INTEGRATED DESIGN OF SUPERCONDUCTING ACCELERATORMAGNETS

S. Russenschuck, C. Paul, S. Ramberger, F. Rodriguez-Mateos, R. WolfCERN,1211Geneva 23,Switzerland

AbstractThis chapterintroducesthe main featuresof the ROXIE programwhich hasbeendevelopedfor the designof the superconductingmagnetsfor the LargeHadronCollider (LHC) at CERN.Theprogramcombinesnumericalfield cal-culationwith areducedvector-potentialformulation,theapplicationof vector-optimizationmethods,and the useof geneticas well as deterministicmini-mizationalgorithms.Togetherwith theappliedconceptof features,thesoft-ware is usedas an approachtowards integrateddesignof superconductingmagnets.The main quadrupolemagnetfor the LHC, waschosenas an ex-amplefor theintegrateddesignprocess.

1 Intr oduction

TheLargeHadronCollider (LHC) projectis a superconductingacceleratorfor protons,heavy ionsandelectron-protoncollisionsin themulti-TeV energy rangeto beinstalledatCERN[1]. In orderto achievethedesignenergy within theconstraintof theexistingLEPtunnel,whichhasacircumferenceof about27km, themagnetsystemmustoperatein superfluidheliumbelow 2 K. Thetunnel’s limited space,aswellascostconsiderations,dictatea two-in-onemagnetdesign,wherethetwo ringsareincorporatedinto thesamecryostat.Themaindipolemagnetswill operateat about0.54T at injectionand8.40T at nominalcurrent.Themainquadrupoles,will operateat a 223T/m field gradient,a magneticlengthof 3.10m, anominalcurrentof 11800A, aninnercoil apertureof 50 mm, andanoperationaltemperatureof 1.8K.For theoptimizationof themagnets,contradictoryparameterssuchasmaximummainfield, minimumcontentof unwantedmultipoles,andsufficient safetymargin for theconductormustbeconsidered.Thekeystoningof the conductorsandthe resultinggradingof the currentdensitiesnecessitatea computa-tional methodthat canmodel the coil andcalculatethe excitationalfield with a higheraccuracy thanrenderedby mostof thecommercialfinite elementpackages.In addition,thecharacteristicdatafor boththecoil andthe iron configurationmustbeparametricfor theapplicationof mathematicaloptimizationtechniques.

Togetherwith themodulefor theaddressinginput andoutputdataasdesignvariablesor designobjectives,theFeature-BasedDesignModule(FBDM) canbeseenastheheartof theROXIE program.This programhasbeendevelopedat CERN for the designandoptimizationof the coil andyoke ge-ometriesfor thesuperconductingmagnetsfor theLarge-Hadron-Collider, LHC. Featuresarefunctionalprimitives that containnot only the geometricalinformation(shape,dimensions,position,orientation,tolerances)of a part,but alsonon-geometricpropertiessuchasmaterialname,properties,partnumber,etc. Designingby featuresis thereforean extensionto parametricprogrammingand is usedtogetherwith mathematicaloptimizationtechniquesandnumericalfield calculationasan approachtowardsanintegrateddesignof superconductingmagnets[5]. Thestepsin thedesignprocessare:

� Feature-basedgeometrycreation.

� Conceptualdesignusinggeneticalgorithms.

18

� Calculationof field errorscausedby persistentcurrents.

� Optimizationof thecoil cross-section.

� Minimization of iron-inducedmultipoles.

� Quenchsimulation.

� 3d coil endgeometryandfield optimization.

� Toleranceanalysis.

� Productionof drawingsby meansof aDXF interface.

� End-spacerdesign.

� Tracingof manufacturingerrors.

