A method for designing a uniform magnetic field coil based on an improved multi-point taylor expansion

Abstract

This invention discloses a design method for uniform magnetic field coils based on an improved multi-point Taylor expansion, applicable to high-precision magnetic field control. Addressing the limitations of traditional Taylor expansion methods in terms of spatial uniformity, this invention employs a cross-uniform partitioning method to divide the target uniform region. Each center point of the divided target uniform region is substituted into the expressions for the magnetic field derivatives of each order. The optimization objective is to minimize the sum of the magnetic field derivatives of each order. An intelligent optimization algorithm is used to solve this constrained multi-objective optimization problem. This method achieves connectivity of the uniformity of each segmented target uniform region, ultimately achieving high magnetic field uniformity across the entire target uniform region, ensuring high-precision magnetic field control.

Description

Technical Field

[0001] This invention relates to a design method for a uniform magnetic field coil based on an improved multi-point Taylor expansion, belonging to the field of magnetic field manipulation technology, and is used in scientific research, metrology and testing, sensor calibration and other fields. Background Technology

[0002] Numerous scientific research and industrial fields require the application of uniform magnetic fields to complete specific processes. Uniform magnetic field coils and their high-precision design methods are crucial for the accuracy of magnetic field control in various applications. The Taylor expansion method is a commonly used approach for designing uniform magnetic field coils. It uses the coil's structural parameters as independent variables, expands the magnetic field expression using a Taylor series, and solves for the structural parameters using the method of undetermined coefficients based on the coil design requirements. The Taylor expansion method requires pre-defining the coil shape and offers advantages such as simplicity, high design accuracy, and minimal error between the magnetic field distribution in the central region and the design target. However, the traditional Taylor expansion method only considers the Taylor expansion of the magnetic field about the coil's center point. As an approximate analysis method, it can only guarantee a uniform magnetic field in the central region and cannot guarantee uniformity over a large area. Therefore, existing uniform magnetic field coil design methods urgently need improvement and refinement to achieve high-precision magnetic field control. Summary of the Invention

[0003] The technical problem this invention aims to solve is that high-precision magnetic field control applications place higher demands on the design accuracy of uniform magnetic field coils. Traditional Taylor expansion methods for designing uniform magnetic field coils essentially consider an approximation of the magnetic field expression in the neighborhood of the coil's center point. Therefore, they have limitations in terms of spatial uniformity and are difficult to apply in situations requiring a large-scale uniform magnetic field region. To address the shortcomings of existing technologies, there is an urgent need to propose an improved uniform magnetic field coil design method to achieve a larger-scale uniform magnetic field distribution, thereby ensuring high-precision magnetic field control in various applications.

[0004] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0005] A method for designing a uniform magnetic field coil based on an improved multi-point Taylor expansion, characterized by the following steps:

[0006] Step 1: Establish a rectangular coordinate system xyz. The center of the uniform magnetic field coil system is located at the origin of the coordinate system. The plane where the coil is located is parallel to the y=0 plane, generating a magnetic field in the y direction. Each pair of coils consists of two identical coils that are symmetrical about the y=0 plane.

[0007] Step 2: The coil shape is square or round. The coil parameters of the coil system include the side length, radius, number of turns, current direction, and absolute value of the y-axis coordinate of each pair of coils.

[0008] Step 3: Establish the expression for the spatial magnetic field generated by the coil system and the expressions for the magnetic field derivatives of each order with respect to the independent variable y using the Biot-Savart law;

[0009] Step 4: Select a target uniform region inside the uniform magnetic field coil system, divide the target uniform region into blocks and determine the center point of the region. The target uniform region is a cube region with a side length of b at the center of the coil. The method of dividing the target uniform region into blocks adopts the cross uniform division method.

[0010] Step 5: Due to the symmetry of the cube region, in the first quadrant of the rectangular coordinate system, consider the center points of each of the target uniform regions, substitute them into the magnetic field derivative expressions of each order in Step 3, and construct a polynomial to approximate the magnetic field function.

[0011] Step 6: Based on the polynomial in Step 5, take the coil parameters determined in Step 2 as the parameters to be solved, take the minimum sum of the derivatives of each order of magnetic field as the optimization objective, determine the constraints on the coil structure parameters according to actual needs, and use an intelligent optimization algorithm to solve the target optimization problem.

