J. Kissel, R. Quitzo-James

In all suspension TOP to TOP transfer functions that we've taken to high frequency (i.e. > 10 Hz), we have found a confusing turn up to a slope of ~f^(1), where we expect the slope to continue to drop off as f^(-2). At the advise of one P. Fritschel, we've taken M1 to M1, OSEM basis, transfer functions of H2SUSFMY both locked and unlocked to try and explain, if not at least identify, the source of this turn up. As Peter suspected, the locked spectra show the same turn up at high frequency. Aside from the mechanical resonances left in the transfer function, this implies that there is some fundamental coupling between the OSEM Coil drive and PD sensor. It has been suggested that this coupling may arise from coil current drive mixing the PD sensing current, either at the OSEM head, or some where along the electronics chain, as these signals are on the same cable.

Note, this is not necessarily of great concern, given that we use these TOP BOSEMs for local damping between ~0.1 and 10 Hz, and perhaps for low frequency ISC offloading; all control authority will be rolled off aggressively by 10 Hz. These transfer functions are taken with basically the maximum amount of current going through the coils (to get the best SNR for the measurement), so with the fast roll off of control authority above 10 Hz, I expect this noise source will be negligible. 

Attached are two sets of plots: 
2011-11-15_H2SUSFMY_hfturnup_M1TFs.pdf -- The collection of M1 to M1, OSEM basis, H2SUSFMY transfer functions comparing the measured, suspended transfer function (GREEN) against a model of that transfer function (CYAN) composed of the production, suspended model transfer function (RED) plus the locked transfer function (BLUE). The model agrees very well with measurement, aside from some scaling factors in the F1, F2, and F3 sensors. (Details below -- **).

2011-11-15_H2SUSFMY_hfturnup_currentcoupling.pdf -- IF the mechanism is sensing current mixing with drive current, this plot shows the locked transfer functions for each degree of freedom calibrated into PD Sensing Current per Coil Drive Current. The calibration assumes the same [m/N] calibration that is always used to scale measurements (a factor of 60 for aLIGO production electronics), and then converted to current assuming a force coefficient of a 10x10mm magnet with a BOSEM coil -- 1.694 [N/A] (see T1000164), and displacement sensitivity of 62.5e-2 [A/m] (see T1100479, or T0900496).


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Details **
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The production matlab models of the aLIGO suspensions are in the EULER basis, so it took a little bit of math to get the model suspended transfer function into the OSEM basis. The math is as follows:

Assume the model, euler basis, mechanical transfer function is a 6 x 6 matrix, Pe. Hence, the 6x1, euler basis, response vector VRe, is related to the 6x1, euler basis, drive vector VDe, by
     VRe = Pe VDe                       (1)

However, we can decompose the euler basis drive and response vectors into their respective OSEM basis, using the known OSEM2EUL, oMe, and EUL2OSEM, eMo matrices:
     VRe = oMe VRo                 (2)
     VDo = eMo VDe
     => VDe = (eMo)-1 VDo         (3)

which means, that we can solve Eq. (1) for the OSEM basis transfer function,
     VRe = Pe VDe
     oMe VRo = Pe (eMo)-1 VDo
     VRo = (oMe)-1 Pe (eMo)-1 VDo  (4)

Now, this would be easy to do in matlab, IF the Pe matrix was a 2D matrix, like the OSEM2 EUL and EUL2OSEM matrices, but because Pe is a 3D matrix, response x drive x frequency, I had to do the matrix multiplication "by hand," i.e.

[ Pe (eMo)-1 ](i,j,f) = Σn  Pe(i,n,f) (eMo)-1(n,j)

[(oMe)-1 Pe (eMo)-1](i,j,f) = Σm (oMe)-1(i,m) [ Pe (eMo)-1 ](m,j,f)


This conversion of the model, i.e. exercise of brute-force math, seems have done awesomely at predicting the < 10 Hz frequency response and DC scaling of the transfer function for LF, RT, and SD, especially after including the normalization factors that are present for each OSEMINF gain to account for the OSEMs sensitivity variations. However, I'm still missing a factor of ~1.5 for the three FACE sensors that I can't figure out from where it comes. I'm not gunna bang my head against the wall about it -- the point of the measurements has been made!

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Data / Analysis
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Raw Data:
${SusSVN}/sus/trunk/BSFM/H2/FMY/SAGM1/Data/111115_H2SUSFMY_WhiteNoise_??_0p1to900Hz_bsfm*locked.xml
Exported Data:
${SusSVN}/sus/trunk/BSFM/H2/FMY/SAGM1/Data/111115_H2SUSFMY_WhiteNoise_??_0p1to900Hz_bsfm*locked_tf.txt

Analysis:
${SusSVN}/sus/trunk/BSFM/H2/FMY/SAGM1/Scripts/analyze_111115_H2SUSFMY_M1_lockedvsunlocked.m