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Analysis of magnetic field harmonics change due to manufacturing error in air-core HTS quadruple magnet
Accepted Manuscript Analysis of Magnetic Field Harmonics Change due to Manufacturing Error in Air-core HTS Quadruple Magnet Junseong Kim, Geonwoo Baek, Jeyull Lee, Yojong Choi, Seunghak Han, Seunghyun Song, Tae Kuk Ko PII: DOI: Reference:
S0011-2275(17)30441-1 https://doi.org/10.1016/j.cryogenics.2018.07.002 JCRY 2831
To appear in:
Cryogenics
Received Date: Revised Date: Accepted Date:
15 December 2017 6 July 2018 9 July 2018
Please cite this article as: Kim, J., Baek, G., Lee, J., Choi, Y., Han, S., Song, S., Ko, T.K., Analysis of Magnetic Field Harmonics Change due to Manufacturing Error in Air-core HTS Quadruple Magnet, Cryogenics (2018), doi: https://doi.org/10.1016/j.cryogenics.2018.07.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of Magnetic Field Harmonics Change due to Manufacturing Error in Air-core HTS Quadruple Magnet
Junseong Kim, Geonwoo Baek, Jeyull Lee, Yojong Choi, Seunghak Han, Seunghyun Song and Tae Kuk Ko Department of Electrical and Electronic Engineering, Yonsei University 50, Yonsei-ro, Seodaemun-gu, Seoul, Republic of Korea (e-mail: [email protected])
Abstract - An high temperature superconducting (HTS) quadruple magnet used to focus and defocus beams in particle accelerator has been studied for overcoming the disadvantages of low temperature superconducting (LTS) quadruple magnet. LTS quadruple magnet is hard to withstand radiation and heat load at hot cell because of small thermal margin of LTS, and has a nonlinear magnetic characteristic due to iron-core. The thermal margin is improved by changing the LTS to HTS. The magnetic field components are linear according to the operation current by removing the iron-core. Therefore, the air-core HTS quadruple magnet was designed in the past research. Field qualities that determine the performance of the quadruple magnet are composed of field gradient, field uniformity and effective length, and those of air-core HTS quadruple magnet can be achieved using harmonic matching (HM) method. However, when the quadruple magnet is fabricated, parameters of the fabricated magnet are different from designed parameters due to the manufacturing errors. This manufacturing error, REBCO conductor’s dimensional variation of thickness and width, and human errors during winding and assembling, could crucially influence to the field qualities of the quadruple magnet. In this paper, we present a draft design of air-core HTS quadruple magnet and describes an analytical method to predict change of field gradient, uniformity and effective length caused by manufacturing errors and 3) an evaluation of the permissible manufacturing error for an air-core HTS quadruple magnet. Index terms – Manufacturing error, Quadrupole, Quadruple magnet, HTS magnet, Harmonic matching method.
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I.
INTRODUCTION
Currently, many accelerator technologies have been developed for heavy-energy beams. One of the various studies is development of high-temperature superconducting quadruple magnets [1]. In general, a quadruple magnet was fabricated using low-temperature superconductor (LTS). However, there is a high radiation region and a neutron radiation heat load in In-flight Fragment (IF) pre-separator of the heavy-ion accelerator. The LTS is not suitable for this part. Therefore, studies about making a quadruple magnet using HTS for withstanding the high heat load are ongoing [2], [3]. The quadruple magnets are used to focus and defocus the ion-beam. Thus this magnet is required a specific shape of magnetic field. In general, this shape of magnetic field has been obtained by using an iron core with a machined surface. So, typical quadruple magnets have iron yoke. For the reasons of mentioned above, the researches about to make the quadruple magnet using iron core and HTS were conducted. Fig. 1 (a) represents schematic of the ironcore HTS quadruple magnet. However, as LTS was replaced with HTS, it was possible to fabricate the quadruple magnet for the heavy ion accelerator even if the iron core was removed. [4] Fig. 1 (b) represents schematic of the air-core HTS quadruple magnet. The study about design and specification characteristic of the air-core HTS quadruple magnet which is removal the iron yoke have been carried out in the previous study [4]. In recently, researches have been conducted on the fabrication of an air-core solenoid-type magnet using HTS. In many studies, the HTS magnets are fabricated differently from the initial design because of manufacturing errors necessarily occurring in actual production [5]. These errors are also considered to affect the magnetic field quality [6]. Therefore, it is necessary to study the change of field quality due to the manufacturing error before manufacturing the air-core HTS quadruple magnet. There are two types of manufacturing error included in the magnet. First, there are width, thickness of HTS tape and winding errors caused by the characteristics of HTS wire. Because the width and thickness error of the HTS tape is larger than that of the LTS, these errors should be considered [4]. Second, there are manufacturing magnet errors such as bobbin, spacer and distortion of axis. These errors are controllable during fabricating the magnet. These manufacturing errors are represented in Fig. 2 and 3. Therefore, in this study, we analyze the influence of the manufacturing error on the field quality by simulation to confirm the feasibility of 12.1[T/m] air-core HTS quadruple magnet which was designed in the past research [4], [7].
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II. THEORY A. Cylindrical harmonic coefficients To describe the magnetic field in good field region of quadruple magnet, it is conventional method to represent the magnetic field in terms of harmonics. The steady state magnetic field is expressed by Maxwell’s equation as follows [8].
(1) Where B, H, J is the magnetic field density, magnetic field intensity and current density, respectively. Under the condition that there is no current source and free space, the magnetic flux density satisfies the following condition.
