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Add analysis of the effect of added resistor

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Thomas Dehaeze 2020-11-04 20:38:59 +01:00
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-10-29 jeu. 10:08 --> <!-- 2020-11-04 mer. 20:38 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Encoder - Test Bench</title> <title>Encoder - Test Bench</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -35,49 +35,50 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#orgfe3d6c5">1. Experimental Setup</a></li> <li><a href="#org2b2f049">1. Experimental Setup</a></li>
<li><a href="#org85c4a97">2. Huddle Test</a> <li><a href="#org9ae614e">2. Huddle Test</a>
<ul> <ul>
<li><a href="#org6f9e775">2.1. Load Data</a></li> <li><a href="#orgc034802">2.1. Load Data</a></li>
<li><a href="#orgd27ba42">2.2. Time Domain Results</a></li> <li><a href="#orgd8ae4ae">2.2. Time Domain Results</a></li>
<li><a href="#org963a243">2.3. Frequency Domain Noise</a></li> <li><a href="#org864c60c">2.3. Frequency Domain Noise</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org0e0c48a">3. Comparison Interferometer / Encoder</a> <li><a href="#org96539c5">3. Comparison Interferometer / Encoder</a>
<ul> <ul>
<li><a href="#orgc15a506">3.1. Load Data</a></li> <li><a href="#orgc6453b2">3.1. Load Data</a></li>
<li><a href="#orgc509698">3.2. Time Domain Results</a></li> <li><a href="#orgc362641">3.2. Time Domain Results</a></li>
<li><a href="#org1b5953a">3.3. Difference between Encoder and Interferometer as a function of time</a></li> <li><a href="#orgb50503c">3.3. Difference between Encoder and Interferometer as a function of time</a></li>
<li><a href="#orgcb56769">3.4. Difference between Encoder and Interferometer as a function of position</a></li> <li><a href="#org5ec03aa">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org8399536">4. Identification</a> <li><a href="#org6b73303">4. Identification</a>
<ul> <ul>
<li><a href="#org4b364a5">4.1. Load Data</a></li> <li><a href="#orga15e9e2">4.1. Load Data</a></li>
<li><a href="#org0dd3820">4.2. Identification</a></li> <li><a href="#org6974653">4.2. Identification</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org4a7e08f">5. Change of Stiffness due to Sensors stack being open/closed circuit</a> <li><a href="#org5be1a0f">5. Change of Stiffness due to Sensors stack being open/closed circuit</a>
<ul> <ul>
<li><a href="#orgba85fb9">5.1. Load Data</a></li> <li><a href="#org00fc76c">5.1. Load Data</a></li>
<li><a href="#org2876c52">5.2. Transfer Functions</a></li> <li><a href="#org757917f">5.2. Transfer Functions</a></li>
</ul> </ul>
</li> </li>
<li><a href="#org1abff1f">6. Generated Number of Charge / Voltage</a> <li><a href="#org8dab16f">6. Generated Number of Charge / Voltage</a>
<ul> <ul>
<li><a href="#org65e8206">6.1. Steps</a></li> <li><a href="#org28149b5">6.1. Steps</a></li>
<li><a href="#org4df253f">6.2. Sinus</a></li> <li><a href="#org900387e">6.2. Add Parallel Resistor</a></li>
<li><a href="#org7cfe5a8">6.3. Sinus</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-orgfe3d6c5" class="outline-2"> <div id="outline-container-org2b2f049" class="outline-2">
<h2 id="orgfe3d6c5"><span class="section-number-2">1</span> Experimental Setup</h2> <h2 id="org2b2f049"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
<p> <p>
The experimental Setup is schematically represented in Figure <a href="#orgb535f32">1</a>. The experimental Setup is schematically represented in Figure <a href="#orgff06236">1</a>.
</p> </p>
<p> <p>
@ -86,21 +87,21 @@ The displacement of the mass (relative to the mechanical frame) is measured both
</p> </p>
<div id="orgb535f32" class="figure"> <div id="orgff06236" class="figure">
<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" /> <p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p> <p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p>
</div> </div>
<div id="org7c9d65a" class="figure"> <div id="orgc6b5ad5" class="figure">
<p><img src="figs/IMG_20201023_153905.jpg" alt="IMG_20201023_153905.jpg" /> <p><img src="figs/IMG_20201023_153905.jpg" alt="IMG_20201023_153905.jpg" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Side View of the encoder</p> <p><span class="figure-number">Figure 2: </span>Side View of the encoder</p>
</div> </div>
<div id="org6ecf3f2" class="figure"> <div id="orgcf538be" class="figure">
<p><img src="figs/IMG_20201023_153914.jpg" alt="IMG_20201023_153914.jpg" /> <p><img src="figs/IMG_20201023_153914.jpg" alt="IMG_20201023_153914.jpg" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Front View of the encoder</p> <p><span class="figure-number">Figure 3: </span>Front View of the encoder</p>
@ -108,8 +109,8 @@ The displacement of the mass (relative to the mechanical frame) is measured both
</div> </div>
</div> </div>
<div id="outline-container-org85c4a97" class="outline-2"> <div id="outline-container-org9ae614e" class="outline-2">
<h2 id="org85c4a97"><span class="section-number-2">2</span> Huddle Test</h2> <h2 id="org9ae614e"><span class="section-number-2">2</span> Huddle Test</h2>
<div class="outline-text-2" id="text-2"> <div class="outline-text-2" id="text-2">
<p> <p>
The goal in this section is the estimate the noise of both the encoder and the intereferometer. The goal in this section is the estimate the noise of both the encoder and the intereferometer.
