Method and system for obtaining mutual inductance of two-bit coils placed randomly in space

Abstract

The invention discloses a method and a system for acquiring mutual inductance of two-bit coils randomly placed in space, and the method comprises the steps: acquiring expressions of two Bitter rings at any position in the space through a space rotation and translation method; vector length infinitesimal elements on the two Bitter rings are subjected to loop integration, mutual inductance expressions of the two Bitter rings are obtained, then numerical discretization expressions of the two Bitter coils at any position in the space are combined, mutual inductance expressions of the two Bitter coils at any position in the space are obtained, and the problem of mutual inductance calculation of the two Bitter coils placed at any position in the space is solved.

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 shown 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) is 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. Usually the position relationship of the two Bitter coils is a combination of three basic position relationships. This combination of position relationships will inevitably affect the calculation of the mutual inductance between the water-cooled magnet coils, and then affect the calculation of the total inductance and energy storage of the water-cooled magnet. The magnitude of the induced voltage between the coils during the excitation process is also directly proportional to the mutual inductance, so the mutual inductance between the Bitter coils is a very important physical quantity.

[0003] Regarding the calculation of the mutual inductance of two Bitter coils, references 2 and 3 focus on the calculation of the mutual inductance of two Bitter coils in the first basic position relationship shown in Figure 1 (a), without considering the second and third cases and the three combined states corresponding to Figures 1 (b) and 1 (c). Reference 4 considers the calculation of the mutual inductance of two Bitter coils in the combined state of the first and second cases, without considering the third case and the general situation of the three combined states.

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

[0005] 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.

[0006] 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.

[0007] 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

[0008] 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 any position in space.

[0009] In order to solve the above technical problems, the present invention provides the following technical solutions: A method for obtaining the mutual inductance of two-bit coils arbitrarily placed in space, comprising: Using the spatial rotation and translation method, we can obtain the expression of two Bitter rings at any position in space. 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.

[0010] In this embodiment, in the mutual inductance expression of two Bitter coils at any position in space, the relationship that the current density distribution of the Bitter coil is inversely proportional to the radius is introduced, and the numerical integration method is used to obtain the calculation result of the mutual inductance of the two Bitter coils; the mutual inductance of the two Bitter coils is obtained by the numerical integration method, and the second-order difference is used instead of the differential to obtain the three-dimensional electromagnetic force of the two Bitter coils in space.

[0011] In this embodiment, the expression of obtaining two Bitter rings at any position in space includes: Place any point on the Bitter ring The coordinates of the circle are expressed in terms of 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 point The relational expression of According to the point With point The relational expression of With point The coordinate relationship of Then according to the point Use the central angle The equation shows that the junction point With point The coordinate relationship of is used to obtain the expression of two Bitter rings at any position in space.

[0012] In this embodiment, the expression of the Bitter ring 2 at any position in space is as follows: ; ; ; In the formula, 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, For Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the example, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the plane normal of the second Bitter ring and C The angle of the axis.

[0013] In this embodiment, the expression for obtaining the mutual inductance of two Bitter rings at any position in space includes: Get a point from 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 contains the mutual inductance about Bitter ring 2; 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 about 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.

[0014] In this embodiment, the expression of the mutual inductance of two Bitter rings at any position in space is as follows: ; in, ; ; ; In the formula, 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 Elliptic integrals of the first and second kind for parameters, For Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the example, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the plane normal of the second Bitter ring and C The angle of the axis, is the radius of Bitter ring 2.

[0015] In this embodiment, obtaining the three-dimensional electromagnetic force of two Bitter coils in space includes: Place the Bitter coil on an axial cross section of a two-dimensional plane A 1 The two axes of the two-dimensional plane are divided into M Share N parts to form multiple mesh rings; similarly, place the Bitter coil two on its own axial section A 2 The two axes of the two-dimensional plane 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-section of Bitter coil 1 and Bitter coil 2, and then combine the mutual inductance expression of the corresponding axial cross-section 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 of the two Bitter coils in space is obtained.

[0016] In this embodiment, the three-dimensional electromagnetic force of the two Bitter coils is obtained by the following formula: ; ; ; In the formula, , , 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 the two Bitter coils, is the numerical expression of the mutual inductance of two Bitter coils.

