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

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP:   ../index.html

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:

* Introduction                                                        :ignore:
In this document is studied how a piezoelectric stack can be used to measured the force.

It is divided in the following sections:
- Section [[sec:open_closed_circuit]]: the effect of the input impedance of the electronics connected to the force sensor stack on the stiffness of the stack is studied
- Section [[sec:parallel_resistor]]: the effect of a resistor in parallel with the sensor stack is studied
- Section [[sec:charge_voltage_estimation]]: the voltage / number of charge generated by the sensor as a function of the displacement is measured

* Change of Stiffness due to Sensors stack being open/closed circuit
:PROPERTIES:
:header-args:matlab+: :tangle matlab/open_closed_circuit.m
:END:
<<sec:open_closed_circuit>>

** Introduction                                                      :ignore:
The experimental Setup is schematically represented in Figure [[fig:exp_setup_schematic]].

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.

#+name: fig:exp_setup_schematic
#+caption: Schematic of the Experiment
[[file:figs/exp_setup_schematic.png]]

When the switch is opened, this correspond of having a measurement electronics with an high input impedance such as a *voltage* amplifier.
When the switch is closed, this correspond of having a measurement electronics with an small input impedance such as a *charge* amplifier.

We wish here to see how the system dynamics is changing in the two extreme cases.

#+begin_note
The equipment used in the test bench are:
- Renishaw Resolution Encoder with 1nm resolution ([[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][doc]])
- Cedrat Amplified Piezoelectric Actuator APA95ML ([[file:doc/APA95ML.pdf][doc]])
- Voltage Amplifier LA75B ([[file:doc/LA75B.pdf][doc]])
- Speedgoat IO131 with 16bits ADC and DAC ([[file:doc/IO130 IO131 OEM Datasheet.pdf][doc]])
#+end_note

** Matlab Init                                             :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  addpath('./matlab/mat/');
#+end_src

#+begin_src matlab :eval no
  addpath('./mat/');
#+end_src

** Load Data
#+begin_src matlab
  oc = load('identification_open_circuit.mat', 't', 'encoder', 'u');
  sc = load('identification_short_circuit.mat', 't', 'encoder', 'u');
#+end_src

** Transfer Functions
#+begin_src matlab
  Ts = 1e-4; % Sampling Time [s]
  win = hann(ceil(10/Ts));
#+end_src

#+begin_src matlab
  [tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
  [co_oc_est, ~] = mscohere(  oc.u, oc.encoder, win, [], [], 1/Ts);

  [tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
  [co_sc_est, ~] = mscohere(  sc.u, sc.encoder, win, [], [], 1/Ts);
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, co_oc_est, '-')
  plot(f, co_sc_est, '-')
  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
  ylabel('Coherence'); xlabel('Frequency [Hz]');
  hold off;
  xlim([0.5, 5e3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/stiffness_force_sensor_coherence.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:stiffness_force_sensor_coherence
#+caption:
#+RESULTS:
[[file:figs/stiffness_force_sensor_coherence.png]]


#+begin_src matlab :exports none
  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');

  ax1 = nexttile;
  hold on;
  plot(f, abs(tf_oc_est), '-', 'DisplayName', 'Open-Circuit')
  plot(f, abs(tf_sc_est), '-', 'DisplayName', 'Short-Circuit')
  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
  hold off;
  ylim([1e-7, 3e-4]);
  legend('location', 'southwest');

  ax2 = nexttile;
  hold on;
  plot(f, 180/pi*angle(tf_oc_est), '-')
  plot(f, 180/pi*angle(tf_sc_est), '-')
  set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
  ylabel('Phase'); xlabel('Frequency [Hz]');
  hold off;
  yticks(-360:90:360);
  axis padded 'auto x'

  linkaxes([ax1,ax2], 'x');
  xlim([0.5, 5e3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/stiffness_force_sensor_bode.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:stiffness_force_sensor_bode
#+caption:
#+RESULTS:
[[file:figs/stiffness_force_sensor_bode.png]]

#+begin_src matlab :tangle no :exports results :results file replace
  xlim([180, 280]);
  exportFig('figs/stiffness_force_sensor_bode_zoom.pdf', 'width', 'small', 'height', 'tall');
#+end_src

#+name: fig:stiffness_force_sensor_bode_zoom
#+caption: Zoom on the change of resonance
#+RESULTS:
[[file:figs/stiffness_force_sensor_bode_zoom.png]]

#+begin_important
  The change of resonance frequency / stiffness is very small and is not important here.
#+end_important

* Effect of a Resistor in Parallel with the Stack Sensor
:PROPERTIES:
:header-args:matlab+: :tangle matlab/parallel_resistor.m
:END:
<<sec:parallel_resistor>>

** Introduction                                                      :ignore:
The setup is shown in Figure [[fig:force_sensor_setup]] 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).