@ -121,8 +122,8 @@ Ideally, a mechanical part would clamp the two together, we here suppose that th
</p> </p>
</div> </div>
<div id="outline-container-org6f9e775" class="outline-3"> <div id="outline-container-orgc034802" class="outline-3">
<h3 id="org6f9e775"><span class="section-number-3">2.1</span> Load Data</h3> <h3 id="orgc034802"><span class="section-number-3">2.1</span> Load Data</h3>
<div class="outline-text-3" id="text-2-1"> <div class="outline-text-3" id="text-2-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_huddle_test.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'t'</span>); <pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_huddle_test.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'t'</span>);
@ -137,11 +138,11 @@ encoder = detrend(encoder, 0);
</div> </div>
</div> </div>
<div id="outline-container-orgd27ba42" class="outline-3"> <div id="outline-container-orgd8ae4ae" class="outline-3">
<h3 id="orgd27ba42"><span class="section-number-3">2.2</span> Time Domain Results</h3> <h3 id="orgd8ae4ae"><span class="section-number-3">2.2</span> Time Domain Results</h3>
<div class="outline-text-3" id="text-2-2"> <div class="outline-text-3" id="text-2-2">
<div id="orgbec33e3" class="figure"> <div id="orgff47510" class="figure">
<p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" /> <p><img src="figs/huddle_test_time_domain.png" alt="huddle_test_time_domain.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Huddle test - Time domain signals</p> <p><span class="figure-number">Figure 4: </span>Huddle test - Time domain signals</p>
@ -153,7 +154,7 @@ encoder = detrend(encoder, 0);
</div> </div>
<div id="org19de732" class="figure"> <div id="org21436f3" class="figure">
<p><img src="figs/huddle_test_time_domain_filtered.png" alt="huddle_test_time_domain_filtered.png" /> <p><img src="figs/huddle_test_time_domain_filtered.png" alt="huddle_test_time_domain_filtered.png" />
</p> </p>
<p><span class="figure-number">Figure 5: </span>Huddle test - Time domain signals filtered with a LPF at 10Hz</p> <p><span class="figure-number">Figure 5: </span>Huddle test - Time domain signals filtered with a LPF at 10Hz</p>
@ -161,8 +162,8 @@ encoder = detrend(encoder, 0);
</div> </div>
</div> </div>
<div id="outline-container-org963a243" class="outline-3"> <div id="outline-container-org864c60c" class="outline-3">
<h3 id="org963a243"><span class="section-number-3">2.3</span> Frequency Domain Noise</h3> <h3 id="org864c60c"><span class="section-number-3">2.3</span> Frequency Domain Noise</h3>
<div class="outline-text-3" id="text-2-3"> <div class="outline-text-3" id="text-2-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
@ -174,7 +175,7 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div> </div>
<div id="orgb6a29d5" class="figure"> <div id="org63bcf43" class="figure">
<p><img src="figs/huddle_test_asd.png" alt="huddle_test_asd.png" /> <p><img src="figs/huddle_test_asd.png" alt="huddle_test_asd.png" />
</p> </p>
<p><span class="figure-number">Figure 6: </span>Amplitude Spectral Density of the signals during the Huddle test</p> <p><span class="figure-number">Figure 6: </span>Amplitude Spectral Density of the signals during the Huddle test</p>
@ -183,8 +184,8 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div> </div>
</div> </div>
<div id="outline-container-org0e0c48a" class="outline-2"> <div id="outline-container-org96539c5" class="outline-2">
<h2 id="org0e0c48a"><span class="section-number-2">3</span> Comparison Interferometer / Encoder</h2> <h2 id="org96539c5"><span class="section-number-2">3</span> Comparison Interferometer / Encoder</h2>
<div class="outline-text-2" id="text-3"> <div class="outline-text-2" id="text-3">
<p> <p>
The goal here is to make sure that the interferometer and encoder measurements are coherent. The goal here is to make sure that the interferometer and encoder measurements are coherent.
@ -192,8 +193,8 @@ We may see non-linearity in the interferometric measurement.