2 Feature basedgeometrycreation

TheROXIE programincludesroutinesfor geometricallydefiningcoil cross-sectionsmadeof Rutherford-typesuperconductingcablesor rectangularshapedbraids.Thegeometricpositionof coil block arrange-mentsin the crosssectionof the magnetsis calculatedfrom given input datasuchas the numberofconductorsperblock,conductortype(specifiedin acabledatabase),radiusof thewindingmandrel,andpositioningandinclination angleof theblocks. The fact that thekeystoning(trapezoidalshape)of thecablesis not sufficient to allow their edgesto bepositionedon thecurvatureof acircle, is takeninto ac-count(c.f. Fig. 1). Thiseffect increaseswith theinclinationof thecoil blocksversustheradialdirection.Thekeystoningof thecablealsoresultsin agradingof thecurrentdensityin theconductorasthecableismorecompacted(lessvoidsbetweenthestrands)towardsthenarrow side.After thegeometricmodellingis done,every feature(strand,cable,block, layer)canbesubjectedto geometrictransformationssuchastranslation,rotation,scaling,imaging,while constraintsaredefinedfor theseoperationsin orderto avoidpenetrationor physicallymeaninglessstructures.Not only canthegeometricpropertiesof themagnetbechangedin theoptimizationprocessbut soalsocanits materialproperties(in our case,for example,thenumberof strands,currentdensityin conductorsandstrands,filling factors,unit priceetc.)

3 Conceptualdesignusing geneticalgorithms.

Theoptimizationproblemassociatedwith superconductingmagnetdesigninvolvesmultiple,conflictingobjectives,which mustbe consideredsimultaneously. The solutionprocessfor theseso-calledvector-optimizationproblems[6] is threefold,basedon decision-makingmethods,methodsfor treatingnonlin-earconstraints,andoptimizationalgorithmsfor minimizing theobjective function[7]. In theconstraintformulationtheoptimizationproblemreads:

��������� (1)�� ������������������ � (2)����� ���(3) "!$#���%'&)(*&��*+-,(4) /.10$#2��%'����&3(*&��*+-,(5) . , #���%'������&4(*&�� +-,(6)

19

For the definition of the relative field errors "!

, �.10

, etc. at a radiusof 10 mm in the aperture,andfor the definition of the quench-margin

���"�referreto the chaptersbelow. �5�6 is the surfaceof

the superconductingmaterialin the cross-sectionto be minimized.�� ������

is the field gradient. Thenumberof coil blockswaslimited to 4. Theresultingdesignvariablesfor theoptimizationproblemarethenumberof turnsperblock, thepositioningandtheinclinationangleof two blocks,andthecurrentineachturn. Theconstrainedoptimizationproblemis transformedby anexact-penaltytransformation.Fortheminimizationof theresultingobjective function,geneticalgorithmsareused.

In geneticalgorithms,eachtrial solutionis codedasa vector(chromosome) 78 with elementsbe-ing describedaschromosomes.Holland[8] suggestsarepresentationof chromosomesby binarystrings.As our problemis a mixtureof continuousandinteger, thedifferentparametersarecombinedby linearsamplingof the floating-pointparametersandGray-encodingof the integersinto a binary string. Thecurrentin theconductorsandtheanglesof thecoil-blockswereencodedby 4 bit stringswhile thenum-berof turnsperblock wereencodedby 3 bit stringseach,thusresultingin chromosomesof 32 bits. Theselectionoperatorguaranteesconvergenceto an optimumby keepingthebetterchromosomesanddis-cardingthelessfit ones.Usingthestandardoperationto retainthebetterhalf of thechromosomesreducesdiversitygenerationby generationthusleadingto gene-poolsunableto profit from furthercrossover (re-combinationof bit stringsof two chromosomesby swappingthestringsat a randompoint). Thereforethediversityof thepopulationis guaranteedby aso-callednichingmechanism.After anew offspringisgeneratedby meansof crossover, thegenewith thesmallesthamming-distance(leastdifferencein bits)is locatedandreplacedif its fitnessis worsethanthatof theoffspring.Niching notonly increasesdiver-sity, but providesa numberof local minima for further investigationssincenot all theobjectives( e.g.manufacturingconsiderationsfor coil windingandcollaring)canbeincludedin theobjective function.Apopulationsizeof 40chromosomeshasthenbeenprovedto besufficient. Thecrossover probabilitywaschosenas0.8. Themutationoperatoravoids preliminaryconvergenceof theentirepopulationtowardsa local minimum. The mutationrate is 0.005. For our parametersetsof 32 bits about3000functionevaluationswereperformed. Fig. 1 shows 3 differentdesignsthat result from this conceptualdesignphase.

Fig. 1 top shows a designthatwould bedifficult to manufactureastheouterblock hasa higheranglethantheinnerblock thusmakingthecollaringdifficult, i.e., requiringspecialshimming.Themostefficientdesignin termsof gradientandconductormaterialis thedesigndisplayedin Fig. 1 bottom,witha conductordistribution of 8,7,8,2. In orderto increasethesizeof the copperwedgebetweenthe twoouterblocks,andin orderto confirmwith thebase-lineSACLAY design,oneconductorin thefirst outerblock wassuppressedfor thefurtherstudies.

