Summary:

I've spent the last week investigating various parts of the DAC -> OSEM coil signal chain in order to add these noises to the MICH NB. Here is what I have thus far.

Current situation:

• Coils are operated with no DAC whitening
• So we expect the DAC noise will dominate any contribution from the electronics noise of the analog De-Whitening and Coil Driver boards
• There is a factor of 3 gain in the analog De-Whitening board

DAC noise measurement:

• I essentially followed the prescription in G1401335 and G1401399
• So far, I only measured one DAC channel (ITMX UL)
• The noise shaping filter in the above documents was adapted for this measurement. The noise used was uniform between DC and 1kHz for this test.
• For the >50Hz bandstops, I used 1 complex pole pair at 5Hz, and 1 compelx zero pair at 50Hz to level off the noise.
• For <50Hz bandstops, I used 1 compelx pole pair at 1Hz and 1 complex zero pair at 5Hz to push the RMS to lower frequencies
• I set the amplitude ("gain" = 10,000 in awggui) to roughly match the Vpp when the ITM local damping loops are on - this is ~300mVpp (measured with a scope). 
• The elliptic bandstops were 6th order, with 50dB stopband attenuation.
• The SR785 input auto-ranging was disabled to allow a fair comparison of the various bandstops - this was fixed to -20 dBVpk for all measurements, and the SR785 noise floor shown is also for this value of the input range. Input was also AC coupled, and since I was using the front-panel LEMO for this test, the signal was effectively single-ended (but the ground of the SR785 was set to "floating" in order to get the differential signal from the DAC) 
• Attachment #1 shows the results of this measurement - I've subtracted the SR785 noise from the other curves. The noise model was motivated by G1401399, but I use an f^-1/2 model rather than an f^-1 model. It seems to fit the measurement alright (though the "fit" is just done by eye and not by systematic optimization of the parameters of the model function).

Noise budget:

• I then tried to translate this result into the noise budget
• The noises for the 4 face coils are added in quadrature, and then the contribution from 3 optics (2 ITMs and BS) are added in quadrature
• To calibrate into metres, I converted the DAC noise spectral density into cts/rtHz, and used the numbers from this elog. I thought I had missed out on the factor of 3 gain in the de-white board, but the cts-to-meters number from the referenced elog already takes into account this factor.
• Just to be clear, the black line for DAC noise in Attachment #2 is computed from the single-channel measurement of Attachment #1 according to the following relation: , where G_act is the coil transfer function from the referenced elog, taken as 5nm/f^2 on average for the 2 ITMs and BS, the factor of 2 comes from adding the noise from 4 coils in quadrature, and the factor of sqrt(6) comes from adding the noise from 3 optics in quadrature (and since the BS has 4 times the noise of the ITMs)
• Using the 0.016N/A number for each coil gave me an answer than was off by more than an order of magnitude - I am not sure what to make of this. But since the other curves in the NB are made using numbers from the referenced elog, I think the answer I get isn't too crazy...
• Attachment #2 shows the noise budget in its current form, with DAC noise added. Except for the 30-70Hz region, it looks like the measured noise is accounted for.

Comments:

• I have made a number of assumptions:
  • All DAC channels have similar noise levels
  • Tried to account for asymmetry between BS and ITMs (BS has 100 ohm resistance in series with the coil driver while the ITMs have 400 ohms) but the individual noises haven't been measured yet
  • This noise estimate holds for the BS, which is the MICH actuator (I didn't attempt to simulate the in-lock MICH control signal and then measure the DAC noise)
• But this seems sensible as a first estimate
• The dmesg logs for C1SUS don't tell me what DACs we are using, but I believe they are 16-bit DACs (I'll have to restart the machine to make sure)
• In the NB, the flattening out of some curves beyond 1kHz is just an artefact of the fact that I don't have data to interpolate in that region, and isn't physical.
• I had a brief chat with ChrisW who told me that the modified EEPROM/Auto-Cal procedure was only required for 18-bit DACs. So if it is true that our DACs are 16-bit, then he advised that apart from the DAC noise measurement above, the next most important thing to be characterized is the quantization noise (by subtracting the calculated digital control signal from the actual analog signal sent to the coils in lock)
• More details of my coil driver electronics investigations to follow...