Parzen, G

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Particle Accelerators1975, Vol. 6, pp. 237-243© Gordon and Breach Science Publishers Ltd.Printed in the United KingdomRANDOM ERRORS IN THE MAGNETIC FIELD OFSUPERCONDUCTING DIPOLES AND QUADRUPOLEStG. PARZENBrookhaven National Laboratory, Upton, New York 11973, USA(Received January 14, 1975; infinalform March 12, 1975)Random errors in the magnetic field of superconducting magnets are likely to be larger than those found in conventionalwarm magnets. The random error multipoles introduced by random errors in the position of the current blocks arecomputed for cosine dipoles and quadrupoles and for window-frame dipoles. Analytical results are compared withcomputed results, and the computed results are compared with the results of measurements made on models of the cosineand window-frame magnets.I INTRODUCTIONRandom errors in the magnetic field of super-conducting magnets are likely to be larger thanthose found in conventional warm magnets. 1In superconducting magnets, the current coils areoften not hidden by the iron pole face, and arerelatively close to the region where the good field isrequired. This is particularly true for circularmagnets with a cosine current distribution and forrectangular window-frame magnets. Randomerrors in the positions of the current blocks intro-duce random multipoles in the field of the magnets.These random multipoles can excite the nonlinearimperfection resonances which appear to be ofconsiderable importance in storage acceleratorshaving intense beams with long lifetimes.This paper computes the random error multi-poles in cosine magnets and in window-framemagnets, which are introduced by random errors inthe current block positions.II COSINE DIPOLES ANDQUADRUPOLESOne method of constructing a cosine dipole isshown in Figure 1, in which the cosine distributionis approximated by current blocks, each of whichcarries a current which is proportional to thecosine of azimuthal angle. The random errormultipole fields introduced by errors in the currentblock position in a cosine dipole or quadrupolemay be computed analytically. This was done byt Work performed under the auspices of the U.S. AtomicEnergy Commission.237IIRONIFIGURE 1 Geometry of a superconducting dipole magnetwith a cosine current distribution.Parzen,2 Randle,3 and Ries.4 The error multipoleshave also been found more exactly by numericalcomputations by Dahl5 and by Parzen and Jellett.In the case of the cosine dipole, the rms value ofthe random error multipoles introduced by an rmserror in the position of the current block is givenby (see Section IV for derivation)_(~)1/2 n + 1f!.bn - NbRn+ 1X 8 [1 + (Rjp)2(n+ 1) - ~no(Rjp)n+ 1]1/21 + (Rip) (2.1)238 G. PARZENwhere the error field ~Bz on the median plane iswritten as~Bz = Bo(~bo + ~blX + ~b2X2 + ...),Bo is the unperturbed field of the dipole, R is theaverage radial position of the current blocks, N bis the total number of current blocks in the magnet,B is the rms error in the position of the currentblocks, p = a2 jR and a is the inner radius of theiron shield.The result Eq. (2.1) assumes that the currentblocks have zero radial thickness. The multipolesin the radial field, ~an, whereBr = BoC~ao + ~alx + ~a2x2 + ...)are roughly equal to the ~bn given by Eq. (2.1)except for n = 0 where - bno is replaced by bno.The result Eq. (2.1) may be generalized to quad-rupoles, sextupoles, etc., and is given by (seeSection IV for derivation)( 2)1/2 n+ 1~b - -n - NbRn + 1 -m[1 + (Rjp)2(n+1) - b (Rjp)n+1J1/2x £ 1 + (R/pt:~ (2.2)where m = 0 for a dipole, m = 1 for a quadrupole,m = 2 for a sextupole, etc. The result is the samefor ~an except - bnm is replaced by bnm .The accuracy of the theoretical result for therandom multipoles as given by Eq. (2.2) may beestimated by comparing this result with the com-puter results for a dipole magnet which has 6blocks per quadrant which are 1.905 cm thick,whose inner radius is 6 cm, and