The field of strong super conducting magnets contains substantial multipole errors. An upper limit of their value is specified on the reference closed orbit. When the dipole is installed in a real machine, its axis cannot be exactly on the reference orbit, and "feed-down multipoles" are introduced. This paper describes an analytical approach to the alignment specification associated with this effect with an example from the LHC at CERN

Verdier, A

Wildner, E

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FEED-DOWN EFFECT IN DIPOLE ALIGNMENTA.Verdier, E.Wildner, CERN, Geneva, SwitzerlandAbstractThe field of strong super conducting  magnets containssubstantial multipole errors. An upper limit of their valueis specified on the reference closed orbit. When the dipoleis installed in a real machine, its axis cannot be exactly onthe reference orbit, and “feed-down multipoles” areintroduced. This paper describes an analytical approach tothe alignment specification associated with this effectwith an example from the LHC at CERN.1 THE LHC DIPOLEThe LHC Dipole has unavoidable field shapeimperfections. For some components of these fieldimperfections there is a spool piece corrector mounted atthe end of the dipole to compensate the average value ofthe field error component. Every dipole has a sextupolarcorrector and every second dipole has a combinedoctupolar/decapolar corrector. The usual expansion of thedipole field reads if we assume a two dimensional fieldalong the dipole axis, x being the horizontal and y thevertical co-ordinate:11)(−∞= ++=+ ∑nrefnnnrefxy RiyxiabBiBB (1)where, in European convention, n=1 identifies a dipolefield n=2 a quadrupole etc. The a and the b indicate theskew and normal components of the field respectively,Rref is the reference radius (17mm) for the LHC. The feeddown harmonics due to (an,bn) harmonics, can becalculated by changing x and y in the formula (1) intox0+x and y0+y respectively. Then the relevant monomialsassociated with a given order k are kept. We obtain:knrefnnkk Riyxknkniabiab− +−−−+=+ 00)!()!1()!1()(where k is the order of the feed-down multipole, withk<n.Figure 1 illustrates the case when there is only onecorrector (b3). The dipole centre is displaced relative tothe reference closed orbit by dz where the z-coordinaterepresents either the horizontal or the verticaldisplacement. dzs is the misalignment of the correctorrelative to the average dipole axis. dzs has onecontribution from the displacement of the correctorrelative to the dipole axis and one contribution thatoriginates from the pitch (yaw) of the dipole. For thisconfiguration dz induces feed-down for components thatare not compensated. For the compensated component thecorrector misalignment (dz-dzs) contributes with -b3 andthe dipole misalignment contributes with b3.z=  x or ysspool piece dz reference closed orbitdzsdipoleFigure 1: Configuration for a dipole with a field errorcorrector. dzs is the displacement of the corrector relativeto the average axis of the dipole, dz is the averagedisplacement of the dipole.For the case where every second dipole has a corrector,the corrector misalignment (dz1-dzs) contributes with –2b4 or –2b5 and the dipole misalignment dz1 or dz2contributes with b4 or b5. For the not compensatedcomponents the contribution comes from dz1 and dz2 fordipole 1 and dipole 2 respectively. The correctors have arandom contribution of 2 times their value. z = x or ysdz1dz2 reference closed orbitspool piecedzs Dipole 2 Dipole 1Figure 2: Configuration where one dipole out of two has acorrector.