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Entry  Wed Jun 9 11:46:01 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF measurement image-6f2923a3-01ce-4d04-bc53-d8db0238e195.jpgimage-72223f4b-3b74-4574-a7ad-de6628a2c5e9.jpgX_Green_ARM_PDH_OLTF.pdf
    Reply  Thu Jun 10 14:01:36 2021, Anchal, Summary, AUX, Xend Green Laser PDH OLTF measurement loop algebra AUX_PDH_LOOP.pdf
       Reply  Mon Jun 14 18:57:49 2021, Anchal, Update, AUX, Xend is unbearably hot. Green laser is loosing lock in 10's of seconds XAUX_PDH_Err_In_ASD.pdfXAUX_PZT_Out_Mon_ASD.pdf
       Reply  Tue Jun 15 15:26:43 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF measurement loop algebra, excitation at control point AuxPDHloop.pdf
          Reply  Fri Jun 18 10:07:23 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF with coherence XEND_PDH_OLTF_with_Coherence.pdfBeta_Amp.pdf
Message ID: 16197     Entry time: Thu Jun 10 14:01:36 2021     In reply to: 16194     Reply to this: 16200   16202
Author: Anchal 
Type: Summary 
Category: AUX 
Subject: Xend Green Laser PDH OLTF measurement loop algebra 

Attachment 1 shows the closed loop of Xend Green laser Arm PDH lock loop. Free running laser noise gets injected at laser head after the PZT actuation as \eta. The PDH error signal at output of miser is fed to a gain 1 SR560 used as summing junction here. Used in 'A-B mode', the B port is used for sending in excitation \nu_e e^{st} where s = i\omega.

We have access to three ports for measurement, marked \alpha at output of mixer, \beta at output of SR560, and \gamma at PZT out monitor port in uPDH box. From loop algebra, we get following:

\large \left[ (\alpha - \nu_e) K(s)A(s) + \eta \right ]C(s)D(s) = \alpha

\large \Rightarrow (\alpha - \nu_e) G_{OL}(s) + \eta C(s)D(s) = \alpha, where \large G_{OL}(s) = C(s) D(s) K(s) A(s) is the open loop transfer function of the loop.

\large \Rightarrow \alpha = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{G_{OL}(s)}{1 - G_{OL}(s)}

\large \Rightarrow \beta = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{1}{1 - G_{OL}(s)}

\large \Rightarrow \gamma = \eta \frac{1}{K(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{K(s)}{1 - G_{OL}(s)}

So measurement of \large G_{OL}(s) can be done in following two ways (not a complete set):

  1. \large G_{OL}(s) \approx \frac{\alpha}{\beta} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{1 - \frac{\eta C(s)D(s)}{\nu_e}}, if excitation amplitude is large enough such that \large \frac{\eta C(s)D(s)}{\nu_e} \ll 1over all frequencies.
    • In this method however, note that SR785 would be taking ratio of unsuppresed excitation at \large \alpha with suppressed excitation at \large \beta.
    • If the closed loop gain (suppression) \large 1/(1 - G_{OL}(s))is too much, the excitation signal might drop below noise floor of SR785 while measuring \large \beta.
    • This would then appear as a flat response in the transfer function.
    • This happened with us when we tried to measure this transfer function using this method. Below few hundered Hz, the measurement will become flat at around 40 dB.
    • Increasing the excitation amplitude where suppression is large should ideally work. We even tried to use Auto level reference option in SR785.
    • But the PDH loop gets unlocked as soon as we put exciation above 35 mV at this point in this loop.
  2. \large \frac{G_{OL}(s)}{K(s)} \approx \frac{\alpha}{\gamma} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{K(s)\left(1 - \frac{\eta C(s)D(s)}{\nu_e}\right )}, if excitation amplitude is large enough such that \large \frac{\eta C(s)D(s)}{\nu_e} \ll 1over all frequencies.
    • In this method, channel 1 (denominator) on SR785 would remain high in amplitude throughout the measurement avoiding the above issue of suppression below noise floor.
    • We can easily measure the feedback transfer funciton \large K(s) with the loop open. Then multiplying the two measurements should give us estimate of open loop transfer function.
    • This is waht we did in 16194. But we still could not increase the excitation amplitude beyond 35 mV at injection point and got a noisy measurement.
    • We checked yesterday coherence of excitation signal with the three measurment points \large \alpha, \beta, \gamma and it was 1 throughout the frequency region of measurement for excitation amplitudes above 20 mV.
    • So as of now, we are not sure why our signal to noise was so poor in lower frequency measurement.
Attachment 1: AUX_PDH_LOOP.pdf  367 kB  Uploaded Thu Jun 10 15:01:43 2021  | Hide | Hide all
AUX_PDH_LOOP.pdf
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