for a random coilposition error of 0.005 cm. The inner radius of theiron shield is 8.25 cm. The analytical and computerresults are given in Table I.The rms random error multipole present in aquadrupole are also given in Table I for a quad-rupole which has 3 blocks per octant whichare 1.905 cm thick, which has an aperture of 12 cm,and with an iron shield inner radius of 8.25 cm.The rms random error in the current block posi-tions is 0.005 cm. Error multipoles are introducedin both the vertical component of the field and theradial component of the field in the median plane,and the error multipoles are approximately thesame for both components.The accuracy of the analytical result for therandom multipole present in a quadrupole may beestimated by comparing this result .with thecomputer results found by Parzen and Jellett. Bothcomputer and analytical results are listed inTable I. The quadrupole multipoles are defined bywriting the error field as~Bz = B1(~bo + ~blX + ~b2X2 + .. '),where B1 is the unperturbed gradient of the quad-rupole.One may note that the error multipoles presentin the quadrupoles are larger than those presentin the dipole by the factor R(l + Rjp)j[l +(Rjp)2J, R being the average radius of the coils.This factor is about 10 for the 12-cm apertureTABLE IThe random multipoles generated by a random error in the current block positions of 0.005 cm for acosine dipole and a cosine quadrupole each having an aperture diameter of 12 cm, and whose currentblocks are 1.905 cm thick radially. Analytic and computed results are given. The an and bn are almostequal.Dipole Quadrupolelibn libn libn libnn Analytic Computed Units n Analytic Computed Units0 1.660 1.772 10- 4 cmo 0 1.356 1.307 10- 3 cmo1 4.016 3.874 10- 5 em- 1 1 3.280 4.019 10- 4 cm2 8.330 8.091 10- 6 em- 2 2 9.345 7.008 10- 5 em- 13 1.572 1.537 10- 6 em- 3 3 1.283 1.302 10- 5 em- 24 2.812 2.739 10- 7 em- 4 4 2.295 2.340 10- 6 cm- 35 4.835 4.919 10- 8 em- 5 5 3.953 4.036 10- 7 cm- 46 8.110 8.194 10- 9 em- 6 6 6.629 6.716 10- 8 cm- 57 1.333 1.319 10- 9 em- 7 7 1.090 1.270 10- 8 cm- 68 2.158 2.270 10- 10 em- 8 8 1.780 1.906 10- 9 cm- 79 3.449 3.640 10- 11 em- 9RANDOM ERRORS IN SUPERCONDUCTING MAGNETS 239magnets computed in Table I. Thus in an accelera-tor whose dipoles are powered to 40 kG, an.d whosequadrupoles go to 10 kG/cm, and for which thecombined length of the dipoles in a cell is 9 timeslarger than the combined length of the quadrupoles,one finds that the error multipoles in the quadru-poles are about one-third as effective as the errormultipoles in the dipoles in disturbing the particleorbits.There are several other ways to approximate acosine current distribution besides the one shownin Figure 1. One can expect that Eq. (2.2) shouldbe roughly valid for any method of approximatingthe cosine current distribution with an error of afactor of 2 or 3.Displacement of the Current Coils as a WholeA particular kind of random error which is notincluded in the above occurs when all the currentblocks in the cosine current winding of a magnetare all displaced by the same amount.One can show (see Section IV) that a horizontaldisplacement, ~x, of the entire coil will introducethe multipoles ~bm-1' and ~bm+ b where m = 0for a dipole magnet, m = 1 for a quadrupolemagnet, etc. A vertical displacement, ~y, intro-duces the multipoles ~am- 1 and ~am+ 1. A hori-zontal displacement ~x produces1f!.bm - 1 = -mf!.x 1 + (Rjpt+l' (2.3a)f!.bm +1 = (m + 2)~ (%r+ 21 + (:jPt+l'(2.3b)p = a2/R, (2.3c)where R is the average radius of the current wind-ing, and p is the radius to the image current,where a is the inner radius of the iron shield.Equations (2.3) are derived for a cosine winding ofzero thickness inside an iron shield of infinitepermeability. The same result in magnitude isfound for the ~am-1 and ~am+ 1 introduced by avertical displacement ~y.The random multipoles in a 12-cm aperturedipole and quadrupole which are generated by arandom error in the position