#+TITLE: Voltage Amplifier PD200 - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP:   ../index.html

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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
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:END:

* 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.

The documentation of the PD200 is accessible [[file:doc/PD200-V7-R1.pdf][here]].

#+name: fig:amplifier_PD200
#+caption: Picture of the PD200 Voltage Amplifier
#+attr_latex: :width 0.8\linewidth
[[file:figs/amplifier_PD200.png]]

* Voltage Amplifier Requirements

#+name: tab:voltage_amplifier_requirements
#+caption: Requirements for the Voltage Amplifier
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX
#+attr_latex: :center t :booktabs t :float t
| <l>                            |        <c>         |
|                                |  *Specification*   |
|--------------------------------+--------------------|
| Continuous Current             |     > 50 [mA]      |
| Output Voltage Noise (1-200Hz) |    < 2 [mV rms]    |
| Voltage Input Range            |     +/- 10 [V]     |
| Voltage Output Range           | -20 [V] to 150 [V] |
| Small signal bandwidth (-3dB)  |     > 5 [kHz]      |

* PD200 Expected characteristics

#+name: tab:pd200_characteristics
#+caption: Characteristics of the PD200
#+attr_latex: :environment tabularx :width \linewidth :align lXX
#+attr_latex: :center t :booktabs t :float t
| <l>                                 |     <c>      |       <c>       |
| *Characteristics*                   |   *Manual*   | *Specification* |
|-------------------------------------+--------------+-----------------|
| Input Voltage Range                 |  +/- 10 [V]  |   +/- 10 [V]    |
| Output Voltage Range                | -50/150 [V]  |   -20/150 [V]   |
| Gain                                |   20 [V/V]   |                 |
| Maximum RMS current                 |   0.9 [A]    |    > 50 [mA]    |
| Maximum Pulse current               |    10 [A]    |                 |
| Slew Rate                           |  150 [V/us]  |                 |
| Noise (10uF load)                   | 0.7 [mV RMS] |  < 2 [mV rms]   |
| Small Signal Bandwidth (10uF load)  |  7.4 [kHz]   |    > 5 [kHz]    |
| Large Signal Bandwidth (150V, 10uF) |   300 [Hz]   |                 |

For a load capacitance of $10\,\mu F$, the expected $-3\,dB$ bandwidth is $6.4\,kHz$ (Figure [[fig:pd200_expected_small_signal_bandwidth]]) and the low frequency noise is $650\,\mu V\,\text{rms}$ (Figure [[fig:pd200_expected_noise]]).

#+name: fig:pd200_expected_small_signal_bandwidth
#+caption:Expected small signal bandwidth
#+attr_latex: :width 0.8\linewidth
[[file:./figs/pd200_expected_small_signal_bandwidth.png]]

#+name: fig:pd200_expected_noise
#+caption: Expected Low frequency noise from 0.03Hz to 20Hz
#+attr_latex: :width 0.8\linewidth
[[file:figs/pd200_expected_noise.png]]

* Voltage Amplifier Model
The Amplifier is characterized by its dynamics $G_a(s)$ 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.

It is also characterized by its output noise $n$.
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.

#+begin_src latex :file pd200-model-schematic.pdf
  \begin{tikzpicture}
    \node[block] (G) at (0,0){$G_a(s)$};
    \node[addb, right=0.8 of G] (add){};

    \draw[<-] (G.west) -- ++(-1.2, 0) node[above right]{$V_{in}$};
    \draw[->] (G.east) -- (add.west);
    \draw[->] (add.east) -- ++(1.2, 0) node[above left]{$V_{out}$};
    \draw[<-] (add.north) -- ++(0, 0.6) node[below right](n){$n$};

    \begin{scope}[on background layer]
      \node[fit={(G.south west) (n.north-|add.east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
      \node[below] at (P.north) {PD-200};
    \end{scope}
  \end{tikzpicture}
#+end_src

#+name: fig:pd200-model-schematic
#+caption: Model of the voltage amplifier
#+RESULTS:
[[file:figs/pd200-model-schematic.png]]

* Noise measurement
** Setup
#+begin_note
Here are the documentation of the equipment used for this test bench:
- Voltage Amplifier [[file:doc/PD200-V7-R1.pdf][PD200]]
- Load Capacitor [[file:doc/0900766b815ea422.pdf][EPCOS 10uF Multilayer Ceramic Capacitor]]
- Low Noise Voltage Amplifier [[file:doc/egg-5113-preamplifier.pdf][EG&G 5113]]
- Speedgoat ADC [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
#+end_note

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:
\begin{equation}
  \phi_{n} \approx \frac{1\,mV}{\sqrt{1\,MHz}} = 1 \frac{\mu V}{\sqrt{Hz}}
\end{equation}

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&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.

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.