(2) Also, magnetic field intensity can be expressed by gradient of a scalar potential, V, as follows. (3) Substituting (2) and (3) into (1), we can be obtained equation as follows. (4) The equation (4) satisfies the Laplace equation. Based on the solution of Laplace equation of cylindrical coordinates, the magnetic flux density is obtained as follows [4].
(5)
Where Bρ and Bφ are the magnetic flux densities for the radial and azimuthal direction, respectively. Bρn and Bφn are the multipole field components of Bρ and Bφ, respectively. The ρ, φ and z are the radius, azimuthal angle and longitudinal direction in the cylindrical coordinates, respectively, and n is a non-negative integer of harmonic components. In general in case of long magnets, Bρn and Bφn are regarded as the same value [9]. Therefore, magnetic field characteristics are expressed by using only radial components. The magnetic field is represented by Fourier series expansion of radial field components, equation (5). The Field quality of quadruple magnet can be evaluated by these equation. Therefore, only the radial component will be considered in this paper. 3
Firstly, the 2-nd component, Bρ2, is the main magnetic field in the quadruple magnet. The reference radius, ρ0, is good field region radius, and Bρ2(ρ0,z=0)/ρ0 is called the field gradient. Secondly, the uniformities, U, of magnetic field in quadruple magnet are represented as follows.
(6)
In equation (6), both of equations are uniformity in two and three dimensions, respectively. The major field inhomogeneity components are 6-th component, and 10-th, 14-th, and so on are the field inhomogeneity components [4]. In three dimensional, the effective length, Leff is a one of the factor of field quality. The effective length means the actual length of beam focusing or defocusing length, and is expressed as (7)
B. Harmonic matching method In the case of the air-core quadrupole, unlike the iron-core magnet, all magnetic fields are induced only by the coil since the iron core is removed. To satisfy the desired field gradient, only one coil is placed on each axis in the iron-core magnet, whereas an air-core magnet requires three or more coils in each axis. Field gradient can be satisfied by increasing the number of coils, but this method is not enough to satisfy uniformity. Therefore, to improve the field uniformity, the change of the field harmonic components according to the shape and arrangement of coil was examined. In Fig. 4, 6-th and 10-th harmonic components have positive and negative sign according to the certain coil position and coil size [9], [10]. By this situation, it is possible to improve the field gradient and uniformity by matching the harmonic components between the coils by determining the proper position and size of the coils. This method is called harmonic matching method that is analyzing the changes of harmonic components according to design parameters and, setting appropriate design parameters and satisfying field gradient and uniformity. In previous studies, we designed a 12.1[T/m] aircore HTS quadruple magnet using this method.
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III. SIMULATION A. Specification of Quadruple magnet In the previous paper, the air-core HTS quadruple magnet designed to have a 12.1[T/m] or more field gradient, 550mm or more effective length and 0.1% or less uniformity of 0.1% using the harmonic matching method. Fig. 5 shows the designed quadruple magnets, DPC, Racetrack coils and the design parameters. The thickness and width of HTS tape is 0.12 mm and 12mm. The detailed design values of the magnet are represented in Table 1. In Fig. 1 (b), three double pancake coils (DPCs) were placed on each axis, total 12 DPCs in the magnet, and 7 km wire is used. Ideally designed magnets have a 12.1[T/m] field gradient at 381.052A operating current. The field inhomogeneity components of 6-th and 10-th are much smaller than the set values. The field characteristic values of the magnet are represented in Table 2. The designed magnet is set as an ideal model in simulation.
B. Manufacturing error The manufacturing errors are inevitably included during making a magnet. The errors are induced by HTS tape and fabricating of magnet. In Fig. 2 and 3, there are error of winding, HTS width, thickness, spacer, bobbin and axis. In the simulation, all errors are considered except for error of spacer and bobbin because these errors are relatively controllable. First, the thickness and width error in HTS wire are included in the wire manufacturing process and cannot be controlled during manufacturing the magnet. The magnitude of this error is very large compared to that of LTS. Therefore, the field inhomogeneity caused by this error should be considered. Second, the winding error is occurred during the HTS wire is wound on the coil. When winding the HTS wire to the coil that have a certain amount of space. At that time, HTS tape can move in this range. So the tapes are not aligned in a reference line and slightly deviate from the reference. This error is called winding error. The last error is an axis error which is occurred during making the magnet and greatly affects the symmetry of the quadruple magnet. This error is represented the distortion in the initial reference axis. Three kinds of errors are considered in the simulation, and the range of the errors are represented in Talbe3. These errors are set to the ideal magnet using the uniform random variable.
C. Sequence of simulation Simulation was performed using Matlab for calculate magnetic field according to the 5
manufacturing errors. Firstly, the air-core HTS quadruple magnet, which is mentioned above, ideally designed and not to include the manufacturing errors is set-up. Secondly, the manufacturing error was applied to the magnet. The manufacturing error, HTS thickness, width, winding, and axis error, was set-up in the magnet each case. Thirdly, the magnetic field of the magnet with the error set was obtained from the specific cylinder surface. In Fig. 6, the reference radius, which is the radius of the cylinder, is 105mm and 68 points on the circle. Also, the height of the cylinder is 1500mm, which is located from the center of the magnet to the end of magnet through the direction of beam. The height was also divided into regular intervals. Finally, the field harmonic components were calculated using the magnetic field obtained at that points. This simulation was conducted hundreds of times for each error case.
