2020-11-10 12:55:51 +01:00
#+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"/>
#+HTML_HEAD : <link rel="stylesheet" type="text/css" href="./css/readtheorg.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
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#+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 ‘ ’
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
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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:
#+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
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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
#+caption : Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor $R_p$
[[file:figs/force_sensor_model_electronics.png ]]
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#+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 ‘ ’
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
