Design Aspects of High-Field Block-Coil

Superconducting Dipole Magnets

E. I. Sfakianakis∗

August 31, 2006

Abstract

Even before the construction of the Large Hadron Collider at CERN is fin-

ished, ideas about the next step in high energy physics experiments are being

examined. The two main directions that are considered worth following are linear

lepton colliders and even bigger hadron storage rings. The latter would require

advances in dipole magnet technology in order to be considered a feasible solution

for the future of high energy physics. This study concentrates on design issues of

the steady-state and transient operation of block-coil dipole magnets in order to

optimize and evaluate their performance.

∗School of Electrical and Computer Engineering, National Technical University of Athens

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1 Introduction and Objectives

The purpose of this project is the study of an alternative approach to the LHC-designof superconducting dipole magnets, called block-coil configuration. The study is donefor a high-field magnet such as the ones that will be required in a future high energyhadron ring accelerator, for example an upgraded version of the LHC. Topics such asfield strength and quality in the steady state operation of the magnet will be considered.In addition to that, the transient behaviour of the magnet will be studied. The resultsobtained will be compared to the ones arising from a cos(θ) design with similar param-eters. At the end we will be able to state the extent to which the block-coil approachcan serve as a replacement for the current cos(θ) design philosophy in future high-fielddipoles for particle accelerators. The simulation and optimization of the dipoles has beencarried out using the ROXIE software package, developed by Stephan Russenschuck etal. at the AT-MEL-EM Section at CERN (for a full documentation see [2]).

2 Specifications

In order to prepare the next generation of high-field superconducting dipole magnets, anEU-funded program referred to as NED (Next European Dipole) has been launched (seefor example [3]). Inside this program salient specifications are stated. The supercon-ducting material that is considered the best solution at this time is Nb3Sn. Although itis more brittle and more strain sensitive than the material used at present ( Nb-Ti), ithas a higher current carrying capability and it can be used to produce the field required,while Nb-Ti cannot operate in the 10 T-15 T region.The cable used in this work is a Rutherford-Type cable without keystoning. The heightand width of the cable are 15.6 mm and 2.175 mm respectively and it consists of 30strands. A similar type of cable was used in a study to optimize the performane oflarge aperture cos(θ) dipole magnets within the NED Program [1]. The main differencesbetween the two cables are the height and the number of strands of the cable used in[1], which were 26 mm and 40 respectively. In addition to that the cos(θ) design usesslightly keystoned cables (with a keystoning angle of about 0.22 degrees) so as to makethe radial alignment of the blocks easier to achieve. Using similar cables facilitates com-parison of the two approaches without having to put a lot of effort to estimate the effectof the shape of the cable itself to our results.

3 Outline

At first the coil cross section for different apertures will be studied and optimized. Thenthe different effects of the nonlinearity of the yoke as well as the superconducting cablemagnetization will be taken into account. Finally the two configurations, block-coil andcos(θ), will be presented in parallel and compared.

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0 21.43 42.86 64.29 85.71 107.14 128.57 150

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Figure 1: Coil cross section for a 40mm bore magnet

4 Steady-State Operation Design

4.1 Small Aperture Magnet

At first the principles were studied for a 40mm bore magnet by using 4 independentblocks distributed at two different y-positions above the midplane. By optimizing thecoil geometry, without taking the yoke into account, it was possible to keep all relativemultipole field errors below 0.05, except b7 and b9. The radius of harmonic analysiswas set to 10 mm. The coil geometry is shown in Figure 1 and the field data from thetwo-dimensional simulation are listed in Tables 1 and 2.

Table 1: Field data for a 40mm bore magnet

Excitation current 18000 AMain field B1 13.634 TPeak field on conductor 14.272 TPeak field / Main field 1.047Percentage on the load line 10.5Excitation current at 0% margin on the load line 20250 APeak field at 0% margin on the load line 16.056 T

By introducing the iron yoke and further optimizing the design of the coil, a peak fieldof over 14.5 T was achieved. The total cross-section of the coil and yoke can be seen in

