With the recent realignment, it was time to re-calibrate the shadow sensor mounts. I wanted to find the conversion factor between the number of counts typed into the "offset" box and the displacement (caused by the coil/magnet actuator) for each blade. This required two steps. First I found the output voltage of the shadow sensors as a function of displacement. Then I found the output voltage of the photodetectors as a function of offset. When I had a conversion factor for each, I could multiply them to get my final conversion factor:

(displacement [microns] / volts)  *  (volts / counts)  = displacement / counts

I took each shadow sensor mount out of the pressure chamber, and measured its output voltage as I moved a razor blade between the diodes. I used a micrometer stage to precisely position the razor at specific displacements. I then fit the output voltages as a function of displacement to the error function using MATLAB's curve fitting app. Here are the parameters for each shadow sensor.

The function fit to is f(x) = a*(erf(-(x-m)/(2*b))+1)

Sensor A: 

Coefficients (with 95% confidence bounds):

a =      0.8901  (0.8782, 0.9019)

b =       172.4  (169, 175.9)

m =       325.7  (319.8, 331.6) 

Goodness of fit:    SSE: 0.000221        R-square: 0.9999        Adjusted R-square: 0.9999     RMSE: 0.004955

Sensor B:

Coefficients (with 95% confidence bounds):

a =      0.9585  (0.9256, 0.9914)

b =         151  (144.1, 157.9)

m =         260  (248.4, 271.6)

Goodness of fit:   SSE: 0.002777     R-square: 0.9991    Adjusted R-square: 0.999     RMSE: 0.01361

Then I plotted output voltage as a function of displacement, and fit a lien to the middle 50% of the voltage output range.  This gave me the following values. My plot with the original data points shown as circles, the fitted error function as a dashed curve, and the line as solid is shown below.

BladeA: 0.0027 ± 3.22e-5 V/micron

BladeB: 0.0034 ± 9.72e-5 V/micron

Next I set the offsets on a range from -15000 to 15000 for each blade, and recorded the output voltages. The relationship is a simple linear fit as you can see in the two plots below. The data points are shown as dots while the fit is a line. The conversion factors are:

BladeA: 1.12e-5 ± 1.55e-7 V/count

BladeB: 7.79e- 6 ± 1.85e-8 V/count

When multiplied, we get the following overall conversion factor:

BladeA: 4.0952e-3 ± 7.59e-5 microns/count

BladeB: 2.27408e-3 ± 6.57e-5 microns/count

Negation of Cross Coupling with Feedback

In my previous post, I commented on how it is possible to negate the coupling from the coils' signal. This can adequately happen only if we know the amount of coupling. To measure this quantitatively, Haixing removed the plate from our setup, so that any signal reported by the sensors would be a product of the magnetic field created directly from the coils; this is what we want to subtract. Also, Haixing believes this technique is sufficient enough so that we no longer need to move the coils further from the sensors. That being said, we switched back to the original behavior of the ACs (for feedback) and DCs coils (DC magnetic offset).

In previous measurements, I calculated the correlation between the coils voltage and the sensors voltage for the AC1 coil and sensor (that are next to each other; we ignore rest of cross-coupling between coils-sensors) and found it to be around -11.28 dB for our DC (f about 0Hz) signal. We also measured the transfer function between the AC1 coil and sensor and found it to be around -11.133 at low frequency; the data are in close agreement. Then, we introduced a factor -G_CG_s in our feedback (G_c was measured to be about 400mV/V and G_s is known from the whitening filter) and measured the transfer function again. The magnitude dropped to -40dB(shown below).

At low frequencies, we need this value to drop even more, to approximately -70dB since the transfer function of the plate is around 2mV/V or -54dB. Further, we only cancelled the coils' coupling in the low frequency range and we should modify our feedback so that we improve the system's behavior over all frequency range.

Cross Coupling between Coils-Sensors

Here I summarize my findings for the calculated cross-coupling: for AC1 (coil-sensor), I found -11.28dB, for AC2 -10.808dB,and for AC3 -11.258dB.

Simulink Model

For some unknown reason, the Simulink Model for the feedback needs at least 2 filter modules and one subsystem in order to work; otherwise it fails to operate. In order to work, we also need to include a time delay so that the coil's output is not at once fed into the feedback. I worked and finished a generic Simulink model for all six degrees of freedom, however all the coefficients are unknown. Even so, I will post it along with some description of what each components does.

 Series resonance still there, but I've gotten a lock with an RMS only a bit higher than my previous best . UGF around 130Hz. 

Was able to keep the Michelson locked for 20 minutes, so I'm making a loop TF measurement, and then setting it up to run as long as it can overnight. Here's a (totally uncalibrated) spectrum! 

I've turned the PD gains up by 10dB since ELOG 677, which is why the RMS seems WAY higher, when in fact it is only somewhat higher. 

Assuming the thing will stay locked for an appreciable amount of time, many things will follow tomorrow....

Eric helped me to figure out why my crackle signal appeared to have more power than my background signal in the last plot from Friday's post. I was using 2 standard deviations from my force Q as the threshold above which we would be able to be 95% sure that we'd seen crackle, but in reality, we don't need that big of a threshold. Instead, I used 2 standard deviations from the time-averaged background noise signal (which converges quickly towards zero). This decreased the alpha value (amount of crackle) necessary for us to see crackle, and now we can clearly notice that this code allows us to detect crackle EVEN when the crackle's power density is less than that of the background noise! I also added a bandpass filter from 50 to 150Hz. The first plot below shows the minimum alpha values necessary to be 95% sure we've seen crackle as a function of background Power Spectral Density with this bandpass filter in place (note that the alpha values are lower than without the bandpass, as we would expect). The second plot shows the power as a function of frequency for the noise and crackle signals in this situation. Both of these have an integration time of 600 seconds.

Next, I checked that the computer was actually spitting out numbers that made sense by visually finding the minimum alpha value at which I could observe crackle in the signal, and comparing that to the results of the first graph above. I found that the observations made sense. Below is a plot of the signal (the sum of the background and crackle noises) as a function of time over a 20 second range. I chose to test at a background power spectral density of 1e-15.

At alpha = 1e-6 you can see that the peaks of the crackle signal have just become visible above the "fuzzy caterpillar" of background noise (there are 5 peaks). This agrees very well with the code. The computer claims we can be sure about seeing crackle at alpha = 0.7e-6 for this power.

