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<title>Piezoelectric Force Sensor - Test Bench</title>
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<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgc978fd7">1. Change of Stiffness due to Sensors stack being open/closed circuit</a>
<li><a href="#org99cbc3b">1. Change of Stiffness due to Sensors stack being open/closed circuit</a>
<ul>
<li><a href="#org806a9c3">1.1. Load Data</a></li>
<li><a href="#orgd3a8e11">1.2. Transfer Functions</a></li>
<li><a href="#org8df2126">1.1. Load Data</a></li>
<li><a href="#orgd97a936">1.2. Transfer Functions</a></li>
</ul>
</li>
<li><a href="#org12cffb3">2. Effect of a Resistor in Parallel with the Stack Sensor</a>
<li><a href="#orgd9adefa">2. Effect of a Resistor in Parallel with the Stack Sensor</a>
<ul>
<li><a href="#org3d3c9c5">2.1. Excitation steps and measured generated voltage</a></li>
<li><a href="#orge91c0f3">2.2. Estimation of the voltage offset and discharge time constant</a></li>
<li><a href="#org45317a1">2.3. Estimation of the ADC input impedance</a></li>
<li><a href="#orgbdebc17">2.4. Explanation of the Voltage offset</a></li>
<li><a href="#orgaf9774b">2.5. Effect of an additional Parallel Resistor</a></li>
<li><a href="#org0931a42">2.6. Obtained voltage offset and time constant with the added resistor</a></li>
<li><a href="#org6cc7b78">2.1. Excitation steps and measured generated voltage</a></li>
<li><a href="#orga1cd4c0">2.2. Estimation of the voltage offset and discharge time constant</a></li>
<li><a href="#orge8fa34a">2.3. Estimation of the ADC input impedance</a></li>
<li><a href="#org96db33c">2.4. Explanation of the Voltage offset</a></li>
<li><a href="#orgf987e1d">2.5. Effect of an additional Parallel Resistor</a></li>
<li><a href="#org1bfcf07">2.6. Obtained voltage offset and time constant with the added resistor</a></li>
</ul>
</li>
<li><a href="#orgf8de1e8">3. Generated Number of Charge / Voltage</a>
<li><a href="#orge72ac2b">3. Generated Number of Charge / Voltage</a>
<ul>
<li><a href="#org1378faa">3.1. Data Loading</a></li>
<li><a href="#org18bda16">3.2. Excitation signal and corresponding displacement</a></li>
<li><a href="#org2647e0c">3.3. Generated Voltage</a></li>
<li><a href="#org74ab9f0">3.4. Generated Charge</a></li>
<li><a href="#org4241ea4">3.5. Generated Voltage/Charge as a function of the displacement</a></li>
<li><a href="#orgd28174f">3.1. Data Loading</a></li>
<li><a href="#orgd39b262">3.2. Excitation signal and corresponding displacement</a></li>
<li><a href="#org0aaf282">3.3. Generated Voltage</a></li>
<li><a href="#orgb349423">3.4. Generated Charge</a></li>
<li><a href="#org16c85ae">3.5. Generated Voltage/Charge as a function of the displacement</a></li>
</ul>
</li>
</ul>
@@ -71,19 +67,19 @@ In this document is studied how a piezoelectric stack can be used to measured th
It is divided in the following sections:
</p>
<ul class="org-ul">
<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>
<li>Section <a href="#orgb1a09bb">2</a>: the effect of a resistor in parallel with the sensor stack is studied</li>
<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>
<li>Section <a href="#org3d57e4e">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>
<li>Section <a href="#org915df51">2</a>: the effect of a resistor in parallel with the sensor stack is studied</li>
<li>Section <a href="#org50b540e">3</a>: the voltage / number of charge generated by the sensor as a function of the displacement is measured</li>
</ul>
<div id="outline-container-orgc978fd7" class="outline-2">
<h2 id="orgc978fd7"><span class="section-number-2">1</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
<div id="outline-container-org99cbc3b" class="outline-2">
<h2 id="org99cbc3b"><span class="section-number-2">1</span> Change of Stiffness due to Sensors stack being open/closed circuit</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="orgede3aab"></a>
<a id="org3d57e4e"></a>
</p>
<p>
The experimental Setup is schematically represented in Figure <a href="#org874417a">1</a>.
The experimental Setup is schematically represented in Figure <a href="#org284a4f0">1</a>.
</p>
<p>
@@ -91,7 +87,7 @@ The dynamics from the voltage \(u\) used to drive the actuator stacks to the enc
</p>
<div id="org874417a" class="figure">
<div id="org284a4f0" class="figure">
<p><img src="figs/exp_setup_schematic.png" alt="exp_setup_schematic.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Schematic of the Experiment</p>
@@ -106,7 +102,7 @@ When the switch is closed, this correspond of having a measurement electronics w
We wish here to see how the system dynamics is changing in the two extreme cases.
</p>
<div class="note" id="orgdfeaa4d">
<div class="note" id="org2dcec72">
<p>
The equipment used in the test bench are:
</p>
@@ -120,8 +116,8 @@ The equipment used in the test bench are:
</div>
</div>
<div id="outline-container-org806a9c3" class="outline-3">
<h3 id="org806a9c3"><span class="section-number-3">1.1</span> Load Data</h3>
<div id="outline-container-org8df2126" class="outline-3">
<h3 id="org8df2126"><span class="section-number-3">1.1</span> Load Data</h3>
<div class="outline-text-3" id="text-1-1">
<div class="org-src-container">
<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>);
@@ -131,8 +127,8 @@ sc = load(<span class="org-string">'identification_short_circuit.mat'</span>, <s
</div>
</div>
<div id="outline-container-orgd3a8e11" class="outline-3">
<h3 id="orgd3a8e11"><span class="section-number-3">1.2</span> Transfer Functions</h3>
<div id="outline-container-orgd97a936" class="outline-3">
<h3 id="orgd97a936"><span class="section-number-3">1.2</span> Transfer Functions</h3>
<div class="outline-text-3" id="text-1-2">
<div class="org-src-container">
<pre class="src src-matlab">Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% Sampling Time [s]</span>
@@ -150,25 +146,25 @@ win = hann(ceil(10<span class="org-type">/</span>Ts));
</div>
<div id="orga9fdab5" class="figure">
<div id="org559e474" class="figure">
<p><img src="figs/stiffness_force_sensor_coherence.png" alt="stiffness_force_sensor_coherence.png" />
</p>
</div>
<div id="org1ce5c9d" class="figure">
<div id="org986b204" class="figure">
<p><img src="figs/stiffness_force_sensor_bode.png" alt="stiffness_force_sensor_bode.png" />
</p>
</div>
<div id="org2d932e9" class="figure">
<div id="org62eed17" class="figure">
<p><img src="figs/stiffness_force_sensor_bode_zoom.png" alt="stiffness_force_sensor_bode_zoom.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Zoom on the change of resonance</p>
</div>
<div class="important" id="org284d89a">
<div class="important" id="orgd5a860c">
<p>
The change of resonance frequency / stiffness is very small and is not important here.
</p>
@@ -178,25 +174,25 @@ The change of resonance frequency / stiffness is very small and is not important
</div>
</div>
<div id="outline-container-org12cffb3" class="outline-2">
<h2 id="org12cffb3"><span class="section-number-2">2</span> Effect of a Resistor in Parallel with the Stack Sensor</h2>
<div id="outline-container-orgd9adefa" class="outline-2">
<h2 id="orgd9adefa"><span class="section-number-2">2</span> Effect of a Resistor in Parallel with the Stack Sensor</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orgb1a09bb"></a>
<a id="org915df51"></a>
</p>
<p>
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.
The setup is shown in Figure <a href="#orgee79898">5</a> where two stacks are used as actuator (in parallel) and one stack is used as sensor.
The voltage amplifier used has a gain of 20 [V/V] (Cedrat LA75B).
</p>
<div id="orgfcaafb7" class="figure">
<div id="orgee79898" class="figure">
<p><img src="figs/force_sensor_setup.png" alt="force_sensor_setup.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Schematic of the setup</p>
</div>
<div class="note" id="org3904317">
<div class="note" id="org61996ca">
<p>
The equipment used in the test bench are:
</p>
@@ -210,8 +206,8 @@ The equipment used in the test bench are:
</div>
</div>
<div id="outline-container-org3d3c9c5" class="outline-3">
<h3 id="org3d3c9c5"><span class="section-number-3">2.1</span> Excitation steps and measured generated voltage</h3>
<div id="outline-container-org6cc7b78" class="outline-3">
<h3 id="org6cc7b78"><span class="section-number-3">2.1</span> Excitation steps and measured generated voltage</h3>
<div class="outline-text-3" id="text-2-1">
<p>
The measured data is loaded.
@@ -222,10 +218,10 @@ The measured data is loaded.
</div>
<p>
The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure <a href="#orgbed6888">6</a>.
The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure <a href="#orgf9c6d37">6</a>.
</p>
<div id="orgbed6888" class="figure">
<div id="orgf9c6d37" class="figure">
<p><img src="figs/force_sen_steps_time_domain.png" alt="force_sen_steps_time_domain.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Time domain signal during the 3 actuator voltage steps</p>
@@ -233,8 +229,8 @@ The excitation signal (steps) and measured voltage across the sensor stack are s
</div>
</div>
<div id="outline-container-orge91c0f3" class="outline-3">
<h3 id="orge91c0f3"><span class="section-number-3">2.2</span> Estimation of the voltage offset and discharge time constant</h3>
<div id="outline-container-orga1cd4c0" class="outline-3">
<h3 id="orga1cd4c0"><span class="section-number-3">2.2</span> Estimation of the voltage offset and discharge time constant</h3>
<div class="outline-text-3" id="text-2-2">
<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.
@@ -323,8 +319,8 @@ The obtained values are shown below.
</div>
</div>
<div id="outline-container-org45317a1" class="outline-3">
<h3 id="org45317a1"><span class="section-number-3">2.3</span> Estimation of the ADC input impedance</h3>
<div id="outline-container-orge8fa34a" class="outline-3">
<h3 id="orge8fa34a"><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:
@@ -346,15 +342,15 @@ The input impedance of the Speedgoat&rsquo;s ADC should then be close to \(1.5\,
</div>
</div>
<div id="outline-container-orgbdebc17" class="outline-3">
<h3 id="orgbdebc17"><span class="section-number-3">2.4</span> Explanation of the Voltage offset</h3>
<div id="outline-container-org96db33c" class="outline-3">
<h3 id="org96db33c"><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="#orgbed6888">6</a>, the voltage across the Piezoelectric sensor stack shows a constant voltage offset.
As shown in Figure <a href="#orgf9c6d37">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="#orgb1098d2">7</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)).
We can explain this offset by looking at the electrical model shown in Figure <a href="#org7c05634">7</a> (taken from (<a href="#citeproc_bib_item_1">Reza and Andrew 2006</a>)).
</p>
<p>
@@ -363,7 +359,7 @@ Note that the impedance of the piezoelectric stack is much larger that that at D
</p>
<div id="orgb1098d2" class="figure">
<div id="org7c05634" 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 7: </span>Model of a piezoelectric transducer (left) and instrumentation amplifier (right)</p>
@@ -383,11 +379,11 @@ The estimated input bias current is then:
</div>
</div>
<div id="outline-container-orgaf9774b" class="outline-3">
<h3 id="orgaf9774b"><span class="section-number-3">2.5</span> Effect of an additional Parallel Resistor</h3>
<div id="outline-container-orgf987e1d" class="outline-3">
<h3 id="orgf987e1d"><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>
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:
Be looking at Figure <a href="#org7c05634">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
@@ -430,11 +426,11 @@ Which is much more acceptable.
<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>.
A resistor \(R_p \approx 100\,k\Omega\) is then added in parallel with the force sensor as shown in Figure <a href="#org435f6c8">8</a>.
</p>
<div id="orga7065ec" class="figure">
<div id="org435f6c8" 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>
@@ -442,8 +438,8 @@ A resistor \(R_p \approx 100\,k\Omega\) is then added in parallel with the force
</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 id="outline-container-org1bfcf07" class="outline-3">
<h3 id="org1bfcf07"><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.
@@ -455,11 +451,11 @@ After the resistor is added, the same steps response is performed.
</div>
<p>
The results are shown in Figure <a href="#org162f5d0">9</a>.
The results are shown in Figure <a href="#org66a3445">9</a>.
</p>
<div id="org162f5d0" class="figure">
<div id="org66a3445" 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>
@@ -571,19 +567,19 @@ This validates the model of the ADC and the effectiveness of the added resistor.
</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 id="outline-container-orge72ac2b" class="outline-2">
<h2 id="orge72ac2b"><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>
<a id="org50b540e"></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 id="outline-container-orgd28174f" class="outline-3">
<h3 id="orgd28174f"><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.
@@ -601,22 +597,22 @@ t = t(t<span class="org-type">&gt;</span>25);
</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 id="outline-container-orgd39b262" class="outline-3">
<h3 id="orgd39b262"><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>).
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure <a href="#orge3dbfb8">10</a>).
</p>
<div id="org2ddf80f" class="figure">
<div id="orge3dbfb8" 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>.
The corresponding displacement as measured by the encoder is shown in Figure <a href="#orgb0594c1">11</a>.
</p>
<p>
@@ -633,7 +629,7 @@ The full stroke is:
<div id="org7dd9b3b" class="figure">
<div id="orgb0594c1" 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>
@@ -641,15 +637,15 @@ The full stroke is:
</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 id="outline-container-org0aaf282" class="outline-3">
<h3 id="org0aaf282"><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>.
The generated voltage by the stack is shown in Figure <a href="#orgc89ecbd">12</a>.
</p>
<div id="orga623ff4" class="figure">
<div id="orgc89ecbd" 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>
@@ -657,8 +653,8 @@ The generated voltage by the stack is shown in Figure <a href="#orga623ff4">12</
</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 id="outline-container-orgb349423" class="outline-3">
<h3 id="orgb349423"><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
@@ -680,11 +676,11 @@ where \(U_C\) is the voltage in Volts, \(Q\) the charge in Coulombs and \(C\) th
<p>
The corresponding generated charge is then shown in Figure <a href="#orgcf66ed5">13</a>.
The corresponding generated charge is then shown in Figure <a href="#orgea72e89">13</a>.
</p>
<div id="orgcf66ed5" class="figure">
<div id="orgea72e89" 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>
@@ -692,11 +688,11 @@ The corresponding generated charge is then shown in Figure <a href="#orgcf66ed5"
</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 id="outline-container-org16c85ae" class="outline-3">
<h3 id="org16c85ae"><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>.
The relation between the generated voltage and the measured displacement is almost linear as shown in Figure <a href="#org2c4050d">14</a>.
</p>
<div class="org-src-container">
@@ -705,7 +701,7 @@ The relation between the generated voltage and the measured displacement is almo
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<p><img src="figs/force_sensor_linear_relation.png" alt="force_sensor_linear_relation.png" />
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<p><span class="figure-number">Figure 14: </span>Almost linear relation between the relative displacement and the generated voltage</p>
@@ -734,7 +730,7 @@ With a 16bits ADC, the resolution will then be equals to (in [nm]):
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<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-10 mar. 13:46</p>
<p class="date">Created: 2020-11-12 jeu. 10:16</p>
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