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Analyze all noise meaurements

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Thomas Dehaeze 2021-01-22 23:44:36 +01:00
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2021-01-19 mar. 23:00 -->
<!-- 2021-01-22 ven. 23:44 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<title>Voltage Amplifier PD200 - Test Bench</title>
<meta name="generator" content="Org mode" />
@ -39,33 +39,41 @@
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org9fe8e60">1. Introduction</a></li>
<li><a href="#org83288a7">2. Voltage Amplifier Requirements</a></li>
<li><a href="#org2725a7d">3. PD200 Expected characteristics</a></li>
<li><a href="#org6748772">4. Voltage Amplifier Model</a></li>
<li><a href="#orgb0f1751">5. Noise measurement</a>
<li><a href="#org8f2862b">1. Introduction</a></li>
<li><a href="#org103c717">2. Voltage Amplifier Requirements</a></li>
<li><a href="#orgf22ce98">3. PD200 Expected characteristics</a></li>
<li><a href="#orge04c2d5">4. Voltage Amplifier Model</a></li>
<li><a href="#org5986efd">5. Noise measurement</a>
<ul>
<li><a href="#org077faf1">5.1. Setup</a></li>
<li><a href="#org8d11397">5.2. Results</a>
<li><a href="#org1515801">5.1. Setup</a></li>
<li><a href="#orgf67652b">5.2. Model of the setup</a></li>
<li><a href="#org109d4fe">5.3. Quantization Noise</a></li>
<li><a href="#org3e7c8ba">5.4. Pre Amplifier noise measurement</a></li>
<li><a href="#orgdd4cdcb">5.5. PD200 noise measurement</a></li>
<li><a href="#org77f4d34">5.6. DAC noise measurement</a></li>
<li><a href="#orgb297da3">5.7. Total noise measurement</a></li>
<li><a href="#org41977eb">5.8. 20bits DAC noise measurement</a></li>
</ul>
</li>
<li><a href="#org311b8b4">6. Transfer Function measurement</a>
<ul>
<li><a href="#org3e569c9">5.2.1. Noise when shunting the input (50 Ohms)</a></li>
<li><a href="#org032d612">6.1. Setup</a></li>
<li><a href="#orgcaa9498">6.2. Maximum Frequency/Voltage to not overload the amplifier</a></li>
<li><a href="#org2323f70">6.3. Results</a>
<ul>
<li><a href="#orge73cc45">6.3.1. First test</a></li>
<li><a href="#orgeb520e4">6.3.2. Results</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgaf96727">6. Transfer Function measurement</a>
<ul>
<li><a href="#org9868c43">6.1. Setup</a></li>
<li><a href="#orgc5c49ee">6.2. Results</a></li>
</ul>
</li>
<li><a href="#org516bcbb">7. Conclusion</a></li>
<li><a href="#orgad3a328">7. Conclusion</a></li>
</ul>
</div>
</div>
<div id="outline-container-org9fe8e60" class="outline-2">
<h2 id="org9fe8e60"><span class="section-number-2">1</span> Introduction</h2>
<div id="outline-container-org8f2862b" class="outline-2">
<h2 id="org8f2862b"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1">
<p>
The goal of this test bench is to characterize the Voltage amplifier <a href="https://www.piezodrive.com/drivers/pd200-60-watt-voltage-amplifier/">PD200</a> from PiezoDrive.
@ -76,7 +84,7 @@ The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here<
</p>
<div id="orga2cd341" class="figure">
<div id="org7aea75d" class="figure">
<p><img src="figs/amplifier_PD200.png" alt="amplifier_PD200.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Picture of the PD200 Voltage Amplifier</p>
@ -84,10 +92,10 @@ The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here<
</div>
</div>
<div id="outline-container-org83288a7" class="outline-2">
<h2 id="org83288a7"><span class="section-number-2">2</span> Voltage Amplifier Requirements</h2>
<div id="outline-container-org103c717" class="outline-2">
<h2 id="org103c717"><span class="section-number-2">2</span> Voltage Amplifier Requirements</h2>
<div class="outline-text-2" id="text-2">
<table id="org6825b69" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="orgf1fdf95" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Requirements for the Voltage Amplifier</caption>
<colgroup>
@ -131,10 +139,10 @@ The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here<
</div>
</div>
<div id="outline-container-org2725a7d" class="outline-2">
<h2 id="org2725a7d"><span class="section-number-2">3</span> PD200 Expected characteristics</h2>
<div id="outline-container-orgf22ce98" class="outline-2">
<h2 id="orgf22ce98"><span class="section-number-2">3</span> PD200 Expected characteristics</h2>
<div class="outline-text-2" id="text-3">
<table id="orgf99d960" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org38f8e47" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Characteristics of the PD200</caption>
<colgroup>
@ -209,18 +217,18 @@ The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here<
</table>
<p>
For a load capacitance of \(10\,\mu F\), the expected \(-3\,dB\) bandwidth is \(6.4\,kHz\) (Figure <a href="#orgf39e37f">2</a>) and the low frequency noise is \(650\,\mu V\,\text{rms}\) (Figure <a href="#org2267cad">3</a>).
For a load capacitance of \(10\,\mu F\), the expected \(-3\,dB\) bandwidth is \(6.4\,kHz\) (Figure <a href="#org2190892">2</a>) and the low frequency noise is \(650\,\mu V\,\text{rms}\) (Figure <a href="#orgeaff484">3</a>).
</p>
<div id="orgf39e37f" class="figure">
<div id="org2190892" class="figure">
<p><img src="./figs/pd200_expected_small_signal_bandwidth.png" alt="pd200_expected_small_signal_bandwidth.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Expected small signal bandwidth</p>
</div>
<div id="org2267cad" class="figure">
<div id="orgeaff484" class="figure">
<p><img src="figs/pd200_expected_noise.png" alt="pd200_expected_noise.png" />
</p>
<p><span class="figure-number">Figure 3: </span>Expected Low frequency noise from 0.03Hz to 20Hz</p>
@ -228,8 +236,8 @@ For a load capacitance of \(10\,\mu F\), the expected \(-3\,dB\) bandwidth is \(
</div>
</div>
<div id="outline-container-org6748772" class="outline-2">
<h2 id="org6748772"><span class="section-number-2">4</span> Voltage Amplifier Model</h2>
<div id="outline-container-orge04c2d5" class="outline-2">
<h2 id="orge04c2d5"><span class="section-number-2">4</span> Voltage Amplifier Model</h2>
<div class="outline-text-2" id="text-4">
<p>
The Amplifier is characterized by its dynamics \(G_a(s)\) from voltage inputs \(V_{in}\) to voltage output \(V_{out}\).
@ -246,11 +254,11 @@ The objective is therefore to determine the transfer function \(G_a(s)\) from th
</p>
<p>
As both \(G_a\) and \(S_n\) depends on the load capacitance, they should be measured when loading the amplifier with a \(\SI{10}{\micro\farad}\) capacitor.
As both \(G_a\) and \(S_n\) depends on the load capacitance, they should be measured when loading the amplifier with a \(10\,\mu F\) capacitor.
