test-bench-pd200/test-bench-pd200.org

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

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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs

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: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

\clearpage

* Introduction                                                        :ignore:
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.

This document is organized as follows:
- Section [[sec:requirements]]: the requirements for the amplifiers and the characteristics of the PD200 amplifiers as advertise in the datasheet are listed.
- Section [[sec:amplifier_model]]: a very simple amplifier model consisting of a transfer function and a noise source is described.
- Section [[sec:tf_meas]]: the transfer function from input voltage to output voltage is identified.
- Section [[sec:noise_meas]]: the power spectral density of the amplifier's noise is measured
- Section [[sec:comp_pi_cedrat]]: the characteristics of the PD200 amplifier are compared to the E.505 amplifier from PI and to the LA75 from cedrat
- Section [[sec:conclusion]]: the measured characteristics of the PD200 amplifier are compared with the requirements

* Requirements PD200 Expected characteristics
<<sec:requirements>>

A picture of the PD200 amplifier is shown in Figure [[fig:amplifier_PD200]].

#+name: fig:amplifier_PD200
#+caption: Picture of the PD200 Voltage Amplifier
#+attr_latex: :width 0.7\linewidth
[[file:figs/amplifier_PD200.png]]

The specifications as well as the amplifier characteristics as shown in the datasheet are summarized in Table [[tab:pd200_characteristics]].

#+name: tab:pd200_characteristics
#+caption: Characteristics of the PD200 compared with the specifications
#+attr_latex: :environment tabularx :width 0.7\linewidth :align lcc
#+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]   |                 |

The most important characteristics are the large (small signal) bandwidth > 5 [kHz] and the small noise (< 2 [mV RMS]).

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]]).

These two characteristics are respectively measured in Section [[sec:tf_meas]] and Section [[sec:noise_meas]].

#+name: fig:pd200_expected_small_signal_bandwidth
#+caption:Expected small signal bandwidth
#+attr_latex: :width 0.7\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 0.7\linewidth
[[file:figs/pd200_expected_noise.png]]

* Voltage Amplifier Model
<<sec:amplifier_model>>

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.

It is also characterized by its *input* noise $n$.

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 input noise.

As $G_p$ depends on the load capacitance, it should be measured when loading the amplifier with a $10\,\mu F$ capacitor.

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

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

    \begin{scope}[on background layer]
      \node[fit={(G.south-|add.west) (n.north-|G.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]]


The input noise of the amplifier $n$ can be further modeled by shaping a white noise with unitary PSD $\tilde{n}$ with a transfer function $G_n(s)$ as shown in Figure [[fig:setup-dynamics-measurement]].

The Amplitude Spectral Density $\Gamma_n$ is then:
\begin{equation}
\Gamma_n(\omega) = |G_n(j\omega)| \Gamma_{\tilde{n}}(\omega)
\end{equation}
with $\Gamma_{\tilde{n}}(\omega) = 1$.


#+begin_src latex :file pd200-model-schematic-normalized.pdf
  \begin{tikzpicture}
    \node[addb] (add) at (0,0) {};
    \node[block, above=0.5 of add] (Gn) {$G_n(s)$};
    \node[block, right=0.8 of add] (G) {$G_p(s)$};

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

    \begin{scope}[on background layer]
      \node[fit={(G.south east) (n.north-|Gn.west)}, 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-normalized
#+caption: Model of the voltage amplifier with normalized noise input
#+RESULTS:
[[file:figs/pd200-model-schematic-normalized.png]]

* Transfer Function measurement
<<sec:tf_meas>>

** Introduction                                                      :ignore:

In this section, the transfer function of the PD200 amplifier is measured:
- Section [[sec:tf_meas_setup]]: the measurement setup is described
- Section [[sec:tf_meas_w_max]]: the maximum sinusoidal excitation frequency is estimated in order to not overload the amplifier
- Section [[sec:meas_small_signal_bandwidth]]: the small signal bandwidth measurement results are shown
- Section [[sec:model_small_signal_bandwidth]]: a model of the small signal dynamics of the amplifier is obtained
- Section [[sec:bandwidth_amplitude]]: the amplifier's transfer function is estimated for several input amplitudes

** Setup
<<sec:tf_meas_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/KEM_F3040_C4G_AXIAL-1104248.pdf][Film Capacitors 600V 10uF 5%]]
- DAC/ADC: [[file:doc/IO131-OEM-Datasheet.pdf][IO313 Speedgoat Interface]]
#+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]]

** 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

** Maximum Frequency/Voltage to not overload the amplifier
<<sec:tf_meas_w_max>>

Then the maximum output current of the amplifier is reached, the amplifier automatically shuts down itself.
We should then make sure that the output current does not reach this maximum specified current.

The maximum current is 1A [rms] which corresponds to 0.7A in amplitude of the sin wave.

The impedance of the capacitance is:
\[ Z_C(\omega) = \frac{1}{jC\omega} \]

Therefore the relation between the output current amplitude and the output voltage amplitude for sinusoidal waves of frequency $\omega$:
\[ V_{out} = \frac{1}{C\omega} I_{out} \]

Moreover, there is a gain of 20 between the input voltage and the output voltage:
\[ 20 V_{in} = \frac{1}{C\omega} I_{out} \]

For a specified voltage input amplitude $V_{in}$, the maximum frequency at which the output current reaches its maximum value is:
\begin{equation}
\boxed{\omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}}}
\end{equation}
with:
- $\omega_{\text{max}}$ the maximum input sinusoidal frequency in Radians per seconds
- $C$ the load capacitance in Farads
- $V_{in}$ the input voltage sinusoidal amplitude in Volts
- $I_{out,\text{max}}$ the specified maximum output current in Amperes

$\omega_{\text{max}}/2\pi$ as a function of $V_{in}$ is shown in Figure [[fig:max_frequency_voltage]].