Finally, I noticed that the integration time used to generate the power graph affected the crackle noise's apparent power. I decided to make a plot of the crackle power (mean of the blue line of the second plot on this page) as a function of integration time. Ideally, we will run this experiment for a very long time, but here I only took a few data points because longer integration times would cause my computer to lock up due to the huge number of calculations. As a result the plot doesn't really converge very well, but it give a good idea of the behavior.

While working on collecting data this weekend, I noticed that a common mode drive would misalign the Michelson over the course of the motion. With the new clamp block that Ben made and we installed, I figure that the leveling of the end mirrors probably had changed, and that fixing this would alleviate some of the misalignment. 

So, I set up a vertical beam outside of the chamber, as I have in the past, and leveled the mirrors. I then reinstalled the blade towers, set up the coils and shadow sensors, and could not lock... 

Fiddled around, recovered a much more unstable lock, and measured some TFs to see what had changed.

Somehow, my plant has developed a series resonance, stealing a ton of phase. (Also, common mode drive still misaligns the arms .)

Just notching the resonance at ~221 Hz results in too much phase loss to stay locked with a UGF any higher than ~90Hz, so my error signal RMS is much larger than I'd like. 

I'm really at a loss as to what could have caused this. I'll tackle it tomorrow. 

I've also done some pen and paper work about analysis via fourier transforms instead of demodulation, and what RMS I really want (the x^3 in the taylor series for the michelson signal means the low-f drive can mix with and broaden noise peaks if the RMS is too high). I'll post more details about these things later.

As I mentioned in my previous post, the signal from the plate's displacement is strongly coupled with the coil's signal, so that our system is unstable. In fact, I calculated the transfer function of this "feedback loop" of the coils and found it to be about 2mV per V, roughly the magnitude of the feedback signal of the plate. We now use DCs coils to provide the feedback loop and want to find the conversion between volts applied from the DC coils and the force and only care about certain readings.

In the above figure, I represent the plate with the circle. Sensors and coils are in black and lie above the plate, while motors are in purple and lie below. As you can see in our arrangement, the DC coils are above the DC motors, so it is safe to ignore readings from the strain gauges that are not at or neighboring with the coils. Then, I calculated the conversion factors between applied V in the coils and applied force on the plate.

If prior post I showed the measurements between Volts in coils and measured mV for the strain gauges [mV=Volts*slope (mV/V)]
I also posted the measurements between weight/force and measured mV for the strain gauges [mV=Force*slope(mV/N)]
I found how volts in the coils correlate to applied force by combinging the two equations:

F=Volts*slope(mV/V) / slope (mV/N) = Volts * slope (N/V)

To give an example, I look at the AC1 coil. I have measured the response of the B1, B2, and B3 strain gauges. I also know how B1 and B2 strain gauges responded to the weights I put on AC1 (here, I ignore the 3rd reading from B3 strain gauge, because it is further away as seen in the above figure). Thus, I will get two readings (one through each, B1 and B2, motor) for how AC1 coil signal correlates to force applied by the AC1 coil. These numbers should in principle agree, or at least be close. Here are my findings:

ΔN (by DC3 coil) = 0.0239 (N/V) * V (by DC3 coil) ; measured through B2
ΔN (by AC3 coil) = 0.00055 (N/V) * V (by AC3 coil) ; measured through B2
ΔN (by AC3 coil) = 0.0023   (N/V) * V (by AC3 coil) ; measured through B3
ΔN (by DC2 coil) = 0.0023 (N/V) * V (by DC2 coil) ; measured through B3
ΔN (by AC2 coil) = 0.0022   (N/V) * V (by AC2 coil) ; measured through B1
ΔN (by AC2 coil) = 0.00198 (N/V) * V (by AC2 coil) ; measured through B3
ΔN (by DC1 coil) = 0.0016   (N/V) * V (by DC1 coil) ; measured through B1
ΔN (by AC1 coil) = 0.00078 (N/V) * V (by AC1 coil) ; measured through B1
ΔN (by AC1 coil) = 0.00363 (N/V) * V (by AC1 coil) ; measured through B2

The coefficient for DC3 seems not to fit the norm shown by the rest data.

Correction of coupling signal

I thought that, knowing the signal from the coils, we could feed its opposite to the sensors to cancel its effect. In practice, we would our feedback loop to look as the picture on the left part in the figure below. I can rearrange it to show it more clearly that the G_c and -G_c would simply add and cancel. We can do this cancellation within our digital feedback loop. Specifically, we can add the term -G_cG_S to cancel the coupling signal of the coils. Haixing agreed and we will try this tomorrow.

Coupling from coils' signal

On Thursday, I measured the HE sensors sensitivity to the magnetic field provided by the feedback coils. Unfortunately, I discovered that there was significant coupling; the sensors feel equally the plate's displacement and the magnetic field from the sensors. This effect changes our feedback loop, which now looks as drawn below. (the line between G_P and G_C boxes is mistakenly showing up after converting from

I calculated the output of the plate based on this configuration and found that .

If our G_Y is big, I can rewrite this equation as x= G_p*δ_F, also assuming that the transfer function of the plant G_p is very small (as desired). In this case, our system will not be stable, because no feedback is essentially used. To reduce the coupling from the coils' signal, we changed the arrangement of the plant. To this point, the ACs feedback coils were at the same place as the ACs sensors. So, we decided to switch the DC and AC coils (not physically) and provide the feedback from the DCs coils instead, which lie further away from the sensors. In this way, we hope that the sensors would not feel as strong of a field from the AC coils as before. Our DC actuation boards did not have the transfer function (low-pass filter) that we included in the AC actuation boards, so we had to make adjustments in the digital feedback. We added dewhitening for the DC coils (now used for feedback) and whitening for the AC coils (now used for DC magnetic offset).

 Today I made several improvements to the code I had created yesterday. I improved my script for finding the lowest amount of crackle (smallest alpha value) at which we can see crackle as a function of background noise. Along with enhancements such as better spaced data points (using log space), and improved efficiency through removal of necessary random number generation, I also converted the x-axis to display power densities of the background noise instead of a meaningless coefficient. The graph is shown below. It has data taken at 50 different background noise power densities. There is also a handy, second order polynomial fit to it.