(2)0-7803-7191-7/01/$10.00 ©2001 IEEE. 1693Proceedings of the 2001 Particle Accelerator Conference, Chicago2 DERIVATION OF EXPRESSIONS FORALIGNMENT TOLERANCESFrom formula (2) follows with n-k=1, first order feed-down, dipole without corrector||)1()1(||||)1()1(||tolrefnrefntoltolrefnrefntolaRdybnRdxanabRdyanRdxbnb≤−+−≤−≤−−−≤−where the suffix tol means tolerance. Solving this sytemgives)1()(||||||)1()(||||||2222−++<−++<nRababbadynRabaabbdxrefnntolntolnrefnntolntolnFor the dipoles with corrector, the contribution from themisalignment of the dipole is dxd and the contributionfrom the corrector is (dxs+dxd) relative to the referenceorbit (similarly in the direction of the y-co-ordinate))()1())())((()()1())((,,,,refdsdsnstolstolrefddnndtoldtolRndydyidxdxbiabRnidydxiabiab−+++−=+−++=+this gives the following, which can be solved only with anassumption about the relation between dxd and dxs, whichis set to α*dxd (the corrector tolerance is a fraction of thedipole tolerance), similarly for the y-direction, dys is setto β*dyd.||)1()1(||||)1()1(||tolrefsnrefdntoltolrefdnrefsntolaRdybnRdxanabRdyanRdxbnb≤−−−≤−≤−−−−≤−Solving for dxd and dyd)1()(||||)1()(||||2222−++<−++<nRababbadynRabaabbdxrefnntolntolndrefnntolntolndαβααββThis is valid for systematic components and is the samefor dipole with one or two correctors.For the random effect we have to distinguish the dipolewith and without corrector for the components b4 and b5.Starting it over again, now scaling x and y by R/(n-1), forsimpler expressions))(( nntoltol iabiyxiab ++=+nntolnntolxaybayaxbb+=−=With xd1 and xd2 taken as the distance from the dipole 1and 2 respectively from the closed orbit and xs,p as thedistance of the spool piece from the dipole)(2)()()(2)()(1212112121sdnddnddntolsdnddnddntolyybxxayybaxxbyyaxxbb+−+++=+−+−+=snddnddntolsnddnddntolybxxayybaxbyyaxxbb2)()(2)()(21122112−++−=−+−−=If we assume the expectation values, bn, an, x1, x2, y1, y2uncorrelated we get222221222212222222122122224)()(4)()(snddnddntolsnddnddntolybxxayybaxbyyaxxbb++++=++++=22222222222222422422ysnxdnydntolxsnydnxdntolbabababbσσσσσσ++>++>The same relation between the misalignment of thecorrector and the misalignment of the dipole as before,dxs+α*dxd and dys+β*dyd2222222222222)21(202)21(20tolxdnydntolydnxdnaabbab<++<<++<σβσσασthis gives],)21(max[],)21(max[2222222222DbaDbaDaaDbbtolnntolydtolnntolxdασβσ+<+<with      4224 )21)(21(2 nn abD −++= αβFor the case all dipoles have a spool piece the factor 2goes away in front of α and β.3 ESTABLISHING ALIGNMENTTOLERANCES FROM “BUDGET”The errors have systematic, systematic per arc(uncertainty) and random error components. Onlysystematic errors combine to contribute to the systematicalignment budget. For the random budget we use thefollowing value of the components causing feed-down2,2,2,2,2,2,rannuncnsysnnrannuncnsysnnaaaabbbb++=++=1694Proceedings of the 2001 Particle Accelerator Conference, Chicagon 2 3 4 5Feed down effect dipole: closed orbit quadrupole: β beat sextupole:resonances, Q’octtupole:anharmonicities  Q’’Tolerable systematican-110% of dipole (0.5u) 1/ 8  unc. (0.3u) 1/ 8 unc. (0.3u) 1/ 8  unc. (0.19u)Tolerable systematicbn-110% of dipole (0.5u) 1/ 8  unc. (0.3u) 1u 0.07u (Q”)Tolerable randoman-114 % of random indipole (1.0u)10% of coupling(0.8u)10% of increase=46% of total (0.2u)10% of increase=46% of total (0.3u)Tolerable randombn-114 % of random indipole (1.0u)7 % β beat (0.36u) 10% of increase=46% of total (0.7u)10% of increase=46% of total (0.3u)Table 1: Summary of feed down budgetFor each multipole we set up a budget for the feed-down harmonic n-1 for systematic, uncertainty andrandom errors separately, for both random and skewcomponents. From this budget we calculate systematic,and random parts of the tolerated alignment. It is  checkedthat second-order (n-2) contributions can be ignored.The