#+begin_src matlab :exports none
Iout_max = 0.57; % Maximum output current [A]
C = 2.7e-6; % Load Capacitance [F]

V_in = linspace(0, 5, 100); % Input Voltage [V]

w_max = 1./(20*C*V_in) * Iout_max; % [rad/s]

figure;
plot(V_in, w_max/2/pi);
xlabel('Input Voltage Amplitude [V]');
ylabel('Maximum Frequency [Hz]');
set(gca, 'yscale', 'log');
#+end_src

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

#+name: fig:max_frequency_voltage
#+caption: Maximum frequency as a function of the excitation voltage amplitude
#+RESULTS:
[[file:figs/max_frequency_voltage.png]]

When doing sweep sine excitation, we make sure not to reach this maximum excitation frequency.

** Small Signal Bandwidth
<<sec:meas_small_signal_bandwidth>>

Here the small signal dynamics of all the 7 PD200 amplifiers are identified.

A (logarithmic) sweep sine excitation voltage is generated by the Speedgoat DAC with an amplitude of 0.1V and a frequency going from 1Hz up to 5kHz.

The output voltage of the PD200 amplifier is measured thanks to the monitor voltage of the PD200 amplifier.
The input voltage of the PD200 amplifier (the generated voltage by the DAC) is measured with another ADC of the Speedgoat.
This way, the time delay related to the ADC will not be apparent in the results.

#+begin_src matlab :exports none
%% Load all the measurements
pd200 = {};
for i = 1:7
    pd200(i) = {load(['tf_pd200_' num2str(i) '_10uF_small_signal.mat'], 't', 'Vin', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
%% Compute sampling Frequency
Ts = (pd200{1}.t(end) - pd200{1}.t(1))/(length(pd200{1}.t)-1);
Fs = 1/Ts;
#+end_src

#+begin_src matlab :exports none
%% Compute all the transfer functions
win = hanning(ceil(0.5*Fs)); % Hannning Windows

for i = 1:length(pd200)
    [tf_est, f] = tfestimate(pd200{i}.Vin, 20*pd200{i}.Vout, win, [], [], 1/Ts);
    pd200{i}.tf = tf_est;
    pd200{i}.f = f;
end
#+end_src

The obtained transfer functions from $V_{in}$ to $V_{out}$ are shown in Figure [[fig:pd200_small_signal_tf]].
#+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))
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([10, 30]);

ax2 = nexttile;
hold on;
for i = 1:length(pd200)
    plot(pd200{i}.f, 180/pi*angle(pd200{i}.tf), 'DisplayName', sprintf('PD200 %i', i))
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:2:360);
ylim([-12, 2]);
legend('location', 'southwest');

linkaxes([ax1,ax2],'x');
xlim([1, 5e3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_small_signal_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:pd200_small_signal_tf
#+caption: Identified dynamics from input voltage to output voltage
#+RESULTS:
[[file:figs/pd200_small_signal_tf.png]]

We can see the very well matching between all the 7 amplifiers.
The amplitude is constant over a wide frequency band and the phase drop is limited to less than 1 degree up to 500Hz.

** Model of the amplifier small signal dynamics
<<sec:model_small_signal_bandwidth>>

The identified dynamics in Figure [[fig:pd200_small_signal_tf]] can very well be modeled this dynamics with a first order low pass filter (even a constant could work fine).

Below is the defined transfer function $G_p(s)$.
#+begin_src matlab
Gp = 20/(1 + s/2/pi/25e3);
#+end_src

Comparison of the model with the identified dynamics is shown in Figure [[fig:pd200_small_signal_tf_model]].
#+begin_src matlab :exports none
colors = get(gca,'colororder');

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), 'color', [colors(1, :), 0.5])
end
set(gca,'ColorOrderIndex',2)
plot(pd200{1}.f, abs(squeeze(freqresp(Gp, pd200{1}.f, 'Hz'))), '--')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e0, 1e2]);

ax2 = nexttile;
hold on;
plot(pd200{1}.f, 180/pi*angle(pd200{1}.tf), 'color', [colors(1, :), 0.5], ...
     'DisplayName', 'Exp. data')
for i = 2:length(pd200)
    plot(pd200{i}.f, 180/pi*angle(pd200{i}.tf), 'color', [colors(1, :), 0.5], ...
         'HandleVisibility', 'off')
end
set(gca,'ColorOrderIndex',2)
plot(pd200{1}.f, 180/pi*angle(squeeze(freqresp(Gp, pd200{1}.f, 'Hz'))), '--', ...
     'DisplayName', '$|G_p|$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
yticks(-360:2:360);
ylim([-12, 2]);
legend('location', 'southwest');

linkaxes([ax1,ax2],'x');
xlim([5, 5e3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_small_signal_tf_model.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:pd200_small_signal_tf_model
#+caption:Bode plot of $G_d(s)$ as well as the identified transfer functions of all 7 amplifiers
#+RESULTS:
[[file:figs/pd200_small_signal_tf_model.png]]

#+begin_src matlab :tangle no :exports none
save('matlab/mat/pd200_model.mat', 'Gp');
#+end_src

And finally this model is saved.
#+begin_src matlab :eval no
save('mat/pd200_model.mat', 'Gp');
#+end_src

** Large Signal Bandwidth
<<sec:bandwidth_amplitude>>

The PD200 amplifiers will most likely not be used for large signals, but it is still nice to see how the amplifier dynamics is changing with the input voltage amplitude.

Several identifications using sweep sin were performed with input voltage amplitude ranging from 0.1V to 4V.
The maximum excitation frequency for each amplitude was limited from the estimation in Section [[sec:tf_meas_w_max]].

