Add test / doc / list of sections
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index.html
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index.html
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
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<head>
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<!-- 2020-11-10 mar. 13:00 -->
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<!-- 2020-11-10 mar. 13:41 -->
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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
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<title>Piezoelectric Force Sensor - Test Bench</title>
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<meta name="generator" content="Org mode" />
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@@ -34,17 +34,29 @@
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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="#orgfa4ffe0">1. Change of Stiffness due to Sensors stack being open/closed circuit</a>
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<li><a href="#orga887666">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="#org664356d">1.1. Load Data</a></li>
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<li><a href="#orgb329298">1.2. Transfer Functions</a></li>
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<li><a href="#org632bb0b">1.1. Load Data</a></li>
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<li><a href="#org06862e1">1.2. Transfer Functions</a></li>
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</ul>
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</li>
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<li><a href="#orgcc12929">2. Generated Number of Charge / Voltage</a>
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<li><a href="#orgd3fb351">2. Effect of a Resistor in Parallel with the Stack Sensor</a>
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<ul>
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<li><a href="#org7a46587">2.1. Steps</a></li>
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<li><a href="#org9938615">2.2. Add Parallel Resistor</a></li>
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<li><a href="#org3e71d2e">2.3. Sinus</a></li>
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||||
<li><a href="#orgf3e29cf">2.1. Excitation steps and measured generated voltage</a></li>
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||||
<li><a href="#org93b0595">2.2. Estimation of the voltage offset and discharge time constant</a></li>
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<li><a href="#orgc749ef4">2.3. Estimation of the ADC input impedance</a></li>
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<li><a href="#org66bf743">2.4. Explanation of the Voltage offset</a></li>
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||||
<li><a href="#org3c86684">2.5. Effect of an additional Parallel Resistor</a></li>
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<li><a href="#org4ae0b53">2.6. Obtained voltage offset and time constant with the added resistor</a></li>
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</ul>
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</li>
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<li><a href="#org81c5db6">3. Generated Number of Charge / Voltage</a>
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<ul>
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<li><a href="#org55eecf7">3.1. Data Loading</a></li>
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<li><a href="#org46592d5">3.2. Excitation signal and corresponding displacement</a></li>
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<li><a href="#org6e6b1f7">3.3. Generated Voltage</a></li>
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<li><a href="#org4ed9610">3.4. Generated Charge</a></li>
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<li><a href="#org69bd84e">3.5. Generated Voltage/Charge as a function of the displacement</a></li>
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</ul>
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</li>
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</ul>
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@@ -55,21 +67,61 @@
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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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<p>
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It is divided in the following sections:
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</p>
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<ul class="org-ul">
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<li>Section <a href="#org574ce5b">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="#org3d96d6c">2</a>:</li>
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<li>Section <a href="#orgcbe0c6f">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="#org72d0a53">2</a>: the effect of a resistor in parallel with the sensor stack is studied</li>
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<li>Section <a href="#org22b743e">3</a>: the voltage / number of charge generated by the sensor as a function of the displacement is measured</li>
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</ul>
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<div id="outline-container-orgfa4ffe0" class="outline-2">
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<h2 id="orgfa4ffe0"><span class="section-number-2">1</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
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<div id="outline-container-orga887666" class="outline-2">
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<h2 id="orga887666"><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="org574ce5b"></a>
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<a id="orgcbe0c6f"></a>
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</p>
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<p>
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The experimental Setup is schematically represented in Figure <a href="#org27d7f75">1</a>.
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</p>
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<p>
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The dynamics from the voltage \(u\) used to drive the actuator stacks to the encoder displacement \(d_e\) is identified when the switch connected to the sensor stack is either open or closed.
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</p>
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<div id="org27d7f75" class="figure">
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<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
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</p>
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||||
<p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p>
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</div>
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<div id="outline-container-org664356d" class="outline-3">
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<h3 id="org664356d"><span class="section-number-3">1.1</span> Load Data</h3>
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<p>
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When the switch is opened, this correspond of having a measurement electronics with an high input impedance such as a <b>voltage</b> amplifier.
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When the switch is closed, this correspond of having a measurement electronics with an small input impedance such as a <b>charge</b> amplifier.
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</p>
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<p>
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We wish here to see how the system dynamics is changing in the two extreme cases.