I also began creating a script to plot the Power Spectral density of the Noise Signal and the Crackle signal separately on the same axes. This should show that the background noise signal at a HIGHER power than the Crackle signal - even though we can detect that crackle. Unfortunately, as you can see in the plot below, something has gone wrong, and my Crackle Spectral Power Density is actually above that of the Background noise when the Background is set to 10^-15. I will continue to look into this to see what is wrong. Once that is complete I will add in a simple bandpass filter in order to simulate how we will only be taking crackle data from a small portion of the sensors' outputs (just like LIGO observatories).

 This is a continuation of my post from yesterday. Today I was able to fit a polynomial to my Mean force Q signal with crackle. Here is the polynomial fit.

Visually, it is hard to tell the 2nd-order polynomial fit with the actual data points on this plot, but I wanted to make sure that there was no trend in the error, so I plotted the residual (polynomial - original points). As you can see below, there is no trend, and the residual is very small, so the 2nd order polynomial is a good approximation.

Satisfied with this, I used MATLAB's "root" function to find the intersection of the 2-sigma boundary line and the Man Qf line.  This is the Minimum value of alpha (so in a sense, the minimum amount of crackle noise) we need in order to be able to be 95% sure that we have seen crackle above the background noise. I then plotted this minimum alpha value as a function of the level of background noise.  The x-axis of this plot is in terms of a coefficient that I am using. The actual range of this plot goes from a power spectral density of  approximately 10^-15 to 10^-12 [sqrt(W/Hz)]. The circular points are the individual data points I found, and the line is a second order polynomial I fit to it.

In the future, I will figure out how to add in Jerk Crackle.

 It was 14.3K om, but as you pointed out, the main issue is that we connect the positive output to the GND. Now, we have a differential amplifier, so everything works fine.

Sensing Matrix
I calculated the entries of our sensing matrix S x = y, where S is the 6x6 sensing matrix, x is 6x1 vector signal from six degrees of freedom and y is the 6x1 signal sensed by six HE sensors. Haixing told me to ignore the 7th sensor (N), because in practice we could levitate the system using only the rest six. The sensing matrix contains many unknown coefficients, which we will find by calibrating the system and adjusting the values. I will post the matrix, once we know the entries are correct.

Calibration

Coil - Force

We want to get an idea of how much force is applied on the plate for different signals from the digital filter. So, we measure the DC motors output (in mV) and we also have previous measurements of how the DC motors voltage correlates to the force applied. Therefore, knowing how the digital filter output corresponds to the DC motors voltage offset, we can infer the relationship between digital signal output and force on the plate. We first worked with the first top coils: AC1, AC2, and AC3. While changing the output for a single coil, we measured the response of all the motors to consider cross-coupling effects. While measuring the response of the bottom motors, we have to make sure there is always contact betweem them and the plate; otherwise, the data will not show how the force on the motors changes.    

When we apply feedback, current runs through the coils and creates/adjust the ambient magnetic field. However, the magnetic force the HE sensors feel from the levitated plate is so coupled with the one created by the coils. We also want to know how sensitive the sensors are to our coils, when we apply feedback and so we again took measurements.

We also want to know how sensitive the sensors are to a vertical displacement of the plate:

In all our measurements, we noticed some hysteresis so we had to follow the same order when recording the data. In case of a mistake, we had to repeat from the beginning.

 Over the past two days I have been busy simulating a signal like the one we might extract from our crackle experiment. This signal is made of two things: crackle and background noise. 

• the traditional spring constant with no crackle
• a force crackle term: this manifests as a random delta-k (I used a normal distribution) multiplied by the Force and a constant Alpha
• a jerk crackle term: this manifests as another random delta-k multiplied by the derivative of the force and a constant Beta

How much was the input impedance of the DAC conditioning board before putting the high-impedance voltage follower?

HE output

We tested the HE output and found that all the outputs were fine, except for the S1 board which saturated. This board was also the one with the odd phase function (when measured earlier in the summer). We realized that some components may have been disconnected (not enough soldering) and we easily fixed the problem.

DAC output - Voltage Follower

The output of DAC was lower than the one we expect. We measured the DAC output in V for different feedback filters and compared it to the value we would expect by converting the digital feedback of our computer from bits to volts. However, when the ribon wires connecting the DAC and the actuation conditioning boards (coils) is disconnected, we get sensible readings; the ribon seems to limit our output. It might be the case that the internal resistance of the DAC is large and that causes a huge voltage drop. We will then use a voltage follower configuration with an OP27, so that the current is not sourced from the DAC. We introduced a voltage follower in the 10 actuation boards (7 coils & 3 DC magnetic offset). Following, you can see the voltage follower configuration.

Simulink Model

I also created a draft model to model our system, using Simulink. The matrix that determines the coupling of the 6 degrees of freedom still remains to be calculated.

ADC & DAC

The bits/volts conversion factor is different for our ADC and DAC. Specifically, I measured the voltage output of the ADC and DAC and, by comparing it to the input and output readings--in bits-- of the computer respectively, I found this relationship to be 1.64bit/mV for the ADC and 3.3bit/mV for the DAC.

HE sensors output range

We also measured the output of the HE to fluctuate at most 100mV in response to the movement of the plate. Given that, a small displacement of the plate that produces roughly 30mV would bring approximately a 18bit change in the ADC output. With the already inherent noise and fluctuation of the bits reading, it is therefore difficult to detect small movements of the plate; it is necessary to boost the HE output after subtracting the HE sensors offset.
The HE sensors signal goes a voltage offset and then a high-pass filter. We will adjust our resistors' values only in the first state, so that the voltage offset more accurately corrects the inherent offset of the sensors and amplifies the output even more. Currently, as described in my first research report, the gain was 1; we will now aim for a gain of 50. I calculated the expression for Vout in the voltage offset configuration to be:

Vout=-(R4/R3)Vin + 5* [R2(R4+R3)/R3(R1+R2)].

A gain of 50 would also increase the inherent offset of the sensors, which would now be about (50*2.5)=12.5V; we also need to fix that. I calculated that if we use R4=50*R3 and R1=19.4*R2, we can get the desired gain, while also appropriately correcting for the offset.

Transfer functions saturation

We measured the transfer function of our damping transfer function: (s+Zo)/[(s+Po)(s+P1)] where Zo<Po<P1. We noticed that the voltage source setting of the spectrum analyzer affected our transfer functions. I extracted and plotted the transfer functions for three different voltage sources: 10mV, 100mV and 500mV which are shown in this order below. We are unsure as to why that happens.