#+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_1_10uF_' Vin_ampl{i} 'V.mat'], 't', 'Vin', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
%% Compute the maximum excitation frequency
Iout_max = 0.57; % Maximum output current [A]
C = 10e-6; % Load Capacitance [F]

V_in = [0.1, 0.5, 1, 2, 4];
f_max = 0.8*Iout_max./(20*C*V_in/sqrt(2))/2/pi;
for i = 1:length(Vin_ampl)
    pd200{i}.notes.pd200.f_max = f_max(i);
    pd200{i}.notes.pd200.Vin = V_in(i);
end
#+end_src

#+begin_src matlab :exports none
%% Compute sampling Frequency
Ts = (pd200{1}.t(end) - pd200{1}.t(1))/(length(pd200{1}.t)-1);
Fs = 1/Ts;
#+end_src

#+begin_src matlab :exports none
%% Compute all the transfer functions
win = hanning(ceil(0.5*Fs)); % Hannning Windows

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

The obtained transfer functions for the different excitation amplitudes are shown in Figure [[fig:pd200_large_signal_tf]].
It is shown that the input voltage amplitude does not affect that much the amplifier dynamics.

#+begin_src matlab :exports none
%% Plot the identified transfer functions
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))
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([1e0, 1e2]);

ax2 = nexttile;
hold on;
for i = 1:length(pd200)
    plot(pd200{i}.f, 180/pi*angle(pd200{i}.tf), 'DisplayName', sprintf('$V_{in} = %.1f [V]$', pd200{i}.notes.pd200.Vin))
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
legend('location', 'southwest');
hold off;
yticks(-360:2:360);
ylim([-12, 2]);

linkaxes([ax1,ax2],'x');
xlim([5, 5e3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_large_signal_tf.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:pd200_large_signal_tf
#+caption: Amplifier dynamics for several input voltage amplitudes
#+RESULTS:
[[file:figs/pd200_large_signal_tf.png]]

* Noise measurement
<<sec:noise_meas>>

** Introduction                                                      :ignore:

In this part, the goal is to measure the noise of the PD200 voltage amplifier.
This noise can be separated into an input voltage noise and an input current noise.
However, the input voltage noise has much larger effects than the input current noise and we will only try to measure the input voltage noise.


In section [[sec:noise_setup]], the measurement setup is described and a model (block diagram) of the setup is given in section [[sec:noise_model]].

Then, the noise contribution of each element is measured:
- Section [[sec:noise_quantization]]: the quantization noise of the ADC is estimated
- Sections [[sec:noise_egg]] and [[sec:noise_femto]]: the noise of the low-noise amplifiers are estimated
- Sections [[sec:low_freq_noise_pd200]] and [[sec:high_freq_noise_pd200]]:: the input voltage noise of the PD200 amplifier is estimated
- Section [[sec:noise_dac]]: the output noise of the DAC is measured
- Section [[sec:noise_full_measurement]]: the noise of the full measurement chain (DAC to PD200 to pre-amplifier to ADC) is measured and it is found that the DAC is the main source of noise
- Section [[sec:noise_ssi2v]]: the noise of an 20bits DAC is measured
- Section [[sec:noise_full_measurement_ssi2v]]: it is shown if using the 20bits DAC could lower the overall noise of the setup

Finally in section [[sec:pd200_noise_model]], a model of the PD200 amplifier's noise is developed.

#+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/KEM_F3040_C4G_AXIAL-1104248.pdf][Film Capacitors 600V 10uF 5%]]
- Low Noise Voltage Amplifiers [[file:doc/egg-5113-preamplifier.pdf][EG&G 5113]] and [[file:doc/de-dlpva-100-b.pdf][Femto DLPVA]]
- ADC: [[file:doc/IO131-OEM-Datasheet.pdf][IO313 Speedgoat card]]
- 16bits DAC: [[file:doc/IO131-OEM-Datasheet.pdf][IO313 Speedgoat card]]
- 20bits DAC: [[file:doc/SSI2V_Datasheet.pdf][SSI2V]]
#+end_note

** 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

** Measurement Setup
<<sec:noise_setup>>

As the output noise of the PD200 voltage amplifier is foreseen to be around 1mV rms in a bandwidth from DC to 1MHz, it is not possible to directly measure it with an ADC.
We need to amplify the noise before digitizing the signal.
To do so, we need to use a low noise voltage amplifier with a noise density much smaller than the measured noise of the PD200 amplifier.

Let's first estimate the noise density of the PD200 amplifier.
If we suppose a white noise, this correspond to an amplitude spectral density:
\begin{equation}
  \Gamma_{n}(\omega) \approx \frac{1\,mV}{\sqrt{1\,MHz}} = 1 \frac{\mu V}{\sqrt{Hz}}
\end{equation}

The input noise of the instrumentation amplifier should be then much smaller than the output noise of the PD200.
We will use either the amplifier EG&G 5113 that has a noise of $\approx 4 nV/\sqrt{Hz}$ referred to its input or the Femto DLPVA amplifier with an input noise of $\approx 3nV/\sqrt{Hz}$.

The gain of the low-noise amplifier is then increased until the full range of the ADC is used.
This gain should be around 1000 (60dB).

A representation of the measurement bench is shown in Figure [[fig:setup-noise-measurement]].

Note that it is quite important to load the amplifier with the "Load Box" including a $10\,\mu F$ capacitor as the (high frequency) noise of the amplifier depends on the actual load being used.

#+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]]

** Model of the setup
<<sec:noise_model>>

As shown in Figure [[fig:noise_meas_procedure]], there are 4 elements 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
- $n_p$: input 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[addb, right=1.2 of addnda] (addnp){};
  \node[block, right=0.4 of addnp] (Gp){$G_p(s)$};

  % Pre Amp
  \node[addb, right=1.2 of Gp] (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) -- (addnp.west);
  \draw[->] (addnp.east) -- (Gp.west);
  \draw[->] (Gp.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={(addnp.west|-bot) (Gp.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]]

In the next sections, we wish to measure all these sources of noise and make sure that we can effectively characterize the noise $n_p$ of the PD200 amplifier.

** Quantization Noise of the ADC
<<sec:noise_quantization>>

The quantization noise is something that can be predicted from the sampling frequency and the quantization of the ADC.
Indeed, 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]

Let's estimate that with the ADC used for the measurements:
#+begin_src matlab
%% ADC Quantization noise
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=)}}} $V/\sqrt{Hz}$.