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</p>
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<div class="note" id="org4171ed0">
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<p>
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The equipment used in the test bench are:
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</p>
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<ul class="org-ul">
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<li>Renishaw Resolution Encoder with 1nm resolution (<a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">doc</a>)</li>
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<li>Cedrat Amplified Piezoelectric Actuator APA95ML (<a href="doc/APA95ML.pdf">doc</a>)</li>
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<li>Voltage Amplifier LA75B (<a href="doc/LA75B.pdf">doc</a>)</li>
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<li>Speedgoat IO131 with 16bits ADC and DAC (<a href="doc/IO130 IO131 OEM Datasheet.pdf">doc</a>)</li>
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</ul>
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</div>
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</div>
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<div id="outline-container-org632bb0b" class="outline-3">
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<h3 id="org632bb0b"><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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@@ -79,8 +131,8 @@ sc = load(<span class="org-string">'identification_short_circuit.mat'</span>, <s
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</div>
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</div>
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<div id="outline-container-orgb329298" class="outline-3">
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<h3 id="orgb329298"><span class="section-number-3">1.2</span> Transfer Functions</h3>
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||||
<div id="outline-container-org06862e1" class="outline-3">
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||||
<h3 id="org06862e1"><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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||||
@@ -98,26 +150,26 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
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</div>
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||||
<div id="org07156da" class="figure">
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||||
<div id="orgaafdea3" 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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||||
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||||
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||||
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||||
<div id="org4e249d7" class="figure">
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||||
<div id="orgc2be608" 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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||||
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||||
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||||
<div id="org503058b" class="figure">
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||||
<div id="org59bf971" 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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<p><span class="figure-number">Figure 4: </span>Zoom on the change of resonance</p>
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</div>
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<div class="important" id="org93bdd08">
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<div class="important" id="org2d5ecc1">
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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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||||
@@ -127,46 +179,76 @@ The change of resonance frequency / stiffness is very small and is not important
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</div>
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</div>
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||||
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||||
<div id="outline-container-orgcc12929" class="outline-2">
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||||
<h2 id="orgcc12929"><span class="section-number-2">2</span> Generated Number of Charge / Voltage</h2>
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||||
<div id="outline-container-orgd3fb351" class="outline-2">
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||||
<h2 id="orgd3fb351"><span class="section-number-2">2</span> Effect of a Resistor in Parallel with the Stack Sensor</h2>
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||||
<div class="outline-text-2" id="text-2">
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||||
<p>
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||||
<a id="org3d96d6c"></a>
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||||
<a id="org72d0a53"></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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||||
The setup is shown in Figure <a href="#org21d0104">5</a> where two stacks are used as actuator (in parallel) and one stack is used as sensor.
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||||
The voltage amplifier used has a gain of 20 [V/V] (Cedrat LA75B).
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||||
</p>
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||||
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||||
<p>
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||||
The amplifier gain is 20V/V (Cedrat LA75B).
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||||
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||||
<div id="org21d0104" class="figure">
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||||
<p><img src="figs/force_sensor_setup.png" alt="force_sensor_setup.png" />
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||||
</p>
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||||
<p><span class="figure-number">Figure 5: </span>Schematic of the setup</p>
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||||
</div>
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||||
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||||
<div id="outline-container-org7a46587" class="outline-3">
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||||
<h3 id="org7a46587"><span class="section-number-3">2.1</span> Steps</h3>
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||||
<div class="note" id="orgdd0ed7a">
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||||
<p>
|
||||
The equipment used in the test bench are:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>Renishaw Resolution Encoder with 1nm resolution (<a href="doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf">doc</a>)</li>
|
||||
<li>Cedrat Amplified Piezoelectric Actuator APA95ML (<a href="doc/APA95ML.pdf">doc</a>)</li>
|
||||
<li>Voltage Amplifier LA75B (<a href="doc/LA75B.pdf">doc</a>)</li>
|
||||
<li>Speedgoat IO131 with 16bits ADC and DAC (<a href="doc/IO130 IO131 OEM Datasheet.pdf">doc</a>)</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
</div>
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||||
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||||
<div id="outline-container-orgf3e29cf" class="outline-3">
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||||
<h3 id="orgf3e29cf"><span class="section-number-3">2.1</span> Excitation steps and measured generated voltage</h3>
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||||
<div class="outline-text-3" id="text-2-1">
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||||
<p>
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||||
The measured data is loaded.
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||||
</p>
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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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||||
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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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||||
<p>
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||||
The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure <a href="#org8a7c66b">6</a>.
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||||
</p>
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||||
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||||
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||||
<div id="orgda3ef57" class="figure">
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||||
<div id="org8a7c66b" 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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||||
<p><span class="figure-number">Figure 6: </span>Time domain signal during the 3 actuator voltage steps</p>
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||||
</div>
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||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-org93b0595" class="outline-3">
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||||
<h3 id="org93b0595"><span class="section-number-3">2.2</span> Estimation of the voltage offset and discharge time constant</h3>
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||||
<div class="outline-text-3" id="text-2-2">
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||||
<p>
|
||||
The measured voltage shows an exponential decay which indicates that the charge across the capacitor formed by the stack is discharging into a resistor.
