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Add exp. fit to estimate thermal time constant

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Thomas Dehaeze 2020-10-29 11:05:45 +01:00
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2 changed files with 84 additions and 22 deletions
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@ -3,7 +3,7 @@
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head> <head>
<!-- 2020-10-29 jeu. 10:41 --> <!-- 2020-10-29 jeu. 11:05 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Attocube - Test Bench</title> <title>Attocube - Test Bench</title>
<meta name="generator" content="Org mode" /> <meta name="generator" content="Org mode" />
@ -35,23 +35,23 @@
<h2>Table of Contents</h2> <h2>Table of Contents</h2>
<div id="text-table-of-contents"> <div id="text-table-of-contents">
<ul> <ul>
<li><a href="#org28fc04b">1. Estimation of the Spectral Density of the Attocube Noise</a> <li><a href="#orge576546">1. Estimation of the Spectral Density of the Attocube Noise</a>
<ul> <ul>
<li><a href="#org8f7a999">1.1. Long and Slow measurement</a></li> <li><a href="#orgbaf4829">1.1. Long and Slow measurement</a></li>
<li><a href="#orgf4f16a8">1.2. Short and Fast measurement</a></li> <li><a href="#org8dbfcef">1.2. Short and Fast measurement</a></li>
<li><a href="#org7bd25e6">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li> <li><a href="#org7683847">1.3. Obtained Amplitude Spectral Density of the measured displacement</a></li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
</div> </div>
<div id="outline-container-org28fc04b" class="outline-2"> <div id="outline-container-orge576546" class="outline-2">
<h2 id="org28fc04b"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2> <h2 id="orge576546"><span class="section-number-2">1</span> Estimation of the Spectral Density of the Attocube Noise</h2>
<div class="outline-text-2" id="text-1"> <div class="outline-text-2" id="text-1">
</div> </div>
<div id="outline-container-org8f7a999" class="outline-3"> <div id="outline-container-orgbaf4829" class="outline-3">
<h3 id="org8f7a999"><span class="section-number-3">1.1</span> Long and Slow measurement</h3> <h3 id="orgbaf4829"><span class="section-number-3">1.1</span> Long and Slow measurement</h3>
<div class="outline-text-3" id="text-1-1"> <div class="outline-text-3" id="text-1-1">
<p> <p>
The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\). The first measurement was made during ~17 hours with a sampling time of \(T_s = 0.1\,s\).
@ -64,17 +64,45 @@ Ts = 0.1; <span class="org-comment">% [s]</span>
</div> </div>
<div id="org7f956e7" class="figure"> <div id="org4f51280" class="figure">
<p><img src="figs/long_meas_time_domain_full.png" alt="long_meas_time_domain_full.png" /> <p><img src="figs/long_meas_time_domain_full.png" alt="long_meas_time_domain_full.png" />
</p> </p>
<p><span class="figure-number">Figure 1: </span>Long measurement time domain data</p> <p><span class="figure-number">Figure 1: </span>Long measurement time domain data</p>
</div> </div>
<p> <p>
We can see in Figure <a href="#org7f956e7">1</a> that there is a transient period where the measured displacement experiences some drifts. Let&rsquo;s fit the data with a step response to a first order low pass filter (Figure <a href="#orgc987453">2</a>).
</p>
<div class="org-src-container">
<pre class="src src-matlab">f = @(b,x) b(1)<span class="org-type">*</span>(1 <span class="org-type">-</span> exp(<span class="org-type">-</span>x<span class="org-type">/</span>b(2)));
y_cur = x(t <span class="org-type">&lt;</span> 17<span class="org-type">*</span>60<span class="org-type">*</span>60);
t_cur = t(t <span class="org-type">&lt;</span> 17<span class="org-type">*</span>60<span class="org-type">*</span>60);
nrmrsd = @(b) norm(y_cur <span class="org-type">-</span> f(b,t_cur)); <span class="org-comment">% Residual Norm Cost Function</span>
B0 = [400e<span class="org-type">-</span>9, 2<span class="org-type">*</span>60<span class="org-type">*</span>60]; <span class="org-comment">% Choose Appropriate Initial Estimates</span>
[B,rnrm] = fminsearch(nrmrsd, B0); <span class="org-comment">% Estimate Parameters &#8216;B&#8217;</span>
</pre>
</div>
<p>
The corresponding time constant is (in [h]):
</p>
<pre class="example">
2.0576
</pre>
<p>
<a href="figs/long_meas_time_domain_fit.pdf">figs/long_meas_time_domain_fit.pdf</a>
</p>
<p>
We can see in Figure <a href="#org4f51280">1</a> that there is a transient period where the measured displacement experiences some drifts.
This is probably due to thermal effects. This is probably due to thermal effects.
We only select the data between <code>t1</code> and <code>t2</code>. We only select the data between <code>t1</code> and <code>t2</code>.
The obtained displacement is shown in Figure <a href="#orgab24552">2</a>. The obtained displacement is shown in Figure <a href="#org24b01ba">2</a>.