IV. RESULTS AND DISCUSSION A. Simulation Results The error of winding, HTS thickness, width, and axis angle were applied to the ideal magnet and simulated for each case. In order to analyze the effect of the errors on the field quality, firstly, the harmonic components obtained from the each error case that components were normalized by the harmonic components of the ideal magnet. The mean and standard deviations value of field gradient and effective length are summarized in Table 4. The values show no significant change for all error cases. The Uniformity of 6-th, 10-th and 14-th are summarized in Table 5, 6 and 7, respectively. The reference value is 100%
based on the
ideal case. In case of error of HTS width and winding, the uniformities are little changed less than 0.3%. On the other hand, the axis angle error is changed about 5% in 6-th uniformity. Also, the maximum change of case of HTS thickness error is almost 1.7%. The all average values of harmonic components to the each error case except for 2-nd are represented in Fig. 7. This result shows that the harmonic components of even-numbered terms change more than the odd-numbered terms. These changes are caused by the distortion of the symmetry of quadruple magnet because manufacturing error is applied. The values of uniformities that are not the normalized is represented in Fig. 8. The all error cases except for the axis angle error have similar uniformities compared with the ideal case.
B. Discussion In this study, the change of the field quality according to the manufacturing error that 6
occurred during the manufacturing process of the air-core HTS quadruple magnet was analyzed through the Matlab simulation. There are four kinds of manufacturing errors which are considered in simulation: error of HTS width, thickness, winding, and axis angle. Each case of these errors was applied to the ideal air-core magnet, then simulated and harmonic component is analyzed. By the analyses in this paper, conclusion may be that: First, the effects of manufacturing error to the field gradient and effective length were analyzed. These components have the greatest effect on axis angle error, and the effect is smaller in order of error of HTS thickness, winding, and HTS width. However, the all normalized mean and standard deviation values of error cases obtained from the simulation are very small. Therefore, the manufacturing errors do not greatly affect the gradient and effective length, and the magnitude of errors are appropriate in terms of gradient and effective length. Second, the inhomogeneity components, magnitude of 6-th, 10-th, and 14-th uniformity, were analyzed according to the each error. The effect of manufacturing error to these components is similar with case of gradient and effective length. Third, we are analyzed the components that were not considered in an ideally designed quadruple magnet. Because ideally designed quadruple magnet have symmetric, only 2-nd, 6th, 10-th, and 14-th and so on harmonic components are considered and the remaining components are neglected. However, according to result of the simulation, the symmetry is distorted due to the manufacturing error, so the magnitude of harmonic components which are not initially considered is drastically increased. These results are represented in Fig. 7. In Fig. 8, for a more realistic comparison analysis, the uniformity value was calculated which is not the normalized value. As a result, the magnitude of asymmetric components are smaller than that of 6-th component. Also, asymmetry occurs in all error cases, but there is little change except axis angle error case. Finally, the manufacturing errors set by this simulation little affect to the main harmonic components, like 2-nd, 6-th, 10-th, and 14-th. However, it was investigated that the harmonic components which was not initially considered are increased because of distortion of symmetry. Although it is a small change, there is no guarantee that it will have a small impact on the actual beam. Therefore, further research is needed to investigated how these components affect beam movement.
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ACKNOWLEDGEMENTS This research was supported by Korea Electric Power Corporation. (Grant number: R17XA05_32). This work was supported by “Human Resources Program in Energy Technology” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea. (No. 20164030201100)
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R. Gupta, M. Anerella, J. Cozzolino, G. Ganetis, A. Ghosh, G. Greene, W. Sampson, Y. Shiroyanagi, P. Wanderer, and A. Zeller, “Second generation HTS quadrupole for FRIB,” IEEE Trans. Appl. Supercond., vol. 21, no. 3 PART 2, pp. 1888–1891, 2011.
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J. Cozzolino, M. Anerella, A. Ghosh, R. Gupta, W. Sampson, Y. Shiroyanagi, and P. Wanderer, “Engineering Design of HTS Quadrupole for FRIB,” pp. 1124–1126, 2011.
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Z. Zhang, S. Wei, and S. Lee, “Design of an Air-Core HTS quadruple triplet for a heavy ion accelerator,” vol. 18, no. 4, pp. 35–39, 2016.
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S. Y. Hahn, M. C. Ahn, E. S. Bobrov, J. Bascuñn, and Y. Iwasa, “An analytical technique to elucidate field impurities from manufacturing uncertainties of an double pancake type hts insert for high field lts/hts nmr magnets,” IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 2281–2284, 2009.
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J. Kim, W. S. Lee, J. Kim, S. Song, and S. Nam, “A simulation study on the variation of virtual NMR signals by winding , bobbin , spacer error of HTS magnet,” vol. 18, no. 3, pp. 21–24, 2016.
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S. Russenschuck, Field Computation for Accelerator Magnets. Weinheim, Germany: Wiley-VCH, 2010.
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Z. Zhang, S. Lee, H. C. Jo, D. G. Kim, and J. Kim, “Harmonic analysis and field quality improvement of an HTS quadrupole magnet for a heavy ion accelerator,” IEEE Trans. Appl. Supercond., vol. 26, no. 4, pp. 21–24, 2016.