of the coil as a wholeof 0.005 cm are given in Table II. Both the analyti-cal results from Eqs. (2.3) and exact computerresults are shown. The error in the analytic resultis about 10%.III WINDOW-FRAME DIPOLEMAGNETSIn computing the random error multipoles, onehas to make some assumption as to how thecurrent-carrying conductors can move. Figure 2shows a window-frame magnet which has 4current blocks. Actually, each of the 4 currentblocks is made up of many conductors. In thiscalculation, it is assumed that each of the 4 currentblocks can have random errors in their positions.This will give a larger error field than what isobtained if one assumes that all the conductorsthat make up each of the 4 current blocks wouldmove in a random way.The calculations given below are for a window-frame magnet with a 10 cm x 10 cm aperture,whose 4 current blocks have a rms random errorin their ,position of e = 0.005 cm. The error multi-poles introduced by moving one of the 4 currentblocks shown in Figure 2 in either the vertical orhorizontal direction was found using the GRACYTABLE IIThe random multipoles generated by a random error of0.005 cm in the position of the current coil as awhole for a cosine dipole and a cosine quadrupole each having an aperture diameter of 12 cm, a circu-lar iron shield with an inner radius of 8.25 cm, and whose current blocks are 1.905-cm thick radially.Analytic and computed results are given. The results for the an are the same as for the bn .Dipole Quadrupole~bn ~bn ~bn ~bnn Analytic Computed Units n Analytic Computed Units0 0.007 0.000 10- 4 cmO 0 ,.-3.513 -3.962 10- 3 cm11 5.540 6.693 10- 5 cm- 1 1 0.005 0.000 10- 4 cmo2 0.031 0.000 10- 6 cm- 2 2 6.425 8.583 1'0- 5 cm- 13 0.000 0.000 10- 7 cm- 3 3 0.007 0.000 10- 5 cm- 2FIGURE 2 Geometry of a window-frame dipole magnet.magnet program. An error in the position of acurrent block destroys the median plane sym-metry and the error field introduced has bothvertical and horizontal components in the medianplane.The rms multipoles introduced by a random rmserror of e = 0.005 em in the position of all 4current blocks in either the horizontal or verticaldirection is given in Table III. Table III also givesthe error multipoles introduced by moving thecurrent block in the upper right corner by theamount i\x = -0.005 em, and i\y = -0.005 em.These results are for a window-frame magnet with240IRON..L..L-'-G. PARZENa 10 x 10 em aperture and whose coils are 2.54 emthick.The results given in Table III may be scaled todifferent values of the rms error in the coil positions,e, since the results are linear in e, and to window-frame magnets with different square apertures sincethe multipoles scale like R n + 1, where R is the widthof the aperture.The rms results given in the first two columns ofTable III are computed by assuming each of thefour coils can move randomly in the x and ydirections with a rms error of 0.005 em. In an idealwindow-frame magnet, the coils appear con-strained so as not to be able to move in certaindirections. However,. in an actual magnet there areusually spaces between the coils and the iron sothat they can move. The rms result given in Table Iis thus twice the square root of the sum of squaresof the multipoles generated by the displacementsi\x = 0.005 em and i\y = 0.005 em which are alsogiven in Table III.IV COMPARISON WITH EXPERIMENTExperimental measurements have been made ofthe random error multipoles present in two identi-cal cosine magnets,8 the 8-cm models ISA I andISA II, and in two identical window-frame mag-nets,9 which are part of the go bend system. Thesemeasurements can be compared with the results oftheoretical calculations of the random error multi-pole .present in cosine magnets,2,5 and in window-TABLE IIIRandom error multipoles in the median plane field of a window-frame magnet with a10 cm x 10 cm aperture and 2.54 cm thick coils. The rms multipoles are for a randomerror in each of the 4 coils of e = 0.005 cm. The