table 2 Comparison of the mutual inductance calculation method of this embodiment and the semi-analytical numerical method in reference 2

[0180] ( M =20; N =20; M / =20; N / =20, and coaxial case)

[0181]

[0182] Table 3 Comparison of the electromagnetic force calculation method of this embodiment with the semi-analytical numerical method in reference 2

[0183] ( M =20; N =20; M / =20; N / =20, coaxial case)

[0184]

[0185] Tables 2 and 3 show that the calculated mutual inductance and electromagnetic force between two Bitter coils, based on the first fundamental relationship in Figure 1(a), are within 1.0E-04 of those reported in Reference 2, validating the accuracy and precision of this embodiment. In the tables, E represents scientific notation. dz represents the axial spacing between the center points of the two coaxial coils.

[0186] (2) The mutual inductance calculation and verification of the two Bitter coils are as follows: the central axes are parallel and the midplanes do not coincide with each other. Figure 7 shown.

[0187] Table 4 Bitter coil parameters in Reference 4 and the calculated mutual inductance results of this embodiment

[0188] ( M =20; N =20; M / =20; N / =20, axis parallel)

[0189]

[0190] From Table 4, we can see that the mutual inductance M of the two Bitter coils in the combination state according to the first and second basic relationships in Figure 1 (a) and (b) is B-B Compared with M in reference 4 GHF The error is on the order of 1.0E-04, which verifies the correctness and high precision of the method of this embodiment.

[0191] (3) Calculate the mutual inductance and electromagnetic force of the two bitter coils according to the three combinations shown in Figure 1 (a), Figure 1 (b), and Figure 1 (c).

[0192] Table 5 Calculation results of mutual inductance and electromagnetic force of two Bitter coils at specific locations in space

[0193] ( for Axis and Angle, for The angle between the projection of the axis on the xoy plane and the x-axis)

[0194]

[0195] Since there is no literature comparing the calculation of the three-dimensional mutual inductance and electromagnetic force between two Bitter coils, Table 5 can be used as a reference correction table for subsequent research.

[0196] In summary, the method of this embodiment can calculate the mutual inductance and electromagnetic force of two Bitter coils at any position in space. By comparing with the calculation results of the examples in References 2 and 5, the correctness and high accuracy of this method can be confirmed.

[0197] See also Figure 8 As shown, the present invention further provides a system for obtaining the mutual inductance of two-bit coils placed arbitrarily in space, and the method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space described above is applied, comprising:

[0198] The position module is used to obtain the expression of two Bitter rings at any position in space using the spatial rotation and translation method.

[0199] The mutual inductance module is used to perform loop integration on the vector length elements on the two Bitter rings to obtain the mutual inductance expression of the two Bitter rings. This is then combined with the numerical discretization expression of the two Bitter coils at any position in space to obtain the mutual inductance expression of the two Bitter coils at any position in space.

[0200] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description. It is intended that all variations within the meaning and range of equivalents of the claims be embraced herein, and any reference signs in the claims should not be construed as limiting the claims to which they relate.

[0201] The above-mentioned embodiments merely represent the implementation methods of the invention. The protection scope of the present invention is not limited to the above-mentioned embodiments. For those skilled in the art, several variations and improvements can be made without departing from the concept of the present invention, which all fall within the protection scope of the present invention.

Claims

1. A method for obtaining the mutual inductance of two bit coils placed arbitrarily in space, characterized in that: include: Divide the two Bitter coils into rings with multiple cross sections; Using the spatial rotation and translation method, the coordinate expressions of two Bitter rings at any position in space are obtained; Perform loop integration on the vector length elements on the two Bitter rings to obtain the mutual inductance expression of the two Bitter rings: Get a point on the Bitter Ring Go to the second point of Bitter Circle The vector length differential expression of ; Combined with the vector length differential expression, the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2 is obtained; Get the Powered Bitter Ring One on Point The resulting gravitational vector potential expression; Substitute the spherical vector potential expression into the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2 to simplify it and obtain the Newman expression that only includes the mutual inductance of Bitter ring 2; Substitute the expression of two Bitter rings at any position in space into the vector length infinitesimal expression, and simplify it again by combining it with the Newman expression that only contains the mutual inductance of the two Bitter rings to obtain the central angle of the two Bitter rings. is the integrand of the parameter, which is the expression of the mutual inductance of two Bitter rings at any position in space; According to the mutual inductance expression of two Bitter rings at any position in space, the mutual inductance expression of the corresponding axial sections of the two Bitter coils is obtained. A 1 and A 2Integrate separately to obtain the expression for the mutual inductance of the two magnet coils.

2. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: Get the coordinate expressions of two Bitter rings at any position in space, including: Place any point on the Bitter ring The coordinates of the circle are expressed using the central angle The equation representation of ; Assume fixed point is any point on the circle with the coordinate origin of Bitter circle 1 as the center, and at the same time uses the central angle The equation representation of ; Assume that the point on Bitter Ring 2 for point After the corresponding points are rotated and translated, the points are obtained. with dot relational expressions; According to the point with dot The relational expression of with dot The coordinate relationship of Then according to the point Use the central angle The equation shows that the junction point with dot The coordinate relationship of two Bitter rings can be obtained by using the coordinate expression of two Bitter rings at any position in space.

3. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 2, characterized in that: The coordinate expression of the second Bitter ring at any position in space is as follows: ； ； ； Where, for point The coordinate value in the three-dimensional coordinate system, is the radius of Bitter Ring 2, are the coordinates of the center of Bitter Ring 2, Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the figure, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the normal of the plane where the second Bitter ring is located and C The angle between the axes.

4. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: The expression for the mutual inductance of two Bitter rings at any position in space is as follows: ； in, ; ； ； Where, is the mutual inductance of two Bitter rings at any position in space, is the magnetic permeability, for point In the coordinate system x Axis and y The coordinate values of the axis, is the radius of Bitter ring 1, for point The distance from the projection point to the origin in the xoy plane, are the z-axis coordinate values of Bitter Ring 1 and Bitter Ring 2 respectively. Contains k Parameterized elliptic integrals of the first and second kind, Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the figure, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the normal of the plane where the second Bitter ring is located and C The angle of the axis, is the radius of Bitter Ring 2.

5. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: Obtain the three-dimensional electromagnetic force of two Bitter coils in space, including: Place the Bitter coil on an axial cross section of a two-dimensional plane A 1 Along the two axes of the two-dimensional plane, they are divided into M Share N parts, forming multiple grid rings; similarly, place the Bitter coil two on its own axial section A 2 Along the two axes of the two-dimensional plane, they are divided into Share parts, forming multiple grid rings; According to the mutual inductance expression of two Bitter coils at any position in space, the mutual inductance expression of the corresponding axial section of the two Bitter coils is obtained; Obtain the current density distribution on the corresponding axial cross-sections of Bitter coil 1 and Bitter coil 2, and then combine the mutual inductance expressions of the corresponding axial cross-sections of the two Bitter coils to obtain the numerical expression of the mutual inductance of the two Bitter coils; According to the numerical expression of the mutual inductance of the two Bitter coils, after calculating the mutual inductance, the three-dimensional electromagnetic force between the two Bitter coils in space is obtained.

6. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 5, characterized in that: The three-dimensional electromagnetic force between two Bitter coils is obtained by the following formula: ； ； ； Where, 、 、 The two Bitter coils in the space after mutual inductance are x, y, z The electromagnetic force in the axial direction, I 1. I 2 are the currents of Bitter coil 1 and Bitter coil 2 respectively, is the numerical expression of the mutual inductance of two Bitter coils.

7. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 5, characterized in that: The numerical expression of the mutual inductance of two Bitter coils is as follows: ； Where, is the numerical expression of the mutual inductance of the two Bitter coils, are the turns of Bitter coil 1 and Bitter coil 2 respectively, is the width of Bitter coil 1, is the width of Bitter coil 2, is the ratio of the outer ring to the inner ring of Bitter coil 1, is the ratio of the outer ring to the inner ring of Bitter coil 2, Bitter coil Mesh Ring and Bitter Coil II Mutual inductance of mesh rings.

8. A system for obtaining the mutual inductance of two bit coils placed arbitrarily in space, characterized in that: The method for obtaining the mutual inductance of two-bit coils arbitrarily placed in space according to any one of claims 1 to 7 comprises: Position module, used to obtain the coordinate expression of two Bitter rings at any position in space using the spatial rotation and translation method; The mutual inductance module is used to perform loop integration on the vector length infinitesimals on the two Bitter rings to obtain the mutual inductance expression of the two Bitter rings. Based on the mutual inductance expression of the two Bitter rings at any position in space, the mutual inductance expression of the corresponding axial sections of the two Bitter coils is obtained. A 1 and A 2Integrate separately to obtain the expression for the mutual inductance of the two magnet coils.

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