Updates:

Yesterday, I stuffed the pcb boards for the hall-effect sensors (Allegro A1301 ),
and also one quadrant photodiode circuit for testing.

   [Hall effect sensor]             [QPD (before stuffed)]          [QPD stuffed (front side)]

The photodiode used is Hammatsu Si Pin photodiode S4349, and the operational amplifier
is Analog Device ADA4004-4 1.8 nV/√Hz, 36 V Precision Amplifiers. Here I attached the
schematics and the pcb layout for the QPD, which might be useful for others in other applications.
The zip file goes as follows:


Issues found (some small modifications are needed):

1. The 1/4-20 tapped holes of the fixed plate is too tight and the screws cannot go through.
2. The 0.5inch hole on the floating plate is a little bit too small and the reflector cannot be fitted in.
3. There is also a tiny problem with the slot (wrong sizing) for fixing the magnet.

Things to be done this week:

1.  Design Signal conditioning circuits (some simple filtering and amplification) for the hall-effect sensors,
   coils and also the linear DC motors.
2. Stuff the chassis power board for the binary input and output.

List of items needed (or to be ordered):

1. A rack for fixing various chassis [the analogy and digital parts].
2. Three 1-u chassis boxes for the signal conditional circuits, the front panel (BNC connectors)
   and the back panel (D-sub connectors slots).

Update:

1. Got the rack settled and tidied up the lab a little bit.
2. Worked on the chassis power regulator for binary input and output.
3. Sketched a timeline for the project before the Christmas.

Near-term timeline:

During the discussion with Koji this afternoon, he suggested to sketch a timeline for coordinating the progress of different modules in order to have a constant flow of progress. Below is a schematic flow chart for the final setup:

Here I first divide the parts in the flow chart into five modules and show who are responsible for each of them:

I then estimate approximately the time needed for completing individual module before Christmas:

This cannot be precise, as there are various uncertainties, but it does give us some feelings about what need to be done in the near term, and how we should coordinate the progress of different modules.

The chassis power part:

The DC power for the binary output box is ±18V. I previously assumed that the power to be ±15V, and naively took the original design by Abbott [LIGO-D1000217]. To correct this mistake, small modifications needs to be made to the resistors for voltage dividing (R2 in the figure below). For positive voltage [we are using adjustable voltage regulator LM2941 (datasheet is attached)],

Here V_REF = 1.275V and R2 was chosen to be 10.7K in the original design to obtain +15V, and it needs to be changed to 13.1K to create +18V.

For the negative voltage [we are using LM2991 (datasheet is attached)]:

Here V_REF=1.21V and R2 was chosen to be 11.3K for -15V, and it needs to be changed to 13.9K to create -18V.

The back panel:

I use the Front Panel Designer (from Front Panel Express) to design the back panel for the box [as shown below].

Apart from the hole for the 37port D-sub connector, the other parts are identical to DC Power Wiring Details for CDS 19 Inch Chassis
[LIGO T-1100079 for the details]. It has a Conec 3W3 connector, two panel-mount LEDs, one E-T-A breaker.

The design file is attached here:

I just found today that the Front Panel Express actually accepts customer provided panel for fabrication. Basically, the chassis box we bought includes both back and front panels, and they fit much better to the box than those we got from Front Panel Express (Normally, one has to drill additional mounting holes on the sides, which are not very nice). We can simply send them to Front Panel Express.

During last few days, I designed the signal conditioning circuit for the hall effect sensors maglev. It mainly contains two parts:

1. The constant gain part.

2. The dewhitening part. It contains two types of high pass filters: one has a zero at 0.5 Hz and pole at 5 Hz, the other one has zero at 5Hz and pole at 50Hz. Due to the uncertainty in the shape of the signal (the floating plate motion), I put them in series (add one additional place holder) and also add jumpers to bypass the intermediate stage if necessary for possible modifications.