Ax = - 0.005 cm and Ay = - 0.005cm columns give the multipoles generated by the indicated displacement of the coil inthe upper right corner. The Abn are the multipoles in the Bz field, and Aan are the multi-poles in the Br field.rms Ax = - 0.005 cm Ay = - 0.005 cmn Abn Aan Abn Aan Abn Aan Units0 0.504 2.418 0.000 0.478 0.252 -1.11 10- 4 cmo1 0.320 0.728 -0.000 0.159 0.160 -0.327 10- 4 cm- 12 0.103 0.109 0.002 0.031 0.052 -0.048 10- 4 cm- 23 2.07 1.30 -0.000 0.045 1.03 -0.384 10- 6 cm- 34 0.322 0.245 -0.005 0.122 0.161 -0.012 10- 6 cm- 45 4.20 3.59 -0.000 1.67 2.106 0.176 10- 8 cm- 56 0.521 0.670 0.04 0.33 0.26 0.0 10- 8 cm- 67 8.8 5.8 -0.000 2.9 4.4 0.0 10- 10 cm- 78 2.3 1.2 0.000 0.6 1.2 0.0 10- 11 cm- 8RANDOM ERRORS IN SUPERCONDUCTING MAGNETS 241frame magnets7 due to a random error in theposition of the current blocks of 0.005 cm.This comparison is shown in Figure 3 where theerror multipoles in the radial component of themagnetic field in the median plane, ~an' are plottedagainst n. In order to make the comparison easier,the results for the cosine magnets, which have aradius to the inner coil surface of 4 cm, were scaledto those of a cosine magnet with a radius of 5 cm.In the go window-frame magnets, the inner surfaceof the coil is 5 cm from the center.ox10-4 X- COSINE, MEASUREDa-WINDOW FRAME,MEASUREDdifferent for the two models of each magnet. Also,in sotpe cases, the origin of the ~an is thought to beknown and these may not be considered as due torandom errors. The experimental results shown onthe graph are the largest values measured for the~an, as one changes the field level and for the twomodels. This gives a pessimistic estimate of themultipoles present.The agreement of the measurement with theoryseems reasonable. One can expect rather largedeviations since the theoretical result is an rmsresult, and a particular measurement can giveresults differing considerably from the rms result.With the limited experimental results available, onemight conclude that the random error multipolespresent for the cosine and window-frame dipolesare about equal, and this is roughly true even forthe higher multipoles.FIGURE 3 A comparison of the measured and computedrandom error multipoles in a cosine dipole and a window-framedipole.The solid lines in Figure 3 are the computedresults for the cosine and window-frame magnets.They indicate that the higher multipoles presentin the window-frame are considerably smaller thanthose of the cosine magnet. This can be understoodby realizing that the Llan ,-....; 1/Rn + 1 where R issome effective radius of the coils, and that theeffective R of the window-frame magnet is J2times larger than the effective R of the cosinemagnet. The computed results should be treatedwith caution, since the assumption of the calcula-tion for the window-frame magnet that each of the4 current blocks moves as a whole is probably notvalid and is worse for the higher multipoles.The experimentally measured results are alsoshown in Figure 3 as crosses for the cosine magnetand circles for the window-frame magnet. Theexperimental results for the Llan vary with the levelof field excitation of the dipoles, and are somewhat(4.4)00A(r,O) = L (A k cos kO + Ck sin kO)r\ (4.3)k=lDerivation ofAnalytical ResultsThis section gives a brief summary of the derivationof the results given in Section III. Further detailsare given in Refs. 2-4.We start from the following expression for thetwo-dimensional vector potential when no iron ispresent.A(r,O) = - ;n fdS' In R j(r', 0'), (4.1a)R = {r,2 - 2rr' cos(O - 0') + r2}1/2 (4.1b)where J1 = 4n/10 and dS' = r' dr' dO'.V sing the expansion00 xkIn{l - 2x cos 0 + X 2 }1/2 = - k~l k cos kO,(4.2)wherewe can find for small r near the center of the magnetthe multipole expansionand a similar expression for Ck obtained by re-placing cos kO by sin kO.If the current carrying conductors are all within acircular iron shield with inner radius a, and withinfinite permeability, then one can show2 that theooCOSINE,COMPUTEDWINDOW FRAME,COMPUTEDohocCf242 G. PARZENDisplacement of the Coil as a wholeIf all the M conductors are displaced a distanceI1x, then I1rn = I1x cos ()n, and rnl1()n = -l1x sin ()mand from Eq. (4.7) we findLlAk= - 11 Llx LIn2n n rnX {~COS(k - 1)On - ~ cos(k + 1Wn}.rn Pn(4.12)For the simple case, where the conductors all lieon the circle rn = R, and In = I cos ko(), thenEq. (4.12) gives only two nonzero multipolesA similar expression can be found for the rmschange in I1Ck ,- J.lI (e)(M)1 /2[ 1 1 1J1 /2LlCk = 2n Ii T R 2k + p2k + e5kko Rkpk. .