ESTIMATE OF SATURATION INDUCED MULTIPOLE ERRORS INTHE LHC MAIN DIPOLES

C. VöllingerCERN,1211Geneva 23,Switzerland

AbstractSincethepublicationof the“Yellow Book” [1] in 1995considerablechangesinthecoil andyokegeometryhavetakenplace.Theaimof thisreportis to giveadetailedsummaryof themultipoleerrorsandsensitivity to manufacturingtol-erancesfor theoptimizedyoke geometryMBP2, andanalternative separatedcollar design.Theresultsarecomparedto theprevious“Yellow Book” designMBP1.

1 Intr oduction

Sincethe publicationof the designstudyof LHC in the “Yellow Book” [1] a numberof changesandmodificationsin theyoke geometryandcoil geometryhave taken place. Theaim of this reportis thusto give a detailedsummaryon the expectedmultipole errors. The studyalsoshows the estimatesforsensitivity of multipoleerrorswith respectto manufacturingtolerancesof thegeometrytypes.

The6-blockcoil designwhichpartlycompensatesthe���������

and���

errorsresultingfrom persistentcurrentsat injectionandwhichis foreseento beusedfor thenext prototypeswasinsertedinto thetwo dif-ferentyoke geometries.For comparison,theyoke geometryasdescribedin the“Yellow Book” (MBP1)andusedfor the last 10 m modelswas calculatedagainwith the 6-block coil in order to distinguishbetweeneffectsfrom thecoil changesandeffectsfrom theyoke modifications.

Theoptimizedyokegeometry(MBP2)includingthe6-blockcoil wascalculated.For bothgeome-tries,expectedrelativemultipoleerrorsfor nominalandinjectioncurrentwereestimated.Thesensitivityof both geometriesdue to manufacturingvariationswas investigated.Finally an alternative yoke ge-ometrywith separatecollarswascalculatedandthe resultscomparedto the first two geometries.Allcalculationsweredoneusingthe programpackageROXIE 6.0 which wasdevelopedat CERN for thedesignandoptimizationof superconductingmagnetsfor theLHC.

TheprogrampackageROXIE solvesthenonlinearfield problemwith thefinite-elementmethod(FEM) implementedin theFEM2D program( c

TU-Graz/Austria).This methodusesa reducedvector

potentialformulation.Theadvantageof thismethodis, thatthecoilsdonothave to berepresentedin thefinite elementmesh,which allows to modelthecoil andiron yoke separately. A detailedintroductiontothereducedvectorpotentialformulationis givenin [2].

2 Coil cross-section

Thecoil usedin thisstudyis the6-blockcoil whichwasintroducedin October97[3] aftertheoptimiza-tion with the programpackageROXIE usinggeneticalgorithmsandthe conceptof niching [4]. Thisoptimizationresultedin two alternative solutionsto replacethe 5-block coil describedin the “WhiteBook” [5]. Figure1 shows thecross-sectionof the6-blockcoil which wasreferredto asV6-1 modelin[1].

3 The optimized yoke geometry(MBP2, July 98)

Figure2 showsadrawing of theyokegeometry(MBP2)with symbolicdimensionsasusedin theROXIEinputfile. All dimensionsshown arelaterusedasdesignparametersfor thesensitivity analysis.Table12

93

0 10 20 30 40 50 60 70

Fig. 1: Coil cross-section

Table1: Geometryspecificationsfor conductorblocksof the6-blockcoil

Block no. Conductor no. Radius (warm ) Pos.angle Incl. anglein spec.block � �

1 9 43.9 0.157 0.2 16 43.9 21.9 27.3 5 28. 0.246 0.4 5 28. 22.02 24.085 3 28. 47.71 486 2 28. 66.71 68.5

in Annex 1 shows thecold dimensionsof theyoke laminationsconsideredfor this study. All valuesaregivenin mm. TheROXIE modelcross-sectionof theyoke geometryis displayedin Fig. 2.

Thegeometry(MBP2) featuresacombinedcollarwith two differentelliptical shapesin thecenterpartandin theouterpart resp.,with minor axis “alel2” and“alel1” asshown in Fig. 2. Using the twodifferentelliptical shapesprovidesmoredegreesof freedomfor theoptimization,sothat thesmallholein thecenterof themagnet(rhol2 in Fig. 4) is no longerneededto control themagneticflux. Figure2shows thattheiron insert,whichwasenclosedin thealuminiumcollar is now takenout. In addition,theheightof thecollar “blel” hasbeenincreasedfrom 96.35mmto 99.8mm.

Comparedto thegeometryof theMBP1 magnet,thegeometryof theMBP2 magnetis lesssensi-tive to manufacturingerrorsasshown in section6. Table2 shows theexpectedrelative multipoleerrorsin unitsof ������ at 17. mm at injectionandnominaloperationandthemaximumvariationof themulti-poles.Figure3 representsthevariationof multipolecomponentsversusexcitation. It canbeseenfromthecurvesandfrom Table2 thatall multipoleshave little variation( � ����� � ����� � � � � � ��� ) while rampingup to 8500A. For currentshigherthan8500A thereis a decreaseof themultipolecomponents

���and� � , while ��� is slowly increasing.

94

Table2: Expectedrelative multipole errorsin units of ������� andmaximumvariationof the multipolesof the MBP2 magnetgeometryat 17. mm radius

At injection At nominal Maximum variation(850A/0.60T) (11800A/8.31T) ���! �#" 0.64631 -0.81680 1.46311�%$ 5.77494 6.17603 0.40109� � 0.21687 0.06213 0.15474�%& -0.91376 -0.91403 0.00027�%' -0.00108 -0.00445 0.00337�#( 0.63236 0.63525 0.00289�%) -0.00025 -0.00030 0.00005�%* 0.10325 0.10388 0.00063�,+.- 0.00033 0.00030 0.00003

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Fig. 2: Sketchof MBP2 magnetgeometry(left), ROXIE modelof thecoldmass(right)

2000 4000 6000 8000 10000 12000

-10

-8

-6

-4

-2

0

2

4

I(A)

bn/

6

b2

b3

b4*10

b5*10

Fig. 3: Multipole field errorsversusexcitationof theMBP2 geometry(July98version)in unitsof ��� �0� at17. mm radius

95

4 Yokegeometryof the MBP1 (yellow book) model

For thesake of completenessthe resultsfor theYellow book yoke geometryaregiven. It is importantto notethat thesame6-blockcoil geometryis usedin orderto comparetheiron saturationeffects.Theresultsare therefore not valid for the industry produced10 and 14 m long dipole modelswith the5-block coil.