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||||
This corresponds to an RC circuit with a time constant \(\tau = RC\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In order to estimate the time domain, we fit the data with an exponential.
|
||||
The fit function is:
|
||||
</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>
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||||
|
||||
<p>
|
||||
@@ -179,14 +261,6 @@ Three steps are performed at the following time intervals:
|
||||
</pre>
|
||||
</div>
|
||||
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||||
<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>
|
||||
@@ -212,6 +286,10 @@ V0 = zeros(size(t_s, 1),1);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained values are shown below.
|
||||
</p>
|
||||
|
||||
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
|
||||
|
||||
@@ -243,7 +321,12 @@ V0 = zeros(size(t_s, 1),1);
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc749ef4" class="outline-3">
|
||||
<h3 id="orgc749ef4"><span class="section-number-3">2.3</span> Estimation of the ADC input impedance</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
With the capacitance being \(C = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows:
|
||||
</p>
|
||||
@@ -261,23 +344,30 @@ Rin = abs(mean(tau))<span class="org-type">/</span>Cp;
|
||||
<p>
|
||||
The input impedance of the Speedgoat’s ADC should then be close to \(1.5\,M\Omega\) (specified at \(1\,M\Omega\)).
|
||||
</p>
|
||||
|
||||
<div class="important" id="org7c6e263">
|
||||
<p>
|
||||
How can we explain the voltage offset?
|
||||
</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org66bf743" class="outline-3">
|
||||
<h3 id="org66bf743"><span class="section-number-3">2.4</span> Explanation of the Voltage offset</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
As shown in Figure <a href="#org1036a3b">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 \(i_n\).
|
||||
As shown in Figure <a href="#org8a7c66b">6</a>, the voltage across the Piezoelectric sensor stack shows a constant voltage offset.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
We can explain this offset by looking at the electrical model shown in Figure <a href="#orgc43b1a9">7</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The differential amplifier in the Speedgoat has some input bias current \(i_n\) that produces a voltage offset across its own internal resistance.
|
||||
Note that the impedance of the piezoelectric stack is much larger that that at DC.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org1036a3b" class="figure">
|
||||
<div id="orgc43b1a9" 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 5: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
|
||||
<p><span class="figure-number">Figure 7: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@@ -291,10 +381,14 @@ The estimated input bias current is then:
|
||||
<pre class="example">
|
||||
1.5119e-06
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org3c86684" class="outline-3">
|
||||
<h3 id="org3c86684"><span class="section-number-3">2.5</span> Effect of an additional Parallel Resistor</h3>
|
||||
<div class="outline-text-3" id="text-2-5">
|
||||
<p>
|
||||
An additional resistor in parallel with \(R_{in}\) would have two effects:
|
||||
Be looking at Figure <a href="#orgc43b1a9">7</a>, we can see that an additional resistor in parallel with \(R_{in}\) would have two effects:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>reduce the input voltage offset
|
||||
@@ -304,7 +398,6 @@ An additional resistor in parallel with \(R_{in}\) would have two effects:
|
||||
\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<p>
|
||||
If we allow the high pass corner frequency to be equals to 3Hz:
|
||||
</p>
|
||||
@@ -335,45 +428,42 @@ With this parallel resistance value, the voltage offset would be:
|
||||
<p>
|
||||
Which is much more acceptable.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9938615" class="outline-3">
|
||||
<h3 id="org9938615"><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="#orgbf8a90f">6</a>.
|
||||
A resistor \(R_p \approx 100\,k\Omega\) is then added in parallel with the force sensor as shown in Figure <a href="#org98410d2">8</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbf8a90f" class="figure">
|
||||
<div id="org98410d2" class="figure">
|
||||
<p><img src="figs/force_sensor_model_electronics.png" alt="force_sensor_model_electronics.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor \(R_p\)</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor \(R_p\)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4ae0b53" class="outline-3">
|
||||
<h3 id="org4ae0b53"><span class="section-number-3">2.6</span> Obtained voltage offset and time constant with the added resistor</h3>
|
||||
<div class="outline-text-3" id="text-2-6">
|
||||
<p>
|
||||
After the resistor is added, the same steps response is performed.
|
||||
</p>
|
||||
|
||||
<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>
|
||||
<p>
|
||||
The results are shown in Figure <a href="#org3c913c7">9</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf9378d8" class="figure">
|
||||
<div id="org3c913c7" 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>
|
||||
<p><span class="figure-number">Figure 9: </span>Time domain signal during the actuator voltage steps</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@@ -391,40 +481,7 @@ Three steps are performed at the following time intervals:
|
||||
</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>
|
||||
The time constant and voltage offset are again estimated using a fit function.
