Global linear coupling has been extensively studied in accelerators and several methods have been developed to compensate the coupling coefficient C using skew quadrupole families scans. However, scanning techniques can become very time consuming especially during the commissioning of an energy ramp. In this paper they illustrate a new technique to measure and compensate, in a single machine cycle, global linear coupling from turn-by-turn BPM data without the need of a skew quadrupole scan. The algorithm is applied to RHIC BPM data using AC dipoles and compared with traditional methods

Calaga, R.

Franchi, A. (Gsi), Tomas, R.(Cern)

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-- , - a  r -  BROOKH EN NAT I 0 N.6A-L LAB 0 R A T 0  RY BNL-75456-2006-CP Fast compensation of global linear coupling in RIIIC using AC dipoles A. Franchi, GSI, Planckstrasse 1, Darmstadt, Germany R. Calaga, BNL, Upton, NY 11973, USA R. Tomas, CERN, Geneva, Switzerland Presented at the.1 Oth Biennial European Particle Accelerator Conference (EPAC) Edinburgh; UK June 26-30,2006 Collider-Accelerator Department Brookhaven National Laboratory Upton,. NY-11973-5000 www.bnl:gov P.0: BOX 5000- Notice: This manuscript hasbeen authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy:The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up; - irrevocable; world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before . publication, it may not be cited or. reproduced without the author's permission. DISCLAIMER This report was prepared as an account of. work sponsored by an agency of the United States Government: Neither the United’ States Government nor any agency thereof; nor any of. their employees, nor any of. their contractors, subcontractors, or their employees, makes any warranty, express or, implied, or assumes any .legal liability or responsibility for the accuracy, completeness, or- any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned 9 rights. Reference herein to any specific commercial *product, process, or. service by trade name,. trademark, manufacturer, or otherwise, does not necessarily constitute or. imply its endorsement, recommendation, or favoring by the United States ’ Government .or any agency thereof I or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state-or reflect those of the United States Government or any agency thereof. . .  FAST COMPENSATION OF GLOBAL LINEAR COUPLING IN RHIC USING AC DIPOLES A. Franchi, GSI, Planckstrasse 1 Darmstadt, Germany R. Calaga, BNL, Uptown Nk: USA R. Tom& CERN, Geneve, Switzerlandj Abstract efficient C,, whose phase Os, is given by .. Global linear.; coupling hasbeen extensively studied.in .*, C' - IC lei% = &Js,x;*e-i(+P-+;,") (2) 2n w . accelerators.. and severalFmethods? have 'been developed ..a to. compensate : the I coupling . coefficient C. using .skew quadrupole families scans. .However; scanning techniques where the sum is over all the skew quadrupoles in.the fam, can: become .very .time .consuming. especially, during the . ily, Jsq is.the integrated strength (assuming a shared power commissioning of:an energy ramp.. In this paper.we illus-:. supply), p" 'and c#P'. are. the-.TwiSs.functions;at the. skew trate a new technique to measure andkompensate, in a sinT . quadrupole locations. . If 0 is unknown a scan of the two gle machine cycle, global linear coupling from turn-by-turn j .  families is necessary to drive an external coupling and min- BPM data without the need of a skew quadrupole scan.-The c.imize C+C;q,l +Cs,,2. .The same,goal is achieved without algorithm isapplied to RHIC BPM data using AC dipoles.. any-scan;.by.measuring- 0; decomposing C on the direc- and compared.with traditional methods.:. tions:.@)g,,l and-0&,2,  and-making .the afamilies drive the opposite strengths as showrin  fig.^ l'.- INTRODUCTION '4 - The complex linear coupling coefficient is defined as [ 11 ~ c =' ICle!?= .: (1) -1. f d s . j ( s ) ~ ~ e - i ( + ' . ( " ) - + . ( " ) ) + i a / R A . ,  2n : where R. is ,  t1ie:machine radius, A. = Q3..-.Qy is the:- the ~~i~~ functions andj(s) is the skew,quadrupolar force along the,ring., Th&coupling strength.lCl, also.hown in the ,American,literature as. AQmin, ,is the tune ,separation.. ).. lines and on. the resonance; whereas 0 denotes: tke phase of the cou- I difference of the,bare tunes.(fractional .part),.,P.and $.are Figure 1: Left: complex plane-witk the measured I C l ' a n d '  the directions>of the, two corrector families.. Right:' case .. .,with,measurement.of- C, its,decomposition along the.two . correction '. FROM FlOOl TO IC1 ( AQMnV 1 pling coefficient: 1 ~ 1 ,  is usually inferred from tune mea-. surement against A (closest-tune approach).. A repeated. tune measurement is therefore needed for different work- ing points. Other techniques such as first turn analysisand beam response after kick are described in [I]. Alternative methods using beam profile monitors are illustrated in [2].  