Figure4 showsadrawing of theyokegeometryof theMBP1magnetwith thesymbolsusedin theROXIE input file. Thevaluesof thedimensionsarelisted in Tab. 11 andthecross-sectionis shown inFig. 4. On thesamepagethemultipolecontentfor both injectionandnominalfield (without persistentcurrenteffects)andthevariationof thelowerordermultipolesversusexcitationaregiven.

5 Alter native yoke geometrywith separatedcollars

Theusageof stainlesssteelasacollarmaterialwouldallow to useseparatedcollars.Dueto theintrinsicsymmetrywith respectto thebeamaxisat low field level (iron yokeactingasperfectlysymmetricscreenof thetwo apertures)basicallyno

���and

� � componentsaregeneratedat injectionfield level. Thereforean alternative collar geometryfeaturingseparatedelliptic collars hasbeenoptimized, the multipoleswerecalculatedandthesensitivity wasinvestigatedandcomparedto theresultsof theMBP1andMBP2dipole cross-sections.A drawing of the separatedcollar geometrycorrespondingto Tab. 13 and theROXIE modelcross-sectionareshown in Fig. 6.

Thegeneralideasusingseparatedcollarsarethereductionof themagneticflux couplingof left andright apertureat injectionlevel andto geta symmetriccollar with respectto thebeamcenterat 97. mm.Thissymmetryallow to flip thecollarsduringassemblysothatpunchingerrorsdonotproduceunwantedasymmetrieswith respectto thebeamaxis.

In orderto balancetheflux in thecenterandouterpartof themagnetandthereforeto reducethe���componentat higherfield levels,a drop-shapedholehasto be punchedinto the iron insertbetween

thecollars. Table4 shows therelative multipoleerrorsin unitsof ������ andthevariationof multipoles,bothat radius17. mm for theseparatecollargeometry.

96

Table 3: Expectedrelative multipole errors in units of ������� and maximumvariation of the multipolesof MBP1 magnetgeometryat 17. mm radius

At injection At nominal Maximum variation(850A/0.61T) (11800A/8.38T) ���! �#" 0.53854 2.40625 2.25045�%$ 1.92057 3.45279 1.59528� � -0.53237 -0.72687 0.19157�%& -0.74902 -0.81090 0.06193�%' 0.02300 0.00649 0.01704�#( 0.61132 0.62349 0.01337�%) 0.00104 0.00070 0.00070�%* 0.10032 0.10184 0.00222�,+.- 0.00016 0.00009 0.00031

01

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Fig. 4: Sketchof yoke geometryMBP1 (left), ROXIE model(right)

2000 4000 6000 8000 10000 12000

-8

-6

-4

-2

0

2

I(A)

bn/

4

b2

b3

b4*10

b5*10

Fig. 5: Multipole field errorsversusexcitationof MBP1 magnetgeometryin unitsof �2� ��� at 17. mm radius

97

Table4: Expectedrelative multipoleerrorsin unitsof ���3�0� andmaximumvariationof themultipolesof theseparatedcollargeometryat 17. mm radius

At injection At nominal Maximum variation(850A/0.62T) (11800A/8.57T) ���! �#" -0.05831 0.03458 0.88402�%$ 11.00844 11.43991 0.41193� � 0.00009 -0.18323 0.16307�%& -1.14665 -1.02441 0.11434�%' 0.00002 -0.01329 0.01227�#( 0.61050 0.61688 0.00577�%) -0.00003 -0.00059 0.00052�%* 0.09865 0.10075 0.00192�,+.- -0.00003 -0.00001 0.00001

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Fig. 6: Sketchof yoke geometrywith separatedcollars(left), ROXIE model(right)

2000 4000 6000 8000 10000 12000

-10

-5

0

5

I(A)

bn/

10

b2

b3

b4*10

b5*10

Fig. 7: Multipole field errorsversusexcitationof separatedcollar geometryat17. mm radius

98

6 Sensitivity analysis

Constrainton computingpower do not allow to calculaterandomerrorsdueto yoke tolerancesin thesameway asit wasdonefor thecoil block deformationswhere500randomlygeneratedcoil structureswerecalculatedandthestandarddeviation wascalculatedfor eachof thefield components.Sensitivityof coil blockdisplacementandrandomerrorscanbefoundin [3].

Thesensistivity analysisfor theyoke is performedby calculatingtheJacobianmatrix containingthederivativesof themultipoleswith respectto all specifieddimensions.

4 �!54�687 9;: -=<

>#?�@AB �,+BDC +FE�E�E B �!GBHC +...

...B � +BHC E�E�EB � GBDC 9;: -=<

E (1)

Thederivativesarecalculatedusingtheupperdifferentialquotient I8J 6LKDM#M ?�NDO :I8J 6 KDM#M ?�N O AQP J

6 KHM#M ?�N OSR P J 6LT�UWV N U O68KHM#M ?�NXR 6 T�UWV N U ETheRMS (root meansquare)valuefor eachof themultipolescanbeusedfor thecomparisonof

thetolerancesensitivity of thedifferentdesigns.For theMBP1 andtheMBP2 geometry, thesensitivityof multipolesis investigatedwith respectto 22resp.27designparameters.In caseof theseparatedcollargeometry, 20 tolerancesareinvestigated.The value

4 ��Ygiven in the tablesindicatesthe sensitivity of

themultipole��Y

in unitsof ������ at thereferenceradius.If for examplethemultipole ��� hasasensitivityvalueof 0.50867E-03to thedesignparameter‘ryoke’ at injectioncurrent,achangein

���of about0.0005

unitsin ������ canbeexpectedin casethattheparameterryoke hasa toleranceof 1 mm.6.1 Comparisonof sensitivity of MBP1 and MPB2 geometry

Table 14 and Tab. 15 show the sensitive parametersas result of the sensitivity analysisfor the yokegeometry(MBP2 magnet)while thesensitive parametersfor theMBP1 magnetgeometryareshown inTab.16 andTab. 17. The resultof thesensitivity analysisof theseparatedcollar geometryis shown inTab. 18 andTab. 19. Thevaluesin boldarethemostsensitive dimensions.

In thesetablesthe leastsensitive designparametersareomitted. It canbe seenthat in generalthe sensitivity to manufacturing tolerancesis higher at a higher field level, as due to iron saturationparametersfurther away from the beampipe, e.g. outer yoke radius,radiusandposition of the heatexchangertubeetc.aregettingmoresensitive.

