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Accurate calculation of field quality in conventional straight dipole magnets

  • Yingshun Zhu
  • Fusan Chen
  • Wen Kang
  • Wan Chen
  • Mei Yang
  • Xi Wu
Original Paper

Abstract

Purpose

In the standard design method of straight dipole magnets, the good field region is symmetric to the magnet mechanical center both in 2D and in 3D, so the obtained field quality is not the actual one because the field integration lines are not consistent with the curved beam paths. In this paper, an improved method for straight dipole magnets aiming at obtaining accurate field quality is proposed.

Methods

The field quality is calculated by taking into account the relationship of the good field region to the magnet straight geometry. General description of the improved method is introduced, and two application examples of straight dipole magnets are presented to investigate the detailed field quality difference between the improved and traditional methods. The result of the improved method is also compared with the field quality calculated along particle trajectory in OPERA-3D.

Results

It is shown that the difference in field quality between the improved and traditional methods cannot be neglected, and the field quality in the improved method is very close to the one calculated along real beam paths.

Conclusion

The field quality in the improved method is accurate enough for practical application in a straight dipole magnet.

Keywords

Accelerator Straight dipole magnet Field simulation Beam sagitta Field quality 

Introduction

Dipole magnet is one of the most fundamental and commonly used magnet type in high energy accelerators [1, 2, 3]. Unlike other magnet types such as quadrupole magnet, sextupole magnet which are usually straight in the longitudinal direction, a dipole magnet can be either straight or curved.

Compared with the curved dipole magnet, straight dipole magnet is easier to be manufactured and assembled, and higher mechanical precision can be achieved [1, 4]. Thus, it is preferred in many applications. So far, a number of conventional dipole magnets (including gradient dipole magnet) in high energy accelerators have been designed and manufactured as straight [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

The general design method of dipole magnet does not distinguish between straight and curved dipole magnets. In existing straight dipole magnets which have been built, the good field region (GFR) is symmetric to the magnet mechanical center both in 2D and 3D [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Relative to the one required by the beam optics, the GFR is enlarged to include the beam sagitta. Furthermore, magnetic field is usually integrated along straight lines longitudinally to obtain the integrated field in both the field simulation and field measurement. The integrated field quality of a straight dipole magnet obtained along straight lines is not accurate because the field integration lines are not consistent with the curved beam paths. The difference in the integrated field quality between the straight line integration and curved line integration is ignored in current straight dipole magnets.

The main purpose of this paper is to improve the accuracy of field quality in conventional straight dipole magnets. The relationship of beam paths with respect to the straight magnet geometry is carefully considered. The good field region is not symmetric to the magnet mechanical center and is not enlarged to include the beam sagitta. The difference in integrated field quality between the improved and traditional methods in the same straight dipole magnet is investigated in two examples, and the result of the improved method is also compared with the field quality calculated along real beam paths in OPERA-3D.
Fig. 1

Beam trajectory passing through a straight dipole magnet

General description of the improved method

The value of field quality is directly related to the width of the good field region. In the traditional design method of straight dipole magnets, the width of total GFR is the sum of beam sagitta and the real GFR required by the beam optics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]; in addition, the center of GFR is consistent with the magnet mechanical center both in 2D and 3D.

The basic idea of the improved method is that the relationship of curved beam paths with respect to the straight magnet geometry is considered, the width of GFR is no longer enlarged to include the beam sagitta, and the real GFR required by the beam optics is used. So the center of GFR does not coincide with the magnet mechanical center both in 2D and 3D. The sketch map of the curved beam trajectory passing through a straight dipole magnet is shown in Fig. 1. The apex of the central beam path (Point A) is offset from the nominal magnet centerline by half the beam sagitta, which is the same as that in the traditional method [5].
Table 1

Design requirements of LR-BB dipole magnets

Item

Unit

Value

Magnet quantity

 

2

Magnet type

 

H-type, DC

Bending angle

Degree

25

Magnetic length

m

1.5

Central field

T

0.384

Pole gap

mm

60

Width of good field region

mm

65

Integrated field uniformity

 

0.1%

It can be seen in Fig. 1 that the transverse relationship of the required GFR with respect to the straight magnet geometry varies at different longitudinal positions, so transverse field uniformity in GFR at different planes also differs. It is convenient to correspond the middle plane of the magnet in longitudinal direction (\(z=0\)) to the plane in 2D design. Denoting the value of beam sagitta as s, the width of required GFR by the beam optics as w, and assuming that the geometric centerline of the magnet correspond to \(x=0\) in the transverse direction, then the range of GFR in 2D field analysis is from \(x_{1}=s/2-w/2\) to \(x_{2}=s/2+w/2\), and the center of GFR is at \(x= s/2\). It should be noted that the center of GFR is not consistent with the magnet mechanical center in the improved method. For comparison, the range of enlarged GFR in the traditional method is from \(x^{{\prime }}_{1}=-s/2-w/2\) to \(x^{{\prime }}_{2}=s/2+w/2\) in 2D field analysis with the center position at \(x=0\).