</p>
<div id="org4313e25" class="figure">
<div id="org5d4d3ab" class="figure">
<p><img src="figs/pd200-model-schematic.png" alt="pd200-model-schematic.png" />
</p>
<p><span class="figure-number">Figure 4: </span>Model of the voltage amplifier</p>
@ -258,14 +266,29 @@ As both \(G_a\) and \(S_n\) depends on the load capacitance, they should be meas
</div>
</div>
<div id="outline-container-orgb0f1751" class="outline-2">
<h2 id="orgb0f1751"><span class="section-number-2">5</span> Noise measurement</h2>
<div id="outline-container-org5986efd" class="outline-2">
<h2 id="org5986efd"><span class="section-number-2">5</span> Noise measurement</h2>
<div class="outline-text-2" id="text-5">
<ul class="org-ul">
<li>Section <a href="#org975f5ab">5.1</a></li>
<li>Section <a href="#org58a7c02">5.2</a></li>
<li>Section <a href="#orgd6eb89a">5.3</a></li>
<li>Section <a href="#orgd67c98a">5.4</a></li>
<li>Section <a href="#orge02d748">5.5</a></li>
<li>Section <a href="#org30c83b3">5.6</a></li>
<li>Section <a href="#org0d900c3">5.7</a></li>
<li>Section <a href="#org576bf2a">5.8</a></li>
</ul>
</div>
<div id="outline-container-org077faf1" class="outline-3">
<h3 id="org077faf1"><span class="section-number-3">5.1</span> Setup</h3>
<div id="outline-container-org1515801" class="outline-3">
<h3 id="org1515801"><span class="section-number-3">5.1</span> Setup</h3>
<div class="outline-text-3" id="text-5-1">
<div class="note" id="org3d87176">
<p>
<a id="org975f5ab"></a>
</p>
<div class="note" id="org0370347">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
@ -298,7 +321,7 @@ This gain should be around 1000.
</p>
<div id="orgb37f1e6" class="figure">
<div id="org451a2c9" class="figure">
<p><img src="figs/setup-noise-measurement.png" alt="setup-noise-measurement.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Schematic of the test bench to measure the Power Spectral Density of the Voltage amplifier noise \(n\)</p>
@ -311,41 +334,192 @@ An high pass filter at low frequency can be added if there is a problem of large
</div>
</div>
<div id="outline-container-org8d11397" class="outline-3">
<h3 id="org8d11397"><span class="section-number-3">5.2</span> Results</h3>
<div id="outline-container-orgf67652b" class="outline-3">
<h3 id="orgf67652b"><span class="section-number-3">5.2</span> Model of the setup</h3>
<div class="outline-text-3" id="text-5-2">
</div>
<div id="outline-container-org3e569c9" class="outline-4">
<h4 id="org3e569c9"><span class="section-number-4">5.2.1</span> Noise when shunting the input (50 Ohms)</h4>
<div class="outline-text-4" id="text-5-2-1">
<p>
The time domain measurements of the amplifier noise are shown in Figure <a href="#org6fb276a">6</a>.
<a id="org58a7c02"></a>
</p>
<div id="org6fb276a" class="figure">
<p><img src="figs/noise_shunt_time_3uF.png" alt="noise_shunt_time_3uF.png" />
<p>
As shown in Figure <a href="#org8801056">6</a>, there are 4 equipment involved in the measurement:
</p>
<p><span class="figure-number">Figure 6: </span>Time domain measurement of the amplifier output noise</p>
<ul class="org-ul">
<li>a Digital to Analog Convert (DAC)</li>
<li>the Voltage amplifier to be measured with a gain of 20 (PD200)</li>
<li>a low noise voltage amplifier with a variable gain and integrated low pass filters and high pass filters</li>
<li>an Analog to Digital Converter (ADC)</li>
</ul>
<p>
Each of these equipment has some noise:
</p>
<ul class="org-ul">
<li>\(q_{da}\): quantization noise of the DAC</li>
<li>\(n_{da}\): output noise of the DAC</li>
<li>\(n_p\): output noise of the PD200 (what we wish to characterize)</li>
<li>\(n_a\): input noise of the pre amplifier</li>
<li>\(q_{ad}\): quantization noise of the ADC</li>
</ul>
<div id="org8801056" class="figure">
<p><img src="figs/noise_meas_procedure.png" alt="noise_meas_procedure.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Sources of noise in the experimental setup</p>
</div>
</div>
</div>
<div id="outline-container-org109d4fe" class="outline-3">
<h3 id="org109d4fe"><span class="section-number-3">5.3</span> Quantization Noise</h3>
<div class="outline-text-3" id="text-5-3">
<p>
<a id="orgd6eb89a"></a>
</p>
<p>
The quantization noise is something that can be predicted.
The Amplitude Spectral Density of the quantization noise of an ADC/DAC is equal to:
</p>
\begin{equation}
\Gamma_q(\omega) = \frac{q}{\sqrt{12 f_s}}
\end{equation}
<p>
with:
</p>
<ul class="org-ul">
<li>\(q = \frac{\Delta V}{2^n}\) the quantization in [V], which is the corresponding value in [V] of the least significant bit</li>
<li>\(\Delta V\) is the full range of the ADC in [V]</li>
<li>\(n\) is the number of bits</li>
<li>\(f_s\) is the sample frequency in [Hz]</li>
</ul>
<div class="org-src-container">
<pre class="src src-matlab">adc = struct();
adc.Delta_V = 20; <span class="org-comment">% [V]</span>
adc.n = 16; <span class="org-comment">% number of bits</span>
adc.Fs = 20e3; <span class="org-comment">% [Hz]</span>
adc.Gamma_q = adc.Delta_V<span class="org-type">/</span>2<span class="org-type">^</span>adc.n<span class="org-type">/</span>sqrt(12<span class="org-type">*</span>adc.Fs); <span class="org-comment">% [V/sqrt(Hz)]</span>
</pre>
</div>
<p>
Obtained low frequency (0.1Hz - 20Hz) noise is shown in Figure <a href="#orgaadf193">7</a> which is very similar to the noise shown in the documentation (Figure <a href="#org2267cad">3</a>).
The obtained Amplitude Spectral Density is <code>6.2294e-07</code> \(V/\sqrt{Hz}\).
</p>
</div>
</div>
<div id="outline-container-org3e7c8ba" class="outline-3">
<h3 id="org3e7c8ba"><span class="section-number-3">5.4</span> Pre Amplifier noise measurement</h3>
<div class="outline-text-3" id="text-5-4">
<p>
<a id="orgd67c98a"></a>
</p>
<p>
First, we wish to measure the noise of the pre-amplifier.
To do so, the input of the pre-amplifier is shunted such that there is 0V at its inputs.
Then, the gain of the amplifier is increase until the measured signal on the ADC is much larger than the quantization noise.
</p>
<p>
The Amplitude Spectral Density \(\Gamma_n(\omega)\) of the measured signal \(n\) is computed.
Finally, the Amplitude Spectral Density of \(n_a\) can be computed taking into account the gain of the pre-amplifier:
</p>
\begin{equation}
\Gamma_{n_a}(\omega) \approx \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
<p>
This is true if the quantization noise \(\Gamma_{q_{ad}}\) is negligible.
