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<h1 class="title">Measurements on the instrumentation</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org9b0dd42">1. Measure of the noise of the Voltage Amplifier</a>
<ul>
<li><a href="#org494f3b2">1.1. Measurement Description</a></li>
<li><a href="#orgf9402f3">1.2. Load data</a></li>
<li><a href="#orgd0212e2">1.3. Time Domain</a></li>
<li><a href="#org18fa6ce">1.4. Frequency Domain</a></li>
<li><a href="#org36eaef7">1.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org15e4c95">2. Measure of the influence of the AC/DC option on the voltage amplifiers</a>
<ul>
<li><a href="#org313898e">2.1. Measurement Description</a></li>
<li><a href="#org86b59a1">2.2. Load data</a></li>
<li><a href="#org22f801c">2.3. Time Domain</a></li>
<li><a href="#orgbd72638">2.4. Frequency Domain</a></li>
<li><a href="#org2647596">2.5. Conclusion</a></li>
</ul>
</li>
<li><a href="#org9623ca1">3. Transfer function of the Low Pass Filter</a>
<ul>
<li><a href="#org921fe36">3.1. First LPF with a Cut-off frequency of 160Hz</a>
<ul>
<li><a href="#org122c7a6">3.1.1. Measurement Description</a></li>
<li><a href="#orgba94437">3.1.2. Load data</a></li>
<li><a href="#org087582c">3.1.3. Transfer function of the LPF</a></li>
<li><a href="#orgf2dc541">3.1.4. Conclusion</a></li>
</ul>
</li>
<li><a href="#orgd6132bd">3.2. Second LPF with a Cut-off frequency of 1000Hz</a>
<ul>
<li><a href="#org76b4e0d">3.2.1. Measurement description</a></li>
<li><a href="#org2941697">3.2.2. Load data</a></li>
<li><a href="#org45462d2">3.2.3. Transfer function of the LPF</a></li>
<li><a href="#org438a172">3.2.4. Conclusion</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-org9b0dd42" class="outline-2">
<h2 id="org9b0dd42"><span class="section-number-2">1</span> Measure of the noise of the Voltage Amplifier</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgbd691e3"></a>
</p>
<p>
The data and matlab files are accessible <a href="data/meas_volt_amp.zip">here</a>.
</p>
</div>
<div id="outline-container-org494f3b2" class="outline-3">
<h3 id="org494f3b2"><span class="section-number-3">1.1</span> Measurement Description</h3>
<div class="outline-text-3" id="text-1-1">
<p>
<b>Goal</b>:
</p>
<ul class="org-ul">
<li>Determine the Voltage Amplifier noise</li>
</ul>
<p>
<b>Setup</b>:
</p>
<ul class="org-ul">
<li>The two inputs (differential) of the voltage amplifier are shunted with 50Ohms</li>
<li>The AC/DC option of the Voltage amplifier is on AC</li>
<li>The low pass filter is set to 1hHz</li>
<li>We measure the output of the voltage amplifier with a 16bits ADC of the Speedgoat</li>
</ul>
<p>
<b>Measurements</b>:
</p>
<ul class="org-ul">
<li><code>data_003</code>: Ampli OFF</li>
<li><code>data_004</code>: Ampli ON set to 20dB</li>
<li><code>data_005</code>: Ampli ON set to 40dB</li>
<li><code>data_006</code>: Ampli ON set to 60dB</li>
</ul>
</div>
</div>
<div id="outline-container-orgf9402f3" class="outline-3">
<h3 id="orgf9402f3"><span class="section-number-3">1.2</span> Load data</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab">amp_off = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_003.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; amp_off = amp_off.data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>,<span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
amp_20d = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_004.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; amp_20d = amp_20d.data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>,<span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
amp_40d = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_005.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; amp_40d = amp_40d.data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>,<span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
amp_60d = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_006.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; amp_60d = amp_60d.data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-type">:</span>, <span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>,<span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
</div>
</div>
<div id="outline-container-orgd0212e2" class="outline-3">
<h3 id="orgd0212e2"><span class="section-number-3">1.3</span> Time Domain</h3>
<div class="outline-text-3" id="text-1-3">
<p>
The time domain signals are shown on figure <a href="#org98f097d">1</a>.
