Lohmann, K D

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ECONOMICAL AND ACOJRATE METHODS FOR MEASURING 1HE MULTIPOLE FIELD COEFFICIENTSOF BENDING MAGNETS AND QUADRUPOLESK.-D. LohmannCERN, Geneva, SwitzerlandAbstractAll relevant field parameters of bendingmagnets and quadrupoles can be measured wi th. longstraight coils, if the curvature of the partIcletrajectories inside these elements is not toogreat.The particular problems related to long coilsare discussed, namely calibration, measuringerrors, and mechanical layout. The measuring ap-paratus used for the CERN Proton Synchrotronbooster magnets is discussed, comprising:1. The long coil system for measuring bendingmagnets of 1.6 m length and 0.07 m gap height,featuring:simultaneous measurement in three planes;coil flipping wi th pneumatic motors;0.003% relative accuracy;simple mechanical construction.2. The harmonic coil with compensation windingfor measuring the quadrupoles.All multipole coefficients up to 28-pole onesare measured with an accuracy of 0.003% (multipolefield referred to the quadrupole field at boreradius) • The displacement of the magnetic centrewi th respect to the geometric centre of the quadru-pole is measured to within 0.01 rom.The system requires neither a precise angUlarposition nor precise incremental rotation.1. IntroductionWhat magnitude of non-linearities in themagnetic field of bending magnets and quadrupolescan be accepted for a particular project (beamtransport or accelerator magnet system)?To answer this question, one has to find amathematical model for the magnets which includesthese non-linearities, and then compute the part-icle trajectories. The mathematical model usedfor the design of the PS Booster 1 ) was the so-called three-thin-lens approach2), where one lenscorresponds to the field inside the magnet andtwo lenses simulate the end fields.If one has a complete field map of the magnet,one can compare the calculated particle trajec-tories in the real magnet to those in the three-len~ model. The differences found could safelybe neglected in the case of the PS Boos ter 3), wherethe bending angle of the magnet is 11. 25 0 •745For the existing beam simulation2 ), the fieldnon-linearities have to be entered into the pro-gram in terms of multipole coefficients of a bi-dimensional field. The three-lens model leads toa measuring method using long straight coils sub-divided longitudinally into three parts. The .measuring points were chosen such that the multl-pole coefficients could be extracted with thehighest accuracy, i.e. at the outer boundary ofthe volume to be measured4 ). For the quadrupole,this field volume is a cylinder; we thereforeused a harmonic coil giving the multipole coeffi-cients directly. For the bending magnet, thefield volume of interest is a rectangular box of0.07 x 0.15 x 1.6 m which had to be explored fol-lowing Cartesian coordinates. The multipole coef-ficients in this case are extracted from themeasurements by fitting the measuring points withpolynomials satisfying the Laplace equation.Using these methods, the number of measuringpoints could be limited to a reasonable amount,and simple mechanical devices with only one degreeof freedom could be used.2. Measurement of the bending magnets2.1 Mechanical construction of themeasuring apparatusFigure I shows the measuring coil assemblyconsisting of nine single coils located on threeparallel axes. Using the three coils of 1.3 mlength inside the iron yoke, we can measure longi-tudinal average values of the bi-dimensional field.By series connection, we can employ three 2.7 mlong coils measuring the bending power JB ds onstraight parallel lines. ZThe coils are flipped simultaneously by meansof pneumatic motors with hydraulic damping. Therotary movement is transmitted to the three coilsby an accurate gear drive. The rotation angle of180 0 can be adjusted on the motor itself.The 2.7 m long coil is carried in five nylonbearings and care has to be taken to avoid vibra-tions during the flipping movement. The bearingsmu~t be short and the whole 2.7 m long coil mustbe fairly straight. Referring to Fig. 2 one cansee that the whole assembly forms a rather stiffrod of 16 kg weight. It simply rests on the poleface of the magnet and needs only to be positionedin the X-direction.This is done by a rack and pinion drive fixedagainst the ma~let (see Fig. 2). The x-positionis encoded by microswitches.Lower Magnet Pole Face/57127Fig. Long coil assembly consisting of nine individual coils. The pneumatic motor is not shown.2.2 CalibrationThe actual relationship is more complicated. Agood approximation for the coil flux is obtainedconsidering a single loop with variable conductordistance w(s):(1)The calibration of a package of long coils isnot as simple as short coil calibrations. What wewould like to have is that the time integral of thecoil voltage u be proportional to the bending powerof the magnet: cFig. 2 Long coil assembly with positioningdevice(2)746For accurate calibrations, the conductor distancew cannot be considered as a constant, especiallyif the long coil consists of several parts withgaps in between. So, a universal coil constantK, independent of B(s), does not exist. Theequivalent average w for a particular field shapeB(s) may be obtained by one of the followingmethods:2. 