Faus-Golfe, A

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Particle Accelerators, 1996, Vol. 55, pp. [449-459J / 203-213Reprints available directly from the publisherPhotocopying permitted by license only© 1996 OPA (Overseas Publishers Association)Amsterdam B.Y. Published in The Netherlands underlicense by Gordon and Breach Science Publishers SAPrinted in MalaysiaMINIMISATION OF THE RAMP-INDUCEDNON-LINEAR FIELDIMPERFECTIONS IN LHCA. FADS-GOLFECERN, Geneva 23, Switzerland(Received 17 January 1996; infinalform 17 January 1996)We consider in this study the consequences of dynamic field imperfections of bn , n ~ 3 on thedynamic aperture. They may be minimised by a proper shaping of the ramp-rate 8B(t)/8t, atthe expense of the ramp time. While the beam stability criterion chosen cannot be met with theMTPIAI prototype, it is shown that MTPIA2 and MTPIA3 prototypes are very suitable forramping. The consequences of bI , b2 and a2 will be studied separately.Keywords: Field imperfection; ramp-rate.1 INTRODUCTIONAt any static level of energy, field imperfections arise from mechanicaltolerances and from the persistent currents in the filaments, which are thememory of the former variations of field. In this study, we assume that thesefield errors are taken care of either by design or proper correction.During the ramp, a new kind of eddy current develops in the loopsformed by the twisted strands. This phenomenon is referred to as inter-strandcoupling. The resulting field imperfections are proportional to the ramp-rateand inversely proportional to inter-strand resistance, Rc .1 Rc depends on thecable design and seems to show a large spread from loop to loop.2 RELATIVE NORMAL FIELD ERRORS IN LHCDIPOLES DURING RAMPINGLet us recall the usual field expansion used to define the multipolecoefficients:2[449]/203[450]/204 A. FADS-GOLFEwhere Bn is the normal coefficient and An is the skew coefficient of the2n-pole, B main is the amplitude of the nominal field of the magnet, Rr is thereference radius of the expansion, bn is the normal relative coefficient and anis the skew relative coefficient of the 2n-pole.In Table I we summarise the values of the relative normal field imperfec-tions at injection, in static and dynamic conditions, given in references:3,4• Static conditions (bno): B(t = 0) = B inj = O.• Dynamic conditions (bn - bno): Binj = 0.4 T/min (nominal value).where B(t) = a~it) is the ramp-rate. The static values were calculatedtaking in account the measured multipoles of the HERA, RHIC and prototypeLHC magnets (MTPIAl, MTPIA2, MTPIA3). Systematic b3 and bs willbe compensated by correctors at each dipole end. The b3 values in thisTABLE I Relative normal field errors in LHC dipoles at injection, all relative field errors areat R r =0.01 m, in units of 10-4Dynamic (bn-bno )Static bno MTP1Al MTP1A2 MTP1A3Expected Aper.l Aper.l Aper. 2 Aper.ln syst. random1 87.931 10.8142 ±0.7 0.4 -0.96 -1.109 0.585 -0.1063 -3.4±0.3 0.5 4.6 0.945 0.683 0.0954 ±O.2 0.1 0.447 -0.129 0.034 0.0245 0.25±0.05 0.08 2.08 0.045 -0.0036 -0.004 0.027 -0.026 0.01 0.0018 0.0059 0.006 0.00310 0.00211 0.008 0.001MINIMISATION RAMP-INDUCED FIELDS IN LHC [451]/205table for static conditions indicate the magnitude of the persistent currenteffect, the coil geometry being designed for b3 == O. The dynamic values arethose measured on the actual LHC magnet prototypes MTPIAl, MTPIA2,MTPIA3.The ramp induced field errors are of the same order as the static errorsin most of these cases. At a constant ramp rate the eddy currents producea constant field, so that the relative field errors are highest at injection. Theerrors are proportional to R;1B. It is probable that bI, b3 and bs would besystematic, whereas b2 and b4 would be random.4 MTPIAI has the lowest Re(Re == 1.6j.LQ) of