Measurements on the instrumentation

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1 Measure of the noise of the Voltage Amplifier

Note

All the files (data and Matlab scripts) are accessible here.

1.1 Measurement Description

Goal:

Determine the Voltage Amplifier noise

Setup:

The two inputs (differential) of the voltage amplifier are shunted with 50Ohms
The AC/DC option of the Voltage amplifier is on AC
The low pass filter is set to 1hHz
We measure the output of the voltage amplifier with a 16bits ADC of the Speedgoat

Measurements:

data_003: Ampli OFF
data_004: Ampli ON set to 20dB
data_005: Ampli ON set to 40dB
data_006: Ampli ON set to 60dB

1.2 Load data

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amp_off = load('mat/data_003.mat', 'data'); amp_off = amp_off.data(:, [1,3]);
amp_20d = load('mat/data_004.mat', 'data'); amp_20d = amp_20d.data(:, [1,3]);
amp_40d = load('mat/data_005.mat', 'data'); amp_40d = amp_40d.data(:, [1,3]);
amp_60d = load('mat/data_006.mat', 'data'); amp_60d = amp_60d.data(:, [1,3]);

1.3 Time Domain

The time domain signals are shown on figure 1.

Figure 1: Output of the amplifier

1.4 Frequency Domain

We first compute some parameters that will be used for the PSD computation.

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dt = amp_off(2, 2)-amp_off(1, 2);

Fs = 1/dt; % [Hz]

win = hanning(ceil(10*Fs));

Then we compute the Power Spectral Density using pwelch function.

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[pxoff, f] = pwelch(amp_off(:,1), win, [], [], Fs);
[px20d, ~] = pwelch(amp_20d(:,1), win, [], [], Fs);
[px40d, ~] = pwelch(amp_40d(:,1), win, [], [], Fs);
[px60d, ~] = pwelch(amp_60d(:,1), win, [], [], Fs);

We compute the theoretical ADC noise.

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q = 20/2^16; % quantization
Sq = q^2/12/1000; % PSD of the ADC noise

Finally, the ASD is shown on figure 2.

Figure 2: Amplitude Spectral Density of the measured voltage at the output of the voltage amplifier

1.5 Conclusion

Important

Questions:

Where does those sharp peaks comes from? Can this be due to aliasing?

Noise induced by the voltage amplifiers seems not to be a limiting factor as we have the same noise when they are OFF and ON.

2 Measure of the influence of the AC/DC option on the voltage amplifiers

Note

All the files (data and Matlab scripts) are accessible here.

2.1 Measurement Description

Goal:

Measure the influence of the high-pass filter option of the voltage amplifiers

Setup:

One geophone is located on the marble.
It’s signal goes to two voltage amplifiers with a gain of 60dB.
One voltage amplifier is on the AC option, the other is on the DC option.

Measurements: First measurement (mat/data_014.mat file):

Column	Signal
1	Amplifier 1 with AC option
2	Amplifier 2 with DC option
3	Time

Second measurement (mat/data_015.mat file):

Column	Signal
1	Amplifier 1 with DC option
2	Amplifier 2 with AC option
3	Time

Figure 3: Picture of the two voltages amplifiers

2.2 Load data

We load the data of the z axis of two geophones.

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meas14 = load('mat/data_014.mat', 'data'); meas14 = meas14.data;
meas15 = load('mat/data_015.mat', 'data'); meas15 = meas15.data;

2.3 Time Domain

The signals are shown on figure 4.

Figure 4: Comparison of the signals going through the Voltage amplifiers

2.4 Frequency Domain

We first compute some parameters that will be used for the PSD computation.

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dt = meas14(2, 3)-meas14(1, 3);

Fs = 1/dt; % [Hz]

win = hanning(ceil(10*Fs));

Then we compute the Power Spectral Density using pwelch function.

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[pxamp1ac, f] = pwelch(meas14(:, 1), win, [], [], Fs);
[pxamp2dc, ~] = pwelch(meas14(:, 2), win, [], [], Fs);

[pxamp1dc, ~] = pwelch(meas15(:, 1), win, [], [], Fs);
[pxamp2ac, ~] = pwelch(meas15(:, 2), win, [], [], Fs);

The ASD of the signals are compare on figure 5.

Figure 5: Amplitude Spectral Density of the measured signals

2.5 Conclusion

Important

The voltage amplifiers include some very sharp high pass filters at 1.5Hz (maybe 4th order)
There is a DC offset on the time domain signal because the DC-offset knob was not set to zero

3 Transfer function of the Low Pass Filter

The computation files for this section are accessible here.