</p> </p>
</div> </div>
<div id="outline-container-orgc15a506" class="outline-3"> <div id="outline-container-orgc6453b2" class="outline-3">
<h3 id="orgc15a506"><span class="section-number-3">3.1</span> Load Data</h3> <h3 id="orgc6453b2"><span class="section-number-3">3.1</span> Load Data</h3>
<div class="outline-text-3" id="text-3-1"> <div class="outline-text-3" id="text-3-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_comp.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>); <pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_comp.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>);
@ -209,18 +210,18 @@ u = detrend(u, 0);
</div> </div>
</div> </div>
<div id="outline-container-orgc509698" class="outline-3"> <div id="outline-container-orgc362641" class="outline-3">
<h3 id="orgc509698"><span class="section-number-3">3.2</span> Time Domain Results</h3> <h3 id="orgc362641"><span class="section-number-3">3.2</span> Time Domain Results</h3>
<div class="outline-text-3" id="text-3-2"> <div class="outline-text-3" id="text-3-2">
<div id="orgdf791be" class="figure"> <div id="orgfd33423" class="figure">
<p><img src="figs/int_enc_one_cycle.png" alt="int_enc_one_cycle.png" /> <p><img src="figs/int_enc_one_cycle.png" alt="int_enc_one_cycle.png" />
</p> </p>
<p><span class="figure-number">Figure 7: </span>One cycle measurement</p> <p><span class="figure-number">Figure 7: </span>One cycle measurement</p>
</div> </div>
<div id="org7e30f32" class="figure"> <div id="orgf82fd3e" class="figure">
<p><img src="figs/int_enc_one_cycle_error.png" alt="int_enc_one_cycle_error.png" /> <p><img src="figs/int_enc_one_cycle_error.png" alt="int_enc_one_cycle_error.png" />
</p> </p>
<p><span class="figure-number">Figure 8: </span>Difference between the Encoder and the interferometer during one cycle</p> <p><span class="figure-number">Figure 8: </span>Difference between the Encoder and the interferometer during one cycle</p>
@ -228,8 +229,8 @@ u = detrend(u, 0);
</div> </div>
</div> </div>
<div id="outline-container-org1b5953a" class="outline-3"> <div id="outline-container-orgb50503c" class="outline-3">
<h3 id="org1b5953a"><span class="section-number-3">3.3</span> Difference between Encoder and Interferometer as a function of time</h3> <h3 id="orgb50503c"><span class="section-number-3">3.3</span> Difference between Encoder and Interferometer as a function of time</h3>
<div class="outline-text-3" id="text-3-3"> <div class="outline-text-3" id="text-3-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4;
@ -250,7 +251,7 @@ d_err_mean = d_err_mean <span class="org-type">-</span> mean(d_err_mean);
</div> </div>
<div id="org95838e4" class="figure"> <div id="org6eb78a0" class="figure">
<p><img src="figs/int_enc_error_mean_time.png" alt="int_enc_error_mean_time.png" /> <p><img src="figs/int_enc_error_mean_time.png" alt="int_enc_error_mean_time.png" />
</p> </p>
<p><span class="figure-number">Figure 9: </span>Difference between the two measurement in the time domain, averaged for all the cycles</p> <p><span class="figure-number">Figure 9: </span>Difference between the two measurement in the time domain, averaged for all the cycles</p>
@ -258,8 +259,8 @@ d_err_mean = d_err_mean <span class="org-type">-</span> mean(d_err_mean);
</div> </div>
</div> </div>
<div id="outline-container-orgcb56769" class="outline-3"> <div id="outline-container-org5ec03aa" class="outline-3">
<h3 id="orgcb56769"><span class="section-number-3">3.4</span> Difference between Encoder and Interferometer as a function of position</h3> <h3 id="org5ec03aa"><span class="section-number-3">3.4</span> Difference between Encoder and Interferometer as a function of position</h3>
<div class="outline-text-3" id="text-3-4"> <div class="outline-text-3" id="text-3-4">
<p> <p>
Compute the mean of the interferometer measurement corresponding to each of the encoder measurement. Compute the mean of the interferometer measurement corresponding to each of the encoder measurement.
