# Accurate calculation of field quality in conventional straight dipole magnets

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## 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.

## 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.

*A*) is offset from the nominal magnet centerline by half the beam sagitta, which is the same as that in the traditional method [5].

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

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 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.

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 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.

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

| \({b}_{n}\) 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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