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Fig. 1. Schematic of (a) iron-core and (b) air-core quadruple magnet. Fig. 2. Schematic of manufacturing errors in double pancake coil. Fig. 3. Schematic of axis error. Fig. 4. Field harmonic component according to the coil width at distance from origin of magnet to center of coil, Rc, (a) 150mm, (b) 170mm, (c) 190mm, (d) 210mm, (e) 230mm and (f) 250mm. Fig. 5. Schematic of (a) air-core quadruple magnet, (b) DPC placed in a quadrant (c) one racetrack coil in DPC. Fig. 6. Mapping point on the surface of cylinder. Fig. 7. Normalized difference value of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case. Fig. 8. Magnitude of uniformity of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case. TABLE I Specifications of air-core quadruple magnet. TABLE Harmonic components of ideal air-core quadruple magnet. TABLE III Reference value and range of error. TABLE IV Gradient and effective length of each error case. TABLE V 6-th uniformity of each error case. TABLE VI 10-th uniformity of each error case. TABLE VII 14-th uniformity of each error case.
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Fig. 1. Schematic of (a) iron-core and (b) air-core quadruple magnet.
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Fig. 2. Schematic of manufacturing errors in double pancake coil.
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Fig. 3. Schematic of axis angle error.
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Fig. 4. Field harmonic component according to the coil width at distance from origin of magnet to center of coil, Rc, (a) 150mm, (b) 170mm, (c) 190mm, (d) 210mm, (e) 230mm and (f) 250mm
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Fig. 5. Schematic of (a) air-core quadruple magnet, (b) DPC placed in a quadrant (c) one racetrack coil in DPC.
15
Fig. 6. Mapping point on the surface of cylinder.
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Fig. 7. Normalized difference value of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case.
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Fig. 8. Magnitude of uniformity of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case.
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TABLE I Specifications of air-core quadruple magnet
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TABLE Harmonic components of ideal air-core quadruple magnet
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TABLE III Reference value and range of error
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TABLE IV Gradient and effective length of each error case
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TABLE V 6-th uniformity of each error case
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TABLE VI 10-th uniformity of each error case
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TABLE VII 14-th uniformity of each error case
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TABLE I Reference value and range of error
Properties
Reference value
Axis angle error
45°, 135°, 225°, 315°
Bobbin bore error
Depend on the design
Range of error 1°
HTS thickness error
0.12 mm
0.005 mm
HTS width error
12.0 mm
0.15 mm
Spacer error
2 mm
mm
Winding error
0 mm
+0.2 mm
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TABLE II Specifications of the designed quadruple magnet Properties
value
HTS width & thickness (Ww & Wt)
12mm, 0.1mm
SUS tape width & thickness
12mm, 0.12mm
Operating Current
381.052 A
Number of turns each coil
200
Number of DPC each axis
3
Distance from origin to center of coil (Rc)
169, 197, 230 (mm)
Outer width of lower DPC (Wi)
302, 252, 206 (mm)
Outer width of upper DPC (Wo)
330, 252, 206 (mm)
Radius of curve in coil (Rcw)
39 mm
Length of Coil (Lc)
621 mm
Gap between Low and upper coil (d) Total length of HTS
2 mm About 7 km
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TABLE III Characteristic of magnetic field of no-error model Properties
value
G
12.1 T/m -2
Uρ6
2.64 x 10 %
Uρ10
7.76 x 10 %
Uρ14
0.37 x 10 %
Leff
555.4 mm
-2
-2
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TABLE IV Results of gradient and effective length each error case Gradient Error case
Absolute value (T/m)
Effective length Normalized value (%)
Absolute value (mm)
Normalized value (%)
Average
S.D -5 (10 )
Average -4 (10 )
Average
S.D -6 (10 )
Average -5 (10 )
No-error
12.0927
*
*
555.4036
*
*
Axis
12.0901
0.90458
210.27
555.4039
2.3933
4.1304
Bobbin
12.0928
234.52
7.2811
555.4046
106.51
16.703
HTS thickness
12.0927
21.105
1.1156
555.4044
9.6228
13.376
HTS width
12.0927
7.1016
0.3067
555.4037
0.20172
0.70382
Spacer
12.0928
90.586
10.536
555.4040
1.2532
6.6823
Winding
12.0933
4.7118
50.607
555.4058
0.17330
39.073
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TABLE V Results of 6th uniformity of each error case 6-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
2.6424
*
*
Axis
2.6389
9.6722
132.34
Bobbin
2.6688
589.49
999.20
HTS thickness
2.6443
36.115
69.299
HTS width
2.6445
8.1190
76.648
Spacer
2.6646
66.175
838.66
Winding
2.7537
5.6709
4211.9
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TABLE VI Results of 10th uniformity of each error case 10-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
7.7566
*
*
Axis
7.7176
14.005
504.55
Bobbin
7.7600
57.633
42.243
HTS thickness
7.7565
3.2932
2.7088
HTS width
7.7567
0.7402
0.1571
Spacer
7.7556
4.9811
14.244
Winding
7.7548
0.4335
24.968
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TABLE VII Results of 14th uniformity each error case 14-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
0.3727
*
*
Axis
0.3690
1.3384
1007.1
Bobbin
0.3735
7.1834
178.6
HTS thickness
0.3728
0.3661
0.71372
HTS width
0.3728
0.0408
0.61206
Spacer
0.3728
0.1181
4.3606
Winding
0.3733
0.0281
132.3
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TABLE VIII Results of sum of uniformities each error model Sum of uniformities (%) Error case
Average -1 (10 )
S.D -3 (10 )
Normalized value
No-error
1.0772
*
*
Axis
11.4907
719.15
966.73
Bobbin
1.7688
33.984
64.205
HTS thickness
1.1347
3.0599
5.3341
HTS width
1.0849
0.4221
0.7140
Spacer
1.1802
4.4921
9.5657
Winding
1.0940
0.3047
1.5562
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Highlights Manufacturing errors are inevitable occurred during making a magnet. HTS tape has a larger error in thickness and width than that of LTS. Investigate the change of field quality of air-core quadruple magnet according to manufacturing error. Gradient, effective length and uniformity of 6-th, 10-th and 14-th harmonic component are little affected by manufacturing error. It was investigated that the harmonic components which was not initially considered are increased because of distortion of symmetry caused by manufacturing errors.
