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<h1 class="title">Voltage Amplifier PD200 - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org8464676">1. Introduction</a></li>
<li><a href="#org377554e">2. Voltage Amplifier Requirements</a></li>
<li><a href="#orgb337f1e">3. PD200 Expected characteristics</a></li>
<li><a href="#orgb5bfcf0">4. Voltage Amplifier Model</a></li>
<li><a href="#orga876e4d">5. Noise measurement</a>
<ul>
<li><a href="#org2c9f60e">5.1. Setup</a></li>
<li><a href="#orgcf8ecc8">5.2. Model of the setup</a></li>
<li><a href="#org5f0cc17">5.3. Quantization Noise</a></li>
<li><a href="#org7c484b7">5.4. Pre Amplifier noise measurement</a></li>
<li><a href="#org4fcdc54">5.5. PD200 noise measurement</a></li>
<li><a href="#orga1b9697">5.6. DAC noise measurement</a></li>
<li><a href="#org6d8f9c2">5.7. Total noise measurement</a></li>
<li><a href="#org59c04f1">5.8. 20bits DAC noise measurement</a></li>
</ul>
</li>
<li><a href="#orga1cb3ef">6. Transfer Function measurement</a>
<ul>
<li><a href="#org2c3812b">6.1. Setup</a></li>
<li><a href="#orge1713a2">6.2. Maximum Frequency/Voltage to not overload the amplifier</a></li>
<li><a href="#org4ff0dfb">6.3. Obtained Transfer Functions</a></li>
</ul>
</li>
<li><a href="#org6becade">7. Conclusion</a></li>
</ul>
</div>
</div>
<div id="outline-container-org8464676" class="outline-2">
<h2 id="org8464676"><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.
</p>
<p>
The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here</a>.
</p>
<div id="org6525f44" 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>
</div>
</div>
</div>
<div id="outline-container-org377554e" class="outline-2">
<h2 id="org377554e"><span class="section-number-2">2</span> Voltage Amplifier Requirements</h2>
<div class="outline-text-2" id="text-2">
<table id="orgf8783f3" 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>
<col class="org-left" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Continuous Current</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Noise (1-200Hz)</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Voltage Input Range</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Voltage Output Range</td>
<td class="org-center">-20 [V] to 150 [V]</td>
</tr>
<tr>
<td class="org-left">Small signal bandwidth (-3dB)</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgb337f1e" class="outline-2">
<h2 id="orgb337f1e"><span class="section-number-2">3</span> PD200 Expected characteristics</h2>
<div class="outline-text-2" id="text-3">
<table id="org6e09271" 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>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><b>Characteristics</b></th>
<th scope="col" class="org-center"><b>Manual</b></th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Input Voltage Range</td>
<td class="org-center">+/- 10 [V]</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Range</td>
<td class="org-center">-50/150 [V]</td>
<td class="org-center">-20/150 [V]</td>
</tr>
<tr>
<td class="org-left">Gain</td>
<td class="org-center">20 [V/V]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Maximum RMS current</td>
<td class="org-center">0.9 [A]</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Maximum Pulse current</td>
<td class="org-center">10 [A]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Slew Rate</td>
<td class="org-center">150 [V/us]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Noise (10uF load)</td>
<td class="org-center">0.7 [mV RMS]</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Small Signal Bandwidth (10uF load)</td>
<td class="org-center">7.4 [kHz]</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
<tr>
<td class="org-left">Large Signal Bandwidth (150V, 10uF)</td>
<td class="org-center">300 [Hz]</td>
<td class="org-center">&#xa0;</td>
</tr>
</tbody>
</table>
<p>
For a load capacitance of \(10\,\mu F\), the expected \(-3\,dB\) bandwidth is \(6.4\,kHz\) (Figure <a href="#org407299e">2</a>) and the low frequency noise is \(650\,\mu V\,\text{rms}\) (Figure <a href="#org94560d5">3</a>).