</p>
<div id="org98f097d" class="figure">
<p><img src="figs/ampli_noise_time.png" alt="ampli_noise_time.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Output of the amplifier</p>
</div>
</div>
</div>
<div id="outline-container-org18fa6ce" class="outline-3">
<h3 id="org18fa6ce"><span class="section-number-3">1.4</span> Frequency Domain</h3>
<div class="outline-text-3" id="text-1-4">
<p>
We first compute some parameters that will be used for the PSD computation.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dt = amp_off<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>amp_off<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Then we compute the Power Spectral Density using <code>pwelch</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pxoff, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>amp_off<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>px20d, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>amp_20d<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>px40d, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>amp_40d<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>px60d, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>amp_60d<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>,<span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
We compute the theoretical ADC noise.
</p>
<div class="org-src-container">
<pre class="src src-matlab">q = <span class="org-highlight-numbers-number">20</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">^</span><span class="org-highlight-numbers-number">16</span>; <span class="org-comment">% quantization</span>
Sq = q<span class="org-type">^</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-highlight-numbers-number">12</span><span class="org-type">/</span><span class="org-highlight-numbers-number">1000</span>; <span class="org-comment">% PSD of the ADC noise</span>
</pre>
</div>
<p>
Finally, the ASD is shown on figure <a href="#orga4734ea">2</a>.
</p>
<div id="orga4734ea" class="figure">
<p><img src="figs/ampli_noise_psd.png" alt="ampli_noise_psd.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Amplitude Spectral Density of the measured voltage at the output of the voltage amplifier</p>
</div>
</div>
</div>
<div id="outline-container-org36eaef7" class="outline-3">
<h3 id="org36eaef7"><span class="section-number-3">1.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-1-5">
<div class="important">
<p>
<b>Questions</b>:
</p>
<ul class="org-ul">
<li>Where does those sharp peaks comes from? Can this be due to aliasing?</li>
</ul>
<p>
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.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org15e4c95" class="outline-2">
<h2 id="org15e4c95"><span class="section-number-2">2</span> Measure of the influence of the AC/DC option on the voltage amplifiers</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="org1e85a61"></a>
</p>
<p>
The data and matlab files are accessible <a href="data/meas_noise_ac_dc.zip">here</a>.
</p>
</div>
<div id="outline-container-org313898e" class="outline-3">
<h3 id="org313898e"><span class="section-number-3">2.1</span> Measurement Description</h3>
<div class="outline-text-3" id="text-2-1">
<p>
<b>Goal</b>:
</p>
<ul class="org-ul">
<li>Measure the influence of the high-pass filter option of the voltage amplifiers</li>
</ul>
<p>
<b>Setup</b>:
</p>
<ul class="org-ul">
<li>One geophone is located on the marble.</li>
<li>It's signal goes to two voltage amplifiers with a gain of 60dB.</li>
<li>One voltage amplifier is on the AC option, the other is on the DC option.</li>
</ul>
<p>
<b>Measurements</b>:
First measurement (<code>mat/data_014.mat</code> file):
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Column</th>
<th scope="col" class="org-left">Signal</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1</td>
<td class="org-left">Amplifier 1 with AC option</td>
</tr>
<tr>
<td class="org-right">2</td>
<td class="org-left">Amplifier 2 with DC option</td>
</tr>
<tr>
<td class="org-right">3</td>
<td class="org-left">Time</td>
</tr>
</tbody>
</table>
<p>
Second measurement (<code>mat/data_015.mat</code> file):
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Column</th>
<th scope="col" class="org-left">Signal</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1</td>
<td class="org-left">Amplifier 1 with DC option</td>
</tr>
<tr>
<td class="org-right">2</td>
<td class="org-left">Amplifier 2 with AC option</td>
</tr>
<tr>
<td class="org-right">3</td>
<td class="org-left">Time</td>
</tr>
</tbody>
</table>
<div id="org6faa1f8" class="figure">
<p><img src="./img/IMG_20190503_170936.jpg" alt="IMG_20190503_170936.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 3: </span>Picture of the two voltages amplifiers</p>
</div>
</div>
</div>
<div id="outline-container-org86b59a1" class="outline-3">
<h3 id="org86b59a1"><span class="section-number-3">2.2</span> Load data</h3>
<div class="outline-text-3" id="text-2-2">
<p>
We load the data of the z axis of two geophones.
