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Design and Performance of thePEP-II B-Factory HER QD4 Quadruple Magnet

J. SwanD.Behne

C. M. KendallR. Yamamoto

T. YokotaJ. Tanabe

This paper was prepared for submittal to the15th International Conference on Magnet Technology

Beijin~ ChinaOctober 20-24,1997

DECLAIMER

This document was prepared as an account of work sponsored by an agency ofthe United States Government. Neither the United StateaGovernment nor theUNversity of Californianor any of theiremployees, makeaany warranty,expressor implied, or assumes any legal Iiability or rqonaibility for the accuracy,completeness, or usefulness of any information apparatus,product or processdiacl~ or represents that its use would not infringe privately owned rights.Rekrence herein to any specific commercialprodu~ process, or service by tradename, trademark,manufacturer,or otherwise, does not necessarily constitute orimply its endorsement, recommendation, or favoring by the United StatesGovernment or the UNversity of California. The views and opiniom of authorsexp-=d herein do not necessarily state or reflect those of the United statesGovernmentor the University of @ifonua“ ,andshallnotbeused foradvertiaing~ @uct endorsement~.

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Design and Performance of the PEP-II B-Factory HER QD4 Quadruple MagnetJohanna Swan, Daniel Behne,

C. Matthew Kendall, Robert Yanmmoto, Ted YokotaLawrence L,vennore National Laboratory, Llvermore, CA 94550

Jack TanabeE. O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720

Abstract-- The High Energy Rkig (HER) in Stanford LinearAccelerator Center’s PEP-11 B-Factory employs two high fieldquality quadrnpole magnets, labeled QD4, for tinal horizontalbeam de-focusing at a gradient of-75.65 kG/m. An Wmmetric,septum quadruple design is required for QD4. Due to spaceconstraint the magnetic field is shaped with both the iron andtbe coil. Each coil has fifteen conductors. A perturbationanalysis was performed using the Poisson code in order to Iucatetbe ideal position of the individual conductors. Manufacturingand assembly tolerances of +/- 0.5 mm of each conductor wererequired to maintain an integrated field quality of multipolecontent of b“lb~s 0.0001 for n =3- 15 at a radius Of 59.0 mm.

The steel core of the magnet is 1.423 m long and is comprised of1.5 mm thick laminations. A cut out in the steel core is requiredto allow the Low Energy Ring beam to pass through the side ofthe magnet. A double shield is in place to allow the LER beamto remain field free. The pole tip shape is a simple hyperbolawithout any end contours. The design and performance of theQD4 magnet is presented.

MAGNET DESIGN

The QD4 quadruple magnet is currently running on theHigh Energy Ring (HER) at Stanford Linear AcceleratorCenters’ PEP-II B-Factory [1]. The septum quadruples arelocated in the Interaction Region (IR) with close proximity toboth the Low Energy Ring (LER) beam and the BabarDetector. These are the final focusing optics before theindividual rings are combined into the same vacuumchamber, The design is unique as a result of the physicsrequirements for the physical beam dynamics and thegeometric constraints of two adjacent beam lines housedwithin a common magnet. One beam is focused in a pure,high quality quadruple field within the magnet aperturewhile the other passes through an essentially field free region.QD4 is shown in Figure 1 and depicts the two beam lineshoused within the septum quadruple. A septum coil isrequired in tbe design because the two beams and theircorresponding vacuum chambers are in close proximity as thebeams near the interaction point (3P). Thk paper will addressthe design and performance of the QD4 quadmpole.

QD4 is the final focusing quadmpole magnet on the HERand it requires exceptionally high field quality, It must alsocontend with extremely limiting physical constraint imposedas the rings join together. It has a 1.5 meter magnetic lengthwith a nominal field gradient of -7,56 Tesla per meter at 9GeV. Field quality requirements for a 15 sigma beam arespecified as a maximum multipole contribution normalized to

This work was performed under the auspices of the U.S.Department of Energy by Lawrence LivennorC National Laboratoryunder comract No, w7405-ENG-48 and DE-AC03-76FW98, LBL.