</p>
<div id="org407299e" 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="org94560d5" 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>
</div>
</div>
</div>
<div id="outline-container-orgb5bfcf0" class="outline-2">
<h2 id="orgb5bfcf0"><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 \(G_p(s)\) from voltage inputs \(V_{in}\) to voltage output \(V_{out}\).
Ideally, the gain from \(V_{in}\) to \(V_{out}\) is constant over a wide frequency band with very small phase drop.
</p>
<p>
It is also characterized by its output noise \(n\).
This noise is described by its Power Spectral Density.
</p>
<p>
The objective is therefore to determine the transfer function \(G_p(s)\) from the input voltage to the output voltage as well as the Power Spectral Density \(S_n(\omega)\) of the amplifier output noise.
</p>
<p>
As both \(G_p\) 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="org388bf47" 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>
</div>
</div>
</div>
<div id="outline-container-orga876e4d" class="outline-2">
<h2 id="orga876e4d"><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="#org56bae9f">5.1</a></li>
<li>Section <a href="#orgc48d4f4">5.2</a></li>
<li>Section <a href="#org2a59e12">5.3</a></li>
<li>Section <a href="#orgff6c06d">5.4</a></li>
<li>Section <a href="#orgdb787b7">5.5</a></li>
<li>Section <a href="#org41a9b46">5.6</a></li>
<li>Section <a href="#org00954a7">5.7</a></li>
<li>Section <a href="#orgbe0f3de">5.8</a></li>
</ul>
</div>
<div id="outline-container-org2c9f60e" class="outline-3">
<h3 id="org2c9f60e"><span class="section-number-3">5.1</span> Setup</h3>
<div class="outline-text-3" id="text-5-1">
<p>
<a id="org56bae9f"></a>
</p>
<div class="note" id="org2bfae12">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Load Capacitor <a href="doc/0900766b815ea422.pdf">EPCOS 10uF Multilayer Ceramic Capacitor</a></li>
<li>Low Noise Voltage Amplifier <a href="doc/egg-5113-preamplifier.pdf">EG&amp;G 5113</a></li>
<li>Speedgoat ADC <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
</ul>
</div>
<p>
The output noise of the voltage amplifier PD200 is foreseen to be around 1mV rms in a bandwidth from DC to 1MHz.
If we suppose a white noise, this correspond to an amplitude spectral density:
</p>
\begin{equation}
\phi_{n} \approx \frac{1\,mV}{\sqrt{1\,MHz}} = 1 \frac{\mu V}{\sqrt{Hz}}
\end{equation}
<p>
The RMS noise begin very small compare to the ADC resolution, we must amplify the noise before digitizing the signal.
The added noise of the instrumentation amplifier should be much smaller than the noise of the PD200.
We use the amplifier EG&amp;G 5113 that has a noise of \(\approx 4 nV/\sqrt{Hz}\) referred to its input which is much smaller than the noise induced by the PD200.
</p>
<p>
The gain of the low-noise amplifier can be increased until the full range of the ADC is used.
This gain should be around 1000.
</p>
<div id="org6c9024e" 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>
</div>
<p>
A low pass filter at 10kHz can be included in the EG&amp;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.
</p>
</div>
</div>
<div id="outline-container-orgcf8ecc8" class="outline-3">
<h3 id="orgcf8ecc8"><span class="section-number-3">5.2</span> Model of the setup</h3>
<div class="outline-text-3" id="text-5-2">
<p>
<a id="orgc48d4f4"></a>
</p>
<p>
As shown in Figure <a href="#org52225dc">6</a>, there are 4 equipment involved in the measurement:
</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="org52225dc" 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-org5f0cc17" class="outline-3">
<h3 id="org5f0cc17"><span class="section-number-3">5.3</span> Quantization Noise</h3>
<div class="outline-text-3" id="text-5-3">
<p>
<a id="org2a59e12"></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>
The obtained Amplitude Spectral Density is <code>6.2294e-07</code> \(V/\sqrt{Hz}\).
</p>
</div>
</div>
<div id="outline-container-org7c484b7" class="outline-3">
<h3 id="org7c484b7"><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="orgff6c06d"></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="org21a64c7" 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>Sources of noise in the experimental setup</p>
</div>
<p>
The gain of the low noise amplifier is set to <code>50000</code>.