</p>
<div class="org-src-container">
<pre class="src src-matlab">meas14 = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_014.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; meas14 = meas14.data;
meas15 = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_015.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; meas15 = meas15.data;
</pre>
</div>
</div>
</div>
<div id="outline-container-org22f801c" class="outline-3">
<h3 id="org22f801c"><span class="section-number-3">2.3</span> Time Domain</h3>
<div class="outline-text-3" id="text-2-3">
<p>
The signals are shown on figure <a href="#org4a464f5">4</a>.
</p>
<div id="org4a464f5" class="figure">
<p><img src="figs/ac_dc_option_time.png" alt="ac_dc_option_time.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Comparison of the signals going through the Voltage amplifiers</p>
</div>
</div>
</div>
<div id="outline-container-orgbd72638" class="outline-3">
<h3 id="orgbd72638"><span class="section-number-3">2.4</span> Frequency Domain</h3>
<div class="outline-text-3" id="text-2-4">
<p>
We first compute some parameters that will be used for the PSD computation.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dt = meas14<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>meas14<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
Then we compute the Power Spectral Density using <code>pwelch</code> function.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>pxamp1ac, f<span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>meas14<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>pxamp2dc, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>meas14<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>pxamp1dc, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>meas15<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-rainbow-delimiters-depth-1">[</span>pxamp2ac, <span class="org-type">~</span><span class="org-rainbow-delimiters-depth-1">]</span> = pwelch<span class="org-rainbow-delimiters-depth-1">(</span>meas15<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
The ASD of the signals are compare on figure <a href="#org73d3360">5</a>.
</p>
<div id="org73d3360" class="figure">
<p><img src="figs/ac_dc_option_asd.png" alt="ac_dc_option_asd.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Amplitude Spectral Density of the measured signals</p>
</div>
</div>
</div>
<div id="outline-container-org2647596" class="outline-3">
<h3 id="org2647596"><span class="section-number-3">2.5</span> Conclusion</h3>
<div class="outline-text-3" id="text-2-5">
<div class="important">
<ul class="org-ul">
<li>The voltage amplifiers include some very sharp high pass filters at 1.5Hz (maybe 4th order)</li>
<li>There is a DC offset on the time domain signal because the DC-offset knob was not set to zero</li>
</ul>
</div>
</div>
</div>
</div>
<div id="outline-container-org9623ca1" class="outline-2">
<h2 id="org9623ca1"><span class="section-number-2">3</span> Transfer function of the Low Pass Filter</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org5eb1733"></a>
</p>
<p>
The computation files for this section are accessible <a href="data/low_pass_filter_measurements.zip">here</a>.