In the 3D field analysis, the center of the GFR is also not consistent with the mechanical centerline. The integral field should be obtained by integrating the magnetic field along curved beam paths, and the transverse offset range of the magnetic field integration path is w. For comparison, the field is integrated along straight lines longitudinally in the traditional method, in which the transverse offset range of the field integration path is equal to \(w+s\). Thus, the field analysis process in the improved method is quite similar to the one in a curved dipole magnet.

Application examples

Development of CSNS LR-BB dipole magnets

As an example, the improved field quality calculation method is applied to the development of LR-BB dipole magnets in the China Spallation Neutron Source (CSNS), which is now under construction in China [14]. The LR-BB dipole magnets transport the beam from linac to Rapid Cycling Synchrotron (RCS), and the design requirements are listed in Table 1.
Fig. 2

2D magnetic flux lines of LR-BB dipole magnet (one half)

The horizontal width of GFR is 65 mm, and the calculated beam sagitta of straight LR-BB dipole magnet is as large as 81.5 mm. The total horizontal GFR will be 146.5 mm if using the traditional method.

Soft iron is used for the core of LR-BB dipole magnet. The excitation coil for each pole is made of two pancake-type coils with a total of four layers, and every pancake is cooled by one water circuit. The coils are wound from 13 mm square OFHC copper conductor with a 7-mm diameter water-cooling channel, and the operation current is 334 A.

Since the magnet central field is modest, a pole root narrower than the magnet pole tip is used. The magnetic field optimization is performed using the software OPERA from Cobham Technical Services. As described in “General description of the improved method” section, the range of GFR is from \(x_{1}=8.25\) mm to \(x_{2}= 73.25\) mm in 2D field analysis. Figure 2 shows the 2D flux lines, and Fig. 3 presents the calculated magnetic field homogeneity in the GFR along the transverse axis, where the center of GFR corresponds to \(d=0\).
Fig. 3

2D field homogeneity of LR-BB dipole magnet

Since the GFR is not symmetric with respect to the magnet mechanical center in the improved method, the field is higher on the right side than on the left side in Fig. 3. However, the peak-to-peak field uniformity of ± 0.1% is achieved in 2D simulation.

In 3D magnetic field simulation, the field is integrated along the ideal beam paths with a constant radius of 3437.2 mm. Pole end chamfer is optimized and determined by the field simulation to meet the integrated field quality requirement and is then directly machined on a CNC machine, which is similar to that in a curved dipole magnet [15]. The whole optimized magnet modeled in OPERA-3D is shown in Fig. 4.
Fig. 4

LR-BB dipole magnet modeled in OPERA-3D (dimensions in mm)

The field homogeneity along the transverse axis in 2D plane at various longitudinal positions inside the magnet gap is investigated and depicted in Fig. 5, where \(z=0\) corresponds to the symmetry plane in the longitudinal direction. It can be seen that the transverse field uniformity in the GFR is within ± 0.1% at all the longitudinal positions.
Fig. 5

Transverse field uniformity at various longitudinal positions

After the fabrication of the two LR-BB dipole magnets, magnetic field measurement was taken using a Hall probe system. The comparison of the measured and simulated integrated field distribution in the midplane (\(y=0\)) is presented in Fig. 6, where the central beam path corresponds to \(d=0\). A theoretically more accurate method of integrating the magnetic field along real particles trajectories in OPERA-3D is used, and the calculated field quality is also shown in Fig. 6. The effect of fringe field on the beam path is fully taken into account. However, in practice it is difficult to measure the magnetic field along the particle trajectory.
Fig. 6

Comparison of integrated field uniformity between the simulation and measurement results

It is shown that the measured integrated field distribution agrees well with the simulation result. The required integrated field uniformity of 0.1% in the GFR is achieved. Furthermore, the field uniformity obtained by integrating the field along ideal beam paths is close to the one calculated along particles trajectories, and the difference is smaller than \(6\times 10^{-5}\). The multipole field coefficients extracted from the measured integrated field distribution in the two LR-BB dipole magnets are big, and the largest ones are the quadrupole and octupole field components, which are about \(-1.2 \times 10 ^{-3}\) and \(1.6 \times 10^{-3}\), respectively. However, there is no requirement for multipole field components for these transport line dipole magnets.

It should be noted that, when the field is integrated along straight lines as in the traditional method, the integrated field uniformity will be 0.27% in the total GFR of 146.5 mm. So there is a large difference in the integrated field quality between the improved and traditional methods for LR-BB dipole magnets.

The magnetic performance of the two LR-BB dipole magnets can well satisfy the field requirement, and these two magnets have been installed in the CSNS tunnel.

Application to BEPCII 67B dipole magnet

The bending angle of CSNS LR-BB dipole magnets is as large as 25 degrees, and the beam sagitta is 81.5 mm. It is meaningful to study the field quality difference between the improved and traditional methods for an existing straight dipole magnet which has a smaller bending angle. So the improved method is applied to 67B dipole magnet in the storage ring of the Beijing Electron Positron Collider Upgrade Project (BEPCII).