</p>
<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="#org0f7d698">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="org0f7d698" 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-org4fcdc54" class="outline-3">
<h3 id="org4fcdc54"><span class="section-number-3">5.5</span> PD200 noise measurement</h3>
<div class="outline-text-3" id="text-5-5">
<p>
<a id="orgdb787b7"></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="orged8fdf4" 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="#org9cb2c72">10</a>.
It is very similar to the one specified in the datasheet in Figure <a href="#org94560d5">3</a>.
</p>
<div id="org9cb2c72" 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="#orgec90acd">3</a>.
</p>
<table id="orgec90acd" 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" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right"><b>RMS [uV]</b></th>
<th scope="col" class="org-right"><b>Peak to Peak [mV]</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Specification [10uF]</td>
<td class="org-right">714.0</td>
<td class="org-right">4.3</td>
</tr>
<tr>
<td class="org-left">PD200_1</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">767.6</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_3</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">615.7</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_5</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">473.2</td>
<td class="org-right">2.7</td>
</tr>
<tr>
<td class="org-left">PD200_7</td>
<td class="org-right">423.1</td>
<td class="org-right">2.3</td>
</tr>
</tbody>
</table>
<p>
The Amplitude Spectral Density of the measured noise is now computed and shown in Figure <a href="#org177c379">11</a>.
</p>
<div id="org177c379" 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-orga1b9697" class="outline-3">
<h3 id="orga1b9697"><span class="section-number-3">5.6</span> DAC noise measurement</h3>
<div class="outline-text-3" id="text-5-6">
<p>
<a id="org41a9b46"></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="orga13ca97" 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="org428ef0b" class="figure">
<p><img src="figs/asd_noise_dac.png" alt="asd_noise_dac.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org6d8f9c2" class="outline-3">
<h3 id="org6d8f9c2"><span class="section-number-3">5.7</span> Total noise measurement</h3>
<div class="outline-text-3" id="text-5-7">
<p>
<a id="org00954a7"></a>
</p>
<p>
Let&rsquo;s now analyze the measurement of the setup in Figure <a href="#org52225dc">6</a>.
</p>
<p>
The PSD of the measured noise is computed and the ASD is shown in Figure <a href="#org18dbc81">14</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(0.5<span class="org-type">/</span>Ts));
<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, 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="org18dbc81" class="figure">
<p><img src="figs/asd_noise_tot.png" alt="asd_noise_tot.png" />
</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="org39d2c06">
<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-org59c04f1" class="outline-3">
<h3 id="org59c04f1"><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="orgbe0f3de"></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="#orga13ca97">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="#orge9f88e0">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="orge9f88e0" 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="org9ccd6d7">
<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-orga1cb3ef" class="outline-2">
<h2 id="orga1cb3ef"><span class="section-number-2">6</span> Transfer Function measurement</h2>
<div class="outline-text-2" id="text-6">
</div>
<div id="outline-container-org2c3812b" class="outline-3">
<h3 id="org2c3812b"><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="#org1a2018d">16</a> is used.
</p>
<div class="note" id="orgb90f0e1">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Load Capacitor <a href="doc/0900766b815ea422.pdf">EPCOS 10uF Multilayer Ceramic Capacitor</a></li>
<li>Speedgoat DAC/ADC <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
</ul>
</div>
<p>
For this measurement, the sampling frequency of the Speedgoat ADC should be as high as possible.
</p>
<div id="org1a2018d" class="figure">
<p><img src="figs/setup-dynamics-measurement.png" alt="setup-dynamics-measurement.png" />
</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-orge1713a2" class="outline-3">
<h3 id="orge1713a2"><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 amplitude and the output voltage amplitude for sinusoidal waves of frequency \(\omega\):
\[ V_{out} = \frac{1}{C\omega} I_{out} \]
</p>
<p>
Moreover, 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 at which the output current reaches its maximum value is:
\[ \omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}} \]
</p>
<p>
\(\omega_max\) as a function of \(V_{in}\) is shown in Figure <a href="#orgfadee8c">17</a>.
