KR101925630B1 - Method of arrangement of measurement points for magnetic field mapping and method of magnetic field mapping using the same - Google Patents

Abstract

We propose a parallel cylindrical line (PCL) arrangement method which maps the magnetic field using the solution of the Laplace equation in the cylindrical coordinate system. The present invention provides a numerical method based on solving Laplace's equations in a cylindrical coordinate system to map the magnetic field in the cylindrical center volume of the magnet. For the mapping procedure, the measurement points can be arranged along a number of circles parallel to the z = 0 plane. The trajectories of the probes become a number of PCLs within the central volume of the magnet. The measurement points are uniformly distributed on each circle.

Description

TECHNICAL FIELD The present invention relates to a method of arranging measurement points for magnetic field mapping and a magnetic field mapping method using the same.

The present invention relates to a method of arranging measurement points for magnetic field mapping to assure magnetic field characteristics of a magnet and a method of using the same to map a magnetic field in a cylindrical central volume of a magnet.

In magnetic applications, it is important to obtain accurate information about the magnetic field. Also, in magnetic field analysis or design, it is important to know the behavior of the magnetic field in the space of interest.

In general, magnetic field mapping is a useful tool for studying magnetic fields. Specifically, a map of the magnetic field distribution is required to guarantee the characteristics of the magnetic field. A simple and straightforward method for magnetic field mapping (mapping) is to measure the magnetic field at each point of the target volume. However, since the number of measurement points is large, the measurement time may take a long time.

An indirect method for magnetic field mapping is to use mathematical tools. The magnetic field of the magnet can be described by some kind of mathematical function in consideration of the magnetic field which must satisfy the Maxwell's equation. The magnetic field of the entire volume can be calculated by applying measurement data to this mathematical function.

Indirect mapping methods based on various mathematical equations are actually being developed. H. Takeda introduced a numerical method for mapping the magnetic field by measuring the 2D magnetic field on the cylindrical surface. This method is flexible but not applicable to solenoids. P. Vernin has published a method for mapping the residual flux, which is the difference between the 3D model and the measured flux. Although this method is a powerful method, it is complicated to construct a three-dimensional model of the magnet when the magnet is not known.

Y. Jing et al., &Quot; Magnetic Field Mapping in the BES III Solenoid, " IEEE Transactions on Applied Superconductivity, vol. 20, no. 3, pp. 324-327, 2010. N. Tan et al., &Quot; Magnetic Field Mapping of the Belle Solenoid, " IEEE Transactions on Nuclear Science, vol. 48, no. 3, pp. 900-907, 2001. H. Takeda et al., &Quot; Extrinsic 3D field maps of magnetic multipoles from 2D surface measurements with applications of the optics calculations of the large-acceptance superconducting fragment separator BigRIPS, " Nucl. Instrum. Meth. Part B, vol.317, pp. 798-809, 2013. P. Vernin et al., &Quot; Field mapping of the high-resolution spectrometers of the Thomas Jefferson National Accelerator Facility (Jefferson Lab), " Nucl. Instrum. Meth. Part A, vol. 449, pp. 505-527, 2000. S. Caspi et al., &Quot; The Use of Harmonics in 3-D Magnetic Fields, " IEEE Transactions on Magnetics, vol. 30, no. 4, pp. 2419-2422, 1994. L. Huang and S. Lee, "Development of a magnetic field calculation program for air-core solenoids which can control the precision of a magnetic field," Progress in Superconductivity and Cryogenics, vol. 16, no. 4, pp. 53-56, 2014. L. Huang and S. Lee, "Study on Magnetic Field Mapping Method in the Center of the Air-Core Solenoid," IEEE Transactions on Applied Superconductivity, Vol. 26, Iss. 4, pp. 4900104, 2016. L. Huang and S. Lee, "A Study on the Effect of the Condition Number in the Magnetic Field Mapping of the Air-Core Solenoid," Progress in Superconductivity and Cryogenics, vol. 17, no. 2, pp. 31-35, 2015. J. S. Arora, Introduction to Optimum Design, Third Ed., Elsevier, Academic Press, 2012.

The present invention solves the disadvantages of the conventional magnetic field mapping as described above, and can be widely applied to the magnetic field mapping in the cylindrical body and the magnetic field mapping generated in various types of magnets by a simple procedure. And a magnetic field mapping method using the same.