The schematics is shown in the attached pdf file: [signal_conditioning_hall_effect_sensor_2channels.pdf].

The Altium file for the schematics and pcb layout is also attached [signal_conditioning_hall_effect_sensor.zip], which uses the multiple channel design idea.

Here is a general signal conditioning circuit for whitening and dewhitening of the signal from the sensor and actuator (multiple channels) in maglev. Previously, I was designing different circuits for whitening and dewhitening. Koji pointed out that by manipulating the zero and poles, we can realize them with the same circuit by choosing the proper values for the resistors and capacitors. In addition, by bypassing certain stages, we can use one type of PCB board for the sensor (the hall-effect sensor and quadrant photodiode), and the actuator (the coil).

The pcb board and the associated alitum file (altium_file.zip) are attached.

This circuit contains five stages, and each can be bypassed by using a three-terminal header and jumper:

The first one is to set the DC offset.

The next three are generic filters, each with one zero and two poles.

(different footprints for the capacitor for a generic purpose)

I will explain the detail in the additional information part appended to this elog.

The last stage is a current boost for coil drive.

-------Additional information----------

In the below, I will briefly explain the idea of the filter part:

The transfer function for this circuit is given by

s0: 1/C1(R1+R2)

s1: 1/(C1 R2)  

s2: 1/(C2 R3)

At very low frequency, the gain is determined by -(R3/R1), which is chosen to be -1 in our case. Given different values for s0, s1, and s2, we can have high-pass filter (s2>s1>s0) [cut-off above s2], or a low-pass filter (s1>s0>s2) [cut-off above s1].

For example, in our case, we have chosen

1. A high-pass filter (more precisely, a bandpass): s0/2π = 0.5 Hz, s1/2π = 5 Hz, and s2/2π = 200 Hz [cut-off].
   The parameters for the components are: R1 = R3 = 28.6 KΩ, R2 = 3.2 KΩ, C1 = 10μF, C2 = 27.8nF
   The amplitude is shown by the figure below:
   
   
2. A low-pass filter: s2/2π = 0.5 Hz, s0/2π = 5 Hz, and s1/2π = 200 Hz [cut off].
   The parameters for the components are: R1 = R3 = 32 KΩ, R2 = 820 Ω, C1 = 0.97 μF, C2 = 10 μF.
   The amplitude is shown by the figure below:
   
   
3. To balance the whitening and dewhitening in the entire loop, we have chosen the zero and poles such that the product of the above two filters is close to unity at low frequencies, as shown by the figure below:
   

- I forgot why you don't have a voltage reference (cf. AD587) for the offset subtraction.

- Don't you want to put an output impedance at the output of the current driver?

 > For your first comment, you are right. I am diverting some voltage from the power, which has a huge noise. This is rather bad.

> For your second comment, it turns out that my coils have quite high impedance, e.g., of the order of hundred Ohm. The maximal current, given the maximal voltage 15V, is smaller than what the buffer can provide. But it is good to add an output impedance for flexibility and also limits the current.

Thank you. I will implement these comments in the next revision of the PCB.

I implemented both of your comments in this revision. The altium file is also attached. Thanks.

Here are the front and rear panels for the signal conditioning boxes. The front-panel files are attached.

For the coils:

For the QPD:

For the Hall-effect sensors:

For the linear motors [using simple DC control]:
Small panel-mounted voltage meter reads out the force gauge signal that indicates the weight of the floating plate, from which we know roughly how far we are away from the working point where the gravity is balanced by the DC magnetic force. We use two on-mom switches to control the linear motor (up and down).

We started by covering the basics of op-amps and their gain. We then talked about differential amplifiers and calculated V_out, in terms of the four resistances of the circuit and Vin₁, Vin₂. We found the gain (-R4/R3) and the offset, which should ideally be zero. For a gain of about 50, we introduced an uncertainty for the R₂ resistor, which in turns leads to an uncertainty for the output voltage:

ΔV/V=(ΔR₂)/R₂ ∙(1/G)

Then, we calculated the resistances for the desired outcome and wired the circuit on the PCB board for signal conditioning, powered it and made sure it gave sensible readings on a digital screen that read the output voltage.