</p>
<div id="orgfadee8c" class="figure">
<p><img src="figs/max_frequency_voltage.png" alt="max_frequency_voltage.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Maximum frequency as a function of the excitation voltage amplitude</p>
</div>
</div>
</div>
<div id="outline-container-org4ff0dfb" class="outline-3">
<h3 id="org4ff0dfb"><span class="section-number-3">6.3</span> Obtained Transfer Functions</h3>
<div class="outline-text-3" id="text-6-3">
<p>
Several identifications using sweep sin were performed with input voltage amplitude ranging from 0.1V to 4V.
</p>
<p>
The obtained frequency response functions are shown in Figure <a href="#orgf2c5d27">18</a>.
As the input voltage increases, the voltage drop is increasing.
</p>
<div id="orgf2c5d27" class="figure">
<p><img src="figs/pd200_tf_voltage.png" alt="pd200_tf_voltage.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Transfer function for the PD200 amplitude between \(V_{in}\) and \(V_{out}\) for multiple voltage amplitudes</p>
</div>
<p>
The small signal transfer function of the amplifier can be approximated by a first order low pass filter.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gp = 19.95<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>35e3);
</pre>
</div>
<p>
The comparison from the model and measurements are shown in Figure <a href="#orga34bca3">19</a>.
</p>
<div id="orga34bca3" class="figure">
<p><img src="figs/tf_pd200_model.png" alt="tf_pd200_model.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Comparison of the model transfer function and the measured frequency response function</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org6becade" class="outline-2">
<h2 id="org6becade"><span class="section-number-2">7</span> Conclusion</h2>
<div class="outline-text-2" id="text-7">
<table id="org9961e87" 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>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><b>Characteristics</b></th>
<th scope="col" class="org-center"><b>Measurement</b></th>
<th scope="col" class="org-center"><b>Manual</b></th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Input Voltage Range</td>
<td class="org-center">-</td>
<td class="org-center">+/- 10 [V]</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Range</td>
<td class="org-center">-</td>
<td class="org-center">-50/150 [V]</td>
<td class="org-center">-20/150 [V]</td>
</tr>
<tr>
<td class="org-left">Gain</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">20 [V/V]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Maximum RMS current</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">0.9 [A]</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Maximum Pulse current</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">10 [A]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Slew Rate</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">150 [V/us]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Noise (10uF load)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">0.7 [mV RMS]</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Small Signal Bandwidth (10uF load)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">7.4 [kHz]</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
<tr>
<td class="org-left">Large Signal Bandwidth (150V, 10uF)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">300 [Hz]</td>
<td class="org-center">-</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-01-23 sam. 15:40</p>
</div>
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</div><div id="content">
<h1 class="title">Voltage Amplifier PD200 - Test Bench</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org6d8ef23">1. Introduction</a></li>
<li><a href="#org7aab2c4">2. Voltage Amplifier Requirements</a></li>
<li><a href="#org15bc799">3. PD200 Expected characteristics</a></li>
<li><a href="#orgbfad261">4. Voltage Amplifier Model</a></li>
<li><a href="#org1e152a3">5. Noise measurement</a>
<ul>
<li><a href="#orgfcd97cd">5.1. Setup</a></li>
<li><a href="#orge5e70c7">5.2. Model of the setup</a></li>
<li><a href="#orge3695f9">5.3. Quantization Noise</a></li>
<li><a href="#org82499a9">5.4. Pre Amplifier noise measurement</a></li>
<li><a href="#orgaee6b6f">5.5. PD200 noise measurement</a></li>
<li><a href="#org191ab71">5.6. DAC noise measurement</a></li>
<li><a href="#org9c8acb0">5.7. Total noise measurement</a></li>
<li><a href="#org4635045">5.8. 20bits DAC noise measurement</a></li>
</ul>
</li>
<li><a href="#orga03c3cb">6. Transfer Function measurement</a>
<ul>
<li><a href="#orgf7753cb">6.1. Setup</a></li>
<li><a href="#org8cb4d98">6.2. Maximum Frequency/Voltage to not overload the amplifier</a></li>
<li><a href="#org69face8">6.3. Obtained Transfer Functions</a></li>
</ul>
</li>
<li><a href="#org3dbe4a7">7. Conclusion</a></li>
</ul>
</div>
</div>
<hr>
<p>This report is also available as a <a href="./test-bench-pd200.pdf">pdf</a>.</p>
<hr>
<div id="outline-container-org6d8ef23" class="outline-2">
<h2 id="org6d8ef23"><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.