In order to solve the above problems, the present invention proposes a parallel cylindrical line (PCL) arrangement method in which a magnetic field is mapped using a solution of a Laplace's equation in a cylindrical coordinate system. That is, the present invention provides a numerical method based on solving the Laplace equation in a cylindrical coordinate system to map the magnetic field in the cylindrical center volume of the magnet.

의 퓨리에 급수와 유사하기 때문이다. 이러한 측정점 배치법을 본 발명에서는 'PCL 배치법(PCL arrangement method)'이라고 부른다.According to the PCL layout method according to the first aspect of the present invention, the measurement points are arranged by a parallel cylindrical line method (PCL method). This places the measurement points along a number of circles parallel to the z = 0 plane in the cylindrical center volume of the magnet. The trajectories of the probes become a number of PCL measurement points within the central volume of the magnet. The measurement points are uniformly distributed on each circle. In this way, the uniformity of the measurement points along a number of circles parallel to the z = 0 plane in the cylindrical central volume of the magnet can be obtained by equations (2) to (4)

Because it is similar to the Fourier series. This method of arranging the measurement points is referred to as a " PCL arrangement method " in the present invention.

In the second aspect of the present invention, the magnetic field mapping procedure according to the PCL placement method will be described as follows.

,

,

을 PCL 배치법에 따라 배치한 측정점들에서 측정한다. Each component of the magnetic field at the center of the magnet

,

,

Are measured at the measurement points arranged according to the PCL arrangement method.

(

: 반경,

: 방위각, z: 높이)에서, 라플라스 방정식은 베셀(Bessel)함수

, 삼각함수

, 지수함수

의 기본해들을 갖는다. 여기서, n 및 m은 변수 분리 과정에서 사용되는 상수이다. 원통 좌표계에서의 라플라스 방정식의 일반해는 이들 기본해를 선형 조합한 것이다.According to the Maxwell equation, the magnetic scalar potential V in a static magnetic field must satisfy the Laplace equation when the volume has no current source or no current sink. Cylindrical coordinates

(

: Radius,

: Azimuth, z: height), the Laplace equation is the Bessel function

, Trigonometric function

, Exponential function

. Where n and m are constants used in the variable separation process. The general solution of the Laplace equation in the cylindrical coordinate system is a linear combination of these fundamental solutions.

의 멱급수(power series)로 전개될 수 있고, 지수함수의 선형 조합은

및

함수로 표현된다. 그리고 자기 스칼라 전위 V는 For the magnetic scalar potential V at the cylindrical center volume of the magnet, the Bessel function is

, And a linear combination of the exponential functions can be developed

And

Function. And the self-scalar potential V

는 기준 반경(reference radius)이다. to be. here,

Is the reference radius.

Therefore, the magnetic flux density

Can be calculated as follows.

및

함수를 알아야 한다. 자기 스칼라 전위 V는 라플라스 방정식을 만족해야 하기 때문에 식 (1)을 라플라스 방정식에 대입해서 In order to calculate each component of magnetic flux density by applying these equations

And

You need to know the function. Since the magnetic scalar potential V must satisfy the Laplace equation, we substitute equation (1) into the Laplace equation

및

함수는

이라고 정할 때 아래와 같이 쓸 수 있다. Can be obtained. In equations (5) and (6)

And

The function

You can write as follows.

및

를 찾기 위한 것이다.Magnetic field mapping is a function of these unknown functions

And

.

성분을

에 대해 퓨리에 변환을 하여

의 삼각함수의 계수들을 구한다. 이들 계수는 아래에서 식 (12)에 나타낸 것과 같은

의 함수, 즉,

로 볼 수 있다. 이와 같이

의 삼각함수의 계수들을 구한 후에는 이를 이용하여

을 풀 수 있다. 이에 대해서 구체적으로 설명한다. Next, in equations (2), (3), and (4)

Ingredients

To Fourier transform

Of the trigonometric function. These coefficients can be expressed as in Equation (12) below

Function, i.e.,

Can be seen as. like this

After obtaining the coefficients of the trigonometric functions of

Can be solved. This will be described in detail.