In fact, for one board Vin=0.012V and Vout=0.058V and another Vin=0.014 and Vout=0.69 (where Vin(2)=5V), showing a gain of roughly fifty. At the end, we connected one strain gauge "wire" that supports the levitated plate to our Wheatstone bridge circuitry and read an output voltage of about 144mV (ideally -and with some feedback- it will be 0). Pressing on the strain gauge wire increased our voltage reading, clearly showing that our system responds to pressuring the strain gauge wire.

Tomorrow, we will replace R₂ with an adjustable resistor, until we get our voltage reading to be 0 (or close to that). After we find our value, we will use a normal resistor as R2 and finish our system.

I am still trying to find how to insert equations

On Friday, we tried to get zero offset for the voltage output of the strain gauge, by replacing our fixed-value R₂ resistors with adjustable ones. At R₂=22.74kΩ, we achieved a 0.01mV offset, which we would then correct with our DC motors.

We measured the voltage required to drive our motors at a slow, steady pace; 5V seemed sufficient. To get that voltage, we used the voltage divider method. The internal resistance of the motors were roughly 100Ω, so we added a 500Ω resistor in series to get close to our desired V_out, using our 15V source. Our motors were "broken" and would however move at all times once connected and pause only when the control buttons were pressed; we ordered new ones.

Since we successfully built one circuit for one strain gauge and DC motor, we wired the rest 5 PCB boards. At the end of the day, we were happy to look at the following:

We wanted to wire up different PCB boards aiming to power our PCB boards that are connected to the strain gauge and amplify the signal. We looked at the schematics and picked resistors' values so that we have negative (using LM2991 adjustable regulator) and positive (with a LM2941 adjustable regulator) voltage inputs of -15V and +15V respectively. We built six of these PCB boards, missing only very few components, which we ordered.

Then, we looked at the box in which we will place the PCB "amplifying" boards. We also figured out how to drill holes on the cover of the box to get all the wires through. Our final measurements are shown below:

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.

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, 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, 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.

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.

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)

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.

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).

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.

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

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)

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.

 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).

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.

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.

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.

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.

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.

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

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.

 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.

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).

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.

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.

Cross-Coupling Measurements

Before measurig the cross-coupling effects, we put the plate back to our setup. Since the plate has magnets attached to it, we expect that it will affect the magnetic field produced by the coils. Therefore, a more realistic measurement of the cross-coupling effect that will resemble the real-time feedback control should take place in the presence of the plate. Of course, we want the plate to be still during the measurements, so that our Hall-effect sensors are not affected by their displacement (if the plate is still, it will only affect the reading of the sensors at 0 frequency, but we are looking at the transfer function over the whole frequency range). To fix the plate, we pressed all motors against it, ignoring any small fluctuations.

VECTOR FITTING

In this configuration, we used Diagnostic Tools software to measure the transfer function beween ACs coils' output and ACs sensors' reading. Then, we used Vector Fitting to obtain a fitting transfer function with zeros, poles, and gain that resembles the behavior of the cross-coupling. In few words, Vector Fitting uses a guess function σ(s)=, whose zeros are the poles of the desired function

. Finding the zeros of σ(s) equals to finding the poles of the function and vector fitting manages to find a linear least-squares fit model by trying values for the poles and solving for the linear coefficients. More information can be found at http://www.sintef.no/Projectweb/VECTFIT/Algorithm/.