</p>
</div>
<div id="outline-container-org921fe36" class="outline-3">
<h3 id="org921fe36"><span class="section-number-3">3.1</span> First LPF with a Cut-off frequency of 160Hz</h3>
<div class="outline-text-3" id="text-3-1">
</div>
<div id="outline-container-org122c7a6" class="outline-4">
<h4 id="org122c7a6"><span class="section-number-4">3.1.1</span> Measurement Description</h4>
<div class="outline-text-4" id="text-3-1-1">
<p>
<b>Goal</b>:
</p>
<ul class="org-ul">
<li>Measure the Low Pass Filter Transfer Function</li>
</ul>
<p>
The values of the components are:
</p>
\begin{aligned}
R &= 1k\Omega \\
C &= 1\mu F
\end{aligned}
<p>
Which makes a cut-off frequency of \(f_c = \frac{1}{RC} = 1000 rad/s = 160Hz\).
</p>
<div class="org-src-container">
<pre class="src src-latex" id="org336f6c8"><span class="org-font-latex-sedate"><span class="org-keyword">\begin</span></span>{<span class="org-function-name">tikzpicture</span>}
<span class="org-font-latex-sedate">\draw</span> (0,2)
to [R=<span class="org-font-latex-sedate"><span class="org-font-latex-math">\(R\)</span></span>] ++(2,0) node[circ]
to ++(2,0)
++(-2,0)
to [C=<span class="org-font-latex-sedate"><span class="org-font-latex-math">\(C\)</span></span>] ++(0,-2) node[circ]
++(-2,0)
to ++(2,0)
to ++(2,0)
<span class="org-font-latex-sedate"><span class="org-keyword">\end</span></span>{<span class="org-function-name">tikzpicture</span>}
</pre>
</div>
<div id="orgda67620" class="figure">
<p><img src="figs/lpf.png" alt="lpf.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Schematic of the Low Pass Filter used</p>
</div>
<p>
<b>Setup</b>:
</p>
<ul class="org-ul">
<li>We are measuring the signal from from Geophone with a BNC T</li>
<li>On part goes to column 1 through the LPF</li>
<li>The other part goes to column 2 without the LPF</li>
</ul>
<p>
<b>Measurements</b>:
<code>mat/data_018.mat</code>:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Column</th>
<th scope="col" class="org-left">Signal</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">1</td>
<td class="org-left">Amplifier 1 with LPF</td>
</tr>
<tr>
<td class="org-right">2</td>
<td class="org-left">Amplifier 2</td>
</tr>
<tr>
<td class="org-right">3</td>
<td class="org-left">Time</td>
</tr>
</tbody>
</table>
<div id="org5022f0b" class="figure">
<p><img src="./img/IMG_20190507_102756.jpg" alt="IMG_20190507_102756.jpg" width="500px" />
</p>
<p><span class="figure-number">Figure 7: </span>Picture of the low pass filter used</p>
</div>
</div>
</div>
<div id="outline-container-orgba94437" class="outline-4">
<h4 id="orgba94437"><span class="section-number-4">3.1.2</span> Load data</h4>
<div class="outline-text-4" id="text-3-1-2">
<p>
We load the data of the z axis of two geophones.
</p>
<div class="org-src-container">
<pre class="src src-matlab">data = load<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'mat/data_018.mat', 'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; data = data.data;
</pre>
</div>
</div>
</div>
<div id="outline-container-org087582c" class="outline-4">
<h4 id="org087582c"><span class="section-number-4">3.1.3</span> Transfer function of the LPF</h4>
<div class="outline-text-4" id="text-3-1-3">
<p>
We compute the transfer function from the signal without the LPF to the signal measured with the LPF.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dt = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Glpf, f<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span>, data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
We compare this transfer function with a transfer function corresponding to an ideal first order LPF with a cut-off frequency of \(1000rad/s\).
