Localization of the SPS Transverse Impedance

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Localization of the SPS Transverse Impedance

• Gianluigi Arduini, Christian Carli,
• Frank Zimmermann

Motivation Optics Perturbation Test of
Algorithm Impedance Localization Robustness Checks
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Motivation
Identify, localize and quantify sources of
transverse impedance Can we see the effect of
five re-installed ferrite extraction
kickers? 3050 impedance increase was
measured in 2003 by H. Burkhardt predicted by
L. Vos Method based on current-dependent b phase
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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, Ngtgt1, 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).
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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)
-gt transverse impedance distribution
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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?
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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
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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
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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 DQ0.1 the error in the quadrupole-strength
• change is 5, for DQ0.01, 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
  DQ0.05
• 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

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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
l0.5
l50
l500
corresponding fit result superimposed on the
measurement
l0.5
l500
l50
intermediate l yields reasonable result
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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

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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
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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
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• 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
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measured phase variation Df(fb-f0) vs. bunch
intensity and linear fit for 7 selected BPMs

1. variation with intensity is indeed linear
2. the 14 GeV/c data are less noisy

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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
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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

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current-dependent phase change predicted by
biased SVD fit compared with the measurements
fair agreement
perfect agreement
based on impedance distribution from previous
slide
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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
QF304, QECD306, QF316, QD317, QD319
QD317
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fit quality only slightly degrades as more and
more quadrupoles are added
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fit with all quadrupoles
fit without 9 quadrupoles in the 3rd arc
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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
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(2) dependence on quadrupole weight
effect on fit quality of varying initial
quadrupole weight
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impedances fitted for different initial values of
l
l0.01
l0.1
l1
l10
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fits for different initial l values and the
measurement
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• 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