</p>
<p>
The documentation of the PD200 is accessible <a href="doc/PD200-V7-R1.pdf">here</a>.
</p>
<div id="org3e7e405" 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>
</div>
</div>
</div>
<div id="outline-container-org7aab2c4" class="outline-2">
<h2 id="org7aab2c4"><span class="section-number-2">2</span> Voltage Amplifier Requirements</h2>
<div class="outline-text-2" id="text-2">
<table id="org930ddef" 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>
<col class="org-left" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Continuous Current</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Noise (1-200Hz)</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Voltage Input Range</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Voltage Output Range</td>
<td class="org-center">-20 [V] to 150 [V]</td>
</tr>
<tr>
<td class="org-left">Small signal bandwidth (-3dB)</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-org15bc799" class="outline-2">
<h2 id="org15bc799"><span class="section-number-2">3</span> PD200 Expected characteristics</h2>
<div class="outline-text-2" id="text-3">
<table id="org2879e1b" 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>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><b>Characteristics</b></th>
<th scope="col" class="org-center"><b>Manual</b></th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Input Voltage Range</td>
<td class="org-center">+/- 10 [V]</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Range</td>
<td class="org-center">-50/150 [V]</td>
<td class="org-center">-20/150 [V]</td>
</tr>
<tr>
<td class="org-left">Gain</td>
<td class="org-center">20 [V/V]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Maximum RMS current</td>
<td class="org-center">0.9 [A]</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Maximum Pulse current</td>
<td class="org-center">10 [A]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Slew Rate</td>
<td class="org-center">150 [V/us]</td>
<td class="org-center">&#xa0;</td>
</tr>
<tr>
<td class="org-left">Noise (10uF load)</td>
<td class="org-center">0.7 [mV RMS]</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Small Signal Bandwidth (10uF load)</td>
<td class="org-center">7.4 [kHz]</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
<tr>
<td class="org-left">Large Signal Bandwidth (150V, 10uF)</td>
<td class="org-center">300 [Hz]</td>
<td class="org-center">&#xa0;</td>
</tr>
</tbody>
</table>
<p>
For a load capacitance of \(10\,\mu F\), the expected \(-3\,dB\) bandwidth is \(6.4\,kHz\) (Figure <a href="#org9cb375d">2</a>) and the low frequency noise is \(650\,\mu V\,\text{rms}\) (Figure <a href="#org4f4605a">3</a>).
</p>
<div id="org9cb375d" 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="org4f4605a" 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>
</div>
</div>
</div>
<div id="outline-container-orgbfad261" class="outline-2">
<h2 id="orgbfad261"><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 \(G_p(s)\) from voltage inputs \(V_{in}\) to voltage output \(V_{out}\).
Ideally, the gain from \(V_{in}\) to \(V_{out}\) is constant over a wide frequency band with very small phase drop.
</p>
<p>
It is also characterized by its output noise \(n\).
This noise is described by its Power Spectral Density.
</p>
<p>
The objective is therefore to determine the transfer function \(G_p(s)\) from the input voltage to the output voltage as well as the Power Spectral Density \(S_n(\omega)\) of the amplifier output noise.