3 ~Ieasuring accuracy\\lith one recalibration every fortnight, thestability of the measuring system was:±O.Ol% for absolute measurements,±0.003% for field homogeneity measurements.Harmonic coilwithcompensationWinding1 Loopx5are:TIle particular features of the harmonic coil3. Measurement of the quadrupoles3.1 Particular features of the measuring coilAs mentioned in Section 1, a harmonic coilsystem was used for measuring the quadrupole fieldparameters. This method is recommended if a) theparticle beam stays within an inscribed circletouching the poles of the quadrupole, and if b)the results of the measurements have to be givenin terms of multipole coefficients.i) For accurate measurements of the higher ordermultipoles, a compensation winding was used, asshown in Fig. 4. A compensation winding, situatedat a smaller radius, has to be more sensitive thanthe measuring winding, so that in a pure quadru-pole field the induced voltage in both of thewindings is the same for any angular position.The difference voltage shows all the multipolefields except the quadrupole one.571291,- Slide the COil into the magnetby a well know amount liS2/Measure the flux variationand oblain W for the correspondingintegral of the coilCalibration in a long magnetby longitudinal displacementof the coilFig. 3z!S~X ~,?//./~~i~~t~i:ld~:~'i~l~05J-' metalscreening t I ...+------,--'lB.~f':.-;I-t-I ~~L.-__~_,SFlux vanatlon ' In~~~~~c) Calibration based on a pointwise measure-ment of B(s), e.g. with hall probes.a) Measurement of the coil surface areaA = n • W• ~c of every individual coil. The coilwinding length ~C' as well as the intercoil gaps,may be determined by mechanical measurements.Assuming a simple function B(s), e.g. a linearfringe field approximation, one can determine was a function of the equivalent length of themagnet.b) Calibration in a long magnet by longi tu-dinal displacement of the coil according toFig. 3. By this method, average values of w(s)may be determined for any longitudinal intervalof the coil. The method requires a long magnetwith uniform field inside and a fringe field asshort as possible.\l'e obtained absolute accuracies of 0.1%,0.02%, and O.OB with methods a, b, and c, res-pectively.Fig. 4 Above: Flux induced in a single loop.Below: Harmonic coil with compensationwinding747one findsAfter integration and the following substitutionThe integration becomes easy by introducing acomplex variable w = x + iz:(5)rnsin n¢ = 1m [ (x + iZ)n] • (7)<P~n = z ~ [an sin ne + bn cos ne] rn sin n¢. (6)Let A be the winding density (nurr~er of wires permm2), then the coil flux due to the 2n-pole fieldis:The loop geometry is given by the parameters r and¢. In order to calculate the flux of a tmiformlywound coil, one has to integrate over a rectangulararea with the boundaries Xl; X2; ZI; Z2.~ B + + Be 00B rorBr 2n-l sin ne + n b n-l cos n0n a r rn n11= 13.2 Theory of the measurementUsing cylindrical coordinates as in Fig. 4,the general field expansion around the origin be-comes:ii) The coil is subdivided into three parts inthe longitudinal sense, so that measurements ofthe field inside the iron yoke and field integralscan be taken with the same measuring apparatus.iii) The magnetic centre measurement may be per-formed either by the (uncompensated) harmonic coilor by a special search coil on the quadrupole axis.The second method turned out to be more accurate.iv) Owing to the compensation method, the mechani-cal layout becomes very simple. There is no needfor accurate angular }iositioning, and slip ringswith the associated troise problems could be avoid-ed.j Br ds ; JBe ds •-00 -00Only the range r ~ RB is considered, RB being thebore radius of the quadrupole. This expansionmay be applied to the bi-dimensional field in thecentre of the lens or to the integrals 5 )B =e Ln=ln an rn- 1 cos ne - n bnrn-lsin ne •(3)~ 2 ~ A [an sin ne + b cos neJ xn nZ2 X2xii 1m (x + iz)n dx dzZ1 Xl~ A [an ne] -12 sin ne + b cos xn (n+l)(n+2)The principle of harmonic coil measurementshas been discussed in several papers 