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S0011-2275(17)30441-1 https://doi.org/10.1016/j.cryogenics.2018.07.002 JCRY 2831
To appear in:
Cryogenics
Received Date: Revised Date: Accepted Date:
15 December 2017 6 July 2018 9 July 2018
Please cite this article as: Kim, J., Baek, G., Lee, J., Choi, Y., Han, S., Song, S., Ko, T.K., Analysis of Magnetic Field Harmonics Change due to Manufacturing Error in Air-core HTS Quadruple Magnet, Cryogenics (2018), doi: https://doi.org/10.1016/j.cryogenics.2018.07.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of Magnetic Field Harmonics Change due to Manufacturing Error in Air-core HTS Quadruple Magnet
Junseong Kim, Geonwoo Baek, Jeyull Lee, Yojong Choi, Seunghak Han, Seunghyun Song and Tae Kuk Ko Department of Electrical and Electronic Engineering, Yonsei University 50, Yonsei-ro, Seodaemun-gu, Seoul, Republic of Korea (e-mail: [email protected])
Abstract - An high temperature superconducting (HTS) quadruple magnet used to focus and defocus beams in particle accelerator has been studied for overcoming the disadvantages of low temperature superconducting (LTS) quadruple magnet. LTS quadruple magnet is hard to withstand radiation and heat load at hot cell because of small thermal margin of LTS, and has a nonlinear magnetic characteristic due to iron-core. The thermal margin is improved by changing the LTS to HTS. The magnetic field components are linear according to the operation current by removing the iron-core. Therefore, the air-core HTS quadruple magnet was designed in the past research. Field qualities that determine the performance of the quadruple magnet are composed of field gradient, field uniformity and effective length, and those of air-core HTS quadruple magnet can be achieved using harmonic matching (HM) method. However, when the quadruple magnet is fabricated, parameters of the fabricated magnet are different from designed parameters due to the manufacturing errors. This manufacturing error, REBCO conductor’s dimensional variation of thickness and width, and human errors during winding and assembling, could crucially influence to the field qualities of the quadruple magnet. In this paper, we present a draft design of air-core HTS quadruple magnet and describes an analytical method to predict change of field gradient, uniformity and effective length caused by manufacturing errors and 3) an evaluation of the permissible manufacturing error for an air-core HTS quadruple magnet. Index terms – Manufacturing error, Quadrupole, Quadruple magnet, HTS magnet, Harmonic matching method.
1
I.
INTRODUCTION
Currently, many accelerator technologies have been developed for heavy-energy beams. One of the various studies is development of high-temperature superconducting quadruple magnets [1]. In general, a quadruple magnet was fabricated using low-temperature superconductor (LTS). However, there is a high radiation region and a neutron radiation heat load in In-flight Fragment (IF) pre-separator of the heavy-ion accelerator. The LTS is not suitable for this part. Therefore, studies about making a quadruple magnet using HTS for withstanding the high heat load are ongoing [2], [3]. The quadruple magnets are used to focus and defocus the ion-beam. Thus this magnet is required a specific shape of magnetic field. In general, this shape of magnetic field has been obtained by using an iron core with a machined surface. So, typical quadruple magnets have iron yoke. For the reasons of mentioned above, the researches about to make the quadruple magnet using iron core and HTS were conducted. Fig. 1 (a) represents schematic of the ironcore HTS quadruple magnet. However, as LTS was replaced with HTS, it was possible to fabricate the quadruple magnet for the heavy ion accelerator even if the iron core was removed. [4] Fig. 1 (b) represents schematic of the air-core HTS quadruple magnet. The study about design and specification characteristic of the air-core HTS quadruple magnet which is removal the iron yoke have been carried out in the previous study [4]. In recently, researches have been conducted on the fabrication of an air-core solenoid-type magnet using HTS. In many studies, the HTS magnets are fabricated differently from the initial design because of manufacturing errors necessarily occurring in actual production [5]. These errors are also considered to affect the magnetic field quality [6]. Therefore, it is necessary to study the change of field quality due to the manufacturing error before manufacturing the air-core HTS quadruple magnet. There are two types of manufacturing error included in the magnet. First, there are width, thickness of HTS tape and winding errors caused by the characteristics of HTS wire. Because the width and thickness error of the HTS tape is larger than that of the LTS, these errors should be considered [4]. Second, there are manufacturing magnet errors such as bobbin, spacer and distortion of axis. These errors are controllable during fabricating the magnet. These manufacturing errors are represented in Fig. 2 and 3. Therefore, in this study, we analyze the influence of the manufacturing error on the field quality by simulation to confirm the feasibility of 12.1[T/m] air-core HTS quadruple magnet which was designed in the past research [4], [7].
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II. THEORY A. Cylindrical harmonic coefficients To describe the magnetic field in good field region of quadruple magnet, it is conventional method to represent the magnetic field in terms of harmonics. The steady state magnetic field is expressed by Maxwell’s equation as follows [8].