3

Figure 2 and the accompanying field data can be found in Tables 3 and 4.In the next step another design for a small aperture magnet was studied which resultedin a 52 mm bore. The idea was to use the intersecting ellipses scheme to create a fieldof better homogeneity, which indeed was achieved. In this design only two independentblocks were used. The field data are shown in Tables 5 and 6 and the design of the coilin Figure 3. The radius of harmonic analysis was set to 15 mm.By inserting the iron yoke and optimizing the coil geometry the peak field exceeded

Table 2: Relative multipole field errors for a 40mm bore magnet

b3 b5 b7 b9 b11 b13 b15

-0.00126 -0.00016 -2.54015 -0.034937 -0.04136 -0.00675 -0.00072

Table 3: Field data for a 40mm bore magnet with the insertion of the iron yoke

Excitation current 15660 AMain field B1 13.918 TPeak field on conductor 14.549Peak field / Main field 1.045Percentage on the load line 10.3Excitation current at 0% margin on the load line 18000 APeak field at 0% margin on the load line 16.471 T

Table 4: Relative Multipole field errors for a 40mm bore magnet with the insertion ofthe iron yoke

b3 b5 b7 b9 b11 b13 b15

-0.11907 0.31166 -2.25847 -0.28734 -0.03461 -0.00504 -0.00047

Table 5: Field data for a 52mm bore magnet

Excitation current 19520 AMain field B1 13.395 TPeak field on conductor 14.158 TPeak field / Main field 1.057Percentage on the load line 10.3Excitation current at 0% margin on the load line 21920 APeak field at 0% margin on the load line 15.899 T

4

0 21.43 42.86 64.29 85.71 107.14 128.57 150

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0 50 100 150 200 250 300 350 400 450 500 550 600

0. 0.403-

0.403 0.807-

0.807 1.210-

1.210 1.613-

1.613 2.017-

2.017 2.420-

2.420 2.824-

2.824 3.227-

3.227 3.630-

3.630 4.034-

4.034 4.437-

4.437 4.840-

4.840 5.244-

5.244 5.647-

5.647 6.051-

6.051 6.454-

6.454 6.857-

6.857 7.261-

7.261 7.664-

|B| flux density (T)

Time (s) : 1.

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Figure 2: Coil cross section for a 40mm bore magnet with the insertion of the iron yoke

0 21.67 43.33 65 86.67 108.33 130

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Figure 3: Coil cross section for a 52mm bore magnet

Table 6: Field data for a 52mm bore magnet

b3 b5 b7 b9 b11 b13 b15

-0.00134 0.01457 -0.71914 -1.43880 -0.10109 -0.01141 -0.00576

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14.5 T while the field quality remained in the desired level. The geometry of the coiland yoke is depicted in Figure 4 and the field data are given in Tables 7 and 8.

Table 7: Field data for a 52mm bore magnet with the insertion of the iron yoke

Excitation current 15200 AMain field B1 14.014 TPeak field on conductor 14.559 TPeak field / Main field 1.039Percentage on the load line 10.5Excitation current at 0% margin on the load line 17120 APeak field at 0% margin on the load line 16.169 T

Table 8: Relative Multipole Field Errors for a 52mm bore magnet with the insertion ofthe iron yoke

b3 b5 b7 b9 b11 b13 b15

0.00131 0.08440 0.13017 -1.60859 -11746 -00993 -0.01085

4.2 Large Aperture Magnet

One of the characteristics of the block-coil model is its scalability. After having studiedthe basic characteristics of the small aperture block coil magnets an attempt was madeto design a large aperture magnet (132 mm bore) in a fast and efficient way, simply byscaling up both the dimension of the aperture and the number of the blocks. By follow-ing the intersecting ellipses scheme and optimizing the coil cross-section two differentdesigns were produced, one having 5 and one having 6 independent blocks.Their main differences are the peak field and of course the amount of superconductingmaterial used. A characteristic that remained unchanged during the design and op-timization phase of both magnets was the fact that the vertical distance between theblocks as well as between the lower block and the midplane was intentionally not zero.In fact it was set to 20 mm and was allowed to variate by a limited amount (≈2 mm)around this value during the optimization phase. This was done assuming that thereis a supporting structure around each coil to guarantee mechanical stability and stressmanagement, as demonstrated in [5]. However, on the one side a somewhat acceptablefield quality was obtained, so there was no serious need for altering the vertical coor-dinates of the blocks dramatically and on the other side the interest was more on theadvantages and drawbacks of the block-coil approach in general, rather than on pro-ducing an accurate design.This also justifies the fact that the magnetic iron yoke was