the fundamental of 1 x 10-4 at a radius of 5.9 cm. QD4 islocated 3.7 m from the 1P and the two beams are onlyseparated by 64 mm at thk location. However, the ellipticalbeam changes shape aa it traverses through the quadruple.In fact, the largest y-dimension is at 4.45 m and the largest x-dimension is at 5.2 m. The QD4 magnet apertureincorporates a 15 sigma beam, a synchrotrons radiation fan

and a cone angle for tbe luminosity monitor. To furthercomplicate the geometry the LER beam tube must be wideenough to accommodate its syncbrotron radiation fan. Thisrequires the Q4 steel yoke to be machined for amPleclearance and prohWs a flux return path on thk side of themagnet. The LER is angled with respect to the HER and thedistance between them changes along the beamline axis.Therefore, an envelope encompmsing the beams largest sizemust be used in conjunction with the LER beams closest

proximity to the HER beam, in order to determine theavailable septum coil width. After alignment tolerances,vacuum chamber and a fringe field shield are added in thisarea, the cross sectional area is 20 x 36 mm for the septumcoil, The horizontal and vertical dimensions of QD4 w

limited to 600 mm by 600 mm because of detector assemblyinterface issues. An additional concern for this magnet isshieIding from external flux generated by the detectorsolenoid field. F] g. 2 illustrates the QD4 magnet cross-section at 2=3.7 m with the HER and LER beam stay clearshapes superposed.

Several fundamental decisions were made in the initialdesign phase which assisted the QD4 in satisfying the fieldquality requirement. Fkst, the septum coil was designed tominimize field perturbations. The coil shape negates the needfor pole shims on the hyperbolic pole tip because it shapes

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the field in the aperture. Second, all of the QD4 coils areidentical to maintain symmetry around the magnet aperture.Maintaining coil symmetry reduces the systematic multipolecontributions in the design, This criterion became the basisfor the final design.

COIL PERTURBATIONSCoil Design i%eory

The two dimensional pole and coil geometry for theseptum quadruple is derived from knowledge of thebehavior of a simple dipole geometry and takes advantage oftechniques of conformal mapping. For infinitely permeableiron the magnetic field distribution in a simple window frameseptum magnet geometry has certain properties which arewell understood. The field is uniform throughout the gap.This uniformity extends to the edge of the coil and does notrequire shaping of the flat pole. In addkion, the field outsideof the septum coil is zero for infinitely permeable iron. Smallvalues for this external field depend only on the level ofsaturation of non-ideal iron in the magnetic circuit.

There are, of course, limitations to the extent that one canachieve the ideal current density in the dipole geometry. Inthe dipole magnet, the current density in the septum coil isnot uniform as demanded by the requirements of the septumgeometry. It consists of several square (or rectangular)conductors, each carrying an approximately uniform currentdensity limited by the hole in the conductor to carry coolingwater. The conductors are separated by gaps required for theturn to turn insulation. Finally, there are mechanicallimitations in the precision the separate conductors can beplaced with respect to their ideal positions.

Analysis of the effects of these small deviations from theuniform current density demonstrates that the fielddkibution is affected in only a small way from the uniformfield distribution in the gap and zero field outside of theseptum predicted by the ideal geometry. The critical lengthfor the decay of errors introduced by these deviations fromthe ideal uniform current density is the dimension of theperturbation [2]. Thus, for instance, the size of the coolingchannel in each conductor and the width of the insulating gapbetween adjacent conductors characterize the I/e dampingdistance of the field perturbations due to these effects. Theeffects of conductor placement errors are a bit more difficult

to visualize. A vertical conductor placement error, e, can becharacterized by a small conductor pair above and below thenominal conductor position. The cument strips will have aheight e and a positive and negative current density equal tothe current density of the conductor. The I/e dampingdistance for the field perturbation due to this effect is theseparation of these two coil strips, or the vertical size of theconductor (similar arguments can be made for thecharacterization of the error fields due to horizontal errors inthe coil placement.) Thus, at a distance of several holediameters, gap widths or conductor heights, the perturbationsdue to cooling holes, conductor to conductor insulation andconductor placement errors are quite small. Conformalmapping of the dipole coil geometry back into the quadruplespace presents some further problems. The square (orrectangular) dipole conductor maps into a curvilinearquadmpOle conductor. Moreover, the required currentdensity for each conductor in the quadruple space isnonuniform.