#+name: fig:force_sensor_setup
#+caption: Schematic of the setup
[[file:figs/force_sensor_setup.png]]

#+begin_note
The equipment used in the test bench are:
- Renishaw Resolution Encoder with 1nm resolution ([[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][doc]])
- Cedrat Amplified Piezoelectric Actuator APA95ML ([[file:doc/APA95ML.pdf][doc]])
- Voltage Amplifier LA75B ([[file:doc/LA75B.pdf][doc]])
- Speedgoat IO131 with 16bits ADC and DAC ([[file:doc/IO130 IO131 OEM Datasheet.pdf][doc]])
#+end_note

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  addpath('./matlab/mat/');
#+end_src

#+begin_src matlab :eval no
  addpath('./mat/');
#+end_src

** Excitation steps and measured generated voltage
The measured data is loaded.
#+begin_src matlab
  load('force_sensor_steps.mat', 't', 'encoder', 'u', 'v');
#+end_src

The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure [[fig:force_sen_steps_time_domain]].
#+begin_src matlab :exports none
  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
  nexttile;
  plot(t, v);
  xlabel('Time [s]'); ylabel('Measured voltage [V]');
  nexttile;
  plot(t, u);
  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sen_steps_time_domain.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:force_sen_steps_time_domain
#+caption: Time domain signal during the 3 actuator voltage steps
#+RESULTS:
[[file:figs/force_sen_steps_time_domain.png]]

** Estimation of the voltage offset and discharge time constant
The measured voltage shows an exponential decay which indicates that the charge across the capacitor formed by the stack is discharging into a resistor.
This corresponds to an RC circuit with a time constant $\tau = RC$.

In order to estimate the time domain, we fit the data with an exponential.
The fit function is:
#+begin_src matlab
  f = @(b,x) b(1).*exp(b(2).*x) + b(3);
#+end_src

Three steps are performed at the following time intervals:
#+begin_src matlab
  t_s = [ 2.5, 23;
         23.8, 35;
         35.8, 50];
#+end_src

We are interested by the =b(2)= term, which is the time constant of the exponential.
#+begin_src matlab
  tau = zeros(size(t_s, 1),1);
  V0  = zeros(size(t_s, 1),1);
#+end_src

#+begin_src matlab
  for t_i = 1:size(t_s, 1)
      t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
      t_cur = t_cur - t_cur(1);
      y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));

      nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
      B0 = [0.5, -0.15, 2.2];                        % Choose Appropriate Initial Estimates
      [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’

      tau(t_i) = 1/B(2);
      V0(t_i)  = B(3);
  end
#+end_src

The obtained values are shown below.

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
#+end_src

#+RESULTS:
| $tau$ [s] | $V_0$ [V] |
|-----------+-----------|
|      6.47 |      2.26 |
|      6.76 |      2.26 |
|      6.49 |      2.25 |

** Estimation of the ADC input impedance

With the capacitance being $C = 4.4 \mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:
#+begin_src matlab
  Cp = 4.4e-6; % [F]
  Rin = abs(mean(tau))/Cp;
#+end_src

#+begin_src matlab :results value replace :exports results
  ans = Rin
#+end_src

#+RESULTS:
: 1494100.0

The input impedance of the Speedgoat's ADC should then be close to $1.5\,M\Omega$ (specified at $1\,M\Omega$).

** Explanation of the Voltage offset
As shown in Figure [[fig:force_sen_steps_time_domain]], the voltage across the Piezoelectric sensor stack shows a constant voltage offset.

We can explain this offset by looking at the electrical model shown in Figure [[fig:force_sensor_model_electronics_without_R]] (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin).

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.