We developed a code for the vector fitting and found the best model with 20 pairs of poles (40 poles, since they are complex conjugates) and about 10 iterations. At the end, we applied compensation feedback given by the modelling transfer function so as to cancel the cross-coupling effects.Below, one can see the cross-coupling measurements before (left figure) and after (right figure) applying the modelled function. We modelled the coupling transfer functions only for ACi_Sensor - ACi_coil (i=1,2,3) pairs, because cross-coupling between e.g. AC1 coil and AC2 sensor was very small (as is evident in the figures). Using this technique, we minimized coupling effects from roughly -10dB to roughly -70dB.

After minimizing the coupling effects, I created a 1 DOF Simulink Model to test whether the setup is stable in the vertical direction. Below appears the model. We have not included yet Eddy Current Damping (simulations showed no great changes).

We create noise with a pulse generator and recorded the step response of the system with the oscilloscope, which appears at Output (1). 

The behavior of the system appears stable and so we tried to levitate the plate starting from AC3. The plate would jiggle back and forth back after some time the DAC would saturate. Going back to the Simulink Model, the relationship between saturation limits and noise amplitude seem to determine the stability of the system as well. Looking at other factors that might affect our system, we consider the following:

• The force of the gain directly depends on the location of the plate. The coils are not strong enough and the magnetic field created is non-uniform around the plate. Prior to this summer, Haixing had acquired measurements for the correlation between the plate's distance from the coils and the force exerted on it and we intend to use this information in our model.
• ADC and DAC background noise may limit the sensitivity of our system
• We need a more precise measurement of the system's rigidity. For a linear degree of freedom, such as the vertical direction, the rigidity can be measured as F/distance. Haixing's measuremts of the force exerted as a function of the distance from the plate will be used.

Instability due to Saturation Issues

Using the Simulink model, I realized that saturation of the ADC & DAC prevents us from acquiring stability, since the signal quickly builds up after a couple of turns within the feedback loop. Explicitly, if the gain of the digital feedback filter is large, either the output signal from the DAC saturates or a big feedback signal is produced, which then -together with the plate- saturates the ADC. To prevent saturation of the DAC, I suggested implementing our gain outside the feedback filter, at the coil signal conditioning stage. Similarly, to avoid saturation of the ADC, the gain of the sensors signal conditioning stage is to be altered. At the end, we are looking for a stable system with balanced signal that effects a maximum vertical displacement of the plate around 0.1mm.

Gains of the feedback loop

There are three different gains in our feedback loop. The gain of the digital feedback filter, the gain of the sensors signal conditioning (also constrained by saturation limits around 12V because of the OP27), and the gain of the coils signal conditioning which acts twice; through the coupling and the plate. I calculated the desired gain for each part, such that the plate will ideally not move beyond 0.1mm. To achieve this, we introduced a gain of 25 in the coils conditioning board (changed resistors to R6=51kΩ and R5=2kΩ) which -along with the 100nF capacitor used- effects a new pole around 30Hz (1/RC=200 rad/s). Modeling the new configuration for very low-noise power(0.003Nm/s), we get stable behavior.

Including bigger noise (0.03Nm/s), however, destabilizes the system once more.

In my previous post, I explained the saturation issues of the ADC and DAC we faced. To prevent saturation of the DAC, we will implement our gain after the feedback filter -we already introduced a gain of 25 at the coil's conditioning stage-and use a gain of approximately 1/25 for the digital filter. In this way, even if the ADC saturates at 10V, the feedback filter will send maximum 400mV (10V/25) to the DAC. However, the possibility that the ADC saturates still exists.

To make matters worse, we cannot even exploit the 10V saturation range of our ADC. The reason is that the undesired cross-coupling between coils and sensors is almost as large as the signal produced by the plate. Although vector fitting was very successful at cancelling cross-coupling, this method can only be implemented inside the digital control, after the ADC. That been said, if we are to stay within the 10V range, only 5V can come from the plate; the rest 5V will inevitably come from cross-coupling. Practically, this means that the signal from the plate is successfully sensed and transferred to the digital control for very small displacements, beyond which the ADC saturates and feedback control is impossible. We use two ways to tackle this obstacle:

1. We revert back to using DC coils for feedback control. They are located further away from the AC sensors and cross-coupling is smaller, such that a larger proportion of the ADC' 10V range can be dedicated to the signal from the plate.
2. We use mu-metal to cover all sensors and coils. We hope that mu metal's high permeability will shield off the magnetic field that comes from cross-coupling. It should also leave the magnetic field changes produced by the plate's movement almost intact, since the plate's magnets are located directly below and above the sensors, the only place we did not cover with mu metal.