We obtain the result on figure <a href="#orgeb8ea3c">8</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gth = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">1000</span><span class="org-rainbow-delimiters-depth-1">)</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
ax1 = subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>Glpf<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>freqresp<span class="org-rainbow-delimiters-depth-4">(</span>Gth, f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</span>,<span class="org-rainbow-delimiters-depth-2">[]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Magnitude'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ax2 = subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, mod<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</span><span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>phase<span class="org-rainbow-delimiters-depth-3">(</span>Glpf<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>unwrap<span class="org-rainbow-delimiters-depth-2">(</span>angle<span class="org-rainbow-delimiters-depth-3">(</span>squeeze<span class="org-rainbow-delimiters-depth-4">(</span>freqresp<span class="org-rainbow-delimiters-depth-5">(</span>Gth, f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
yticks<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-type">-</span><span class="org-highlight-numbers-number">90</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">90</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Phase'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
linkaxes<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>ax1,ax2<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-string">'x'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div id="orgeb8ea3c" class="figure">
<p><img src="figs/Glpf_bode.png" alt="Glpf_bode.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Bode Diagram of the measured Low Pass filter and the theoritical one</p>
</div>
</div>
</div>
<div id="outline-container-orgf2dc541" class="outline-4">
<h4 id="orgf2dc541"><span class="section-number-4">3.1.4</span> Conclusion</h4>
<div class="outline-text-4" id="text-3-1-4">
<div class="important">
<p>
As we want to measure things up to \(500Hz\), we chose to change the value of the capacitor to obtain a cut-off frequency of \(1kHz\).
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgd6132bd" class="outline-3">
<h3 id="orgd6132bd"><span class="section-number-3">3.2</span> Second LPF with a Cut-off frequency of 1000Hz</h3>
<div class="outline-text-3" id="text-3-2">
</div>
<div id="outline-container-org76b4e0d" class="outline-4">
<h4 id="org76b4e0d"><span class="section-number-4">3.2.1</span> Measurement description</h4>
<div class="outline-text-4" id="text-3-2-1">
<p>
This time, the value are
</p>
\begin{aligned}
R &= 1k\Omega \\
C &= 150nF
\end{aligned}
<p>
Which makes a low pass filter with a cut-off frequency of \(f_c = 1060Hz\).
</p>
</div>
</div>
<div id="outline-container-org2941697" class="outline-4">
<h4 id="org2941697"><span class="section-number-4">3.2.2</span> Load data</h4>
<div class="outline-text-4" id="text-3-2-2">
<p>
We load the data of the z axis of two geophones.
</p>
<div class="org-src-container">
<pre class="src src-matlab">data = load<span class="org-rainbow-delimiters-depth-1">(</span>'mat<span class="org-type">/</span>data_019.mat', <span class="org-string">'data'</span><span class="org-rainbow-delimiters-depth-1">)</span>; data = data.data;
</pre>
</div>
</div>
</div>
<div id="outline-container-org45462d2" class="outline-4">
<h4 id="org45462d2"><span class="section-number-4">3.2.3</span> Transfer function of the LPF</h4>
<div class="outline-text-4" id="text-3-2-3">
<p>
We compute the transfer function from the signal without the LPF to the signal measured with the LPF.
</p>
<div class="org-src-container">
<pre class="src src-matlab">dt = data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span><span class="org-type">-</span>data<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">3</span><span class="org-rainbow-delimiters-depth-1">)</span>;
Fs = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span>dt; <span class="org-comment">% [Hz]</span>
win = hanning<span class="org-rainbow-delimiters-depth-1">(</span>ceil<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">10</span><span class="org-type">*</span>Fs<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-rainbow-delimiters-depth-1">[</span>Glpf, f<span class="org-rainbow-delimiters-depth-1">]</span> = tfestimate<span class="org-rainbow-delimiters-depth-1">(</span>data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-2">)</span>, data<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-type">:</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-2">)</span>, win, <span class="org-rainbow-delimiters-depth-2">[]</span>, <span class="org-rainbow-delimiters-depth-2">[]</span>, Fs<span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<p>
We compare this transfer function with a transfer function corresponding to an ideal first order LPF with a cut-off frequency of \(1060Hz\).
