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">
<head>
<!-- 2020-10-29 jeu. 10:08 -->
<!-- 2020-11-04 mer. 20:38 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Encoder - Test Bench</title>
<meta name="generator" content="Org mode" />
@ -35,49 +35,50 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgfe3d6c5">1. Experimental Setup</a></li>
<li><a href="#org85c4a97">2. Huddle Test</a>
<li><a href="#org2b2f049">1. Experimental Setup</a></li>
<li><a href="#org9ae614e">2. Huddle Test</a>
<ul>
<li><a href="#org6f9e775">2.1. Load Data</a></li>
<li><a href="#orgd27ba42">2.2. Time Domain Results</a></li>
<li><a href="#org963a243">2.3. Frequency Domain Noise</a></li>
<li><a href="#orgc034802">2.1. Load Data</a></li>
<li><a href="#orgd8ae4ae">2.2. Time Domain Results</a></li>
<li><a href="#org864c60c">2.3. Frequency Domain Noise</a></li>
</ul>
</li>
<li><a href="#org0e0c48a">3. Comparison Interferometer / Encoder</a>
<li><a href="#org96539c5">3. Comparison Interferometer / Encoder</a>
<ul>
<li><a href="#orgc15a506">3.1. Load Data</a></li>
<li><a href="#orgc509698">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="#orgcb56769">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
<li><a href="#orgc6453b2">3.1. Load Data</a></li>
<li><a href="#orgc362641">3.2. Time Domain Results</a></li>
<li><a href="#orgb50503c">3.3. Difference between Encoder and Interferometer as a function of time</a></li>
<li><a href="#org5ec03aa">3.4. Difference between Encoder and Interferometer as a function of position</a></li>
</ul>
</li>
<li><a href="#org8399536">4. Identification</a>
<li><a href="#org6b73303">4. Identification</a>
<ul>
<li><a href="#org4b364a5">4.1. Load Data</a></li>
<li><a href="#org0dd3820">4.2. Identification</a></li>
<li><a href="#orga15e9e2">4.1. Load Data</a></li>
<li><a href="#org6974653">4.2. Identification</a></li>
</ul>
</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>
<li><a href="#orgba85fb9">5.1. Load Data</a></li>
<li><a href="#org2876c52">5.2. Transfer Functions</a></li>
<li><a href="#org00fc76c">5.1. Load Data</a></li>
<li><a href="#org757917f">5.2. Transfer Functions</a></li>
</ul>
</li>
<li><a href="#org1abff1f">6. Generated Number of Charge / Voltage</a>
<li><a href="#org8dab16f">6. Generated Number of Charge / Voltage</a>
<ul>
<li><a href="#org65e8206">6.1. Steps</a></li>
<li><a href="#org4df253f">6.2. Sinus</a></li>
<li><a href="#org28149b5">6.1. Steps</a></li>
<li><a href="#org900387e">6.2. Add Parallel Resistor</a></li>
<li><a href="#org7cfe5a8">6.3. Sinus</a></li>
</ul>
</li>
</ul>
</div>
</div>
<div id="outline-container-orgfe3d6c5" class="outline-2">
<h2 id="orgfe3d6c5"><span class="section-number-2">1</span> Experimental Setup</h2>
<div id="outline-container-org2b2f049" class="outline-2">
<h2 id="org2b2f049"><span class="section-number-2">1</span> Experimental Setup</h2>
<div class="outline-text-2" id="text-1">
<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>
@ -86,21 +87,21 @@ The displacement of the mass (relative to the mechanical frame) is measured both
</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>
<p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p>
</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>
<p><span class="figure-number">Figure 2: </span>Side View of the encoder</p>
</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>
<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 id="outline-container-org85c4a97" class="outline-2">
<h2 id="org85c4a97"><span class="section-number-2">2</span> Huddle Test</h2>
<div id="outline-container-org9ae614e" class="outline-2">
<h2 id="org9ae614e"><span class="section-number-2">2</span> Huddle Test</h2>
<div class="outline-text-2" id="text-2">
<p>
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>
</div>
<div id="outline-container-org6f9e775" class="outline-3">
<h3 id="org6f9e775"><span class="section-number-3">2.1</span> Load Data</h3>
<div id="outline-container-orgc034802" class="outline-3">
<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="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>);
@ -137,11 +138,11 @@ encoder = detrend(encoder, 0);
</div>
</div>
<div id="outline-container-orgd27ba42" class="outline-3">
<h3 id="orgd27ba42"><span class="section-number-3">2.2</span> Time Domain Results</h3>
<div id="outline-container-orgd8ae4ae" class="outline-3">
<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 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>
<p><span class="figure-number">Figure 4: </span>Huddle test - Time domain signals</p>
@ -153,7 +154,7 @@ encoder = detrend(encoder, 0);
</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>
<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 id="outline-container-org963a243" class="outline-3">
<h3 id="org963a243"><span class="section-number-3">2.3</span> Frequency Domain Noise</h3>
<div id="outline-container-org864c60c" class="outline-3">
<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="org-src-container">
<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 id="orgb6a29d5" class="figure">