 Today we finished installing both shadow sensor mounts, and everything is wired up. The system successfully locked with the new hardware, although there is some tweaking that needs to be done to the transfer functions.  Adjusting the shadow sensor heights was a piece of cake compared with how it was before. Here is a photo of the new setup from outside of one of the windows.

Afterwards, I recorded the output voltage of each channel (A and B) at offsets from -15000 to 15000, and I used MATLAB to find a line of best fit. The plots with the two best fit lines are attachments 2 & 3. The slopes are printed on the plots.

Plot A: 

Plot B: 

Eric used the ratio of the slopes to compensate for the fact that a given input voltage into one coil will not necessarily lead to the same force on the on the blade spring as it will on the other. With the new proper offset ration (SlopeA/SlopeB = 1.9298), we can control the two blades as if they were identical.

Computer Feedback Filter

I installed Linux on my office computer today, because Windows was sometimes crashing. Then, using secure shell (SSH) I remotely accessed the supercomputer that receives the input signal from the HE sensors. With the Foton software, I created a preliminary feedback filter with an integrator and a damping factor. Assuming a normal response function R(s)=1/[m(s²+ω_n²-γs)] for our plate, the transfer function is very large near the resonance frequency (where s² and ω_n² cancel) and at small frequencies (where only the resonance frequency term, presumably small, remains). Therefore, we need our feedback filter to add to the response function of the plate; the integrator --proportional to s--adds a large term near small frequencies, and the damping factor--proportional to s--adds another factor near the resonance frequency. I designed the filter so that the cut-off frequencies occur at roughly 2, 5, 20, and 200Hz. Below are the results. WIth the correct gain factor, we have a unity gain from 2Hz to 5Hz.

DC Motors

We tried to test the motors, but they did not move as fast. Apparently, we had include a 3.6kΩ resistor in series with them, using a 15V source; no wonder they did not work. We replaced the 3.6kΩ resistors with 1kΩ ones and achieved a better movement.

HE sensors output

We compared the bits of the input signal digitally diagnosed with the output signal of the HE signal measured manually with a voltage meter to check whether the correspondence made any sense. I plotted the pair of data ( {millivolts, bits}, ..) and found the best fit for the data; the slope was 1.64 (1mV corresponds to 1.64bits). For our ADC converters, 20V correspond to 2¹⁶ or (65536) bits, so 1mV corresponds roughly to 3.3bits. However, the bits correspond to the voltage difference, so the actual readings for the bits should be half (1mV=2¹⁵/20=1.638bits). Our conversion works. 

Simulink

I started using Simulink and looked at Rana's examples. I will keep building our setup with Simulink to ultimately simulate the behavior of our plate.

 Today I modified part 1 from my July 16 elog by couterboring the two rear coil mounting holes as shown in the photo below. This allows us to forgo the extra spacer, and it lets us use the same, short screws that we had already been using to attach the coil.

Next Eric removed one of the old shadow sensor apparatuses from the chamber, and I soldered the power and sensor cables in place (as described in my July 23 elog). Thus, the top half of the assembly was complete and ready to be installed. The following photo shows it mounted in place with the one remaining old mount in the background.

Another view (note that the screws are not yet tightened down):

Finally, I epoxied the magnet onto the new razor blade mount. It shouldn't take too much time to get this new assembly up and running tomorrow.

DC motors - Tuning Adjustable Resistors

I first tuned the resistors of our DC motors circuits, such that the voltage meters read close to 0V when the strain gauge sensors are not stretched by the plate. A zero voltage reading would later help us know when the plate is at equilibrium. The equilibrium is unstable and so the plate moves either up or down. I tuned the resistors for the bottom sensors while the plate was stuck against the top ones and vice versa. The BE adjustable resistor (R2 in the picture) was very sensitive and acting strangely; the voltage reading would change as long as the screw driver touched the resistor, making it impossible to know whether we are close to 0V while adjusting. The resistances also drifted away from their values throughout the day. Whereas the initial offset was set within 5-10mV, at the end of the day it had grown much larger as is evident in the following image:    

Bode Plots in Matlab

I developed a code in Matlab to read .dat files and create Body plots. For future reference, Matlab files should not include a "." in their name, since Matlab recognized whatever comes after the period as the extension of the file. Following is the code for the phase plot and the results:

>> cd('C:\Users\Γιώργος\Dropbox\maglev\SURF\progress_report\Transfer Functios of Coils')
>> load 008ASC.dat
>> G = semilogx(X008ASC(:,1), X008ASC(:,2))
>> axis ([0.5 1000 -160 0])
colorss = {[0.5  0.5  0.5],
           [0.8  0.3  0.7],
           [0.0  0.0  1.0],
           [0.97  0.1  0.0],
           [0.1  0.9  1.0],
           [0.2  0.8  0.1],
           [0.4  0.4  1.0]};
for k = 1:length(G)
    set(G(k), 'Color', colorss{k});
end
grid
grid minor
%axis tight
hXLabel = xlabel('Frequency [Hz]');
hYLabel = ylabel('Phase [degrees]');
title('Phase of the Low-Pass filter', 'FontWeight', 'bold', 'FontSize',12)
set( gca                       , ...
    'FontName'   , 'Times'     , ...
    'FontSize'   , 15          );
set([hXLabel, hYLabel], ...
    'FontName'   , 'Times',...
    'FontSize'   , 15          );
set([Legend, gca]             , ...
    'FontSize'   , 15          );
%set( hTitle                    , ...
%    'FontSize'   , 12          , ...
%    'FontWeight' , 'bold'      );

set(gca, ...
  'Box'         , 'on'     , ...
  'TickDir'     , 'in'     , ...
  'TickLength'  , [.02 .0] , ...
  'XMinorTick'  , 'on'      , ...
  'YMinorTick'  , 'on'      , ...
  'YGrid'       , 'on'      , ...
  'XColor'      , .1*[.3 .3 .3], ...
  'YColor'      , .1*[.3 .3 .3], ...
  'FontSize'    , 25, ...
  'LineWidth'   , 1.5        );

 HE boards and coils test

I also measured the output of the HE sensors to see whether there were working fine:

AC1=105mv, AC2=128mV, AC3=137mV, S1=1.73V, S2=82mV, N=16mV, W=7mV. In the same way, I measured the resistances of the coils (since we have not yet created a signal with the computer) and they worked fine, too (same values as before, slightly higher resistances, possibly because of the long ribon wires attached).