4 Calculation of field errors due to persistentcurrentsand coupling currents

Besidesgeometricalfield errorsdueto theconductorplacement,additionalfield distortionsarecausedby persistentcurrentsin thesuperconductingfilamentsandcouplingcurrentsbetweenthestrandsin thecable(ISCC).ThepersistentcurrentsandISCCsin eachstrandof eachturn canbedeterminedfrom thelocal field distribution in thecoil cross-section.Theresultingfield errorsdueto thesecurrentsin all theturnsof thecoilscanthenbecalculatedusingBiot-Savart’s law.

In the outerregion of a superconductingfilament,subjectto an appliedmagneticfield, currentswith density9�: aregenerated,whichscreentheinteriorof thefilament.At full penetrationof thescreen-ing currentsin thefilament,themagnetization

�(i.e. themagneticmomentperunit volume)is

�<;�= ��> 0��? 9�:�@ �A"�CBD� (7)

20

0 10 20 30 40 50 60

0 10 20 30 40 50 60

0 10 20 30 40 50 60

Fig. 1: Resultingcoil block distributionsfrom optimizationusinggeneticalgorithmsandnichingmechanism

with�

thefilamentdiameterandthecritical currentdensity 9�: . Protonswill be injectedinto theLHCat ratherlow field levels, where 9�: andhence

�arelarge. A typical valuefor

�at ; ��%'E4�

and� ; &�%'FHGis about30 mT for

� ;�I > � [9].

Inter-strandcouplingcurrents(ISCCs)flowing in andbetweenthe strandsof the cablearegen-eratedif the cableis subjectto a varying field, i.e. during rampingof the magnet. The ISCCsareproportionalto:

J :LK M�NLO�PQSR�T�UV�W :

�� ��X (8)

withQSR

thetwist lengthof thestrands,T

thecablewidth,UV�

thenumberof strandsin thecableandW :

thecontactresistancebetweentwo crossingstrands.Usinganetwork modelof theRutherfordtypecable,in which thestrandsareconnectedthroughcontactresistances,thecouplingcurrentscanbecalculatedusingKirchhoffs laws. Typicalvaluesin thestrandsneartheedgesof thecablearea few amperesfor theforeseenramp-ratefor LHC (dB/dt=6mT/s)and

W : =10>�Y

[10].

The resultingfield errorsdue to thesecurrentsaregiven in Table1. Both typesof field errorsfollow thesymmetryof thequadrupolefield so thatonly thenormalevenharmonics

"Z, "!

, �.10

, ... aregenerated.Dueto uncertaintiesin themagnetizationof thefilamentsandin

W : over thecross-sectionofthecoils,additionalfield distortionsarepresent.In Table2 anestimateis givenof theseerrorsassuminga3%variationin thepersistentcurrentsanda30%variationin

W : .

21

Order Persistentcurr. Couplingcurr.d=6

>m, T=1.8K 0.18T/m/s,

W : =14>�Y "Z

-5.6 16.70 "!-0.54 0.020 �.10

0.0018 -0.0008

Table1. Thepredictederrorsat injectiongradient(14.5T/m) of theLHC latticequadrupoledueto themagneticmomentof thepersistentcurrentsandtheinter-strandcouplingcurrentsduringramping(in unitsof

&�� +-,of themainfield at r=10mm).

Order Persistentcurr. Couplingcurr.d=6

> � , T=1.8K 0.18T/m/s,W : ; &�[�>�Y\ � � �

=0.03\ W : ; [�>�Y

n "] ^_] `] ^_]

1 1.9 1.9 1.6 1.62 0.75 - 2.7 0.53 0.32 0.32 1. 1.4 - 0.2 0.2 0.45 0.04 0.04 0.05 0.056 0.01 0.012 0.012

Table2. Thepredictedrandomerrorsin theLHC latticequadrupoledueto uncertaintiesin M andRc (in unitsof

&�� +-,of themainfield at r=10mm).