6

0 21.43 42.86 64.29 85.71 107.14 128.57 150

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0 50 100 150 200 250 300 350 400 450 500 550 600

0.002 0.456-

0.456 0.909-

0.909 1.363-

1.363 1.817-

1.817 2.270-

2.270 2.724-

2.724 3.178-

3.178 3.632-

3.632 4.085-

4.085 4.539-

4.539 4.993-

4.993 5.446-

5.446 5.900-

5.900 6.354-

6.354 6.807-

6.807 7.261-

7.261 7.715-

7.715 8.168-

8.168 8.622-

|B| flux density (T)

Time (s) : 1.

ROXIE9.0BEMFEM *

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Figure 4: Coil cross section for a 52mm bore magnet with the insertion of the iron yoke

not taken into account, thus only the coil itself was examined. However, as far as thefield strength and quality of the configuration and the comparison between them, areconcerned, the effects of the yoke can be neglected.The coil cross-section for the 5 block design is given in Figure 5 along with the dataabout the produced field (Tables 9 and 10).

Table 9: Field data for a 132 mm bore magnet using 5 independent blocks

Excitation current 14400 AMain field B1 13.642Peak field on conductor 14.624 TPeak field / Main field 1.072Percentage on the load line 10.6Excitation current at 0% margin on the load line 16200 APeak field at 0% margin on the load line 16.452 T

As one can see the field at a margin on the load line of roughly 10% is close to 15 T andall relative multipole field errors (with the exception of b9) are kept under 0.2.On the other hand by adding a sixth block on top of the previous 5 ones, it is possibleto increase the peak field of the coil and one would expect the multipole errors to bemanipulated in a more efficient way, whish is indeed proved to be true by the simulation.The cross-section of the coil and the field data are shown in Figure 6 and Tables 11 and12 respectively.

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0 21.43 42.86 64.29 85.71 107.14 128.57 150ROXIE9.0

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Figure 5: Coil cross section for a 132 mm bore magnet using 5 independent blocks

Table 10: Relative multipole field errors for a 132 mm bore magnet using 5 independentblocks

b3 b5 b7 b9 b11 b13 b15

-0.00010 -0.00019 -0.00001 -4.66079 -0.35117 -0.30851 -0.21595

Table 11: Field data for a 132 mm bore magnet using 6 independent blocks

Excitation current 12900 AMain field B1 16.077 TPeak field on conductor 14.997 TPeak field / Main field 1.072Percentage on the load line 9.4Excitation current at 0% margin on the load line 14250 APeak field at 0% margin on the load line 16566 T

Table 12: Relative multipole field errors for a 132mm bore magnet using 6 independentblocks

b3 b5 b7 b9 b11 b13 b15

0.00001 -0.00080 0.17097 -0.157934 0.72681 0.02720 -0.22920

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0 20 40 60 80 100 120 140 160ROXIE9.0

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Figure 6: Coil cross section for a 132 mm bore magnet using 6 independent blocks

5 Multipole Field Error Variation

5.1 Theoretical Baseline

Until now we have been interested in the nominal and maximum field of the magnetand in the corresponding relative multipole errors. This analysis has been done for theso-called steady state operation of the dipole. However in a real accelerator the particlesend up in a high energy circular collider after a (usually long) sequence of linear acceler-ators (linacs), smaller storage rings and boosters. In each of these steps the particles areaccelerated to the nominal energy of each machine before being extracted and injectedinto the next one. From that it is clear that a high energy accelerator will operate at anenergy far below its nominal energy during the first rounds after injection. For circularcolliders this means that the bending dipoles will have to produce a magnetic field farlower than their nominal one. Otherwise the particles will be bent beyond the curvatureof the ring and hence hit the beam pipe and be lost while at the same time damagingthe beam pipe itself. After injection the particles will be accelerated until they reach thenominal energy of the ring, where the collisions and experiments will have to be carriedout. During this procedure of accelerating the particles the field of the dipoles will haveto be constantly increased to bend the particles of increasing energy to the same angle.It is therefore essential to have a highly homogeneous field not only for the nominal fieldstrength, but also for operation below this point. If this is not the case the multipoleerrors can cause a degradation of the beam quality during the time after injection andbefore the nominal energy is reached which could even lead to a loss of a part of the in-jected particles. Two phases of a magnet’s transient response are distinguished: the one