|
||||
And indeed, we obtain a much smaller offset voltage and a much faster time constant.
|
||||
</p>
|
||||
|
||||
@@ -513,10 +570,26 @@ This validates the model of the ADC and the effectiveness of the added resistor.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org81c5db6" class="outline-2">
|
||||
<h2 id="org81c5db6"><span class="section-number-2">3</span> Generated Number of Charge / Voltage</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org22b743e"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, we wish to estimate the relation between the displacement performed by the stack actuator and the generated voltage/charge on the sensor stack.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org55eecf7" class="outline-3">
|
||||
<h3 id="org55eecf7"><span class="section-number-3">3.1</span> Data Loading</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
The measured data is loaded and the first 25 seconds of data corresponding to transient data are removed.
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org3e71d2e" class="outline-3">
|
||||
<h3 id="org3e71d2e"><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>);
|
||||
|
||||
@@ -526,20 +599,29 @@ encoder = encoder(t<span class="org-type">></span>25) <span class="org-type">
|
||||
t = t(t<span class="org-type">></span>25);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org46592d5" class="outline-3">
|
||||
<h3 id="org46592d5"><span class="section-number-3">3.2</span> Excitation signal and corresponding displacement</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#org2ae5f53">8</a>).
|
||||
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#org4706469">10</a>).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org2ae5f53" class="figure">
|
||||
<div id="org4706469" 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>
|
||||
<p><span class="figure-number">Figure 10: </span>Driving Voltage</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The full stroke as measured by the encoder is:
|
||||
The corresponding displacement as measured by the encoder is shown in Figure <a href="#org1d38208">11</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The full stroke is:
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">max(encoder)<span class="org-type">-</span>min(encoder)
|
||||
@@ -551,28 +633,34 @@ The full stroke as measured by the encoder is:
|
||||
</pre>
|
||||
|
||||
|
||||
<p>
|
||||
Its signal is shown in Figure <a href="#org3458f57">9</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3458f57" class="figure">
|
||||
<div id="org1d38208" 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>
|
||||
<p><span class="figure-number">Figure 11: </span>Encoder measurement</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6e6b1f7" class="outline-3">
|
||||
<h3 id="org6e6b1f7"><span class="section-number-3">3.3</span> Generated Voltage</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
The generated voltage by the stack is shown in Figure
|
||||
The generated voltage by the stack is shown in Figure <a href="#org267434e">12</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga69980e" class="figure">
|
||||
<div id="org267434e" 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>
|
||||
<p><span class="figure-number">Figure 12: </span>Voltage measured on the stack used as a sensor</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4ed9610" class="outline-3">
|
||||
<h3 id="org4ed9610"><span class="section-number-3">3.4</span> Generated Charge</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
The capacitance of the stack is
|
||||
</p>
|
||||
@@ -582,18 +670,34 @@ The capacitance of the stack is
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The corresponding generated charge is then shown in Figure <a href="#orgeafdaf5">11</a>.
|
||||
The voltage and charge across a capacitor are related through the following equation:
|
||||
</p>
|
||||
|
||||
<div id="orgeafdaf5" class="figure">
|
||||
<p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" />
|
||||
\begin{equation}
|
||||
U_C = \frac{Q}{C}
|
||||
\end{equation}
|
||||
<p>
|
||||
where \(U_C\) is the voltage in Volts, \(Q\) the charge in Coulombs and \(C\) the capacitance in Farads.
|
||||
</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="#org83e4a49">12</a>.
|
||||
The corresponding generated charge is then shown in Figure <a href="#org0331333">13</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org0331333" class="figure">
|
||||
<p><img src="figs/force_sensor_sin_charge.png" alt="force_sensor_sin_charge.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Generated Charge</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org69bd84e" class="outline-3">
|
||||
<h3 id="org69bd84e"><span class="section-number-3">3.5</span> Generated Voltage/Charge as a function of the displacement</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
The relation between the generated voltage and the measured displacement is almost linear as shown in Figure <a href="#org6b7aee0">14</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@@ -602,10 +706,10 @@ The relation between the generated voltage and the measured displacement is almo
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org83e4a49" class="figure">
|
||||
<div id="org6b7aee0" 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>
|
||||
<p><span class="figure-number">Figure 14: </span>Almost linear relation between the relative displacement and the generated voltage</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
@@ -631,7 +735,7 @@ 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: 2020-11-10 mar. 13:00</p>
|
||||
<p class="date">Created: 2020-11-10 mar. 13:41</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user