Method for full superconducting cyclotron magnetic field shim based on target first harmonic

Abstract

The application discloses a method for magnetic field padding of a full superconducting cyclotron based on a target first harmonic, and comprises the following steps: determining distribution requirements of the target first harmonic; determining superconducting coil groups for generating the target first harmonic; forming the superconducting coil groups by a group of coils; obtaining an adjustment scheme of each superconducting coil group and a total adjustment scheme number m; determining positions and numbers of radial measurement points and a total harmonic component number n; based on Fourier analysis, establishing a padding matrix A of the first harmonic based on a unit current adjustment amount, the total harmonic component n and the total adjustment scheme number m; based on Fourier analysis, establishing a target first harmonic amplitude matrix B based on the total harmonic component n; listing a padding equation B=AX, and solving X, wherein X is a matrix of superposition coefficients of the adjustment schemes; and the method can perform the magnetic field padding by a coil group as a unit, and a first harmonic formed by a group of coils can better express a distribution shape of the magnetic field, and can better replace a conventional strip padding method.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of accelerators, and particularly relates to a method for magnetic field padding of a full superconducting cyclotron based on a target first harmonic. BACKGROUND

[0002] An important goal of the development of cyclotron technology is the miniaturization and lightening of accelerators, and the use of full superconducting technology further reduces the weight of accelerators. The lightening and coil-based magnetic field adjustment method are important requirements for the magnetic field adjustment of full superconducting accelerators. Full superconducting accelerators adopt a coreless design, so it is more convenient to use a coil-based magnetic field adjustment method. At present, the traditional accelerator with a core generally uses an additional installation method to control the generation of the first harmonic. Due to the limitation of the coil structure, there is no way to generate a first harmonic by controlling the current of the existing coil in the structure. Therefore, based on the structural characteristics of the full superconducting accelerator with multiple adjustment coils, a method for generating a first harmonic by adjusting the coils has great value in the application of full superconducting accelerators.

[0003] At present, the method of generating a first harmonic in a non-full superconducting accelerator (with a core) generally uses a padding strip. The padding strip needs to be additionally installed on the main magnet of the accelerator, and the shape of the padding strip is adjusted to generate a first harmonic.

[0004] Although the non-full superconducting accelerator can generate a first harmonic, it is not suitable for the structural characteristics of the full superconducting accelerator. The coil replaces the core of the full superconducting cyclotron as shown in Figure 1a , and the coil has already occupied a lot of space in the full superconducting cyclotron. It is impossible to add a padding strip on both sides of each core as in the cyclotron with a core.

[0005] The prior art generally uses a single coil as a unit for adjusting the magnetic field of the full superconducting accelerator. However, a single coil cannot form a complete first harmonic, so it cannot accurately describe the change of the magnetic field and solve the problem of magnetic field padding of the target first harmonic. SUMMARY

[0006] To solve the problems in the prior art, the present application provides a method for magnetic field padding of a full superconducting cyclotron based on a target first harmonic. The conventional method for magnetic field padding of a full superconducting accelerator generally uses a single coil as a unit for adjustment. Since a single coil cannot form a complete first harmonic, it cannot accurately describe the change of the magnetic field and solve the problem of magnetic field padding of the target first harmonic.

[0007] To solve the technical problems, the present application adopts the following technical solutions:

[0008] A method for full superconducting cyclotron magnetic field shim based on target first harmonic

[0009] The method has the characteristics that:

[0010] Step one, determining the distribution requirement of the target first harmonic, which is the target first harmonic for magnetic field shim using superconducting coils; the superconducting coils are multiple, and they are arranged along the radial and angular directions of the full superconducting cyclotron;

[0011] Step two, determining the superconducting coil group for generating the target first harmonic: a group of coils with the same radial direction and different angular directions in the full superconducting cyclotron is used to form a superconducting coil group;

[0012] Step three, obtaining the adjustment scheme of each superconducting coil group and the total adjustment scheme number m: the adjustment scheme is determined as one adjustment scheme according to the peak position of the harmonic of the coil group; assuming that a group of coils has 3 coils, any one of the 3 coils is the peak position of the first harmonic, and each coil group can generate 3 different adjustment schemes; the total adjustment scheme number m = coil group number × 3;

[0013] Step four, determining the position and number of radial measurement points and the total harmonic component number n, assuming that the number of radial measurement points is k, and each radial measurement point has 2 harmonic components, sin and cos, so n = 2k;

[0014] Step five, based on Fourier analysis, establishing a shim matrix A of the first harmonic based on the unit current adjustment amount and the total harmonic component n and the total adjustment scheme number m;

[0015] Step six, based on Fourier analysis, establishing a target first harmonic amplitude matrix B based on the total harmonic component n;

[0016] Step seven, listing the shim equation B = AX, and solving X, X is the matrix of the superposition coefficients of the adjustment scheme; wherein the element is Xi, Xi corresponds to the superposition coefficient of the current adjustment amount in the i-th adjustment scheme, which is equivalent to adjusting several unit currents.