In [5] it is shown that the linear coupling resonance driv- ing term (RDT) f1001. = If1001 lezq:is an observable, as it is measurable from the FFT of turn-by-turn BPM data 0f.a transversely excited beam according to None of the above technique provides any information on the phase of C. In [3] both amplitude and phase were beam,profile after exciting the beam with a fast.horizon- -/- 1 IV(h O ) l l H ( O ,  111 (3) measured observing the time evolution. of the transverse (4) tal,kick.>* In [4]. the same measurement was carried out. from turn-by-turn-BPM data and a fit of the corresponding where H(1,O) and V ( 0 , l )  are the horizontal and ver- PoincarC map., tical tune peaks respectively .(i.e. the .fundamental har- In machine with unsplit tunes and working point close to monic in the horizontal and' vertical FFT space respec- the difference resonance, q ! ~ ~ ( s )  - &(s) N 0 along the en- tively), whereas H(0,l) and3V(1, 0) are the secondary tire ring. This results in 0 N 0 and a real C: one family of harmonics excited by linear coupling,close to the differ- skew quadrupole is therefore enough for the correction. In. ence resonance; Amplitude and phase of each harmonic, machine with split tunes instead &(s) - &(s) varies from< in the horizontal spectrum for example, read H(m, n) = 0 to 2n along the ring,.resultingin.O # 0. The correc- IH(m,n)leZ@'H(m,n). tion is performedby means of at least two families of skew In [2] it is shown how to measure IC1 from two measure- quadrupoles; each one them represented by a complex co- ments of If1001 I (at any location) at two different working 2 IH(LO)llV(O, 111 lfl0OlI' = n 4 = @H(O,l) - @V(O,l) + 5 1 points. Introducing the difference of the tunes (fractional I part) A = Qz - QY, it can be shown that A linear fit against A is therefore enough to infer IC[. In the upper plot of Fig. 2 the inferred IC/ from two simulated measurements of lf100lI according to Eq. (5) is shown. -1 lrommodsl : [GI = 0.00401 - 0--e from FFT(a= 0 m) :.IC1 =0.00402 - +--+ lrom FFT(s=38 m) : IC1 n0.00401 0.0044 -. %?-++$ Irom FFT(a=72 m) : IC1 - 0 . W O Z  +--+ from FFT (a=144 m) : IC1 = 0.00402 -0.06 , -0.04 -0.02. 0 I 0.02 0.04 . 0.06 A:  Figure 2: Examples of. IC1 inferred from two simulated measurements of If1ooll'according to Eq. (5) at one loca- tion (top).and at several locations (bottom). Variation of lf100lI along the ring due to localized cou-. pling .sources results, in different ,lines having nearly the same values at A = 0, as shown.in the bottomtplot of Fig. 2 [5, 61. There single particle simulations,of the SIS- 18 of GSI with distributed skew quadrupole kicks along the ring are run for two working points and 4lf1oolAl is com- puted from the FFT of turn-by-turn data. The picture in- dicates that a single measurement of 4lfl00lAl at differentA locations provides already a good estimation of IC/, being ' where the normalized sum denotes the average over the BPMs (Le. along the ring), N is the number of available monitors, and<the latter condition is required for making use of Eq. (5). In simulations shown in the bottom plot of Fig. 2 the accuracy in computing,lCI is of,about 0.5%. For an effective measurement the available BPMs should be distributed uniformly along ring: regions with large cou- pling uncovered by BPMs would prevent the average from describing the global amount of coupling. FROM 3'1001 TO PHASE OF C ( 0 ) In [2] it is shown how the phase of the coupling strength 0 is related to the phase of f ioo l  where the normalized sum denotes the same average over the BPMs of Eq. (61, qw is the phase of f100l measured at the wtb BPM, and $w are the BPM betatron phases (taken from ,the lattice model). The above approximation has.a reminder proportional to A. It can be shown that on the resonance (A = 0) both lf100lI and q - (& - $y) are constant along the ring. Note that also the betatron phases could be in principle inferred from the BPM spectra after choosing one BPM as reference'. Nevertheless the correc- tion relies on the model skew quadrupole Twiss parameters as shown in Eq. (2). The latters are not measurable in a straightforward and reasonably fast way. .Therefore a more refined measurement of 0 would not be of help finding the best corrector setting. In Fig. 3 the variation along the RHIC Yellow ring (in- jection energy) of qw - (+: - $:) + n (1 - sgn(A)+) was simulatedl(MADX tracking) and is plotted for differ- ent working points. Five skew quadrupoles generate a cou- pling of IC) = 0.01. For A > IC1 jumps are visible in correspondence of skew quadrupolar kicks. As you move closer to the difference resonance, A -+ 0, the jumps be- come less visible. From simulations at A = f0.05, Eq. (7) provides O = 2.79 and O = 3.10 respectively. The devia- tion from the correct value 0 = 3.0 is therefore N 5% as expected. I I 1 I ,  I I 1  I I 1  1 %  1000. 