[0012] Step 7: Calculate the magnetic field generated by the designed uniform magnetic field coil system based on the Biot-Savart law, and evaluate the uniformity of the magnetic field distribution. If the design effect does not meet the design target, return to step 2; if the design effect meets the design target, determine the required current by the coil constant to achieve high-precision uniform magnetic field control.

[0013] The expression for the spatial magnetic field in step 3 is as follows:

[0014] ,

[0015] In the formula, I is the current in the coil, C is the coil path, r0 is an arbitrary field point, B(r0) is the magnetic induction intensity generated by the current in the coil at the arbitrary field point r0, μ0 is the permeability of free space, dl is the infinitesimal element of the closed loop on the coil path C, r' is the radius vector between the current source point and the field point, and r is the distance between the current source point and the field point.

[0016] The expressions for the magnetic field derivatives of each order in step 3 are derived from the expression for the spatial magnetic field, and the expressions for the magnetic field derivatives of each order are as follows:

[0017]

[0018] In the formula, B(k)y is the magnetic field component B in the y-direction. y With respect to the k-th derivative of the independent variable y, for a coil system symmetric about the plane at center z = 0, all odd-order derivatives of B(k)y are zero.

[0019] The method for dividing the target uniform region in step 4 includes: dividing the side length of the cube target uniform region with a central side length of b into M parts, where M is a positive integer ≥ 2, forming several cubes of equal size that intersect each other. The side length of each cube is b / M. The first cube is centered on the cube target uniform region with a side length of b. The second to fifth cubes are constructed with the four vertices of the first cube as the center, and so on, until the entire target uniform region is filled.

[0020] The polynomial in step 5 is as follows:

[0021] ,

[0022] in Let be an approximate expression of the magnetic field function for the y-direction magnetic field component. This is the y-coordinate of the center of the cube.

[0023] The optimization objective in step 6 is expressed by the following function:

[0024] ,

[0025] Where f is the objective function and k is the order of the derivative.

[0026] The formula for evaluating the uniformity of the magnetic field distribution in step 7 is as follows:

[0027] ,

[0028] In the formula, δ represents the relative deviation of the magnetic field inside the coil. It is the magnetic field component in the y-direction of the origin. The smaller δ is, the higher the magnetic field uniformity in the target area.

[0029] The technical effects of this invention are as follows: This invention solves the limitation of traditional Taylor expansion method in the spatial uniformity range of uniform magnetic field coil design, and proposes a uniform magnetic field coil design method based on improved multi-point Taylor expansion. This method can achieve high magnetic field uniformity within each segmented target uniform region. Simultaneously, the target uniform region is divided using a cross-uniform division method, which can achieve connectivity of uniformity among each segmented target uniform region, ultimately achieving high magnetic field uniformity across the entire target uniform region and ensuring high-precision magnetic field control. Attached Figure Description

[0030] Figure 1This is a schematic diagram of the distributed coil system involved in implementing the present invention. The center of the coil system is located at the origin O of the rectangular coordinate system xyz. The distributed coil system consists of multiple pairs of coils located at different absolute values ​​of the y-axis coordinates. Each pair of coils consists of two identical coils that are symmetrical about the y=0 plane, and the plane containing the coils is parallel to the y=0 plane.

[0031] Figure 2 This is a schematic diagram of a planar coil involved in implementing the present invention. The center of the coil system is located at the origin O of the rectangular coordinate system xyz. The planar coil system consists of multiple pairs of coils located on two identical and symmetrical y-planes. A pair of coils consists of two identical coils that are symmetrical about the y=0 plane, and the plane containing the coils is parallel to the y=0 plane. Each pair of coils in the planar coil system has a different side length.

[0032] Figure 3 This is a schematic diagram of the target uniform region inside the coil involved in implementing the present invention. The target uniform region is located at the center of the coil system and is a cube-shaped region with a side length of b.

[0033] Figure 4 This is a schematic diagram illustrating the cross-uniform division method involved in this invention. Taking M=2 as an example, M is the number of divisions of the side length. The side length of the target uniform region of the cube with a central side length of b is divided into 2 parts, forming 5 equal-sized and intersecting cubes. The side length of each cube is uniformly b / 2. The first cube is located at the center of the target uniform region of the cube with a side length of b. The second to fifth cubes are constructed with the four vertices of the first cube as their centers. Detailed Implementation

[0034] The following is in conjunction with the attached diagram ( Figures 1-4 The present invention will be described in conjunction with the embodiments.