(4.9b)For the cos ko() winding, the unperturbedmultipoles are given by Eq. (4.1) asJ.lI [1 1JMAko = 2nkoRko + pko T. (4.10)Using Eqs. (4.9) and (4.10) for the multipoles inthe vector potential, one can find the perturbedmultipoles in the field asI1b = (~)1/2 (n + 1)en M R k + 1 - k o[1 + (Rlp)2k - bkko(Rlp)k]1 /2x [1 + (Rjp)kO] (4. 11a)where k = n + 1. A similar expression can bederived for the l1an , the error multipole in theradial field in the median planel1a = (~)1/2 (n + 1)en M Rk + 1- ko[1 + (RIp)2k + bkko(RIP)k] 1/2x [1 + (Rjp)ko] . (4.11b)expression (4.4) for the multipole Ak is replaced byAk = 2~k fdS [~ + (;2y}(r, O)cos kO. (4.5)A current-carrying filament located at r, () andcarrying a current I will produce the multipolesJ.lI [1 1JAk = 2nk ?' + l cos kO, (4.6)where p = a21r and a displacement of the filamentby I1r, 11() will cause the multipoles to change byI1Ak •LlAk = III { - ~ [~ + ~Jsin k() r 11()2n r rk pk- ~ [~ - ~Jcos k() I1r}, (4.7)r rk pkwith a similar expression for I1Ck •Random DisplacementsIf there are M such filaments each situated atrn, ()n carrying the current In' undergoing dis-placements I1rn and rn l1()n which each have therms value of e, then one finds for the rms changein I1A kI1A~ = (!!-)2e2 L I;2n n r nx [(r;k + p;k) - cos 2kO p~,J<J (4.8)where I1Ak is the rms value of I1A k and Pn = a2Irn.For the simple case where one assumes that Invaries like I cos ko(), and rn = R where ko = 1for a dipole, ko = 2 for a quadrupole, etc., thenone can approximate the sum over n by"22 MLJIn =I 2 ,n" 22MLJ In cos 2k() = bkkoI 4'nand find that- J.lI e (M)1 /2[ 1 1 1 J1 /2LlAk = 2n Ii 2 R 2k + p2k - e5kko Rkl '(4.9a)where p = a2/R.pIl1x 1 MI1A = - ----.-kO-l 2n R Rko-l 2'I1A =pI I1x _1_ MkO+l 2n R pkO + 1 2 .(4.13a)(4.13b)RANDOM ERRORS IN SUPERCONDUCTING MAGNETS 243Using Eq. (4.10) for the unperturbed field Akaone finds the multipoles in the perturbed field1Abm- 1 = -Ax m 1 + (R/p)m+l' (4. 14a)Abm+1 = ~~ (m + 2)(%r+ 2 1+ (:/Pt+l'(4. 14b)where m = ko - 1.A similar analysis shows that a displacement ~yof the entire coil produces ~am-l and ~am+ 1 whichare equal in magnitude to the ~bm- 1 and ~bm+ 1caused by the ~x displacement.ACKNOWLEDGMENTI wish to thank K. Jellett for his help with the computations, andP. F. Dahl and W. B. Sampson for helpful discussions.REFERENCES1. G. Parzen, Accelerator Dept. Report CRISP 72-48,Brookhaven National Laboratory (1972).2. M. Month and G. Parzen, Particle Accelerators, 2, 227(1971); G. Parzen, Accelerator Dept. Report AADD-187,Brookhaven National Laboratory (1972).3. T. C. Randle, Rutherford Laboratory Report RHEL/R197(1970).4. G. Ries, Kernforschungszentrum Karlsruhe Report 3/71-8(1971).5. P. F. Dahl, Accelerator Dept. Report CRISP 73-2,Brookhaven National Laboratory (1973).6. G. Parzen, Accelerator Dept. Report AADD 73-10,Brookhaven National Laboratory (1973).7. G. Parzen and K. Jellett, Accelerator Dept. Report AADD73-12, Brookhaven National Laboratory (1973).8. W. B. Sampson, P. F. Dahl, A. D. McInturff, and K. E.Robins, Proc. IX Intern. Con! High-Energy. Accelerators,SLAC, 1974, p. 170.9. J. Allinger, G. Danby, B. DeVito, H. Foelsche, S. Hsieh,J. Jackson, and A. Prodell, Proc. IX Intern. Con! High-Energy Accelerators, SLAC, 1974, p. 198.

Random Errors In The Magnetic Field Of Superconducting dipoles And quadrupoles

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