</p>
<p>
As both \(G_p\) 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="orgc28c573" 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>
</div>
</div>
</div>
<div id="outline-container-org1e152a3" class="outline-2">
<h2 id="org1e152a3"><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="#orgc202318">5.1</a></li>
<li>Section <a href="#orgcd974a5">5.2</a></li>
<li>Section <a href="#org958b946">5.3</a></li>
<li>Section <a href="#orgc07c075">5.4</a></li>
<li>Section <a href="#org856bc0c">5.5</a></li>
<li>Section <a href="#org9650828">5.6</a></li>
<li>Section <a href="#org04bd7f3">5.7</a></li>
<li>Section <a href="#org26fd0cc">5.8</a></li>
</ul>
</div>
<div id="outline-container-orgfcd97cd" class="outline-3">
<h3 id="orgfcd97cd"><span class="section-number-3">5.1</span> Setup</h3>
<div class="outline-text-3" id="text-5-1">
<p>
<a id="orgc202318"></a>
</p>
<div class="note" id="orgad07dbf">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Load Capacitor <a href="doc/0900766b815ea422.pdf">EPCOS 10uF Multilayer Ceramic Capacitor</a></li>
<li>Low Noise Voltage Amplifier <a href="doc/egg-5113-preamplifier.pdf">EG&amp;G 5113</a></li>
<li>Speedgoat ADC <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
</ul>
</div>
<p>
The output noise of the voltage amplifier PD200 is foreseen to be around 1mV rms in a bandwidth from DC to 1MHz.
If we suppose a white noise, this correspond to an amplitude spectral density:
</p>
\begin{equation}
\phi_{n} \approx \frac{1\,mV}{\sqrt{1\,MHz}} = 1 \frac{\mu V}{\sqrt{Hz}}
\end{equation}
<p>
The RMS noise begin very small compare to the ADC resolution, we must amplify the noise before digitizing the signal.
The added noise of the instrumentation amplifier should be much smaller than the noise of the PD200.
We use the amplifier EG&amp;G 5113 that has a noise of \(\approx 4 nV/\sqrt{Hz}\) referred to its input which is much smaller than the noise induced by the PD200.
</p>
<p>
The gain of the low-noise amplifier can be increased until the full range of the ADC is used.
This gain should be around 1000.
</p>
<div id="org5e97209" 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>
</div>
<p>
A low pass filter at 10kHz can be included in the EG&amp;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.
</p>
</div>
</div>
<div id="outline-container-orge5e70c7" class="outline-3">
<h3 id="orge5e70c7"><span class="section-number-3">5.2</span> Model of the setup</h3>
<div class="outline-text-3" id="text-5-2">
<p>
<a id="orgcd974a5"></a>
</p>
<p>
As shown in Figure <a href="#org44e4b22">6</a>, there are 4 equipment involved in the measurement:
</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="org44e4b22" 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-orge3695f9" class="outline-3">
<h3 id="orge3695f9"><span class="section-number-3">5.3</span> Quantization Noise</h3>
<div class="outline-text-3" id="text-5-3">
<p>
<a id="org958b946"></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>
The obtained Amplitude Spectral Density is <code>6.2294e-07</code> \(V/\sqrt{Hz}\).
</p>
</div>
</div>
<div id="outline-container-org82499a9" class="outline-3">
<h3 id="org82499a9"><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="orgc07c075"></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="org32efc97" 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>Sources of noise in the experimental setup</p>
</div>
<p>
The gain of the low noise amplifier is set to <code>50000</code>.
</p>
<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="#org2b5121b">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="org2b5121b" 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-orgaee6b6f" class="outline-3">
<h3 id="orgaee6b6f"><span class="section-number-3">5.5</span> PD200 noise measurement</h3>
<div class="outline-text-3" id="text-5-5">
<p>
<a id="org856bc0c"></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="org62ef1e2" 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="#orgba385cf">10</a>.
It is very similar to the one specified in the datasheet in Figure <a href="#org4f4605a">3</a>.
</p>
<div id="orgba385cf" 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="#org295193f">3</a>.