및

를 찾기 위한 것이라고 하였다. 함수

및

를 얻기 위해서는 이들 함수의 형태를 먼저 결정하여야 한다. 자기 스칼라 전위 V에서, 미지 함수

및

는 z 방향으로 연속적이며, 무한 차수의 도함수(infinite order derivative)를 갖는다. 또한, 함수

및

의 값은 z가 무한대일 때에 영(0)에 접근한다. 로렌츠 곡선의 형태로 인하여 미지 함수

및 는 로렌츠 곡선들의 합으로 근사할 수 있다. In the above, the magnetic field mapping is an unknown function

And

. function

And

The type of these functions must be determined first. At the magnetic scalar potential V, the unknown function

And

Is continuous in the z direction and has an infinite order derivative. Also,

And

The value of zero approaches zero when z is infinite. Because of the form of the Lorenz curve, the unknown function

And Can be approximated by the sum of Lorenz curves.

The Lorenz curve f (x)

및

는 . Where x ₀ is the center of the Lorenz curve and a and L determine the size and width of the Lorenz curve. Assuming that N Lorentz curves are used during the mapping procedure, the unknown function

And

The

와

는 미지의 상수이고,

는 측정점의 높이에 의해 결정될 수 있고,

은 상수이다.

와

의 값은 측정 데이터에 의해 최적화될 수 있다. PCL 배치법의 파라미터(아래의 표 1 참조)로부터 알 수 있는

와 z 값을 고려하여서 미지의 상수

와

를 풀 수 있다. . here

Wow

Is an unknown constant,

Can be determined by the height of the measurement point,

Is a constant.

Wow

Can be optimized by the measurement data. From the parameters of the PCL placement method (see Table 1 below)

And the z value, the unknown constant

Wow

Can be solved.

와

의 합임을 고려하여, 이산 퓨리에 변환(DFT)에 의해서

와

의 계수를 구할 수 있다. 따라서

성분의

항에 대해서 On the other hand, when the magnetic flux density component

Wow

(DFT), considering the sum of

Wow

Can be obtained. therefore

Constituent

About the Port

Can be obtained.

에 대한 선형 방정식들의 집합을 얻을 수 있고, 아래와 같은 매트릭스 형식으로 식 (12)를 표기할 수 있다.If we obtain N coefficients at several elevations z, we can use Eq. (12)

(12) can be expressed in the following matrix form.

은 DFT에 따른

항의 계수들의 벡터,

은 식 (12)로 계산된 성분들을 갖는 매트릭스,

은 미지 상수들

의 벡터이다. 또한, 미지의 상수

는

항의 계수를 사용하여 풀 수 있다. 일단 모든

와

를 알면,

성분을 식 (2)를 써서 계산할 수 있다. here

According to DFT

The vector of the coefficients of the term,

Is a matrix having components calculated by equation (12)

Unknown constants

. Also, unknown constants

The

Can be solved by using a coefficient of interest. Once all

Wow

If you know,

The component can be calculated using equation (2).

및

를 구할 수 있다.As described above, the unknown functions (10) and (11)

And

Can be obtained.

및

성분도 계산할 수 있다.Finally, each component of the magnetic field at different measurement points can be calculated using equations (2), (3), and (4)

And

The composition can also be calculated.

The objects and construction of the present invention described above will become more apparent from the following description of specific embodiments with reference to the drawings.

Using the solution of the Laplace equation in a cylindrical coordinate system, a PCL arrangement is applied to map the magnetic field within the central volume of the magnet. The method of the present invention is suitable for magnetic field mapping in a cylindrical volume and can be widely applied to magnetic field mapping occurring in various kinds of magnets. From the mapping results according to the present invention, it can be seen that the PCL placement method is a useful and convenient method of mapping the magnetic field in the cylindrical volume of the magnet center. And, the PCL configuration method can be applied to magnetic fields generated from various types of magnets.

계산시 절대 오차의

값에 따른 변화
도 7.

성분의 계산 항들
도 8.