It canbeseenfrom thetables,that theMBP1 geometryis muchmoresensitive to manufacturingtolerancesthantheMBP2 geometry, e.g.for theyoke geometryMBP1, the

���componentis very sen-

sitive to five out of the22 designparametersat injectioncurrentwhile thereareonly four of 27 designparametersthat arevery sensitive in caseof the MBP2 geometry(very sensitive definedhereasbeingaboutanorderof magnitudemoresensitive thenall theotherparameters).In caseof theMBP1geometrytheworstcasevalueat injectionfield level for

���is 10.098units/ mmat17. mmradius.This is oneorder

of magnitudehigherthantheworstcasevaluefor���

(7.2568units/ mm) in caseof theMBP2geometry.

6.2 Comparisonof sensitivity of MBP2 geometryand separatedcollar geometry

Table18 andTab. 19 show thesensitive parametersof theseparatecollar geometry. Comparisonwiththesensitivity investigationof MBP1 andMBP2geometryshow thattheseparatecollar geometryis theleastsensitive designboth at injection andnominalcurrent. Becauseof the separatedcollarsandthesymmetricellipsewith respectto the beamcenterat 97. mm, the RMS valueof multipoleswith evenindices(

���,� � , ��Z ) areordersof magnitudelower thanfor theMBP1andMBP2geometries.

99

Themultipole���

in caseof theMBP2geometryis sensitive to four of 27designparameterswhile���in caseof theseparatedcollar geometryis sensitive to noneof the30 designparameters.Theworst

casevaluefor���

in caseof theseparatedcollar geometryis threeordersof magnitudelower thanit is incaseof theMBP2geometryandthesensitivity of themultipoles

� � and ��Z in caseof theseparatedcollargeometryarenegligible.

7 Conclusion

Table5 shows themostimportantresultsat a glance.In the following subsectiontheoptimizedMBP2geometryis comparedto the MBP1 geometry. All objectivesareimproved in the new design.Subse-quentlythemeritsof theseparatedcollargeometryarecomparedto theMBP2geometry.

Table5: Comparisonof mainparametersof MBP1 andMBP2 magnet,andseparatedcollar geometry

MBP1 MBP2 Separatedcollar

Curr ent @ 8.36T 11772A 11871A 11500Aryoke(cold), (mm) 274.5 274.5 274.5

collar hight in y-dir ection (cold), (mm) 96.35 99.80 90.82[�\�\(T) 9.69 9.68 9.75]�^

@ inj. 0.539 0.646 -0.058_ ] ^2.250 1.463 0.884] ^

(RMS) @ inj. 4.59 3.71 0.002]�`@ inj. 1.921 5.775 11.008_ ] `

1.595 0.401 0.412]�`(RMS) @ inj. 1.024 0.926 1.600]�a

@ inj. -0.532 0.217 0.00009_ ]2a0.192 0.155 0.163] a

(RMS) @ inj. 0.070 0.061 —]�b@ inj. -0.749 -0.914 -1.147_ ] b

0.062 0.0003 0.114]�b(RMS) @ inj. 0.030 0.026 0.047

mostsensitivedesignparam.]2^

race1 d1,alel1 —mostsensitivedesignparam.

] `colly houdh,blel1 houdh

7.1 ComparisonbetweenMBP1 and MBP2 yokegeometries

Comparisonof theoptimizedMBP2yokegeometryandtheMBP1geometryshowsthefollowing meritsfor theMBP2version:

c Althoughthemultipole ��� of theMBP2 yoke geometryat injectionis about0.1unitshigherthanfor theMBP1 yoke geometry, thevariationis 0.4units lower. At thesametime theamountof the� � componentcouldbereducedby thefactorof 2. Also thevariationof the � � componentcouldbeslightly reduced.

c Themultipole ��� hasa highernumericalvalueat injection,but this is desiredbecauseof thepartcompensationof thepersistentcurrents.Thevariationof

� �wasagainconsiderablyreduced.

c slight reductionsof theamountof themultipoles ��Z (reducedfrom 0.023to -0.00108at injectionand0.0065to -0.0045at nominalcurrent)andof

��d(reducedfrom 0.001at injectionto -0.00025

and0.0007to -0.0003at nominalcurrent)wereachieved.

100

7.2 ComparisonbetweenMBP2 and separatedcollar geometries

Comparisonof theseparatedcollar geometryto theMBP2yoke designshows thefollowing merits:

It can be seenfrom Tab. 4, that the multipolesat injection field level in caseof the separatedcollargeometryareextremelylow comparedto theothergeometries,exceptfor thevalueof

���. However, the���

componentcaneasilybecorrectedby thecoil design.Themeritsare:

c Theamountof multipole ��� and � � is considerablylower at injectionfield level. At thesametimethe sensitivity to manufacturingtolerancesis basicallynegligible. Also the variation of the

���multipoleerrorsversusexcitationcouldbereduced.All theothervariationsareaboutthesameorsmallerfor theseparatedcollar design.

c Thenominalcurrentis reducedfor about370A.c The e T�T shortsamplefield increasesby about0.07T to 9.75T.

7.3 Payoff-tables for MBP2 and SeparateCollar Geometries

Onepossibilityto provide thehiddenresourcesof differentgeometriesis apayoff-table. Themathemat-ical principleandtheuseof payoff-tablesareshown in [6]. Table6 andTab. 7 show thepayoff-tablesof theMBP2magnetgeometryandseparatecollargeometry, respectively. Thepayoff-tablesarecreatedafterrunningthegeometriesfor six differentoptimizationobjectives f 7 (here:max. e T�T , min � ��� , min� ��� , min � � � , min � ��g #h andmin ryoke). Thevaluesin bold typeindicatea kind of ”utopia point” andthereforegive additionalinformationaboutthe geometries.Of courseit is not possibleto achieve allvaluesgivenasutopiapointat thesametime. However it shows theusertheattainablevaluesfor certainobjectivesandhow thecorrelatedparameterschangeaccordingly.

Table6: Payoff-tablefor theMBP2 magnetgeometry(top)andthedesignparameters(lower)[i\2\ _ ] ^ _ ] ` _ ]2a ]2akjml#nryoke oqpr

8.2T s 4.0 s 3.0 s 0.3 s 0.15 s 275.mmmax.