(1) Where B, H, J is the magnetic field density, magnetic field intensity and current density, respectively. Under the condition that there is no current source and free space, the magnetic flux density satisfies the following condition.
(2) Also, magnetic field intensity can be expressed by gradient of a scalar potential, V, as follows. (3) Substituting (2) and (3) into (1), we can be obtained equation as follows. (4) The equation (4) satisfies the Laplace equation. Based on the solution of Laplace equation of cylindrical coordinates, the magnetic flux density is obtained as follows [4].
(5)
Where Bρ and Bφ are the magnetic flux densities for the radial and azimuthal direction, respectively. Bρn and Bφn are the multipole field components of Bρ and Bφ, respectively. The ρ, φ and z are the radius, azimuthal angle and longitudinal direction in the cylindrical coordinates, respectively, and n is a non-negative integer of harmonic components. In general in case of long magnets, Bρn and Bφn are regarded as the same value [9]. Therefore, magnetic field characteristics are expressed by using only radial components. The magnetic field is represented by Fourier series expansion of radial field components, equation (5). The Field quality of quadruple magnet can be evaluated by these equation. Therefore, only the radial component will be considered in this paper. 3
Firstly, the 2-nd component, Bρ2, is the main magnetic field in the quadruple magnet. The reference radius, ρ0, is good field region radius, and Bρ2(ρ0,z=0)/ρ0 is called the field gradient. Secondly, the uniformities, U, of magnetic field in quadruple magnet are represented as follows.
(6)
In equation (6), both of equations are uniformity in two and three dimensions, respectively. The major field inhomogeneity components are 6-th component, and 10-th, 14-th, and so on are the field inhomogeneity components [4]. In three dimensional, the effective length, Leff is a one of the factor of field quality. The effective length means the actual length of beam focusing or defocusing length, and is expressed as (7)
B. Harmonic matching method In the case of the air-core quadrupole, unlike the iron-core magnet, all magnetic fields are induced only by the coil since the iron core is removed. To satisfy the desired field gradient, only one coil is placed on each axis in the iron-core magnet, whereas an air-core magnet requires three or more coils in each axis. Field gradient can be satisfied by increasing the number of coils, but this method is not enough to satisfy uniformity. Therefore, to improve the field uniformity, the change of the field harmonic components according to the shape and arrangement of coil was examined. In Fig. 4, 6-th and 10-th harmonic components have positive and negative sign according to the certain coil position and coil size [9], [10]. By this situation, it is possible to improve the field gradient and uniformity by matching the harmonic components between the coils by determining the proper position and size of the coils. This method is called harmonic matching method that is analyzing the changes of harmonic components according to design parameters and, setting appropriate design parameters and satisfying field gradient and uniformity. In previous studies, we designed a 12.1[T/m] aircore HTS quadruple magnet using this method.
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III. SIMULATION A. Specification of Quadruple magnet In the previous paper, the air-core HTS quadruple magnet designed to have a 12.1[T/m] or more field gradient, 550mm or more effective length and 0.1% or less uniformity of 0.1% using the harmonic matching method. Fig. 5 shows the designed quadruple magnets, DPC, Racetrack coils and the design parameters. The thickness and width of HTS tape is 0.12 mm and 12mm. The detailed design values of the magnet are represented in Table 1. In Fig. 1 (b), three double pancake coils (DPCs) were placed on each axis, total 12 DPCs in the magnet, and 7 km wire is used. Ideally designed magnets have a 12.1[T/m] field gradient at 381.052A operating current. The field inhomogeneity components of 6-th and 10-th are much smaller than the set values. The field characteristic values of the magnet are represented in Table 2. The designed magnet is set as an ideal model in simulation.
B. Manufacturing error The manufacturing errors are inevitably included during making a magnet. The errors are induced by HTS tape and fabricating of magnet. In Fig. 2 and 3, there are error of winding, HTS width, thickness, spacer, bobbin and axis. In the simulation, all errors are considered except for error of spacer and bobbin because these errors are relatively controllable. First, the thickness and width error in HTS wire are included in the wire manufacturing process and cannot be controlled during manufacturing the magnet. The magnitude of this error is very large compared to that of LTS. Therefore, the field inhomogeneity caused by this error should be considered. Second, the winding error is occurred during the HTS wire is wound on the coil. When winding the HTS wire to the coil that have a certain amount of space. At that time, HTS tape can move in this range. So the tapes are not aligned in a reference line and slightly deviate from the reference. This error is called winding error. The last error is an axis error which is occurred during making the magnet and greatly affects the symmetry of the quadruple magnet. This error is represented the distortion in the initial reference axis. Three kinds of errors are considered in the simulation, and the range of the errors are represented in Talbe3. These errors are set to the ideal magnet using the uniform random variable.
C. Sequence of simulation Simulation was performed using Matlab for calculate magnetic field according to the 5
manufacturing errors. Firstly, the air-core HTS quadruple magnet, which is mentioned above, ideally designed and not to include the manufacturing errors is set-up. Secondly, the manufacturing error was applied to the magnet. The manufacturing error, HTS thickness, width, winding, and axis error, was set-up in the magnet each case. Thirdly, the magnetic field of the magnet with the error set was obtained from the specific cylinder surface. In Fig. 6, the reference radius, which is the radius of the cylinder, is 105mm and 68 points on the circle. Also, the height of the cylinder is 1500mm, which is located from the center of the magnet to the end of magnet through the direction of beam. The height was also divided into regular intervals. Finally, the field harmonic components were calculated using the magnetic field obtained at that points. This simulation was conducted hundreds of times for each error case.