</p>
<div id="orgaadf193" class="figure">
<p><img src="figs/low_noise_time_domain_3uF.png" alt="low_noise_time_domain_3uF.png" />
<div id="org0e8db56" class="figure">
<p><img src="figs/noise_measure_setup_preamp.png" alt="noise_measure_setup_preamp.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Low Frequency Noise (0.1Hz - 20Hz)</p>
<p><span class="figure-number">Figure 7: </span>Sources of noise in the experimental setup</p>
</div>
<p>
The obtained RMS and peak to peak values of the measured noises are shown in Table <a href="#orgd174c39">3</a>.
The gain of the low noise amplifier is set to <code>50000</code>.
</p>
<table id="orgd174c39" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> RMS and Peak to Peak measured noise</caption>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-comment">% Hanning window</span>
win = hanning(ceil(0.5<span class="org-type">/</span>Ts));
<span class="org-comment">% Power Spectral Density</span>
[pxx, f] = pwelch(preamp.Vn, win, [], [], Fs);
<span class="org-comment">% Save the results inside the struct</span>
preamp.pxx = pxx;
preamp.f = f;
</pre>
</div>
<p>
The obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier is shown in Figure <a href="#org2880354">8</a>.
The obtained noise amplitude is very closed to the one specified in the documentation of \(4nV/\sqrt{Hz}\) at 1kHZ.
</p>
<div id="org2880354" class="figure">
<p><img src="figs/asd_preamp.png" alt="asd_preamp.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier</p>
</div>
</div>
</div>
<div id="outline-container-orgdd4cdcb" class="outline-3">
<h3 id="orgdd4cdcb"><span class="section-number-3">5.5</span> PD200 noise measurement</h3>
<div class="outline-text-3" id="text-5-5">
<p>
<a id="orge02d748"></a>
</p>
<p>
The input of the PD200 amplifier is shunted such that there is 0V between its inputs.
Then the gain of the pre-amplifier is increased in order to measure a signal much larger than the quantization noise of the ADC.
We compute the Amplitude Spectral Density of the measured signal \(\Gamma_n(\omega)\).
The Amplitude Spectral Density of \(n_p\) can be computed taking into account the gain of the pre-amplifier:
</p>
\begin{equation}
\Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
<p>
And we verify that this is indeed the noise of the PD200 and not the noise of the pre-amplifier by checking that:
</p>
\begin{equation}
\Gamma_{n_p} \ll \Gamma_{n_a}
\end{equation}
<div id="org5660b1a" class="figure">
<p><img src="figs/noise_measure_setup_pd200.png" alt="noise_measure_setup_pd200.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Sources of noise in the experimental setup</p>
</div>
<p>
The measured low frequency noise \(n_p\) of one of the amplifiers is shown in Figure <a href="#org99c1c8c">10</a>.
It is very similar to the one specified in the datasheet in Figure <a href="#orgeaff484">3</a>.
</p>
<div id="org99c1c8c" class="figure">
<p><img src="figs/pd200_noise_time_lpf.png" alt="pd200_noise_time_lpf.png" />
</p>
<p><span class="figure-number">Figure 10: </span>Measured low frequency noise of the PD200 from 0.01Hz to 20Hz</p>
</div>
<p>
The obtained RMS and peak to peak values of the measured noises are shown in Table <a href="#org904c283">3</a>.
</p>
<table id="org904c283" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 3:</span> RMS and Peak to Peak measured low frequency noise (0.01Hz to 20Hz)</caption>
<colgroup>
<col class="org-left" />
@ -370,87 +544,202 @@ The obtained RMS and peak to peak values of the measured noises are shown in Tab
<tr>
<td class="org-left">PD200_1</td>
<td class="org-right">524.9</td>
<td class="org-right">4.5</td>
<td class="org-right">565.1</td>
<td class="org-right">3.7</td>
</tr>
<tr>
<td class="org-left">PD200_2</td>
<td class="org-right">807.7</td>
<td class="org-right">6.7</td>
<td class="org-right">767.6</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_3</td>
<td class="org-right">630.3</td>
<td class="org-right">5.4</td>
<td class="org-right">479.9</td>
<td class="org-right">3.0</td>
</tr>
<tr>
<td class="org-left">PD200_4</td>
<td class="org-right">619.7</td>
<td class="org-right">5.5</td>
<td class="org-right">615.7</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_5</td>
<td class="org-right">630.8</td>
<td class="org-right">5.6</td>
<td class="org-right">651.0</td>
<td class="org-right">2.4</td>
</tr>
<tr>
<td class="org-left">PD200_6</td>
<td class="org-right">517.3</td>
<td class="org-right">4.9</td>
<td class="org-right">473.2</td>
<td class="org-right">2.7</td>
</tr>
<tr>
<td class="org-left">PD200_7</td>
<td class="org-right">393.8</td>
<td class="org-right">3.7</td>
<td class="org-right">423.1</td>
<td class="org-right">2.3</td>
</tr>
</tbody>
</table>
<p>
The PSD of the measured noise is computed and the ASD is shown in Figure <a href="#org17a3769">8</a>.
The Amplitude Spectral Density of the measured noise is now computed and shown in Figure <a href="#org7bcb803">11</a>.
</p>
<div id="org7bcb803" class="figure">
<p><img src="figs/asd_noise_3uF_warmup.png" alt="asd_noise_3uF_warmup.png" />
</p>
<p><span class="figure-number">Figure 11: </span>Amplitude Spectral Density of the measured noise</p>
</div>
</div>
</div>
<div id="outline-container-org77f4d34" class="outline-3">
<h3 id="org77f4d34"><span class="section-number-3">5.6</span> DAC noise measurement</h3>
<div class="outline-text-3" id="text-5-6">
<p>
<a id="org30c83b3"></a>
</p>
<p>
In order not to have any quantization noise, we impose the DAC to output a zero voltage.
The gain of the low noise amplifier is adjusted to
</p>
<p>
The Amplitude Spectral Density \(\Gamma_n(\omega)\) of the measured signal is computed.
The Amplitude Spectral Density of \(n_{da}\) can be computed taking into account the gain of the pre-amplifier:
</p>
\begin{equation}
\Gamma_{n_{da}}(\omega) = \frac{\Gamma_m(\omega)}{|G_a(\omega)|}
\end{equation}
<p>
And it is verify that the Amplitude Spectral Density of \(n_{da}\) is much larger than the one of \(n_a\):
</p>
\begin{equation}
\Gamma_{n_{da}} \gg \Gamma_{n_a}
\end{equation}
<div id="org744e44a" class="figure">
<p><img src="figs/noise_measure_setup_dac.png" alt="noise_measure_setup_dac.png" />
</p>
<p><span class="figure-number">Figure 12: </span>Sources of noise in the experimental setup</p>
</div>
<div id="orgc0933a7" class="figure">
<p><img src="figs/asd_noise_dac.png" alt="asd_noise_dac.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-orgb297da3" class="outline-3">
<h3 id="orgb297da3"><span class="section-number-3">5.7</span> Total noise measurement</h3>
<div class="outline-text-3" id="text-5-7">
<p>
<a id="org0d900c3"></a>
</p>
<p>
Let&rsquo;s now analyze the measurement of the setup in Figure <a href="#org8801056">6</a>.