@ -278,7 +279,7 @@ i_mean_error = (i_mean <span class="org-type">-</span> e_sorted);
</div> </div>
<div id="orga34e36f" class="figure"> <div id="org2aecb58" class="figure">
<p><img src="figs/int_enc_error_mean_position.png" alt="int_enc_error_mean_position.png" /> <p><img src="figs/int_enc_error_mean_position.png" alt="int_enc_error_mean_position.png" />
</p> </p>
<p><span class="figure-number">Figure 10: </span>Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles</p> <p><span class="figure-number">Figure 10: </span>Difference between the two measurement as a function of the measured position by the encoder, averaged for all the cycles</p>
@ -299,7 +300,7 @@ e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init<span class="org-typ
</div> </div>
<div id="org064a7da" class="figure"> <div id="orgd0cea4e" class="figure">
<p><img src="figs/int_non_linearity_period_wavelength.png" alt="int_non_linearity_period_wavelength.png" /> <p><img src="figs/int_non_linearity_period_wavelength.png" alt="int_non_linearity_period_wavelength.png" />
</p> </p>
<p><span class="figure-number">Figure 11: </span>Non-Linearity of the Interferometer over the period of the wavelength</p> <p><span class="figure-number">Figure 11: </span>Non-Linearity of the Interferometer over the period of the wavelength</p>
@ -308,12 +309,12 @@ e_sorted_mean_over_period = mean(reshape(i_mean_error(i_init<span class="org-typ
</div> </div>
</div> </div>
<div id="outline-container-org8399536" class="outline-2"> <div id="outline-container-org6b73303" class="outline-2">
<h2 id="org8399536"><span class="section-number-2">4</span> Identification</h2> <h2 id="org6b73303"><span class="section-number-2">4</span> Identification</h2>
<div class="outline-text-2" id="text-4"> <div class="outline-text-2" id="text-4">
</div> </div>
<div id="outline-container-org4b364a5" class="outline-3"> <div id="outline-container-orga15e9e2" class="outline-3">
<h3 id="org4b364a5"><span class="section-number-3">4.1</span> Load Data</h3> <h3 id="orga15e9e2"><span class="section-number-3">4.1</span> Load Data</h3>
<div class="outline-text-3" id="text-4-1"> <div class="outline-text-3" id="text-4-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_id_noise_bis.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>); <pre class="src src-matlab">load(<span class="org-string">'mat/int_enc_id_noise_bis.mat'</span>, <span class="org-string">'interferometer'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'t'</span>);
@ -329,8 +330,8 @@ u = detrend(u, 0);
</div> </div>
</div> </div>
<div id="outline-container-org0dd3820" class="outline-3"> <div id="outline-container-org6974653" class="outline-3">
<h3 id="org0dd3820"><span class="section-number-3">4.2</span> Identification</h3> <h3 id="org6974653"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2"> <div class="outline-text-3" id="text-4-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span> <pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
@ -348,14 +349,14 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div> </div>
<div id="org4e10071" class="figure"> <div id="org37fcd05" class="figure">
<p><img src="figs/identification_dynamics_coherence.png" alt="identification_dynamics_coherence.png" /> <p><img src="figs/identification_dynamics_coherence.png" alt="identification_dynamics_coherence.png" />
</p> </p>
</div> </div>
<div id="orgfa48d56" class="figure"> <div id="orge65c21c" class="figure">
<p><img src="figs/identification_dynamics_bode.png" alt="identification_dynamics_bode.png" /> <p><img src="figs/identification_dynamics_bode.png" alt="identification_dynamics_bode.png" />
</p> </p>
</div> </div>
@ -363,12 +364,12 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div> </div>
</div> </div>
<div id="outline-container-org4a7e08f" class="outline-2"> <div id="outline-container-org5be1a0f" class="outline-2">
<h2 id="org4a7e08f"><span class="section-number-2">5</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2> <h2 id="org5be1a0f"><span class="section-number-2">5</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
<div class="outline-text-2" id="text-5"> <div class="outline-text-2" id="text-5">
</div> </div>
<div id="outline-container-orgba85fb9" class="outline-3"> <div id="outline-container-org00fc76c" class="outline-3">
<h3 id="orgba85fb9"><span class="section-number-3">5.1</span> Load Data</h3> <h3 id="org00fc76c"><span class="section-number-3">5.1</span> Load Data</h3>
<div class="outline-text-3" id="text-5-1"> <div class="outline-text-3" id="text-5-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">oc = load(<span class="org-string">'./mat/identification_open_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>); <pre class="src src-matlab">oc = load(<span class="org-string">'./mat/identification_open_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>);
@ -378,8 +379,8 @@ sc = load(<span class="org-string">'./mat/identification_short_circuit.mat'</spa
</div> </div>
</div> </div>
<div id="outline-container-org2876c52" class="outline-3"> <div id="outline-container-org757917f" class="outline-3">
<h3 id="org2876c52"><span class="section-number-3">5.2</span> Transfer Functions</h3> <h3 id="org757917f"><span class="section-number-3">5.2</span> Transfer Functions</h3>
<div class="outline-text-3" id="text-5-2"> <div class="outline-text-3" id="text-5-2">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span> <pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
@ -397,26 +398,26 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div> </div>
<div id="org525ff65" class="figure"> <div id="orgcba7c3a" class="figure">
<p><img src="figs/stiffness_force_sensor_coherence.png" alt="stiffness_force_sensor_coherence.png" /> <p><img src="figs/stiffness_force_sensor_coherence.png" alt="stiffness_force_sensor_coherence.png" />
</p> </p>
</div> </div>
<div id="orge72a5de" class="figure"> <div id="orgfdef3b4" class="figure">
<p><img src="figs/stiffness_force_sensor_bode.png" alt="stiffness_force_sensor_bode.png" /> <p><img src="figs/stiffness_force_sensor_bode.png" alt="stiffness_force_sensor_bode.png" />
</p> </p>
</div> </div>
<div id="orgd8a00e8" class="figure"> <div id="org655038d" class="figure">
<p><img src="figs/stiffness_force_sensor_bode_zoom.png" alt="stiffness_force_sensor_bode_zoom.png" /> <p><img src="figs/stiffness_force_sensor_bode_zoom.png" alt="stiffness_force_sensor_bode_zoom.png" />
</p> </p>
<p><span class="figure-number">Figure 16: </span>Zoom on the change of resonance</p> <p><span class="figure-number">Figure 16: </span>Zoom on the change of resonance</p>
</div> </div>
<div class="important" id="org5981596"> <div class="important" id="orga5e301b">
<p> <p>
The change of resonance frequency / stiffness is very small and is not important here. The change of resonance frequency / stiffness is very small and is not important here.
