TI8 analysis / J. Wenninger Coupling analysis Analysis is based on the high statistics measurements in the last night of the first MD : For each plane, kicks of +- 40 mrad were applied to 2 correctors 90˚ out of phase in the first part of TI8 (~ 1 hour at each setting). The analysis is based on autosave data from steering program. Each data sample is averaged and then re-converted into new steering program file using UNIX shell and gawk scripts. The difference orbits are analysed with the steering program to get a first idea of the possible source of coupling (if any). The data is then converted to LOCO input and fitted with LOCO in a version that handles the coupled response. 02.12.2004 TI8 analysis / J. Wenninger

Coupling – steering test / V1 Coupling from 80 mrad kick @ MCIAV.813 Correction with a single kick in H plane … MICADO picks QIF.824 : 1.4 mrad Before Difference After 02.12.2004 TI8 analysis / J. Wenninger

Coupling – steering test / V2 Coupling from 80 mrad kick @ MCIAV.815 Correction with a single kick in H plane … MICADO picks again QIF.824 : 1.4 mrad Before Difference After 02.12.2004 TI8 analysis / J. Wenninger

Coupling – steering test / H1 Coupling from 80 mrad kick @ MCIAH.816 Correction with a single kick in V plane … MICADO picks now QID.821 : -1.1 mrad Before Difference After 02.12.2004 TI8 analysis / J. Wenninger

Coupling – steering test / H2 Coupling from 80 mrad kick @ MCIAH.818 Correction with a single kick in V plane … MICADO picks again QIF.824 : -3.3 mrad Before Difference After 02.12.2004 TI8 analysis / J. Wenninger

‘Graphical’ analysis : V  H Have a look at the trajectories associated to the V correctors… From a naïve point of view the result from the steering prgram analysis can be explained : same kick because the trajectory phase & amplitude is the same ! A kick at 821 should appear for one corrector – does not fit with simple analysis of the trajectories… 813 814 815 816 817 818 819 820 821 822 823 824 825 826 02.12.2004 TI8 analysis / J. Wenninger

Graphical analysis : H  V Have a look at the trajectories associated to the H correctors… A kick at 821 should appear for both correctors – does not fit with simple analysis of the trajectories… Also it does not fit with the V  H ! From 824 there can only be a kick for corr. 818 – trajectory goes through 0 for 816. This is consistent with the steering program results… 813 814 815 816 817 818 819 820 821 822 823 824 825 826  the situation may be more complex : more than 1 or 2 sources ! 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Coupling fit The fit is done with the LOCO – full analysis of the response including cross-plane response. The fit includes BPM and corrector gains. To handle the coupling, short (1 cm) skew quads are installed in the MADX sequence file in front of suspicious quadrupoles. A full fit is performed, including skews. Very small skews are removed and the fit is iterated… The skew strength Ks is converted into a quadrupole tilt f using : The fit is redone with the tilts and skew strengths until the Ks  0. Two iterations are sufficient. 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Fit results Coupling : QIF.824 is ‘outstanding’ with a tilt of 5 mrad corresponding to a2/b2 = 0.01. The values for the tilts can be plugged directly into the TI8 sequence. Coupling is at the level of 2% (defined as amplitude ratio). Other fit results : The spread of the BPM calibrations is 1.5%. This better than before – better statistics… The average BPM gain is consisted with previous fits : G = 0.898 H plane G = 0.880 V plane Quadrupole f (mrad) QID.819 1.2 QIF.822 2.6 QIF.824 5.0 QID.855 -1.2 QID.859 The residual fit-measurement is ~ 250 mm for the in-plane data. This is a factor ~ 10 higher than the statistical error on the data of ~ 30 mm : Local errors ? BPM effects ? non-linearities… 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Fit results - plots Fit residual are slightly larger than errors (~ 30 mm) HV 816 HV 818 VH 813 VH 815 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Stability analysis / 1 The BPMs are high correlated in a line/ring, and one can profit from this property through a Singular Value Decomposition (SVD) of the BPM data. This decomposition is a powerful way of extracting modes out of the data. The BPM data is put in the form of a matrix B. The reading of BPM i for measurement j represented by bij . Each trajectory corresponds to a line of B : N = no. of BPMs M = no. measurements assume M > N ! time space 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Stability analysis / 2 The matrix B is decomposed by SVD into : where : U is a matrix of normalized and orthogonal time patterns. V is a matrix of normalized and orthogonal space patterns. W is a diagonal matrix with the eigenvalues of the space patterns. Each ‘mode’ / spacial pattern has an associated weight / eigenvalue that characterizes its amplitude and an associated time pattern. This method is often referred to as MIA (Model Independent Analysis). It is an interesting and (sometimes) powerfull way to extract characteristic ‘behaviours’ from the data. 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Stability data The analysis is based on the stability measurement from the first night (SatSun) of the first MD (pilots). For each plane, the trajectories are combined into a file suitable to be read in by the SPS multi-turn acquisition program where the MIA method is implemented. SVD is applied and the eigenvectors are scanned for ‘meaningful’ information. Except for the ‘static’ trajectory (one mode), there is nothing of interest (just noise and BPM junk) except for one mode in the horizontal plane. The ‘meager’ outcome is probably due to the fact that the noise of 200-300 mm and bad BPM readings tend to dominate. 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Eigenvalues When the analysis is applied to TI8 data (H and V) there is only one interesting signal : the 5th horizontal eigenvalue shows a clean betatron oscillation starting at the beginning of the line… time structure spacial structure (BPMs) 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger MSE candidate The spacial vector is fed back into steering program for analysis : Search for most efficient corrector with MICADO. MICADO picks MSE (or MDMH.4001 that is ~ at the same phase) as possible source of the perturbation ! Before corr. Difference After corr. with MSE 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Cross-check Pick 2 extreme trajectories (large MSE contribution according to MIA) : Compute trajectory difference to get largest possible signal. Search for most efficient corrector with MICADO. MICADO picks MSE – OK ! H traj difference Before corr. Difference After corr. RMS is ~ consistent with noise… 02.12.2004 TI8 analysis / J. Wenninger

MSE contribution to (in)stability Largest kicks : ± 4.5 mrad  ± 3.8×10-4 relative error. RMS kick : 1.4 mrad  ± 1.2×10-4 relative error. The oscillation amplitude corresonding to the RMS kick is ~ 100 mm which corresponds to ~ s/8 for a nominal emittance. The expected RMS stability at the location of the TCS (Dm = 158˚) is ~ 40 mm. BUT : I cannot see a correlation with the logged MSE currents !! ?? 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Long term average Trajectory difference (average) after 5 hours : RMS ~ 50 mm – practicaly consistent with the noise. No significant coherent oscillation visible. 02.12.2004 TI8 analysis / J. Wenninger

TI8 analysis / J. Wenninger Further work… One could search for coherent oscillations by fitting betatron oscillations to the BPM data : A fit to the entire line should find the signal observed with MIA and find/set a limit of the signal that is 90˚ out of phase  sensitive to ‘incoming’ signals. A fit to the end of the line (last 4-5 BPMs ?) may give and indication on contributions coming from the line itself (+ incoming). Could resurect / adapt my LEP fit program… 02.12.2004 TI8 analysis / J. Wenninger