[ \�\-9.7334 0.7580 1.4638 0.0859 0.1029 284.83 out

min_ ]�^

-9.6630 0.0365 0.5513 0.0769 -0.1163 284.16 o ^min

_ ]�`-9.6536 3.8471 0.0022 0.2670 0.0850 266.08 o `

min_ ]�a

-9.6601 0.8226 0.7214 0.0575 -0.0469 284.99 o amin

]2a,jvl%n-9.6593 1.2652 0.3064 0.1142 0.0000 277.33 o b

min ryoke -9.6820 1.0999 0.4827 0.2388 0.0948 265.53 oxwoyp alel1 blel1 alel2 lh2r leang houdh z \�\

(mm) (mm) (mm) (mm) (deg) (mm) (A)

out 75.84 95.00 76.08 7.22 22.14 4.04 13646o ^ 92.94 102.20 84.44 9.15 33.92 9.21 13894o ` 86.73 102.60 88.23 17.24 41.69 4.65 13935o a 86.39 104.99 95.94 7.03 34.26 11.99 13962o b 88.52 103.47 89.02 13.43 39.75 6.91 13922oxw 82.96 97.40 79.31 16.14 29.88 10.54 13836

It canbeseenfrom the tables,that therearehiddenresourcesboth for theMBP2 andfor theseparatedcollargeometry:E. g.anincreaseof thequenchmargin e T�T is possiblefor bothgeometriesif at thesametimeahighermultipolevariationin particularfor � ��� is admitted.

101

For theMBP2 magnetgeometry, Table6 showsthatthequenchmargin canbeincreasedupto 9.7331T.This increaseis payedfor by therelatively high variationof multipole

���. Thefact,thatthereis a corre-

lation betweenthemultipolevariationof � ��� and � ��� canalsobeseenfrom runningtheminimizationfor � ��� . This minimizationyieldsa heavy increaseof � ��� up to 3.8471unitsof �� ��� andactivatedtheboundsbeingsetfor � ��� .

Table7: Payoff-tablefor theseparatedcollar geometry(top)andthedesignparameters(lower)[i\2\ _ ] ^ _ ] ` _ ]2a ]2akjml#nryoke oqpr

8.2T s 4.0 s 3.0 s 0.3 s 0.15 s 275.mmmax.

[�\2\-9.9172 1.3829 2.5146 0.1645 -0.0003 282.10 o t

min_ ] ^

-9.7785 0.7679 2.4343 0.1988 -0.00002 275.00 o ^min

_ ] `-9.7409 2.2116 0.0455 0.1168 0.0002 280.11 o `

min_ ]2a

-9.7680 3.9071 1.6269 0.0288 -0.00005 284.80 o amin

]�a,jvl%n-9.7677 1.4132 0.2748 0.2223 0.0000 272.30 o b

min ryoke -9.7509 1.2166 2.2148 0.2238 0.0000 270.00 oxwo p alel1 blel1 sstax sendx z \2\

(mm) (mm) (mm) (mm) (A)

o t 70.27 73.06 2.63 7.31 13074o ^ 88.86 84.97 4.40 12.40 13467o ` 92.47 94.35 4.22 12.94 13630o a 84.59 99.25 4.55 11.10 13625o b 87.83 89.78 4.36 10.10 13546oxw 85.80 97.85 6.78 11.94 13656

In caseof theseparatecollar geometry, themaximumvalueof thequenchmargin e T�T canbeincreasedup to 9.9172T. It canbeseenfrom thetable,thatthereis alsoapayoff with respectto thevariation � � �which increasesto 1.3829unitsof �� ��� . For this geometrya correlationbetween� ��� and � � � exists:Theminimizationof multipolevariation � � � activatedtheboundsbeingsetfor thevariationof multipole���

.

ComparisonbetweentheMBP2magnetgeometryandtheseparatedcollargeometryshow thatthemaxi-mumattainablequenchmargin of theseparatedcollargeometryis considerablyhigherandthereforetherearemorehiddenresourceswith respectto quenchmargin for thisgeometry. Themaximumquenchmarginof 9.9172T wasachievedwithout activating theboundsbeingsetfor themultipolevariationandat thevery low valueof 13074A for { T�T .Both geometriesshow thepossibility to reducemultipole

� ��g #h at injectionfield level to zero.Sincetheseparatedcollar geometryalreadyhasa certainsymmetrywith respectto thebeamcenter, thevaluesof� ��g =h remainsvery small in the payoff-table andthe minimizationof � ��g #h mustnot be payedfor withhigh increasesin multipolevariation.

8 Numerical accuracyof the calculations

In thischapterthenumericalaccuracy of thecalculationswith respectto meshsizeandfarfield boundaryconditionis investigatedusingtheMBP2magnetgeometryascasestudy.

Table8 andTable9 show both theestimatedmultipolescalculatedfor theMBP2 yoke geometryataradiusof 17mmfor differentnumbersof nodesin thefinite elementmesh.Theresultsarecalculatedat a current level of 850 A and 11800A, respectively. The meshis shown in Fig. 8. Creatingthefundamentalmeshfolloweddifferentcriterion:

102

Fig. 8: Finiteelementmeshof yoke area,MBP2 geometry

Table8: Relative multipoleerrorsasa functionof meshsizeat injectioncurrent(850A)

Number of nodesin the finite elementmesh1052 5000 7205 12374 15054

n � � � � � 2 0.04213 0.67104 0.66161 0.64631 0.640223 7.31497 5.77796 5.77433 5.77494 5.774684 0.26855 0.21855 0.21796 0.21687 0.216575 -1.43042 -0.91466 -0.91419 -0.91376 -0.913676 0.10758 -0.00097 -0.00111 -0.00108 -0.000987 0.77981 0.63242 0.63262 0.63236 0.632318 -0.08235 -0.00156 -0.00155 -0.00025 -0.000229 0.23976 0.10342 0.10310 0.10325 0.10334

10 -0.03666 0.00276 0.00319 0.00033 -0.0000511 0.52352 0.61315 0.61183 0.61215 0.61207

Table9: Relative multipoleerrorsasa functionof meshsizeatnominalcurrent(11800A)

Number of nodesin the finite elementmesh1052 5000 7205 12374 15054

n �! �! �! �! �! 2 -1.56275 -0.87332 -0.81775 -0.81680 -0.815513 7.37051 6.19499 6.17812 6.17603 6.175884 0.11453 0.05955 0.06286 0.06213 0.062035 -1.26775 -0.91509 -0.91444 -0.91403 -0.913976 0.07685 -0.00447 -0.00447 -0.00445 -0.004367 0.77056 0.63538 0.63549 0.63525 0.635218 -0.05241 -0.00148 -0.00147 -0.00030 -0.000289 0.09319 0.10406 0.10373 0.10388 0.10396

10 -0.03717 0.00248 0.00287 0.00030 -0.0000411 0.68906 0.61674 0.61538 0.61568 0.61561

c A symmetricmeshin theapertureis desiredin orderto limit errorsonthequadrupoleandoctupolecalculationdueto meshasymmetries.