</p> </p>
@ -426,8 +427,8 @@ The change of resonance frequency / stiffness is very small and is not important
</div> </div>
</div> </div>
<div id="outline-container-org1abff1f" class="outline-2"> <div id="outline-container-org8dab16f" class="outline-2">
<h2 id="org1abff1f"><span class="section-number-2">6</span> Generated Number of Charge / Voltage</h2> <h2 id="org8dab16f"><span class="section-number-2">6</span> Generated Number of Charge / Voltage</h2>
<div class="outline-text-2" id="text-6"> <div class="outline-text-2" id="text-6">
<p> <p>
Two stacks are used as actuator (in parallel) and one stack is used as sensor. Two stacks are used as actuator (in parallel) and one stack is used as sensor.
@ -438,8 +439,8 @@ The amplifier gain is 20V/V (Cedrat LA75B).
</p> </p>
</div> </div>
<div id="outline-container-org65e8206" class="outline-3"> <div id="outline-container-org28149b5" class="outline-3">
<h3 id="org65e8206"><span class="section-number-3">6.1</span> Steps</h3> <h3 id="org28149b5"><span class="section-number-3">6.1</span> Steps</h3>
<div class="outline-text-3" id="text-6-1"> <div class="outline-text-3" id="text-6-1">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/force_sensor_steps.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>); <pre class="src src-matlab">load(<span class="org-string">'./mat/force_sensor_steps.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
@ -459,7 +460,7 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
</div> </div>
<div id="org0233cf4" class="figure"> <div id="org0c7c950" class="figure">
<p><img src="figs/force_sen_steps_time_domain.png" alt="force_sen_steps_time_domain.png" /> <p><img src="figs/force_sen_steps_time_domain.png" alt="force_sen_steps_time_domain.png" />
</p> </p>
<p><span class="figure-number">Figure 17: </span>Time domain signal during the 3 actuator voltage steps</p> <p><span class="figure-number">Figure 17: </span>Time domain signal during the 3 actuator voltage steps</p>
@ -468,11 +469,12 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
<p> <p>
Three steps are performed at the following time intervals: Three steps are performed at the following time intervals:
</p> </p>
<ul class="org-ul"> <div class="org-src-container">
<li>2.5, 23</li> <pre class="src src-matlab">t_s = [ 2.5, 23;
<li>23.8, 35</li> 23.8, 35;
<li>35.8, 50</li> 35.8, 50];
</ul> </pre>
</div>
<p> <p>
Fit function: Fit function:
@ -485,52 +487,25 @@ Fit function:
<p> <p>
We are interested by the <code>b(2)</code> term, which is the time constant of the exponential. We are interested by the <code>b(2)</code> term, which is the time constant of the exponential.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">tau = zeros(3,1); <pre class="src src-matlab">tau = zeros(size(t_s, 1),1);
V0 = zeros(3,1); V0 = zeros(size(t_s, 1),1);
</pre> </pre>
</div> </div>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">t_cur = t(2.5 <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> 23); <pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name">t_i</span> = <span class="org-constant">1:size(t_s, 1)</span>
t_cur = t_cur <span class="org-type">-</span> t_cur(1); t_cur = t(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
y_cur = v(2.5 <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> 23); t_cur = t_cur <span class="org-type">-</span> t_cur(1);
y_cur = v(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span> nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
B0 = [0.5, <span class="org-type">-</span>0.15, 2.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span> B0 = [0.5, <span class="org-type">-</span>0.15, 2.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span> [B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
tau<span class="org-type">(1) </span>= 1<span class="org-type">/</span>B(2); tau(t_i) = 1<span class="org-type">/</span>B(2);
V0<span class="org-type">(1) </span>= B(3); V0(t_i) = B(3);
</pre> <span class="org-keyword">end</span>
</div>
<div class="org-src-container">
<pre class="src src-matlab">t_cur = t(23.8 <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> 35);
t_cur = t_cur <span class="org-type">-</span> t_cur(1);
y_cur = v(23.8 <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> 35);
nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
B0 = [0.5, <span class="org-type">-</span>0.15, 2.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
tau<span class="org-type">(2) </span>= 1<span class="org-type">/</span>B(2);
V0<span class="org-type">(2) </span>= B(3);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">t_cur = t(35.8 <span class="org-type">&lt;</span> t);
t_cur = t_cur <span class="org-type">-</span> t_cur(1);
y_cur = v(35.8 <span class="org-type">&lt;</span> t);
nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
B0 = [0.5, <span class="org-type">-</span>0.15, 2.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
tau<span class="org-type">(3) </span>= 1<span class="org-type">/</span>B(2);
V0<span class="org-type">(3) </span>= B(3);
</pre> </pre>
</div> </div>
@ -584,7 +559,7 @@ Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
The input impedance of the Speedgoat&rsquo;s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)). The input impedance of the Speedgoat&rsquo;s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)).
</p> </p>
<div class="important" id="org8ca234e"> <div class="important" id="org60ccf75">
<p> <p>
How can we explain the voltage offset? How can we explain the voltage offset?
