J. Kissel

While it has reported that the electrostatic drive is under what is expected by factors of 2, 4 and 8, repeated attempts at measuring and understanding the actuation strength of the ESD have at least garnered quite a bit of understanding on my part. The message: A factor of 8 is wrong -- a result of me misinterpreting how the actuation coefficient was calculated. A factor of 2 and 4 are still plausible, because all measurements have been made using the green ALS system, whose optical gain can vary significantly over the course of the measurement from alignment fluctuations. The latest comparisons of all H1 ETMX and H1 ETMY swept sine transfer functions report a quantitatively rigorous factor discrepancy (with uncertainty) below the expected actuation strength by factors that are indeed between 2 and 4. Many details below.

Details
-------
All published efforts to-date that attempt to quantify the actuation strength the ESDs:
aLOG             Meas Date        Optic       DOF             Drive                Bias [ct_DAC]/[V_ESD]         Response Channel [units]              Factor below 4.2e-10 [N/V^2] 
----             ---------        -----       ---       ---------------            ---------------------         ------------------------              ----------------------------
This aLOG        2014-04-28       ETMX        L         1.0-20 [Hz] Swept Sine      +125e3 / +381                ALS-C_COMM_PLL_CTRL_OUT [Hz]               2.26 +/- 0.012
                 2014-04-30       ETMX        L         0.1-10 [Hz] Swept Sine      +125e3 / +381                ALS-X_REFL_CTRL_OUT     [kHz]              3.66 +/- 0.007
                 2014-05-29       ETMY        L         0.1-10 [Hz] Swept Sine      -125e3 / -381                ALS-Y_REFL_CTRL_OUT     [um]               2.97 +/- 0.005
LHO aLOG 11929   2014-05-16       ETMX        L         11 [Hz] Sine Wave           +125e3 / +381                ALS-X_REFL_CTRL_OUT     [kHz]              8    INCORRECT ANALYSIS
LHO aLOG 12109   2014-05-16       ETMX                    Same measurement, reprocessed with better understanding                                           2    CORRECTED ANALYSIS
LHO aLOG 11676   2014-04-30       ETMX        L           Same measurement in this aLOG, reprocessed with uncertainty analysis                              4    (consistent with above)
LHO aLOG 11581   2014-04-28       ETMX        P,Y       1.0-20 [Hz] Swept Sine      +125e3 / +381                        EX Oplev        [urad]             2ish (consistent with above)

Possibly contains actuation strength, but quadrant information is too cross-coupled to obtain a pure P/Y number in terms of force:
LHO aLOG 12026   2014-05-22       ETMX   UL, LL, UR, LR  3 [Hz] Sine Wave   (+124e3 0 -124e3) / (+377 0 - 377)           EX Oplev           [urad]             Difficult to say
LHO aLOG 12027   2014-05-22       ETMY   UL, LL, UR, LR  3 [Hz] Sine Wave   (+124e3 0 -124e3) / (+377 0 - 377)           EY Oplev           [urad]             Difficult to say

Unpublished attempts:
L1/ETMX/SAGL3/Data/2014-05-30_0800_L1SUSETMX_L3_L2LPY_SweptSine.xml -- coherence deemed too poor because ALS noise too high.
L1/ETMY/SAGL3/Data/2014-05-30_0800_L1SUSETMY_L3_L2LPY_SweptSine.xml -- coherence deemed too poor because ALS noise too high.
H1/ETMX/SAGL3/Data/2014-05-15_H1SUSETMX_L3_ESD_P_2Hz.xml -- attempted to process, but got a factor 8 to 10, and gave up deciding torque was too hard
H1/ETMX/SAGL3/Data/2014-05-15_H1SUSETMX_L3_ESD_P_3Hz.xml --     ""
H1/ETMX/SAGL3/Data/2014-05-15_H1SUSETMX_L3_ESD_Y_2Hz.xml --     ""
H1/ETMX/SAGL3/Data/2014-05-16_H1SUSETMX_L3_ChargeTest_LL.xml -- attempted to take quadrant-by-quadrant data, but did not process before 2014-05-22 measurements, then gave up
H1/ETMX/SAGL3/Data/2014-05-16_H1SUSETMX_L3_ChargeTest_LR.xml -- ""
H1/ETMX/SAGL3/Data/2014-05-16_H1SUSETMX_L3_ChargeTest_UL.xml -- ""
H1/ETMX/SAGL3/Data/2014-05-16_H1SUSETMX_L3_ChargeTest_UR.xml -- ""

On the linear dependence of longitudinal Force, F on Control Voltage, V_CTRL:
We know 
F = a (V_CTRL - V_BIAS)^2
  = a (V_CTRL^2 - 2*V_CTRL*V_BIAS + V_BIAS^2)
for a given bias voltage V_BIAS.