[0017] Further, assuming that a group of coils has 3 coils, each adjustment scheme corresponds to 3 unit current adjustment amounts I1, I2, I3, with the unit being ampere (A); I1, I2, I3 are determined in advance.

[0018] Further, the unit current is 1000 ampere-turns for each coil adjustment, that is, for an x-turn coil, the current adjustment amount is 1000 / x.

[0019] Further, the step five based on Fourier analysis, to establish based on the unit current adjustment amount and the total harmonic component n and the total adjustment scheme number m, the first harmonic of the padding matrix A, the specific process is as follows:

[0020] A, to establish based on the unit current amount of the first harmonic of the padding matrix A

[0021]

[0022] Wherein, n represents the first harmonic of the cos and sin components on the k radial measurement points, and m represents the number of coil adjustment schemes;

[0023] B, to obtain each group of coil current adjustment amount [I1, I2, I3]

[0024] Adjust the superposition coefficient of the adjustment scheme corresponding to the i-th coil group, Xi, Xi is equivalent to adjust several unit current, the coefficient can be written as an m-dimensional vector X, written as:

[0025]

[0026] Corresponding to each adjustment scheme of each coil group in the total adjustment scheme; assuming that the subscript X1, X2, X3 coefficient in formula (2) corresponds to the first group of coils, the total current adjustment amount can be expressed as:

[0027] [I1 I2 I3]=X1[I 11 I 12 I 13 ]+X2[I 21 I 22 I 23 ]+X3[I 31 I 32 I 33 ]#(3)

[0028] The left side of the equation [I1, I2, I3] represents the total amount of current that needs to be adjusted in the three coils in the same group of coils;

[0029] Further, the step six based on Fourier analysis, to establish based on the total harmonic component n, the target first harmonic amplitude matrix B, the specific process is as follows:

[0030] 1) determine the amplitude of the target first harmonic two components B r,cos ,B r,sin , wherein r is the corresponding radius measurement point, 1, 2 is two harmonic components, the difference can be written as an n-dimensional vector B, its unit is T.

[0031] 2) the component matrix of the target first harmonic is:

[0032]

[0033] Further, the step seven is to list the padding equation B=AX and solve X, the specific process is as follows:

[0034] 1) list the padding equation:

[0035] B=AX#(5)

[0036] 2) solve the coefficient vector X by using the solving formula of the padding equation:

[0037] X=(A T A) -1 A T B#(6)

[0038] Formula (6) is the matrix moving of formula (5). The left side of formula (6) represents the X to be solved, and the right side of the equal sign of formula (6), A T represents the transpose of the A matrix, A T A represents the product of the transpose of the A matrix and the A matrix, (A T A) -1 represents the inverse of the A T A matrix, B represents the component matrix of the target first harmonic, and each matrix product (A T A) -1 A T B on the right side can calculate X.

[0039] Further, because the number k of radial measurement points can be arbitrarily taken, and the number of adjustment schemes is m, if the number n of elements in B (n=2k) is equal to the number of elements in X, formula (6) can be solved uniquely; if the number n of elements in B is less than the number of elements in X, formula (6) is an over-determined equation group, which can be solved by the least square method; if the number n of elements in B is greater than the number of elements in X, formula (6) is an under-determined equation group, and there are infinitely many solutions, but the unique closest solution can still be solved by mathematical means.

[0040] Advantages and effects of the present application

[0041] 1. The present application carries out magnetic field padding in units of a coil group instead of a coil. A group of coils can form a first harmonic, and since the first harmonic formed by a group of coils can better express the distribution shape of the magnetic field, it can better replace the conventional slot padding method.

[0042] 2. No additional equipment needs to be installed: the existing method of generating a first harmonic often needs to install additional equipment, and installing additional equipment requires additional labor and equipment costs. The present application takes advantage of the design structure of the full superconducting accelerator, only adjusts the existing coil current, uses a program to calculate the adjustment amount, and does not need to install additional equipment, saving the use cost. BRIEF DESCRIPTION OF DRAWINGS

[0043] Figure 1a This is a layout diagram of the magnetic field coils of a fully superconducting cyclotron accelerator.

