1. Quadrupole error localization using Response Fits
Trajectory (or orbit) response measurements have become a widely used method to verify machine models to various levels from very rough BPM and corrector checks to detailed optics fits.
Basic principle :
Pick a number of correctors.
For each corrector :
Increment the corrector kick by some amount and record the trajectory/orbit.
Repeat with a kick of opposite sign.
Compute the difference of the two measurements (removes static offsets).
Fit the measured data to the machine model. Fit variables :
BPM calibration factors.
Corrector calibration factors.
Selected model parameters, like quadrupole strengths……
PRO of this method : simple & non-destructive !
LHC-OP / J. Wenninger

2. SPS Studies
Since I considered optics measurements and corrections to be crucial for the LHC, I started tests based on orbit response in the SPS in 2002.
For the test I have (slightly) adapted a program called LOCO, written for orbit response analysis by J. Safranek (then BNL). The program principle and results from the SPS are documented in CERN-AB
LOCO is coupled to MAD(X) and every MAD variable can be used as fit parameter.
The program was used for the first time ‘for real’ during the TI8 tests in The steering application provided simple functionality for response measurements. Online fit results were available ~ 20 minutes after the measurements.
LHC-OP / J. Wenninger

3. TI8 – the ‘surprise’ Problem within the first 100-200 m of the line…
First measurement :
Fit parameters : BPM & COD calibrations, main QD/F quadrupole string strength.
Individually powered matching quads not used in fit.
First horizontal corrector response
does not fit model at all !!
Horizontal correctors further down
do not exhibit significant problems!!
Histogram : data
* + Line : model fit
Problem within the first m of the line…
LHC-OP / J. Wenninger

4. Fit & Correct Correct strength & re-measure
Fit improvement :
Fitting one additional quad at a time, the fit gives a consistent/reasonable result only for QTLF4004 and predicts a strength error of -20%.
The limited number of BPM samplings prevents a to fit all Q strengths at the same time.
Correct strength
& re-measure
Second measurement :
QTLF4004 strength increased by 20%.
Model fits now …
It turned out there was another quadrupole with a 20% error that was not detected because it was too close to the beginning of the line  ~ no effect of the error on the MEASURED trajectories.
LHC-OP / J. Wenninger

5. TI8 phase advance
In addition to the individual powering problems, the data also revealed a 1% strength error of the vertically focusing arc quadrupole string.
V plane :
1% error on phase advance (larger than nominal).
Clearly visible on the plot : the V phase advance / cell is not 90 deg.
H plane :
phase advance is right on !
TI8 arc cells
TI8 arc cells
LHC-OP / J. Wenninger

6. Lessons from TI8
Response measurements based on trajectories were successful in localizing ‘coarse’ Q errors (here 20%). The actual detection threshold is ~ few % - depends on the available time and effort.
Embedding & (semi-)automating the response measurements within the steering program was very important  ‘operational’ measurement on day 1.
Proper interfacing and online analysis of the data (based on tools developed for the SPS test) was crucial to obtain fast feedback – in fact within < 1 hour.
For the LHC :
It is possible to localize and quantify coarse Q errors (~ few %) without having to establish the closed orbit.
It is possible to measure/check the machine tune to better than ~ 0.05/0.1 based on the trajectory response fit.
Note : the BPM density in the LHC is 4 times higher than in TI8 !
LHC-OP / J. Wenninger

7. Example 1 Very clear signal !! Isolated quadrupole gradient error :
In practice one can take advantage of many such trajectories, recorded for different correctors located up- and downstream of the error source !
Isolated quadrupole gradient error :
5% or 10% gradient error on MQ.30r2.B1.
Trajectory measurement with ± 20 mrad.
The expected BPM resolution (noise) for pilot is ~ 200 mm.
Very clear signal !!
LHC-OP / J. Wenninger

8. Example 2 Very clear signal on phase !!
Arc/sector quadrupole gradient error :
1% gradient error on circuit associated to KQF.A23.
Trajectory measurement with ± 20 mrad.
The expected BPM resolution (noise) for pilot is ~ 200 mm.
Very clear signal on phase !!
LHC-OP / J. Wenninger

9. Outlook
Response measurements based on trajectories provide a simple and powerful tool to localize ‘coarse’ Q errors. The detection threshold is
~ 5% on isolated errors (distance between errors ~ 2 betatron wavelengths ?)
<<1% on the strength of the arc circuits.
The only elements that may be difficult to pick are the first quadrupole(s) behind the injection point  sampling & lever arm…
It is possible to provide more accurate figures and study lot’s of scenarios, but I’m not convinced that this is useful. Really IMPORTANT points :
The LHC steering application (which will be identical for LHC, SPS, SPS TLs and LEIR) provides now fully automated response measurement procedures (to be tested at LEIR this summer).
The online analysis had to rely on the presence of an experienced person because a fully automated fit with 100’s of quadrupole gradients as free fit parameters will not work … Ad-hoc guidance to the fit is required. Still, results can be available in short time ~ hour(s).
LHC-OP / J. Wenninger