Summary

The PMC servo was analysed. OLTF was measured and modeled by ZPK (Attachment 1). The error and actuator signals were calibrated in m/rtHz (Attachment 2)

Measurement methods

OLTF:

- The PMC servo board does not have dedicated summing/monitor points for the OLTF measurement. Moreover the PZT HV output voltage is monitored with 1/49.6 attenuation.
  Therefore we need a bit of consideration.

- The noise injection can be done at EXT DC.
- Quantity (A): Transfer function between HV OUT MON and MIX OUT MON with the injection.
  We can measure the transfer function between the HV OUT (virtual) and the MIX OUT. (HV OUT->MIX OUT). In reality, HV OUT is attenuated by factor of 49.6.
  i.e. A = (HV_OUT->MIX_OUT)*49.6
- Quantity (B): Transfer function between HV OUT MON and MIX OUT MON without the injection.
  This is related to the transfer function between the MIX OUT and HV OUT. In reality, HV OUT is attenuated. 
  i.e. B = 1/((MIX_OUT->HV_OUT)/49.6)

- What we want to know is HV_OUT->MIX_OUT->HV_OUT. i.e. A/B = (HV_OUT->MIX_OUT*49.6)*((MIX_OUT->HV_OUT)/49.6) = HV_OUT->MIX_OUT->HV_OUT

PSD:

- The MIX OUT and HV OUT spectra have been measured. The MIX OUT was calibrated with the calibration factor in the previous entry. This is the inloop stability estimation.
  From the calibrated MIX OUT and HV OUT, the free running stability of the cavity was estimated, by mutiplying with |1-OLTF| and |1-1/(1-OLTF)|, respectively, in order to recover
  the free running motion.

OLTF Modeling

Here is the model function for the open loop TF. The first line comes from the circuit diagram. The overall factor was determined by eye-fit.
The second and third lines are to reproduce the peak/notch feature at 12kHz. The fourth line is to reproduce 28kHz feature.
The LPF right after the mixer was analyzed by a circuit simulation (Circuit Lab). It can be approximated as 150kHz LPF as the second pole
seems to come at 1.5MHz.

The sixth line comes from the LPF formed by the output resistance and the PZT capacitance.

The seventh line is to reproduce the limit by the GBW product of OP27. As the gain is 101 in one of the stages,
it yields the pole freq of ~80kHz. But it is not enough to explain the phase delay at low frequency. Therefore this
discrepancy was compensated by empirical LPF at 30kHz.

function cmpOLTFc = PMC_OLTF_model(freqOLTFc)

cmpOLTFc = -7e5*pole1(freqOLTFc,0.162).*zero1(freqOLTFc,491)... % from the circuit diagram
    .*zero2(freqOLTFc,12.5e3,100)... % eye-fit
    .*pole2(freqOLTFc,12.2e3,6)... % eye-fit
    .*pole2(freqOLTFc,27.8e3, 12)... % eye-fit
    .*pole1(freqOLTFc,150e3)... % Mixer LPF estimated from Circuit Lab Simulation
    .*pole1(freqOLTFc,11.3)... % Output Impedance + PZT LPF
    .*pole1(freqOLTFc,8e6/101)... % GBW OP27
    .*pole1(freqOLTFc,3e4); % Unknown

end

Result

Attachment 1:

The nominal OLTF (Nov 17 data) shows the nominal UGF is ~1.7kHz and the phase margin of ~60deg.

The measured OLTF was compared with the modelled OLTF. In the end they show very sufficient agreement for further calibration.
The servo is about to be instable at 28kHz due to unknown series resonance. Later in the same day, the gain of the PMC loop had to be
reduced from 7dB to 3dB to mitigate servo oscillation. It is likely that this peak caused the oscillation. The notch frequency was measured
next day and it showed no sign of frequncy drift. That's good.

We still have some phase to reduce the high freq peaks by an LPF in order to increase the over all gain.

Attachment 2:

The red curve shows the residual floor displacement of 2~10x10^-15 m/rtHz. Below 4Hz there is a big peak. I suspect that I forgot to close
the PSL shutter and the IMC was locked during the measurement. Then does this mean the measured noise corresponds to the residual laser
freq noise or the PMC cavity displacement? This is interesting to see.

The estimated free running motion from the error and actuation signals agrees very well. This ensures the precision of the caibration in the precious entries.