<div id="org63bcf43" class="figure">
<p><img src="figs/huddle_test_asd.png" alt="huddle_test_asd.png" />
</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 id="outline-container-org0e0c48a" class="outline-2">
<h2 id="org0e0c48a"><span class="section-number-2">3</span> Comparison Interferometer / Encoder</h2>
<div id="outline-container-org96539c5" class="outline-2">
<h2 id="org96539c5"><span class="section-number-2">3</span> Comparison Interferometer / Encoder</h2>
<div class="outline-text-2" id="text-3">
<p>
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>
</div>
<div id="outline-container-orgc15a506" class="outline-3">
<h3 id="orgc15a506"><span class="section-number-3">3.1</span> Load Data</h3>
<div id="outline-container-orgc6453b2" class="outline-3">
<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="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>);
@ -209,18 +210,18 @@ u = detrend(u, 0);
</div>
</div>
<div id="outline-container-orgc509698" class="outline-3">
<h3 id="orgc509698"><span class="section-number-3">3.2</span> Time Domain Results</h3>
<div id="outline-container-orgc362641" class="outline-3">
<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 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>
<p><span class="figure-number">Figure 7: </span>One cycle measurement</p>
</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>
<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 id="outline-container-org1b5953a" 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>
<div id="outline-container-orgb50503c" class="outline-3">
<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="org-src-container">
<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 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>
<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 id="outline-container-orgcb56769" 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>
<div id="outline-container-org5ec03aa" class="outline-3">
<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">
<p>
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 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>
<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 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>
<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 id="outline-container-org8399536" class="outline-2">
<h2 id="org8399536"><span class="section-number-2">4</span> Identification</h2>
<div id="outline-container-org6b73303" class="outline-2">
<h2 id="org6b73303"><span class="section-number-2">4</span> Identification</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org4b364a5" class="outline-3">
<h3 id="org4b364a5"><span class="section-number-3">4.1</span> Load Data</h3>
<div id="outline-container-orga15e9e2" class="outline-3">
<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="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>);
@ -329,8 +330,8 @@ u = detrend(u, 0);
</div>
</div>
<div id="outline-container-org0dd3820" class="outline-3">
<h3 id="org0dd3820"><span class="section-number-3">4.2</span> Identification</h3>
<div id="outline-container-org6974653" class="outline-3">
<h3 id="org6974653"><span class="section-number-3">4.2</span> Identification</h3>
<div class="outline-text-3" id="text-4-2">
<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>
@ -348,14 +349,14 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</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>
</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>
</div>
@ -363,12 +364,12 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div>
</div>
<div id="outline-container-org4a7e08f" 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>
<div id="outline-container-org5be1a0f" class="outline-2">
<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>
<div id="outline-container-orgba85fb9" class="outline-3">
<h3 id="orgba85fb9"><span class="section-number-3">5.1</span> Load Data</h3>
<div id="outline-container-org00fc76c" class="outline-3">
<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="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>);
@ -378,8 +379,8 @@ sc = load(<span class="org-string">'./mat/identification_short_circuit.mat'</spa
</div>
</div>
<div id="outline-container-org2876c52" class="outline-3">
<h3 id="org2876c52"><span class="section-number-3">5.2</span> Transfer Functions</h3>
<div id="outline-container-org757917f" class="outline-3">
<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="org-src-container">
<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 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>
</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>
</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>
<p><span class="figure-number">Figure 16: </span>Zoom on the change of resonance</p>
</div>
<div class="important" id="org5981596">
<div class="important" id="orga5e301b">
<p>
The change of resonance frequency / stiffness is very small and is not important here.