** EG&G - Amplifier noise measurement
<<sec:noise_egg>>

We now wish to measure the noise of the pre-amplifier.
To do so, the input of the pre-amplifier is shunted with a 50Ohms resistor such that the pre-amplifier input voltage is just its input noise.
Then, the gain of the amplifier is increased 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}

#+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
%% EG&G Input Voltage Noise
egg = load('mat/noise_preamp_5113.mat', 't', 'Vn', 'notes'); % Load Data
#+end_src

The gain of the low noise amplifier is set to src_matlab[:exports results :results value replace]{ans = egg.notes.pre_amp.gain} {{{results(=50000=)}}} for the measurement.

#+begin_src matlab :exports none
% Compute the equivalent voltage at the input of the amplifier
egg.Vn = egg.Vn/egg.notes.pre_amp.gain;
egg.Vn = egg.Vn - mean(egg.Vn);
#+end_src

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (egg.t(end) - egg.t(1))/(length(egg.t) - 1);
Fs = 1/Ts;
#+end_src

#+begin_src matlab :exports none
% Hanning window
win = hanning(ceil(0.5/Ts));

% Power Spectral Density
[pxx, f] = pwelch(egg.Vn, win, [], [], Fs);

% Save the results inside the struct
egg.pxx = pxx;
egg.f = f;
#+end_src

The obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier is shown in Figure [[fig:asd_egg]].
The obtained noise amplitude is very closed to the one specified in the documentation of $4nV/\sqrt{Hz}$ at 1kHZ.

It is also verified that the quantization noise of the ADC is much smaller and what we are measuring is indeed the noise of the pre-amplifier.

#+begin_src matlab :exports none
figure;
hold on;
plot(egg.f, sqrt(egg.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./egg.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]); ylim([1e-11, 1e-7]);
#+end_src

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

#+name: fig:asd_egg
#+caption: Obtained Amplitude Spectral Density of the EG&G Low Noise Voltage Amplifier
#+RESULTS:
[[file:figs/asd_egg.png]]

** Femto - Amplifier noise measurement
<<sec:noise_femto>>

Similarly to Section [[sec:noise_egg]], the noise of the Femto amplifier is identified.

#+begin_src matlab :exports none
%% Femto Input Voltage Noise
femto = load('mat/noise_femto.mat', 't', 'Vout', 'notes'); % Load Data
#+end_src

#+begin_src matlab :exports none
% Compute the equivalent voltage at the input of the amplifier
femto.Vout = femto.Vout/femto.notes.pre_amp.gain;
femto.Vout = femto.Vout - mean(femto.Vout);
#+end_src

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (femto.t(end) - femto.t(1))/(length(femto.t) - 1);
Fs = 1/Ts;
#+end_src

#+begin_src matlab :exports none
% Hanning window
win = hanning(ceil(0.5/Ts));

% Power Spectral Density
[pxx, f] = pwelch(femto.Vout, win, [], [], Fs);

% Save the results inside the struct
femto.pxx = pxx;
femto.f = f;
#+end_src

The obtained Amplitude spectral density is shown in Figure [[fig:asd_femto]].

#+begin_src matlab :exports none
figure;
hold on;
plot(femto.f, sqrt(femto.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./femto.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]); ylim([1e-11, 1e-7]);
#+end_src

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

#+name: fig:asd_femto
#+caption: Obtained Amplitude Spectral Density of the Femto Low Noise Voltage Amplifier
#+RESULTS:
[[file:figs/asd_femto.png]]

** PD200 - Low frequency noise measurement
<<sec:low_freq_noise_pd200>>

The measurement setup is shown in Figure [[fig:noise_measure_setup_pd200]].
The input of the PD200 amplifier is shunted with a 50 Ohm resistor such that there in no voltage input expected the PD200 input voltage noise.
The gain of the pre-amplifier is increased in order to measure a signal much larger than the quantization noise of the ADC.

#+begin_src latex :file noise_measure_setup_pd200.pdf
\begin{tikzpicture}
  \node[block={0.6cm}{0.6cm}] (const) {$0$};

  % PD200
  \node[addb, right=0.6 of const] (addnp){};
  \node[block, right=0.4 of addnp] (Gp){$G_p(s)$};

  % Pre Amp
  \node[addb, right=1.2 of Gp] (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) -- (addnp.west);
  \draw[->] (addnp.east) -- (Gp.west);
  \draw[->] (Gp.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={(addnp.west|-bot) (Gp.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
%% PD200 Input Voltage Noise
% 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 and PD200 Gain
for i = 1:7
    pd200w{i}.Vn = pd200w{i}.Vn/pd200w{i}.notes.pre_amp.gain/20;
end
#+end_src

The measured low frequency (<20Hz) *output* noise of one of the PD200 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), 20*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 *output* noise are shown in Table [[tab:rms_pkp_noise]] and found to be very similar to the specified ones.
#+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(20*pd200w{i}.Vn);
    Vn_lpf = 20*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 [$10\,\mu F$]', 'PD200 1', 'PD200 2', 'PD200 3', 'PD200 4', 'PD200 5', 'PD200 6', 'PD200 7'}, {'*RMS [$\mu V$]*', '*Peak to Peak [$mV$]*'}, ' %.1f ');
#+end_src

#+name: tab:rms_pkp_noise
#+caption: RMS and Peak to Peak measured low frequency output noise (0.01Hz to 20Hz)
#+attr_latex: :environment tabularx :width 0.5\linewidth :align lcc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|                             | *RMS [$\mu V$]* | *Peak to Peak [$mV$]* |
|-----------------------------+-----------------+-----------------------|
| Specification [$10\,\mu F$] |           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 |

** PD200 - High frequency noise measurement
<<sec:high_freq_noise_pd200>>

The measurement setup is the same as in Figure [[fig:noise_measure_setup_pd200]].