</p> </p>
@ -592,11 +567,11 @@ How can we explain the voltage offset?
</div> </div>
<p> <p>
As shown in Figure <a href="#org6a0d23a">18</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)), an input voltage offset is due to the input bias current \(i_n\). As shown in Figure <a href="#org39e1694">18</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)), an input voltage offset is due to the input bias current \(i_n\).
</p> </p>
<div id="org6a0d23a" class="figure"> <div id="org39e1694" class="figure">
<p><img src="figs/piezo_sensor_model_instrumentation.png" alt="piezo_sensor_model_instrumentation.png" /> <p><img src="figs/piezo_sensor_model_instrumentation.png" alt="piezo_sensor_model_instrumentation.png" />
</p> </p>
<p><span class="figure-number">Figure 18: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p> <p><span class="figure-number">Figure 18: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
@ -632,7 +607,7 @@ If we allow the high pass corner frequency to be equals to 3Hz:
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">fc = 3; <pre class="src src-matlab">fc = 3;
Ra = Rin<span class="org-type">/</span>(fc<span class="org-type">*</span>C<span class="org-type">*</span>Rin <span class="org-type">-</span> 1); Ra = Rin<span class="org-type">/</span>(fc<span class="org-type">*</span>Cp<span class="org-type">*</span>Rin <span class="org-type">-</span> 1);
</pre> </pre>
</div> </div>
@ -660,9 +635,178 @@ Which is much more acceptable.
</div> </div>
</div> </div>
<div id="outline-container-org4df253f" class="outline-3"> <div id="outline-container-org900387e" class="outline-3">
<h3 id="org4df253f"><span class="section-number-3">6.2</span> Sinus</h3> <h3 id="org900387e"><span class="section-number-3">6.2</span> Add Parallel Resistor</h3>
<div class="outline-text-3" id="text-6-2"> <div class="outline-text-3" id="text-6-2">
<p>
A resistor of \(\approx 100\,k\Omega\) is added in parallel with the force sensor and the same kin.
</p>
<div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/force_sensor_steps_R_82k7.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-type">figure</span>;
tiledlayout(2, 1, <span class="org-string">'TileSpacing'</span>, <span class="org-string">'None'</span>, <span class="org-string">'Padding'</span>, <span class="org-string">'None'</span>);
nexttile;
plot(t, v);
xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Measured voltage [V]'</span>);
nexttile;
plot(t, u);
xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Actuator Voltage [V]'</span>);
</pre>
</div>
<div id="orgaee44e2" class="figure">
<p><img src="figs/force_sen_steps_time_domain_par_R.png" alt="force_sen_steps_time_domain_par_R.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Time domain signal during the actuator voltage steps</p>
</div>
<p>
Three steps are performed at the following time intervals:
</p>
<div class="org-src-container">
<pre class="src src-matlab">t_s = [1.9, 6;
8.5, 13;
15.5, 21;
22.6, 26;
30.0, 36;
37.5, 41;
46.2, 49.5]
</pre>
</div>
<p>
Fit function:
</p>
<div class="org-src-container">
<pre class="src src-matlab">f = @(b,x) b(1)<span class="org-type">.*</span>exp(b(2)<span class="org-type">.*</span>x) <span class="org-type">+</span> b(3);
</pre>
</div>
<p>
We are interested by the <code>b(2)</code> term, which is the time constant of the exponential.
</p>
<div class="org-src-container">
<pre class="src src-matlab">tau = zeros(size(t_s, 1),1);
V0 = zeros(size(t_s, 1),1);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">for</span> <span class="org-variable-name">t_i</span> = <span class="org-constant">1:size(t_s, 1)</span>
t_cur = t(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
t_cur = t_cur <span class="org-type">-</span> t_cur(1);
y_cur = v(t_s(t_i, 1) <span class="org-type">&lt;</span> t <span class="org-type">&amp;</span> t <span class="org-type">&lt;</span> t_s(t_i, 2));
nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
B0 = [0.5, <span class="org-type">-</span>0.2, 0.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
tau(t_i) = 1<span class="org-type">/</span>B(2);
V0(t_i) = B(3);
<span class="org-keyword">end</span>
</pre>
</div>
<p>
And indeed, we obtain a much smaller offset voltage and a much faster time constant.
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">\(tau\) [s]</th>
<th scope="col" class="org-right">\(V_0\) [V]</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">0.43</td>
<td class="org-right">0.15</td>
</tr>
<tr>
<td class="org-right">0.45</td>
<td class="org-right">0.16</td>
</tr>
<tr>
<td class="org-right">0.43</td>
<td class="org-right">0.15</td>
</tr>
<tr>
<td class="org-right">0.43</td>
<td class="org-right">0.15</td>
</tr>
<tr>
<td class="org-right">0.45</td>
<td class="org-right">0.15</td>
</tr>
<tr>
<td class="org-right">0.46</td>
<td class="org-right">0.16</td>
</tr>
<tr>
<td class="org-right">0.48</td>
<td class="org-right">0.16</td>
</tr>
</tbody>
</table>
<p>
Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of \(\approx 1\,M\Omega\)):
</p>
<div class="org-src-container">
<pre class="src src-matlab">Cp = 4.4e<span class="org-type">-</span>6; <span class="org-comment">% [F]</span>
Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
</pre>
</div>
<pre class="example">
101200.0
</pre>
<p>
And we can verify that the bias current estimation stays the same:
</p>
<div class="org-src-container">
<pre class="src src-matlab">in = mean(V0)<span class="org-type">/</span>Rin;
</pre>
</div>
<pre class="example">
1.5305e-06
</pre>
<p>
This validates the model of the ADC and the effectiveness of the added resistor.