</p>
@ -426,8 +427,8 @@ The change of resonance frequency / stiffness is very small and is not important
</div>
</div>
<div id="outline-container-org1abff1f" class="outline-2">
<h2 id="org1abff1f"><span class="section-number-2">6</span> Generated Number of Charge / Voltage</h2>
<div id="outline-container-org8dab16f" class="outline-2">
<h2 id="org8dab16f"><span class="section-number-2">6</span> Generated Number of Charge / Voltage</h2>
<div class="outline-text-2" id="text-6">
<p>
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>
</div>
<div id="outline-container-org65e8206" class="outline-3">
<h3 id="org65e8206"><span class="section-number-3">6.1</span> Steps</h3>
<div id="outline-container-org28149b5" class="outline-3">
<h3 id="org28149b5"><span class="section-number-3">6.1</span> Steps</h3>
<div class="outline-text-3" id="text-6-1">
<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>);
@ -459,7 +460,7 @@ xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-stri
</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>
<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>
Three steps are performed at the following time intervals:
</p>
<ul class="org-ul">
<li>2.5, 23</li>
<li>23.8, 35</li>
<li>35.8, 50</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">t_s = [ 2.5, 23;
23.8, 35;
35.8, 50];
</pre>
</div>
<p>
Fit function:
@ -485,52 +487,25 @@ Fit function:
<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(3,1);
V0 = zeros(3,1);
<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">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);
t_cur = t_cur <span class="org-type">-</span> t_cur(1);
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);
<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>0.15, 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>
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">(1) </span>= 1<span class="org-type">/</span>B(2);
V0<span class="org-type">(1) </span>= B(3);
</pre>
</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);
tau(t_i) = 1<span class="org-type">/</span>B(2);
V0(t_i) = B(3);
<span class="org-keyword">end</span>
</pre>
</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 \(1\,M\Omega\)).
</p>
<div class="important" id="org8ca234e">
<div class="important" id="org60ccf75">
<p>
How can we explain the voltage offset?
</p>
@ -592,11 +567,11 @@ How can we explain the voltage offset?
</div>
<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>
<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>
<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>
<div class="org-src-container">
<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>
</div>
@ -660,9 +635,178 @@ Which is much more acceptable.
</div>
</div>
<div id="outline-container-org4df253f" class="outline-3">
<h3 id="org4df253f"><span class="section-number-3">6.2</span> Sinus</h3>
<div id="outline-container-org900387e" class="outline-3">
<h3 id="org900387e"><span class="section-number-3">6.2</span> Add Parallel Resistor</h3>
<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">
<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>
<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>
<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>
<p><span class="figure-number">Figure 19: </span>Driving Voltage</p>
<p><span class="figure-number">Figure 20: </span>Driving Voltage</p>
</div>
<p>
@ -698,14 +842,14 @@ The full stroke as measured by the encoder is:
<p>
Its signal is shown in Figure <a href="#orgde5c100">20</a>.
Its signal is shown in Figure <a href="#orgcda82ea">21</a>.
</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>
<p><span class="figure-number">Figure 20: </span>Encoder measurement</p>
<p><span class="figure-number">Figure 21: </span>Encoder measurement</p>
</div>
<p>
@ -713,10 +857,10 @@ The generated voltage by the stack is shown in Figure
</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>
<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>
<p>
@ -728,18 +872,18 @@ The capacitance of the stack is
</div>
<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>
<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>
<p><span class="figure-number">Figure 22: </span>Generated Charge</p>
<p><span class="figure-number">Figure 23: </span>Generated Charge</p>
</div>
<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>
<div class="org-src-container">
@ -748,10 +892,10 @@ The relation between the generated voltage and the measured displacement is almo
</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>
<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>
<p>
@ -777,7 +921,7 @@ With a 16bits ADC, the resolution will then be equals to (in [nm]):
</div>
<div id="postamble" class="status">
<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>
</body>
</html>

169
index.org
View File

@ -551,9 +551,11 @@ The amplifier gain is 20V/V (Cedrat LA75B).
[[file:figs/force_sen_steps_time_domain.png]]
Three steps are performed at the following time intervals:
- 2.5, 23
- 23.8, 35
- 35.8, 50
#+begin_src matlab
t_s = [ 2.5, 23;
23.8, 35;
35.8, 50];
#+end_src
Fit function:
#+begin_src matlab
@ -561,49 +563,24 @@ Fit function:
#+end_src
We are interested by the =b(2)= term, which is the time constant of the exponential.
#+begin_src matlab
tau = zeros(3,1);
V0 = zeros(3,1);
tau = zeros(size(t_s, 1),1);
V0 = zeros(size(t_s, 1),1);
#+end_src
#+begin_src matlab
t_cur = t(2.5 < t & t < 23);
t_cur = t_cur - t_cur(1);
y_cur = v(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);
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, -0.15, 2.2]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters B
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(1) = 1/B(2);
V0(1) = B(3);
#+end_src
#+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);
tau(t_i) = 1/B(2);
V0(t_i) = B(3);
end
#+end_src
#+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:
#+begin_src matlab
fc = 3;
Ra = Rin/(fc*C*Rin - 1);
Ra = Rin/(fc*Cp*Rin - 1);
#+end_src
#+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.
** 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
#+begin_src matlab
load('./mat/force_sensor_sin.mat', 't', 'encoder', 'u', 'v');