The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal by the ADC is computed.
The Amplitude Spectral Density of the input voltage noise of the PD200 amplifier $n_p$ is then computed taking into account the gain of the pre-amplifier and the gain of the PD200 amplifier:
\begin{equation}
\Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_p(j\omega) G_a(j\omega)|}
\end{equation}

And we verify that we are indeed measuring the noise of the PD200 and not the noise of the pre-amplifier by checking that:
\begin{equation}
\Gamma_{n_p}(\omega) |G_p(j\omega)| \ll \Gamma_{n_a}
\end{equation}

#+begin_src matlab :exports none
%% PD200 Input Voltage Noise
% Load all the measurements
pd200 = {};
for i = 1:7
    pd200(i) = {load(['mat/noise_PD200_' num2str(i) '_10uF.mat'], 't', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
% Take into account the pre-amplifier gain and PD200 Gain
for i = 1:7
    pd200{i}.Vout = pd200{i}.Vout/pd200{i}.notes.pre_amp.gain/20;
end
#+end_src

#+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

#+begin_src matlab :exports none
% Compute the PSD of the measured noise
win = hanning(ceil(2/Ts));

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

The Amplitude Spectral Density of the measured *input* noise is computed and shown in Figure [[fig:asd_noise_pd200_10uF]].

It is verified that the contribution of the PD200 noise is much larger than the contribution of the pre-amplifier noise of the quantization noise.
#+begin_src matlab :exports none
colors = get(gca,'colororder');

figure;
hold on;
plot(femto.f, sqrt(femto.pxx)/20, 'DisplayName', '$\Gamma_{n_a}/|G_p|$');
plot(pd200{1}.f, sqrt(pd200{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
    plot(pd200{i}.f, sqrt(pd200{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./pd200{1}.notes.pre_amp.gain/20, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_p G_a|$');
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_pd200_10uF.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:asd_noise_pd200_10uF
#+caption: Amplitude Spectral Density of the measured input voltage noise of the PD200 amplifiers
#+RESULTS:
[[file:figs/asd_noise_pd200_10uF.png]]

#+begin_note
The Amplitude Spectral Density of the input noise of the PD200 amplifiers present sharp peaks.
It is not clear yet what causes such peaks and if these peaks have high influence on the total RMS noise of the amplifiers.
#+end_note

** 16bits DAC noise measurement
<<sec:noise_dac>>

In order not to have any quantization noise and only measure the output voltage noise of the DAC, we "ask" the DAC to output a zero voltage.

The measurement setup is schematically represented in Figure [[fig:noise_measure_setup_dac]].
The gain of the pre-amplifier is adjusted such that the measured amplified noise is much larger than the quantization noise of the ADC.

The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal is computed.
The Amplitude Spectral Density of the DAC output voltage noise $n_{da}$ 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 verified 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 Output Voltage Noise
dac = load('mat/noise_preamp_5113_dac.mat', 't', 'Vn', 'notes');
#+end_src

#+begin_src matlab :exports none
% Take input acount the gain of the pre-amplifier
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
% Compute the PSD of the measured noise
win = hanning(ceil(0.5/Ts));

[pxx, f] = pwelch(dac.Vn, win, [], [], Fs);
dac.pxx = pxx;
dac.f = f;
#+end_src

The obtained Amplitude Spectral Density of the DAC's output voltage is shown in Figure [[fig:asd_noise_dac]].

#+begin_src matlab :exports none
colors = get(gca,'colororder');

figure;
hold on;
plot(egg.f, sqrt(egg.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: Amplitude Spectral Density of the measured output voltage noise of the 16bits DAC
#+RESULTS:
[[file:figs/asd_noise_dac.png]]

** Noise of the full setup with 16bits DAC
<<sec:noise_full_measurement>>

Let's now measure the noise of the full setup in Figure [[fig:noise_meas_procedure_bis]] and analyze the results.

#+name: fig:noise_meas_procedure_bis
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_meas_procedure.png]]

#+begin_src matlab :exports none
%% Full measurement chain from DAC to ADC
pd200dac = {};
for i = 1:7
    pd200dac(i) = {load(['mat/noise_PD200_' num2str(i) '_10uF_DAC.mat'], 't', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
% Take into account the pre-amplifier gain
for i = 1:7
    pd200dac{i}.Vout = pd200dac{i}.Vout/pd200dac{i}.notes.pre_amp.gain/20;
    pd200dac{i}.Vout = pd200dac{i}.Vout - mean(pd200dac{i}.Vout);
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 Amplitude Spectral Density of the measured noise is computed and the shown in Figure [[fig:asd_noise_tot]].

We can very well see that to total measured noise is the sum of the DAC noise and the PD200 noise.
#+begin_src matlab :exports none
% Compute the PSD of the measured noise
win = hanning(ceil(2/Ts));

for i = 1:7
    [pxx, f] = pwelch(pd200dac{i}.Vout, 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(egg.f, sqrt(egg.pxx)/20, 'DisplayName', '$\Gamma_{n_a}$');
plot(pd200{1}.f, sqrt(pd200{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}/|G_p|$');
for i = 2:7
    plot(pd200{i}.f, sqrt(pd200{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',3)
plot(dac.f, 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/20, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_p G_a|$');
plot(pd200dac{1}.f, sqrt(pd200dac{1}.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]); ylim([1e-11, 1e-4]);
#+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 input noise of the PD200 amplifier is limited by the output voltage noise of the DAC.
Having a DAC with lower output voltage noise could lower the overall noise of the setup.
SSI2V 20bits DACs are used in the next section to verify that.
#+end_important

** 20bits DAC noise measurement
<<sec:noise_ssi2v>>

Let's now measure the output voltage noise of another DAC called the "SSI2V" ([[file:doc/\[SSI2V\]Datasheet.pdf][doc]]).
It is a 20bits DAC with an output voltage range of +/-10.48 V and a very low output voltage noise.

The measurement setup is the same as the one in Figure [[fig:noise_measure_setup_dac]].