</p>
</div>
</div>
<div id="outline-container-org7cfe5a8" class="outline-3">
<h3 id="org7cfe5a8"><span class="section-number-3">6.3</span> Sinus</h3>
<div class="outline-text-3" id="text-6-3">
<div class="org-src-container"> <div class="org-src-container">
<pre class="src src-matlab">load(<span class="org-string">'./mat/force_sensor_sin.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>); <pre class="src src-matlab">load(<span class="org-string">'./mat/force_sensor_sin.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>, <span class="org-string">'v'</span>);
@ -674,14 +818,14 @@ t = t(t<span class="org-type">&gt;</span>25);
</div> </div>
<p> <p>
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#orga819354">19</a>). The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#orga42b4f8">20</a>).
</p> </p>
<div id="orga819354" class="figure"> <div id="orga42b4f8" class="figure">
<p><img src="figs/force_sensor_sin_u.png" alt="force_sensor_sin_u.png" /> <p><img src="figs/force_sensor_sin_u.png" alt="force_sensor_sin_u.png" />
</p> </p>
<p><span class="figure-number">Figure 19: </span>Driving Voltage</p> <p><span class="figure-number">Figure 20: </span>Driving Voltage</p>
</div> </div>
<p> <p>
@ -698,14 +842,14 @@ The full stroke as measured by the encoder is:
<p> <p>
Its signal is shown in Figure <a href="#orgde5c100">20</a>. Its signal is shown in Figure <a href="#orgcda82ea">21</a>.
</p> </p>
<div id="orgde5c100" class="figure"> <div id="orgcda82ea" class="figure">
<p><img src="figs/force_sensor_sin_encoder.png" alt="force_sensor_sin_encoder.png" /> <p><img src="figs/force_sensor_sin_encoder.png" alt="force_sensor_sin_encoder.png" />
</p> </p>
<p><span class="figure-number">Figure 20: </span>Encoder measurement</p> <p><span class="figure-number">Figure 21: </span>Encoder measurement</p>
</div> </div>
<p> <p>
@ -713,10 +857,10 @@ The generated voltage by the stack is shown in Figure
</p> </p>
<div id="org1f6aabd" class="figure"> <div id="orgdd5fd9b" class="figure">
<p><img src="figs/force_sensor_sin_stack.png" alt="force_sensor_sin_stack.png" /> <p><img src="figs/force_sensor_sin_stack.png" alt="force_sensor_sin_stack.png" />
</p> </p>
<p><span class="figure-number">Figure 21: </span>Voltage measured on the stack used as a sensor</p> <p><span class="figure-number">Figure 22: </span>Voltage measured on the stack used as a sensor</p>
</div> </div>
<p> <p>
@ -728,18 +872,18 @@ The capacitance of the stack is
</div> </div>
<p> <p>
The corresponding generated charge is then shown in Figure <a href="#orgcfb4c95">22</a>. The corresponding generated charge is then shown in Figure <a href="#org15cf433">23</a>.
</p> </p>
<div id="orgcfb4c95" class="figure"> <div id="org15cf433" class="figure">
<p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" /> <p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" />
</p> </p>
<p><span class="figure-number">Figure 22: </span>Generated Charge</p> <p><span class="figure-number">Figure 23: </span>Generated Charge</p>
</div> </div>
<p> <p>
The relation between the generated voltage and the measured displacement is almost linear as shown in Figure <a href="#orgd584dd5">23</a>. The relation between the generated voltage and the measured displacement is almost linear as shown in Figure <a href="#orge276186">24</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -748,10 +892,10 @@ The relation between the generated voltage and the measured displacement is almo
</div> </div>
<div id="orgd584dd5" class="figure"> <div id="orge276186" class="figure">
<p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" /> <p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" />
</p> </p>
<p><span class="figure-number">Figure 23: </span>Almost linear relation between the relative displacement and the generated voltage</p> <p><span class="figure-number">Figure 24: </span>Almost linear relation between the relative displacement and the generated voltage</p>
</div> </div>
<p> <p>
@ -777,7 +921,7 @@ With a 16bits ADC, the resolution will then be equals to (in [nm]):
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-10-29 jeu. 10:08</p> <p class="date">Created: 2020-11-04 mer. 20:38</p>
</div> </div>
</body> </body>
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@ -551,9 +551,11 @@ The amplifier gain is 20V/V (Cedrat LA75B).