[0035] Figure 1 This is a schematic diagram of the distributed coil involved in implementing the present invention. Figure 2 This is a schematic diagram of a planar coil involved in implementing the present invention. Figure 3 This is a schematic diagram of the target uniform region inside the coil involved in implementing the present invention. Figure 4 This is a schematic diagram of the cross-uniform division method involved in implementing the present invention. (Reference) Figures 1 to 4 This invention proposes a design method for a uniform magnetic field coil based on an improved multi-point Taylor expansion, in order to achieve high magnetic field uniformity throughout the target uniform region.

[0036] This invention discloses a design method for uniform magnetic field coils based on an improved multi-point Taylor expansion, applicable to high-precision magnetic field control. Addressing the limitations of traditional Taylor expansion methods in terms of spatial uniformity, this invention employs a cross-uniform partitioning method to divide the target uniform region. Each center point of the divided target uniform region is substituted into the expressions for the magnetic field derivatives of each order. The optimization objective is to minimize the sum of the magnetic field derivatives of each order. An intelligent optimization algorithm is used to solve this constrained multi-objective optimization problem. This method achieves connectivity of the uniformity of each segmented target uniform region, ultimately achieving high magnetic field uniformity across the entire target uniform region, ensuring high-precision magnetic field control.

[0037] The coil shape is formed by simple configurations such as square, circle, or combinations thereof. A coil system consists of multiple coil pairs of the selected shapes arranged in a specific configuration. The main types of coil configurations include distributed coils and planar coils, such as… Figure 1 and Figure 2 As shown. Establish a rectangular coordinate system xyz, with the center of the uniform magnetic field coil system located at the origin of the coordinate system. The plane containing the coil is parallel to the y=0 plane, generating a magnetic field in the y direction. A pair of coils consists of two identical coils that are symmetrical about the y=0 plane.

[0038] After determining the shape and configuration of the coil, the parameters of the coil system are determined. These include, but are not limited to, the side length, radius, number of turns, current direction, and absolute values ​​of the y-axis coordinates of each pair of coils. Then, the expression for the spatial magnetic field generated by the coil system and the expressions for the derivatives of the magnetic field with respect to the independent variable y are established using the Biot-Savart law.

[0039] For any current-carrying magnetic field coil in space, with current I and path C, the magnetic induction intensity produced by the current at any field point r0 in the coil can be derived by the Biot-Savart law. for

[0040] ,

[0041] In the formula, μ0 is the permeability of free space, dl is the infinitesimal element on the closed loop C, r' is the radius vector between the current source point and the field point, and r is the distance between the current source point and the field point.

[0042] The derivative expressions of each order derived from the expression for the spatial magnetic field are as follows:

[0043] ,

[0044] In the formula, B(k)y is the magnetic field component B. y The k-th derivative with respect to the independent variable y. For a coil system symmetric about the central plane z = 0, all odd-order derivatives of B(k)y are zero.

[0045] Select a target uniform region within the uniform magnetic field coil system, such as... Figure 3 As shown. The target uniform region is divided into blocks, and the center point of each region is determined. The target uniform region is generally a cube-shaped region with a side length of b at the center of the coil. For example... Figure 4 As shown, the method for dividing the target uniform region into blocks can be a cross-uniform division method: the side length of the cube-shaped target uniform region with a central side length of b is divided into M parts, forming several equal-sized and intersecting cubes, each with a side length of b / M. The first cube is located at the center of the cube-shaped target uniform region with a side length of b. The second to fifth cubes are constructed with the four vertices of the first cube as the center, and so on, until the entire target uniform region is filled.

[0046] Due to the symmetry of the cube region, in the first quadrant of the rectangular coordinate system, considering each center point of the target uniform region, and substituting the expressions for the derivatives of the magnetic field of various orders, a polynomial is constructed to approximate the magnetic field function:

[0047] ,

[0048] Using the coil parameters as the parameters to be determined, the optimization objective is to minimize the sum of the derivatives of all magnetic fields.

[0049] ,

[0050] Based on actual needs, the constraints on the coil structure parameters are determined, and an intelligent optimization algorithm is used to solve the multi-objective optimization problem.

[0051] Finally, the magnetic field generated by the designed uniform magnetic field coil system was calculated based on the Biot-Savart law, and the uniformity of the magnetic field distribution was evaluated. The uniformity evaluation method is as follows:

[0052]

[0053] In the formula, δ represents the relative deviation of the magnetic field inside the coil. The smaller δ is, the higher the uniformity of the magnetic field within the target area.