#+begin_src matlab :exports none
%% SSI2V Output Voltage Noise
ssi2v = load('mat/noise_preamp_5113_SSI2V.mat', 't', 'Vn', 'notes');
#+end_src

#+begin_src matlab :exports none
% Take into account the pre-amplifier gain
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 :exports none
% Compute the Power Spectral Density of the measured noise
win = hanning(ceil(0.5/Ts));

[pxx, f] = pwelch(ssi2v.Vn, win, [], [], Fs);
ssi2v.pxx = pxx;
ssi2v.f = f;
#+end_src

The obtained Amplitude Spectral Density of the output voltage noise of the SSI2V DAC is shown in Figure [[fig:asd_ssi2v_noise]] and compared with the output voltage 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(egg.f, sqrt(egg.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

** TODO Noise of the full setup with 20bits DAC
<<sec:noise_full_measurement_ssi2v>>

** PD200 Amplifier noise model
<<sec:pd200_noise_model>>

Let's design a transfer function $G_n(s)$ whose norm represent the Amplitude Spectral Density of the input voltage noise of the PD200 amplifier as shown in Figure [[fig:pd200-model-schematic-normalized-bis]].

#+name: fig:pd200-model-schematic-normalized-bis
#+caption: Model of the voltage amplifier with normalized noise input
[[file:figs/pd200-model-schematic-normalized.png]]


A simple transfer function that allows to obtain a good fit is defined below.
#+begin_src matlab
%% Model of the PD200 Input Voltage Noise
Gn = 1e-5 * ((1 + s/2/pi/20)/(1 + s/2/pi/2))^2 /(1 + s/2/pi/5e3);
#+end_src

The comparison between the measured ASD of the modeled ASD is done in Figure [[fig:pd200_asd_noise_model]].
#+begin_src matlab :exports none
% Plot the ASD of both the measured noise and the modelled one
freqs = logspace(-1, 4, 1000);

figure;
hold on;
plot(pd200{1}.f, sqrt(pd200{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
    plot(pd200{i}.f, sqrt(pd200{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gn, freqs, 'Hz'))), 'k-', 'DisplayName', '$|G_n(j\omega)|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]');
xlim([1, Fs/2]);
ylim([1e-8, 1e-4]);
legend('location', 'northeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_asd_noise_model.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:pd200_asd_noise_model
#+caption: ASD of the measured input voltage noise and modeled noise using $G_n(s)$
#+RESULTS:
[[file:figs/pd200_asd_noise_model.png]]

Let's now compute the Cumulative Amplitude Spectrum corresponding to the measurement and the model and compare them.

The integration from low to high frequency and from high to low frequency are both shown in Figure [[fig:pd200_cas_noise_model]].

The fit between the model and the measurements is rather good considering the complex shape of the measured ASD and the simple model used.
#+begin_src matlab :exports none
% Compute the cumulative power spectrum from high to low and low to high frequencies
for i = 1:7
    pd200{i}.CPS_f = flip(-cumtrapz(flip(pd200{i}.f), flip(pd200{i}.pxx)));
    pd200{i}.CPS = cumtrapz(pd200{i}.f, pd200{i}.pxx);
end

CPS_Gn_f = flip(-cumtrapz(flip(freqs), flip(abs(squeeze(freqresp(Gn, freqs, 'Hz'))).^2)));
CPS_Gn = cumtrapz(freqs, abs(squeeze(freqresp(Gn, freqs, 'Hz'))).^2);
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(pd200{1}.f, sqrt(pd200{1}.CPS), 'color', [colors(1, :), 0.5], 'DisplayName', '$CAS$');
for i = 2:7
    plot(pd200{i}.f, sqrt(pd200{i}.CPS), 'color', [colors(1, :), 0.5], 'HandleVisibility', 'off');
end
for i = 1:7
    plot(pd200{i}.f, sqrt(pd200{i}.CPS_f), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end

set(gca,'ColorOrderIndex',1)
plot(freqs, sqrt(CPS_Gn), '--', 'DisplayName', 'model');
set(gca,'ColorOrderIndex',2)
plot(freqs, sqrt(CPS_Gn_f), '--', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('CAS [V rms]');
xlim([1, Fs/2]);
ylim([1e-6, 1e-4]);
legend('location', 'northeast');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_cas_noise_model.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:pd200_cas_noise_model
#+caption: Cumulative Amplitude Spectrum of the measured input voltage noise and modeled noise using $G_n(s)$
#+RESULTS:
[[file:figs/pd200_cas_noise_model.png]]

The obtained RMS noise of the model is src_matlab[:exports results :results value replace]{ans = 1e6*20*sqrt(CPS_Gn(end))} {{{results(=286.74=)}}} uV RMS which is not that far from the specifications.

#+begin_src matlab :tangle no :exports none
save('matlab/mat/pd200_model.mat', 'Gn', '-append');
#+end_src

Finally the model of the amplifier noise is saved.
#+begin_src matlab :eval no
save('mat/pd200_model.mat', 'Gn', '-append');
#+end_src

** 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>>
#+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

*** 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

#+begin_src matlab
preamp.Vn = preamp.Vn/preamp.notes.pre_amp.gain;
preamp.Vn = preamp.Vn - mean(preamp.Vn);
#+end_src

#+begin_src matlab
figure;
plot(preamp.t, preamp.Vn);
xlabel('Time [s]');
ylabel('Voltage [V]');
#+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
win = hanning(ceil(0.5/Ts));

[pxx, f] = pwelch(preamp.Vn, win, [], [], Fs);

preamp.pxx = pxx;
preamp.f = f;
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx));
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

*** 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

#+begin_src matlab
dac.Vn = dac.Vn/dac.notes.pre_amp.gain;
#+end_src

#+begin_src matlab
dac.Vn = dac.Vn - mean(dac.Vn);
#+end_src

#+begin_src matlab
figure;
plot(dac.t, 1e6*dac.Vn);
xlabel('Time [s]');
ylabel('Voltage [$\mu V$]');
#+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
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
figure;
hold on;
plot(dac.f, sqrt(dac.pxx), 'DisplayName', 'DAC');
plot(dac.f, ones(size(dac.f))*(10/2^16)/sqrt(12*Fs)/dac.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

*** 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
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 time domain measurements of the amplifier noise are shown in Figure [[fig:noise_shunt_time_3uF_warmup]].