</p>
<p>
The PSD of the measured noise is computed and the ASD is shown in Figure <a href="#org929789a">14</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(0.5<span class="org-type">/</span>Ts));
[pxx, f] = pwelch(pd200{1}.Vn, win, [], [], Fs);
pxx = zeros(length(pxx), 7);
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:7</span>
pxx(<span class="org-type">:</span>, <span class="org-constant">i</span>) = pwelch(pd200{<span class="org-constant">i</span>}.Vn, win, [], [], Fs);
[pxx, f] = pwelch(pd200dac{<span class="org-constant">i</span>}.Vn, win, [], [], Fs);
pd200dac{<span class="org-constant">i</span>}.f = f;
pd200dac{<span class="org-constant">i</span>}.pxx = pxx;
<span class="org-keyword">end</span>
</pre>
</div>
<div id="org17a3769" class="figure">
<p><img src="figs/asd_noise_3uF.png" alt="asd_noise_3uF.png" />
<div id="org929789a" class="figure">
<p><img src="figs/asd_noise_tot.png" alt="asd_noise_tot.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Amplitude Spectral Density of the measured noise</p>
<p><span class="figure-number">Figure 14: </span>Amplitude Spectral Density of the measured noise and of the individual sources of noise</p>
</div>
<div class="important" id="org623c3d1">
<p>
The output noise of the PD200 amplifier is limited by the noise of the DAC.
Having a DAC with lower noise could lower the output noise of the PD200.
SSI2V DACs will be used to verify that.
</p>
</div>
</div>
</div>
<div id="outline-container-org41977eb" class="outline-3">
<h3 id="org41977eb"><span class="section-number-3">5.8</span> 20bits DAC noise measurement</h3>
<div class="outline-text-3" id="text-5-8">
<p>
<a id="org576bf2a"></a>
Let&rsquo;s now measure the noise of another DAC called the &ldquo;SSI2V&rdquo; (<a href="doc/[SSI2V]Datasheet.pdf">doc</a>).
It is a 20bits DAC with an output of +/-10.48 V and a very low noise.
</p>
<p>
The measurement setup is the same as the one in Figure <a href="#org744e44a">12</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(0.5<span class="org-type">/</span>Ts));
[pxx, f] = pwelch(ssi2v.Vn, win, [], [], Fs);
ssi2v.pxx = pxx;
ssi2v.f = f;
</pre>
</div>
<p>
The obtained noise of the SSI2V DAC is shown in Figure <a href="#orgd5ecb95">15</a> and compared with the noise of the 16bits DAC.
It is shown to be much smaller (~1 order of magnitude).
</p>
<div id="orgd5ecb95" class="figure">
<p><img src="figs/asd_ssi2v_noise.png" alt="asd_ssi2v_noise.png" />
</p>
<p><span class="figure-number">Figure 15: </span>Amplitude Spectral Density of the SSI2V DAC&rsquo;s noise</p>
</div>
<div class="important" id="org3ec30db">
<p>
Using the SSI2V as the DAC with the PD200 should give much better noise output than using the 16bits DAC.
The limiting factor should then be the noise of the PD200 itself.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgaf96727" class="outline-2">
<h2 id="orgaf96727"><span class="section-number-2">6</span> Transfer Function measurement</h2>
<div id="outline-container-org311b8b4" class="outline-2">
<h2 id="org311b8b4"><span class="section-number-2">6</span> Transfer Function measurement</h2>
<div class="outline-text-2" id="text-6">
</div>
<div id="outline-container-org9868c43" class="outline-3">
<h3 id="org9868c43"><span class="section-number-3">6.1</span> Setup</h3>
<div id="outline-container-org032d612" class="outline-3">
<h3 id="org032d612"><span class="section-number-3">6.1</span> Setup</h3>
<div class="outline-text-3" id="text-6-1">
<p>
In order to measure the transfer function from the input voltage \(V_{in}\) to the output voltage \(V_{out}\), the test bench shown in Figure <a href="#org472ad71">9</a> is used.
In order to measure the transfer function from the input voltage \(V_{in}\) to the output voltage \(V_{out}\), the test bench shown in Figure <a href="#orga5c58e5">16</a> is used.
</p>
<div class="note" id="org5cbd7bf">
<div class="note" id="org44386ba">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
@ -467,22 +756,166 @@ For this measurement, the sampling frequency of the Speedgoat ADC should be as h
</p>
<div id="org472ad71" class="figure">
<div id="orga5c58e5" class="figure">
<p><img src="figs/setup-dynamics-measurement.png" alt="setup-dynamics-measurement.png" />
</p>
<p><span class="figure-number">Figure 9: </span>Schematic of the test bench to estimate the dynamics from voltage input \(V_{in}\) to voltage output \(V_{out}\)</p>
<p><span class="figure-number">Figure 16: </span>Schematic of the test bench to estimate the dynamics from voltage input \(V_{in}\) to voltage output \(V_{out}\)</p>
</div>
</div>
</div>
<div id="outline-container-orgc5c49ee" class="outline-3">
<h3 id="orgc5c49ee"><span class="section-number-3">6.2</span> Results</h3>
<div id="outline-container-orgcaa9498" class="outline-3">
<h3 id="orgcaa9498"><span class="section-number-3">6.2</span> Maximum Frequency/Voltage to not overload the amplifier</h3>
<div class="outline-text-3" id="text-6-2">
<p>
The maximum current is 1A [rms] which corresponds to 0.7A in amplitude of the sin wave.