</p>
<table id="org295193f" 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" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">&#xa0;</th>
<th scope="col" class="org-right"><b>RMS [uV]</b></th>
<th scope="col" class="org-right"><b>Peak to Peak [mV]</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Specification [10uF]</td>
<td class="org-right">714.0</td>
<td class="org-right">4.3</td>
</tr>
<tr>
<td class="org-left">PD200_1</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">767.6</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_3</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">615.7</td>
<td class="org-right">3.5</td>
</tr>
<tr>
<td class="org-left">PD200_5</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">473.2</td>
<td class="org-right">2.7</td>
</tr>
<tr>
<td class="org-left">PD200_7</td>
<td class="org-right">423.1</td>
<td class="org-right">2.3</td>
</tr>
</tbody>
</table>
<p>
The Amplitude Spectral Density of the measured noise is now computed and shown in Figure <a href="#org134d5de">11</a>.
</p>
<div id="org134d5de" 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-org191ab71" class="outline-3">
<h3 id="org191ab71"><span class="section-number-3">5.6</span> DAC noise measurement</h3>
<div class="outline-text-3" id="text-5-6">
<p>
<a id="org9650828"></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="org0b30e65" 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="orga93a397" class="figure">
<p><img src="figs/asd_noise_dac.png" alt="asd_noise_dac.png" />
</p>
</div>
</div>
</div>
<div id="outline-container-org9c8acb0" class="outline-3">
<h3 id="org9c8acb0"><span class="section-number-3">5.7</span> Total noise measurement</h3>
<div class="outline-text-3" id="text-5-7">
<p>
<a id="org04bd7f3"></a>
</p>
<p>
Let&rsquo;s now analyze the measurement of the setup in Figure <a href="#org44e4b22">6</a>.
</p>
<p>
The PSD of the measured noise is computed and the ASD is shown in Figure <a href="#org7914e7c">14</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab">win = hanning(ceil(0.5<span class="org-type">/</span>Ts));
<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, 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="org7914e7c" class="figure">
<p><img src="figs/asd_noise_tot.png" alt="asd_noise_tot.png" />
</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="org91abcb6">
<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-org4635045" class="outline-3">
<h3 id="org4635045"><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="org26fd0cc"></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="#org0b30e65">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="#org55f8fb8">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="org55f8fb8" 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="org3b900fc">
<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-orga03c3cb" class="outline-2">
<h2 id="orga03c3cb"><span class="section-number-2">6</span> Transfer Function measurement</h2>
<div class="outline-text-2" id="text-6">
</div>
<div id="outline-container-orgf7753cb" class="outline-3">
<h3 id="orgf7753cb"><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="#orgbe81f0c">16</a> is used.
</p>
<div class="note" id="org5728bc6">
<p>
Here are the documentation of the equipment used for this test bench:
</p>
<ul class="org-ul">
<li>Voltage Amplifier <a href="doc/PD200-V7-R1.pdf">PD200</a></li>
<li>Load Capacitor <a href="doc/0900766b815ea422.pdf">EPCOS 10uF Multilayer Ceramic Capacitor</a></li>
<li>Speedgoat DAC/ADC <a href="doc/IO131-OEM-Datasheet.pdf">IO313</a></li>
</ul>
</div>
<p>
For this measurement, the sampling frequency of the Speedgoat ADC should be as high as possible.
</p>
<div id="orgbe81f0c" class="figure">
<p><img src="figs/setup-dynamics-measurement.png" alt="setup-dynamics-measurement.png" />
</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-org8cb4d98" class="outline-3">
<h3 id="org8cb4d98"><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 amplitude and the output voltage amplitude for sinusoidal waves of frequency \(\omega\):
\[ V_{out} = \frac{1}{C\omega} I_{out} \]
</p>
<p>
Moreover, 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 at which the output current reaches its maximum value is:
\[ \omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}} \]
</p>
<p>
\(\omega_max\) as a function of \(V_{in}\) is shown in Figure <a href="#org4f3d09d">17</a>.