[0054] If the design effect does not meet the design goal, the design result can be optimized again by changing the specific implementation details of steps 1-6; if the design effect meets the design goal, a high-stability current source can be used to drive the coil system, and the magnitude of the required applied current can be determined by the coil constant to achieve high-precision uniform magnetic field control.

[0055] A method for designing a uniform magnetic field coil based on an improved multi-point Taylor expansion includes the following steps:

[0056] Step 1: Establish a rectangular coordinate system xyz, with the center of the uniform magnetic field coil system located at the origin of the coordinate system. The plane containing the coil is parallel to the y=0 plane, generating a magnetic field in the y direction. A pair of coils consists of two identical coils that are symmetrical about the y=0 plane.

[0057] Step 2: Select the coil shape, coil configuration, and coil parameters according to the actual application requirements. The coil shape can be formed by simple configurations such as square, circle, or combinations thereof. The coil system consists of multiple coil pairs of the selected shapes arranged together. The types of coil configurations mainly include distributed coils and planar coils. After determining the coil shape and configuration, determine the parameters of the coil system. These include, but are not limited to, the side length, radius, number of turns, current direction, and absolute values ​​of the y-axis coordinates of each coil pair.

[0058] Step 3: Based on the coil shape, configuration, and parameters determined in Step 2, establish the expression for the spatial magnetic field generated by the coil system and the expressions for the derivatives of the magnetic field with respect to the independent variable y using the Biot-Savart law;

[0059] Step 4: Select a target uniform region within the uniform magnetic field coil system, divide the target uniform region into blocks, and determine the center point of each block. The target uniform region is generally a cube-shaped region with a side length of b at the center of the coil. A method for dividing the target uniform region into blocks can be a cross-uniform division method.

[0060] Step 5: Due to the symmetry of the cube region, in the first quadrant of the rectangular coordinate system, consider the center points of each of the target uniform regions, substitute them into the magnetic field derivative expressions of each order in Step 3, and construct a polynomial to approximate the magnetic field function.

[0061] Step 6: Based on the multiple polynomials constructed in Step 5, the coil parameters determined in Step 2 are used as the parameters to be solved, and the minimum sum of the derivatives of each order of magnetic field is taken as the optimization objective. The constraints on the coil structure parameters are determined according to actual needs, and the multi-objective optimization problem is solved by intelligent optimization algorithm.

[0062] Step 7: Calculate the magnetic field generated by the designed uniform magnetic field coil system based on the Biot-Savart law, and evaluate the uniformity of the magnetic field distribution. If the design effect does not meet the design target, the design result can be optimized again by changing the specific implementation details of steps 1-6; if the design effect meets the design target, a high-stability current source can be used to drive the coil system, and the required current magnitude can be determined by the coil constant to achieve high-precision uniform magnetic field control.

[0063] The expression for the spatial magnetic field in step 3 is as follows:

[0064] ,

[0065] In the formula, I is the current in the coil, C is the coil path, r0 is an arbitrary field point, B(r0) is the magnetic induction intensity generated by the current in the coil at the arbitrary field point r0, μ0 is the permeability of free space, dl is a infinitesimal element on the closed loop C, r' is the radius vector between the current source point and the field point, and r is the distance between the current source point and the field point.

[0066] The derivative expressions of each order derived from the expression for the spatial magnetic field are as follows:

[0067] ,

[0068] In the formula, B(k)y is the magnetic field component B. y The k-th derivative with respect to the independent variable y. For a coil system symmetric about the central plane z = 0, all odd-order derivatives of B(k)y are zero.

[0069] The specific method for dividing the target uniform region in step 4 is as follows: Divide the side length of the cube-shaped target uniform region with a central side length of b into M parts, forming several equal-sized and intersecting cubes, each with a side length of b / M. The first cube is located at the center of the cube-shaped target uniform region with a side length of b. The second to fifth cubes are constructed with the four vertices of the first cube as their centers, and so on, until the entire target uniform region is filled.

[0070] The polynomial that approximates the magnetic field function in step 5 is as follows:

[0071] ,

[0072] The optimization objective function in step 6 is as follows:

[0073] ,

[0074] The formula for evaluating the uniformity of the magnetic field distribution in step 7 is as follows:

[0075] ,

[0076] In the formula, δ represents the relative deviation of the magnetic field inside the coil. The smaller δ is, the higher the uniformity of the magnetic field within the target area.