IV. RESULTS AND DISCUSSION A. Simulation Results The error of winding, HTS thickness, width, and axis angle were applied to the ideal magnet and simulated for each case. In order to analyze the effect of the errors on the field quality, firstly, the harmonic components obtained from the each error case that components were normalized by the harmonic components of the ideal magnet. The mean and standard deviations value of field gradient and effective length are summarized in Table 4. The values show no significant change for all error cases. The Uniformity of 6-th, 10-th and 14-th are summarized in Table 5, 6 and 7, respectively. The reference value is 100%
based on the
ideal case. In case of error of HTS width and winding, the uniformities are little changed less than 0.3%. On the other hand, the axis angle error is changed about 5% in 6-th uniformity. Also, the maximum change of case of HTS thickness error is almost 1.7%. The all average values of harmonic components to the each error case except for 2-nd are represented in Fig. 7. This result shows that the harmonic components of even-numbered terms change more than the odd-numbered terms. These changes are caused by the distortion of the symmetry of quadruple magnet because manufacturing error is applied. The values of uniformities that are not the normalized is represented in Fig. 8. The all error cases except for the axis angle error have similar uniformities compared with the ideal case.
B. Discussion In this study, the change of the field quality according to the manufacturing error that 6
occurred during the manufacturing process of the air-core HTS quadruple magnet was analyzed through the Matlab simulation. There are four kinds of manufacturing errors which are considered in simulation: error of HTS width, thickness, winding, and axis angle. Each case of these errors was applied to the ideal air-core magnet, then simulated and harmonic component is analyzed. By the analyses in this paper, conclusion may be that: First, the effects of manufacturing error to the field gradient and effective length were analyzed. These components have the greatest effect on axis angle error, and the effect is smaller in order of error of HTS thickness, winding, and HTS width. However, the all normalized mean and standard deviation values of error cases obtained from the simulation are very small. Therefore, the manufacturing errors do not greatly affect the gradient and effective length, and the magnitude of errors are appropriate in terms of gradient and effective length. Second, the inhomogeneity components, magnitude of 6-th, 10-th, and 14-th uniformity, were analyzed according to the each error. The effect of manufacturing error to these components is similar with case of gradient and effective length. Third, we are analyzed the components that were not considered in an ideally designed quadruple magnet. Because ideally designed quadruple magnet have symmetric, only 2-nd, 6th, 10-th, and 14-th and so on harmonic components are considered and the remaining components are neglected. However, according to result of the simulation, the symmetry is distorted due to the manufacturing error, so the magnitude of harmonic components which are not initially considered is drastically increased. These results are represented in Fig. 7. In Fig. 8, for a more realistic comparison analysis, the uniformity value was calculated which is not the normalized value. As a result, the magnitude of asymmetric components are smaller than that of 6-th component. Also, asymmetry occurs in all error cases, but there is little change except axis angle error case. Finally, the manufacturing errors set by this simulation little affect to the main harmonic components, like 2-nd, 6-th, 10-th, and 14-th. However, it was investigated that the harmonic components which was not initially considered are increased because of distortion of symmetry. Although it is a small change, there is no guarantee that it will have a small impact on the actual beam. Therefore, further research is needed to investigated how these components affect beam movement.
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ACKNOWLEDGEMENTS This research was supported by Korea Electric Power Corporation. (Grant number: R17XA05_32). This work was supported by “Human Resources Program in Energy Technology” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea. (No. 20164030201100)
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Z. Zhang, S. Wei, and S. Lee, “Design of an Air-Core HTS quadruple triplet for a heavy ion accelerator,” vol. 18, no. 4, pp. 35–39, 2016.
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S. Y. Hahn, M. C. Ahn, E. S. Bobrov, J. Bascuñn, and Y. Iwasa, “An analytical technique to elucidate field impurities from manufacturing uncertainties of an double pancake type hts insert for high field lts/hts nmr magnets,” IEEE Trans. Appl. Supercond., vol. 19, no. 3, pp. 2281–2284, 2009.
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S. Russenschuck, Field Computation for Accelerator Magnets. Weinheim, Germany: Wiley-VCH, 2010.
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Fig. 1. Schematic of (a) iron-core and (b) air-core quadruple magnet. Fig. 2. Schematic of manufacturing errors in double pancake coil. Fig. 3. Schematic of axis error. Fig. 4. Field harmonic component according to the coil width at distance from origin of magnet to center of coil, Rc, (a) 150mm, (b) 170mm, (c) 190mm, (d) 210mm, (e) 230mm and (f) 250mm. Fig. 5. Schematic of (a) air-core quadruple magnet, (b) DPC placed in a quadrant (c) one racetrack coil in DPC. Fig. 6. Mapping point on the surface of cylinder. Fig. 7. Normalized difference value of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case. Fig. 8. Magnitude of uniformity of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case. TABLE I Specifications of air-core quadruple magnet. TABLE Harmonic components of ideal air-core quadruple magnet. TABLE III Reference value and range of error. TABLE IV Gradient and effective length of each error case. TABLE V 6-th uniformity of each error case. TABLE VI 10-th uniformity of each error case. TABLE VII 14-th uniformity of each error case.
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Fig. 1. Schematic of (a) iron-core and (b) air-core quadruple magnet.
