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20 KiB
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638 lines
20 KiB
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<!-- 2020-11-10 mar. 12:55 -->
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<title>Piezoelectric Force Sensor - Test Bench</title>
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<meta name="author" content="Dehaeze Thomas" />
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<body>
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<div id="org-div-home-and-up">
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<a accesskey="h" href="../index.html"> UP </a>
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<a accesskey="H" href="../index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Piezoelectric Force Sensor - Test Bench</h1>
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<div id="table-of-contents">
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<h2>Table of Contents</h2>
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<div id="text-table-of-contents">
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<ul>
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<li><a href="#orga1465ad">1. Change of Stiffness due to Sensors stack being open/closed circuit</a>
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<ul>
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<li><a href="#orgd924c73">1.1. Load Data</a></li>
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<li><a href="#org59cc20a">1.2. Transfer Functions</a></li>
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</ul>
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</li>
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<li><a href="#org76a1832">2. Generated Number of Charge / Voltage</a>
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<ul>
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<li><a href="#org1fa991d">2.1. Steps</a></li>
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<li><a href="#org5e9eb44">2.2. Add Parallel Resistor</a></li>
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<li><a href="#org15676e1">2.3. Sinus</a></li>
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</ul>
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</li>
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</ul>
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</div>
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</div>
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<p>
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In this document is studied how a piezoelectric stack can be used to measured the force.
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org887b61a">1</a>: the effect of the input impedance of the electronics connected to the force sensor stack on the stiffness of the stack is studied</li>
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<li>Section <a href="#org2b5f630">2</a>:</li>
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</ul>
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<div id="outline-container-orga1465ad" class="outline-2">
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<h2 id="orga1465ad"><span class="section-number-2">1</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
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<div class="outline-text-2" id="text-1">
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<p>
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<a id="org887b61a"></a>
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</p>
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</div>
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<div id="outline-container-orgd924c73" class="outline-3">
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<h3 id="orgd924c73"><span class="section-number-3">1.1</span> Load Data</h3>
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<div class="outline-text-3" id="text-1-1">
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<div class="org-src-container">
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<pre class="src src-matlab">oc = load(<span class="org-string">'identification_open_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>);
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sc = load(<span class="org-string">'identification_short_circuit.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'encoder'</span>, <span class="org-string">'u'</span>);
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</pre>
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</div>
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</div>
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</div>
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<div id="outline-container-org59cc20a" class="outline-3">
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<h3 id="org59cc20a"><span class="section-number-3">1.2</span> Transfer Functions</h3>
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<div class="outline-text-3" id="text-1-2">
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<div class="org-src-container">
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<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
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win = hann(ceil(10<span class="org-type">/</span>Ts));
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab">[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
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[co_oc_est, <span class="org-type">~</span>] = mscohere( oc.u, oc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
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[tf_sc_est, <span class="org-type">~</span>] = tfestimate(sc.u, sc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
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[co_sc_est, <span class="org-type">~</span>] = mscohere( sc.u, sc.encoder, win, [], [], 1<span class="org-type">/</span>Ts);
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</pre>
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</div>
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<div id="org3c75143" class="figure">
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<p><img src="figs/stiffness_force_sensor_coherence.png" alt="stiffness_force_sensor_coherence.png" />
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</p>
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</div>
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<div id="org4424b1c" class="figure">
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<p><img src="figs/stiffness_force_sensor_bode.png" alt="stiffness_force_sensor_bode.png" />
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</p>
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</div>
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<div id="org216fcc3" class="figure">
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<p><img src="figs/stiffness_force_sensor_bode_zoom.png" alt="stiffness_force_sensor_bode_zoom.png" />
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</p>
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<p><span class="figure-number">Figure 3: </span>Zoom on the change of resonance</p>
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</div>
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<div class="important" id="org9ea3712">
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<p>
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The change of resonance frequency / stiffness is very small and is not important here.
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</p>
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</div>
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</div>
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</div>
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</div>
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<div id="outline-container-org76a1832" class="outline-2">
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<h2 id="org76a1832"><span class="section-number-2">2</span> Generated Number of Charge / Voltage</h2>
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<div class="outline-text-2" id="text-2">
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<p>
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<a id="org2b5f630"></a>
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</p>
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<p>
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Two stacks are used as actuator (in parallel) and one stack is used as sensor.
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</p>
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<p>
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The amplifier gain is 20V/V (Cedrat LA75B).
