741 lines
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741 lines
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<title>Piezoelectric Force Sensor - Test Bench</title>
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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="#orgc978fd7">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="#org806a9c3">1.1. Load Data</a></li>
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<li><a href="#orgd3a8e11">1.2. Transfer Functions</a></li>
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</ul>
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</li>
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<li><a href="#org12cffb3">2. Effect of a Resistor in Parallel with the Stack Sensor</a>
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<ul>
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<li><a href="#org3d3c9c5">2.1. Excitation steps and measured generated voltage</a></li>
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<li><a href="#orge91c0f3">2.2. Estimation of the voltage offset and discharge time constant</a></li>
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<li><a href="#org45317a1">2.3. Estimation of the ADC input impedance</a></li>
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<li><a href="#orgbdebc17">2.4. Explanation of the Voltage offset</a></li>
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<li><a href="#orgaf9774b">2.5. Effect of an additional Parallel Resistor</a></li>
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<li><a href="#org0931a42">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="#orgf8de1e8">3. Generated Number of Charge / Voltage</a>
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<ul>
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<li><a href="#org1378faa">3.1. Data Loading</a></li>
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<li><a href="#org18bda16">3.2. Excitation signal and corresponding displacement</a></li>
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<li><a href="#org2647e0c">3.3. Generated Voltage</a></li>
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<li><a href="#org74ab9f0">3.4. Generated Charge</a></li>
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<li><a href="#org4241ea4">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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</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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<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="#orgede3aab">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="#orgb1a09bb">2</a>: the effect of a resistor in parallel with the sensor stack is studied</li>
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<li>Section <a href="#org49d4bd9">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-orgc978fd7" class="outline-2">
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<h2 id="orgc978fd7"><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="orgede3aab"></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="#org874417a">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="org874417a" 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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<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="orgdfeaa4d">
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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-org806a9c3" class="outline-3">
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<h3 id="org806a9c3"><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-orgd3a8e11" class="outline-3">
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<h3 id="orgd3a8e11"><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="orga9fdab5" 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="org1ce5c9d" 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="org2d932e9" 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 4: </span>Zoom on the change of resonance</p>
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</div>
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<div class="important" id="org284d89a">
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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-org12cffb3" class="outline-2">
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<h2 id="org12cffb3"><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="orgb1a09bb"></a>
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</p>
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<p>
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The setup is shown in Figure <a href="#orgfcaafb7">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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<div id="orgfcaafb7" 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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<div class="note" id="org3904317">
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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">
|
|
<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-org3d3c9c5" class="outline-3">
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<h3 id="org3d3c9c5"><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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<p>
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The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure <a href="#orgbed6888">6</a>.
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</p>
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<div id="orgbed6888" 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 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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<div id="outline-container-orge91c0f3" class="outline-3">
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<h3 id="orge91c0f3"><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>
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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 \(\tau = RC\).
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</p>
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<p>
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In order to estimate the time domain, we fit the data with an exponential.
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The fit function is:
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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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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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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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<p>
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The obtained values are shown below.
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</p>
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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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</div>
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</div>
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<div id="outline-container-org45317a1" class="outline-3">
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<h3 id="org45317a1"><span class="section-number-3">2.3</span> Estimation of the ADC input impedance</h3>
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<div class="outline-text-3" id="text-2-3">
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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>
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</div>
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<div id="outline-container-orgbdebc17" class="outline-3">
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<h3 id="orgbdebc17"><span class="section-number-3">2.4</span> Explanation of the Voltage offset</h3>
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<div class="outline-text-3" id="text-2-4">
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<p>
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As shown in Figure <a href="#orgbed6888">6</a>, the voltage across the Piezoelectric sensor stack shows a constant voltage offset.
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</p>
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<p>
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We can explain this offset by looking at the electrical model shown in Figure <a href="#orgb1098d2">7</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)).
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</p>
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<p>
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The differential amplifier in the Speedgoat has some input bias current \(i_n\) that produces a voltage offset across its own internal resistance.
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Note that the impedance of the piezoelectric stack is much larger that that at DC.
