Vertical dispersion fitting

[murilobalves]

Script to fit the vertical dispersion with chromatic skew quadrupoles.

This script was used to obtain the strengths of normal and skew quadrupoles of "pymodels.fitted_models". According to its docstring:

"This model was built via fitting of the measured vertical dispersion function with the chromatic skew quadrupoles of the machine. The measurement used for fitting was the matrix measurement of folder: shared/screens-iocs/data_by_day/2022-05-24-SI_LOCO/ with name: respmat_endofmay22_bpms_03sector_switching_issue.pickle. The transverse tunes were also adjusted to match the measured ones."

There are 60 chromatic QS. Thus, the "etay/KsL" response matrix has 60 singular values. A scan of the number of singular values used in the fitting was performed, resulting in:

Note that after 41 singular values, the fitting residue is stable while the KsL strengths and beta-beating keep increasing. Therefore it does not make much sense to include more singular values than 41 to improve the fitting. Note that there is a peak in the emittance ratio calculated with the beam envelope formalism at this singular value and also in the minimum tune separation. I still do not understand the reason for this. The emittance ratio calculated with the radiation integrals (which is not adequate for coupled lattices and it is mainly due to the vertical emittance created by the vertical dispersion, not from betatron coupling) evolves smoothly as the singular value number increases. Also, for the singular values matrix, there is a step at SV 40 as can be seen in the plot below:

Note that the minimum tune separation with 41 SVs is about 1%, the same value that we measured. This model reproduces the measured vertical dispersion and the measured global betatron coupling.

This was the process to define 41 as the number of singular values to be used in the fitting. Initially, fitting the measured vertical dispersion and the global coupling was also the motivation to use 35 SVs, but at the time I did not notice the strange behavior around the SV 35.

The fitted dispersion as compared to the measured one is shown below:

The non-negligible difference between the QS strengths obtained with 35 or 41 is shown bellow:

The QN and QS strengths can be printed to then be plugged in "pymodels.fitted_models" with the function print_strengths_fitted_model. Related to the PR: https://github.com/lnls-fac/pymodels/pull/85

Maybe we should update the QS strengths in the current fitted mode, what do you think @fernandohds564 ?

For the question raised by @xresende related to comparison between ORM of this model and measured. I calculated the ORM of the fitted model with 41 singular values and compared with the measured ORM. The correlation between off-diagonal blocks of measured and fitted matrices is plotted bellow for CHs (blue) and CVs (orange) signatures.

Taking a look on CH and CVs with maximum correlation:

Now correctors signatures with minimum correlation:

Remember that in this fit only chromatic skew quadrupoles were used. In principle, one can use the achromatic QS (the remaining 40 magnets) to match the coupling blocks of ORM and/or control the global betatron coupling to the measured value.

[xresende]

thanks, @murilobalves . If the largest 35 singular values area included there is clearly a spike in emittance ratio in the plot. Was it just a coincidence that there is no gain in including correction modes above this particular singular value with this spike? if it is a coincidence, did it not bother you to pick 35 modes, as for they lead to a higher emittance ratio compared to other cases ? why not 34 or 36 correction modes ?

[murilobalves]

thanks, @murilobalves . If the largest 35 singular values area included there is clearly a spike in emittance ratio in the plot. Was it just a coincidence that there is no gain in including correction modes above this particular singular value with this spike? if it is a coincidence, did it not bother you to pick 35 modes, as for they lead to a higher emittance ratio compared to other cases ? why not 34 or 36 correction modes ?

Hi @xresende, @fernandohds564 also made this observation. I must confess that in the first time I did this vertical dispersion fitting, I only observed the residue evolution with the singular values, and 35 appeared to be a good number. During the code organization for this PR, I made this SV-scan also plotting other properties as the emittance ratios and it was a weird coincidence that right at 35 singular values there is a step in the emittance ratio. I did not understand why is that. I would like to investigate this issue but I really have to focus on the harmonic cavity topic now.

You guys are right, 34 or 36 modes seem to be less problematic. Now I have chosen 41, since there is a slight drop in the residue, the emittances are "well behaved" around this SV, the QS strengths std do not increase much as compared to the SV 35 (although the strengths for some magnets change considerably, as can be seen in the plot above) and the minimum tune separation is about 0.01 (1%), which is close to the measured value. At SV 40 there is a step in the SV distribution, the QS strengths increase without considerably reducing the residue so I decided not going beyond SV 41.