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</p>
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</div>
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<div id="outline-container-org1fa991d" class="outline-3">
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<h3 id="org1fa991d"><span class="section-number-3">2.1</span> Steps</h3>
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<div class="outline-text-3" id="text-2-1">
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<div class="org-src-container">
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<pre class="src src-matlab">load(<span class="org-string">'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>);
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</pre>
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</div>
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<div class="org-src-container">
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<pre class="src src-matlab"><span class="org-type">figure</span>;
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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>);
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nexttile;
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plot(t, v);
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xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Measured voltage [V]'</span>);
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nexttile;
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plot(t, u);
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xlabel(<span class="org-string">'Time [s]'</span>); ylabel(<span class="org-string">'Actuator Voltage [V]'</span>);
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</pre>
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</div>
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<div id="orgf889803" class="figure">
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<p><img src="figs/force_sen_steps_time_domain.png" alt="force_sen_steps_time_domain.png" />
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</p>
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<p><span class="figure-number">Figure 4: </span>Time domain signal during the 3 actuator voltage steps</p>
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</div>
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<p>
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Three steps are performed at the following time intervals:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">t_s = [ 2.5, 23;
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23.8, 35;
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35.8, 50];
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</pre>
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</div>
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<p>
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Fit function:
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</p>
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<div class="org-src-container">
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<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);
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</pre>
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</div>
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<p>
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We are interested by the <code>b(2)</code> term, which is the time constant of the exponential.
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">tau = zeros(size(t_s, 1),1);
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V0 = zeros(size(t_s, 1),1);
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</pre>
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</div>
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<div class="org-src-container">
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<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>
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t_cur = t(t_s(t_i, 1) <span class="org-type"><</span> t <span class="org-type">&</span> t <span class="org-type"><</span> t_s(t_i, 2));
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t_cur = t_cur <span class="org-type">-</span> t_cur(1);
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y_cur = v(t_s(t_i, 1) <span class="org-type"><</span> t <span class="org-type">&</span> t <span class="org-type"><</span> t_s(t_i, 2));
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nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
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B0 = [0.5, <span class="org-type">-</span>0.15, 2.2]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
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[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters ‘B’</span>
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tau(t_i) = 1<span class="org-type">/</span>B(2);
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V0(t_i) = B(3);
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<span class="org-keyword">end</span>
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</pre>
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</div>
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<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
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<colgroup>
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<col class="org-right" />
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<col class="org-right" />
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</colgroup>
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<thead>
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<tr>
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<th scope="col" class="org-right">\(tau\) [s]</th>
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<th scope="col" class="org-right">\(V_0\) [V]</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td class="org-right">6.47</td>
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<td class="org-right">2.26</td>
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</tr>
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<tr>
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<td class="org-right">6.76</td>
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<td class="org-right">2.26</td>
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</tr>
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<tr>
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<td class="org-right">6.49</td>
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<td class="org-right">2.25</td>
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</tr>
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</tbody>
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</table>
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<p>
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With the capacitance being \(C = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">Cp = 4.4e<span class="org-type">-</span>6; <span class="org-comment">% [F]</span>
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Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
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</pre>
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</div>
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<pre class="example">
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1494100.0
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</pre>
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<p>
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The input impedance of the Speedgoat’s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)).
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</p>
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<div class="important" id="org572654b">
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<p>
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How can we explain the voltage offset?
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</p>
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</div>
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<p>
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As shown in Figure <a href="#org7c2c57f">5</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 \(i_n\).