[0044] Figure 1b A schematic diagram of magnetic field padding using a superconducting coil in this invention;

[0045] Figure 1c This is a schematic diagram of magnetic field padding on both sides of the main magnetic pole using conventional methods;

[0046] Figure 2 This is a schematic diagram of the magnetic field distribution of a set of coils according to the present invention;

[0047] Figure 3 This is a schematic diagram of the radial distribution of the first harmonic generated by the unit current change of the peak region coil group used in this invention.

[0048] Figure 4 This is a schematic diagram of the radial distribution of the first harmonic generated by the unit current change of the valley coil group used in this invention.

[0049] Figure 5 This is a schematic diagram illustrating the adjustment of the first harmonic using a set of coils according to the present invention;

[0050] Figure 6 This is a schematic diagram illustrating the use of first harmonics to represent a circular magnetic field in this invention;

[0051] Figure 7 This is a flowchart of the method for magnetic field padding of a fully superconducting cyclotron accelerator based on the first harmonic of the target, according to the present invention. Detailed Implementation

[0052] Design principle of the invention

[0053] 1. Design challenges of this invention: One of the challenges lies in the limited space available for a fully superconducting cyclotron accelerator, such as... Figure 1a As shown, it lacks an iron core and cannot compensate for the magnetic field by adding additional padding strips around the iron core, as is done in traditional accelerators. Traditional accelerators, such as... Figure 1c As shown, the iron core is the main magnetic pole, which consists of two layers, with four main magnetic pole iron cores in each layer. The magnetic field padding method involves adding padding strips around each main magnetic pole. In a fully superconducting cyclotron accelerator, coils are used instead of the iron core, as shown in the example. Figure 1aAs shown, the coils have taken up a lot of space of the full superconducting cyclotron, and it is impossible to add additional shim strips on both sides of each iron core as in the iron-core accelerator; Conclusion: in the traditional accelerator, due to the use of the shim strip method, there is a lot of free design space for the method, so it can easily generate the target magnetic field. The coil shape and position of the full superconducting accelerator have been determined in the accelerator physical design, and the only space that can be freely adjusted is the ampere-turn number, so it is difficult to adjust the magnetic field by adjusting the ampere-turn number. The second difficulty lies in: it is difficult to express the shape of the magnetic field with a superconducting coil and then to pad the magnetic field. As shown, Figure 1b As shown, the conventional method adopts the adjustment method of independent coils, that is, a full superconducting cyclotron magnetic field adjustment scheme is formed in units of each coil. Each coil can only express half of the magnetic field distribution of the circular coil and cannot express the full magnetic field distribution of the circular coil. As shown, Figure 5 As shown, when only one coil is used to adjust the magnetic field, one coil can only adjust the peak value of the sine curve or the positive half cycle of the sine curve, and the negative half cycle is 0, so the circular magnetic field can only express half, and therefore, the difficulty lies in that the magnetic field padding with one coil is not accurate enough.

[0054] 2, The innovation point of the application: The innovation point lies in that a group of coils (a group of coils with the same radius) are used for magnetic field padding, instead of using one coil as a unit for magnetic field padding (the conventional method uses one coil as a unit for magnetic field padding, instead of using one coil group as a unit for magnetic field padding). A group of coils can better express the distribution shape of the first harmonic, and can better replace the conventional shim padding method. Specifically as follows:

[0055] First, the application uses a group of coils (3 coils) for magnetic field padding as shown in the following formula (3), although there are three adjustment schemes on the right side of the equal sign, the three adjustment schemes are three adjustment schemes formed when the peak value is located in different coils in a group of coils, but the three adjustment schemes and one scheme using three coils for adjustment are two concepts, and the three schemes do not represent that each scheme uses only one coil, but each scheme uses three coils.

[0056] [I1 I2 I3]=X1[I 11 I 12 I 13 ]+X2[I 21 I 22 I 23 ]+X3[I 31 I 32 I 33 ]#(3)Second, the principle of using three coils for magnetic field padding is as follows Figure 5As shown, ① the first coil is used to adjust the curve of the upper half of the sine curve, and the last two coils form the curve of the lower half, although the more coils the smoother the sine curve, but the original intention of the design is not to increase the coil. ② the relationship between the magnetic field and the first harmonic (sine curve) is as shown in the figure Figure 2 、 Figure 6 As shown, the direction of the magnetic field is a clockwise circular curve, the upper half of which is from left to right, and the lower half is from right to left, the upper half corresponds to the upper half of the sine curve, and the lower half corresponds to the lower half of the sine curve. Since three coils (discrete mode) can basically express a sine curve, and one coil can only express the upper half of the sine curve, therefore, using three coils to pad the first harmonic is more accurate than using one coil to pad the first harmonic.