For all of the above analysis, as a given frequency, we apply a sinusoidal control voltage at some frequency, w and amplitude V_0. As such, because of the quadratic nature of the actuator,
F = a (V_0^2*sin^2(wt) - 2*V_0*V_BIAS sin(wt) + V_BIAS^2)
           [sin^2(wt) = 1/2 - 1/2*cos(2wt) + O(4w)]
  = a (V_0^2*[1/2 - 1/2*cos(2wt)] - 2*V_0*V_BIAS sin(wt) + V_BIAS^2)
  = [(a/2)*V_0^2 + a*V_BIAS^2] - [2*V_0*V_BIAS*sin(wt)] + [(a/2)*V_0^2*cos(2wt)]
F = [        DC term         ]   [    linear term     ]   [    bilinear term   ]

Now, for a transfer function (i.e. a linear cross-correlation function), only the linear term will show up, such that
F_lin = - 2*V_0*V_BIAS*sin(wt),
or in the frequency domain, simply,
F_lin = - 2*V_0*V_BIAS.
Thus, at a single frequency, the linear transfer function coefficient magnitude between force and control voltage is

F_lin
----- = 2*a*V_BIAS
 V_0


And this is exactly the function one must use to calibrate the drive component of the electrostatic drive chain, assuming the above equation is defined where V_CTRL and V_BIAS are in units of [V] on the electrodes in-vacuum. Tracing these back to the digital drive points for the bias voltage, ct_BIAS, and for control voltage ct_LOCK, where
V_0 = G_ESD * G_DAC * EUL2ESD * ct_LOCK
V_BIAS = G_ESD * G_DAC * ct_BIAS 
where G_ESD = 40 [V/V] is the gain of the ESD driver, G_DAC = 20/2^18 [Vpp/ct] is the DAC gain, and EUL2ESD = 0.25 is the coefficient in the Euler to ESD basis transformation output matrix. This leaves the calibration to be

 F_lin
-------  = 2 * a * G_ESD^2 * G_DAC^2 * EUL2OSEM * ct_BIAS = 2.44e-10 [N/ct]
ct_LOCK

assuming a bias equivalent voltage of 125e3 [ct_BIAS], as all of the above mentioned values are quoted. As a reminder, the control voltage used in this calculation is the control voltage on one quadrant, even though you're drive all four quadrants. As a is defined above, the longitudinal force is generated by the potential difference between the ring of control voltage and the ring of bias voltage. That ring of control voltage is held at the same voltage for all four quadrants, and hence one quadrant is representative of the ring (see LHO aLOG 12109 for further description).

Calibration of each measurement's response channel newly described in this aLOG
Since three different people took swept sine TFs, with sensors each having different calibrations that were changing over time, it took a bit of work to calibrate all the transfer functions into [m].
- The calibration for ALS-C_COMM_PLL_CTRL_OUT, though it claimed to be in (green) [Hz] had not had the VCO response removed (as confirmed by a trend of the H1:ALS-C_COMM_PLL_CTRL_SWSTAT, which showed that the compensation filter in FM3, "antiVCO" was OFF during the measurement), so my calibration from [Hz/ct] to [m/ct] was
[m/ct] = L * (lambda_g / c) * zpk(-2*pi*40,-2*pi*1.6,1.6/40) * (H1:ALS-C_COMM_PLL_CTRL_OUT_DQ / H1:SUS-ETMX_L3_LOCK_L_EXC)
- The calibration for ALS-X_REFL_CTRL_OUT was similar, but this had the VCO response removed, so the only difference is the order of magnitude,
[m/ct] = L * (lambda_g / c) * (H1:ALS-X_REFL_CTRL_OUT_DQ / H1:SUS-ETMX_L3_LOCK_L_EXC)
- Finally, ALS-Y_REFL_CTRL_OUT had been calibrated graciously into displacement units, so one merely had to adjust the order of magnitude,
[m/ct] = (H1:ALS-X_REFL_CTRL_OUT_DQ / H1:SUS-ETMX_L3_LOCK_L_EXC)

Uncertainty Estimation
I've folded in the coherence to assess the uncertainty at frequency point for the magnitude and phase of the transfer function as described in LHO aLOG 12109,
meas.unc.radians = sqrt( (1 - meas.coh ./ (2*nAvgs*meas.coh );
meas.unc.m_per_N = abs(meas.tf.m_per_N) * meas.unc.radians;

Once these are obtain, I find the residual between the model and measurement, propagating the uncertainty assuming the model has no uncertainty,
residual.tf  = model.tf / meas.tf;
residual.unc = abs(residual.tf) * meas.unc.radians;

And then compute the weighted mean of each frequency point to arrive at the factor under the expected force coefficient value of 4.2e-10 [N/V^2],
residual.weightedmean = sum( abs(residual.tf) ./ residual.unc.^2 ) / sum( 1 / residual.unc.^2 )
with uncertainty
residual.weightedmean = sqrt( 1 ./ sum( 1 / residual.unc.^2 )
Now, one can justifiably argue that the reducing the obviously frequency-dependent residual down two one number, assuming that each frequency point is an independent measure of the actuation coefficient with merely statistical variations about some true mean value is not strictly correct. I agree -- there are still plenty of systematics at play that cause even a single measurement sweep to vary by as much as a factor of 2 over the frequency span of the measurement. There are several systematic errors that have *not* been accounted for in the model:
(1) The undamped dynamical model is not strictly correct at the first pitch mode at ~0.5 [Hz]. This means, that -- though the current damping filters are used in the model -- the resulting closed-loop model of the damped QUAD is not perfect. However, this should only be a discrepancy right around the resonances, and certainly not a source for an overall scale factor
(2) The optical gain of the interferometric sensors varies as much as a factor two during the 2014-04-30 H1 EX measurement (see attachment 2014-04-30_H1ALS_X_NormalizedTransmission.png). The sweep was performed from high to low frequency, and is likely the source of the drops in coherence / increase in uncertainty, especially at low frequency. This also may be the source of some apparent frequency dependence.

Welp. It's 2am. I'm impressed you read this far. Go get some coffee, you deserve it. G'morning!