성분 계산시 항들의 최대 크기
도 9. PCL 법을 사용한 경우 ECRIS 자석의 자속 밀도 매핑의 성분들의 절대 오차
도 10. PCL 법을 사용한 경우 ECRIS 자석의 자속 밀도 매핑의 성분들의 절대 오차
도 11. 반경 70mm인 ECRIS 자석의 중심에서의 자속 밀도 성분들의 분포. (a)

성분, (b)

성분, (c)

성분
도 12. 4극 자석의 시뮬레이션을 위한 4개의 동일한 전류 루프 (네 개의 루프를 표현하기 위해 그들 사이에 약간의 틈새를 표시하였음)
도 13. 4극 자석에서의 측정점들의 위치와 매핑된 볼륨
도 14. 130mm의 반경을 갖는 원통 표면에서의 4극 자석의 균일도
도 15. PCL 배치법을 사용할 때의 4극 자석에서의 자속 밀도 매핑의 성분들의 절대 오차
도 16. 퓨리에 변환을 이용한 매핑법 사용시의 매핑 오차Figure 1. Measurement points in the PCL batch method. (a) location of measurement points, (b) plan view, (c) side view
Figure 2. Magnetic field mapping procedure using PCL placement
Figure 3. Structure of ECRIS magnet composed of four solenoids and six pole magnets
Figure 4. Six identical current loops for simulating a six pole magnet in an ECRIS magnet.
Figure 5. The location of the measurement points in the ECRIS magnet and the position of the mapped volume when using the PCL method
6.

Absolute error of calculation

Change by value
7.

Calculation terms of components
Figure 8.

The maximum size of the terms in the component calculation
Figure 9. Absolute error of the components of the magnetic flux density mapping of the ECRIS magnet using the PCL method
10. The absolute error of the components of the magnetic flux density mapping of the ECRIS magnet when using the PCL method
Figure 11. Distribution of magnetic flux density components at the center of an ECRIS magnet with a radius of 70 mm. (a)

(B)

(C)

ingredient
Figure 12. Four identical current loops for simulating a four pole magnet (with a small gap between them to represent the four loops)
Figure 13. Position of measurement points and mapped volume in 4 pole magnets
Figure 14. Uniformity of quadrupole magnets on a cylindrical surface with a radius of 130 mm
15. The absolute error of the components of the magnetic flux density mapping in the quadrupole magnet when using the PCL arrangement
Figure 16. Mapping error when using mapping method using Fourier transform

의 퓨리에 급수와 유사하다는 것을 고려하여 측정점(10)들을 자석의 원통형 중심 볼륨에서 z=0 평면에 평행한 다수의 원을 따라 균일하게 배치한 것이다. 자석의 중심 볼륨 내에 있는 다수의 측정점들이 곧 프로브들의 궤적이 된다. 이러한 PCL 배치법의 파라미터들을 표 1에 열거하였다.Fig. 1 shows measurement points in the PCL layout method, where (a) is a position of measurement points, (b) is a plan view, and (c) is a side view. Measurement points 10 were arranged in a parallel cylindrical line fashion. (2) to

The measurement points 10 are uniformly arranged along a plurality of circles parallel to the z = 0 plane in the cylindrical central volume of the magnet. A number of measurement points within the central volume of the magnet are now the trajectories of the probes. The parameters of this PCL batch method are listed in Table 1.

이며, 각 원에 있는 측정점의 수는 다음과 같이 계산할 수 있다.In the above table, the radius of the cylinder surface is R and the height of the surface is H in the PCL arrangement. The azimuth angle between two adjacent points of each circle is

, And the number of measurement points in each circle can be calculated as follows.

When the distance between each circle in the axial direction is h, the number of circles in this method can be calculated as follows.

The radius R and height H normalized using the reference radius r ₀ at the time of analysis can be obtained as follows.

을 만들 수 있고 미지의 상수

와

를 풀 수 있다. Once the parameters of the PCL batch method are determined as shown in the above table, the positions of the measurement points can be found by their definition. Using the coordinates of Eq. (12) and measurement points, the matrix of Eq. (13)

And an unknown constant

Wow

Can be solved.

Figure 2 illustrates a magnetic field mapping procedure using PCL placement.

First, a measurement point is arranged at the center volume of the magnet according to the PCL arrangement method, and each component of the magnetic field is measured at a specific measurement point (100).

를 기준으로 퓨리에 변환을 적용해서 식 (2), (3), (4)의

의 삼각함수의 계수를 구한다(200). 이들 계수는 식 (12)에 나타낸 바와 같이

의 함수로 볼 수 있다. And,

(2), (3), and (4) by applying the Fourier transform on the basis of

(200) of the trigonometric function. These coefficients are expressed as shown in equation (12)

Can be seen as a function of.