[[file:figs/force_sen_steps_time_domain.png]] [[file:figs/force_sen_steps_time_domain.png]]
Three steps are performed at the following time intervals: Three steps are performed at the following time intervals:
- 2.5, 23 #+begin_src matlab
- 23.8, 35 t_s = [ 2.5, 23;
- 35.8, 50 23.8, 35;
35.8, 50];
#+end_src
Fit function: Fit function:
#+begin_src matlab #+begin_src matlab
@ -561,49 +563,24 @@ Fit function:
#+end_src #+end_src
We are interested by the =b(2)= term, which is the time constant of the exponential. We are interested by the =b(2)= term, which is the time constant of the exponential.
#+begin_src matlab #+begin_src matlab
tau = zeros(3,1); tau = zeros(size(t_s, 1),1);
V0 = zeros(3,1); V0 = zeros(size(t_s, 1),1);
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab
t_cur = t(2.5 < t & t < 23); for t_i = 1:size(t_s, 1)
t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
t_cur = t_cur - t_cur(1); t_cur = t_cur - t_cur(1);
y_cur = v(2.5 < t & t < 23); y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’ [B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
tau(1) = 1/B(2); tau(t_i) = 1/B(2);
V0(1) = B(3); V0(t_i) = B(3);
#+end_src end
#+begin_src matlab
t_cur = t(23.8 < t & t < 35);
t_cur = t_cur - t_cur(1);
y_cur = v(23.8 < t & t < 35);
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
tau(2) = 1/B(2);
V0(2) = B(3);
#+end_src
#+begin_src matlab
t_cur = t(35.8 < t);
t_cur = t_cur - t_cur(1);
y_cur = v(35.8 < t);
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
tau(3) = 1/B(2);
V0(3) = B(3);
#+end_src #+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
@ -665,7 +642,7 @@ An additional resistor in parallel with $R_{in}$ would have two effects:
If we allow the high pass corner frequency to be equals to 3Hz: If we allow the high pass corner frequency to be equals to 3Hz:
#+begin_src matlab #+begin_src matlab
fc = 3; fc = 3;
Ra = Rin/(fc*C*Rin - 1); Ra = Rin/(fc*Cp*Rin - 1);
#+end_src #+end_src
#+begin_src matlab :results value replace :exports results #+begin_src matlab :results value replace :exports results
@ -689,6 +666,116 @@ With this parallel resistance value, the voltage offset would be:
Which is much more acceptable. Which is much more acceptable.
** Add Parallel Resistor
A resistor of $\approx 100\,k\Omega$ is added in parallel with the force sensor and the same kin.
#+begin_src matlab
load('./mat/force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');
#+end_src
#+begin_src matlab
figure;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
plot(t, v);
xlabel('Time [s]'); ylabel('Measured voltage [V]');
nexttile;
plot(t, u);
xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/force_sen_steps_time_domain_par_R.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:force_sen_steps_time_domain_par_R
#+caption: Time domain signal during the actuator voltage steps
#+RESULTS:
[[file:figs/force_sen_steps_time_domain_par_R.png]]
Three steps are performed at the following time intervals:
#+begin_src matlab
t_s = [1.9, 6;
8.5, 13;
15.5, 21;
22.6, 26;
30.0, 36;
37.5, 41;
46.2, 49.5]
#+end_src
Fit function:
#+begin_src matlab
f = @(b,x) b(1).*exp(b(2).*x) + b(3);
#+end_src
We are interested by the =b(2)= term, which is the time constant of the exponential.
#+begin_src matlab
tau = zeros(size(t_s, 1),1);
V0 = zeros(size(t_s, 1),1);
#+end_src
#+begin_src matlab
for t_i = 1:size(t_s, 1)
t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
t_cur = t_cur - t_cur(1);
y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
B0 = [0.5, -0.2, 0.2]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
tau(t_i) = 1/B(2);
V0(t_i) = B(3);
end
#+end_src
And indeed, we obtain a much smaller offset voltage and a much faster time constant.
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
#+end_src
#+RESULTS:
| $tau$ [s] | $V_0$ [V] |
|-----------+-----------|
| 0.43 | 0.15 |
| 0.45 | 0.16 |
| 0.43 | 0.15 |
| 0.43 | 0.15 |
| 0.45 | 0.15 |
| 0.46 | 0.16 |
| 0.48 | 0.16 |
Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1\,M\Omega$):
#+begin_src matlab
Cp = 4.4e-6; % [F]
Rin = abs(mean(tau))/Cp;
#+end_src
#+begin_src matlab :results value replace :exports results
ans = Rin
#+end_src
#+RESULTS:
: 101200.0
And we can verify that the bias current estimation stays the same:
#+begin_src matlab
in = mean(V0)/Rin;
#+end_src
#+begin_src matlab :results value replace :exports results
ans = in
#+end_src
#+RESULTS:
: 1.5305e-06
This validates the model of the ADC and the effectiveness of the added resistor.
** Sinus ** Sinus
#+begin_src matlab #+begin_src matlab
load('./mat/force_sensor_sin.mat', 't', 'encoder', 'u', 'v'); load('./mat/force_sensor_sin.mat', 't', 'encoder', 'u', 'v');