567 lines
16 KiB
Org Mode
567 lines
16 KiB
Org Mode
#+TITLE: Piezoelectric Force Sensor - Test Bench
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:DRAWER:
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/custom.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results raw replace :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports both
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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:END:
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* Introduction :ignore:
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In this document is studied how a piezoelectric stack can be used to measured the force.
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- 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
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- Section [[sec:charge_voltage_estimation]]:
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* Change of Stiffness due to Sensors stack being open/closed circuit
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/open_closed_circuit.m
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:END:
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<<sec:open_closed_circuit>>
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/mat/');
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#+end_src
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#+begin_src matlab :eval no
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addpath('./mat/');
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#+end_src
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** Load Data
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#+begin_src matlab
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oc = load('identification_open_circuit.mat', 't', 'encoder', 'u');
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sc = load('identification_short_circuit.mat', 't', 'encoder', 'u');
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#+end_src
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** Transfer Functions
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#+begin_src matlab
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Ts = 1e-4; % Sampling Time [s]
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win = hann(ceil(10/Ts));
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#+end_src
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#+begin_src matlab
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[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
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[co_oc_est, ~] = mscohere( oc.u, oc.encoder, win, [], [], 1/Ts);
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[tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
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[co_sc_est, ~] = mscohere( sc.u, sc.encoder, win, [], [], 1/Ts);
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(f, co_oc_est, '-')
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plot(f, co_sc_est, '-')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Coherence'); xlabel('Frequency [Hz]');
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hold off;
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xlim([0.5, 5e3]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/stiffness_force_sensor_coherence.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:stiffness_force_sensor_coherence
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#+caption:
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#+RESULTS:
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[[file:figs/stiffness_force_sensor_coherence.png]]
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#+begin_src matlab :exports none
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figure;
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tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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plot(f, abs(tf_oc_est), '-', 'DisplayName', 'Open-Circuit')
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plot(f, abs(tf_sc_est), '-', 'DisplayName', 'Short-Circuit')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-7, 3e-4]);
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legend('location', 'southwest');
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ax2 = nexttile;
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hold on;
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plot(f, 180/pi*angle(tf_oc_est), '-')
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plot(f, 180/pi*angle(tf_sc_est), '-')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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yticks(-360:90:360);
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axis padded 'auto x'
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linkaxes([ax1,ax2], 'x');
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xlim([0.5, 5e3]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/stiffness_force_sensor_bode.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:stiffness_force_sensor_bode
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#+caption:
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#+RESULTS:
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[[file:figs/stiffness_force_sensor_bode.png]]
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#+begin_src matlab :tangle no :exports results :results file replace
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xlim([180, 280]);
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exportFig('figs/stiffness_force_sensor_bode_zoom.pdf', 'width', 'small', 'height', 'tall');
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#+end_src
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#+name: fig:stiffness_force_sensor_bode_zoom
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#+caption: Zoom on the change of resonance
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#+RESULTS:
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[[file:figs/stiffness_force_sensor_bode_zoom.png]]
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#+begin_important
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The change of resonance frequency / stiffness is very small and is not important here.
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#+end_important
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* Generated Number of Charge / Voltage
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/charge_voltage_estimation.m
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:END:
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<<sec:charge_voltage_estimation>>
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** Introduction :ignore:
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Two stacks are used as actuator (in parallel) and one stack is used as sensor.
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The amplifier gain is 20V/V (Cedrat LA75B).
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/mat/');
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#+end_src
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#+begin_src matlab :eval no
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addpath('./mat/');
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#+end_src
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** Steps
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#+begin_src matlab
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load('force_sensor_steps.mat', 't', 'encoder', 'u', 'v');
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#+end_src
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#+begin_src matlab
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figure;
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tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
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nexttile;
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plot(t, v);
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xlabel('Time [s]'); ylabel('Measured voltage [V]');
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nexttile;
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plot(t, u);
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xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/force_sen_steps_time_domain.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:force_sen_steps_time_domain
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#+caption: Time domain signal during the 3 actuator voltage steps
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#+RESULTS:
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[[file:figs/force_sen_steps_time_domain.png]]
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Three steps are performed at the following time intervals:
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#+begin_src matlab
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t_s = [ 2.5, 23;
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23.8, 35;
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35.8, 50];
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#+end_src
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Fit function:
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#+begin_src matlab
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f = @(b,x) b(1).*exp(b(2).*x) + b(3);
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#+end_src
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We are interested by the =b(2)= term, which is the time constant of the exponential.
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#+begin_src matlab
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tau = zeros(size(t_s, 1),1);
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V0 = zeros(size(t_s, 1),1);
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#+end_src
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#+begin_src matlab
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for t_i = 1:size(t_s, 1)
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t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
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t_cur = t_cur - t_cur(1);
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y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
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nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
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B0 = [0.5, -0.15, 2.2]; % Choose Appropriate Initial Estimates
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[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
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tau(t_i) = 1/B(2);
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V0(t_i) = B(3);
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end
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
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#+end_src
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#+RESULTS:
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| $tau$ [s] | $V_0$ [V] |
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|-----------+-----------|
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| 6.47 | 2.26 |
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| 6.76 | 2.26 |
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| 6.49 | 2.25 |
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With the capacitance being $C = 4.4 \mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:
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#+begin_src matlab
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Cp = 4.4e-6; % [F]
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Rin = abs(mean(tau))/Cp;
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = Rin
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#+end_src
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#+RESULTS:
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: 1494100.0
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The input impedance of the Speedgoat's ADC should then be close to $1.5\,M\Omega$ (specified at $1\,M\Omega$).
