J. Kissel

I got sick of trying to debug the high-frequency affects UIM driver (LHO aLOG 21142), so I've moved on to the more important ESD Driver for H1 SUS ETMY. Instead of fitting the results of the remote measurements of the driver with the monitor circuit (LHO aLOG 20846), I've gone back to the measurements originally taken just after the LVLN driver was installed (LHO aLOG 18579), and fit them more precisely than Evan used to inform the current compensation filter (LHO aLOG 18596). Below are the details. 

The messages: 
- Each of the quadrants are *differently* frequency dependent at the ~5% level ~ 5 [deg] level; since we desire higher precision than this, they cannot be modeled with the same set of poles and zeros for each quadrant.
- In order to model this correctly, will have to 
    - break out the quadrants in the actuation chain model, 
    - de-compensate for the installed compensation filter, 
    - use the fit poles and zeros from each quadrant, and 
    - then sum the quadrants together (and divide by four as necessary)

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The answer:
Here're the precise poles and zeros for each quadrant:

       Diff Receiver      Summing Node      LP1            LP2      Overall DC Gain                         Summary
      (one pole, [Hz])    (z:p) [Hz]      (z:p) [Hz]   (z:p) [Hz]       [V/V]                      z [Hz]: p [Hz]: k [V/V]
UL         24e3            3200:165       48.8:2.160   48.8:2.160      1.8868         (48.8 48.8 3200) : (2.16  2.16  165 24e3) : 1.8868 
LL         23e3            3050:160       47.8:2.105   47.8:2.105      1.8868         (47.8 47.8 3050) : (2.105 2.105 160 23e3) : 1.8868 
UR         24e3            3050:157       48.8:2.160   48.8:2.160      1.8868         (48.8 48.8 3050) : (2.16  2.16  157 24e3) : 1.8868 
LR         24e3            3125:160       47.8:2.130   47.8:2.130      1.8868         (47.8 47.8 3125) : (2.13  2.13  160 23e3) : 1.8868 

These poles and zeros result in a frequency dependent residal, which represents a systematic error in the fit. This systematic error is smaller that 1% and 1 [deg] over the entire frequency band for all quadrants (save for where we know the measurement is incoherent noise at ~30 and ~100 [Hz]). Rather than bothering to carry around the precise systematic error and figuring out how to propagate it correctly through the uncertainty analysis, we shall just represent this as this a statistical, 1-sigma, uncertainty of 1% and 1 [deg]. We used to call this being "conservative," but really it's a "perfect is the enemy of good enough" decision.

Open the attachment for plots demonstrating the goodness of fit (pg 1) and comparison of each quadrant against the fit and other models (pg 2-5), and a comparison of the models when the LP filter is OFF (pg 6).

The fitting process:
There are four models that I show: "Spice", "Simple Circuit", "(Compensation Filter)^-1", and PZ fit. 
"Spice:" Rich had graciously provided us with a spice model of the entire driver with both the low-pass off and engaged: 
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/PreER7/H1/Scripts/Spice/
LVLN_Driver_v8_PZ_Off.txt
LVLN_Driver_v8_PZ_On.txt
"Simple Circuit:" For my edification, to determine what parts of the box were determining which parts of the frequency response, I also did some simple analytical analysis of the driver accounting for all of the parts of the driver that contribute to the frequency response. 
"(Compensation Filter)^-1" These are the poles and zeroes currently used in the ESDOUTF bank, under the "antiAcq" and "antiLP" filters, a "good enough" compensation developed by Evan back when the driver was first installed.

I had started off hoping, like we always hope, that one can compensate each quadrant of the ESD driver with the same set of poles and zeros, based on those pole and zero frequency derived from some simple RC analysis of circuit. Turns out this is not the case. The frequency-depednent residual between the Spice, Simple Circuit, and (Compensation Filter)^-1 models and each quadrants' measurement for the following models is upwards of 10% and 5 [deg], different for each quadrant. 

A such, I created the PZfit model, where each quadrant's residual is tuned to as small as possible using the same number of poles and zeros as in the simple circuit model. That results in the poles and zeros quoted above.

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Analysis script lives here:
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/ER8/H1/Scripts/Electronics/model_LVLN_driver_20150902.m