</p>
<p>
The impedance of the capacitance is:
\[ Z_C(\omega) = \frac{1}{jC\omega} \]
</p>
<p>
Therefore the relation between the output current and the output voltage is (in amplitude):
\[ V_{out} = \frac{1}{C\omega} I_{out} \]
</p>
<p>
There is a gain of 20 between the input voltage and the output voltage:
\[ 20 V_{in} = \frac{1}{C\omega} I_{out} \]
</p>
<p>
For a specified voltage input amplitude \(V_{in}\), the maximum frequency is then:
\[ \omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}} \]
</p>
<div class="org-src-container">
<pre class="src src-matlab">Iout_max = 0.57; <span class="org-comment">% Maximum output current [A]</span>
C = 2.7e<span class="org-type">-</span>6; <span class="org-comment">% Load Capacitance [F]</span>
V_in = linspace(0, 5, 100); <span class="org-comment">% Input Voltage [V]</span>
w_max = 1<span class="org-type">./</span>(20<span class="org-type">*</span>C<span class="org-type">*</span>V_in) <span class="org-type">*</span> Iout_max; <span class="org-comment">% [rad/s]</span>
<span class="org-type">figure</span>;
plot(V_in, w_max<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span>);
xlabel(<span class="org-string">'Input Voltage Amplitude [V]'</span>);
ylabel(<span class="org-string">'Maximum Frequency [Hz]'</span>);
<span class="org-type">set</span>(<span class="org-variable-name">gca</span>, <span class="org-string">'yscale'</span>, <span class="org-string">'log'</span>);
</pre>
</div>
</div>
<div id="outline-container-org516bcbb" class="outline-2">
<h2 id="org516bcbb"><span class="section-number-2">7</span> Conclusion</h2>
</div>
<div id="outline-container-org2323f70" class="outline-3">
<h3 id="org2323f70"><span class="section-number-3">6.3</span> Results</h3>
<div class="outline-text-3" id="text-6-3">
</div>
<div id="outline-container-orge73cc45" class="outline-4">
<h4 id="orge73cc45"><span class="section-number-4">6.3.1</span> First test</h4>
<div class="outline-text-4" id="text-6-3-1">
<div class="org-src-container">
<pre class="src src-matlab">pd200_1V_1 = load(<span class="org-string">'mat/tf_pd200_7_1V.mat'</span>, <span class="org-string">'t'</span>, <span class="org-string">'Vin'</span>, <span class="org-string">'Vout'</span>, <span class="org-string">'Iout'</span>);
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">Ts = (pd200_1V_1.t(end) <span class="org-type">-</span> pd200_1V_1.t(1))<span class="org-type">/</span>(length(pd200_1V_1.t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(1<span class="org-type">*</span>Fs));
[tf_1, f] = tfestimate(pd200_1V_1.Vin, pd200_1V_1.Vout, win, [], [], 1<span class="org-type">/</span>Ts);
</pre>
</div>
</div>
</div>
<div id="outline-container-orgeb520e4" class="outline-4">
<h4 id="orgeb520e4"><span class="section-number-4">6.3.2</span> Results</h4>
<div class="outline-text-4" id="text-6-3-2">
<div class="org-src-container">
<pre class="src src-matlab">Ts = (pd200{1}.t(end) <span class="org-type">-</span> pd200{1}.t(1))<span class="org-type">/</span>(length(pd200{1}.t)<span class="org-type">-</span>1);
Fs = 1<span class="org-type">/</span>Ts;
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(0.5<span class="org-type">*</span>Fs));
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(pd200)</span>
[tf_est, f] = tfestimate(pd200{<span class="org-constant">i</span>}.Vin, 20<span class="org-type">*</span>pd200{<span class="org-constant">i</span>}.Vout, win, [], [], 1<span class="org-type">/</span>Ts);
pd200{<span class="org-constant">i</span>}.tf = tf_est(f <span class="org-type">&lt;</span> 0.99<span class="org-type">*</span>pd200{<span class="org-constant">i</span>}.notes.pd200.f_max);
pd200{<span class="org-constant">i</span>}.f = f(f <span class="org-type">&lt;</span> 0.99<span class="org-type">*</span>pd200{<span class="org-constant">i</span>}.notes.pd200.f_max);
<span class="org-keyword">end</span>
</pre>
</div>
<div class="org-src-container">
<pre class="src src-matlab">f_max = zeros(1, length(pd200));
Vin_ampl = zeros(1, length(pd200));
<span class="org-keyword">for</span> <span class="org-variable-name"><span class="org-constant">i</span></span> = <span class="org-constant">1:length(pd200)</span>
f_max(<span class="org-constant">i</span>) = pd200{<span class="org-constant">i</span>}.notes.pd200.f_max;
Vin_ampl(<span class="org-constant">i</span>) = pd200{<span class="org-constant">i</span>}.notes.pd200.Vin;
<span class="org-keyword">end</span>
</pre>
</div>
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-right">Vin</th>
<th scope="col" class="org-right">Fmax</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-right">0.1</td>
<td class="org-right">5000.0</td>
</tr>
<tr>
<td class="org-right">0.5</td>
<td class="org-right">3801.3</td>
</tr>
<tr>
<td class="org-right">1.0</td>
<td class="org-right">1900.7</td>
</tr>
<tr>
<td class="org-right">2.0</td>
<td class="org-right">950.3</td>
</tr>
<tr>
<td class="org-right">4.0</td>
<td class="org-right">475.2</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div id="outline-container-orgad3a328" class="outline-2">
<h2 id="orgad3a328"><span class="section-number-2">7</span> Conclusion</h2>
<div class="outline-text-2" id="text-7">
<table id="orgcddfe96" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<table id="org4bb2717" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 4:</span> Measured characteristics, Manual characterstics and specified ones</caption>
<colgroup>
@ -572,7 +1005,7 @@ For this measurement, the sampling frequency of the Speedgoat ADC should be as h
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-19 mar. 23:00</p>
<p class="date">Created: 2021-01-22 ven. 23:44</p>
</div>
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@ -105,7 +105,7 @@ This noise is described by its Power Spectral Density.
The objective is therefore to determine the transfer function $G_a(s)$ from the input voltage to the output voltage as well as the Power Spectral Density $S_n(\omega)$ of the amplifier output noise.
As both $G_a$ and $S_n$ depends on the load capacitance, they should be measured when loading the amplifier with a $\SI{10}{\micro\farad}$ capacitor.
As both $G_a$ and $S_n$ depends on the load capacitance, they should be measured when loading the amplifier with a $10\,\mu F$ capacitor.
#+begin_src latex :file pd200-model-schematic.pdf
\begin{tikzpicture}
@ -130,7 +130,38 @@ As both $G_a$ and $S_n$ depends on the load capacitance, they should be measured
[[file:figs/pd200-model-schematic.png]]
* Noise measurement
** Introduction :ignore:
- Section [[sec:noise_setup]]
- Section [[sec:noise_model]]
- Section [[sec:noise_quantization]]
- Section [[sec:noise_preamp]]
- Section [[sec:noise_pd200]]
- Section [[sec:noise_dac]]
- Section [[sec:noise_full_measurement]]
- Section [[sec:noise_ssi2v]]
** 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/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
#+end_src
** Setup
<<sec:noise_setup>>
#+begin_note
Here are the documentation of the equipment used for this test bench:
- Voltage Amplifier [[file:doc/PD200-V7-R1.pdf][PD200]]
@ -160,7 +191,683 @@ This gain should be around 1000.
A low pass filter at 10kHz can be included in the EG&G amplifier in order to limit aliasing.
An high pass filter at low frequency can be added if there is a problem of large offset.
** Results
** Model of the setup
<<sec:noise_model>>
As shown in Figure [[fig:noise_meas_procedure]], there are 4 equipment involved in the measurement:
- a Digital to Analog Convert (DAC)
- the Voltage amplifier to be measured with a gain of 20 (PD200)
- a low noise voltage amplifier with a variable gain and integrated low pass filters and high pass filters
- an Analog to Digital Converter (ADC)
Each of these equipment has some noise:
- $q_{da}$: quantization noise of the DAC
- $n_{da}$: output noise of the DAC
- $n_p$: output noise of the PD200 (what we wish to characterize)
- $n_a$: input noise of the pre amplifier
- $q_{ad}$: quantization noise of the ADC
#+begin_src latex :file noise_meas_procedure.pdf
\begin{tikzpicture}
% DAC
\node[DAC] (DAC) at (0,0) {DAC};
\node[addb, right=0.4 of DAC] (addqda){};
\node[addb, right=0.4 of addqda] (addnda){};
% PD200
\node[block, right=1.2 of addnda] (Gp){$G_p(s)$};
\node[addb, right=0.4 of Gp] (addnp){};
% Pre Amp
\node[addb, right=1.2 of addnp] (addna){};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
% \draw[->] (const.east) -- node[sloped]{$/$} (DAC.west);
\draw[<-] (DAC.west) -- node[sloped]{$/$} ++(-1.0, 0);
\draw[->] (DAC.east) -- (addqda.west);
\draw[->] (addqda.east) -- (addnda.west);
\draw[->] (addnda.east) -- (Gp.west);
\draw[->] (Gp.east) -- (addnp.west);
\draw[->] (addnp.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addnda.north) -- ++(0, 0.6) node[below right](nda){$n_{da}$};
\draw[<-] (addqda.north) -- ++(0, 0.6) node[below right](qda){$q_{da}$};
\draw[<-] (addnp.north) -- ++(0, 0.6) node[below right](np){$n_{p}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (nda.north);
\coordinate[] (bot) at (Ga.south);
% DAC
\begin{scope}[on background layer]
\node[fit={(DAC.west|-bot) (addnda.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {DAC};
\end{scope}
% PD200
\begin{scope}[on background layer]
\node[fit={(Gp.west|-bot) (addnp.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {PD200};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_meas_procedure
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_meas_procedure.png]]
** Quantization Noise
<<sec:noise_quantization>>
The quantization noise is something that can be predicted.
