#+TITLE: Amplifier Piezoelectric Actuator APA300ML - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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* Introduction                                                        :ignore:

The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML.

This include:
- Stroke
- Stiffness
- Hysteresis
- Gain from the applied voltage $V_a$ to the generated Force $F_a$
- Gain from the sensor stack strain $\delta L$ to the generated voltage $V_s$
- Dynamical behavior

* Model of an Amplified Piezoelectric Actuator and Sensor

Consider a schematic of the Amplified Piezoelectric Actuator in Figure [[fig:apa_model_schematic]].

#+name: fig:apa_model_schematic
#+caption: Amplified Piezoelectric Actuator Schematic
[[file:figs/apa_model_schematic.png]]

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 $g_s$.

The block-diagram model of the piezoelectric actuator is then as shown in Figure [[fig:apa-model-simscape-schematic]].

#+begin_src latex :file apa-model-simscape-schematic.pdf
  \begin{tikzpicture}
    \node[block={2.0cm}{2.0cm}, align=center] (model) at (0,0){Simscape\\Model};
    \node[block, left=1.0 of model] (ga){$g_a(s)$};
    \node[block, right=1.0 of model] (gs){$g_s(s)$};

    \draw[<-] (ga.west) -- node[midway, above]{$V_a$} node[midway, below]{$[V]$} ++(-1.0, 0);
    \draw[->] (ga.east) --node[midway, above]{$F_a$} node[midway, below]{$[N]$} (model.west);
    \draw[->] (model.east) --node[midway, above]{$dl$} node[midway, below]{$[m]$} (gs.west);
    \draw[->] (gs.east) -- node[midway, above]{$V_s$} node[midway, below]{$[V]$} ++(1.0, 0);
  \end{tikzpicture}
#+end_src

#+name: fig:apa-model-simscape-schematic
#+caption: Model of the APA with Simscape/Simulink
#+RESULTS:
[[file:figs/apa-model-simscape-schematic.png]]

* Test-Bench Description

#+begin_note
Here are the documentation of the equipment used for this test bench:
- Voltage Amplifier: [[file:doc/PD200-V7-R1.pdf][PD200]]
- Amplified Piezoelectric Actuator: [[file:doc/APA300ML.pdf][APA300ML]]
- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
- Encoder: [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Renishaw Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]]
- Interferometer: [[https://www.attocube.com/en/products/laser-displacement-sensor/displacement-measuring-interferometer][Attocube IDS3010]]
#+end_note

#+name: fig:test_bench_apa_alone
#+caption: Schematic of the Test Bench
[[file:figs/test_bench_apa_alone.png]]

* Measurement Procedure
** Introduction                                                      :ignore:

** 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.

** Dynamical Behavior

Perform a system identification from $V_a$ to the measured displacement $d$ by the interferometer and by the encoder, and to the general voltage $V_s$.

This can be performed using different excitation signals.

This can also be performed with and without the encoder fixed to the APA.

** Compare the results obtained for all 7 APA300ML

Compare all the obtained parameters for all the test APA.

* Measurement Results