#+name: fig:force_sensor_model_electronics_without_R
#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
[[file:figs/force_sensor_model_electronics_without_R.png]]

The estimated input bias current is then:
#+begin_src matlab
  in = mean(V0)/Rin;
#+end_src

#+begin_src matlab :results value replace :exports results
   ans = in
#+end_src

#+RESULTS:
: 1.5119e-06

** Effect of an additional Parallel Resistor
Be looking at Figure [[fig:force_sensor_model_electronics_without_R]], we can see that an additional resistor in parallel with $R_{in}$ would have two effects:
- reduce the input voltage offset
  \[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
- increase the high pass corner frequency $f_c$
  \[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
  \[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]

If we allow the high pass corner frequency to be equals to 3Hz:
#+begin_src matlab
  fc = 3;
  Ra = Rin/(fc*Cp*Rin - 1);
#+end_src

#+begin_src matlab :results value replace :exports results
  ans = Ra
#+end_src

#+RESULTS:
: 79804

With this parallel resistance value, the voltage offset would be:
#+begin_src matlab
  V_offset = Ra*Rin/(Ra + Rin) * in;
#+end_src

#+begin_src matlab :results value replace :exports results
  ans = V_offset
#+end_src

#+RESULTS:
: 0.11454

Which is much more acceptable.


A resistor $R_p \approx 100\,k\Omega$ is then added in parallel with the force sensor as shown in Figure [[fig:force_sensor_model_electronics]].

#+name: fig:force_sensor_model_electronics
#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor $R_p$
[[file:figs/force_sensor_model_electronics.png]]

** Obtained voltage offset and time constant with the added resistor
After the resistor is added, the same steps response is performed.

#+begin_src matlab
  load('force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');
#+end_src

The results are shown in Figure [[fig:force_sen_steps_time_domain_par_R]].

#+begin_src matlab :exports none
  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
  nexttile;
  plot(t, v);
  xlabel('Time [s]'); ylabel('Measured voltage [V]');
  nexttile;
  plot(t, u);
  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sen_steps_time_domain_par_R.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:force_sen_steps_time_domain_par_R
#+caption: Time domain signal during the actuator voltage steps
#+RESULTS:
[[file:figs/force_sen_steps_time_domain_par_R.png]]

Three steps are performed at the following time intervals:
#+begin_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]
#+end_src

The time constant and voltage offset are again estimated using a fit function.
#+begin_src matlab :exports none
  f = @(b,x) b(1).*exp(b(2).*x) + b(3);

  tau = zeros(size(t_s, 1),1);
  V0  = zeros(size(t_s, 1),1);

  for t_i = 1:size(t_s, 1)
      t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
      t_cur = t_cur - t_cur(1);
      y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));

      nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
      B0 = [0.5, -0.2, 0.2];                        % Choose Appropriate Initial Estimates
      [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’

      tau(t_i) = 1/B(2);
      V0(t_i)  = B(3);
  end
#+end_src

And indeed, we obtain a much smaller offset voltage and a much faster time constant.

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
#+end_src

#+RESULTS:
| $tau$ [s] | $V_0$ [V] |
|-----------+-----------|
|      0.43 |      0.15 |
|      0.45 |      0.16 |
|      0.43 |      0.15 |
|      0.43 |      0.15 |
|      0.45 |      0.15 |
|      0.46 |      0.16 |
|      0.48 |      0.16 |

Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1\,M\Omega$):

#+begin_src matlab
  Cp = 4.4e-6; % [F]
  Rin = abs(mean(tau))/Cp;
#+end_src

#+begin_src matlab :results value replace :exports results
  ans = Rin
#+end_src

#+RESULTS:
: 101200.0

And we can verify that the bias current estimation stays the same:
#+begin_src matlab
  in = mean(V0)/Rin;
#+end_src

#+begin_src matlab :results value replace :exports results
   ans = in
#+end_src

#+RESULTS:
: 1.5305e-06

This validates the model of the ADC and the effectiveness of the added resistor.

* Generated Number of Charge / Voltage
:PROPERTIES:
:header-args:matlab+: :tangle matlab/charge_voltage_estimation.m
:END:
<<sec:charge_voltage_estimation>>

** Introduction                                                      :ignore:
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.