및

는 The Lorenz curve is used to determine these functions (300). Specifically, assuming that N Lorenz curves are used during the mapping procedure, the unknown function

And

The

와

는 미지의 상수이고,

는 측정점의 높이에 의해 결정될 수 있고,

은 상수이다.

와

의 값은 측정 데이터에 의해 최적화될 수 있다. 이제, 미지의 함수

및

를 구하기 위해서는 PCL 배치법의 파라미터로부터 알 수 있는

와 z 값을 고려하여서 미지의 상수

와

를 풀면 된다.. here

Wow

Is an unknown constant,

Can be determined by the height of the measurement point,

Is a constant.

Wow

Can be optimized by the measurement data. Now, the unknown function

And

The PCL allocation method

And the z value, the unknown constant

Wow

.

와

를 식 (10) 및 (11)에 대입해서 미지의 함수

및

를 구한다(400). The constant from which to get the value

Wow

(10) and (11) to calculate an unknown function

And

(400).

Finally, each component of the magnetic field at different measurement points can be calculated using Eqs. (2), (3), and (4) (500).

Hereinafter, an embodiment will be described in which the magnetic flux density of the center of the ECRIS magnet and the quadrupole magnet is mapped using the PCL arrangement method according to the present invention.

Example 1: ECRIS magnet

3 shows the structure of an ECRIS magnet 30 composed of four solenoids and six pole magnets. An electron cyclotron resonance ion source (ECRIS) magnet is used to intensively provide a highly charged ion beam to a linear accelerator. The structure of the ECRIS magnet consists of four axial coils 32 and six radial coils 34. The axial coil 32 is comprised of four coaxial solenoids of different magnitudes and currents. And the six radial coils 34 are in the form of a sextupole. The six-pole magnet is disposed inside the solenoid, as shown in Fig.

This magnetic system confines the axial magnetic field from the four solenoid coils to a radial magnetic field from the six pole magnets to confine the ECR plasma stream.

The design parameters of the coils in the magnetic system are listed in Table 2.

In this embodiment, the magnetic field generated by the solenoid is simulated with a high-precision " rzBI " program (L. Huang and S. Lee, " Development of a magnetic field calculation program for an air- , Progress in Superconductivity and Cryogenics , vol. 16, no. 4, pp. 53-56, 2014.).

Since the magnetic field generated in the straight wire can be calculated in a closed form, the magnetic field generated in the six pole magnet was simulated with six identical current loops as shown in FIG. Figure 4 shows six identical current loops 40 for simulating a six pole magnet in an ECRIS magnet. The magnetic field can then be calculated by superimposing the lines and the magnetic fields generated in the solenoid. In this embodiment, it is not necessary to use the FEM (finite element method) in the calculation of the magnetic field.

Assuming that the height of the mapped volume 52 is 1,000 mm and the radius of the cylindrical body 50 at the center of the ECRIS magnet is 70 mm, the measurement points 54 when using the PCL placement method are shown in FIG. 5, The parameters are shown in Table 3. Figure 5 shows the location of the measurement points and the mapped volume in the ECRIS magnet when using the PCL method.

로 계산될 수 있다. ECRIS 자석이 6중 대칭 구조임을 고려하여, 퓨리에 급수의 성분들은 n이 3, 9, 12, ... 로 커질수록 더 큰 값을 갖게 된다. 매핑 과정에서, 크기가 작은 퓨리에 급수의 성분들은 무시할 수 있다.

값은 측정 데이터로부터 최적화될 수 있다. By using the PCL layout method, the components of the magnetic flux density at the measurement point of the center of the ECRIS magnet can be obtained. According to the discrete Fourier transform, the maximum value of n in the expressions (2), (3), and (4)

Lt; / RTI > Considering that the ECRIS magnet is a six-symmetric structure, the components of the Fourier series become larger as n becomes 3, 9, 12, .... In the mapping process, the components of the small Fourier series can be ignored.

The value can be optimized from the measurement data.

성분에서의 n=3의 경우를 나타낸 것과 같이 매핑 과정에서의 오차는 도 6에 나타낸 바와 같이 산출되었다. 도 6에서

값에 따른

계산시의 절대 오차(absolute error)가 변동됨을 알 수 있다.

가 150mm일 때 계산시 오차가 최소값이 될 수 있다.

As shown in the case of n = 3 in the component, the error in the mapping process was calculated as shown in Fig. 6

Depending on the value

It can be seen that the absolute error at the time of calculation changes.

