Localization of the SPS Transverse Impedance

1. Localization of the SPS Transverse Impedance Motivation Optics Perturbation Test of Algorithm Impedance Localization Robustness Checks Gianluigi Arduini, Christian Carli, Frank Zimmermann

2. Motivation Identify, localize and quantify sources of transverse impedance Can we see the effect of five re-installed ferrite extraction kickers? 30~50% impedance increase was measured in 2003 by H. Burkhardt & predicted by L. Vos Method based on current-dependent b phase

3. betatron phase from multi-turn BPM reading Harmonic analysis of x(n) gives betatron phase at the pick up (J. Borer et al., EPAC92, for LEP!). Oscillation at kth BPM is with m the turn number and Ak the measured amplitude. For many turns, N>>1, the betatron phase p0,k at the kth BPM is Phase advances (f0,k-f0,k-1) for several currents gives localized transverse impedance. Maximizing as a function of Qx also determines Qx (J. Klem, 1999).

4. betatron phase shift with bunch current in LEP (D. Brandt, P. Castro, K. Cornelis, A. Hofmann, G. Morpurgo, G. Sabbi, J. Wenninger, B. Zotter, et al., PAC1995) -> transverse impedance distribution

5. 2000/2001 attempt to look for steps in Df/DN no error estimate step? betatron phase shift with current in the SPS (J. Klem, G. Arduini, G. Morpurgo, EPAC2000): indication for step?

6. Optics Perturbation considering only vertical plane, impedance acts like a current-dependent quadrupole of effective gradient bunch population effective impedance beam energy note that vertical impedance is defocusing

7. impedance will introduce beta & phase beating (C. Carli); explains oscillations in 2003 analysis response matrix was derived in 1st order perturbation theory: rewritten as adding 87x237 matrix cut-off of singular values in SVD inversion is another free parameter + SVD solution is stabilized by introducing additional equations with weights l M computed from MAD optics solution obtained by matrix pseudo-inversion, e.g., SVD

8. Tests of the Algorithm 1) simulation test • varied the strength of single quadrupole QE603 in MAD, and took calculated phase change as input • correct quadrupole was identified either by determining the best single quadrupole or by SVD pseudo-inversion • since we use 1st order perturbation matrix, agreement between actual and fitted change worsens for large perturbations; for DQ=0.1 the error in the quadrupole-strength change is ~5%, for DQ=0.01, it is 0.3% 2) experimental test • procedure can only function if model is close to real optics • varied single quadrupole to give tune change DQ~0.05 • correct choice of l is important • largest change is found for next quadrupole (QD603 vs QE603), • alternatively, QE603 was identified when looking for most efficient single change; fitted strength differs by 6%

9. experimental test DK obtained by fitting measured phase change induced by change in QE603 at BPMs by SVD inversion with SVD cut-off 0.1 and three different weights l: l=0.5 l=50 l=500 corresponding fit result superimposed on the measurement: l=0.5 l=500 l=50 intermediate l yields ~reasonable result

10. Impedance Localization • data sets were taken at • 26 GeV/c on 04.09. and 30.09.2003 • 14 GeV/c on 27.10.2003 beam was kicked transversely and 1000-turn BPM readings were recorded for all BPMs intensity of single p bunch was varied in 4-6 steps from 2x1010 to 1.2x1011 chromaticity was held at lowest value compatible with beam stability helped by increase of e||

11. typical BPM raw data at high and low intensity vertical position in arbitrary units vs. turn number 26 GeV/c 14 GeV/c decoherence time and closed orbit vary with beam intensity

12. for each data set we compute average tune and rms tune spread over all BPMs data sets with large tune error (spread) are discarded in the harmonic analysis, since a large variation of the tune from BPM to BPM implies a large uncertainty in the phase

13. analysis of “good” data • for each BPM & data set we determine the phase of • oscillation with respect to the start of the line; we constrain f to lie within +/-p from MAD model then for each BPM we fit f vs. Nb to a straight line optics phase error at 0 current (?) effect of impedance

14. measured phase variation Df=(fb-f0) vs. bunch intensity and linear fit for 7 selected BPMs • variation with intensity is indeed linear • the 14 GeV/c data are less noisy

15. fit results for all BPMs fitted f0 fitted Df/DNb DN/Df decreases similarly; effect is larger for lower beam energy as expected monotonic increase due to difference between real 0-current & model tune 14 GeV/c data show beating super- imposed on gradual decline – less evident at 26 GeV/c

16. determine impedance sources at positions of 237 quadrupoles quads # 11, 24, 88, 102, 139, 164 (QD111, QDA119, QD307, QD319, QF420 and QD507) w. large impedance quads # 24, 80, 136, 140, 157, 164, 236 (QDA119, QD301, QDA417, QD421, QD501, QD507, and QD635) with large impedance • SVD minimization recipe: • singular-value cut-off = 5 • initial weight l = 1 for all quadrupoles; increased by factors of 10 for DK values of the wrong sign in 10 iterations, to make impedance defocusing

17. current-dependent phase change predicted by biased SVD fit compared with the measurements fair agreement “perfect” agreement based on impedance distribution from previous slide

18. Robustness checks (1) selection of quadrupoles could impedance located near QD301 in reality be due to rf cavities (at quadrupoles 316-317)? [G. Arduini] • re-processed only 14 GeV/c data which have higher quality • successively eliminated quadrupoles in the 300s region with large fitted impedance from subsequent fits until rf cavity location is found QD301 QF300 QD305, QD309 QE302, QF308, QD313 QD317 QF304, QECD306, QF316, QD317, QD319

19. fit quality only slightly degrades as more and more quadrupoles are added

20. fit with all quadrupoles fit without 9 quadrupoles in the 3rd arc

21. results from full fit & from fit w/o 9 quads compared with data difference between the two fits is much smaller than the remaining discrepancy to the measurement

22. (2) dependence on quadrupole weight effect on fit quality of varying initial quadrupole weight

23. impedances fitted for different initial values of l l=0.01 l=0.1 l=1 l=10

24. fits for different initial l values and the measurement

25. Summary • From multi-turn BPM data at different bunch intensities & two beam energies, the current-dependent phase advance was obtained at each BPM of the SPS ring • We determined the impedance locations and strengths giving rise to the measured phase shifts, using the theoretical phase response matrix, which we “pseudo- inverted” by an SVD algorithm with adaptive weight factors. In an iterative approach, we suppressed large values as well as focusing impedances, yielding a consistent solution at both • SPS regions 119 (near MKP kickers), ~301-307 (arc, rf?), 417-421 (near MKE kickers), and 507 (arc?) identified at both beam energies as locations with high impedance. • Data at 14 GeV/c show much cleaner signal & less noise • The exact location may be missed as illustrated • Accurate optics model & good data quality essential