We measured cross-coupling effects before and after the use of mu metal and noticed a drastic reduction in cross-coupling (about a factor of 3). The figures below show the measured transfer function before and after the use of mu-metal.

One of the things to also note is that the bottom strain gauge motors readings are damaged and no longer worth looking at to determine when the plate is near equilibrium. This is probably the result of excessive weight on them from the plate. The top strain gauge motors seem to still be working fine.

DCs/ACs Matrices

In the previous post ("Feedback through DC coils"), I described the reasons for using DC coils for our feedback control. This trick significantly decreases cross-coupling before the feedback control and so allows us to afford a larger signal from the plate, without the ADC ever saturating. However, it comes with a cost; the feedback force is not applied directly at our sensors. Specifically, the DC coils greatly affect two of the sensors surrounding it -that are equally far. That creates the need to develop a matrix to calculate how the force from each DC coil affects each AC sensor and, further, how each sensor should "distribute" its signal to different DC coils in the feedback control. Using the geometry of our plate and the arrangement of the coils and sensors, I calculated these two matrices, which should also be inverse of each other. Here is what I found:

To measure how much of the coils' signal each sensor reads: [A][DC1;DC2;DC3]=[AC1;AC2;AC3], where A=[2 -1 2; 2 2 -1; -1 2 2]. Similarly, to convert each sensor's signal to the feedback channels of the DC coils, [B][AC1;AC2;AC3]=[DC1;DC2;DC3], where B=[2 2 -1; -1 2 2;2 -1 2].

Simulink for 3 DoF

Then, I created a Simulink model for three degrees of freedom (sensors: AC1-3, coils: DC1-3). The effect of noise was catastrophical for the stability of the model. So, I first tried to replicate 1 DoF stability, by "nulling" DC2 and DC3 coils. Below, one can see the Bode Plots of the model (resonance around 2Hz) and the displacement of the plate in 1DoF (by AC1).

Attached is also the ADC input, which clearly shows how the signal is well below saturation. Three different colous represent the signal from the three sensors AC1, AC2, AC3 (they are all affected by DC1 coil).

Today, I achieved stability in Simulink for 3DoF, including noise to the hall-effect sensors and the coil's conditioning. We had measured the noise at the ADC to be max 20mV, but that value is amplified by the gain (91) of the HE conditioning boards. So, I included noise of 20/91 mV. I attached the final model and the script.   .

I also used vector fitting to find the transfer functions of the coupling between DC1 coil to all sensors. An example of the successful resemblance is shown in the figures below (DC1 coil to W sensor). The figure on the left shows the modelling of the coupling and the deviation between the fitting and the data. The right figure shows a body plot of the modelled and measured transfer function.

I also calculated the amount of cross-coupling and noise inside our system in order to find the allowed gain to avoid any saturation.

Since the OP27 in the coil conditioning board also saturates at 10V, the DAC should provide no more than 400mV; beyond that point, the gain of 25 we introduced in the coils would saturate the OP27.

We had also found the cross coupling to be around 0.01 for two nearby sensors and 0.001 for the third one (in the 3DoF case we ignore all others). If our DAC never exceeds 400mV, cross coupling would get at most 0.0084V (8.4mV) at the ADC.
Similarly, if the coils get at most 10V, the maximum force they provide is 0.02N, which translates to 0.0113m (1.13cm) maximum displacement of the plate. Such displacement would produce 0.1989V at the ADC. Adding noise to these, our signal is only 0.2273V, well below the saturation of the ADC.
Inside the feedback filter, the cross coupling is cancelled down to -60dB (0.001V/V), so only 0.0012V remains, given a 400mV DAC output). The signal is thus 0.2201V. To avoid saturation of the DAC, we can afford a maximum gain of 1.8.