A decreaseof thefield errorsdueto the ISCCscanbe obtainedby increasingW : , which canbe

achieved by applyinga propercoatingonto the strandsurface. Decreasingthe filamentdiameterpro-videstheonly possibilityto reduceM since9�: hasto beaslargeaspossiblein orderto obtainacompacthigh-field magnet.Sincethe lower limit of d is about5-7

> � , importanterrorsstill remainandhaveto be correctedfor with the coil-block geometry. The final optimizationof the coil cross-sectionwithpartial compensationfor thepersistentcurrentmultipoleerrorsis performedusingdeterministicsearchmethods,asdescribedin thenext section.

5 Optimization of the coil cross-section

The aim of this designstepis to find a coil cross-sectionwith a part compensationof the persistentcurrenteffectsby an appropriateplacementof thecoil blocks. As thenumberof ampereturnsarenotvariedin thisdesignphase,a relatively simpleleast-squaresobjective function

����� @ Xa. @ �! = ��%'E�B Z3b X`Z @ �.10 b ��%'����&�B Z4b X`c @ /. , = ��B Z B(9)

is minimizedusingthedeterministicoptimizationalgorithmEXTREM [11]. For the4 remainingdegreesof freedom(inclination andpositioningangleof the two outerblocks)about100 function evaluationshave to becarriedout. Theweightingfactors

Xa.-X`c

hadto beadjustedin aniterative processtoXa.

= 10.,X`Z= 100.,

X"c= 8000.Theoptimizedquadrupolecross-sectionis shown in Fig. 2.

6 Optimization of the ir on cross-section

The influenceof the iron magnetizationis taken into accountby meansof a finite elementcalculationbasedon a reducedvectorpotentialformulation. As explainedin the contribution by Biro, Preisand

22

0 10 20 30 40 50 60

Fig. 2: Optimizedquadrupolecross-section.Displayof thequadrupolefield andthefield modulusin thecoils.

Paul, this avoids themeshingof thecoil, which is advantageousasthefield errorsarevery sensitive tomodellingerrorsof theconductorplacement.Themethodalsoallows usto distinguishbetweenthecoilandtheiron effectsandto accuratelycalculatethemultipolecontentasa functionof theexcitation.Thepeakfield in thecoil, whichis calculatedneglectingtheselffield of thestrands,determinesthemargin toquenchatthenominalfield (seealsoAppendixB). As canbeobservedin generalin quadrupolemagnets,thevariationof multipoleerrorsin theapertureis relatively insensitive to theexcitationalcurrentin thecoils asthe saturationeffectsarelesspronouncedthanin the dipole magnets.The iron magnetizationin the twin-aperturemagnetcross-sectionasdesignedby SACLAY, a variantof that describedin [4],causesthe additionalfield errorsasgiven in Table3. A further minimizationof theseeffectswasnotnecessary. Fig. 3 shows thereducedfield in thecross-section.Themargin to quenchis 80.3% resultingin a gradientat quenchof 277T/m.

Order Coil field Coil in yokeInjection Nominal Injection Nominal �.

- - -10.7 -20.8 "c- - -0.37 -0.41 , - - -0.018 -0.028 "d- - - - "!

-0.07 0.47 -0.07 0.48 �e- - - - "f- - - - "g- - - - �.10

-0.0011 -0.0029 -0.0011 -0.0029

Table3. Thepredictedfield errorsdueto coil geometryandpersistentcurrenteffectsin theLHClatticequadrupoleat injectionfield level andnominaloperation.Theeffect of thetwo-in-oneiron yokewith saturationcanbe seenfrom the right handcolumns. Note that the valuesgiven at injection field

23

0.5966h

0.0985h

-

0.0985h

0.1969h

-

0.1969h

0.2953h

-

0.2953h

0.3938h

-

0.3938h

0.4922h

-

0.4922h

0.5906h

-

0.5906h

0.6891h

-

0.6891h

0.7875h

-

0.7875h

0.8859h

-

0.8859h

0.9844h

-

0.9844h

1.0828-

1.0828 1.1813-

1.1813i

1.2797i

-

1.2797i

1.3781i

-

1.3781 1.4766-

1.4766i

1.5750i

-

1.5750 1.6734-

1.6734i

1.7719i

-

1.7719i

1.8703i

-

1.8703i

1.9688i

-

|B| red. (T)

ROXIE4.2j

27/02/97 14.41main quadrupole

Fig. 3: Magnetcross-sectionwith reducedfield. Notethatthecenterof thequadrupoleis shiftedtowardsthecenterof theyoke.