The Amplitude Spectral Density of the quantization noise of an ADC/DAC is equal to:
\begin{equation}
\Gamma_q(\omega) = \frac{q}{\sqrt{12 f_s}}
\end{equation}
with:
- $q = \frac{\Delta V}{2^n}$ the quantization in [V], which is the corresponding value in [V] of the least significant bit
- $\Delta V$ is the full range of the ADC in [V]
- $n$ is the number of bits
- $f_s$ is the sample frequency in [Hz]
#+begin_src matlab
adc = struct();
adc.Delta_V = 20; % [V]
adc.n = 16; % number of bits
adc.Fs = 20e3; % [Hz]
adc.Gamma_q = adc.Delta_V/2^adc.n/sqrt(12*adc.Fs); % [V/sqrt(Hz)]
#+end_src
The obtained Amplitude Spectral Density is src_matlab[:exports results :results value replace]{adc.Gamma_q} {{{results(=6.2294e-07=)}}} $V/\sqrt{Hz}$.
** Pre Amplifier noise measurement
<<sec:noise_preamp>>
First, we wish to measure the noise of the pre-amplifier.
To do so, the input of the pre-amplifier is shunted such that there is 0V at its inputs.
Then, the gain of the amplifier is increase until the measured signal on the ADC is much larger than the quantization noise.
The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal $n$ is computed.
Finally, the Amplitude Spectral Density of $n_a$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_a}(\omega) \approx \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
This is true if the quantization noise $\Gamma_{q_{ad}}$ is negligible.
#+begin_src latex :file noise_measure_setup_preamp.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% Pre Amp
\node[addb, right=0.6 of const] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_preamp
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_preamp.png]]
#+begin_src matlab :exports none
% Load Data
preamp = load('mat/noise_preamp_5113.mat', 't', 'Vn', 'notes');
#+end_src
The gain of the low noise amplifier is set to src_matlab[:exports results :results value replace]{ans = preamp.notes.pre_amp.gain} {{{results(=50000=)}}}.
#+begin_src matlab :exports none
% Compute the equivalent voltage at the input of the amplifier
preamp.Vn = preamp.Vn/preamp.notes.pre_amp.gain;
preamp.Vn = preamp.Vn - mean(preamp.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (preamp.t(end) - preamp.t(1))/(length(preamp.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab
% Hanning window
win = hanning(ceil(0.5/Ts));
% Power Spectral Density
[pxx, f] = pwelch(preamp.Vn, win, [], [], Fs);
% Save the results inside the struct
preamp.pxx = pxx;
preamp.f = f;
#+end_src
The obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier is shown in Figure [[fig:asd_preamp]].
The obtained noise amplitude is very closed to the one specified in the documentation of $4nV/\sqrt{Hz}$ at 1kHZ.
#+begin_src matlab :exports none
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./preamp.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'northeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_preamp.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:asd_preamp
#+caption: Obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier
#+RESULTS:
[[file:figs/asd_preamp.png]]
** PD200 noise measurement
<<sec:noise_pd200>>
The input of the PD200 amplifier is shunted such that there is 0V between its inputs.
Then the gain of the pre-amplifier is increased in order to measure a signal much larger than the quantization noise of the ADC.
We compute the Amplitude Spectral Density of the measured signal $\Gamma_n(\omega)$.
The Amplitude Spectral Density of $n_p$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
And we verify that this is indeed the noise of the PD200 and not the noise of the pre-amplifier by checking that:
\begin{equation}
\Gamma_{n_p} \ll \Gamma_{n_a}
\end{equation}
#+begin_src latex :file noise_measure_setup_pd200.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% PD200
\node[block, right=0.6 of const] (Gp){$G_p(s)$};
\node[addb, right=0.4 of Gp] (addnp){};
% Pre Amp
\node[addb, right=1.2 of addnp] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- (Gp.west);
\draw[->] (Gp.east) -- (addnp.west);
\draw[->] (addnp.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addnp.north) -- ++(0, 0.6) node[below right](np){$n_{p}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% PD200
\begin{scope}[on background layer]
\node[fit={(Gp.west|-bot) (addnp.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {PD200};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_pd200
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_pd200.png]]
#+begin_src matlab :exports none
%% Load all the measurements
pd200w = {};
for i = 1:7
pd200w(i) = {load(['mat/noise_PD200_' num2str(i) '_3uF_warmup.mat'], 't', 'Vn', 'notes')};
end
#+end_src
#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
pd200w{i}.Vn = pd200w{i}.Vn/pd200w{i}.notes.pre_amp.gain;
end
#+end_src
The measured low frequency noise $n_p$ of one of the amplifiers is shown in Figure [[fig:pd200_noise_time_lpf]].
It is very similar to the one specified in the datasheet in Figure [[fig:pd200_expected_noise]].
#+begin_src matlab :exports none
% Compute the low frequency noise
G_lpf = 1/(1 + s/2/pi/20);
t_max = 40;
figure;
hold on;
plot(pd200w{1}.t(1:t_max/Ts), lsim(G_lpf, 1e3*pd200w{1}.Vn(1:t_max/Ts), pd200w{1}.t(1:t_max/Ts)))
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
ylim([-3, 3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_noise_time_lpf.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:pd200_noise_time_lpf
#+caption: Measured low frequency noise of the PD200 from 0.01Hz to 20Hz
#+RESULTS:
[[file:figs/pd200_noise_time_lpf.png]]
The obtained RMS and peak to peak values of the measured noises are shown in Table [[tab:rms_pkp_noise]].