[0017] In this embodiment, the numerical expression of the mutual inductance of the two Bitter coils is as follows: ; In the formula, is the numerical expression of the mutual inductance of 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.

[0018] 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: Position module, used to obtain the expression of two Bitter rings at any position in space by 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, and then combine 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.

[0019] Compared with the prior art, the beneficial effect of the present invention is that it solves the problem of calculating the mutual inductance of two Bitter coils placed at any position in space. It first uses the space rotation and translation method to obtain the expression of two circular rings at any position in space, then applies the Newman formula of the mutual inductance of any two closed loops in space to the calculation of the mutual inductance of two circular rings at any position, 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, The local angle integral is replaced by a 20-node Gaussian integral, thus obtaining a method for calculating the mutual inductance of two bitter coils at arbitrary positions in space. That is, the two bitter coils are divided into many circular rings with different cross sections, and the loop integration is performed on each circular ring in the two bitter coils one by one using the vector length differential element according to the Newman formula, and the calculation results of the mutual inductance of all the circular rings are summed up to obtain the mutual inductance expression of the bitter coil and the superconducting coil.

[0020] By comparing the mutual inductance calculation of two Bitter coils with the first and second position relationships in references 2 and 4, the correctness and high accuracy of this method are verified.

[0021] 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

[0022] FIG. 1( a ) is a schematic diagram of the first basic position relationship of two Bitter coils according to an embodiment of the present invention.

[0023] FIG. 1( b ) is a schematic diagram of the second basic position relationship of two Bitter coils according to an embodiment of the invention.

[0024] FIG. 1( c ) is a schematic diagram of the third basic position relationship of two Bitter coils according to an embodiment of the invention.

[0025] 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.

[0026] Figure 3 Schematic diagram of the spatial position relationship of two Bitter coils according to an embodiment of the present invention.

[0027] Figure 4 A schematic diagram of any two closed loops in space according to an embodiment of the present invention.

[0028] Figure 5 Positional relationship and cross-sectional mesh division diagram of any two Bitter coils in space according to an embodiment of the present invention.

[0029] Figure 6 Parameter diagram of the position relationship of two coaxial Bitter coils according to an embodiment of the present invention.

[0030] Figure 7 Axial cross-sectional position relationship diagram of two Bitter coils in space according to an embodiment of the present invention.

[0031] Figure 8 A block diagram of a system for obtaining mutual inductance of two-bit coils arbitrarily placed in space according to an embodiment of the present invention. DETAILED DESCRIPTION

[0032] In order 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 in conjunction with the accompanying drawings of the specification.

[0033] The terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of this application, the meaning of "plurality" is two or more, unless otherwise clearly and specifically defined.

[0034] 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: S10, using the spatial rotation and translation method, obtain the expression of two Bitter rings at any position in space.

[0035] In one embodiment of the present invention, in the expression method of Bitter rings at two arbitrary positions in space, for the convenience of expression, it is established 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, it means that Bitter Ring 2 needs to know three elements: The coordinates of the center of Bitter ring 2 in the three-dimensional coordinate system xyz .

[0036] The radius of the Bitter ring 2 .

[0037] Bitter ring two normals Azimuth of direction .

[0038] See Figure 3 As shown, Parallel to the x-axis, Parallel to the y-axis, Parallel to the z-axis. Normal In plane Projection on. is the two normals of the Bitter ring exist Plane projection and The angle between the axes ( ), is the normal of the plane where the second Bitter ring lies The angle with the z-axis (i.e. ). Then Bitter ring 2 can always be solved by changing the radius to Ring Rotate around the y axis , rotate around the z axis , then pan to point The following is the specific derivation process of the expression of any position in the two-ring space.

[0039] See Figure 2 As shown, in one embodiment of the present invention, the expression of obtaining two Bitter rings at any position in space includes: S11, move the Bitter ring to any point The coordinates of the circle are expressed in terms of the central angle The equation represents .

[0040] In this embodiment, any point on the Bitter ring The coordinates can be used with angle parameters The equations (1), (2), and (3) are expressed as follows, where the angle parameter Also the central angle.

[0041] , (1); , (2); , (3); In the formula, 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.

