#+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
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#+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.

- 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:charge_voltage_estimation]]:

* 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:

** 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

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

** Introduction                                                      :ignore:
Two stacks are used as actuator (in parallel) and one stack is used as sensor.

The amplifier gain is 20V/V (Cedrat LA75B).

** 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

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

#+begin_src matlab
  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]]

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

Fit function:
#+begin_src matlab
  f = @(b,x) b(1).*exp(b(2).*x) + b(3);
#+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

#+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 |

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 $1\,M\Omega$).

#+begin_important
  How can we explain the voltage offset?
#+end_important

As shown in Figure [[fig:force_sensor_model_electronics_without_R]] (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin), an input voltage offset is due to the input bias current $i_n$.

#+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

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.

** Add Parallel Resistor
A resistor $R_p \approx 100\,k\Omega$ is 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]]

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

#+begin_src matlab
  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

Fit function:
#+begin_src matlab
  f = @(b,x) b(1).*exp(b(2).*x) + b(3);
#+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, -0.2, 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.

** Sinus
#+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

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 full stroke as measured by the encoder is:
#+begin_src matlab :results value replace
  max(encoder)-min(encoder)
#+end_src

#+RESULTS:
: 5.005e-05

Its signal is shown in Figure [[fig:force_sensor_sin_encoder]].

#+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]]

The generated voltage by the stack is shown in Figure

#+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]]

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

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]]


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