[0077] The design method for uniform magnetic field coils based on the improved multi-point Taylor expansion can achieve high magnetic field uniformity within each segmented target uniform region. At the same time, the method of dividing the target uniform region adopts a cross-uniform division method, which can achieve the connectivity of the uniformity of each segmented target uniform region, and finally achieve high magnetic field uniformity of the entire target uniform region.

[0078] Contents not described in detail in this specification are prior art known to those skilled in the art. It is hereby indicated that the above description is intended to help those skilled in the art understand this invention, but does not limit the scope of protection of this invention. Any equivalent substitutions, modifications, improvements, and / or simplifications of the above descriptions that do not depart from the essential content of this invention fall within the scope of protection of this invention.

Claims

1. A method for designing a uniform magnetic field coil based on an improved multipoint Taylor expansion, characterized by, Includes the following steps: Step 1: Establish a rectangular coordinate system xyz. The center of the uniform magnetic field coil system is located at the origin of the coordinate system. The plane where the coil is located is parallel to the y=0 plane, generating a magnetic field in the y direction. Each pair of coils consists of two identical coils that are symmetrical about the y=0 plane. Step 2: The coil shape is square or round. The coil parameters of the coil system include the side length, radius, number of turns, current direction, and absolute value of the y-axis coordinate of each pair of coils. Step 3: Establish the expression for the spatial magnetic field generated by the coil system and the expressions for the magnetic field derivatives of each order with respect to the independent variable y using the Biot-Savart law; Step 4: Select a target uniform region inside the uniform magnetic field coil system, divide the target uniform region into blocks and determine the center point of the region. The target uniform region is a cube region with a side length of b located at the center of the coil system. The method of dividing the target uniform region into blocks adopts the cross uniform division method. Step 5: Due to the symmetry of the cube region, in the first quadrant of the rectangular coordinate system, consider the center points of each of the target uniform regions, substitute them into the magnetic field derivative expressions of each order in Step 3, and construct a polynomial based on Taylor expansion to approximate the magnetic field function. Step 6: Based on the polynomial in Step 5, take the coil parameters determined in Step 2 as the parameters to be solved, take the minimum sum of the derivatives of each order of magnetic field as the optimization objective, determine the constraints on the coil structure parameters according to actual needs, and use an intelligent optimization algorithm to solve the target optimization problem. Step 7: Calculate the magnetic field generated by the designed uniform magnetic field coil system based on the Biot-Savart law, and evaluate the uniformity of the magnetic field distribution. If the design effect does not meet the design target, return to step 2; if the design effect meets the design target, determine the required current by the coil constant to achieve uniform magnetic field control. The method for dividing the target uniform region in step 4 includes: dividing the side length of the cube target uniform region located at the center of the coil system and with a side length of b into M parts, where M is a positive integer ≥ 2, forming several cubes of equal size that intersect each other. The side length of each cube is b / M. The first cube is centered on the cube target uniform region with a side length of b. The second to fifth cubes are constructed with the four vertices of the first cube as the center, and so on, until the entire target uniform region is filled.

2. The improved multipoint Taylor expansion based uniform magnetic field coil design method of claim 1, wherein, The expression for the spatial magnetic field in step 3 is as follows: ， In the formula, I is the current in the coil, C is the coil path, r0 is an arbitrary field point, B(r0) is the magnetic induction intensity generated by the current in the coil at the arbitrary field point r0, μ0 is the permeability of free space, dl is the infinitesimal element of the closed loop on the coil path C, r' is the radius vector between the current source point and the field point, and r is the distance between the current source point and the field point.

3. The improved multipoint Taylor expansion based uniform magnetic field coil design method of claim 2, wherein, The expressions for the magnetic field derivatives of each order in step 3 are derived from the expression for the spatial magnetic field, and the expressions for the magnetic field derivatives of each order are as follows: ， where B(k) y is the y-component of the magnetic field B y With respect to the k-th derivative of the argument y, for a coil system that is symmetric with respect to the central z = 0 plane, all odd derivatives of B(k) y are zero.

4. The improved multipoint Taylor expansion based uniform magnetic field coil design method of claim 3, wherein, The optimization objective in step 6 is expressed by the following function: ， where f is the optimization objective function, k is the derivative order, is the y coordinate value of the center of the block cube.

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