Tomorrow, I will start working on Simulink and learn how to use the computer to provide the feedback filter.

You should add the flags in the model so that the error and control signals are written to the disk (as _DQ channels). Also make some filters so that the calbration of the signals can be loaded into them. In this way the data written to disk will be calibrated in some units (e.g. meters or Newtons).

 Still tweaking the loop; although the UGF appears to be around 150, and a fair amount noise suppression is present, if I jump even softly near the table it knocks out of lock. 

Also, looking at higher frequencies, I've found that there seems to be a series resonance at around 700Hz in the blade, as the phase jumps down a ton, and doesn't come back. (In contrast, the feature around 150-200Hz was in parallel, since the phase lost around it comes back afterwards.)

Here is the TF unwrapped and wrapped. 

Increasing the gain until instability right now injects a broad chunk of noise centered around 700Hz and its harmonics. Is this worth trying to compensate/control, or should I just stay in the stable regime and not worry about it?

Got it!

I tweaked the compensator, and now there is no more instability in this 150-200Hz range. Turned the gain up to 300 now, in comparison to 170 yesterday. RMS of the locked error signal is about a factor of five smaller than yesterday, to boot. (With the increased stability, I was able to throw on another integrator, too.) Plots!

Here the blue trace is yesterday's best spectrum, red is today's. This is the strongest lock the crackle IFO has seen. 

Time series trace of the error signal, as I disengage the servo:

Today, the front panels for the HE and Coils Board arrived. I spent the day preparing the last power lines for the DC coil boards, marking and drilling holes, cutting ribon wires for the ADC and DAC conversion, grouping the wires together using cable ties, testing the boards. We are officially finished with the first stage. Tomorrow, we will measure the transfer function of the system to start creating the feedback filter with the computer.

I also installed MatLab. At the beginning, the setup.exe file could not load the installation. Apparently, there was a problem with the saving directory, because it included non-English(Greek) characters. I saved the data in a folder inside the C disk: "C:\MatLab" to make it work.  Tomorrow, I will create the Bode Plots and use MatLab to simulate the behavior of our system.

 Today I finished wiring one emitter/receiver set (as shown in yesterday's post), and I mounted the two components in their location inside the shadow sensor mount. I coated the surface they sit on in nail polish in order to prevents shorts, and epoxied the diodes in place as shown below.

I tested the setup on a solderless breadboard. The wiring I used can be seen in the photo below, with one exception. In reality I used an oscilloscope to measure the output voltage across the 50 ohm resistor (which is sitting in the upper, right corner of the board). The scope read approximately 1.75 V above the dark level when nothing was between the two diodes.

Finally, I cut out a piece of bread board for the purpose of mounting that circuitry onto the face of the assembly. Here is a photo of the first completed assembly. The four wires that will be soldered in place along the top of the board already exist in the experiment. At time of installation, we will solder these wires in place. From left to right, they are Sensor +, Sensor -, +15V, Ground. I used the power supply and scope to check that the circuit still worked, and it did.

The thumb screw and spring system for adjusting the sensor height feels strong, and it will not move when you take your hand off of the screw. Tightening the set screws locks the height completely.

Objective: raise UGF by making a compensator for the oddities in the loop TF (presumably in the blade) that were causing instability. 

TF measurement in hand, I was having trouble fitting poles/zeros to it, until Nic told me about vectfit. A google search of "vectfit" resulted in code in a folder of Rana's as number 1, the official website as number 2.

With this tool, I was able to produce a reasonable fit to the ideal compensator, which is given by the desired loop TF divided by the measured loop TF, with 4 pairs of poles and zeros. I weighted the data so that the area around the big phase gain peak was of utmost importance, the others less so, and things outside of the 150-250Hz range irrelevant. 

I stuck the numbers into foton, and gave it a gain such that outside the region of interest, the compensators gain is unity. The TF of this filter looks like the dotted red trace in teh above plots.

Here are the loop TF measurements with and without the compensator. (The net phase offset is not due to the compensator, I talk about it below)

At first, I saw a lot of noise being injected at 1.6kHz and its multiples. My main Servo TF had a 2-pole cutoff at ~4.4kHz, but since its gain was pretty high at 1.6kHz, I figured that may be responsible for the injection, so I lowered the cutoff to a pole each at 1k and 2k, thus taking away some of the phase in the regime in the plot above.

In any case, the main annoyance (phase loss around 180Hz) is completely gone. But there is a little glitchy dude around 160...

Now, I am able to turn the gain up a few dB higher without losing lock, but there starts to be a lot of noise injected right under 160. I need a better fit in that region, and then I should be done compensating.

 Today I figured out the proper orientation for the SME2470 Emitter and the SMD2420 Detector which we will be using for the shadow sensor. A very simple diagram is shown in the first attachment (a photo from my notebook). My system is to always use a black wire for the ground side, which is in both cases the side with a flat (not half-moon-shaped) conductor. The emitter gets a white wire on the curved side, while the detector will get a red wire (so we can tell the two apart). Note that the wiring can be confusing because the Cathode and Anode side of the components are opposite for each, but the fact that they are placed in the reverse orientation un-does this.

The second attachments shows the emitter with two wires soldered onto it. This proved to be difficult. As noted in my notebook, the soldering iron can only be set to 500° F for 5 seconds before the part is ruined (according to its documentation), so making these connections strong took work. The final attachment shows the emitter mounted in the part I designed to hold it. (See my 7/16/13 ELOG).

The solid works model for the experimental apparatus has been set up and the parts have been ordered. Today, I cleaned the cryostat with solvents and delivered the clamp and insulator schematics to the machine specialist in Church.

Progress on HE & Coil Boxes

I used the labeling device to print labels for the HE sensors and coils' boxes. We use medium font and line spacing of 1.

I screwed the boards and panels together and prepared power lines for the coils' box; we have 10 boards (7 for the actuator coils and 3 for the DC magnetic field offset), each with an input and output, so I made 20 power lines. For the connections on the fit-through board, we follow the same order as the physical setup:

 We use A's for the input and B's for the output signal on the fit-through board. 

Tomorrow

Tomorrow, we will test our HE and Coils' boxes and see whether they work. We are still waiting for some front panels and some missing components to complete our circuits (such asholders and resistors for the coils' boards)

I have made new drawings which better abide by engineering conventions. Here they are.