[0042] S12, assuming fixed point is a circle with the coordinate origin of Bitter circle 1 as its center Any point on the circle, and use the central angle The equation represents .

[0043] In this embodiment, it is assumed that The radius r 2 Ring Any point on It can be expressed by formulas (4)-(6).

[0044] , (4); , (5); , (6); In the formula, is the radius of Bitter ring 2.

[0045] S13, assuming that the point on Bitter ring 2 For point After the corresponding points are rotated and translated, the points are obtained. With point The relational expression of .

[0046] In this embodiment, it is assumed that the point on the second Bitter ring is For point Corresponding points after rotation and translation. Rotation around the y axis The unit rotation matrix of is shown in formula (7). Rotation around the z axis The unit rotation matrix of is as shown in formula (8), then the point With point The relationship expression of can be expressed by formula (9).

[0047] , (7); , (8); , (9); In this embodiment, For point The coordinate value in the three-dimensional coordinate system, are the coordinates of the center of Bitter ring 2, For Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the example, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the plane normal of the second Bitter ring and C The angle of the axis.

[0048] S14, according to point With point The relational expression of With point The coordinate relationship of .

[0049] In this embodiment, it can be deduced from formula (9) that With point The specific coordinate relationship is (10)-(12).

[0050] , (10); , (11); , (12); S15, then according to point Use the central angle The equation shows that the junction point With point The coordinate relationship of is used to obtain the expression of two Bitter rings at any position in space.

[0051] In this embodiment, substituting formulas (4)-(6) into formulas (10)-(12) yields specific expressions (13)-(15) of the spatial Bitter ring II.

[0052] , (13); , (14); , (15); At this point, the specific expression of two Bitter rings at two arbitrary positions in space is completed.

[0053] S20, loop integration is performed on the vector length infinitesimals on the two Bitter rings respectively 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, the mutual inductance expression of the two Bitter coils at any position in space is obtained.

[0054] In one embodiment of the present invention, establishing 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 Two on Point The distance between is the magnetic permeability of vacuum or air, and are the vector length elements on Bitter ring 1 and Bitter ring 2 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 generated spherical vector potential calculation expression (19) and the first and second elliptic integral expressions (22)-(23) can easily solve formula (18). The following is the specific formula reasoning process, which includes: S21, obtain the data from Bitter Ring 1 Go to the second point of Bitter Circle The vector length differential expression of .

[0055] In this embodiment, it is assumed that and Respectively represent a point from the Bitter ring Two points with Bitter Ring 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.

[0056] , (16); , (17); In the formula, is the differential symbol.

[0057] S22, combining the vector length differential expression, obtain the Newman expression of the mutual inductance between Bitter ring 1 and Bitter ring 2.

[0058] In this embodiment, the Newman formula for calculating the mutual inductance of the Bitter ring 1 and the Bitter ring 2 is shown in formula (18): , (18); In the formula, is the mutual inductance of two Bitter rings at any position in space.

[0059] S23, get the energized Bitter ring at point The resulting gravitational vector potential expression.

[0060] In this embodiment, the Bitter ring is powered at point The resulting spherical vector potential is calculated as shown in formula (19): , (19); , (20); ,(twenty one); ,(twenty two); ,(twenty three); ,(twenty four); In the formula, For the Bitter Ring to be powered on, one is at the point The resulting vector potential is 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 Elliptic integrals of the first and second kind for parameters, 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 coordinates of the unit vector in cylindrical coordinates.

[0061] S24, the expression of the spherical vector potential is substituted 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 about Bitter ring 2.

[0062] In this embodiment, formula (19) is simplified to obtain formula (25): , (25); Substituting formula (25) into formula (18) yields formula (26): , (26); S25, 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 about 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.

[0063] In this embodiment, formulas (13), (14), and (15) are substituted into formula (17) to obtain formulas (27)-(29).

[0064] , (27); , (28); , (29); make: , (30); , (31); , (32); Substituting formula (30)-(32) into formula (27)-(29) yields formula (33)-(35): , (33); , (34); , (35); Formula (17) can be transformed into formula (36): , (36); Substituting formula (36) and formula (24) into formula (26) and simplifying it, we can get formula (37): , (37); 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.