#+begin_src matlab :exports none
%% Compute the RMS and Peak to Peak noise for the low frequency noise
Vn_rms = zeros(7,1); % RMS value [uV rms]
Vn_pkp = zeros(7,1); % Peak to Peak Value in 20Hz bandwidth [mV]
for i = 1:7
Vn_rms(i) = 1e6*rms(pd200w{i}.Vn);
Vn_lpf = lsim(1/(1 + s/2/pi/20), pd200w{i}.Vn, pd200w{i}.t);
Vn_pkp(i) = 1e3*(max(Vn_lpf)-min(Vn_lpf));
end
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([[714; Vn_rms], [4.3; Vn_pkp]], {'Specification [10uF]', 'PD200_1', 'PD200_2', 'PD200_3', 'PD200_4', 'PD200_5', 'PD200_6', 'PD200_7'}, {'*RMS [uV]*', '*Peak to Peak [mV]*'}, ' %.1f ');
#+end_src
#+name: tab:rms_pkp_noise
#+caption: RMS and Peak to Peak measured low frequency noise (0.01Hz to 20Hz)
#+attr_latex: :environment tabularx :width \linewidth :align lXX
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| | *RMS [uV]* | *Peak to Peak [mV]* |
|----------------------+------------+---------------------|
| Specification [10uF] | 714.0 | 4.3 |
| PD200_1 | 565.1 | 3.7 |
| PD200_2 | 767.6 | 3.5 |
| PD200_3 | 479.9 | 3.0 |
| PD200_4 | 615.7 | 3.5 |
| PD200_5 | 651.0 | 2.4 |
| PD200_6 | 473.2 | 2.7 |
| PD200_7 | 423.1 | 2.3 |
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200w{1}.t(end) - pd200w{1}.t(1))/(length(pd200w{1}.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab :exports none
win = hanning(ceil(0.5/Ts));
for i = 1:7
[pxx, f] = pwelch(pd200w{i}.Vn, win, [], [], Fs);
pd200w{i}.f = f;
pd200w{i}.pxx = pxx;
end
#+end_src
The Amplitude Spectral Density of the measured noise is now computed and shown in Figure [[fig:asd_noise_3uF_warmup]].
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot(pd200w{1}.f, sqrt(pd200w{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./pd200w{1}.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_3uF_warmup.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_3uF_warmup
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_3uF_warmup.png]]
** DAC noise measurement
<<sec:noise_dac>>
In order not to have any quantization noise, we impose the DAC to output a zero voltage.
The gain of the low noise amplifier is adjusted to
The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal is computed.
The Amplitude Spectral Density of $n_{da}$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_{da}}(\omega) = \frac{\Gamma_m(\omega)}{|G_a(\omega)|}
\end{equation}
And it is verify that the Amplitude Spectral Density of $n_{da}$ is much larger than the one of $n_a$:
\begin{equation}
\Gamma_{n_{da}} \gg \Gamma_{n_a}
\end{equation}
#+begin_src latex :file noise_measure_setup_dac.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% DAC
\node[DAC, right=0.6 of const] (DAC) {DAC};
\node[addb, right=0.4 of DAC] (addnda){};
% Pre Amp
\node[addb, right=1.2 of addnda] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- node[sloped]{$/$} (DAC.west);
\draw[->] (DAC.east) -- (addnda.west);
\draw[->] (addnda.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0);
\draw[<-] (addnda.north) -- ++(0, 0.6) node[below right](nda){$n_{da}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% DAC
\begin{scope}[on background layer]
\node[fit={(DAC.west|-bot) (addnda.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {DAC};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_dac
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_dac.png]]
#+begin_src matlab :exports none
dac = load('mat/noise_preamp_5113_dac.mat', 't', 'Vn', 'notes');
#+end_src
#+begin_src matlab :exports none
dac.Vn = dac.Vn/dac.notes.pre_amp.gain;
dac.Vn = dac.Vn - mean(dac.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (dac.t(end) - dac.t(1))/(length(dac.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab :exports none
win = hanning(ceil(0.5/Ts));
[pxx, f] = pwelch(dac.Vn, win, [], [], Fs);
dac.pxx = pxx;
dac.f = f;
#+end_src
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
set(gca,'ColorOrderIndex',3)
plot(dac.f, sqrt(dac.pxx), 'DisplayName', '$\Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./dac.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_dac.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_dac
#+caption:
#+RESULTS:
[[file:figs/asd_noise_dac.png]]
** Total noise measurement
<<sec:noise_full_measurement>>
Let's now analyze the measurement of the setup in Figure [[fig:noise_meas_procedure]].
#+begin_src matlab :exports none
%% Load all the measurements
pd200dac = {};
for i = 1:7
pd200dac(i) = {load(['mat/noise_PD200_' num2str(i) '_3uF_DAC.mat'], 't', 'Vn', 'notes')};
end
#+end_src
#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
pd200dac{i}.Vn = pd200dac{i}.Vn/pd200dac{i}.notes.pre_amp.gain;
pd200dac{i}.Vn = pd200dac{i}.Vn - mean(pd200dac{i}.Vn);
end
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200dac{1}.t(end) - pd200dac{1}.t(1))/(length(pd200dac{1}.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_tot]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
for i = 1:7
[pxx, f] = pwelch(pd200dac{i}.Vn, win, [], [], Fs);
pd200dac{i}.f = f;
pd200dac{i}.pxx = pxx;
end
#+end_src
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot(pd200w{2}.f, sqrt(pd200w{2}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',3)
plot(dac.f, 20*sqrt(dac.pxx), 'DisplayName', '$|G_p| \cdot \Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./dac.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
plot(pd200dac{2}.f, sqrt(pd200dac{2}.pxx), 'color', [colors(4, :), 0.5], 'DisplayName', '$\Gamma_{tot}$');
for i = 2:7
plot(pd200dac{i}.f, sqrt(pd200dac{i}.pxx), 'color', [colors(4, :), 0.5], 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_tot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_tot
#+caption: Amplitude Spectral Density of the measured noise and of the individual sources of noise
#+RESULTS:
[[file:figs/asd_noise_tot.png]]
#+begin_important
The output noise of the PD200 amplifier is limited by the noise of the DAC.
Having a DAC with lower noise could lower the output noise of the PD200.
SSI2V DACs will be used to verify that.
#+end_important
** 20bits DAC noise measurement
<<sec:noise_ssi2v>>
Let's now measure the noise of another DAC called the "SSI2V" ([[file:doc/\[SSI2V\]Datasheet.pdf][doc]]).
It is a 20bits DAC with an output of +/-10.48 V and a very low noise.
The measurement setup is the same as the one in Figure [[fig:noise_measure_setup_dac]].
#+begin_src matlab :exports none
ssi2v = load('mat/noise_preamp_5113_SSI2V.mat', 't', 'Vn', 'notes');
#+end_src
#+begin_src matlab :exports none
ssi2v.Vn = ssi2v.Vn/ssi2v.notes.pre_amp.gain;
ssi2v.Vn = ssi2v.Vn - mean(ssi2v.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (ssi2v.t(end) - ssi2v.t(1))/(length(ssi2v.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
[pxx, f] = pwelch(ssi2v.Vn, win, [], [], Fs);
ssi2v.pxx = pxx;
ssi2v.f = f;
#+end_src
The obtained noise of the SSI2V DAC is shown in Figure [[fig:asd_ssi2v_noise]] and compared with the noise of the 16bits DAC.
It is shown to be much smaller (~1 order of magnitude).
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
set(gca,'ColorOrderIndex',3)
plot(dac.f, sqrt(dac.pxx), 'DisplayName', '$\Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./ssi2v.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
set(gca,'ColorOrderIndex',5)
plot(ssi2v.f, sqrt(ssi2v.pxx), 'DisplayName', '$\Gamma_{n_{SSI2V}}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_ssi2v_noise.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_ssi2v_noise
#+caption: Amplitude Spectral Density of the SSI2V DAC's noise
#+RESULTS:
[[file:figs/asd_ssi2v_noise.png]]
#+begin_important
Using the SSI2V as the DAC with the PD200 should give much better noise output than using the 16bits DAC.
The limiting factor should then be the noise of the PD200 itself.