Digital Voltage Meters

We need 5V to operate our digital voltage meters that indicate the offset from the strain gauge circuit in our DC motors configuration. One of the two power boards in the DC motors box had an unused 4-pin holder (+24V, GND, GND, -24V) and we will use that. To find the internal resistance of the digital voltage meters, we read the manual and the specifics of it: for 5V, the voltage meters source 400μA.

So, the internal resistance of the digital meters is R=V/I=12.5kΩ.We have two boards with 3 digital voltage meters each. They are all connected to the power supply and are in series with R1, whose value we want to determine so that we source enough current for all the resistors. We have six parallel voltage meters of 12.5kΩ each, so R_eff=1/ (6/12.5kΩ)=2.08kΩ.

Each voltage meter needs 400mA, so we need in total I=2.4mA. That means that the resistance of the total system R_tot (=R_eff+R₁) needs to be 24V/2.4mA=10kΩ. Then, R₁=7.92kΩ (we used 7.5kΩ). The meters worked fine.

Labeling

We labeled all the wires in our DC Motors Box:

• the wires connecting the 6 PCB boards to the fit-through board; either Motor_orientation (such as TS-top south), or Gauge_orientation.
• the wires connecting the push-buttons to the PCB boards, receiving strain gauge output and sending the signal

Rest

After completing the power wires for the DC Motors Box, we were almost finished; we only need very few components for the testing of the LEDs.

We also started fitting everything in our HE sensors box. We drilled holes/cut parts of our panels so that the PCB boards fit and then screwed the panels together.

Tomorrow

Tomorrow, we need to do the following:

• make the labels for the HE sensors and Coils' boxes
• finish fit-through boards for HE & coils' boxes
• cut 4 panels for coils, so that the PCB boards can fit inside the box

Some further details; here's what lock acquisition looks like on the error signal. Just before t=10, the main servo is switched on. At t ~ 12, the low frequency boost is engaged to squash the ~6Hz motion. At t ~ 14 I turn off the shadow sensor damping, which takes out some broadband-ish noise above 100Hz. 

My model of the Loop TF doesn't correspond too well right now. Foton has some rolloff in the phase of the servo in the hundreds of Hz regime, which MATLAB doesn't show with the same arguments. I'm presuming this is due to RCG limitations, and figure this is the reason for the phase discrepancy above ~200Hz. However, I'm not really sure about the pretty big disagreement at 50Hz. I'll check it out tomorrow. 

 I have designed the following new shadow sensor mount, and the designs have been given to the machine shop - the parts should be here soon.

 We needed a new design because currently it is very difficult to adjust the position of the shadow sensors without messing with the position of the coil relative to the magnet. The whole apparatus is also currently too flimsy. The new assembly is going to be machined out of 6061 aluminum, and it will not have large thin surfaces, so it should be much less susceptible to vibrations. A simple thumb screw will drive the shadow sensors up and down relative to the razor blade.

The assembly is made of the 4 parts (shown in the first 4 attachments below). There is an assembly picture as well as an exploded view afterwards.

• Part 2 replaces the upper half of the current blade spring clamp. It is modified to be larger and to house a groove where the razor blade mounts. As usual, the magnet will be epoxied directly between the two holes.
• Part 1 has a hole pattern with which we will mount the coil (note that the coil is simply modeled as a large grey cylinder. This part also has a large track cut out in the middle which will house the mobile sensor mount piece (part 4). A through-hole on the top will allow for the thumb screw. The bread board with the electronics will sit on the front face of the part. The four front-facing holes are tapped. standoffs will separate the circuit board from the metal part.
• Part 4 is the sensor mount. The diodes sit on the inside of the cavity with their leads coming out of the small holes (two holes per side). The large threaded hole allows the thumb screw to drive the part up and down. Two set screws can be tightened down through the holes in part 1 onto part 4 so as to prevent further adjustment.
• Part 3 simply attaches the assembly to the large optical mounting post. It allows well over 180 degrees of freedom so that we can easily change the position of the mounting post relative to the optics when necessary.

Note that a spring on the thumb screw between parts 1 and 4 will maintain a force between those parts - this should make adjustment smoother. Also note that I will order washers to stack up on the thumb screw on top of part 1 to allow for an extended range.

The Crackle IFO has been locked with a CYMAC digital loop 

UGF is a little under 100Hz. This can and will be improved tomorrow. Damping was fairly straightforward to get working nicely. Getting the right TF for michelson locking took some fiddling. Now that I can mess around with arbitrary filters, the 38Hz wiggle that annoyed me before was easy to deal with. With some integrators, the RMS of the error signal is much, much, smaller than ever seen with the analog loop. 

There's still this odd feature around 150-200Hz that I think is the current culprit for instability as I try to up the gain (last seen in ELOG 636), and I plan on tackling it head on tomorrow. 

In the meantime, I took a spectrum to compare with results from June, as a sanity check. It shows some excess noise under 100Hz, but corresponds well after that. 

Once I get the UGF to a few hundred Hz, it's full steam ahead, driving the blades in common mode and hunting for crackle!

LED connections

We use LEDs to indicate whether our power boards work. For one power board, we need two LEDs, one for the positive and one for the negative voltage. For boxes that contain more than one power board, we will still use onlyy one pair of LEDs, since we only care to test whether our power supply works. We have six LEDs and we will use 2 for the HE sensors board, 2 for the coil-actuation box and 2 for the strain gauge box. Today, we made the connections for our LEDs; on our power boards, there are two letters: A and K. A is for the signal wires (positive/red or negative/black) and K is for the ground wires.

Power Boards

The missing components for our power boards have arrived, so we finished our power board circuits, drilled the holes on the box panels and screwed the power boards. We also created more power lines, such that we have enough for the 7 coil-actuator and 3 DC offset boards, as well as the HE sensor boards. We prepared power lines for all HE sensor boards and 6 coil-actuator ones, but ran out of components; we still need to create power lines for one coil-actuator board and 3 DC offset ones.

DC magnetic field offset

For the boards that will provide tuning of the magnetic field, we only use a current booster circuit (configuration with a current buffer in a feedback loop with a low-noise op-amp). We built all three boards.

 Minimizing the coherence of the intensity noise signal.