9

where the excitation current and accordingly the magnetic field are increased, which isreferred to as ”up-ramp” and the opposite one where the excitation current is decreased,which is referred to as ”down-ramp”. It is obvious that the most important of the twophases is the up-ramp phase. That is because the up-ramp phase takes place after theinjection where it is crucial that the beam quality is kept at a good level.The fact that different values of relative multipole errors appear for different values ofthe excitation current points to some source of nonlinearity in the response of the mag-net. Here the two main sources of nonlinearities -the iron yoke and the superconductingmagnetization- are defined and studied.It is known that the magnetic iron exhibits a nonlinear response and is fully saturatedat a field strength of about 2 T and above. This indicates a highly nonlinear behaviourwhich results unavoidably in a variation of both the rise of the main field and the mul-tipole errors depending on the excitation current. For the relative multipole errors thisvariation can be significant and can produce degradation of the beam quality if it isinjected into the storage ring at a low energy.Another source of nonlinearity and hence multipole error variation are the so-calledpersistent currents that are produced on the superconducting cables and form anothersource of magnetic field that alters the field profile in the aperture. For hard super-conductors like the ones used for building the coils of superconducting magnets, thepersistent currents also show a hysteresis, which means that the response is differentbetween up-and down ramp. The relation between these two errors and certain yokegeometries has been studied in order to identify possible compensation techniques.

5.2 Proposed Approach and Computer Simulation

It has been proposed (in [4]) that inserting a magnetically permeable blade (which will bereferred to as ”flux-blade”) between the two sets of blocks parallel to the horizontal axiswould suppress the persistent-current-induced field errors and hence provide a methodfor persistent current compensation. We will try to examine the effect of such a structureand evaluate its performance.

5.2.1 Total Transient Response Analysis

In the following analysis only the first coil geometry of the 40mm bore magnet has beenexamined (see Figure 1), because the interest lies only in the origin of these errors andthe proposed compensation technique and not in the field quality itself. The geometryof the iron yoke chosen for this study is shown in Figures 7 and 8. We should note thatthe yoke is placed very close to the coil. This has to be done if one wishes to achievethe maximum field with a given number of cables.It is not an important result in this case to evaluate the steady-state value of the relativemultipole error, since that can change by rearranging the blocks in the coil. Howeverthat would mainly shift the whole curve up or down. What is more important is thevariation of the curve -from the minimum to the maximum value during the up-ramp-as well as the ”width” of the hysteresis loop. One can see from Figure 9 that the lower

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Figure 7: Iron yoke used for simulating the uncompensated magnet

Figure 8: Iron yoke after the inclusion of the ”flux-blade”

11

Figure 9: Transient response for the two yoke geometries(The upper curve represents the compensated yoke geometry)

Figure 10: Width of the hysteresis curves for the two yoke geometries(The upper curve represents the compensated yoke geometry)

curve (which corresponds to the ”uncompensated” yoke geometry) shows a bigger spreadof values during the up-ramp than the upper curve. From this point of view one couldsay that the flux-blade seems to improve the transient response of the magnet.By comparing the width of the two hysteresis curves, that is the difference of the valuesfor the same excitation current during up- and down-ramp, one ends up with Figure 10.From these two figures one can see that the flux-blade in this geometry suppresses thevariation of the relative sextupole error during the up-ramp, thus making injection atlower energies less damaging for the quality of the beam. However the width of thehysteresis curve is increased by the insertion of the flux-blade by more than 100% forlow excitation current.