The hall effect sensor is quite noisy, and I am trying to find where the noise comes from. The first I tried today is to measure the power spectrum and coherence among three hall effect sensors (in the vertical direction). Here is the result:

(the unit for the power spectrum density is in digital volt per root hertz.)

I do not quite understand why there is almost no coherence (apart from the 60Hz power line), even though the power spectra are almost identical among these sensors.

Can someone shed some light where the issue is? Is the noise non-stationary or what?

--------Another measurement with fewer average---------------------

It seems that when the number of average is small, the coherence is large, an indication of non-stationary?

Yesterday, we tried levitating for one degree of freedom; we failed. The plate would move back and forth the equilibrium, but not settle there for more than a second. Haixing suggested moving the sensors closer to the plate, so that our signal is larger. This way, we can be less sensitivity to noise. However, that entails that we recalibrate the HE offset, the cross coupling measurement and the gains of our feedback loop.

So, today we moved the AC1, AC2, AC3 sensors closer to the plate and removed the DC1, DC2, DC3 sensors from our setup, since they are of no use. We also replaced the 91 gain of the HE sensors with a gain of 11 (11k and 1k are the resistors we used).
Then, we measured the response of the HE sensors and found that the signal produced changes by about 120mV for each mm; that is about a factor of 6 better than before.

Since the sensors are now at a different location, they have a different voltage offset; this should be larger since they are now closer to the levitated plate. To measure the new offset, we moved the motors such that the equilibrium would lie between the top and bottom motors. Then, we measured the offset reading of the sensors when the plate was stuck at the top and at the bottom and averaged the two. We used our measurements and, by taking into account the offset we had already applied in the first place, we replaced the resistors in our circuit such that the offset of the AC1-3 sensors would be no greater than 50mV. For AC1: R1=6.2K, R2=5.1K, for AC2: R1=13K, R2=11K, for AC3: R1=20K, R2=16K.

NEW HE OFFSET

We moved all 7 HE sensors closer to the plate. The AC sensors were burnt out and we replaced them with the DC ones, which we had removed yesterday because they were of no use.
The sensors in their current position feel different magnetic field and their offset is also different (around 100mV, compared to 2.4V before). The smaller value of the voltage offset sounds counterintuitive--because the sensors are now closer and the field around them larger--but it only indicates the direction of the field. The HE sensors read 2.5 at the presence of no field; far away their offset was 2.4V, so they only produced 100mV (2.5V - 2.4V) signal. Now that they are closer, their offset is huge (roughly 2.5V). We are worried they might actually saturate, once the plate starts moving.

For future reference, here are the values of the resistors we use:

AC1: R1= 110K, R2=2.2K

AC2: R1=130K, R2=2.2K

AC3: R1=62K, R2=1.2K

S1: R1=7.5K, R2=2.2K

S2: R1=20K, R2=7.5K

N: R1=20K, R2=6.2K

E: R1=4.3K, R2=1.5K

Noise reduction and debated Cross-Croupling

The new gain for all sensors conditioning board is now 11. The noise would therefore be amplified less (by a factor of 9). Indeed, the improve noise effects were apparent, since signal fluctuated only a couple of mV (see Figure below).

The cross coupling between sensors would now also be smaller (200mV/V for feedback through AC coils would now become almost 20mV; 20mV/V for feedback through DC coils would now be about 2mV/V).
We will measure cross coupling and try to levitate tomorrow for all six degrees of freedom

We levitate the plate by controlling three vertical degrees of freedom---one translational, and two tilts. Right now even without applying controls to the horizontal direction, the system is quite stable.