103

c In theaperturea rectangularshapedregularmeshis chosento reducetheinfluenceof badcondi-tionedfinite elements;for thesamepurpose,thediscretizationin thebeamcenterareais high.

Theresultsshow convergencefor mesheswith morethan12000nodes.It canbeseenfrom Tables8 and9 thatthemultipoles

���and

���arerelatively sensitive to themeshsizewhile themultipoles

���,��Z

,���,� d

and���

arelesssensitive. Notethattherelatively highvaluesfor the���

,���

,���

multipolecomponentsarearisingfrom thefactthatthepersistentcurrentsat injectionfield level arepartly compensatedby thecoil designandarethereforeintentional.

Whensolving differential equationswith approximative solutions,two different typesof equa-tionsexist: initial valueproblemsandboundaryvalueproblems. Calculatingthefield quantities| and}

yieldsaboundaryvalueproblem,wherein additionto thedifferentialequation,informationaboutthesolutionand/orsomederivatesis specifiedat theboundary~ of thecalculationarea . Theboundaryconditionsarisefrom physicallaws andmustbefulfilled by thefield quantities| and } . In caseof thefinite elementsolutionfor the given two-in-oneaperture,the vectorpotentialhasto fulfil the Dirichletboundaryconditionat the coordinateaxis andat the far field boundary. Sincefinite elementcalcula-tions needa closedsurfaceboundingthe calculationarea,ideal boundaryconditionsare regardedasinfinitely far away. Thereforethe chosenradiusfor settingthe real boundarymustbe sufficiently faraway from theyoke radius. Becauseof theusedsymmetry, the far field boundarycorrespondsto a 90degreesarc. Table10 shows therelative multipoleerrorsin unitsof ������ calculatedat nominalcurrent(11800A) dependingon thefar field boundarycondition.Thefar field boundaryis chosenat 1.3,2., 5.,7. and10. timestheouteryoke radius.It canbeseenthat themultipoles

��,���

,���

,� � vary with respect

to thedistanceof thefar field boundary. Thehigherordermultipoles���

,��Z

,���

,� d

arehardlyinfluencedby the boundarycondition. The strongdependenceof the dipole, quadrupole,sextupoleandoctupolecomponentson theboundaryconditionshows that thechosenprogrampackagemustbeableto handlefar field boundariesat least5 of theouteryoke radius.If theboundaryconditionis not sufficiently faraway, solutionsusingfinite elementmethodsshouldbecheckedcarefully.

Table10: Influenceof far field boundaryconditionatnominalcurrent(11800A)

Distanceof far field boundary1.3 ryoke 2. ryoke 5. ryoke 7. ryoke 10. ryoke

n � � � � � 1 -8.32827T -8.31830T -8.31474T -8.31466T -8.31463T2 0.30050 -0.52900 -0.81551 -0.82188 -0.824173 6.36948 6.22494 6.17588 6.17479 6.174394 0.12384 0.07834 0.06203 0.06166 0.061535 -0.91243 -0.91356 -0.91397 -0.91398 -0.913986 -0.00280 -0.00394 -0.00436 -0.00437 -0.004377 0.63420 0.63494 0.63521 0.63521 0.635228 -0.00026 -0.00027 -0.00028 -0.00028 -0.000289 0.10380 0.10392 0.10396 0.10396 0.10396

10 -0.00004 -0.00004 -0.00004 -0.00004 -0.0000411 0.61461 0.61535 0.61561 0.61562 0.61562

REFERENCES

[1] The LHC studygroup. TheYellow Book,LHC, TheLarge Hadron Collider - ConceptualDesign.CERN/AC/95-5(LHC),Geneva,Switzerland,1995.

[2] Oszḱar Bı́ró. Theuseof a reducedvectorpotentialformulationfor thecalculationof iron inducedfield errors. In Proceedingsto the1st InternationalROXIE Users MeetingandWorkshop, Geneva,Switzerland,1998.

104

[3] StephanRussenschuck.Comparative studyof differentcoils for the LHC main dipoles. In LHC-Project-Report-159, Geneva,Switzerland,1997.

[4] SuitbertRamberger and StephanRussenschuck.Geneticalgorithmswith niching for conceptualdesignstudies.In ProceedingsM. GrecoINFN, Frascati,Italy, 1997.

[5] The LHC study group. LHC - The Large Hadron Collider Accelerator Project. CERN/AC/93-03(LHC),Geneva,Switzerland,1993.

[6] StephanRussenschuck.Mathematicaloptimizationtechniques.In Proceedingsto the1st Interna-tional ROXIE Users MeetingandWorkshop, Geneva,Switzerland,1998.

9 Annex 1: GeometricData for the ROXIE Models

Table11: Colddimensionsfor theyoke laminationsof theMBP1 magnet

Cold dim. (mm) Descriptionbeamd 97. Beamseparationdistance/2ryoke 274.48 Radiusof iron yokerace1 100.62 Collarouterdiameter(from centerof coil)alel 90.23 Minor axisof ellipseabst 6.99 Dist. of iron insertfrom collar outerdiam.colly 96.35 Major axisof ellipserei 9.98 Radiusof insertspl1 15.17 Width 1 of insertspl2 10.08 Width 2 of insertpklx3 26.95 Heightof inserthy 10.98 Depthof gaphx1 18.96 Upperwidth of gaphx2 21.96 Lowerwidth of gaph11 25.95 Depthof busbarslotpklx1,pkly1 0., 124.76 x,y-positionof holerhol2 9.98 Radiusof holepklx2,pkly2 0.,179.6 x,y-positionof hole(Heatexchanger)rhole 29.94 Radiusof halfhole(Heatexchanger)pklx3,pkly3 36.93,229.56 x,y-positionof holerhole1 14.97 Radiusof holepklx4,pkly4 120.27,171.67 x,y-positionof holerhole2 10.28 Radiusof hole