[0057] Based on the above principle, the present application designs a method for full superconducting cyclotron magnetic field padding based on target first harmonic, as shown in the figure Figure 7

[0058] The characteristics are:

[0059] Step one, determine the distribution requirements of the target first harmonic, which is the target first harmonic for magnetic field padding using superconducting coils; the superconducting coils are multiple, which are arranged along the radial and angular directions of the full superconducting cyclotron;

[0060] Supplementary note 1:

[0061] As shown in the figure Figure 1a 、 1b The coil to be adjusted by the present application refers to the padding coil. The superconducting coil includes the outermost annular coil, the 3 or 4 magnetic pole coils inside the annular coil, and the padding coil inside the 3 or 4 magnetic pole coils. The padding coil is divided into inside and outside the magnetic pole, there are 4 padding coils inside the magnetic pole, and there are 3 padding coils outside the magnetic pole. The 3 padding coils outside the magnetic pole and the 4 padding coils inside the magnetic pole are not on the same radius, according to the radius division, there are a total of 7 positions of padding coils along the radial direction. Since the magnetic pole coil of the present embodiment has 3, the present embodiment can adjust a total of 21 padding coils.

[0062] Step two, determine the superconducting coil group for generating the target first harmonic: a group of coils with the same radial direction and different angular directions in the full superconducting cyclotron is used to form a superconducting coil group;

[0063] Supplementary note 2:

[0064] As shown in the figure Figure 1b ​The magnetic field compensation by the coil group is the creative point of the present application. The coil group is a group of coils with the same radius, and there are 7 groups in total. Because the three coils of the coil group can form a first harmonic, and the single coil expresses the first harmonic with a large error, although the expression is discrete, if the single coil is used as the unit for adjustment, the magnetic field error will be increased.

[0065] Step three, obtaining the adjustment scheme of each superconducting coil group and the total adjustment scheme number m: the adjustment scheme is determined as one adjustment scheme according to the different peak positions of the harmonic of the coil group; assuming that a group of coil groups has three coils, any one of the three coils is the peak position of the first harmonic, and each coil group can generate three different adjustment schemes; the total adjustment scheme number m = coil group number x 3;

[0066] Supplementary note 3:

[0067] One adjustment scheme of the present application does not mean that only one coil is adjusted, but one adjustment scheme uses three coils for adjustment, and one group of coils has three adjustment schemes, which means that the peak positions are different, but each scheme cannot be separated from the other two coils except the peak coil.

[0068] Step four, determining the position and number of the radial measurement points and the total harmonic component number n, assuming that the number of radial measurement points is k, each radial measurement point has two harmonic components sin and cos, so n = 2k;

[0069] Supplementary note 4:

[0070] A. The position of the radial measurement point can be arbitrary, which does not represent the position of the coil.

[0071] B. The position of the magnetic field is expressed by two harmonic components sin and cos, which is like expressing the position of a point on a plane by X and Y.

[0072] Step five, based on Fourier analysis, establishing a compensation matrix A of the first harmonic based on the unit current adjustment amount and the total harmonic component number n and the total adjustment scheme number m;

[0073] The specific process is as follows:

[0074] A. Establishing a compensation matrix A of the first harmonic based on the unit current amount

[0075]

[0076] Wherein, n represents the cos and sin components of the first harmonic at k radial measurement points, and m represents the number of coil adjustment schemes;

[0077] Supplementary note 5:

[0078] Figure 3 The peak value of the first harmonic generated by the peak zone coil set is distributed along the radial direction; Figure 4 The peak value of the first harmonic generated by the valley zone coil set is distributed along the radial direction; the valley zone has 3 coils, so there are 3 curves; these two figures are to describe the peak value of the first harmonic generated by the coil set in the radial direction, and intuitively show part of the content in matrix A. Because A describes the distribution of the first harmonic generated by a regulation scheme.

[0079] B. Obtain the current regulation amount [I1, I2, I3] of each coil set

[0080] Assume that a group of coil sets has 3 coils, and each regulation scheme corresponds to 3 unit current regulation amounts I1, I2, I3, with units of amperes (A); I1, I2, I3 are determined in advance.