6) In ourcase, the multipole coefficients an and bn havebeen determined by a pointwise measurement of thefunction Br(e).Let us first calculate the flux ~£ induced inone loop of the coil (see Fig. 4). If ~ is theloop length, the flux contribution of the 2n-polefield becomes:+ n bn~- 1 cos neJ r de •e2~ J [n an rn- I sin ne +elHaving measured the periodic function <p(e~ ~= L~=l ~(e) in a certain number m of eqUIdIstantpoints one can find an and bn up to the order ofml2 by'discrete Fourier analysis using Eq. (8~.Two methods for measuring ~(e) have been appl1ed:a) by pulsing the quadrupole fie~d withvarious positions e of the COI~; .b) by stepwise rotation of the COlI In asteady-state field.3.3 Advantage of the compensation windingMeasuring errors may be introduced by the~-measurement (voltmeter errors and noise) and bythe error in positioning or measuring of e. .Furthermore a radial displacement 6r of the COlIaxis (due t~ play, vibration and non-c::ylindricbearing surfaces) will produce measurIng errors.In many cases, the error due to 6r is the mostimportant.(8)(4)B r der02<P~n = ! B ciS = ~ Je1748Let us cOQsider the diJrensionless error ~quantities M/epz; Llr/rn and Lla [rad], where epzis the ampli tude of the quadrupole flux and I"B isthe quadrupole bore radius.We assUJre Gaussian distribution and denotethe standard deviations as aep, or and aa' For asimplified harmonic coil consisting of a singleloop, we then calculate the measuring error(standard deviation) of the harmonic coefficients.Here also, we define diJrensionless harmonic coef-ficients an, an representing field errors withrespect to the quadrupole field at bore radius:(9)After treatment with the discrete Fourier trans-fOI1ll and Eq. (6) one obtains for the standarddeviation of an and an :n-za~n (J = ~B) n s~n 24> x~. an LTc 2 SIn n4>(10)rc is the distance of the coil from the quadrupolecentre (measuring radius) and p the number of(equidistant) measuring points.Equation (10) illustrates several designprinciples:1) Try to make rB/rc '" l.2) Avoid that sin n<j> '" 0 for any n ofinterest.3) Introducing typical numerical values(rB/rc = 1.1); n 5 10; P = 72) we find~hat the measuring error on an and anIS of the same order of magnitude as theindividual er:ors (Jep, ar and Ga. This isof course valId for the uncompensated coilsince we did the calculation for a singleloop.With the compensation winding, the measuringaccuracy was improved by a factor of 100 in thepresent case. The reproducibility for measuringthe coefficients an and an was better than ±0.003%up to n = 14. To achieve this result, it was im-portant to keep the radial distance small betweenmeasuring winding and compensation winding. Thislayout reduces the influence of the error due toradial coil movements (in the present case by afactor of 8). Figure 5 gives a plot of the com-pensated coil voltage versus the angular positiona, showing that compensated quadrupole flux isabout O.H of the uncompensated value so that themultipoles may be clearly distinguished.749......iT r!.., ~ lid" ~H"t 1-4~q ,.J~ ~d...J f-t H·r li." I'~f. ,~~ ~~Fig. 5 Compensated coil voltage versusangular position as measured ona booster quadrupole3.4 The measuring frameFigure 6 shows the measuring frame with oneof the booster quadrupoles. The particular featureof these quad[Upoles is the four gap, single-coreconstruction1 ) • One of the most important parameterto be measured is the position of the four magneticaxes with respect to the top alignment targets.Using rotated or flipped coils for this measure-ment, the positions of the magnetic axis are foundwith respect to the coil rotation axes. Thismeans that the bearings of the measuring coils haveto be at a well known position with respect to thequadrupole targets and outside reference surfaces.Fig. 6 Measuring frame for thebooster quadrupoles. ~e adopte~ solution is a free standing frameWIth eIght bearIngs which are aligned parallel andat the nominal distances. The alignment itselfwas done by means of an alignment telescope andtwo pentaprisms on the right-hand framework ofFig. 