5.2.2 Persistent-Current-Induced Field Errors

In the above analysis the total field error was considered. This means that one couldnot study the effect of the persistent-current-induced field errors and the field errorsthat are due to the non-linearities of the iron yoke separately. An easy way to do so is

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Figure 11: Comparison between purely geometric and total errors for the two yoke ge-ometries(The two upper curves represent the compensated yoke geometry)

to calculate the variation of the field errors with respect to the excitation current, butwithout taking one of the two sources of non-linear errors into account. Using ROXIE,one can calculate the transient response of the magnet ignoring the persistent currents.This means that one uses an idealized model of the superconducting cables, where thereis no superconductor magnetization while at the same time all the other properties areunchanged.The outcome of this calculation (ignoring the superconducting magnetization) is plottedin Figure 11, along with the total field errors (the ones calculated by taking the persistentcurrents as well as the non-linear behaviour of the yoke into account).One can see that after a certain value of the excitation current the two curves (purelygeometric and total field error) tend to converge. This means that the relative persistent-current-induced field errors decrease with inceasing excitation current (and thus withincreasing overall magnetic field) and that they play an important role only at lowexcitation current (i.e. low magnetic field). To study the different behaviour of the twogeometries with respect to each one of the two sources of non-linear relative multipolefield errors we plotted the difference between the geometric and the total sextupole errorwith respect to excitation current for both cases and ended up with Figure 12.As one can see, the difference between geometric and total sextupole error is twice aslarge for the ”compensated” geometry, until of course the persistent currents becometoo small in both cases.After seeing their results on the field quality it can be useful to have an overview ofthe nature and the effects of the persistent currents by looking at the magnetization inthe region of the coil. In Figures 13 to 16 we have plotted the magnetization of thecoil for low and high excitation current and for both geometries. It is obvious thatthe magnetization profile changes with excitation current. However this change doesn’tapply only to the magnetization strength, but also to its distribution on the coil. Thisproves the claim that superconductor magnetization in magnet coils is indeed a nonlineareffect. Therefore it has to be studied and taken into account if the magnets have to beused with a varying excitation current, for example during the up-ramp phase after

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Figure 12: Difference between the purely geometrical and the total sextupole errors forthe two yoke geometries(The upper curve represents the compensated yoke geometry)

0 21.43 42.86 64.29 85.71 107.14 128.57 150

-4.66 -2.59-

-2.59 -0.52-

-0.52 1.547-

1.547 3.618-

3.618 5.688-

5.688 7.759-

7.759 9.829-

9.829 11.9-

11.9 13.97-

13.97 16.04-

16.04 18.11-

18.11 20.18-

20.18 22.25-

22.25 24.32-

24.32 26.39-

26.39 28.46-

28.46 30.53-

30.53 32.60-

32.60 34.67-

(*104)

M (A/m)

Time (s) : 1.

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Figure 13: Magnetization in the coil for the uncompensated geometry at low excitationcurrent

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0 21.43 42.86 64.29 85.71 107.14 128.57 150

-28.7 -27.2-

-27.2 -25.8-

-25.8 -24.3-

-24.3 -22.9-

-22.9 -21.5-

-21.5 -20.0-

-20.0 -18.6-

-18.6 -17.1-

-17.1 -15.7-

-15.7 -14.2-

-14.2 -12.8-

-12.8 -11.3-

-11.3 -9.91-

-9.91 -8.46-

-8.46 -7.01-

-7.01 -5.57-

-5.57 -4.12-

-4.12 -2.67-

-2.67 -1.22-

(*104)

M (A/m)

Time (s) : 1.

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Figure 14: Magnetization in the coil for the uncompensated geometry at high excitationcurrent

0 21.43 42.86 64.29 85.71 107.14 128.57 150

-22.7 -19.7-

-19.7 -16.8-

-16.8 -13.8-

-13.8 -10.9-

-10.9 -7.94-

-7.94 -4.98-

-4.98 -2.02-

-2.02 0.933-

0.933 3.892-

3.892 6.852-

6.852 9.812-

9.812 12.77-

12.77 15.73-

15.73 18.69-

18.69 21.65-

21.65 24.61-

24.61 27.57-

27.57 30.53-

30.53 33.49-

(*104)

M (A/m)

Time (s) : 1.

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Figure 15: Magnetization in the coil for the compensated geometry at low excitationcurrent

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0 21.43 42.86 64.29 85.71 107.14 128.57 150

-28.6 -27.3-

-27.3 -25.9-

-25.9 -24.5-

-24.5 -23.2-

-23.2 -21.8-

-21.8 -20.4-

-20.4 -19.1-

-19.1 -17.7-

-17.7 -16.3-

-16.3 -15.-

-15. -13.6-

-13.6 -12.2-

-12.2 -10.8-

-10.8 -9.52-

-9.52 -8.16-

-8.16 -6.79-

-6.79 -5.42-

-5.42 -4.05-

-4.05 -2.68-

(*104)

M (A/m)

Time (s) : 1.