Here we describe the major steps for achieving levitation:

1. Move the plate close to the equilibrium point by using the DC motors and strain gauge.

We started off from the point where the magnetic force is stronger than the gravity force. Basically, the plate is touching tips of DC motors mounted on the top fixed platform. By using the strain gauge attached to the tips, we can tell how how far we are away from the equilibrium point where the strain gauges are not stretched. We slowly actuate the motors to push the plate close to the equilibrium point.

2. Lock one degree of freedom

We started from some generic PID controller. After many random trials, we end up with a controller that was barely doing the work, and the system was marginally stable. We then measured the closed-loop transfer function of the system and use simulink to model this one degree of freedom: and

After tuning various parameters, we got a reasonable match between the model and the measurement. In particular, the parameters we found go as follows:

mass    = 0.5 kg        % mass
K          = -180 N/m   % negative spring constant
Vm       = 250 V/m    % hall effect sensitivity
Gby      = -27dB       % the bypass gain from the coil to the hall effect sensor
Gcomp = -39.5dB    % the residue bypass gain after compensation

To obtain a better controller, we tried to use "sltunable" class in the matlab, in particular using the functions "systune" and "looptune" by specifing the target phase and gain margin. Somehow, it did not produce the desired result. We found out that the calculated phase margin and gain margin are not correct (we will explain this in more details with another elog entry). We then used "nyquist" function to design the feedback loop and we used the following parameters for the controller, which gives a reasonable phase and gain margins.

DGain = -4*pi*13;                 % the constant gain
Dzs     = [-2*pi*0.5; -2*pi*3]; % zeros for the controller
Dps     = [0;-2*pi*13];            % poles for the controller

After applying this new controller, we got a quite stable levitation. We redid the open-loop transfer function measurement. The agreement between the model and the system is shown by the figure below (the blue line is the model and the green line is the measurement data):

The difference around 4Hz and 20Hz could arise from a simplified model for the compensation (assuming a constant compensation gain, and the reality is more complicated), and we need to refine the model.

2. Repeat the same process and tune the feedback controllers for the other two degrees of freedom.

3. Slowly ramp up the gain for the other two controllers and finally levitate all three DOFs.

We found that if we engage the controllers abruptly, the system will rapidly destabilized. Instead, we had to slowly ramp up the gain of the controllers so as to approach the final stable state softly.

The figure below shows the signal from all seven sensors (one redundant in the horizontal direction).

We can see the resonant frequency is around 2Hz (to be confirmed by future TF measurements), which is quite high. After fully characterizing the system, we will need to tune the DC magnet force with DC control coils to make the equilibrium point closer to the force maximum, where the rigidity is low.

------------------------

Here is the link to the video of the levitating plate:  link

Since the plate is levitating, we are now in the position for real work. Here are the two major issues to investigate in the plan.

1. TF of the plate and cross coupling among different degrees of freedom (important in order to optimize the control servo)

We will measure various transfer functions to characterize the plate TF and cross couplings. We will build six degrees of freedom simulink model based on Georgios's work of three DOFs, and try to make a match between the model and the system.

2. Noise budget (to pin down the major noise source)

(a) sensing and actuation noise

We will calibrate the noise from the hall effect sensor. If it is confirmed to be the major noise, we can switch to the optical lever sensing scheme as planned. The coil is quite weak, in terms of voltage to force conversion factor, and it is 5mN/V. The thermal noise of the coil may not be important (to be confirmed with more rigorous analysis).

(c) acoustic noise

Right now the system is exposed in air, and it is anticipated that the acoustic noise is quite significant. To mitigate this noise, we can use a bell jar to cover it which can give a reasonable level of noise isolation.

(b) seismic noise

We will make a correlation measurement between the sensor output and the seismometer (or accelerometer) to see where the seismic noise dominates.