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</p>
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<div id="org7c2c57f" class="figure">
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<p><img src="figs/force_sensor_model_electronics.png" alt="force_sensor_model_electronics.png" />
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</p>
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<p><span class="figure-number">Figure 5: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
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</div>
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<p>
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The estimated input bias current is then:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">in = mean(V0)<span class="org-type">/</span>Rin;
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</pre>
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</div>
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<pre class="example">
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1.5119e-06
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</pre>
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<p>
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An additional resistor in parallel with \(R_{in}\) would have two effects:
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</p>
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<ul class="org-ul">
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<li>reduce the input voltage offset
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\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]</li>
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<li>increase the high pass corner frequency \(f_c\)
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\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
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\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]</li>
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</ul>
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<p>
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If we allow the high pass corner frequency to be equals to 3Hz:
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</p>
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<div class="org-src-container">
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<pre class="src src-matlab">fc = 3;
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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>
|
||
|
|
|
||
|
|
<pre class="example">
|
||
|
|
79804
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
With this parallel resistance value, the voltage offset would be:
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">V_offset = Ra<span class="org-type">*</span>Rin<span class="org-type">/</span>(Ra <span class="org-type">+</span> Rin) <span class="org-type">*</span> in;
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<pre class="example">
|
||
|
|
0.11454
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Which is much more acceptable.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div id="outline-container-org5e9eb44" class="outline-3">
|
||
|
|
<h3 id="org5e9eb44"><span class="section-number-3">2.2</span> Add Parallel Resistor</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-2">
|
||
|
|
<p>
|
||
|
|
A resistor \(R_p \approx 100\,k\Omega\) is added in parallel with the force sensor as shown in Figure <a href="#org1fac5a7">6</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org1fac5a7" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_model_electronics_without_R.png" alt="force_sensor_model_electronics_without_R.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 6: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with added resistor \(R_p\)</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">load(<span class="org-string">'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="org29964b5" 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 7: </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"><</span> t <span class="org-type">&</span> t <span class="org-type"><</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"><</span> t <span class="org-type">&</span> t <span class="org-type"><</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 ‘B’</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-org15676e1" class="outline-3">
|
||
|
|
<h3 id="org15676e1"><span class="section-number-3">2.3</span> Sinus</h3>
|
||
|
|
<div class="outline-text-3" id="text-2-3">
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">load(<span class="org-string">'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>);
|
||
|
|
|
||
|
|
u = u(t<span class="org-type">></span>25);
|
||
|
|
v = v(t<span class="org-type">></span>25);
|
||
|
|
encoder = encoder(t<span class="org-type">></span>25) <span class="org-type">-</span> mean(encoder(t<span class="org-type">></span>25));
|
||
|
|
t = t(t<span class="org-type">></span>25);
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#org1fbf89d">8</a>).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org1fbf89d" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_sin_u.png" alt="force_sensor_sin_u.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 8: </span>Driving Voltage</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The full stroke as measured by the encoder is:
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">max(encoder)<span class="org-type">-</span>min(encoder)
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<pre class="example">
|
||
|
|
5.005e-05
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
|
||
|
|
<p>
|
||
|
|
Its signal is shown in Figure <a href="#org1d74efa">9</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org1d74efa" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_sin_encoder.png" alt="force_sensor_sin_encoder.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 9: </span>Encoder measurement</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The generated voltage by the stack is shown in Figure
|
||
|
|
</p>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org077a6d7" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_sin_stack.png" alt="force_sensor_sin_stack.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 10: </span>Voltage measured on the stack used as a sensor</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The capacitance of the stack is
|
||
|
|
</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>
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
The corresponding generated charge is then shown in Figure <a href="#org4baf062">11</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div id="org4baf062" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 11: </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="#org8b9df34">12</a>.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">b1 = encoder<span class="org-type">\</span>(v<span class="org-type">-</span>mean(v));
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
<div id="org8b9df34" class="figure">
|
||
|
|
<p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" />
|
||
|
|
</p>
|
||
|
|
<p><span class="figure-number">Figure 12: </span>Almost linear relation between the relative displacement and the generated voltage</p>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<p>
|
||
|
|
With a 16bits ADC, the resolution will then be equals to (in [nm]):
|
||
|
|
</p>
|
||
|
|
<div class="org-src-container">
|
||
|
|
<pre class="src src-matlab">abs((20<span class="org-type">/</span>2<span class="org-type">^</span>16)<span class="org-type">/</span>(b1<span class="org-type">/</span>1e9))
|
||
|
|
</pre>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
<pre class="example">
|
||
|
|
3.9838
|
||
|
|
</pre>
|
||
|
|
|
||
|
|
|
||
|
|
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><h2 class='citeproc-org-bib-h2'>Bibliography</h2>
|
||
|
|
<div class="csl-bib-body">
|
||
|
|
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Reza, Moheimani, and Fleming Andrew. 2006. <i>Piezoelectric Transducers for Vibration Control and Damping</i>. London: Springer.</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
</div>
|
||
|
|
<div id="postamble" class="status">
|
||
|
|
<p class="author">Author: Dehaeze Thomas</p>
|
||
|
|
<p class="date">Created: 2020-11-10 mar. 12:55</p>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|