** Matlab Init                                            :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  addpath('./matlab/mat/');
#+end_src

#+begin_src matlab :eval no
  addpath('./mat/');
#+end_src

** Data Loading
The measured data is loaded and the first 25 seconds of data corresponding to transient data are removed.

#+begin_src matlab
  load('force_sensor_sin.mat', 't', 'encoder', 'u', 'v');

  u       = u(t>25);
  v       = v(t>25);
  encoder = encoder(t>25) - mean(encoder(t>25));
  t       = t(t>25);
#+end_src

** Excitation signal and corresponding displacement
The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure [[fig:force_sensor_sin_u]]).

#+begin_src matlab :exports none
  figure;
  plot(t, u)
  xlabel('Time [s]'); ylabel('Control Voltage [V]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sensor_sin_u.pdf', 'width', 'normal', 'height', 'small');
#+end_src

#+name: fig:force_sensor_sin_u
#+caption: Driving Voltage
#+RESULTS:
[[file:figs/force_sensor_sin_u.png]]

The corresponding displacement as measured by the encoder is shown in Figure [[fig:force_sensor_sin_encoder]].

The full stroke is:
#+begin_src matlab :results value replace
  max(encoder)-min(encoder)
#+end_src

#+RESULTS:
: 5.005e-05

#+begin_src matlab :exports none
  figure;
  plot(t, encoder)
  xlabel('Time [s]'); ylabel('Encoder [m]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sensor_sin_encoder.pdf', 'width', 'normal', 'height', 'small');
#+end_src

#+name: fig:force_sensor_sin_encoder
#+caption: Encoder measurement
#+RESULTS:
[[file:figs/force_sensor_sin_encoder.png]]

** Generated Voltage
The generated voltage by the stack is shown in Figure [[fig:force_sensor_sin_stack]].

#+begin_src matlab :exports none
  figure;
  plot(t, v)
  xlabel('Time [s]'); ylabel('Force Sensor Output [V]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sensor_sin_stack.pdf', 'width', 'normal', 'height', 'small');
#+end_src

#+name: fig:force_sensor_sin_stack
#+caption: Voltage measured on the stack used as a sensor
#+RESULTS:
[[file:figs/force_sensor_sin_stack.png]]

** Generated Charge
The capacitance of the stack is
#+begin_src matlab
  Cp = 4.4e-6; % [F]
#+end_src

The voltage and charge across a capacitor are related through the following equation:
\begin{equation}
  U_C = \frac{Q}{C}
\end{equation}
where $U_C$ is the voltage in Volts, $Q$ the charge in Coulombs and $C$ the capacitance in Farads.


The corresponding generated charge is then shown in Figure [[fig:force_sensor_sin_charge]].

#+begin_src matlab :exports none
  figure;
  plot(t, 1e6*Cp*(v-mean(v)))
  xlabel('Time [s]'); ylabel('Generated Charge [$\mu C$]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sensor_sin_charge.pdf', 'width', 'normal', 'height', 'small');
#+end_src

#+name: fig:force_sensor_sin_charge
#+caption: Generated Charge
#+RESULTS:
[[file:figs/force_sensor_sin_charge.png]]

** Generated Voltage/Charge as a function of the displacement
The relation between the generated voltage and the measured displacement is almost linear as shown in Figure [[fig:force_sensor_linear_relation]].

#+begin_src matlab
  b1 = encoder\(v-mean(v));
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(encoder, v-mean(v), 'DisplayName', 'Measured Voltage');
  plot(encoder, encoder*b1, 'DisplayName', sprintf('Linear Fit: $U_s \\approx %.3f [V/\\mu m] \\cdot d$', 1e-6*abs(b1)));
  hold off;
  xlabel('Measured Displacement [m]'); ylabel('Generated Voltage [V]');
  legend();
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/force_sensor_linear_relation.pdf', 'width', 'normal', 'height', 'small');
#+end_src

#+name: fig:force_sensor_linear_relation
#+caption: Almost linear relation between the relative displacement and the generated voltage
#+RESULTS:
[[file:figs/force_sensor_linear_relation.png]]

With a 16bits ADC, the resolution will then be equals to (in [nm]):
#+begin_src matlab :results value replace
  abs((20/2^16)/(b1/1e9))
#+end_src

#+RESULTS:
: 3.9838