Coil Design ImplementationIn order to implement the coil design theory a practical

place to begin was with the limited available coil space. Asmall square conductor, 0.255” with a slightly over sized holediameter (0.158”) was laid out geometrically in a 15 turn (5columns by 3 rows) design in the quadmpole space. Fromthe perturbation theory, it was decided to allow the 9conductors furthest from the good field region to be inuniform, straight rows. The idea was that the perturbationeffects would decay before reaching the good field region.The remaining conductors were given an initial positionbased on approximating the same shape as lines of inductionin the quadruple field. These positions were then mappedinto dipole space as shown in Figure 3. The three conductorsshown were used as knobs within the magnetic code Poissonto determine the location of minimum perturbation. Aninitial dipole model was computed with all the conductorsand the individual harmonics were normalized to thefundamental at the aperture. Then a model of an individualconductor at positions located +/-3 mm from the nominallocation was computed. The odd harmonic contributions(n=3,5,7) was then plotted with respect to the distancemoved. From these perturbation runs, an ideal location wasselected based on a minimum harmonic contribution. Thiswas determined graphically by locating where the curvescrossed the zero point (ordinate axis) as shown in Figure 4.For the QD4 magnet this occurred at approximately 2.6 mmfrom the initial position. Only the first three turns, or fUStcolumn were required to be located in this manner. Thesecond column of conductor perturbations decayed farenough away from the good field region. The conductorlocations were then mapped backed into quadruple space. Afull model of the quadruple magnet was then computedutilizing Poisson. The next step was to compute the effectsof manufacturing tolerances on the good field region. Themodeling indicated that tolerances of +/- 0.5 mm on the firstcolumn of conductors were tolerable to the design. Theremaining conductors were held to these same toleranceshowever, the modeling showed that the design could be moreforgiving.

,

--- .. . . . .

FJ?iConformally mappeddipole geometry from initial quadmpole

Fig.4 Perturbadoneffectsfrom conductor1displaced+/- 3mm.

Steel Core Design

The steel yoke is a two piece design. The minimumaperture for the magnet is dictated by the beam and fan stayclear zone, alignment and machining tolerances, the coneangle for a luminosity monitor and the vacuum chamber. Themaximum aperture is dictated by the physical constraints ofthe adjacent hardware and detector assembly interfaces.Therefore the pole tip was designed with a simple hyperbolaat a radius of 6,65 cm. Since the coil was effectively shapingthe field it was not necessary to add any pole tip contours. Acut out through the yoke was required to allow for the bypasschannel containing the adjacent LER beam pipe. This forcedan asymmetric magnet design. In order for the LER toremain field free, it was required to have the support piecebetween the two halves non-magnetic. The shieldsurrounding the LER channel however was magnetic toprotect the beam from stray fringe fields. In order to balancethe flux on the LER side with the flux on the opposite side, agap spacing was determined using Poisson. The final design

closely resembles a Collins quadruple with very little fluxcarried by the side pieces.

The steel core is 1.425 meter long. It is comprised of 1.6mm thick, stamped laminations and 76.2 mm thick endplates.The pole tips on the endplates are 25.4 mm thick and areremovable. This was required in order to empiricallychamfer the end pieces to remove the n=6, 10 and 14 realterms which are caused by magnetic end effects. Thelaminations were stacked and pressed together with aresulting interlaminar pressure of approximately 100 psi.The half core stack was then secured by pinning longitudinaltension bars to the endplates. A pre-designed gap betweenthe bars and the half core was then filled with epoxy. Theepoxy was used in place of welding in order to minimizepotential heat induced dktortions on the laminations.

MAGNETPARAMETERS

Table I summarizes the magnet requirements. The coiloperates nominally for 9,0 GeV energy but is also run at12GeV. The ampere-turns required are 13305 and 17740.

TABLE IMAG~ REQUIREMENTS

Gradient @ 9 CW TimGradient@ 12Ge~, T/m

756:10.09

Magneticlength, m 1.5Max. beam offwt Xaxis,mm 25Max.beamstayclearwidth,mm 99Max.beamstayclearheight,mm 103Allowed field stnmgth error lXIA-3

.4”

Integratedmultipoleerror@59mm 1X1O-4

The coil has 15 turns per pole and was fabricated fromstandard square, hollow copper conductor by EversonElectric Company. The nominal size of the conductor is 6.48mm (0.255”) square with a 4.013 mm (0.158”) diameter hole.The 9 and 12 GeV operations generate respective currentdensities of 30 and 40 ampshnm2 . The developed length ofeach coil is 50.3 m and the resistance is calculated to be 32.1milliohms. The magnet power requirements for bothoperating energies are listed in Table II.

