Phase advances ( 0,k - 0,k-1 ) for several currents gives localized. also determines Q x (J. Klem, 1999). Impedance Localization Robustness Checks

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1 Localization of the SPS Tranvere Impedance Gianluigi Arduini, Chritian Carli, Fran Zimmermann Motivation Optic Perturation Tet of Algorithm Impedance Localization Routne Chec Motivation Identify, localize and quantify ource of tranvere impedance Can we ee the effect of five re-intalled ferrite etraction icer? 3~5% impedance increae wa meaured in 23 y H. Burhardt & predicted y L. Vo Method aed on current-dependent phae etatron phae from multi-turn BPM reading Harmonic analyi of n give etatron phae at the pic up J. Borer et al., EPAC92, for LEP!. Ocillation at th BPM i 2 m A co mq, with m the turn numer and A the meaured amplitude. For many turn, >>1, the etatron phae p, at the th BPM i C S /, arctan C m co 2 mq S m in 2 mq m 1 m 1 Phae advance, -,-1 for everal current give localized 2 2 tranvere impedance. Maimizing A a a function of Q C S alo determine Q J. Klem, etatron phae hift with unch current in LEP D. Brandt, P. Catro, K. Corneli, A. Hofmann, G. Morpurgo, G. Sai, J. Wenninger, B. Zotter, et al., PAC1995 -> tranvere impedance ditriution

2 2/21 attempt to loo for tep in / tep? no error etimate etatron phae hift with current in the SPS J. Klem, G. Arduini, G. Morpurgo, EPAC2: indication for tep? Optic Perturation conidering only vertical plane, impedance act lie a current-dependent quadrupole of effective gradient e K eff Im Z, eff 2 E / e z unch population eam energy effective impedance note that vertical impedance i defocuing impedance will introduce eta & phae eating C. Carli; eplain ocillation in 23 analyi repone matri wa derived in 1t order perturation theory: co 2 2in 2 co 2 in 2 2in 2 Q M K 2 Q Q in rewritten a matri adding 2 Q K + K M computed from MAD optic olution otained y matri peudo-inverion, e.g., SVD for K for cut-off of ingular value in SVD inverion i another free parameter SVD olution i tailized y introducing additional equation with weight Tet of the Algorithm 1 imulation tet varied the trength of ingle quadrupole QE63 in MAD, and too calculated phae change a input correct quadrupole wa identified either y determining the et ingle quadrupole or y SVD peudo-inverion ince we ue 1 t order perturation matri, agreement etween actual and fitted change woren for large perturation; for Q=.1 the error in the quadrupole-trength change i ~5%, for Q=.1, it i.3% 2 eperimental tet procedure can only function if model i cloe to real optic varied ingle quadrupole to give tune change Q~.5 correct choice of i important larget change i found for net quadrupole QD63 v QE63, alternatively, QE63 wa identified when looing for mot efficient ingle change; fitted trength differ y 6%

3 eperimental tet K otained y fitting meaured phae change induced y change in QE63 at BPM y SVD inverion with SVD cut-off.1 and three different weight =.5 =5 =5 correponding fit reult uperimpoed on the meaurement: =.5 =5 =5 intermediate yield ~reaonale reult Impedance Localization data et were taen at 26 GeV/c on 4.9. and GeV/c on eam wa iced tranverely and 1-turn BPM reading were recorded for all BPM intenity of ingle p unch wa varied in 4-6 tep from 21 1 to chromaticity wa held at lowet value compatile with eam taility helped y increae of typical BPM raw data at high and low intenity vertical poition in aritrary unit v. turn numer 26 GeV/c 14 GeV/c decoherence time and cloed orit vary with eam intenity for each data et we compute average tune and rm tune pread over all BPM data et with large tune error pread are dicarded in the harmonic analyi, ince a large variation of the tune from BPM to BPM implie a large uncertainty in the phae

4 analyi of good data for each BPM & data et we determine the phae of ocillation with repect to the tart of the line; we contrain to lie within +/- from MAD model then for each BPM we fit v. to a traight line optic phae error at current? effect of impedance meaured phae variation v. unch intenity and linear fit for 7 elected BPM 1 variation with intenity i indeed linear 2 the 14 GeV/c data are le noiy fit reult for all BPM fitted fitted monotonic increae due to difference etween real -current & model tune / decreae imilarly; effect i larger for lower eam energy a epected 14 GeV/c data how eating uperimpoed on gradual decline le evident at 26 GeV/c determine impedance ource at poition of 237 quadrupole quad # 11, 24, 88, 12, 139, 164 QD111, QDA119, QD37, QD319, QF42 and QD57 w. large impedance quad # 24, 8, 136, 14, 157, 164, 236 QDA119, QD31, QDA417, QD421, QD51, QD57, and QD635 with large impedance SVD minimization recipe: ingular-value cut-off = 5 initial weight = 1 for all quadrupole; increaed y factor of 1 for K value of the wrong ign in 1 iteration, to mae impedance defocuing

5 current-dependent phae change predicted y iaed SVD fit compared with the meaurement fair agreement perfect agreement aed on impedance ditriution from previou lide Routne chec 1 election of quadrupole could impedance located near QD31 in reality e due to rf cavitie at quadrupole ? [G. Arduini] re-proceed only 14 GeV/c data which have higher quality ucceively eliminated quadrupole in the 3 region with large fitted impedance from uequent fit until rf cavity location i found QD31 QF3 QD35, QD39 QE32, QF38, QD313 QD317 QF34, QECD36, QF316, QD317, QD319 fit quality only lightly degrade a more and more quadrupole are added fit with all quadrupole fit without 9 quadrupole in the 3 rd arc

6 reult from full fit & from fit w/o 9 quad compared with data difference etween the two fit i much maller than the remaining dicrepancy to the meaurement 2 dependence on quadrupole weight effect on fit quality of varying initial quadrupole weight impedance fitted for different initial value of =.1 =.1 =1 =1 fit for different initial value and the meaurement

7 Summary From multi-turn BPM data at different unch intenitie & two eam energie, the current-dependent phae advance wa otained at each BPM of the SPS ring We determined the impedance location and trength giving rie to the meaured phae hift, uing the theoretical phae repone matri, which we peudoinverted y an SVD algorithm with adaptive weight factor. In an iterative approach, we uppreed large value a well a focuing impedance, yielding a conitent olution at oth SPS region 119 near MKP icer, ~31-37 arc, rf?, near MKE icer, and 57 arc? identified at oth eam energie a location with high impedance. Data at 14 GeV/c how much cleaner ignal & le noie The eact location may e mied a illutrated Accurate optic model & good data quality eential