table of field imperfections used is the “9901”table. It gives values of the field components expressed inunits (u) of 10-4 relative to  the main field.Table 1 gives the summary of the budget for calculationof the tolerances. For n=2 the feed down component isonly a dipolar effect. Random b1 and a1 allowed is 14 %of random b1and a1 in the field table, which correspondsto 1% of aperture budget, i.e. noise.For the calculation of the systematic feed-down froma3/b3 we do not take the value for the a3 and b3,  which aretoo small, from the error table but the value is divided by8 (random distribution over the 8 arcs) which gives 0.3.For the a2/b2 budget (feed down from a3/b3) a tune shift orcoupling coefficient of 0.03 has to be accommodated. Aβ-beating of 7% gives us a random b2 of 0.36. As randoma2, an increase of the random component of 10% (allowedby the correction system) is taken which gives 46% of thea2 in the field table.For the systematic a4/b4 feed-down the values in thefield table is divided by 8  (same argument as for a3/b3).The systematic and uncertainty for the feed-down budgetare taken as 10% of the a3/b3 components in the field tableand the random a3/b3 are taken as 46% of the randomcomponents (increases the random component by 10%).For the a5/b5 feed down we take the values of a5/b5 fromthe table. The Q” budget requires the systematic b4 to beless than 0.1. If we assume that the systematic and theuncertainty errors of the field contribute equally1.02424 =+ us bb uthen the systematic b4 is 07.02/1.0 = u. The systematica4 is set to the rms of the acceptable uncertainty whichgives 0.55/ 8 =0.19. The b4 uncertainty is taken as82/1.0 ⋅ =0.2 u. The a4 uncertainty is defined by thedynamic aperture and is 0.55u.The random b4/a4 is taken as 46% of the randomcomponent of b4, i.e. it increases the random componentby  10%.For a6/ b6 and a7/ b7 feed down the budget is supposedto be 10% of the allowed component in the error tableexcept for the a5 and a6 budget where the value issupposed to be of the same order as in the error table.4 SUMMING UP RESULTSUsing the formulas above with the established budget,taking the most constraining misalignment for the dipoleitself and for the b3, b4, and b5 correctors separately, givesthe result shown in Table 2. For the dipole itself themisalignment in the x–direction is limited by thecontribution from the b3 multipole feed down and for they-direction the b3 and the b5 are the most constraining.The misalignments of each spool piece corrector are takenrelative to the dipole average axis.SUMMARYSystematic Random (rms)dx, dipole (b3) 0.14 0.29dy, dipole (b3) 0.14 0.93dy, dipole (b5) 1.6 0.42dx, b3-corrector 0.09 0.18dy, b3-corrector 0.09 0.61dx, b4-corrector 31.9 1.9dy, b4-corrector 20.1 1.6dx, b5-corrector 0.41 0.35dy, b5-corrector 1.07 0.28Table 2: Summary of the alignment tolerances in mm forthe dipole (w.r.t. reference c.o.) and the correctors (w.r.t.the dipole axis.The random alignment tolerance of the b3 spool piece of0.18 mm is probably not achievable. A complete aperturebalance has to be remade (the β beating budget has to bere-evaluated). The random tolerance of 0.29 mm for thedipole is smaller than the r.m.s. closed orbit distributionand this would also imply a revision of the aperturebudget.  For the b5 corrector the tolerance is pessimistic asthe effect of random a4/b4on dynamic aperture is probablynegligible (should be evaluated).5 REFERENCES[1] LHC Alignment Workshop, CERN Dec. 2000,http://wwwslap.cern.ch/~vrd/aplaw/page.html1695Proceedings of the 2001 Particle Accelerator Conference, Chicago

Feed-down effect in dipole alignment

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