105

Table12: Colddimensionsfor theyoke laminationsof theMBP2 magnetCold dim. (mm) Description

beamd 97. Beamseparationdistance/2ryoke 274.45 Radiusof iron yokealel1 82.9338 Minor axisof right ellipseblel1 99.8 Major axisof ellipsesalel2 86.6763 Minor axisof left ellipsed1 17.76 Centerof right ellipse(from centerof coil)d2 14.471 Centerof left ellipse(from centerof coil)leang 39. deg Startangleof left ellipsehoudw 26.946 Width of houdhoudh 5.998 Heightof houdhoudsh 5.998 Width of houdshoulderlh1r 2.495 Radiusof holelh1x,lh1y 17.,90. x,y-positionof holelh2r 14.97 Radiusof holelh2x,lh2y 57.,220. x,y-positionof holelh3r 8.7824 Radiusof holelh3x,lh3y 90.32,176. x,y-positionof holelh4r 8.7824 Radiusof holelh4x, lh4y 180.2,120.2 x,y-positionof holelhhr 29.94 Radiusof halfhole(Heatexchanger)lhhx, lhhy 0.,179.6 x,y-positionof hole(Heatexchanger)bbr 244. Radiusof busbargroundplatebbw 51. Width of busbarslotphim 53.6 middleangleof busbarslot

Table13: Colddimensionsfor theyoke laminationsof theseparatecollar geometry

Cold dim. (mm) Descriptionbeamd 97. Beamseparationdistance/2ryoke 274.45 Radiusof iron yokealel 90.27 Minor axisof ellipseblel 90.82 Major axisof ellipsehoudw 22.95 Width of houdhoudh 4.23 Heightof houdhoudsh 9.98 Width of houdshoulderhx1 17.96 Upperwidth of gaphx2 20.96 Lowerwidth of gaphy 12.97 Depthof gaplh2r 11.98 Radiusof holelh2x, lh2y 90.32,176. x,y-positionof holelh3r 7.05 Radiusof holelh3x, lh3y 43.,237. x,y-positionof holelhhr 31.19 Radiusof halfhole(Heatexchanger)lhhx, lhhy 0., 179.6 x,y-positionof hole(Heatexchanger)ins1x,ins1y 0., 89.85 coordinateof dropshapedholeins4x,ins4y 0., 41.14 coordinateof dropshapedholeins5x,ins5y 8.5,83. coordinateof dropshapedholesstax,sstay 4.59,45.95 coordinateof dropshapedholesendx 11.72 coordinateof dropshapedholesendy 63.77 coordinateof dropshapedholerins1 90. radiusof dropshapedholerins2 23.04 radiusof dropshapedholebbr 243.5 Radiusof busbargroundplatebbw 50.9 Width of busbarslotphim 53.6 Middle angleof busbarslot

10 Annex 2: Sensitivity Matrices

106

Table14: Valuesof sensitivity of MBP2 yoke geometryat injectioncurrent(850A); leastsensitive parametersexcludedfrom

thetable

MBP2 geometryat 850A> � " > �%$ > � � > � & > � 'RYOKE 0.43060E-03 0.66874E-05 0.64597E-06 0.37610E-06 -0.16209E-08ALEL1 -0.17187E+01 -0.34323E+00 -0.39158E-01 -0.25296E-02 0.40231E-04BLEL1 0.12352E+00 0.51020E+00 -0.24492E-02 0.51842E-02 -0.32220E-03ALEL2 0.10364E+01 -0.11866E+00 -0.12671E-01 0.65850E-02 -0.11960E-02

D1 -0.25412E+01 -0.39042E+00 -0.18185E-01 0.19920E-02 0.21993E-03D2 0.18087E+01 -0.14504E+00 -0.38870E-01 0.11996E-01 -0.14324E-02

HOUDW 0.80632E-02 0.10358E+00 -0.86091E-03 -0.19790E-02 0.64039E-04HOUDH -0.14335E-01 0.52543E+00 0.17377E-02 -0.20714E-01 -0.47325E-04HOUDSH -0.44187E-02 0.75311E-01 -0.26568E-03 -0.13520E-02 0.59354E-04

LH1R 0.31983E-03 0.15364E-05 -0.11973E-04 0.24864E-05 -0.23344E-06LH2R -0.14324E-03 0.23032E-04 0.23249E-06 -0.31024E-06 0.91708E-08LH3R -0.13868E-03 0.14456E-04 0.20099E-05 -0.31486E-06 -0.26854E-07LHHR -0.15078E-03 0.58484E-04 -0.42726E-05 -0.48879E-06 0.84046E-07

worstcase 0.72568E+01 0.22120E+01 0.11423E+00 0.52337E-01 0.33820E-02RMS value 0.37112E+01 0.92635E+00 0.59542E-01 0.25678E-01 0.19094E-02

Table 15: Valuesof sensitivity analysisof MBP2 yoke geometryat nominal current(11800A); leastsensitive parameters

excludedfrom thetable

MBP2 geometryat 11800A> �#" > � $ > � � > �%& > �%'RYOKE 0.25780E-01 0.83564E-02 0.52436E-04 -0.82369E-05 -0.28638E-04ALEL1 -0.17486E+01 -0.31549E+00 -0.39436E-01 -0.20825E-02 0.21958E-04BLEL1 0.23907E-01 0.49339E+00 -0.53400E-02 0.40965E-02 -0.40299E-03ALEL2 0.84815E+00 -0.87962E-01 -0.19325E-01 0.57854E-02 -0.12939E-02

D1 -0.26332E+01 -0.34523E+00 -0.28180E-01 0.15519E-02 0.41850E-04D2 0.14042E+01 -0.87139E-01 -0.50230E-01 0.99027E-02 -0.18553E-02

HOUDW -0.18493E+00 0.12593E+00 -0.15544E-01 -0.19794E-02 -0.38590E-03HOUDH -0.58836E+00 0.68527E+00 -0.50313E-01 -0.22822E-01 -0.14148E-02HOUDSH -0.73558E+00 0.19722E+00 -0.49863E-01 -0.36686E-02 -0.16623E-02

HX1 -0.38882E+00 0.82415E-01 -0.26972E-01 -0.19178E-02 -0.90179E-03HX2 -0.27301E+00 0.58894E-01 -0.19100E-01 -0.13420E-02 -0.63275E-03HY -0.44404E+00 0.92628E-01 -0.30823E-01 -0.21259E-02 -0.10034E-02