[0065] See also Figure 5 As shown, this embodiment also includes: S30, in the mutual inductance expression of two Bitter coils at any position in space, the relationship that the current density distribution of the Bitter coil is inversely proportional to the radius is introduced, and the calculation result of the mutual inductance of the two Bitter coils is obtained by numerical integration method, the mutual inductance of the two Bitter coils is obtained by numerical integration method, and the three-dimensional electromagnetic force of the two Bitter coils in space is obtained by using second-order difference instead of differential.

[0066] In this embodiment, obtaining the three-dimensional electromagnetic force of two Bitter coils in the space behind the mutual inductance includes: S31, Bitter coil axial section on a two-dimensional plane A 1 The two axes of the two-dimensional plane are divided into M Share Nparts to form multiple mesh rings; similarly, place the Bitter coil two on its own axial section A 2 The two axes of the two-dimensional plane are divided into Share parts, forming multiple grid rings.

[0067] 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 Divide into M , divided equally along the z direction N Thus, the Bitter coil is discretized into NM Each ring is represented by the cross-section coordinates Coordinate pair identification . Similarly, put the Bitter coil 2 in Axial section on the surface A 2 along Direction is divided into Servings, along Direction is divided into Thus, Bitter coil 2 is discretized into grid rings, each ring uses Coordinate pairs to identify.

[0068] S32, according to the mutual inductance expression of the two Bitter coils at any position in space, obtain the mutual inductance expression of the corresponding axial cross-section of the two Bitter coils.

[0069] In this embodiment, Bitter coil Mesh Ring and Bitter Coil II The mutual inductance of the mesh ring, 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).

[0070] , (38); , (39); , (40); , (41); , (42); , (43); In the formula, J 1 , J 2 They are the current density distribution 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, For 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.

[0071] S33, obtaining the current density distribution on the axial cross-section corresponding to the Bitter coil 1 and the Bitter coil 2, and then combining the mutual inductance expression of the axial cross-section corresponding to the two Bitter coils to obtain the numerical expression of the mutual inductance of the two Bitter coils.

[0072] In this embodiment, formula (39) is substituted into formula (38) to obtain formula (44): , (44); , (45); S34, after calculating the mutual inductance according to the numerical expression of the mutual inductance of the two Bitter coils, the three-dimensional electromagnetic force of the two Bitter coils in space is obtained.

[0073] In this embodiment, formula (45) is substituted into formula (44) and simplified to obtain formula (46): , (46); In the formula, is the numerical expression of the mutual inductance of two Bitter coils.

[0074] The calculation formula of the three-dimensional electromagnetic force of the two coils in the space after mutual inductance is shown in (47)-(49): , (47); , (48); , (49); In the formula, , , The two Bitter coils in the space behind the mutual inductance are x, y, z Electromagnetic force in the axial direction.

[0075] See also Figure 6 As shown, in one embodiment of the present invention, by comparing the mutual inductance calculation of two Bitter coils in the first and second position relationships in references 2 and 5, the correctness and high precision of this embodiment are verified.

[0076] (1) Mutual inductance calculation verification when the central axes of two Bitter coils coincide with each other, such as Figure 6 shown.

[0077] Table 1 Parameters of the position relationship between the two Bitter coils in reference 2

[0078] Table 2 Comparison between the mutual inductance calculation method of this embodiment and the semi-analytical numerical method in reference 2 ( M =20; N =20; M / =20; N / =20, and coaxial case)

[0079] Table 3 Comparison between the electromagnetic force calculation method of this embodiment and the semi-analytical numerical method in reference 2 ( M =20; N =20; M / =20; N / =20, coaxial case)

[0080] From Table 2 and Table 3, it can be seen that the calculation of the mutual inductance and electromagnetic force between the two Bitter coils according to the first basic relationship combination state in Figure 1 (a) is at the order of 1.0E-04 compared with that in Reference 2, which verifies the correctness and high accuracy of the method of this embodiment. In the table, E represents scientific notation. dz represents the axial spacing between the center points of the two coaxial coils. (2) The mutual inductance calculation of two Bitter coils is verified in two combinations according to Figure 1 (a) and Figure 1 (b): the central axes are parallel and the midplanes do not coincide. Figure 7 shown.