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</p>
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<div id="orgb1098d2" class="figure">
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<p><img src="figs/force_sensor_model_electronics_without_R.png" alt="force_sensor_model_electronics_without_R.png" />
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</p>
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<p><span class="figure-number">Figure 7: </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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</div>
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</div>
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<div id="outline-container-orgaf9774b" class="outline-3">
|
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<h3 id="orgaf9774b"><span class="section-number-3">2.5</span> Effect of an additional Parallel Resistor</h3>
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<div class="outline-text-3" id="text-2-5">
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<p>
|
|
Be looking at Figure <a href="#orgb1098d2">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
|
|
\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]</li>
|
|
<li>increase the high pass corner frequency \(f_c\)
|
|
\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
|
|
\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]</li>
|
|
</ul>
|
|
|
|
<p>
|
|
If we allow the high pass corner frequency to be equals to 3Hz:
|
|
</p>
|
|
<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>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>
|
|
|
|
|
|
<p>
|
|
A resistor \(R_p \approx 100\,k\Omega\) is then added in parallel with the force sensor as shown in Figure <a href="#orga7065ec">8</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga7065ec" class="figure">
|
|
<p><img src="figs/force_sensor_model_electronics.png" alt="force_sensor_model_electronics.png" />
|
|
</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-org0931a42" class="outline-3">
|
|
<h3 id="org0931a42"><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>
|
|
|
|
<p>
|
|
The results are shown in Figure <a href="#org162f5d0">9</a>.
|
|
</p>
|
|
|
|
|
|
<div id="org162f5d0" 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 9: </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>
|
|
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>
|
|
|
|
<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>
|
|
|
|
<div id="outline-container-orgf8de1e8" class="outline-2">
|
|
<h2 id="orgf8de1e8"><span class="section-number-2">3</span> Generated Number of Charge / Voltage</h2>
|
|
<div class="outline-text-2" id="text-3">
|
|
<p>
|
|
<a id="org49d4bd9"></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-org1378faa" class="outline-3">
|
|
<h3 id="org1378faa"><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 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>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org18bda16" class="outline-3">
|
|
<h3 id="org18bda16"><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="#org2ddf80f">10</a>).
|
|
</p>
|
|
|
|
|
|
<div id="org2ddf80f" class="figure">
|
|
<p><img src="figs/force_sensor_sin_u.png" alt="force_sensor_sin_u.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 10: </span>Driving Voltage</p>
|
|
</div>
|
|
|
|
<p>
|
|
The corresponding displacement as measured by the encoder is shown in Figure <a href="#org7dd9b3b">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)
|
|
</pre>
|
|
</div>
|
|
|
|
<pre class="example">
|
|
5.005e-05
|
|
</pre>
|
|
|
|
|
|
|
|
<div id="org7dd9b3b" class="figure">
|
|
<p><img src="figs/force_sensor_sin_encoder.png" alt="force_sensor_sin_encoder.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 11: </span>Encoder measurement</p>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
|
|
<div id="outline-container-org2647e0c" class="outline-3">
|
|
<h3 id="org2647e0c"><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 <a href="#orga623ff4">12</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orga623ff4" class="figure">
|
|
<p><img src="figs/force_sensor_sin_stack.png" alt="force_sensor_sin_stack.png" />
|
|
</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-org74ab9f0" class="outline-3">
|
|
<h3 id="org74ab9f0"><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>
|
|
<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 voltage and charge across a capacitor are related through the following equation:
|
|
</p>
|
|
\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>
|
|
The corresponding generated charge is then shown in Figure <a href="#orgcf66ed5">13</a>.
|
|
</p>
|
|
|
|
|
|
<div id="orgcf66ed5" 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-org4241ea4" class="outline-3">
|
|
<h3 id="org4241ea4"><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="#org2f61fcf">14</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="org2f61fcf" class="figure">
|
|
<p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" />
|
|
</p>
|
|
<p><span class="figure-number">Figure 14: </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. 13:46</p>
|
|
</div>
|
|
</body>
|
|
</html>
|