LH1R -0.59824E+00 0.12837E+00 -0.46020E-01 -0.27091E-02 -0.15148E-02LH1X -0.51234E+00 0.10255E+00 -0.34677E-01 -0.24196E-02 -0.11405E-02LH1Y -0.23320E+00 0.54295E-01 -0.18338E-01 -0.10688E-02 -0.57227E-03LH2R -0.42176E+00 0.89258E-01 -0.29010E-01 -0.20090E-02 -0.94091E-03LH2X -0.34333E+00 0.76825E-01 -0.24094E-01 -0.16669E-02 -0.79431E-03LH2Y -0.21475E+00 0.48040E-01 -0.15031E-01 -0.10332E-02 -0.49413E-03LH3R -0.54845E+00 0.12254E+00 -0.37791E-01 -0.26427E-02 -0.12493E-02LH3X -0.54558E+00 0.12500E+00 -0.38217E-01 -0.26429E-02 -0.12613E-02LH3Y -0.36260E+00 0.85043E-01 -0.25602E-01 -0.17694E-02 -0.84655E-03LH4R -0.55290E+00 0.12951E+00 -0.38586E-01 -0.26596E-02 -0.12807E-02LH4X -0.36813E+00 0.87533E-01 -0.25984E-01 -0.18019E-02 -0.85980E-03LH4Y -0.36897E+00 0.88747E-01 -0.26137E-01 -0.18187E-02 -0.86603E-03LHHR -0.60832E+00 0.13084E+00 -0.41838E-01 -0.28239E-02 -0.13459E-02LHHX -0.45625E+00 0.10677E+00 -0.32304E-01 -0.22043E-02 -0.10691E-02LHHY -0.37784E+00 0.90484E-01 -0.26763E-01 -0.18493E-02 -0.89041E-03

worstcase 0.15811E+02 0.40457E+01 0.79557E+00 0.88402E-01 0.24772E-01RMS value 0.41319E+01 0.10834E+01 0.16668E+00 0.27793E-01 0.53707E-02

107

Table16: Valuesof sensitivity analysisof MBP1 geometryat injection current(850 A); leastsensitive parametersexcluded

from thetable

MBP1 geometryat 850A> �#" > � $ > � � > �%& > �%'RYOKE 0.50867E-03 -0.13134E-04 0.22965E-05 0.88754E-06 -0.14793E-06RACE1 -0.26049E+01 -0.40061E+00 -0.10518E-01 0.47839E-02 0.40609E-03ALEL 0.91026E+00 0.47575E-01 -0.26988E-01 -0.55364E-02 -0.39699E-04ABST -0.19574E+01 0.40998E+00 -0.30413E-01 -0.75095E-02 0.32068E-02

COLLY 0.36979E+00 0.69802E+00 0.45685E-01 -0.12309E-01 -0.11389E-02SPL1 -0.26056E+01 0.43307E+00 0.65350E-02 -0.18790E-01 0.45295E-02SPL2 -0.16479E+01 0.20816E+00 0.31793E-01 -0.17027E-01 0.29388E-02

RHOLE -0.38215E-03 0.13135E-03 -0.87511E-05 -0.22950E-05 0.50275E-06RHOL2 -0.21162E-03 0.14799E-03 -0.16627E-04 -0.19752E-05 0.69937E-06

RHOLE1 -0.27995E-03 0.61302E-04 -0.20226E-05 -0.11813E-05 0.19860E-06RHOLE2 -0.17313E-03 0.94667E-05 0.25429E-05 -0.18653E-06 -0.24324E-07

worstcase 0.10098E+02 0.21979E+01 0.15197E+00 0.65963E-01 0.12262E-01RMS value 0.45920E+01 0.10242E+01 0.70033E-01 0.30073E-01 0.63953E-02

Table17: Valuesof sensitivity analysisof MBP1 geometryat nominalcurrent(11800A); leastsensitive parametersexcluded

from thetable

MBP1 geometryat 11800A> �#" > � $ > � � > �%& > �%'RYOKE 0.82681E-01 0.26881E-01 0.52692E-02 0.24871E-03 -0.72670E-04RACE1 -0.26288E+01 -0.31776E+00 0.12248E-02 0.45903E-02 0.29677E-03ALEL 0.88408E+00 0.11934E+00 -0.28945E-01 -0.56039E-02 -0.62854E-04ABST -0.13755E+01 0.39191E+00 -0.13864E-01 -0.37344E-02 0.14208E-02

COLLY -0.22916E-01 0.59306E+00 0.40860E-01 -0.36811E-02 -0.94528E-03SPL1 -0.19743E+01 0.43829E+00 0.26292E-01 -0.11323E-01 0.18576E-02SPL2 -0.12598E+01 0.25812E+00 0.34543E-01 -0.98281E-02 0.97704E-03HX1 -0.35063E+00 0.25364E+00 0.18615E-01 0.12172E-02 -0.12527E-02HX2 -0.57415E+00 0.33111E+00 0.19883E-01 -0.59710E-04 -0.11396E-02HY -0.65443E+00 0.35780E+00 0.20442E-01 -0.41578E-03 -0.11294E-02

RHOLE -0.59904E+00 0.25090E+00 0.98529E-02 -0.79264E-03 -0.76607E-03PKLX2 -0.17396E+00 0.11036E+00 0.74134E-02 -0.11907E-03 -0.30516E-03PKLY2 -0.29845E+00 0.17683E+00 0.11370E-01 -0.28083E-03 -0.45719E-03RHOL2 -0.74976E+00 0.29995E+00 0.11266E-01 -0.11104E-02 -0.82109E-03PKLX1 -0.24708E+00 0.12158E+00 0.66340E-02 -0.26364E-03 -0.33373E-03PKLY1 -0.31737E+00 0.23668E+00 0.71447E-02 -0.18333E-02 -0.37795E-03

RHOLE1 -0.76939E+00 0.29444E+00 0.15450E-01 -0.26368E-03 -0.92733E-03PKLX3 -0.39532E+00 0.21882E+00 0.15393E-01 -0.39275E-04 -0.59325E-03PKLY3 -0.38682E+00 0.19489E+00 0.11943E-01 -0.27173E-03 -0.49528E-03

RHOLE2 -0.96909E+00 0.49266E+00 0.30465E-01 -0.72407E-03 -0.12342E-02PKLX4 -0.38646E+00 0.19938E+00 0.12233E-01 -0.29998E-03 -0.49044E-03PKLY4 -0.38947E+00 0.20111E+00 0.12311E-01 -0.31461E-03 -0.48971E-03

worstcase 0.15490E+02 0.58855E+01 0.36141E+00 0.47016E-01 0.16446E-01RMS value 0.43929E+01 0.13945E+01 0.90133E-01 0.17685E-01 0.41032E-02