</p>
<div id="org4f3d09d" class="figure">
<p><img src="figs/max_frequency_voltage.png" alt="max_frequency_voltage.png" />
</p>
<p><span class="figure-number">Figure 17: </span>Maximum frequency as a function of the excitation voltage amplitude</p>
</div>
</div>
</div>
<div id="outline-container-org69face8" class="outline-3">
<h3 id="org69face8"><span class="section-number-3">6.3</span> Obtained Transfer Functions</h3>
<div class="outline-text-3" id="text-6-3">
<p>
Several identifications using sweep sin were performed with input voltage amplitude ranging from 0.1V to 4V.
</p>
<p>
The obtained frequency response functions are shown in Figure <a href="#org7412e65">18</a>.
As the input voltage increases, the voltage drop is increasing.
</p>
<div id="org7412e65" class="figure">
<p><img src="figs/pd200_tf_voltage.png" alt="pd200_tf_voltage.png" />
</p>
<p><span class="figure-number">Figure 18: </span>Transfer function for the PD200 amplitude between \(V_{in}\) and \(V_{out}\) for multiple voltage amplitudes</p>
</div>
<p>
The small signal transfer function of the amplifier can be approximated by a first order low pass filter.
</p>
<div class="org-src-container">
<pre class="src src-matlab">Gp = 19.95<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>35e3);
</pre>
</div>
<p>
The comparison from the model and measurements are shown in Figure <a href="#orgb4e728b">19</a>.
</p>
<div id="orgb4e728b" class="figure">
<p><img src="figs/tf_pd200_model.png" alt="tf_pd200_model.png" />
</p>
<p><span class="figure-number">Figure 19: </span>Comparison of the model transfer function and the measured frequency response function</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org3dbe4a7" class="outline-2">
<h2 id="org3dbe4a7"><span class="section-number-2">7</span> Conclusion</h2>
<div class="outline-text-2" id="text-7">
<table id="org66913d3" 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>
<col class="org-left" />
<col class="org-center" />
<col class="org-center" />
<col class="org-center" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><b>Characteristics</b></th>
<th scope="col" class="org-center"><b>Measurement</b></th>
<th scope="col" class="org-center"><b>Manual</b></th>
<th scope="col" class="org-center"><b>Specification</b></th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">Input Voltage Range</td>
<td class="org-center">-</td>
<td class="org-center">+/- 10 [V]</td>
<td class="org-center">+/- 10 [V]</td>
</tr>
<tr>
<td class="org-left">Output Voltage Range</td>
<td class="org-center">-</td>
<td class="org-center">-50/150 [V]</td>
<td class="org-center">-20/150 [V]</td>
</tr>
<tr>
<td class="org-left">Gain</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">20 [V/V]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Maximum RMS current</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">0.9 [A]</td>
<td class="org-center">&gt; 50 [mA]</td>
</tr>
<tr>
<td class="org-left">Maximum Pulse current</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">10 [A]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Slew Rate</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">150 [V/us]</td>
<td class="org-center">-</td>
</tr>
<tr>
<td class="org-left">Noise (10uF load)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">0.7 [mV RMS]</td>
<td class="org-center">&lt; 2 [mV rms]</td>
</tr>
<tr>
<td class="org-left">Small Signal Bandwidth (10uF load)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">7.4 [kHz]</td>
<td class="org-center">&gt; 5 [kHz]</td>
</tr>
<tr>
<td class="org-left">Large Signal Bandwidth (150V, 10uF)</td>
<td class="org-center">&#xa0;</td>
<td class="org-center">300 [Hz]</td>
<td class="org-center">-</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2021-02-02 mar. 18:47</p>
</div>
</body>
</html>

7
index.org → test-bench-pd200.org
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@ -16,6 +16,7 @@
#+LaTeX_CLASS: scrreprt
#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
#+EXPORT_FILE_NAME: test-bench-pd200.tex
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
@ -40,6 +41,12 @@
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
#+begin_export html
<hr>
<p>This report is also available as a <a href="./test-bench-pd200.pdf">pdf</a>.</p>
<hr>
#+end_export
* Introduction
The goal of this test bench is to characterize the Voltage amplifier [[https://www.piezodrive.com/drivers/pd200-60-watt-voltage-amplifier/][PD200]] from PiezoDrive.

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