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Fig. 2. Schematic of manufacturing errors in double pancake coil.
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Fig. 3. Schematic of axis angle error.
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Fig. 4. Field harmonic component according to the coil width at distance from origin of magnet to center of coil, Rc, (a) 150mm, (b) 170mm, (c) 190mm, (d) 210mm, (e) 230mm and (f) 250mm
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Fig. 5. Schematic of (a) air-core quadruple magnet, (b) DPC placed in a quadrant (c) one racetrack coil in DPC.
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Fig. 6. Mapping point on the surface of cylinder.
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Fig. 7. Normalized difference value of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case.
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Fig. 8. Magnitude of uniformity of harmonic components about (a) Axis angle, (b) HTS thickness, (c) HTS width and (d) Winding error case.
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TABLE I Specifications of air-core quadruple magnet
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TABLE Harmonic components of ideal air-core quadruple magnet
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TABLE III Reference value and range of error
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TABLE IV Gradient and effective length of each error case
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TABLE V 6-th uniformity of each error case
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TABLE VI 10-th uniformity of each error case
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TABLE VII 14-th uniformity of each error case
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TABLE I Reference value and range of error
Properties
Reference value
Axis angle error
45°, 135°, 225°, 315°
Bobbin bore error
Depend on the design
Range of error 1°
HTS thickness error
0.12 mm
0.005 mm
HTS width error
12.0 mm
0.15 mm
Spacer error
2 mm
mm
Winding error
0 mm
+0.2 mm
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TABLE II Specifications of the designed quadruple magnet Properties
value
HTS width & thickness (Ww & Wt)
12mm, 0.1mm
SUS tape width & thickness
12mm, 0.12mm
Operating Current
381.052 A
Number of turns each coil
200
Number of DPC each axis
3
Distance from origin to center of coil (Rc)
169, 197, 230 (mm)
Outer width of lower DPC (Wi)
302, 252, 206 (mm)
Outer width of upper DPC (Wo)
330, 252, 206 (mm)
Radius of curve in coil (Rcw)
39 mm
Length of Coil (Lc)
621 mm
Gap between Low and upper coil (d) Total length of HTS
2 mm About 7 km
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TABLE III Characteristic of magnetic field of no-error model Properties
value
G
12.1 T/m -2
Uρ6
2.64 x 10 %
Uρ10
7.76 x 10 %
Uρ14
0.37 x 10 %
Leff
555.4 mm
-2
-2
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TABLE IV Results of gradient and effective length each error case Gradient Error case
Absolute value (T/m)
Effective length Normalized value (%)
Absolute value (mm)
Normalized value (%)
Average
S.D -5 (10 )
Average -4 (10 )
Average
S.D -6 (10 )
Average -5 (10 )
No-error
12.0927
*
*
555.4036
*
*
Axis
12.0901
0.90458
210.27
555.4039
2.3933
4.1304
Bobbin
12.0928
234.52
7.2811
555.4046
106.51
16.703
HTS thickness
12.0927
21.105
1.1156
555.4044
9.6228
13.376
HTS width
12.0927
7.1016
0.3067
555.4037
0.20172
0.70382
Spacer
12.0928
90.586
10.536
555.4040
1.2532
6.6823
Winding
12.0933
4.7118
50.607
555.4058
0.17330
39.073
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TABLE V Results of 6th uniformity of each error case 6-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
2.6424
*
*
Axis
2.6389
9.6722
132.34
Bobbin
2.6688
589.49
999.20
HTS thickness
2.6443
36.115
69.299
HTS width
2.6445
8.1190
76.648
Spacer
2.6646
66.175
838.66
Winding
2.7537
5.6709
4211.9
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TABLE VI Results of 10th uniformity of each error case 10-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
7.7566
*
*
Axis
7.7176
14.005
504.55
Bobbin
7.7600
57.633
42.243
HTS thickness
7.7565
3.2932
2.7088
HTS width
7.7567
0.7402
0.1571
Spacer
7.7556
4.9811
14.244
Winding
7.7548
0.4335
24.968
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TABLE VII Results of 14th uniformity each error case 14-th Uniformity (%) Error case
Average -2 (10 )
S.D -5 (10 )
Normalized -3 value (10 )
No-error
0.3727
*
*
Axis
0.3690
1.3384
1007.1
Bobbin
0.3735
7.1834
178.6
HTS thickness
0.3728
0.3661
0.71372
HTS width
0.3728
0.0408
0.61206
Spacer
0.3728
0.1181
4.3606
Winding
0.3733
0.0281
132.3
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TABLE VIII Results of sum of uniformities each error model Sum of uniformities (%) Error case
Average -1 (10 )
S.D -3 (10 )
Normalized value
No-error
1.0772
*
*
Axis
11.4907
719.15
966.73
Bobbin
1.7688
33.984
64.205
HTS thickness
1.1347
3.0599
5.3341
HTS width
1.0849
0.4221
0.7140
Spacer
1.1802
4.4921
9.5657
Winding
1.0940
0.3047
1.5562
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Highlights Manufacturing errors are inevitable occurred during making a magnet. HTS tape has a larger error in thickness and width than that of LTS. Investigate the change of field quality of air-core quadruple magnet according to manufacturing error. Gradient, effective length and uniformity of 6-th, 10-th and 14-th harmonic component are little affected by manufacturing error. It was investigated that the harmonic components which was not initially considered are increased because of distortion of symmetry caused by manufacturing errors.
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