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#+begin_important
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How can we explain the voltage offset?
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#+end_important
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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 $i_n$.
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#+name: fig:force_sensor_model_electronics_without_R
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#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right)
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[[file:figs/force_sensor_model_electronics_without_R.png]]
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The estimated input bias current is then:
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#+begin_src matlab
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in = mean(V0)/Rin;
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = in
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#+end_src
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#+RESULTS:
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: 1.5119e-06
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An additional resistor in parallel with $R_{in}$ would have two effects:
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- reduce the input voltage offset
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\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
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- increase the high pass corner frequency $f_c$
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\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
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\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]
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If we allow the high pass corner frequency to be equals to 3Hz:
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#+begin_src matlab
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fc = 3;
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Ra = Rin/(fc*Cp*Rin - 1);
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = Ra
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#+end_src
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#+RESULTS:
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: 79804
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With this parallel resistance value, the voltage offset would be:
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#+begin_src matlab
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V_offset = Ra*Rin/(Ra + Rin) * in;
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = V_offset
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#+end_src
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#+RESULTS:
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: 0.11454
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Which is much more acceptable.
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** Add Parallel Resistor
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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]].
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#+name: fig:force_sensor_model_electronics
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#+caption: Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor $R_p$
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[[file:figs/force_sensor_model_electronics.png]]
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#+begin_src matlab
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load('force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');
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#+end_src
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#+begin_src matlab
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figure;
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tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
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nexttile;
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plot(t, v);
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xlabel('Time [s]'); ylabel('Measured voltage [V]');
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nexttile;
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plot(t, u);
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xlabel('Time [s]'); ylabel('Actuator Voltage [V]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/force_sen_steps_time_domain_par_R.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:force_sen_steps_time_domain_par_R
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#+caption: Time domain signal during the actuator voltage steps
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#+RESULTS:
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[[file:figs/force_sen_steps_time_domain_par_R.png]]
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Three steps are performed at the following time intervals:
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#+begin_src matlab
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t_s = [1.9, 6;
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8.5, 13;
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15.5, 21;
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22.6, 26;
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30.0, 36;
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37.5, 41;
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46.2, 49.5]
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#+end_src
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Fit function:
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#+begin_src matlab
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f = @(b,x) b(1).*exp(b(2).*x) + b(3);
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#+end_src
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We are interested by the =b(2)= term, which is the time constant of the exponential.
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#+begin_src matlab
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tau = zeros(size(t_s, 1),1);
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V0 = zeros(size(t_s, 1),1);
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#+end_src
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#+begin_src matlab
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for t_i = 1:size(t_s, 1)
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t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
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t_cur = t_cur - t_cur(1);
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y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));
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nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
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B0 = [0.5, -0.2, 0.2]; % Choose Appropriate Initial Estimates
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[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
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tau(t_i) = 1/B(2);
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V0(t_i) = B(3);
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end
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#+end_src
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And indeed, we obtain a much smaller offset voltage and a much faster time constant.
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([abs(tau), V0], {}, {'$tau$ [s]', '$V_0$ [V]'}, ' %.2f ');
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#+end_src
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#+RESULTS:
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| $tau$ [s] | $V_0$ [V] |
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|-----------+-----------|
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| 0.43 | 0.15 |
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| 0.45 | 0.16 |
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| 0.43 | 0.15 |
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| 0.43 | 0.15 |
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| 0.45 | 0.15 |
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| 0.46 | 0.16 |
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| 0.48 | 0.16 |
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Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1\,M\Omega$):
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#+begin_src matlab
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Cp = 4.4e-6; % [F]
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Rin = abs(mean(tau))/Cp;
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = Rin
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#+end_src
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#+RESULTS:
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: 101200.0
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And we can verify that the bias current estimation stays the same:
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#+begin_src matlab
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in = mean(V0)/Rin;
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#+end_src
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#+begin_src matlab :results value replace :exports results
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ans = in
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#+end_src
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#+RESULTS:
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: 1.5305e-06
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This validates the model of the ADC and the effectiveness of the added resistor.
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** Sinus
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#+begin_src matlab
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load('force_sensor_sin.mat', 't', 'encoder', 'u', 'v');
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u = u(t>25);
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v = v(t>25);
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encoder = encoder(t>25) - mean(encoder(t>25));
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t = t(t>25);
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#+end_src
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The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure [[fig:force_sensor_sin_u]]).
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#+begin_src matlab :exports none
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figure;
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plot(t, u)
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xlabel('Time [s]'); ylabel('Control Voltage [V]');
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#+end_src
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#+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
|