</p> </p>
<div class="org-src-container"> <div class="org-src-container">
@ -88,7 +116,7 @@ t = t <span class="org-type">-</span> t(1);
</div> </div>
<div id="orgab24552" class="figure"> <div id="org24b01ba" class="figure">
<p><img src="figs/long_meas_time_domain_zoom.png" alt="long_meas_time_domain_zoom.png" /> <p><img src="figs/long_meas_time_domain_zoom.png" alt="long_meas_time_domain_zoom.png" />
</p> </p>
<p><span class="figure-number">Figure 2: </span>Kept data (removed slow drifts during the first hours)</p> <p><span class="figure-number">Figure 2: </span>Kept data (removed slow drifts during the first hours)</p>
@ -105,8 +133,8 @@ The Power Spectral Density of the measured displacement is computed
</div> </div>
</div> </div>
<div id="outline-container-orgf4f16a8" class="outline-3"> <div id="outline-container-org8dbfcef" class="outline-3">
<h3 id="orgf4f16a8"><span class="section-number-3">1.2</span> Short and Fast measurement</h3> <h3 id="org8dbfcef"><span class="section-number-3">1.2</span> Short and Fast measurement</h3>
<div class="outline-text-3" id="text-1-2"> <div class="outline-text-3" id="text-1-2">
<p> <p>
An second measurement is done in order to estimate the high frequency noise of the interferometer. An second measurement is done in order to estimate the high frequency noise of the interferometer.
@ -120,11 +148,11 @@ Ts = 1e<span class="org-type">-</span>4; <span class="org-comment">% [s]</span>
</div> </div>
<p> <p>
The time domain measurement is shown in Figure <a href="#orge8034c6">3</a>. The time domain measurement is shown in Figure <a href="#org06abe6b">3</a>.
</p> </p>
<div id="orge8034c6" class="figure"> <div id="org06abe6b" class="figure">
<p><img src="figs/short_meas_time_domain.png" alt="short_meas_time_domain.png" /> <p><img src="figs/short_meas_time_domain.png" alt="short_meas_time_domain.png" />
</p> </p>
<p><span class="figure-number">Figure 3: </span>Time domain measurement with the high sampling rate</p> <p><span class="figure-number">Figure 3: </span>Time domain measurement with the high sampling rate</p>
@ -141,15 +169,15 @@ The Power Spectral Density of the measured displacement is computed
</div> </div>
</div> </div>
<div id="outline-container-org7bd25e6" class="outline-3"> <div id="outline-container-org7683847" class="outline-3">
<h3 id="org7bd25e6"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3> <h3 id="org7683847"><span class="section-number-3">1.3</span> Obtained Amplitude Spectral Density of the measured displacement</h3>
<div class="outline-text-3" id="text-1-3"> <div class="outline-text-3" id="text-1-3">
<p> <p>
The computed ASD of the two measurements are combined in Figure <a href="#org060925e">4</a>. The computed ASD of the two measurements are combined in Figure <a href="#orgbe2f5ce">4</a>.
</p> </p>
<div id="org060925e" class="figure"> <div id="orgbe2f5ce" class="figure">
<p><img src="figs/psd_combined.png" alt="psd_combined.png" /> <p><img src="figs/psd_combined.png" alt="psd_combined.png" />
</p> </p>
<p><span class="figure-number">Figure 4: </span>Obtained Amplitude Spectral Density of the measured displacement</p> <p><span class="figure-number">Figure 4: </span>Obtained Amplitude Spectral Density of the measured displacement</p>
@ -160,7 +188,7 @@ The computed ASD of the two measurements are combined in Figure <a href="#org060
</div> </div>
<div id="postamble" class="status"> <div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p> <p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-10-29 jeu. 10:41</p> <p class="date">Created: 2020-10-29 jeu. 11:05</p>
</div> </div>
</body> </body>
</html> </html>

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@ -70,6 +70,40 @@ The first measurement was made during ~17 hours with a sampling time of $T_s = 0
#+RESULTS: #+RESULTS:
[[file:figs/long_meas_time_domain_full.png]] [[file:figs/long_meas_time_domain_full.png]]
Let's fit the data with a step response to a first order low pass filter (Figure [[fig:long_meas_time_domain_fit]]).
#+begin_src matlab
f = @(b,x) b(1)*(1 - exp(-x/b(2)));
y_cur = x(t < 17*60*60);
t_cur = t(t < 17*60*60);
nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
B0 = [400e-9, 2*60*60]; % Choose Appropriate Initial Estimates
[B,rnrm] = fminsearch(nrmrsd, B0); % Estimate Parameters ‘B’
#+end_src
The corresponding time constant is (in [h]):
#+begin_src matlab :results value replace :exports results
B(2)/60/60
#+end_src
#+RESULTS:
: 2.0576
#+begin_src matlab :exports none
figure;
hold on;
plot(t_cur/60/60, 1e9*y_cur);
plot(t_cur/60/60, 1e9*f(B, t_cur));
hold off;
xlabel('Time [h]'); ylabel('Displacement [nm]');
#+end_src
#+name: fig:long_meas_time_domain_fit
#+caption: Fit of the measurement data with a step response of a first order low pass filter
[[file:figs/long_meas_time_domain_fit.pdf]]
We can see in Figure [[fig:long_meas_time_domain_full]] that there is a transient period where the measured displacement experiences some drifts. We can see in Figure [[fig:long_meas_time_domain_full]] that there is a transient period where the measured displacement experiences some drifts.
This is probably due to thermal effects. This is probably due to thermal effects.
We only select the data between =t1= and =t2=. We only select the data between =t1= and =t2=.