We need to use the difference between the the output voltages of the PDs at the symmetric and asymmetric michelson ports. This lets us eliminate intensity noise while not messing with the displacement data. The gains of these two signals (Vas & Vsy) may not be quite the same though, so a small level of coherence may still exist. I created a MATLAB script which takes in Vas and Vsy signals, and find the coefficient by which we should multiply one of the input voltages in order to minimize the coherence.

To do this, I first created a simple function (integralCoherence.m) to find the integral of the coherence for a given Coefficient and given Vas and Vsy inputs. The coherence is found using MATLAB's mscohere function.

Next, I created a function which uses Newton's root-finding method to detect which coefficient value would give integralCoherence the smallest value possible. In other words, this function (rootFinder.m) minimizes integralCoherence(Coefficient). To do this, I seed the function with a coefficient of 1, then ask MATLAB to find the first and second derivatives of integralCoherence at 1. The first divided by the second derivative is a quantity I call the "jump." The best coefficient must lie somewhere in the direction of the jump according to Newton's method. We add the jump to the original coefficient and start over. The process iterates until the jump size is less than 1e-6, which means that we are converging on a best value for the coefficient. That coefficient is returned.

Finally, I made a test script which loops over several possible offset values for the Vas signal (relative to the Vsy signal), and returns the best coefficient to minimize the hypothetical intensity noise in each case. I have included a plot of the percent error of each output coefficient with respect to the input value. Although we are seeing some trends in the error, this does not seem to be too much of an issue, as the relative deviation is generally well below 1e-3.

Tomorrow, I will create a function that takes real-life Vas and Vsy values as input and outputs the best coefficient.

Strain Gauge Boards

Our conditioning boards did not have a low-pass filter. That is a problem, since these circuits were designed to amplify a DC voltage offset, but the op-amp cannot provide that gain at very high frequencies. We introduced a capacitor to create a low-pass filter and made sure the cut-off frequency of our setup was lower than the one of the op-amp: f= 1/(RC*2pi). For our R=24kΩ, we chose C=0.1μF. So, we built a low-pass filter for our 6 strain gauge boards and then measured the DC voltage offset. Our digital voltage meter can read up to 200mV, so we adjusted the one adjustable resistor to get the offset voltage as low as a few mV. As we slightly pussed on the strain gauge sensors, the voltage increased indicating that our circuits work fine.

Panels for Coil Actuators & Hall-Effect sensors' power boards

At the end of the day, I marked the holes for 5 power boards on the panels of our coil (3 power boards) and Hall-effect sensors' box (2 power boards).

Today, we first talked about the connection of the HE conditioning boards to the HE sensors and the arrangement of the wires on the connector. There are 7 HE sensors named after their position (e.g. W=West). Starting from the right, the first four pins denote the sensors that lie above the plate.The bottom row is the bottom part of the connector to the HE signal. X denotes the pins not used and the last three pin places on the left are for the sensors imbedded in the coils, which are though--for the time being--not used.

We tested the transfer functions (TF) of our 7 HE sensor conditioning boards. Six of them had identical TF, same as the ones we expected and one of them (S₁) had a similar TF, but a totally different phase. We extracted them to a floppy disk and inserted them to a computer, where we created files that contain the data of the TF plots. Tomorrow, we need to plot the data in Mathematica. We also measured the offset for our Hall-effect sensors on the oscilloscope. We used V_in=0 to measure the actual offset and then adjust R₂ to null it. Here are the recorded offsets:

AC₁:2.37V, AC₂: not working, AC₃:2.34V, S₁:2.5V, S₂:2.5V, N:2.5V, W:2.46V.

We also looked at the connector for the Strain Gauge (S.G.) and DC motors (M). We have six connections for each. We named our S.G. boards, depending on the location of the corresponding--in our setup--strain gauge. IMoving from the right to the left, the strain gauge sensors correspond to: TS (top south), TW, TE, BN (bottom north), BS, BE. We found a problem with the BE op-amp; it must be broken. We tested the output signal of some boards and we did not find a steady DC amplified voltage we expected; we thought of introducing a low-pass filter (since DC signals have ideally a 0Hz frequency)before the signal reaches the strain gauge op-amp.

Tomorrow, we will measured the TF of the Strain Gauge boards to see what is wrong. We will also insert a low-pass filter with a cut-off frequency around 10Hz.

Today, we completed the power boards and tested to make sure they work. We had a problem with the LM2991 negative voltage regulator; its tab at the back is connected to the input voltage and should therefore not touch our panel, because it is in this way grounded. We will fix that problem by placing spacors below the power boards.

We also built the connections for our DAC and ADC controllers and designed our DAC & ADC panels so that we can order a company to drill holes for our wires.

To this point, we have to finish the circuits for some of our Hall-effect sensors boards (our missing components arrived) and test all of our circuits to make sure they work as desired. After that, we will move to the second stage of this summer research, taking measurements, debugging problems, improving computer processing, and ultimately successfully isolating the levitated plate up to very low frequencies (starting from 5Hz and hopefully going down to 0.01Hz)

On Monday and Tuesday, we talked about the ADC & DAC controller and the time signal input from the function generator.

We also designed the boxes, drilled the holes and wired up both the ADC and DAC controller.

In the meantime, we received the components that were missing from the power boards (precise resistors). We started wiring more power boards and we will finish them tomorrow morning.

We went into the Ogin  future Crackle Lab and found a few things that looked like vacuum pumps. There are two roughing pumps (one has the words LEAKS written on it), and most of a turbo pump. They all look like replacement parts for the presumably working pump setup that is hooked up the the thermal noise chamber in that room.

link

I include comsol models of 6 different eigenfrequencies for a certain silicon flexure. In addition,  the expected graphs of the thermoelastic noise and phonon-phonon loss are also presented for the various mode shapes.

Today, we determined our low-pas filter circuits and its components. To achieve our cut-off filtering, we need to change the values of our zeroes and poles of the transfer function. Specifically, we saw that only if P₁<Z₀<P₀ can we achieve low-pass filtering.

It is essential that are cut-off frequencies are very close to the values we used in our high-pass circuits, so that we achieve perfect de-whitening, especially for the two lowest cut-off values; we computed the values of our resistors and capacitors that yield such transfer functions and started wiring our 7 actuator PCB boards. However, the low-pass circuits cannot source a lot of current and so our PCB boards, apart from the low-pass filter, also include a current-booster circuit, which consists of the high-noise current buffer that is in a feedback loop with a low-noise op-amp. Later, we designed the box panels where we will place our 7 PCB actuator boards, along with 3 DC control boards, and 3 power boards.