[0081] The unit current is 1000 ampere-turns for each coil regulation, that is, for an x-turn coil, the current regulation amount is 1000 / x.

[0082] Adjust the superposition coefficient X of the regulation scheme corresponding to the ith coil set i ,X i When adjusting several unit currents, the coefficients can be written as an m-dimensional vector X, written as:

[0083]

[0084] Corresponding to each regulation scheme of each coil set in the total regulation scheme; assume that the subscripts X1, X2, X3 coefficients in formula (2) all correspond to the first coil set, then the total current regulation amount can be represented as:

[0085] [I1 I2 I3]=X1[I 11 I 12 I 13 ]+X2[I 21 I 22 I 23 ]+X3[I 31 I 32 I 33 ]#(3)

[0086] The [I1, I2, I3] on the left side of the equal sign represents the total amount of current that needs to be regulated in the three coils in the same coil set;

[0087] Supplementary note 6:

[0088] A. X1, X2, X3 of formula (3) are unknown numbers, not known numbers;

[0089] B. I1, I2, I3 of formula (3) are known numbers;

[0090] Step six, based on Fourier analysis, establish the target first harmonic amplitude matrix B based on the total harmonic component n;

[0091] The specific process is as follows:

[0092] 1) Determine the amplitude of the two components of the target first harmonic B r,cos ,B r,sin , where r is the corresponding radius measurement point, 1, 2 are two harmonic components, the difference can be written as an n-dimensional vector B, whose unit is T.

[0093] 2) The component matrix of the target first harmonic is:

[0094]

[0095] Supplementary note 7:

[0096] A, B in formula (6) is the amplitude of the target first harmonic, that is, the amplitude of the target magnetic field compensation to be reached, which is a known number;

[0097] B, n = 2k, k is the radial measurement point, and there are 2 amplitudes for each radial measurement point, sin and con amplitude.

[0098] Step seven, list the compensation equation B = AX, and solve X, X is the matrix of the superposition coefficient of the adjustment scheme; Where, the element is Xi, Xi corresponds to the superposition coefficient of the current adjustment amount in the i-th adjustment scheme, which is equivalent to adjusting several units of current.

[0099] The specific process is as follows:

[0100] 1) List the compensation equation:

[0101] B = AX#(5)

[0102] 2) Use the solution of the compensation equation to solve the coefficient vector X:

[0103] X = (A T A) -1 A T B#(6)

[0104] Formula (6) is the matrix shift of formula (5). The left side of formula (6) represents the solution of X, and the right side of formula (6) is equal to A T , which represents the transpose of A matrix, A T A represents the product of the transpose of A matrix and A matrix, (A T A) -1 represents the inverse of A T A matrix, B represents the component matrix of the target first harmonic, and the right side of each matrix product (A TA) -1 A T B can calculate X.

[0105] Because the radial measurement point number k can be arbitrarily taken, and the number of adjustment schemes is m, if the number of elements n (n = 2k) in B is equal to the number of elements in X, formula (6) can be solved uniquely; if the number of elements n in B is less than the number of elements in X, formula (6) is an over-determined equation group, which can be solved by the least square method; if the number of elements n in B is greater than the number of elements in X, formula (6) is an under-determined equation group, there are infinitely many solutions, but the unique closest solution can still be solved by mathematical means.

[0106] It should be emphasized that the above specific embodiments are only an explanation of the present application, and are not a limitation of the present application. Those skilled in the art can make modifications to the above embodiments without creative contribution after reading the present specification, but as long as it is within the scope of the claims of the present application, it is protected by the patent law.

Claims

1. A method for full superconducting cyclotron magnetic field shimming based on target first harmonic; Characterized in that: Step one, determine the distribution requirements of the target first harmonic, which is the target first harmonic for magnetic field shimming using superconducting coils; the superconducting coils are multiple, which are arranged along the radial and angular directions of the full superconducting cyclotron; Step two, determine the superconducting coil group for generating the target first harmonic: a group of coils with the same radial direction and different angular directions in the full superconducting cyclotron is used to form a superconducting coil group; Step three, obtain the adjustment scheme of each superconducting coil group and the total number of adjustment schemes m: the adjustment scheme is determined by the peak position of the harmonic of the group of coils; a group of coils has three coils, and any one of the three coils is the peak position of the first harmonic; each coil group can generate three different adjustment schemes; the total number of adjustment schemes m = coil group number × 3; an adjustment scheme is adjusted by three coils, and a group of coils has three adjustment schemes, which means that the peak positions are different, but each scheme cannot be separated from the other two coils except the peak coil; Step four, determine the position and number of radial measurement points and the total number of harmonic components n, the number of radial measurement points is k, and each radial measurement point has two harmonic components sin and cos, so n = 2k; Step five, based on Fourier analysis, establish a shimming matrix A of the first harmonic based on the unit current adjustment amount and the total number of harmonic components n and the total number of adjustment schemes m; Step six, based on Fourier analysis, establish a target first harmonic amplitude matrix B based on the total number of harmonic components n; Step seven, list the shimming equation B = AX, and solve X, X is the matrix of the superposition coefficients of the adjustment scheme; wherein, the element is Xi, Xi corresponds to the superposition coefficient of the current adjustment amount in the i th adjustment scheme, which is equivalent to adjusting several unit currents.