3.1 First LPF with a Cut-off frequency of 160Hz

3.1.1 Measurement Description

Goal:

Measure the Low Pass Filter Transfer Function

The values of the components are:

\begin{aligned} R & = 1 k Ω \\ C & = 1 μ F \end{aligned}

Which makes a cut-off frequency of $f_{c} = \frac{1}{R C} = 1000 r a d / s = 160 H z$ .

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\begin{tikzpicture}
  \draw (0,2)
        to [R=\(R\)] ++(2,0) node[circ]
        to ++(2,0)
        ++(-2,0)
        to [C=\(C\)] ++(0,-2) node[circ]
        ++(-2,0)
        to ++(2,0)
        to ++(2,0)
\end{tikzpicture}

Figure 6: Schematic of the Low Pass Filter used

Setup:

We are measuring the signal from from Geophone with a BNC T
On part goes to column 1 through the LPF
The other part goes to column 2 without the LPF

Measurements: mat/data_018.mat:

Column	Signal
1	Amplifier 1 with LPF
2	Amplifier 2
3	Time

Figure 7: Picture of the low pass filter used

3.1.2 Load data

We load the data of the z axis of two geophones.

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data = load('mat/data_018.mat', 'data'); data = data.data;

3.1.3 Transfer function of the LPF

We compute the transfer function from the signal without the LPF to the signal measured with the LPF.

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dt = data(2, 3)-data(1, 3);

Fs = 1/dt; % [Hz]

win = hanning(ceil(10*Fs));

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[Glpf, f] = tfestimate(data(:, 2), data(:, 1), win, [], [], Fs);

We compare this transfer function with a transfer function corresponding to an ideal first order LPF with a cut-off frequency of $1000 r a d / s$ . We obtain the result on figure 8.

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Gth = 1/(1+s/1000)

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figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(f, abs(Glpf));
plot(f, abs(squeeze(freqresp(Gth, f, 'Hz'))));
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude');

ax2 = subplot(2, 1, 2);
hold on;
plot(f, mod(180+180/pi*phase(Glpf), 360)-180);
plot(f, 180/pi*unwrap(angle(squeeze(freqresp(Gth, f, 'Hz')))));
hold off;
set(gca, 'xscale', 'log');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase');

linkaxes([ax1,ax2],'x');
xlim([1, 500]);

Figure 8: Bode Diagram of the measured Low Pass filter and the theoritical one

3.1.4 Conclusion

Important

As we want to measure things up to $500 H z$ , we chose to change the value of the capacitor to obtain a cut-off frequency of $1 k H z$ .

3.2 Second LPF with a Cut-off frequency of 1000Hz

3.2.1 Measurement description

This time, the value are

\begin{aligned} R & = 1 k Ω \\ C & = 150 n F \end{aligned}

Which makes a low pass filter with a cut-off frequency of $f_{c} = 1060 H z$ .

3.2.2 Load data

We load the data of the z axis of two geophones.

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data = load('mat/data_019.mat', 'data'); data = data.data;

3.2.3 Transfer function of the LPF

We compute the transfer function from the signal without the LPF to the signal measured with the LPF.

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dt = data(2, 3)-data(1, 3);

Fs = 1/dt; % [Hz]

win = hanning(ceil(10*Fs));

Copy
[Glpf, f] = tfestimate(data(:, 2), data(:, 1), win, [], [], Fs);

We compare this transfer function with a transfer function corresponding to an ideal first order LPF with a cut-off frequency of $1060 H z$ . We obtain the result on figure 9.

Copy
Gth = 1/(1+s/1060/2/pi);

Copy
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(f, abs(Glpf));
plot(f, abs(squeeze(freqresp(Gth, f, 'Hz'))));
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude');

ax2 = subplot(2, 1, 2);
hold on;
plot(f, mod(180+180/pi*phase(Glpf), 360)-180);
plot(f, 180/pi*unwrap(angle(squeeze(freqresp(Gth, f, 'Hz')))));
hold off;
set(gca, 'xscale', 'log');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
xlabel('Frequency [Hz]'); ylabel('Phase');

linkaxes([ax1,ax2],'x');
xlim([1, 500]);

Figure 9: Bode Diagram of the measured Low Pass filter and the theoritical one

3.2.4 Conclusion

Important

The added LPF has the expected behavior.