Summary:

In order to estimate the free-running DARM displacement noise, I measured the DARM OLTF using the usual IN1/IN2 prescription. The measured data was then used to fit some model paramters for a loop model that can be used over a larger frequency range.

Details:

• Attachment #1 shows an overlay of the measured and modelled TFs.
• Attachment #2 shows the various components that went into building up this model. 
  • The digital AA and AI filter coefficients were taken from the RTCDS code.
  • The analog AA and AI filter zpks were taken from here and here respectively.
  • CDS filters taken from the banks enabled. The 20Hz : 0Hz z:p filter in the CARM_B path is also accounted for, as have the violin-mode notches.
  • Pendulum TF is just 1/f^2, the overall scaling is unimportant because it will be fitted (in combination with the overall scaling uncertainty on the DC optical gain), but I used a value of 10 nm/f^2 which should be in the right ballbark.
  • The optical gain includes the DARM pole at ~4.5 kHz for this config.
• With all these components, to make the measurement and fit line up, I added two free parameters - an overall gain, and a delay. 
  • NLSQ minimizer was used to find the best-fit values for these parameters.
  • I'm not sure what to make of the relatively large disagreement between measurement and model below 100 Hz - I'm pretty sure I got all the CDS filters included...
  • Moreover, I don't have a good explanation for why the best-fit delay is 400 us. One RTCDS clock cycle is onyl 60 us, and even with an extra clock cycle for the RFM transfer, I still can't get up to such a high delay...

In summary, the UGF is ~150 Hz and phase margin is ~30 deg. This loop would probably benefit from some low-pass filter being turned on.