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Figure 16: Magnetization in the coil for the compensated geometry at high excitationcurrent

injection.

6 Conclusions

The conclusions that can be derived from the present work can be divided in two maincategories. On the one hand a comparison can be done between the different kinds ofblock-coil design geometries which were tested. At the same time the overall capabilitiesand potentials of the block-coil approach to magnet design can be evaluated and com-pared to some extend with similar results derived for cos(θ) dipole magnets. In orderto be fair to both designs, a fully optimized final design of a block-coil magnet shouldbe produced before the comparison between the two philosophies can lead to a finaldecision. On the other hand the proposed flux-blade compensation scheme has beenanalyzed and can be further evaluated.

6.1 Steady-State Operation

Four different coil geometries have been chosen and optimized in order to produce twosmall- and two large-aperture dipole magnets. The two small-aperture magnets haveshown a relatively high field of over 14 T, which is further increased when the iron yokeis included. The scaling of the design to larger apertures was done in a fast way withoutthe need of many time-consuming optimization rounds. However the final design ofthe 5-block magnet showed a good behaviour as far as the relative multipole errors areconsidered with the exception of b9. By adding the sixth block all errors were kept at

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a low level. The only exception is b11 with a value of about 0.7. However one shouldnot forget that the NED design proposed a cos(θ) magnet with a b11 of more than 1.In addition to that the value of the lower order multipole (in this case the sextupole) isextremely low. The peak field of the coil approaches 15 T at the operation point andexceeds 16.5 T when the load-line-margin vanishes. This field strength has not yet beenachieved using cos(θ) magnets. A point that would require further investigation is theexcitation current. In this design the excitation current used is small relative to otherdesigns. This results unavoidably to a larger amount of cables in order to produce thedesired field. However the spread of the current over a larger area lowers the Lorentzforce per area which can be crucial for the mechanical robustness of the magnet in high-field operation.Summarizing the above one can say that the block-coil configuration has the potentialto provide the field strength and quality needed for future storage rings. However inorder to be able to come to a decision a final design has to be carried out includingboth an optimized core geometry, as well as taking into account issues of cost and storedmagnetic energy arising from the low excitation current and high number of windings.

6.2 Transient Response

From calculating the relative sextupole error variation for the uncompensated and com-pensated yoke geometry it is seen that the flux-blade suppresses the variation, thusforcing the curve to become more ”flat”. However this seems not to be because ofpersistent-current compensation. When subtracting the overall error from the purelygeometric one (due to the non-linear iron yoke) one can isolate the contribution of thepersistent currents. By doing that we found out that the persistent-current-induced fielderrors increase with the insertion of the blade. This means that although the flux-bladescheme suppresses much of the variation of the sextupole error this suppression doesnot come from a persistent-current compansation, but seems to be a purely geometricaleffect.

Acknowledgement

This work was carried out during the 2006 CERN Official Summer Student Program,within the AT Department, MEL Group, EM Section.

References

[1] J. N. Schwerg, Electromagnetic Design Study for a 15 T Large Bore Superconduct-ing Dipole Magnet, Berlin, November 2005

[2] S. Russenschuck, Electromagnetic Design and Mathematical Optimization Methodsin Magnet Technology, Version 3.3, April 2006

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[3] F. Toral, Progress in Comparison of Different High Field Magnet Designs for NED,CIEMAT, Madrid, Spain

[4] R. Blackburn, T. Elliott, W. Henchel , L. McInturff, P. McIntyre, and A. Sattarov,Construction of Block-Coil High-Field Model Dipoles for Future Hadron Colliders,IEEE Transactions on Applied Superconductivity, Vol. 13, No. 2, June 2003

[5] P. Noyes, R. Blackburn, N. Diaczenko, T. Elliott, W. Henchel, A. Jaisle, A. McIn-turff, P. McIntyre, and A. Sattarov, Construction of a Mirror-Configuration Stress-Managed Nb3Sn Block-Coil Dipole, IEEE Transactions on Applied Superconduc-tivity, Vol. 16, No. 2, June 2006

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