#+name: fig:setup-noise-measurement
#+caption: Schematic of the test bench to measure the Power Spectral Density of the Voltage amplifier noise $n$
#+attr_latex: :width \linewidth
[[file:figs/setup-noise-measurement.png]]

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
*** 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

*** Noise when shunting the input (50 Ohms)

#+begin_src matlab :exports none
%% Load all the measurements
pd200 = {};
for i = 1:7
    pd200(i) = {load(['mat/noise_PD200_' num2str(i) '.mat'], 't', 'Vn', 'notes')};
end

%% Take into account the pre-amplifier gain
for i = 1:7
    pd200{i}.Vn = pd200{i}.Vn/pd200{i}.notes.pre_amp.gain;
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;
for i = 1:7
    plot(pd200{i}.t, 1e3*pd200{i}.Vn)
end
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/noise_shunt_time_3uF.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:noise_shunt_time_3uF
#+caption: Time domain measurement of the amplifier output noise
#+RESULTS:
[[file:figs/noise_shunt_time_3uF.png]]

Obtained low frequency (0.1Hz - 20Hz) noise is shown in Figure [[fig:low_noise_time_domain_3uF]] which is very similar to the noise shown in the documentation (Figure [[fig:pd200_expected_noise]]).

#+begin_src matlab :exports none
figure;
hold on;
plot(pd200{1}.t, lsim(1/(1 + s/2/pi/20), 1e3*pd200{1}.Vn, pd200{1}.t))
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
xlim([0, 40]); ylim([-3, 3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/low_noise_time_domain_3uF.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:low_noise_time_domain_3uF
#+caption: Low Frequency Noise (0.1Hz - 20Hz)
#+RESULTS:
[[file:figs/low_noise_time_domain_3uF.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
Vn_rms = zeros(7,1); % RMS value [uV rms]
Vn_pkp = zeros(7,1); % Peak to Peak Value [mV]
for i = 1:7
    Vn_rms(i) = 1e6*rms(pd200{i}.Vn);
    Vn_pkp(i) = 1e3*(max(pd200{i}.Vn)-min(pd200{i}.Vn));
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 noise
#+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              |      524.9 |                 4.5 |
| PD200_2              |      807.7 |                 6.7 |
| PD200_3              |      630.3 |                 5.4 |
| PD200_4              |      619.7 |                 5.5 |
| PD200_5              |      630.8 |                 5.6 |
| PD200_6              |      517.3 |                 4.9 |
| PD200_7              |      393.8 |                 3.7 |

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200{1}.t(end) - pd200{1}.t(1))/(length(pd200{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));

[pxx, f] = pwelch(pd200{1}.Vn, win, [], [], Fs);

pxx = zeros(length(pxx), 7);

for i = 1:7
    pxx(:, i) = pwelch(pd200{i}.Vn, win, [], [], Fs);
end
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
    plot(f, sqrt(pxx(:, i)), 'DisplayName', sprintf('PD200-%i', i));
end
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

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_3uF.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:asd_noise_3uF
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_3uF.png]]

* 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.

#+begin_note
Here are the documentation of the equipment used for this test bench:
- Voltage Amplifier [[file:doc/PD200-V7-R1.pdf][PD200]]
- Load Capacitor [[file:doc/0900766b815ea422.pdf][EPCOS 10uF Multilayer Ceramic Capacitor]]
- Speedgoat DAC/ADC [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
#+end_note

For this measurement, the sampling frequency of the Speedgoat ADC should be as high as possible.

#+name: fig:setup-dynamics-measurement
#+caption: Schematic of the test bench to estimate the dynamics from voltage input $V_{in}$ to voltage output $V_{out}$
[[file:figs/setup-dynamics-measurement.png]]

** Results
* Conclusion

#+name: tab:table_name
#+caption: Measured characteristics, Manual characterstics and specified ones
#+attr_latex: :environment tabularx :width \linewidth :align lXXX
#+attr_latex: :center t :booktabs t :float t
| <l>                                 |      <c>      |     <c>      |       <c>       |
| *Characteristics*                   | *Measurement* |   *Manual*   | *Specification* |
|-------------------------------------+---------------+--------------+-----------------|
| Input Voltage Range                 |       -       |  +/- 10 [V]  |   +/- 10 [V]    |
| Output Voltage Range                |       -       | -50/150 [V]  |   -20/150 [V]   |
| Gain                                |               |   20 [V/V]   |        -        |
| Maximum RMS current                 |               |   0.9 [A]    |    > 50 [mA]    |
| Maximum Pulse current               |               |    10 [A]    |        -        |
| Slew Rate                           |               |  150 [V/us]  |        -        |
| Noise (10uF load)                   |               | 0.7 [mV RMS] |  < 2 [mV rms]   |
| Small Signal Bandwidth (10uF load)  |               |  7.4 [kHz]   |    > 5 [kHz]    |
| Large Signal Bandwidth (150V, 10uF) |               |   300 [Hz]   |        -        |