2. The method of claim 1, wherein the method is based on a target first harmonic for full superconducting cyclotron magnetic field shim. A group of coils has three coils, and each adjustment scheme corresponds to three unit current adjustment amounts I1, I2, I3, with the unit being ampere; I1, I2, I3 are determined in advance.

3. The method for full superconducting cyclotron magnetic field shimming based on target first harmonic according to claim 2, wherein the unit current is 1000 ampere-turns for each coil adjustment, that is, for an x-turn coil, the current adjustment amount is 1000 / x.

4. The method of claim 1, wherein the method is based on target first harmonic for full superconducting cyclotron magnetic field shim. The Fourier analysis based shimming matrix A of the first harmonic based on the unit current adjustment amount and the total number of harmonic components n and the total number of adjustment schemes m in step five is established as follows: A, establish a shimming matrix A of the first harmonic based on the unit current amount Wherein, n represents the cos and sin components of the first harmonic at k radial measurement points, and m represents the number of coil adjustment schemes; B, obtain the current adjustment amount [I1, I2, I3] of each coil group The superposition coefficient of the adjustment scheme corresponding to the i th coil group is Xi, Xi is equivalent to adjusting several unit currents, and these coefficients can be written as an m-dimensional vector X, which is written as: The corresponding each adjustment scheme of each group of coils in the total adjustment scheme; the subscript X1, X2, X3 in formula (2) all correspond to the first group of coils, and the total current adjustment amount can be represented as: [I I I2 I3]=X1[I 11 I 12 I 13 ]+X2[I 21 I 22 I 23 ]+X3[I 31 I 32 I 33 ]#(3) The [I1, I2, I3] on the left side of the equal sign represents the total amount of current that needs to be adjusted in the three coils in the same group.

5. The method of claim 1, wherein the method is based on a target first harmonic for full superconducting cyclotron magnetic field shim. The step six is based on Fourier analysis, and the target first harmonic amplitude matrix B based on the total harmonic component n is established, and the specific process is as follows: 1) Determine the amplitude B of the target first harmonic two components r,cos ,B r,sin , where r is the corresponding radius measurement point, 1, 2 are two harmonic components, the difference can be written as an n-dimensional vector B, whose unit is T; 2) The component matrix of the target first harmonic is:

6. The method of claim 1, wherein the method is based on a target first harmonic for full superconducting cyclotron magnetic field shimming. The step seven is to list the padding equation B=AX and solve X, and the specific process is as follows: 1) List the padding equation: B=AX#(5) 2) Use the solving formula of the padding equation to solve the coefficient vector X: X = (A T A) -1 A T B#(6) Equation (6) is a matrix transposition of equation (5), the left side of equation (6) represents X to be solved, and the right side of equation (6) represents A T represents the transpose of the A matrix, A T represents the transpose of the A matrix and the product of the A matrix, (A T A) -1 represents the transpose of the A matrix and the product of the A matrix, (A T represents the inverse of the A matrix, and B represents the component matrix of the target first harmonic, the matrix products (A T A) -1 A T B can be calculated X.

7. The method of claim 6, wherein the target first harmonic is a first harmonic of a target frequency of the superconducting cyclotron. Because the number of radial measurement points k is arbitrarily selected, and the number of adjustment schemes is m, if the number of elements n in B is equal to the number of elements in X, formula (6) can be solved uniquely, n=2k; if the number of elements n in B is less than the number of elements in X, formula (6) is an overdetermined equation group, which can be solved by the least square method; if the number of elements n in B is greater than the number of elements in X, formula (6) is an underdetermined equation group, and there are infinitely many solutions, but the unique closest solution can still be solved through mathematical means. ​

Images