(d) ambient magnetic field noise

We will use two low-noise honeywell hall effect sensors [link] to measure the ambient magnetic field. To get a better sensitivity, we will use differential measurement by shielding one (together with instrumentation amplifier for amplifying the readout).

(e) thermal noise of the magnet

The major noise source comes from the random jitering of the magnetic moments due to thermal excitation. We can find the literature on how to analyze this kind of noise.

(f) long-term drift

We know little about the long stability of the magnets and also how the temperature drift affects the magnets. This needs to be investigated.

I used the matlab function margin() to plot the phase and gain margins for the open-loop transfer function for maglev. It seems to give an incorrect answer. Here is what I got:

As the gain margin is negative, this indicates that the system (plant + controller) is unstable. However, this is not the case.

I used the matlab function nyquist() to make a Nyquist plot, and this is what I got:

The contour circles -1 counter-clock wise once, and this satisfies the Nyquist stability criterion, as the plant (in my case the plate can be modeled as a mechanical object attached to a negative spring) has one pole on the right-half complex plane. Basically, my plant together with the controller in indeed stable, which is also the reality.

Therefore, this seems to indicate that nyquist(), instead of margin() is the right way to examine the stability in the case with an unstable plant in matlab.

I attempted a matlab simulation for closing the loops on the triple suspensions. The system (hsts) was imported from https://git.ligo.org/jenne-driggers/SUS.
The active damping filters were read for the PR3 optic at LLO. 
There were a few glitches which were accounted for. 

- The complex pairs of poles and zeros differed ever so slightly in their conjugate parts. An example is given in Attachment 3. The real and imaginary values were rounded off to the 10th place
- The closed loop system was unstable. Brett suggested accounting for the electronic gains which were obtained from https://svn.ligo.caltech.edu/svn/sus/trunk/HSTS/L1/MC1/SAGM1/Results/2012-05-08_1700_L1SUSMC1_M1_ALL_TFs.pdf. A factor of 34.1 was accounted for all DOFs.
- The matlab version uses connect for closing the loops. Even after accounting for the gain, there were a few RHP poles (smaller now). 
- A simulink version for the system was created which essentially did the function of "connect" in matlab and it was simulated. This closed loop system was stable. A comparison of the Bode plots (m1->drive->DOF -------> m1->disp->DOF) is given in Attachment 1.
(forgot to wrap the phase for the Bode plots, will change to the corrected version tomorrow)
- The step responses fo the simulink plant for (m1->drive->DOF -------> m1->disp->DOF) were generated (Attachment 2).

Next Steps:

- Need to check for the inconsistencies between the two simulation methods
- Can proceed to add ports for noise inputs
- Need to verify the current electronic gain calibrations at LLO for the optic (if available)
 

Closer look at the damping siutation.
tldr: Corrected electronics gain, Signs of filters flipped, AND THE CORRECT MODEL (HLTS) WAS USED.

- I looked around the SVN to find the recent electronics gain. To begin with, the gains applied previous to the simulation were incorrect (MC1, 2012, e_k = 34.1). This was corrected to (PR3, 2023, e_k = 1.5404): https://svn.ligo.caltech.edu/svn/sus/trunk/HLTS/L1/PR3/SAGM1/Results/2023_07_20_1100_L1SUSPR3_M1_ALL_TFs.pdf

- With the new gain, the system was unstable. I wanted to check the closed loop behavior of each individual filter to see where the instabilty was rising from. I checked the matlab and simulink simulations for each filter turned on individually and flipping signs of feedback sequentially.

(0 = unstable, 1 = stable)

While writing this elog, I realized that PR3 was a large suspension and I needed to use the HLTS model instead of the HSTS model. I updated the model and poof, the closed loop system is stable. Matlab and simulink agree with each other now so I'll proceed with using Matlab now.

 (m1->drive->DOF -------> m1->disp->DOF) 
- Bode Plots: Attachment 1
- Step Responses: Attachment 2