We obtain the result on figure <a href="#org19f2e2d">9</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gth = <span class="org-highlight-numbers-number">1</span><span class="org-type">/</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">1</span><span class="org-type">+</span>s<span class="org-type">/</span><span class="org-highlight-numbers-number">1060</span><span class="org-type">/</span><span class="org-highlight-numbers-number">2</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
ax1 = subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">1</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>Glpf<span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, abs<span class="org-rainbow-delimiters-depth-2">(</span>squeeze<span class="org-rainbow-delimiters-depth-3">(</span>freqresp<span class="org-rainbow-delimiters-depth-4">(</span>Gth, f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; set</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">gca, 'yscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'XTickLabel'</span>,<span class="org-rainbow-delimiters-depth-2">[]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Magnitude'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ax2 = subplot<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-highlight-numbers-number">2</span>, <span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">2</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold on;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, mod<span class="org-rainbow-delimiters-depth-2">(</span><span class="org-highlight-numbers-number">180</span><span class="org-type">+</span><span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>phase<span class="org-rainbow-delimiters-depth-3">(</span>Glpf<span class="org-rainbow-delimiters-depth-3">)</span>, <span class="org-highlight-numbers-number">360</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-1">)</span>;
plot<span class="org-rainbow-delimiters-depth-1">(</span>f, <span class="org-highlight-numbers-number">180</span><span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">*</span>unwrap<span class="org-rainbow-delimiters-depth-2">(</span>angle<span class="org-rainbow-delimiters-depth-3">(</span>squeeze<span class="org-rainbow-delimiters-depth-4">(</span>freqresp<span class="org-rainbow-delimiters-depth-5">(</span>Gth, f, <span class="org-string">'Hz'</span><span class="org-rainbow-delimiters-depth-5">)</span><span class="org-rainbow-delimiters-depth-4">)</span><span class="org-rainbow-delimiters-depth-3">)</span><span class="org-rainbow-delimiters-depth-2">)</span><span class="org-rainbow-delimiters-depth-1">)</span>;
hold off;
<span class="org-type">set</span><span class="org-rainbow-delimiters-depth-1">(</span><span class="org-variable-name">gca</span>, <span class="org-string">'xscale', 'log'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
ylim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
yticks<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-type">-</span><span class="org-highlight-numbers-number">180</span>, <span class="org-type">-</span><span class="org-highlight-numbers-number">90</span>, <span class="org-highlight-numbers-number">0</span>, <span class="org-highlight-numbers-number">90</span>, <span class="org-highlight-numbers-number">180</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlabel<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-string">'Frequency </span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">[</span></span><span class="org-string">Hz</span><span class="org-string"><span class="org-rainbow-delimiters-depth-2">]</span></span><span class="org-string">'</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">)</span></span><span class="org-string">; ylabel</span><span class="org-string"><span class="org-rainbow-delimiters-depth-1">(</span></span><span class="org-string">'Phase'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
linkaxes<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span>ax1,ax2<span class="org-rainbow-delimiters-depth-2">]</span>,<span class="org-string">'x'</span><span class="org-rainbow-delimiters-depth-1">)</span>;
xlim<span class="org-rainbow-delimiters-depth-1">(</span><span class="org-rainbow-delimiters-depth-2">[</span><span class="org-highlight-numbers-number">1</span>, <span class="org-highlight-numbers-number">500</span><span class="org-rainbow-delimiters-depth-2">]</span><span class="org-rainbow-delimiters-depth-1">)</span>;
</pre>
</div>
<div id="org19f2e2d" class="figure">
<p><img src="figs/Glpf_bode_bis.png" alt="Glpf_bode_bis.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Bode Diagram of the measured Low Pass filter and the theoritical one</p>
</div>
</div>
</div>
<div id="outline-container-org438a172" class="outline-4">
<h4 id="org438a172"><span class="section-number-4">3.2.4</span> Conclusion</h4>
<div class="outline-text-4" id="text-3-2-4">
<div class="important">
<p>
The added LPF has the expected behavior.
</p>
</div>
</div>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-05-10 ven. 10:07</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
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