all magnets measured and could therefore be consideredas a worst case. On the contrary MTPIA2 (Re == 6.5j.LQ) and MTPIA3(Re == 14.0j.LQ) have a value of Re close to the design value.3 MINIMISATION OF RELATIVE NORMAL FIELD ERRORS INLHCDIPOLESAn increase of the field imperfections during the ramp is unavoidable. It willdistort the linear optics and reduce the dynamic aperture. In this study, weonly consider the second effect. It is still possible to limit it to fulfil somecriterion on beam stability by initially reducing the ramp rate iJ. At the end ofthis initial phase Battains its nominal value and the rest of the ramp is linear.In this study the maximum ramp rate Bmax is assumed to be a parameterwhich may allow to control the total ramp time.3.1 Criterion for Beam StabilityThere is so far no reliable law to predict the dynamic aperture for agiven configuration of magnetic field imperfections. One normally resortsto tracking, which is much too heavy for this kind of study as it requiresabout 1 day of CPU for each and any field configuration under study.A method often employed is to compute one parameter of the non-linearsystem which lends itself to a faster calculation: onset ofchaos, "norm" of thenon-linear part of the map, amplitude detuning, resonance strength, e.g.S,6and demonstrate that it carries a reasonable correlation with the dynamicaperture in the domain of interest.In this study we rather use "loosely" an exact scaling law of the dynamicaperture. We know from the invariance of the equation of motion that, if[452]1206 A. FADS-GOLFEeach 2n-pole bn is multiplied by kn- 2 , k being an arbitrary number, thenthe dynamic aperture is reduced by k. For example, the dynamic apertureis reduced by 2 if: b3, b4, bs, b6, ... are respectively multiplied by 2, 4, 8,16... .7,8 We now make the assumption that, if the dynamic aperture is to besignificantly reduced by a small increase of a single multipole bn , then thismultipole must dominate; we can thus apply the scaling law to this singlemultipole. If we require the dynamic aperture A (expressed in amplitude) todecrease by not more than (1 - ~a):(2)the dominant multipole bn shall not exceed:(3)A tight control of the dynamic aperture being required, we take ~a = 5%.3.2 Ramp at Constant Field ImperfectionsThe theory and the measurements show that the absolute field error, Bn ,related to the relative field error bn = R~-l BBn. ,depends linearly on B like:mam(4)where Bno is the absolute static field error and kn is a constant which dependson the cable design. In this expression we assume that Bno does not dependon B(t). This assumption is valid around injection, where it is most critical.The condition for a constant perturbation equal to the maximum allowedby beam dynamics is, assuming that bn is dominant:From Equations (4) and (5) we can obtain the differential equation:. bn BnoB(t) - --B(t) + - = O.knR~-l kn(5)(6)MINIMISATION RAMP-INDUCED FIELDS IN LHC [453]/207The solution of this differential equation is an exponential increase of theguide field:The constants Bno and kn are computed from observables (Table I):bnoBno =B in1 --1;J R~-Binjn-l' (bn - bno)Rr B inj(7)(8)(9)In the following when we refer to this part of the ramp we put the subscript"exp" in the variables.3.3 Ramp Profile and ParametersThe ramp profile and parameters are given in Figure 1.Taking as parameter bn,exp, knowing that B inj =0.58 T, B col =8.4 T andassuming a maximum ramp-rate, Brnax , we can calculate the parameters ofthe ramp from the formulas above.B(t)BcolLinear B'maxBmax,exp B'maxExponentialBinjB'injtmax,exp tmax,linFIGURE 1 Ramp profile (B' is equivalent to B).