[0081] Table 4 Bitter coil parameters in reference 4 and calculation results of mutual inductance in this embodiment ( M =20; N =20; M / =20; N / =20, axis parallel)

[0082] From Table 4, we can see that the mutual inductance M of the two Bitter coils in the combination state of the first and second basic relations in Figure 1 (a) and (b) is B-B Compared with M in reference 4 GHF The error is in the order of 1.0E-04, which verifies the correctness and high precision of the method in this embodiment.

[0083] (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).

[0084] Table 5 Calculation results of mutual inductance and electromagnetic force of two Bitter coils at specific locations in space ( for Axis and Angle, for The angle between the projection of the axis on the xoy plane and the x-axis)

[0085] Since there is no literature comparing the calculation of the three-dimensional mutual inductance and electromagnetic force of two Bitter coils, Table 5 can be used as a reference correction table for subsequent research by scholars.

[0086] 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.

[0087] See also Figure 8 As shown, the present invention also 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 as described above is applied, comprising: 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.

[0088] 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, and then combine 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.

[0089] It is obvious 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 present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, the embodiments should be regarded as exemplary and non-limiting from any point of view, and the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that all changes falling within the meaning and scope of the equivalent elements of the claims are included in the present invention, and any reference numerals in the claims should not be regarded as limiting the claims involved.

[0090] The above-described embodiments merely represent implementation methods of the invention. The protection scope of the present invention is not limited to the above-described embodiments. For those skilled in the art, several modifications and improvements may be made without departing from the concept of the present invention, which all belong to 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: Using the spatial rotation and translation method, we can obtain the expression of two Bitter rings at any position in space. 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.

2. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: In the mutual inductance expression of two Bitter coils at any position in space, the relationship that the current density distribution of the Bitter coil is inversely proportional to the radius is introduced, and the numerical integration method is used to obtain the calculation results of the mutual inductance of the two Bitter coils; the mutual inductance of the two Bitter coils is obtained by the numerical integration method, and the second-order difference is used instead of the differential to obtain the three-dimensional electromagnetic force of the two Bitter coils in space.

3. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: Get the expression of two Bitter rings at any position in space, including: Place any point on the Bitter ring The coordinates of the circle are expressed in terms of 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 point The relational expression of According to the point With point The relational expression of With point The coordinate relationship of Then according to the point Use the central angle The equation shows that the junction point With point The coordinate relationship of is used to obtain the expression of two Bitter rings at any position in space.

4. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 3, characterized in that: The expression of Bitter ring 2 at any position in space is as follows: ； ； ； In the formula, 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, For Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the example, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the plane normal of the second Bitter ring and C The angle of the axis.

5. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 1, characterized in that: Get the expression of the mutual inductance of two Bitter rings at any position in space, including: Get a point from 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 contains the mutual inductance about Bitter ring 2; 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 about 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.

6. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 5, characterized in that: The expression of the mutual inductance of two Bitter rings at any position in space is as follows: ； in, ; ； ； In the formula, 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 Elliptic integrals of the first and second kind for parameters, For Bitter Ring 2 The coordinate system where the center of the circle is located ABC In the example, the plane normal of Bitter ring 2 is Plane projection and The angle of the axis, is the plane normal of the second Bitter ring and C The angle of the axis, is the radius of Bitter ring 2.

7. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 2, 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 equally into M Share N parts to form multiple mesh 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-section of Bitter coil 1 and Bitter coil 2, and then combine the mutual inductance expression of the corresponding axial cross-section 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 of the two Bitter coils in space is obtained.

8. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 7, characterized in that: The three-dimensional electromagnetic force of the two Bitter coils is obtained by the following formula: ； ； ； In the formula, , , The two Bitter coils in the space behind the 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.

9. The method for obtaining the mutual inductance of two-bit coils placed arbitrarily in space according to claim 7, characterized in that: The numerical expression of the mutual inductance of two Bitter coils is as follows: ； In the formula, is the numerical expression of the mutual inductance of 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.

10. 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 9 comprises: Position module, used to obtain the expression of two Bitter rings at any position in space by 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, and then combine 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.

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