6. By displacing the lower pentaprism, fiveoptical axes can be established. Distances andparallelism of the axes are obtained by a systemof ten optical targets built into the two outerframes. The tilt of the quadrupole, with respectto vertical target axes, is controlled by an elec-tronic level.M:lst of the measurements were performed by pulsingthe quadrupole, and the coil was rotated by oneincrement between the pulses. This incrementingwas made manually, the angular position being re-corded by means of a microswitch and an electroniccounter.The measuring tube rests on four accuratelymachined rollers. The tube may be put into posi-tion very quickly and the reproducibility of thetube location is of the order of a few ]JIll.Figure 7 shows the connection end of themeasuring tube. lAle can distinguish the opticaltarget, a toothed positioning gear, and the har-JlJJnic coil fixed against the outside of the tube.Fig. 7 Measuring tube supportingthe harmonic coil3.5 Performance of the systemEmphasis was put on accuracy and reliabilityrather than on measuring speed and sophisticateddata handling, because the pure measuring timewas, in any case, small compared wi. th the time forinstallation and alignment of the quadrupoles.For the magnetic centre neasurement, we ob-tained an absolute accuracy of ±0.02 rrun and a repro-ducibility of ±0.01 rrun in an air-conditioned labo-ratory.For the multipole coefficients up to n = 14,the reproducibility is better than 0.003% in termsof normalized coefficients an, 8n (see Eq. 9).This figure depends on the order of magni tude ofthe errors to be measured (see Fig. 5). Systematicerrors due to coil imperfections and coil calibra-tion .errors were certainly smaller than 3% of themeasured an, 8n. A better calibration (to be donein both dipole and quadrupole fields) is possible.Table 1 gives a typical measuring result fora booster quadrupole. The spectrum of the ctn andan is in good agreement ~ith theory. ~ perfectlysymmetric quadrupole ha~lllg a h~er~o~lc pole pro-file which is not contlnued to lnf1ll1ty but cutoff ~omewhere, contains the series of coefficientsct2(2k+l) (k = 1, 2, 3 ••• ). Syrrunetry dis~ortionsproduce a continuous spectrum of an, an Wlth de-creasing amplitude, i.e. mainly sextupole and octu-pole fields.TABLE I Multipole coefficients of a booster quadrupolen 3 4 5 6 7an [%] -0.006 0.021 -0.003 -0.009 -0.002en [%] 0.062 0.024 0.024 0.000 0.011n 8 9 10 11 12an [%] -0.010 0.001 -0.132 0.003 -0.0028 [\] 0.002 0.002 0.012 -0.003 0.002nn 13 14 15 16 17 18an [%] 0.001 -0.030 0.000 -0.001 0.004 -0.084en [\] -0.002 0.001 0.009 0.001 0.002 -0.006750QUADRUPOLE. 9 co11 inputsBEND~t.ET. 6 coil inputsMAGNET SHUNTI«lllE ACQUISITION~ data (BCD)~ analog signalstri gger pulsesI«lDE ACQUISITIONLOGICCONTROLUNIT PneumaticmotorsCOILSELECTOR2 ~ Cl""g -«..u«c. IBM....CARD=> ~ .... PUNCH=w~ w~ :>:frOlll up-down counter.position encoder.IlIlnual inputFig. 8 Simplified block diagram of the data acquisition systemFrom the mUltipole coefficients, a field ploton any quadrupole axis (x, z, r) may be computed.This was done by our standard evaluation computerprogram, and the result was compared to gradientcoil measurements performed on the same quadrupoleooit. Taking into accotmt multipoles up to theorder n = 19, the differences are smaller than0.015% within 90% of the quadrupole aperture.4. Data handlingThe measured data is stored on plIDched cards.The card pooch is coupled directly to the integra-ting digital voltmeter, the coordinate transducers,and a measurement mode encoder giving infonnationon:the measuring coil circuit (long coil/shortcoil, compensated/oocompensated, field coil,gradient coil, harmonic coil etc.);the trigger point on the current cycle.Figure 8 shows the simplified block diagramof the data acquisition system for bending magnetsand quadrupoles. As the system is fairly standard,we only mention a few particular features.The measurement data is plIDched onto cardsby words of 20 characters. Each word contains:voltmeter reading, measurement mode, and coordi-nate. In total there are 75 possible combinationsof measurement modes and to avoid errors, it wasessential to generate the mode information auto-751matically from the coil selectors and the timingpulse selector.In order to record the voltage of three coilscorresponding to three z levels either three volt-meters or an analog scarmer is required. Foreconomical reasons, an analog scarmer was usedwhich is driven from a central logic control unit.This unit also controls the pneumatic flipping.References1) A. Asner et al., Proc. 3rd Int. Conf. onMagnet Technology. Hamburg. 1970 (DESY,Hamburg, 1970), p.418.2) C. Bovet and K. Schindl, PSB emittance blow-up in non-linear fields, CERN Report SI/Int.DL/70-4 (1970).3) C. Bovet et a1., Proc. 8th Int. Conf onHigh-Energy Accelerators, Geneva, 19Z1 (CERN,Geneva, 1971), p. 380.4) H. Wind, Nuclear lnstrum. Methods 84, 117(1970) •5) W.C. Elmore and M.W. Garret, Rev. Sci. Instrtm1.25, 480 (1954).6) J.K. Cobb and D. Horelick, Proc. 3rd Int. Conf.on Magnet Technology. Hamburg ]970 (DESY,Hamburg, 1970), p. 1439.

Economical and accurate methods for measuring the multipole field coefficients of bending magnets and quadrupoles

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