[454]/208 A. FADS-GOLFE4 BEHAVIOUR OF OTHER NORMAL FIELD ERRORS WHENTHE EFFECT OF ONE OF THEM IS MINIMISEDThe aim of this section is to calculate the behaviour of the other fieldimperfections when we minimise the effect of one of them by adjustingthe ramp parameters.4.1 Exponential Part of the RampFrom Equations (4) and (5), two different multipolar coefficients are relatedby:R~-l(BnO + knB(t))B(t) - ---bn-(t-)--Rr;-l(Bmo+ kmB(t))bm(t) (10)From Equation (10) and assuming that we minimise bn (t) = bn ,exp adjustingthe ramp parameters, one can write for bm (t) the expression:B + km (bn,expBinj - B ) exp ( bn,exp t)mO k Rn- 1 nO k Rn- 1b () - Rm - n b n r n r (11)m t - r n,exp B + (bn,expB inj _ B ) exp ( bn,exp t)nO R~-l nO knR~-lbm (t) is a decreasing function if we can meet the condition:(12)4.2 Linear Part of the RampFor the linear part of the ramp we have the same behaviour for all thecoefficients.An example of the profiles of bn (t) and bm (t) during the ramp is given inFigure 2. bn (t) and bm (t) decrease in the linear ramp if:(13)MINIMISATION RAMP-INDUCED FIELDS IN LHCbm,n(t)Exponentialbm,max[455]/209bn(t)bn,exp f-----'-,----------.-bm(t)Linearbn,min I- , c~_- - - - - - - - -~---,bm,min f--------------------------------,-------------------"::~~---,tmax,exp t,max,linFIGURE 2 Profile of bn(t) and bm(t) during the ramp adjusted taking as parameter bn,exp.5 MINIMISATION OF RELATIVE NORMAL FIELDCOMPONENTS IN MTPIAl, MTPIA2 AND MTPIA3 DIPOLESAll the parameters and relations mentioned so far have been implemented ina Mathematica program. It will be used for the following case studies.Usually the larger field imperfections produced by the inter-strand couplingcurrent in dipoles are the normal sextupole (b3) and decapole (bs) terms. Wehave three possible cases:• b3 is the most important imperfection, in that case from beam dynamicswe know that the dynamic aperture is proportional to llb3. The criterionthat we take is that b3 should not increase by more than 5% in dynamicconditions just after t=O.• bs is the most important imperfection, in that case from beam dynamicswe know that the dynamic aperture is proportional to 1/(bs)3. The criterionis that bs should not increase by more than 15% in dynamic conditionsjust after t=O.• b3 and bs both contribute, hence the two conditions should be satisfiedsimultaneously.Sometimes the important imperfection is b7. In that case the criterion isthat b7 should not increase by more than 27% in dynamic conditions justafter t=O.[456]/210 A. FADS-GOLFETABLE II Ramp performances for MTPIAI dipole, adjusted takingas parameter bS,expBmax 0.400 [T/min]Binj 0.012 [T/min]Texp 72.225 [min]Bmax,exp 8.650 [T]T 72.225 [min]~b(n=3) 28.676 %~b(n=S) 81.041 %5.1 MTPIAI (10 Meter Dipole)From Table I both b3 and bs show a large dynamic increase; we should try tominimise the two components at the same time. Condition (12) shows thatif we adjust the ramp-parameters to minimise bs(t = 0) = bS,exp, then b3(t)is a decreasing function in the exponential part of the ramp. Hence we takebs (t = 0) = bS,exp as parameter to adjust the ramp.It may be that no exponential ramp is found which satisfies the criterion onbeam stability. The minimum increase ~b is obtained for a purely exponentialramp (no linear part) as summarised in Table II. The increase of bS,exp by81 % is unacceptable, as well as the ramp time of over 70 minutes.5.2 MTPIA2 (10 Meter Dipole)5.2.1 Aperture 1 From Table I both b3 and bs show a large dynamicincrease; we should try to minimise the two components at the same time.Condition (12) shows that if we adjust the ramp-parameters to minimiseb3(t = 0) = b3,exp, then bs(t) is a decreasing function in the exponentialpart of the ramp. Hence we take b3 (t = 0) = b3,exp as parameter to adjustthe ramp. The solution found is summarised in Table III. The ramp time Tis increased from to 20 to 27 min. The nominal time may be recovered byincreasing