Theiron magnetizationis thereforescreeningthetwo-in-oneeffectvisible in theexcitationalfield.

level containtheeffectsfrom persistentcurrents.

7 Quenchsimulation

Studyingthebehaviour in theeventof transitionto theresistive state(quench)is alsoanimportantcon-siderationduring the designphaseof superconductingmagnets.The aim of thesestudiesis to knowwhetheror not themagnetis self-protectedagainstresistive transitions,andhow to protectit in casethequenchesshouldthreatenthe integrity of themagnet.A simulationpackagecalledQUABER [12] hasbeendevelopedat CERNin orderto investigatethebehaviour of acceleratorsuperconductingmagnetsin theeventof a quench.This packagehasbeenbuilt up in theenvironmentof thecommercialnetworkanalysisprogramSABER[13], usingits associatedhigh level designlanguageMAST.

Thenetwork solver dealswith a setof differentthermo-electricalmodelscontainedin templates(simulatorsubprograms),whichmodelthequenchspreadthroughoutthecoilsaccordingto thedifferentpropagationmechanisms(original quenchandits propagation,heater-provoked quenchandits propaga-tion, etc.). In the definition of the electricalcircuit the differentblocksof the magnetarerepresentedby coupledinductancesandvariablequenchresistances.In the calculationof both quenchresistancesandtemperaturemaps,the simulatortakes into accountthe magneticfield distribution throughoutthecoils. The distribution of the magneticfield aswell asthe mutualandself-inductancesarecalculatedwith ROXIE andinterfacedinto QUABER.

Thebasicequationthatlinks thetemperatureof theconductorwith thecurrentis givenby theheatbalanceperunit of volumeunderadiabaticconditions[14]:k

kml �Z @ X�BD��X ; �onp�5q q

q lr @ �sBt on @ �A"�VA W�W�W B

���4A(10)

24

(K

)

0.0

50.0

100.0

150.0

200.0

250.0

(A

)

0.0

2500.0

5000.0

7500.0

10000.0

12500.0

(V

)

0.0

10.0

20.0

30.0

40.0

50.0

t(s)0.0 0.1 0.2 0.3 0.4 0.5 0.6

Hot spot temperatureMagnetcurrentu

Maximum layer voltagev

Fig. 4: Resultsof quenchsimulation

where �5on is the sectionof copperof a conductor, �q the whole sectionof the conductor, C(T) theaveragespecificheatof theconductor, andtwon @ sA"�4A W�W�W B is theresistivity of thecopperasa functionof magneticfield, temperatureandtheresidualresistivity ratio (i.e. theratio betweencopperresistivityat300K andat4.2K in absenceof magneticfield). Thetermontheleft sideof theequationexpressedinunitsof

&�� !is calledMIIT andrepresentsthequenchload. Thetermon theright representsthequench

capacityof agivensuperconductingcable.Fromthisequationthemapof temperatureT in thecoilsasafunctionof time canbecalculated.

It is assumedthataquenchstartsin thehighfield regionof theouterlayeratnominalcurrent(11.8kA) andpropagateslongitudinallyandtransversally to theneighbouringturns. The initial longitudinalquenchvelocity is 15m/sandthetransitionpropagatestransversallywith a turn-to-turndelayof 30mil-liseconds(valuesaccordingto experimentalmeasurements).The magnetis protectedby strip heaterscoveringthefull lengthof 6 turnsperpolein theouterlayershells.Thequenchheatersareeffectiveaftera delayof 80 millisecondsfrom thequenchonset.Sincethemagnetis by-passedthrougha cold diode,thecoils will dissipatethe full energy storedin themagnet.Themaximumtemperature(hot spot)andthemaximumvoltageacrossonelayer in onepole (seeFig. 4) areacceptableandqualify this magnetfrom thesimulationpoint of view for theoperationundertheLHC machineconditions.