Is 150mm, the error in calculation can be the minimum value.

성분은 로렌츠 곡선들의 고차 도함수의 합으로 계산될 수 있다. (12)에 있는 항들의 크기는 도 7에 표시된 것과 같이 m이 증가함에 따라 감소하게 될 것이고, 각 m에서의 최대값을 도 8에 도시하였다. m이 10일 때,

계산시의 절대 오차는 0.01 가우스(Gauss)보다 작을 것이다. 도 7은 상이한 m에서의

성분 계산시의 항들을 나타내고, 도 8은

성분 계산시 항들의 최대 크기를 나타낸다.According to equation (12)

The component can be calculated as the sum of the higher order derivatives of the Lorenz curves. The magnitude of the terms in (12) will decrease as m increases as shown in Fig. 7, and the maximum value at each m is shown in Fig. When m is 10,

The absolute error in the calculation will be less than 0.01 Gauss. Figure 7 shows the

Components in calculation of the components, and Fig. 8

It represents the maximum size of the terms in the component calculation.

성분의

항 및

항 모두를 구했으면, 식 (2)를 이용하여 매핑된 볼륨의 모든 지점에서의

성분을 계산할 수 있다. 동일한 방식으로, 해당 볼륨에서의

및

성분을 구할 수 있다.First

Constituent

And

If we have all of the terms, we can use Eq. (2)

The components can be calculated. In the same way,

And

Components can be obtained.

,

,

성분에서의 절대 오차는 도 9에 표시하였다. 도 9는 PCL법을 사용한 경우 ECRIS 자석의 자속 밀도 매핑의 성분들의 절대 오차를 나타낸다.

의 매핑 결과가 다른 것들보다 더 나쁘며 매핑 결과의 절대 오차는 거의 0.1 가우스보다 작다. 그러나 홀 효과를 이용한 전계강도 측정기의 범위가 0.1 mT ~ 3×10⁴ mT인 것을 고려하면, 이 매핑 결과는 받아들일만 하다.

,

,

The absolute error in the components is shown in Fig. 9 shows the absolute error of the components of the magnetic flux density mapping of the ECRIS magnet when the PCL method is used.

Is worse than the others and the absolute error of the mapping result is less than about 0.1 Gauss. However, considering that the range of the electric field intensity measuring device using the Hall effect is 0.1 mT to 3 x 10 ⁴ mT, the mapping result is acceptable.

The present inventor's previous research, HCL method (refer to L. Huang and S. Lee, "Study on Magnetic Field Mapping Method in the Center of the Air-Core Solenoid," IEEE Transactions on Applied Superconductivity , Vol. 26, Iss. 4, pp. 4900104, 2016 and L. Huang and S. Lee, "A Study on the Effect of the Condition Number in the Air-Core Solenoid," Progress in Superconductivity and Cryogenics , vol. 17, no. 2, pp. 31-35, 2015] is shown in Fig. 10. The absolute error of each component in the mapped volume is shown in Fig. 10 shows the absolute error of the components of the magnetic flux density mapping of the ECRIS magnet when the PCL method is used.

및 방위각 성분

가 HCL 배치법을 이용한 매핑 결과보다 더 우수하다. 그리고, HCL 배치법을 이용한 매핑 결과는 매핑된 볼륨의 양단부에서 더 나빠진다.Compared with the mapping results using the HCL batch method,

And an azimuth component

Is better than the mapping result using the HCL batch method. And, the mapping result using the HCL batch method gets worse at both ends of the mapped volume.

성분, (b)는

성분, (c)는

성분을 나타낸다.The distribution of the three components of the magnetic field at the portion of the mapped volume is shown in FIG. 11 shows the distribution of magnetic flux density components at the center of an ECRIS magnet having a radius of 70 mm, (a)

The component, (b)

Component, (c)

Lt; / RTI >

Example 2: Four-pole magnet

FIG. 12 shows four identical current loops 120 (showing a small gap between them to represent four loops) for the simulation of a quadropole magnet. Four-pole magnets are useful for focusing the particle beam because the magnitude of the magnetic field generated by the quadrupole magnet quickly spreads radially from the longitudinal axis. In this embodiment, the quadrupole magnet is simulated with four identical current loops 120 as shown in Fig. The parameters of these current loops are listed in Table 4.