the nominal ramp-rate by 50%.5.2.2 Aperture 2 From Table I we can see that the most importantimperfection in this aperture is b3. We should try to minimise this componentwithout disturbing very much bs which in this case is lower than thestatic value. Condition (12) is the same than in Aperture 1. Hence we takeb3 (t = 0) = b3,exp as parameter to adjust the ramp. The solution found isMINIMISATION RAMP-INDUCED FIELDS IN LHC [457]/211TABLE III Ramp performances for MTP~A2 (Aperture 1 and 2) dipole adjusted taking asparameter b3,exp, for the different values of Bmax consideredAperture 1 Aperture 2Emax 0.400 0.600 0.800 0.400 0.600 0.800 [T/min]Einj 0.011 0.011 0.011 0.015 0.015 0.015 [T/min]Texp 9.480 10.539 11.289 6.239 7.004 7.547 [min]Bmax,exp 1.596 2.118 2.640 1.307 1.684 2.061 [T]T 27.114 21.425 18.802 24.597 18.614 15.783 [min]~b(n=3) 5.000 5.000 5.000 5.000 5.000 5.000 %~b(n=S) 1.488 1.488 1.488 -0.137 -0.137 -0.137 %summarised in Table III. The increase in ramp time for the nominal iJ seemsacceptable.5.3 MTPIA3 (10 Meter Dipole)From Table I we can see that the most important imperfection is b3. We donot have a contribution from bs but we have a contribution from b7. Then weshould try to minimise b3 without disturbing very much b7. Condition (12)shows that if we adjust the ramp-parameters to minimise b3(t = 0) = b3,exp,then b7(t) decreases.The solution, which only requires 20 seconds ofexponential ramp is shownon Figures 3 and 4 and Table IV.6 CONCLUSIONIn this study, we considered the case of three individual magnets. Whatmatters is really the spread of imperfections for the whole population ofdipoles; the average b3 and bs can indeed be corrected with the spool pieces.A spread comparable to the imperfections of MTPIAI would significantlyreduce the LHC dynamic aperture during the ramp. A spread comparable tothe imperfections of MTPIA2 or even better MTPIA3 would not degradesignificantly the dynamic aperture. MTPIA2 might be seen as an acceptancelimit, as higher imperfections could be manageable, but at the expense of asignificant increase of the already long ramp time. These conclusions shouldbe confirmed by tracking. studies.[458]1212 A. FAUS-GOLFEB (T) Ramp (b3,exp 0.525 10"(-4))10 15 20 t (min)FIGURE 3 B as a function of t, b3,exp == 0.525 10-4 CBmax == 0.4 T/min), for MTP1A3 dipole.b3 (1 0 " (R~b (b3, exp O. 525 10" ( - 4 ) )0.50.40.30.20.110 15 20 t (min)FIGURE 4 Profile of b3 and b7 as a function of t, for MTPlA3 dipole adjusted taking asparameter b3,exp.TABLE IV Ramp performances for MTPlA3 dipole adjustedtaking as parameter b3,expBmax 0.400 [T/min]Binj 0.105 [T/min]Texp 0.350 [min]Bmax,exp 0.657 [T]T 20.332 [min];b(n=3) 5.000 %;b(n=7) -1.645 %AcknowledgementsI am grateful to J.P. Koutchouk and R. Wolf for guidance and fruitfuldiscussions.ReferencesMINIMISATION RAMP-INDUCED FIELDS IN LHC [459]/213[l] A.P. Verweij and R. Wolf, "Field errors due to inter-strand coupling current in the LHCdipole and quadrupole", (CERN AT-MAlAV, Internal Note 94-97, March 1994).[2] R. Wolf, "Field errors definitions for LHC version I", (CERN AT-MAlAV, Internal Note94-102, October 1994).[3] "Expected field errors in LHC (version I)", (CERN AT-MAlAV, Internal Note).[4] "The Large Hadron Collider, Accelerator Project", (LHC Study Group, CERN/AC/95-05(LHC), August 1995).[5] R. Kleiss, et aI., (Particle Accelerators, 41, 117-32, 1993).[6] M. Giovannozzi, et aI., "A sorting of approach to the magnetic random errors", (CERN SL95-11, (AP)).[7] V. Ziemann, "Crude Scaling Laws for the Dynamic Aperture of LHC from RandomNon-Linear Errors", (CERN SL/Note 95-20, February 1995).[8] J.P. Koutchouk, private communication.

Minimisation of the ramp-induced non-linear field imperfections in LHC

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