8 3d coil endgeometryand field optimization

Whendesigningthe3d coil geometry, theshapeof thecoil is determinedby theobjectivesof maximiz-ing the radiusof curvaturein the end,applyingas little hard-way strainaspossibleto the cable,andoptimizingthemultipolecontentof theintegratedfield. Theinputparametersfor thecoil endgenerationarethez positionof thefirst conductorof eachcoil block, its inclinationangle,thestraightsectionandthesizeof theinter turn spacersbetweentheconductors.It is assumedthat theupperedgesof thecon-ductorsfollow ellipses,super-ellipsesor circlesin thedevelopedplanedefinedby their radialpositioninthestraightsection.A de-keystoningfactorcanbedefinedfor thepurposesof consideringacable-shapechangein the endscausedby the winding processand the fact that a Rutherford-typecablemadeofstrandsdoesnot have thepropertiesof a solid beam.By shifting therelative positionof thecoil blockstheintegratedmultipolefield canbeoptimized.For thispurposeanobjective weightingfunctionis usedandthealgorithmEXTREM is applied.As the3dcalculationsarevery timeconsumingonly 60functionevaluationsareperformed.

25

-11.242 -8.5124-

-8.5124 -5.7823-

-5.7823 -3.0522-

-3.0522 -0.3221-

-0.3221 2.40799-

2.40799 5.13809-

5.13809 7.86819-

7.86819 10.598-

10.598 13.328-

13.328 16.058-

16.058 18.789-

18.789 21.519-

21.519 24.249-

24.249 26.979-

-0.6958 -0.3366-

-0.3366 0.02258-

0.02258 0.38177-

0.38177 0.74097-

0.74097 1.10017-

1.10017 1.45937-

1.45937 1.81857-

1.81857 2.17777-

2.17777 2.53697-

2.53697 2.89617-

2.89617 3.25537-

3.25537 3.61457-

3.61457 3.97377-

3.97377 4.33297-

Pressure on broad face

of cable, positive away

from the pole (N/mm**2)x

Pressure on narrow face

of cable, positive in out-

ward direction (N/mm**2)y

Fig. 5: Artist view of thecoil endtogetherwith theelectromagneticforcesparallelandperpendicularto thebroadfaceof the

cable,displayedasa grayscale.

The averagefield errorsalongthe coil endfrom the onsetof the endtowardsz=80 mm arefortheoptimizeddesign

!=-3.7,

.10=-0.066,

.10=-0.0035(all in unitsof

&�� +-,) for anequivalentmagnetic

lengthof theendof 32mm. As themagnetis long(3.10m magneticlength)with respectto lengthof thecoil end,thefield errorsresultingfrom thecoil enddesignareacceptable.Fig. 5 shows anartistview ofthecoil endtogetherwith theelectromagneticforcesdisplayedin thelocalconductorcoordinatesystem,i.e.,parallelandperpendicularto thebroadfaceof thecable.Fig. 6 shows acut throughthecoil end(yzplane)andtheelectromagneticforcesin azimuthaldirectionactingonthecoil end.Thesizeof theshimsplacedat themid-planebeforecollaring theendhasto bechosensuchthat theprestressis higherthantheelectromagneticforcesat full excitation.

9 Toleranceand manufacturablity analysis.

From the sensitivity matrix (which canbe transferredvia a CSV interfaceinto spread-sheetprograms,e.g.,EXCEL) themultipolecontentcanbeevaluatedasa functionof the tolerancesin coil block posi-tioning, coil size,asymmetriesresultingfrom thecollaringprocedureetc. This matrix is too big to beshown here,however, importantconclusionscanbedrawn suchas:

� For thecoil optimizationit is assumedthattheshapeis determinedby thewindingmandrelon theinnerdiameterof thecoil. If, however, aftercuringandcollaringthecoil shapeis determinedbythecollaron theouterdiameterof thecoil, thedisplacementswouldcausea

"!of 0.37unitsanda �.10

of 0.006units.Thatis virtually aslargeastheeffect of thepersistentcurrents!

� Onecoil beingradially displacedby only 0.01mm would cause �.

and^w.

of z 0.78units, "c

and^ cof z 0.36unitsanda

!of 0.47units.

26

0{

20 40 60|

80}

100

-700

-600

-500

-400

-300

-100

0{

z (mm)~

F /z (N/mm)

Fig. 6: Electromagneticforcesin azimuthaldirectionat nominalfield actingon thecoil end. Thesizeof theshimsplacedat

themid-planebeforecollaringtheendhasto bechosensuchthattheprestressis higherthantheelectromagneticforcesat full

excitation.

Thesetwo examplesshow how importantit is to control thecoil positioningduringtheassemblyprocessof themagnet.