The BEPCII 67B dipole magnet is a laminated C-type straight dipole magnet with a pole gap of 67 mm, magnetic length of 1.4135 m and nominal central field of 0.6892 T (corresponding to the optimized beam energy of 1.89 GeV) [9, 16], which was developed using the tradition method. The pole end chamfer was determined by the field measurement result of prototype magnet in 2004. The batch field measurement of forty 67B dipole magnets was finished in 2005 using a straight translating long coil system, and they have been in operation since 2006.

The bending angle of BEPCII 67B dipole magnet is 8.8628 degrees, and the beam sagitta is 27.3 mm. The total width of horizontal GFR including the beam sagitta was 135.3 mm in the development of 67B dipole magnets, whereas the GFR required by the beam optics of 108 mm is used in the improved method, and the field will be integrated along the ideal beam paths with a constant radius of 9149.6 mm.

The whole 67B dipole magnet including the pole end chamfer is remodeled, and the magnetic field analysis is performed using OPERA-3D. Then, the field quality is investigated using the traditional and improved methods, respectively. Figure 7 shows the whole 67B dipole magnet modeled in OPERA-3D.
Fig. 7

3D model of 67B dipole magnet (dimensions in mm)

The calculated integrated field distribution using the traditional method is very close to the batch field measurement results of straight translating long coil system in 2005. The measured integrated field uniformity in the midplane of the batch 67B magnets using the traditional method at the nominal excitation current was within \(2\times 10^{-4 }\) [16], whereas the calculated one is \(1\times 10^{-4}\) in OPERA-3D.

Figure 8 presents the comparison of the integrated field homogeneity in the two methods, where the central beam path corresponds to \(d=0\) in the improved method and the straight mechanical centerline denotes \(x=0\) in the traditional method, respectively. For comparison, the theoretically more accurate field quality obtained by integrating the magnetic field along real particles trajectories in OPERA-3D is also shown in Fig. 8.
Fig. 8

Calculated integrated field uniformity of 67B dipole magnet

Table 2 lists the multipole field coefficients extracted from the integrated field distribution of BEPCII 67B dipole magnet in the improved method and traditional method by polynomial fitting, respectively. For example, \(b_{2}\) and \(b_{4}\) denote the quadrupole and octupole field components normalized to the dipole field, respectively.
Table 2

Multipole field coefficients in unit of \(10^{-4}\)

n

\({b}_{n}\) in improved method

\(b_{n}\) in traditional method

2

2.152

0.643

3

0.693

1.794

4

3.556

\(-\) 0.116

5

\(-\) 0.181

\(-\) 0.683

6

\(-\) 0.030

0.073

It is shown that there is a significant difference in the integrated field quality between the two methods for BEPCII 67B dipole magnet. The calculated integrated field uniformity is only \(1\times 10^{-4}\) using the traditional method, but it is as large as \(6\times 10^{-4}\) in the improved method. In addition, there is also a large difference in the multipole field coefficients between the two methods.

Figure 8 shows that the integrated field uniformity in the improved method is very close to the one calculated along particle trajectory in OPERA-3D, and the difference is smaller than \(3 \times 10 ^{-5}\). The field uniformity in the improved method represents the accurate theoretical integrated field quality in the BEPCII 67B dipole magnet. In Table 2, it is shown that there is an integrated quadrupole field component in BEPCII 67B dipole magnet, which will cause the betatron tune shift of the accelerator. For a total of 40 dipole magnets in BEPCII, the induced betatron tune shift in the horizontal direction is smaller than 0.005. The multipole field contents are still within the specification of less than \(5\times 10^{-4}\).

The improved method can be modified to further improve the accuracy of the field quality. For example, the field integration path outside the magnet iron core can be modified to be a straight line which is tangential to the arc inside the magnet. Then, the integrated field distribution can be closer to the one calculated along particle trajectory. However, this modified version of the improved method makes the field measurement process complicated. Since the difference in the integrated field quality between the improved method and the most accurate one calculated along particle trajectory is already very small, the improved method is accurate enough for actual application.

Conclusion

The traditional method of field quality calculation in straight dipole magnets is improved. The improved method is applied to the CSNS LR-BB straight dipole magnets and the existing straight 67B dipole magnet in BEPCII storage ring to illustrate the field quality difference between the improved and traditional methods. The discrepancy in integrated field quality between the two methods cannot be neglected. The field quality in the improved method is very close to the one calculated along real beam paths, so it is accurate enough for practical use in a straight dipole magnet. The method can be applied to future straight dipole magnets.

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Copyright information

© Institute of High Energy Physics, Chinese Academy of Sciences; China Nuclear Electronics and Nuclear Detection Society and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Particle Acceleration Physics and Technology, Institute of High Energy PhysicsChinese Academy of SciencesBeijingChina
  2. 2.University of Chinese Academy of ScienceBeijingChina
  3. 3.Dongguan Campus, Institute of High Energy PhysicsChinese Academy of SciencesDongguanChina

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