The story of trying to understand time constants for the QUaD detectors has been a long and sad one. Jamie and co measured timeconstants in the lab using a copper wheel. I measured on the sky using RCW38 blip crossings - see here and here. Jamie Hinderks found a good correlation with his lab values here. I did my original analysis for 2006 data here. I changed my method but got very similar results here.
It became clear that some of the detectors have a significant second (slow) timeconstant. Jamie fit the lab data to a dual tc model here and found improved fit. I then followed up trying to fit the RCW38 data to dual model here. This was probably pushing things too far... the data just won't support the number of parameters required.
In Feb 2007 myself, Tom Culverhouse and Mike Zemcov used John Kovac's Gunn oscillator setup to inject modulated RF power through the hole in the top cap. A Gunn oscillator drives through a pin diode switch which has micro second response time. The modulation signal is brought on coax from the compressor room along with power and heater current. Waveguide attenuators are used to adjust the RF level. The Gunn can be adjusted between 90GHz and 150GHz (using a doubler to reach 150GHz). Here is a picture of the assembly mounted on the top-cap. Clearly since the secondary hole is blocked with 300K material loading is increased a bit versus normal observing.
Because the Gunn is a narrow band source there are weird standing wave effects leading to some channels seeing dramatically lower received power versus others. In addition these effects cause the power on any given channel to be highly unstable. We partially mitigated the second of these problems by dithering the Gunn bias voltage and hence frequency, but the available frequency range is not enough to cure the first problem.
We took data with several modulation schemes. Here I use only 2 periods where the Gunn was switched on/off at 0.01Hz. The first period is within the time range 09-Feb-2007:09:50:24 to 09-Feb-2007:12:19:24. The good 0.01Hz section is marked by d.frame.features==2^12+2^6. The Gunn was adjusted to 100GHz. The second period is 10-Feb-2007:08:46:03 to 10-Feb-2007:11:30:15. Again good 0.01Hz section is marked by d.frame.features==2^12+2^6 and this time Gunn was adjusted to 150GHz. Here is the raw data from the second period. Note that the 100Ghz channels are insensitive to the 150GHz radiation and that the 150GHz channels show a wide range of amplitudes (intrinsic gain of our channels varies by order factor 2).
We recorded the modulation waveform as the second lockin.spare channel. As mentioned above the on state is not very stable. I therefore follow an analysis technique developed by Ki Won Yoon of BICEP:
Here are pagers showing the 100 and 150GHz results. For the "edges" plot we have from left to right: 1) processed transitions as colors with reference waveform overplotted in black. 2) the real part of the Fourier transforms of these. 3) the amplitude of ratio of ft's (the transfer func.) with the mean now shown as black line. 4) the phase of the above.
In the "transfunc" plot I show the mean amp and phase versus the single and dual tc models I extracted from RCW38 data.
We can see that the models are in general pretty sad. For some reason when I switched from single to dual the first timeconstant started coming out as zero even when no second timeconstant was found to be needed - this is clearly not right so I'll need to look further why this happened.
Here is a plot showing some example amp/phase for 4 different filters. Note that the effect on the amplitude from the adding Butterworth to 20ms exp is negligible but the effect on the phase is strong (red to green). I note this because the observed phase modulation is always stronger than the models in the pagers above and I think this is because the Butterworth electronics filter is not being modeled properly. I use the following Matlab code to compute the Butterworth response:
[bb,ab]=butter(6,20/50); [hb,f2]=freqz(bb,ab,5000,100);
Further to the problem noted above with phase at higher freq appearing to indicate problem with modelling of Butterworth. Matlab "butter" function can return the filter coefficients in analog or digital domains and freqs/freqz functions allow to calculate the corresponding responses. Here is a plot comparing these - it does appear that the analog version predicts a stronger phase change then the digital version, but doesn't appear to be a strong enough effect to explain the problems.
I made this plot using:
% plot response of analog and digital butterworth
f=linspace(0,pi,5000);
% analog Butterworth
[bba,aba]=butter(6,pi*20/50,'s');
% analog freq response
hba=freqs(bba,aba,f);
% digital Butterworth - this is bilinear transform of analog filter
[bbd,abd]=butter(6,20/50);
% digital freq response
hbd=freqz(bbd,abd,f);
f=f*50/pi;
% compare
clf; setwinsize(gcf,1000,600);
subplot(1,2,1)
semilogx(f,abs(hbd));
hold on
plot(f,abs(hba),'r');
hold off
xlim([0.01,20]);
ylim([0,1.2]);
legend({'digital Butterworth','analog Butterworth'},'Location','South');
subplot(1,2,2)
semilogx(f,unwrap(angle(hbd))*180/pi);
hold on
plot(f,unwrap(angle(hba))*180/pi,'r');
hold off
xlim([0.01,20]);
ylim([-400,20]);
Here are pictures of Clem, Tom and Mike at Pole in Feb 2007:
