Add APA model + explain measurement procedure

figs/apa_model_schematic.svg

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index.html

@@ -3,7 +3,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-12-15 mar. 23:33 -->
+<!-- 2020-12-16 mer. 11:07 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>Amplifier Piezoelectric Actuator APA300ML - Test Bench</title>
 <meta name="generator" content="Org mode" />
@@ -30,8 +30,18 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#org35c5c6b">1. Model of an Amplified Piezoelectric Actuator and Sensor</a></li>
+<li><a href="#org1313e99">1. Model of an Amplified Piezoelectric Actuator and Sensor</a></li>
-<li><a href="#org1e8593e">2. Test-Bench Description</a></li>
+<li><a href="#org3c114e3">2. Test-Bench Description</a></li>
+<li><a href="#orgeef8a7b">3. Measurement Procedure</a>
+<ul>
+<li><a href="#orgf5f4de4">3.1. Stroke Measurement</a></li>
+<li><a href="#orgc6a7f40">3.2. Stiffness Measurement</a></li>
+<li><a href="#orgf924c27">3.3. Hysteresis measurement</a></li>
+<li><a href="#org8dd84d4">3.4. Piezoelectric Actuator Constant</a></li>
+<li><a href="#org133086b">3.5. Piezoelectric Sensor Constant</a></li>
+<li><a href="#org6d5e309">3.6. Capacitance Measurement</a></li>
+</ul>
+</li>
 </ul>
 </div>
 </div>
@@ -47,24 +57,48 @@ This include:
 <li>Stroke</li>
 <li>Stiffness</li>
 <li>Hysteresis</li>
-<li>Gain from \(V_a\) to \(d\) (in [m/V]), then it should be possible to compute the gain from \(V_a\) to \(F_a\) using the Simscape model</li>
+<li>Gain from the applied voltage \(V_a\) to the generated Force \(F_a\)</li>
-<li>Gain from \(\delta L\) to \(V_s\) (in [V/m])</li>
+<li>Gain from the sensor stack strain \(\delta L\) to the generated voltage \(V_s\)</li>
 </ul>
 
-<div id="outline-container-org35c5c6b" class="outline-2">
+<div id="outline-container-org1313e99" class="outline-2">
-<h2 id="org35c5c6b"><span class="section-number-2">1</span> Model of an Amplified Piezoelectric Actuator and Sensor</h2>
+<h2 id="org1313e99"><span class="section-number-2">1</span> Model of an Amplified Piezoelectric Actuator and Sensor</h2>
 <div class="outline-text-2" id="text-1">
-<ul class="org-ul">
+<p>
-<li class="off"><code>[&#xa0;]</code> Schematic of the APA with values of displacement, voltages, forces, etc.</li>
+Consider a schematic of the Amplified Piezoelectric Actuator in Figure <a href="#org5dd279e">1</a>.
-<li class="off"><code>[&#xa0;]</code> Required things to measure</li>
+</p>
-</ul>
+
+<p>
+A voltage \(V_a\) applied to the actuator stacks will induce an actuator force \(F_a\):
+</p>
+\begin{equation}
+  F_a = g_a \cdot V_a
+\end{equation}
+
+<p>
+A change of length \(dl\) of the sensor stack will induce a voltage \(V_s\):
+</p>
+\begin{equation}
+  V_s = g_s \cdot dl
+\end{equation}
+
+<p>
+We wish here to experimental measure \(g_a\) and \(g_s\).
+</p>
+
+
+<div id="org5dd279e" class="figure">
+<p><img src="figs/apa_model_schematic.png" alt="apa_model_schematic.png" />
+</p>
+<p><span class="figure-number">Figure 1: </span>Amplified Piezoelectric Actuator Schematic</p>
+</div>
 </div>
 </div>
 
-<div id="outline-container-org1e8593e" class="outline-2">
+<div id="outline-container-org3c114e3" class="outline-2">
-<h2 id="org1e8593e"><span class="section-number-2">2</span> Test-Bench Description</h2>
+<h2 id="org3c114e3"><span class="section-number-2">2</span> Test-Bench Description</h2>
 <div class="outline-text-2" id="text-2">
-<div class="note" id="org5ab97da">
+<div class="note" id="org84f08a9">
 <p>
 Here are the documentation of the equipment used for this test bench:
 </p>
@@ -79,17 +113,161 @@ Here are the documentation of the equipment used for this test bench:
 </div>
 
 
-<div id="org2908959" class="figure">
+<div id="org9ba13fa" class="figure">
 <p><img src="figs/test_bench_apa_alone.png" alt="test_bench_apa_alone.png" />
 </p>
-<p><span class="figure-number">Figure 1: </span>Schematic of the Test Bench</p>
+<p><span class="figure-number">Figure 2: </span>Schematic of the Test Bench</p>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgeef8a7b" class="outline-2">
+<h2 id="orgeef8a7b"><span class="section-number-2">3</span> Measurement Procedure</h2>
+<div class="outline-text-2" id="text-3">
+</div>
+<div id="outline-container-orgf5f4de4" class="outline-3">
+<h3 id="orgf5f4de4"><span class="section-number-3">3.1</span> Stroke Measurement</h3>
+<div class="outline-text-3" id="text-3-1">
+<p>
+Using the PD200 amplifier, output a voltage:
+\[ V_a = 65 + 85 \sin(2\pi \cdot t) \]
+To have a quasi-static excitation between -80 and 150V.
+</p>
+
+<p>
+As the gain of the PD200 amplifier is 20, the DAC output voltage should be:
+\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \]
+</p>
+
+<p>
+Verify that the voltage offset is zero!
+</p>
+
+<p>
+Measure the output vertical displacement \(d\) using the interferometer.
+</p>
+
+<p>
+Then, plot \(d\) as a function of \(V_a\), and perform a linear regression.
+Conclude on the obtained stroke.
+</p>
+</div>
+</div>
+
+<div id="outline-container-orgc6a7f40" class="outline-3">
+<h3 id="orgc6a7f40"><span class="section-number-3">3.2</span> Stiffness Measurement</h3>
+<div class="outline-text-3" id="text-3-2">
+<p>
+Add some (known) weight \(\delta m g\) on the suspended mass and measure the deflection \(\delta d\).
+This can be tested when the piezoelectric stacks are open-circuit.
+</p>
+
+<p>
+As the stiffness will be around \(k \approx 10^6 N/m\), an added mass of \(m \approx 100g\) will induce a static deflection of \(\approx 1\mu m\) which should be large enough for a precise measurement using the interferometer.
+</p>
+
+<p>
+Then the obtained stiffness is:
+</p>
+\begin{equation}
+  k = \frac{\delta m g}{\delta d}
+\end{equation}
+</div>
+</div>
+
+<div id="outline-container-orgf924c27" class="outline-3">
+<h3 id="orgf924c27"><span class="section-number-3">3.3</span> Hysteresis measurement</h3>
+<div class="outline-text-3" id="text-3-3">
+<p>
+Supply a quasi static sinusoidal excitation \(V_a\) at different voltages.
+</p>
+
+<p>
+The offset should be 65V, and the sin amplitude can range from 1V up to 85V.
+</p>
+
+<p>
+For each excitation amplitude, the vertical displacement \(d\) of the mass is measured.
+</p>
+
+<p>
+Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes.
+</p>
+</div>
+</div>
+
+<div id="outline-container-org8dd84d4" class="outline-3">
+<h3 id="org8dd84d4"><span class="section-number-3">3.4</span> Piezoelectric Actuator Constant</h3>
+<div class="outline-text-3" id="text-3-4">
+<p>
+Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\).
+Use a quasi static (1Hz) excitation signal \(V_a\) on the piezoelectric stack and measure the vertical displacement \(d\).
+Perform a linear regression to obtain:
+</p>
+\begin{equation}
+  d = g_{d/V_a} \cdot V_a
+\end{equation}
+
+<p>
+Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force \(F_a\) to the induced displacement \(d\):
+</p>
+\begin{equation}
+  d = g_{d/F_a} \cdot F_a
+\end{equation}
+
+<p>
+From the two gains, it is then easy to determine \(g_a\):
+</p>
+\begin{equation}
+  g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}}
+\end{equation}
+</div>
+</div>
+
+<div id="outline-container-org133086b" class="outline-3">
+<h3 id="org133086b"><span class="section-number-3">3.5</span> Piezoelectric Sensor Constant</h3>
+<div class="outline-text-3" id="text-3-5">
+<p>
+From a quasi static (1Hz) excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\):
+</p>
+\begin{equation}
+  V_s = g_{V_s/V_a} V_a
+\end{equation}
+
+<p>
+Using the simscape model, compute the static gain from the actuator force \(F_a\) to the strain of the sensor stack \(dl\):
+</p>
+\begin{equation}
+  dl = g_{dl/F_a} F_a
+\end{equation}
+
+<p>
+Then, the static gain from the sensor stack strain \(dl\) to the general voltage \(V_s\) is:
+</p>
+\begin{equation}
+  g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}}
+\end{equation}
+
+<p>
+Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain.
+This external force can be some weight added, or a piezo in parallel.
+</p>
+</div>
+</div>
+
+<div id="outline-container-org6d5e309" class="outline-3">
+<h3 id="org6d5e309"><span class="section-number-3">3.6</span> Capacitance Measurement</h3>
+<div class="outline-text-3" id="text-3-6">
+<p>
+Measure the capacitance of the 3 stacks individually using a precise multi-meter.
+</p>
 </div>
 </div>
 </div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Dehaeze Thomas</p>
-<p class="date">Created: 2020-12-15 mar. 23:33</p>
+<p class="date">Created: 2020-12-16 mer. 11:07</p>
 </div>
 </body>
 </html>

index.org

@@ -48,13 +48,28 @@ This include:
 - Stroke
 - Stiffness
 - Hysteresis
-- Gain from $V_a$ to $d$ (in [m/V]), then it should be possible to compute the gain from $V_a$ to $F_a$ using the Simscape model
+- Gain from the applied voltage $V_a$ to the generated Force $F_a$
-- Gain from $\delta L$ to $V_s$ (in [V/m])
+- Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$
 
 * Model of an Amplified Piezoelectric Actuator and Sensor
 
-- [ ] Schematic of the APA with values of displacement, voltages, forces, etc.
+Consider a schematic of the Amplified Piezoelectric Actuator in Figure [[fig:apa_model_schematic]].
-- [ ] Required things to measure
+
+A voltage $V_a$ applied to the actuator stacks will induce an actuator force $F_a$:
+\begin{equation}
+  F_a = g_a \cdot V_a
+\end{equation}
+
+A change of length $dl$ of the sensor stack will induce a voltage $V_s$:
+\begin{equation}
+  V_s = g_s \cdot dl
+\end{equation}
+
+We wish here to experimental measure $g_a$ and $g_s$.
+
+#+name: fig:apa_model_schematic
+#+caption: Amplified Piezoelectric Actuator Schematic
+[[file:figs/apa_model_schematic.png]]
 
 * Test-Bench Description
 
@@ -70,3 +85,85 @@ Here are the documentation of the equipment used for this test bench:
 #+name: fig:test_bench_apa_alone
 #+caption: Schematic of the Test Bench
 [[file:figs/test_bench_apa_alone.png]]
+
+* Measurement Procedure
+** Stroke Measurement
+
+Using the PD200 amplifier, output a voltage:
+\[ V_a = 65 + 85 \sin(2\pi \cdot t) \]
+To have a quasi-static excitation between -80 and 150V.
+
+As the gain of the PD200 amplifier is 20, the DAC output voltage should be:
+\[ V_{dac}(t) = 3.25 + 4.25\sin(2\pi \cdot t) \]
+
+Verify that the voltage offset is zero!
+
+Measure the output vertical displacement $d$ using the interferometer.
+
+Then, plot $d$ as a function of $V_a$, and perform a linear regression.
+Conclude on the obtained stroke.
+
+** Stiffness Measurement
+
+Add some (known) weight $\delta m g$ on the suspended mass and measure the deflection $\delta d$.
+This can be tested when the piezoelectric stacks are open-circuit.
+
+As the stiffness will be around $k \approx 10^6 N/m$, an added mass of $m \approx 100g$ will induce a static deflection of $\approx 1\mu m$ which should be large enough for a precise measurement using the interferometer.
+
+Then the obtained stiffness is:
+\begin{equation}
+  k = \frac{\delta m g}{\delta d}
+\end{equation}
+
+** Hysteresis measurement
+
+Supply a quasi static sinusoidal excitation $V_a$ at different voltages.
+
+The offset should be 65V, and the sin amplitude can range from 1V up to 85V.
+
+For each excitation amplitude, the vertical displacement $d$ of the mass is measured.
+
+Then, $d$ is plotted as a function of $V_a$ for all the amplitudes.
+
+** Piezoelectric Actuator Constant
+
+Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage $V_a$ to the induced displacement $d$.
+Use a quasi static (1Hz) excitation signal $V_a$ on the piezoelectric stack and measure the vertical displacement $d$.
+Perform a linear regression to obtain:
+\begin{equation}
+  d = g_{d/V_a} \cdot V_a
+\end{equation}
+
+Using the Simscape model of the APA, it is possible to determine the static gain between the actuator force $F_a$ to the induced displacement $d$:
+\begin{equation}
+  d = g_{d/F_a} \cdot F_a
+\end{equation}
+
+From the two gains, it is then easy to determine $g_a$:
+\begin{equation}
+  g_a = \frac{F_a}{V_a} = \frac{F_a}{d} \cdot \frac{d}{V_a} = \frac{g_{d/V_a}}{g_{d/F_a}}
+\end{equation}
+
+** Piezoelectric Sensor Constant
+
+From a quasi static (1Hz) excitation of the piezoelectric stack, measure the gain from $V_a$ to $V_s$:
+\begin{equation}
+  V_s = g_{V_s/V_a} V_a
+\end{equation}
+
+Using the simscape model, compute the static gain from the actuator force $F_a$ to the strain of the sensor stack $dl$:
+\begin{equation}
+  dl = g_{dl/F_a} F_a
+\end{equation}
+
+Then, the static gain from the sensor stack strain $dl$ to the general voltage $V_s$ is:
+\begin{equation}
+  g_s = \frac{V_s}{dl} = \frac{V_s}{V_a} \cdot \frac{V_a}{F_a} \cdot \frac{F_a}{dl} = \frac{g_{V_s/V_a}}{g_a \cdot g_{dl/F_a}}
+\end{equation}
+
+Alternatively, we could impose an external force to add strain in the APA that should be equally present in all the 3 stacks and equal to 1/5 of the vertical strain.
+This external force can be some weight added, or a piezo in parallel.
+
+** Capacitance Measurement
+
+Measure the capacitance of the 3 stacks individually using a precise multi-meter.