TABLEHQ4POWZRIuWumFMWTs

9.0 ~vCurrent A 887 1183voltage, v 28.5 38.7Power, kW 25.3 45.7

Each coil has five parallel water circuits for cooling. Table IIIlists the required water for the magnet at each operatingcondition. The pressure drop was selected to limit themaximum water velocity to approximately 4.5 nds (15 fds).

TABLE111QdWATERSPECIPKATIONS

u

.

MAGNETICMAPPING TABLEIVKLAUS HALBACH COEFFICIENTSPORTWO PIECEQUADRUPOU2

Several methods were utilized in order to bring the--individual, higher order harmonics to within the requk-mentof ~bn/b2 S1e-04~First, a table~ Ients aeveloped by~tilt

~IUaus Halbach and adapted for a two piece quadruple wasIS chodn used to make rigid body motions to one half core with respectih@5S to the other half core. Using the standard expression for the

,

function of a complex v;iable which satisfies the twodimensional Maxwell’s equations, the expressions for theratio of the nth harmonic with the fundamental is:

*

~. L. Cnat the pole tip radius. The displacement variable‘N N

Lk

k nOITI’i&ZedtO thej)Ok tipHithS where &. —, AY = %mdh h

.~. Where 5X represents shear, by is up and‘P “ ‘~

down and e= change in rotation. The imaginary term, i. ~

in the coefficient for the horizontal assembly error indicatesthat the multipole skew with respect to the fundamental. Themultipole error values vary with the radius

l:l=l:lp,,tiw*[:)The formula was adjusted

since the coil measured at 4.3 cm radius, the region of interestis 5.9 cm and the aperture is 6.65 cm. The n=3 real term andnd real and skew terms were all reduced by applying theserigid body mechanics. Firs$ the non-magnetic c-support androd diameter were machined smaller in order to bring thetwo halves closer together. Next, a shim was added on theinside edge of the v-notch in order to translate the top half ofthe magnet away from the LER side. Finally, a double shimwas used in the notch in order to lift the rod and pivot aboutthe c-support. Since there are interdependencies whenmaking these changes it was important to follow the signs ofthe odd and even higher order multiples to ensure that theeffect did not substantially increase the multipolemagnitudes.

An empirical approach was used to reduce the n=3 skewterm. Several pole contours were machined off of the sparelaminations. Then each was ground down to severaldifferent thickness. Holes were drilled such that thelamination pole piece could be installed on the same pins asthe removable pole tips on the endplates. ‘l’he skew termrequires adding steel to the two poles either on the upper halfcore or lower half core. The location and quantity wasdetermined experimentally. During the process, it was alsoevident that the n=3 real term could be reduced by addingsteel either on the left side lower and upper cores or rightside. The measurements indicated that the additional steelbehaves orthogonal to each term.

Pole tip end chamfers on quadruple magnets arecommonly used to correct errors in the n=6, 10, 14introduced by three dimensional fringe fields. The size,location, and angle of the chamfers effect the relative changein these allowed harmonics. a single angle end chamfer wasused on the QD4 magnet to correct the n=6 term withouteffecting the n=10 and 14 terms.

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Fig. 5 Isoerror plot of measured QD4 harrnoNcs at r=5.9

CONCLUSIONS

The theoretical perturbation analysis correctly predctedthe ideal location for the individual conductors.Manufacturing tolerances of the coil appear to have beenachieved. However, some errors were introduced during thestacking of the steel laminations which affected the coilplacement within the magnet. These assembly errors wereremoved with shimming during the magnetic measurements.All requirements of the magnet were achieved including anintegrated field quality of multipole content of bn/b2 <0.0001 for n = 3-15 at a radius of 59.0 mm.

ACKNOWLEDGME~

The authors would like to thank Klaus Halbach for hisideas which have guided the design of this magnet.

REFERENCES

[1] PEP-IIAnAsymmetricB Factory Conceptual Design Report, June 1993

[2] USPAS Msgnetic Systems cow no=. * UrriVemitYtJ~usr’Y, 1*5

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(69.52 X 103:51)

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Technical Inform

ation Departm

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University of C

alifornia • Livermore, C

alifornia 94551