 I converted the raw Volts/rootHz data retrieved from the asymmetric Michelson experiment to frequency noise using the proportionality constant derived in Eric's June 19 Post. I then plotted frequency noise as a function of frequency on a log log plot. The noise for the asymmetric setup is shown in red (it is the noisiest). The other three data sets can be thought of as components of the total signal. The symmetric setup (in green) theoretically give us the noise for everything besides frequency noise from the asymmetry. The red line was found by taking the spectrum of the laser shining directly into the detector (the intensity noise component), and cyan curve is from when the laser was turned off.

The bottom of the cryostat contains a chamber where the main components of the experiment will be contained. This space is cylindrical in nature with heighth = 5.6cm and diameter = 13cm. In addition, the cylinder has two inner lips which create an inner diamter = 11.4cm. Furthermore, hole spacing for the screws that will attach the apparatus to the cryostat is approximately 1cm.

Also, we are working with a square window through which a laser will pass. The opening is an embedded circle with diamter 2.7cm and the square itself having length = 5.5cm.

 Yesterday was very productive. I began by continuing to align the two output beams horizontally of the asymmetric Michelson. Unfortunately, it wasn't working - I was limited greatly in my range of motion by the fact that the long arm had such a narrow window in which it could both come back in perpendicularly, and still not be reflected straight back into the laser (which would be bad for the laser). Eric suggested that instead, I try to use the laser he had been using for his crackle experiment which travels through a long, fiber optic cable. This gave us two benefits. First, the fiber optic cable is supposed to act as a collimator, so our lens would be unnecessary. The spot size would automatically be the same everywhere along the beam, including at the detector. Also, the tiny radius of the fiber optic "groove" meant that there was basically no chance of shining the laser directly back into the tube and into the laser. This allowed me to align the beam out and back along the same path, which was less challenging than before. After some fiddling, it worked! We could see fringing effects clearly. (slightly hard to see in this photo)

After replacing some of the mirror mounts with more sturdy "Polaris" mounts, we were ready to take some data. Eric positioned the beam so that we were sitting right on a fringe (the PD output voltage would wander around with slight shakes of the table or loud noises), and he let the spectrum analyzer take its data. There was a lot of unexpected noise in the signal. We wondered if this was acoustic noise, so I covered the setup with a large blue plastic box (with a hole cut out for the long arm). Suddenly low frequency noise (less than 10Hz or so) dropped significantly, so we decided to use the box for our measurements. Eric took a noise spectrum both for the asymmetrical setup, and for a symmetrical setup with both arms just about 8.8 cm in length.

Here is an overhead view of the asymmetric setup. The source is at the bottom, and the two grey mounts are the end mirrors of the arms. The detector is on the right, and the other two mirrors are used to bring the beam to the correct vertical level. The small iris in the bottom right is irrelevant.

Here is the long arm of the Michelson which passes outside of the blue box in the asymmetric setup.

 Additionally, yesterday I learned how to plot the noise spectrum density using MATLAB's pwelch function. I also learned to make good-looking plots. Here is the PSD for 4 random signals that Eric came up with.

Today, we analyzed the transfer function of our circuit:

 Specifically, Vout/Vin= -((R_1 〖+R〗_2)R_3)/(R_1 R_2 ) {((s+Z_o)P_1)/((s+P_o)(s+P_1))} where Z₀, P₀, and P₁ are important parameteres of our transfer function.

Were we to graph it against frequency:

So, we determined the values of R₁, R₂, R₃, C₁, and C₂ so that Zo occurs at 0.1Hz (close to the estimated natural frequency of the levitated plate), Po=50Hz and P₁=200Hz Specifically, R₁=R₃13kΩ, R₂=1.5kΩ, C₁=2.2μF, and C₂=61nF. We did not have a 61nF capacitor, so instead we used a 47nF one (slightly changing our P₁ point).

Our Hall-effect sensors give a constant 2.5V when no magnetic field is present. Therefore, we need to include an offset of 2.5V. We will achieve that with a voltage divider with R₁=13kΩ, R₂=4.3kΩ, R₃=R₄=2.2kΩ.

In the afternoon, I wired up 7 PCB boards for the Hall-effect sensors circuits. These include the -2.5 offset and the high-pass filter. We used the spectrum analyser to see whether the transfer function of our circuits are as predicted; the experimental and theoretical data agreed.

Today, I wired up the power PCB boards for the DC motors and then talked and read about Hall-effect sensors. The box that will hold the sensors is too small, so we spent some time figuring a way to combine two boxes into one to fit all of our components. Then, we spent a lot of time talking about transfer functions, Laplace transforms and how they help us disentangle linear, time-dependent equations from their time-relationship and yield convenient equations that describe the behavior of a system at different frequencies.

We want to apply a high-pass filter to "remove" the low-frequency noise from the signal in our Hall-effect sensors and achieve low frequency seismic noise isolation. So, we used the following high-pass filter circuit and found its transfer function V_out/V_in. It is my homework to find the values for R₁, R₂, R₃, C₁, and C₂, so that our transfer function gives large output at high frequencies (above 50Hz) and small output at low ones (below 5Hz).

 Today I only had a little bit of time to work in the lab. With the current set up, A La Mode indicated that there was an offset between the beams from the short and long arms as can be seen in this graphic.

Using guess and check, I found that by decreasing the length of the short arm by 5 cm, this could be solved.

We realigned the michelson, and much to our dismay, no fringing appeared at the detector. To rectify this, we tried a few things. First, we replaced the round beam splitter with a cubical one. The round splitter had to be housed in a black mount which meant it was very susceptible to clipping. The cube did not have this problem. After changing this, we still could not see any fringing, but we noticed another problem. Horizontally, the two output beams were not parallel. I began trying to fix this today, and will continue tomorrow.

Today, we started designing the box that will hold all of our circuits.

Particularly, we used a software (Front Panel Designer) to design where to drill the holes on the front and rare panels of our box for our wires to pass through; we will order the designs from a company. Our designs had to be very precise for the plugs and connectors on the PCB boards to fit through the holes.

In the afternoon, we drilled holes on the bottom panel of our box for the "amplifying" PCB boards. At the end of the day, we were almost ready with the amplifying boards. Now, we need to order more spacers and screw the PCB boards on the box.