10 Production of drawings by meansof the DXF interface.

TheDXF interfacecreatesfilesfor thedrawing of thecross-sectionin thexy andyz planesof themagnet,thedevelopedview andthepolygonsfor theend-spacermanufacturethuseliminatingdraughtingworkof about2 man-days.

11 End-spacerdesignand data transfer for the CNC machining

Theshapeof theend-spacersis determinedby theshapeandpositionof thecoil blocksasfoundin thefield optimizationprocess.Thesurfacesto bemachinedaredescribedby 9 polygons,which aretrans-ferredinto a CAM system,e.g.,CATIA, for thecalculationandemulationof thecuttermovementsformachiningthepiece.As aninterfaceanASCII file, a VDA file anda DXF file is available.Thespacersaremachinedby meansof a 5-axisCNC machinefrom glass-epoxytubes(G11). Becauseof theabra-sive natureof theglassdust,diamondtoolsmustbeused.Fig. 7 shows anartistsview of theendspacers(outerlayercoil) for thequadrupole.

12 Inversefield calculation for the tracing of manufacturing errors.

The dimensionsof the active partsof the coils are impossibleto verify undertheir operationalcondi-tions after their deformationdue to manufacture,warm pre-stressing,cool-down andexcitation. Theinverseproblemsolvingconsistsof usingoptimizationroutinesto find distortedcoil geometrieswhichexactly producethemultipolecontentmeasured[16]. Thefunctionto beminimizedin theinversefield

27

Fig. 7: Artists view of theendspacersfor thequadrupole,outerlayercoil

computationproblemyields

�����.10��� .�� �

( @ /�� @V78 B = � B Z b�� � ( @ ^-�� @V78 B = ^ � B Z (11)

where �� @V78 B

,^ �� @478 B

arethecalculatedand � , ^ � arethemeasuredmultipoles. 78 is thevectorof the

designvariablesfor the inverseproblem. The � � and� � areweighing factorsthat compensatefor the

differentnumericalvaluesof theresiduals.Becauseof thenon-symmetricnatureof thegeometricalcoilpositioningerrors,a largenumberof designvariablesresultfor theinversefield problem.It is thereforeassumedthat the positioningerrorshold for an entirecoil block ratherthanfor individual conductors.The designvariablesare the possibleperturbations,in radial direction, of all 24 coil-blocks plus 16azimuthaldisplacementsof theblocks.It is assumedthattheblocksthatareconnectedat themid-planearefreeto moveonly by thesameamount.Becauseof thefactthattherearefarmoredegreesof freedomthanobjectivestheproblemis ill-posed.Thereforea regularization termis addedto Eq. (11)cD0

��� .� � ( @ � � = ��B Z

(12)

to make surethat thecoil-block displacementsstayassmall aspossible.As a minimizationalgorithmtheLevenberg-Marquardmethodis applied.

Table4 gives the measuredmultipole distribution [15] in the straightpart of a previously builtquadrupolemagnetwith slightly differentconductordistribution [2] togetherwith theexpected(intrin-sic) values.

28

0 10 20 30 40 50 60

ROXIE4.4

06/03/97 15.18�

Fig. 8: Coil-blockdisplacementof a quadrupoleprototype,biggestvectorrepresentsa 0.25mmdisplacement

Measured Intrinsicn Normal Skew Normal Skew1 - - - -3 0.27 0.36 - -4 0.01 -0.21 - -5 -0.03 0. - -6 -0.23 -0.02 0.107 -7 0.01 0. - -8 0. 0. - -9 0. 0. - -10 -0.01 0. -0.0087 -

Table4: Measuredmultipolesin thestraightpartof oneapertureandintrinsic valuesasexpectedfrom thecoil design,unitsof

&�� +-,at10 mm,current10000A

Theresultof theinverseproblemcomputationcanbeseenin Fig. 8. Thearrow lengthof themostimportantdisplacementcorrespondsto 0.25mm in block no. 24. All otherdisplacementarrows aretoscale.Theseresultsalsoallow to checktheneedfor amandrelinsidethecoil aperturein thefinal collar-ing phase(in fact,thecoil assemblymandrelwasextractedbeforethefinal compressionof thecollars).Thedisplacementsin Fig. 8 show someinwardmovementof thesmallcoil blockswhichpossiblycouldhave beenavoided.

REFERENCES

[1] CERN, EuropeanOrganisationfor NuclearResearch.:The Large HadronCollider, ConceptualDesign,CERN/AC/95-05

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