2000 ' 3000 4000 v 2.40 longitudinal position [m] Figure 3: MADX tracking simulation of the RHIC Yellow ring at injection energy with betatron coupling ]CI = 0.01: the quantity qw - (4: - 4;) + 7r (1 - sgn(A)$) at the BPM locations is plotted for different working points. MEASUREMENT AND CORRECTION OF C IN RHIC DURING 2005 5 Eq. (6) is applied to RHIC BPM data taken during 2005 [7]. The beam,was transversely excited by.two AC dipoles. The inferred ICl,are listed in Tab. 1 and compared with the one obtained applying the N-turn map algorithm described in [8]. The standard deviation of lf1oolI along the ring is used as error indicator:.The use for several BPMs and the average make the formula be robust against failure of few BPMs (isolated large jumps in upper plot of Fig. 4). During the measurement of May 30 and June 13 2005 two date- < lf1oolI >,. IC1 Eq. (6) IC1 [81 May30 2 0 f 8  13 1.1 f 0 . 4  1.6 May30 5 0 & 9  48 1 0 f 1 . 7 .  10 June8 '  2 5 f 9  39 4 f l .  3.1 June 13 3 0 f 9  41 4 . 9 f 1 .  4.4 June 13 4 0 f 1 0 -  30 4 . 8 f 1  4.4 June 13 2 5 5 8  45 4 . 5 5 1  4.4 Table 1: IC1 of the RHIC yellow ring at4njection energy from f100l measurement during 2005wsing Eq. (6) com-. pared with- the ones obtained with the N-turn map algo- rithm [8]. All the numbers are in units of lor3. Infer. ox " 0.745 Infer. 0" D 0.B0,  Ohnln =o,(1088 RHIC-YELLOW-INJ ,30 may 2005 TBT FILES adc04-adcll 0.12 0.08 +2- 0.06 0.04 0.02 -. - 0 - .  0 lwo 2wo 3000 4ow longitudinal pasition [m] I"f.3,. ox-0.745 I d e r .  0" = 0.I18, co.wBe RHIGYELLOW-INJ ,30 may 2005 '. TBT FILES: ndc04-odc12 (1 E.i.56 /rad/ = .? l7 /<Ic~/  m . .  2 ,  0 I IWO 2wo 3000 4wo longitudinal position [m] sets of BPM data were taken turning off the three families Of corrector skew qUadrUpOleS. The corresponding IC1 and+ 0 are therefore the amplitude and the Phase respectively of the natural global coupling coefficient C (see Fig. 4 for the data of May 30). Eqs. (6) and (7) yield ~i~~~~ 4: Variation along the ;yellow ring at injection of lf loo l  I (top) and p - (4: - 4;) -sgn(A) ; (bottom). The corrector skew.. Data points (Le. BPMs) -with fluctuation larger than 30% were rejected. IC1 = (9.6 f 1.7) x May30 (8) SClOlCPY (116O) b, IC( = ( 1 6  f 1) x June 13 (9) 0. =. (5.56 f 0.19) rad = (319 f 16) May 30 (IO) SQllCPY (165') '>, - ------- 0 = (5.58 Lt 0.17) rad = (320 f 10) June 13 (1 1) - - - - - _ _ _ _ _  The standard deviation of qe- (&.- q&) along the ring is.. used to estimate the error. , In May 30 a traditional scan using two independent skew 2 quadrupoles (SQliC2Y and SQOlC2Y) was performed to minimize IC[, providing the following best setting J1;SQOIC2y si. 6.x rn-' (scan) C,,,,,,v= -0.00571 0048 J1,SQllCzy si 7 x 10-~  m-' (scan).. Figure 5:  Scketch of the measured C decomposed on the. axis defined by the two skew quadrupoles used for correc- tion. A similar. result, is obtained without any scan 1 by just decomposing C on. the. axes defined by the skew quadrupoles. Their directions in the complex plane, de- fined byse-z(#=-#n) according to Eq. (2), are plotted in REFERENCES Fig. 5 togetherswith C and its decomposition on the two axes. The gradient Jsq are obtained by C,, using the model beta functions and inverting Eq. (2). The best corrector set- ting eventually reads J1, SQOlC2y 2: (5.2 f 1:9))x m-' (RDT) 51, SQllC2y N (6.3 f 1.9) X loW4 mY1 (RDT) . The error is inferred from the decompositions of the upper and lower values of both')C) and 0 defined by their error- bars. * [l] N. Minty and E Zimmermann, Measurernetit and Control of [Z] A. Franchi, PhD thesis, Goethe Universitat Franfurt am Main. [3] P. E Tavares et al., Phys. Rev. STAB 1 (1998) 054001. [4] J. Y. Liu et al., Phys. Rev. E 49/2347 (1994). [5]  R. Tornas Garcfa, PhD thesis, CERN-THESIS-2003-010. [6] .R. Calaga et a]:, Phys. Rev. ST Accel. Beams 8,34001 (2005). [7] R. Calaga et al., proceedings of PACO5, Knoxwille (2005). [8] W. Fischer, Phys. Rev. STAB 6 (2003) 062801. Charged Particle Beams, Springer 2003. 

Fast Compensation of Global Linear Coupling in Rhic Using Ac Dipoles.

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Fast Compensation of Global Linear Coupling in Rhic Using Ac Dipoles.

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