Suppose the mapping volume is a cylindrical volume with a height of 1,000 mm and a radius of 170 mm in the center of the quadrupole magnet.

The measurement points at the time of using the PCL arrangement method are shown in Fig. 13, and the parameters of the PCL arrangement method are shown in Table 5. 13 shows the position of the measurement points in the quadrupole magnet and the mapped volume.

At the center of the quadrupole magnet, the components of the magnetic field are quadruply symmetrical. For n = 2, 6, 10, ..., the components of the Fourier series are increasingly large. The uniformity of the magnetic field in the quadrupole magnet

은 z에 대한 퓨리에 급수에서의 n=6 항을 의미하고,

은 n=2 항을 의미한다.. here

Means n = 6 in the Fourier series for z,

Means n = 2.

Fig. 14 shows the uniformity of quadrupole magnets on a cylindrical surface having a radius of 130 mm. According to the mapping result and calculation result of the present invention, uniformity of a radius of 130 mm can be obtained as shown in Fig. The uniformity obtained from the mapping results is almost the same as the uniformity obtained from the simulation results.

The mapping error is identified by the difference between the mapping result and the simulation result. The maximum absolute error for z at the surface of the circle is shown in Fig. 15 shows the absolute error of the components of the magnetic flux density mapping in the quadrupole magnet when using the PCL arrangement method.

16 shows a mapping error when using the mapping method using the Fourier transform. The mapping method using Fourier transform [H. Takeda et al ., &Quot; Extraction of 3D field maps of magnetic multipoles from 2D surface measurements with applications to the optics calculations of the large-acceptance superconducting fragment separator BigRIPS, " Nucl. Instrum. Meth . Part B, vol.317, pp. 798-809, 2013], it is possible to obtain a more accurate mapping result as shown in FIG. 16 from the PCL layout method.

As a result of the magnetic field mapping in the ECRIS magnet and the quadrupole magnet as described above, it has been proved that the PCL layout mapping method according to the present invention is feasible and convenient.

The above-described embodiments are merely examples for actual implementation of the technical idea of the present invention. The technical scope of the present invention is determined by the rational interpretation of the claims set forth below.

Claims (5)

delete delete To map the magnetic field in the cylindrical center volume of the magnet, three magnetic field components for the radius, azimuth, and height of the cylinder

,

,

(here,

: Radius,

: Azimuth angle, z: height) are arranged along a plurality of spaced apart circles parallel to the z = 0 plane, the method comprising the steps of:
The respective components of the magnetic field measured at the measurement points

,

,

Is calculated by the following equations (2), (3), and (4).

(In the formulas (2), (3), and (4)

Is the reference radius,

,

ego,
In equations (7) and (8)

,

,

ego,
In equations (10) and (11)

Wow

Is an unknown constant,

Is a constant determined by the height of the measurement point, and

, n , m are constants)
Each component of the magnetic field measured at the measurement points of the cylindrical center of the magnet

,

,

(Where each magnetic field component is equal to the following equations (2) to (4)).

(here

Is the reference radius)),
In the formulas (2) to (4)

And

Function,

,

To know the unknown function

And

A magnetic field mapping method for calculating a magnetic field,
(here,

ego,

,

Lt;
In equations (10) and (11)

Wow

Is an unknown constant,

Is a constant determined by the height of the measurement point, and

, n , m are constants)
1) arranging the measurement points uniformly along a plurality of circles parallel to the z = 0 plane in the cylindrical central volume of the magnet according to the method of arranging the measuring points of claim 3,
2) Each component of the magnetic field at the center of the magnet

,

,

At the placed measurement points,
3) The magnetic field

Ingredients

(2) < / RTI >< RTI ID = 0.0 >

Trigonometric function

Wow

, And the Lorenz curve is applied

(Equation (10) above) and

(Formula (11)),
4) Unknown function

And

Of the measurement points placed in step 1)

And the z value.

Wow

, ≪ / RTI >
5) Constant to get the value

Wow

(10) and (11) to obtain the unknown function

And

, ≪ / RTI >
6) Other components of the magnetic field

And

Repeating steps 3) to 4) above to calculate < RTI ID = 0.0 >

(3) in the step 3)

Of the trigonometric function,

(4) in the step 3)

To obtain a coefficient of a trigonometric function of the magnetic field of the magnet. delete

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