108

Table18: Valuesof sensitivity analysisof separatedcollar geometryat injection current(850 A); leastsensitive parameters

excludedfrom thetable

Separatedcollar geometryat 850A> �#" > � $ > � � > �%& > �%'RYOKE 0.49129E-03 0.11870E-04 -0.81080E-06 0.13864E-06 0.45985E-07ALEL1 0.11063E-03 -0.11322E+01 0.35820E-04 -0.93185E-03 0.35511E-05BLEL1 0.14786E-03 0.69924E+00 0.89889E-05 0.21551E-01 0.28628E-05INS1X -0.40759E-03 0.17415E-04 0.95805E-05 -0.24815E-05 0.40450E-06INS1Y 0.49630E-03 -0.38987E-04 -0.81366E-05 0.29835E-05 -0.58621E-06INS4X -0.31820E-03 0.83393E-04 -0.16159E-04 0.24606E-05 -0.22079E-06INS4Y -0.78687E-03 0.18324E-03 -0.28877E-04 0.27792E-05 0.21381E-06INS5X 0.45493E-03 -0.62149E-04 -0.68845E-06 0.22113E-05 -0.64653E-06INS5Y 0.39895E-03 -0.51203E-04 -0.14882E-05 0.20163E-05 -0.55375E-06SSTAX 0.64071E-03 -0.11747E-03 0.94774E-05 0.14217E-05 -0.87279E-06SSTAY -0.39643E-03 0.77203E-04 -0.78091E-05 -0.39127E-06 0.45457E-06SENDX 0.19315E-03 -0.21744E-04 -0.29714E-05 0.17956E-05 -0.45457E-06HOUDW -0.63540E-04 0.22333E+00 -0.22946E-05 -0.45704E-02 -0.15676E-05HOUDH -0.40201E-03 0.85525E+00 0.19420E-04 -0.41406E-01 -0.57556E-06HOUDSH 0.86317E-05 0.12599E+00 -0.47975E-05 -0.20358E-02 -0.16582E-05

LH2R -0.18261E-03 0.21248E-04 0.19799E-05 -0.34238E-06 -0.42673E-07LHHR -0.10086E-03 0.51480E-04 -0.56633E-05 -0.48717E-08 -0.90930E-08

worstcase 0.59058E-02 0.30368E+01 0.17201E-03 0.70515E-01 0.14944E-04RMS value 0.16134E-02 0.16025E+01 0.56862E-04 0.46956E-01 0.53734E-05

Table19: Valuesof sensitivity analysisof separatedcollar geometryat nominalcurrent(11800A); leastsensitive parameters

excludedfrom thetable

Separatedcollar geometryat 11800A> � " > �%$ > � � > � & > � 'RYOKE 0.18512E-01 0.28112E-01 0.70269E-02 0.14917E-02 0.70407E-04ALEL1 -0.38167E-02 -0.10370E+01 0.50643E-02 -0.19060E-02 0.70597E-03BLEL1 0.15623E+00 0.68419E+00 -0.27799E-02 0.12774E-01 -0.26567E-03INS1X -0.21918E-01 0.47808E-01 0.15498E-02 0.19278E-02 0.39031E-05INS1Y 0.73755E-01 0.24892E-01 -0.39340E-03 0.27113E-02 -0.23435E-03INS4X -0.16539E+00 0.97889E-01 -0.49832E-02 0.40215E-02 -0.13749E-03INS4Y -0.64373E-01 0.14015E+00 -0.45769E-02 0.87834E-02 -0.57788E-03INS5X 0.80034E-01 0.50660E-01 -0.12669E-02 0.57897E-02 -0.59117E-03INS5Y 0.83825E-01 0.50847E-01 -0.18796E-02 0.51653E-02 -0.44544E-03SSTAX 0.21464E+00 0.23360E-01 -0.44694E-03 0.56352E-02 -0.64442E-03SSTAY -0.45283E-01 0.89485E-01 -0.36508E-02 0.55877E-02 -0.30054E-03SENDX 0.13040E+00 0.50142E-01 -0.42882E-03 0.58965E-02 -0.51349E-03SENDY 0.71849E-02 0.82720E-01 -0.24152E-02 0.60584E-02 -0.36404E-03RINS1 0.38538E-01 0.76393E-01 -0.38252E-03 0.60645E-02 -0.36160E-03RINS2 0.30509E-01 0.79366E-01 -0.62254E-03 0.61817E-02 -0.35602E-0

HOUDW 0.10924E+00 0.19718E+00 -0.24026E-02 0.61132E-02 -0.22314E-03HOUDH 0.65783E+00 0.11244E+01 -0.20092E-01 -0.21154E-01 -0.45719E-03HOUDSH 0.32912E-01 0.10451E+00 0.85173E-02 0.18173E-01 -0.91943E-03

HX1 -0.14323E-01 0.10165E+00 0.64566E-02 0.15063E-01 -0.93389E-03HX2 0.50067E-02 0.78155E-01 0.26736E-02 0.96346E-02 -0.56596E-03HY -0.15186E-01 0.80441E-01 0.16619E-03 0.92562E-02 -0.57917E-03

LH2X 0.89218E-02 0.87234E-01 0.38296E-03 0.91675E-02 -0.55107E-03LH2Y 0.20505E-01 0.58374E-01 0.13846E-03 0.54775E-02 -0.31309E-03LH2R 0.18548E-01 0.15042E+00 -0.54484E-03 0.13532E-01 -0.79594E-03LH3X 0.34418E-01 0.10241E+00 -0.29678E-03 0.91135E-02 -0.52630E-03LH3Y 0.23338E-01 0.62371E-01 -0.14760E-03 0.54886E-02 -0.31530E-03LH3R 0.29759E-01 0.10296E+00 -0.82015E-03 0.91233E-02 -0.54271E-03LHHX 0.31625E-01 0.78795E-01 -0.24475E-03 0.68922E-02 -0.39964E-03LHHY 0.35401E-01 0.80412E-01 -0.11007E-03 0.69111E-02 -0.39713E-03LHHR -0.21560E-01 0.14374E+00 -0.47729E-02 0.13481E-01 -0.92463E-03

worstcase 0.21930E+01 0.51161E+01 0.85235E-01 0.23857E+00 0.14017E-01RMS value 0.77175E+00 0.17437E+01 0.26670E-01 0.50340E-01 0.28635E-02

109