#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
    plot(pd200w{i}.t, 1e3*pd200w{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_warmup.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:noise_shunt_time_3uF_warmup
#+caption: Time domain measurement of the amplifier output noise
#+RESULTS:
[[file:figs/noise_shunt_time_3uF_warmup.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 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 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              |      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
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

#+begin_src matlab :exports none
figure;
hold on;
for i = 1:7
    plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'DisplayName', sprintf('PD200W-%i', i));
end
plot(preamp.f, sqrt(preamp.pxx), 'k-', 'DisplayName', 'Pre Amp');
plot(dac.f, ones(size(dac.f))*(10/2^16)/sqrt(12*Fs)/pd200w{1}.notes.pre_amp.gain, 'k--', 'DisplayName', 'ADC quant.');
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]);
% ylim([5e-7, 1e-3]);
#+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]]

#+begin_src matlab
Gn = 1e-6*(s + 2*pi*40)^2/(s + 2*pi)^2;
#+end_src

#+begin_src matlab :exports none
freqs = logspace(0, 4, 1000);
figure;
hold on;
for i = 1:7
    plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'DisplayName', sprintf('PD200W-%i', i));
end
plot(freqs, abs(squeeze(freqresp(Gn, freqs, 'Hz'))), 'k--', 'DisplayName', '$|G_n(j\omega)|$');
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

*** Load / No Load                                                :noexport:
#+begin_src matlab
pd200_load = load('noise_PD200_7_3uF_warmup.mat');
pd200_no_load = load('noise_PD200_7_no_load.mat');
#+end_src

#+begin_src matlab
pd200_load.Vn = pd200_load.Vn/pd200_load.notes.pre_amp.gain;
pd200_no_load.Vn = pd200_no_load.Vn/pd200_no_load.notes.pre_amp.gain;
#+end_src

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200_load.t(end) - pd200_load.t(1))/(length(pd200_load.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_load, f] = pwelch(pd200_load.Vn, win, [], [], Fs);
[pxx_no_load, ~] = pwelch(pd200_no_load.Vn, win, [], [], Fs);
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
plot(f, sqrt(pxx_load), 'DisplayName', 'Load');
plot(f, sqrt(pxx_no_load), 'DisplayName', 'No Load');
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)                       :noexport:

#+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]]

*** 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 = {};
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
figure;
hold on;
for i = 1:7
    plot(pd200dac{i}.t, 1e3*pd200dac{i}.Vn)
end
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
xlim([0, 0.1]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/noise_shunt_time_3uF_dac.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:noise_shunt_time_3uF_dac
#+caption: Time domain measurement of the amplifier output noise
#+RESULTS:
[[file:figs/noise_shunt_time_3uF_dac.png]]

#+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_3uF_dac]].
#+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
figure;
hold on;
for i = 1:7
    plot(pd200dac{i}.f, sqrt(pd200dac{i}.pxx), 'DisplayName', sprintf('PD200DAC-%i', i));
end
plot(preamp.f, sqrt(preamp.pxx), 'k-', 'DisplayName', 'Pre Amp');
plot(dac.f, 20*sqrt(dac.pxx), 'k-', 'DisplayName', 'ADC noise');
plot(dac.f, ones(size(dac.f))*(10/2^16)/sqrt(12*Fs)/pd200dac{1}.notes.pre_amp.gain, 'k--', 'DisplayName', 'ADC quant.');
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]);
% ylim([5e-7, 1e-3]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_3uF_dac.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:asd_noise_3uF_dac
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_3uF_dac.png]]

#+begin_src matlab :exports none
figure;
hold on;
plot(pd200dac{1}.f, sqrt(pd200dac{1}.pxx), 'DisplayName', 'PD200 + DAC');
plot(pd200w{1}.f, sqrt(pd200w{1}.pxx), 'DisplayName', 'PD200');
plot(dac.f, 20*sqrt(dac.pxx), 'k-', 'DisplayName', 'DAC');
plot(preamp.f, sqrt(preamp.pxx), 'k-', 'DisplayName', 'Pre Amp');
plot(dac.f, ones(size(dac.f))*(10/2^16)/sqrt(12*Fs)/pd200dac{1}.notes.pre_amp.gain, 'k--', 'DisplayName', 'ADC quant.');
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]);
% ylim([5e-7, 1e-3]);
#+end_src

#+begin_important
The output noise of the PD200 amplifier is limited by the noise of the DAC.
#+end_important

*** With 10uF load and Femto pre-amplifier                        :noexport:

#+begin_src matlab :exports none
% Load Data
preamp = load('mat/noise_femto.mat', 't', 'Vout', 'notes');
#+end_src

#+begin_src matlab :exports none
% Compute the equivalent voltage at the input of the amplifier
preamp.Vout = preamp.Vout/preamp.notes.pre_amp.gain/20;
preamp.Vout = preamp.Vout - mean(preamp.Vout);
#+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.Vout, win, [], [], Fs);

% Save the results inside the struct
preamp.pxx = pxx;
preamp.f = f;
#+end_src

#+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 :exports none
%% Load all the measurements
pd200b = {};
for i = 1:7
    pd200b(i) = {load(['mat/noise_PD200_' num2str(i) '_10uF.mat'], 't', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
    pd200b{i}.Vout = pd200b{i}.Vout/pd200b{i}.notes.pre_amp.gain/20;
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(pd200b{1}.t(1:t_max/Ts), 20*lsim(G_lpf, 1e3*pd200b{1}.Vout(1:t_max/Ts), pd200b{1}.t(1:t_max/Ts)))
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
ylim([-3, 3]);
#+end_src

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(6,1); % RMS value [uV rms]
Vn_pkp = zeros(6,1); % Peak to Peak Value in 20Hz bandwidth [mV]
for i = 1:7
    Vn_rms(i) = 1e6*rms(pd200b{i}.Vout);
    Vn_lpf = lsim(1/(1 + s/2/pi/20), pd200b{i}.Vout, pd200b{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              |      281.9 |                 1.6 |
| PD200_2              |      665.6 |                 2.0 |
| PD200_3              |      314.8 |                 2.1 |
| PD200_4              |      360.1 |                 2.2 |
| PD200_5              |      563.2 |                 1.7 |
| PD200_6              |      323.3 |                 2.0 |
| PD200_7              |      212.5 |                 1.1 |