#+end_important
** Tests :noexport:
*** 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>>
@ -179,7 +886,8 @@ addpath('./matlab/');
addpath('./mat/');
#+end_src
*** Pre-Amp Noise
*** DONE Pre-Amp Noise
CLOSED: [2021-01-22 ven. 22:51]
#+begin_src matlab
preamp = load('mat/noise_preamp_5113.mat', 't', 'Vn', 'notes');
#+end_src
@ -222,7 +930,8 @@ legend('location', 'southwest');
xlim([1, Fs/2]);
#+end_src
*** DAC (16bits) Noise
*** DONE DAC (16bits) Noise
CLOSED: [2021-01-22 ven. 23:13]
#+begin_src matlab
dac = load('mat/noise_preamp_5113_dac.mat', 't', 'Vn', 'notes');
#+end_src
@ -248,7 +957,6 @@ Ts = (dac.t(end) - dac.t(1))/(length(dac.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_3uF]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
@ -270,53 +978,8 @@ legend('location', 'southwest');
xlim([1, Fs/2]);
#+end_src
*** SSI2V DAC (20bits) Noise
#+begin_src matlab
ssi2v = load('mat/noise_preamp_5113_SSI2V.mat', 't', 'Vn', 'notes');
#+end_src
#+begin_src matlab
ssi2v.Vn = ssi2v.Vn/ssi2v.notes.pre_amp.gain;
ssi2v.Vn = ssi2v.Vn - mean(ssi2v.Vn);
#+end_src
#+begin_src matlab
figure;
plot(ssi2v.t, 1e6*ssi2v.Vn);
xlabel('Time [s]');
ylabel('Voltage [$\mu V$]');
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (ssi2v.t(end) - ssi2v.t(1))/(length(ssi2v.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_3uF]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
[pxx, f] = pwelch(ssi2v.Vn, win, [], [], Fs);
ssi2v.pxx = pxx;
ssi2v.f = f;
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(dac.f, sqrt(dac.pxx), 'DisplayName', 'DAC');
plot(ssi2v.f, sqrt(ssi2v.pxx), 'DisplayName', 'SSI2V');
plot(ssi2v.f, ones(size(ssi2v.f))*(10/2^16)/sqrt(12*Fs)/ssi2v.notes.pre_amp.gain, 'k--', 'DisplayName', 'ADC quant.');
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', 'Pre Amp');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southwest');
xlim([1, Fs/2]);
#+end_src
*** Noise when shunting the input (50 Ohms) - After Warmup
*** DONE Noise when shunting the input (50 Ohms) - After Warmup
CLOSED: [2021-01-22 ven. 23:09]
#+begin_src matlab :exports none
%% Load all the measurements
@ -333,7 +996,7 @@ for i = 1:7
end
#+end_src
The time domain measurements of the amplifier noise are shown in Figure [[fig:noise_shunt_time_3uF]].
The time domain measurements of the amplifier noise are shown in Figure [[fig:noise_shunt_time_3uF_warmup]].
#+begin_src matlab :exports none
figure;
@ -393,7 +1056,6 @@ Ts = (pd200w{1}.t(end) - pd200w{1}.t(1))/(length(pd200w{1}.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_3uF]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
@ -616,7 +1278,8 @@ exportFig('figs/asd_noise_3uF.pdf', 'width', 'wide', 'height', 'tall');
#+RESULTS:
[[file:figs/asd_noise_3uF.png]]
*** Noise with DAC at the input of the PD200
*** DONE Noise with DAC at the input of the PD200
CLOSED: [2021-01-22 ven. 23:39]
#+begin_src matlab :exports none
%% Load all the measurements
pd200dac = {};
@ -633,8 +1296,6 @@ for i = 1:7
end
#+end_src
The time domain measurements of the amplifier noise are shown in Figure [[fig:noise_shunt_time_3uF]].
#+begin_src matlab :exports none
figure;
hold on;
@ -662,7 +1323,7 @@ Ts = (pd200dac{1}.t(end) - pd200dac{1}.t(1))/(length(pd200dac{1}.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_3uF]].
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_3uF_dac]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
@ -719,8 +1380,6 @@ xlim([1, Fs/2]);
The output noise of the PD200 amplifier is limited by the noise of the DAC.
#+end_important
*** TODO Noise with SSI2V at the input of the PD200
* Transfer Function measurement
** Setup
In order to measure the transfer function from the input voltage $V_{in}$ to the output voltage $V_{out}$, the test bench shown in Figure [[fig:setup-dynamics-measurement]] is used.
@ -755,8 +1414,8 @@ For a specified voltage input amplitude $V_{in}$, the maximum frequency is then:
\[ \omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}} \]
#+begin_src matlab
Iout_max = 0.7; % Maximum output current [A]
C = 3e-6; % Load Capacitance [F]
Iout_max = 0.57; % Maximum output current [A]
C = 2.7e-6; % Load Capacitance [F]
V_in = linspace(0, 5, 100); % Input Voltage [V]
@ -830,6 +1489,85 @@ linkaxes([ax1,ax2],'x');
xlim([1, 2e3]);
#+end_src
*** Results
#+begin_src matlab :exports none
%% Load all the measurements
Vin_ampl = {'0_1', '0_5', '1', '2', '4'};
pd200 = {};
for i = 1:length(Vin_ampl)
pd200(i) = {load(['tf_pd200_7_' Vin_ampl{i} 'V.mat'], 't', 'Vin', 'Vout', 'notes')};
end
#+end_src
#+begin_src matlab
Ts = (pd200{1}.t(end) - pd200{1}.t(1))/(length(pd200{1}.t)-1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab
win = hanning(ceil(0.5*Fs));
for i = 1:length(pd200)
[tf_est, f] = tfestimate(pd200{i}.Vin, 20*pd200{i}.Vout, win, [], [], 1/Ts);
pd200{i}.tf = tf_est(f < 0.99*pd200{i}.notes.pd200.f_max);
pd200{i}.f = f(f < 0.99*pd200{i}.notes.pd200.f_max);
end
#+end_src
#+begin_src matlab :exports none
figure;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
for i = 1:length(pd200)
plot(pd200{i}.f, abs(pd200{i}.tf), 'DisplayName', sprintf('$V_{in} = %.1f [V]$', pd200{i}.notes.pd200.Vin))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([19, 21]);
legend('location', 'northeast');
ax2 = nexttile;
hold on;
for i = 1:length(pd200)
plot(pd200{i}.f, 180/pi*angle(pd200{i}.tf))
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:5:360);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
ylim([-15, 5]);
linkaxes([ax1,ax2],'x');
xlim([10, 5e3]);
#+end_src
#+begin_src matlab
f_max = zeros(1, length(pd200));
Vin_ampl = zeros(1, length(pd200));
for i = 1:length(pd200)
f_max(i) = pd200{i}.notes.pd200.f_max;
Vin_ampl(i) = pd200{i}.notes.pd200.Vin;
end
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Vin_ampl; f_max]', {}, {'Vin', 'Fmax'}, ' %.1f ');
#+end_src
#+RESULTS:
| Vin | Fmax |
|-----+--------|
| 0.1 | 5000.0 |
| 0.5 | 3801.3 |
| 1.0 | 1900.7 |
| 2.0 | 950.3 |
| 4.0 | 475.2 |
* Conclusion
#+name: tab:table_name

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