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200b{1}.t(end) - pd200b{1}.t(1))/(length(pd200b{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(pd200b{i}.Vout, win, [], [], Fs);
    pd200b{i}.f = f;
    pd200b{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(pd200b{1}.f, sqrt(pd200b{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
    plot(pd200b{i}.f, sqrt(pd200b{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./pd200b{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_10uF.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:asd_noise_10uF
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_10uF.png]]

#+begin_src matlab :exports none
%% Load all the measurements
pd200t = {};
for i = 1:7
    pd200t(i) = {load(['mat/noise_PD200_' num2str(i) '_10uF_DAC.mat'], 't', 'Vout', 'notes')};
end
#+end_src

#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
    pd200t{i}.Vout = pd200t{i}.Vout/pd200t{i}.notes.pre_amp.gain/20;
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(pd200t{1}.t(1:t_max/Ts), lsim(G_lpf, 1e3*pd200t{1}.Vout(1:t_max/Ts), pd200t{1}.t(1:t_max/Ts)))
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
ylim([-3, 3]);
#+end_src

#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200t{1}.t(end) - pd200t{1}.t(1))/(length(pd200t{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(pd200t{i}.Vout, win, [], [], Fs);
    pd200t{i}.f = f;
    pd200t{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(pd200b{1}.f, sqrt(pd200b{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
    plot(pd200b{i}.f, sqrt(pd200b{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(pd200t{1}.f, sqrt(pd200t{1}.pxx), 'color', [colors(4, :), 0.5], 'DisplayName', '$\Gamma_{tot}$');
for i = 2:7
    plot(pd200t{i}.f, sqrt(pd200t{i}.pxx), 'color', [colors(4, :), 0.5], 'HandleVisibility', 'off');
end
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./pd200t{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_10uF.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:asd_noise_10uF
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_10uF.png]]

* Comparison to other commercial amplifiers
<<sec:comp_pi_cedrat>>
** Introduction
In this section, three similar voltage amplifiers are compared:
- the [[file:doc/PD200-V7-R1.pdf][PD200]] from PiezoDrive
- the [[file:doc/LA75B.pdf][LA75B]] from CedratTechnologies
- the [[file:doc/E-505-Datasheet.pdf][E-505.00]] from PI

These are compared in term of dynamic from input voltage to output voltage for a load of $10\,\mu F$ in Section [[sec:tf_comp]] and then in term of input voltage noise in Section [[sec:noise_comp]].

The characteristics that I could find for the three amplifiers are summarized in Table [[tab:amplifiers_characteristics]].

#+name: tab:amplifiers_characteristics
#+caption: Characteristics of the three tested voltage amplifiers
#+attr_latex: :environment tabularx :width 0.9\linewidth :align lccc
#+attr_latex: :center t :booktabs t :float t
| <l>                    |       <c>        |         <c>         |     <c>      |
| *Characteristics*      |     *PD200*      |       *LA75B*       |   *E-505*    |
|------------------------+------------------+---------------------+--------------|
| Gain                   |     20 [V/V]     |      20 [V/V]       |   10 [V/V]   |
| Maximum RMS current    |     0.9 [A]      |       0.4 [A]       |              |
| Maximum Pulse current  |      10 [A]      |        1 [A]        |    2 [A]     |
| Slew Rate              |    150 [V/us]    |                     |              |
| Noise (10uF load)      |   0.7 [mV RMS]   |    3.4 [mV RMS]     | 0.6 [mV RMS] |
| Small Signal Bandwidth | 7.4 [kHz] (10uF) | 30 [kHz] (unloaded) |              |

#+begin_note
The documentation for the three amplifiers can be found here: [[file:doc/PD200-V7-R1.pdf][PD200]], [[file:doc/LA75B.pdf][LA75B]], [[file:doc/E-505-Datasheet.pdf][E-505.00]].
#+end_note

** 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

** Transfer functions
<<sec:tf_comp>>

#+begin_src matlab
la75 = load('tf_la75_10uF_small_signal.mat', 't', 'Vin', 'Vout');
pd200 = load('tf_pd200_1_10uF_small_signal.mat', 't', 'Vin', 'Vout', 'notes');
#+end_src

#+begin_src matlab :exports none
%% Compute sampling Frequency
Ts = (pd200.t(end) - pd200.t(1))/(length(pd200.t)-1);
Fs = 1/Ts;
#+end_src

#+begin_src matlab :exports none
%% Compute all the transfer functions
win = hanning(ceil(0.5*Fs)); % Hannning Windows

[tf_pd200, f] = tfestimate(pd200.Vin, 20*pd200.Vout, win, [], [], 1/Ts);
[tf_la75,  ~] = tfestimate(la75.Vin,  la75.Vout,     win, [], [], 1/Ts);
#+end_src

#+begin_src matlab :exports none
figure;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
hold on;
plot(f, abs(tf_pd200))
plot(f(f<900), abs(tf_la75(f<900)))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([10, 30]);

ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(tf_pd200), 'DisplayName', 'PD200')
plot(f(f<900), 180/pi*angle(tf_la75(f<900)),  'DisplayName', 'LA75')
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:2:360);
ylim([-12, 2]);
legend('location', 'southwest');

linkaxes([ax1,ax2],'x');
xlim([1, 5e3]);
#+end_src

** Noise Characteristics
<<sec:noise_comp>>

#+begin_src matlab :exports none
pd200 = load('mat/noise_PD200_1_10uF.mat', 't', 'Vout', 'notes');
la75  = load('mat/noise_la75_10uF.mat', 't', 'Vout', 'notes');
#+end_src

#+begin_src matlab
pd200.Vout = pd200.Vout/pd200.notes.pre_amp.gain;
la75.Vout  = la75.Vout/la75.notes.pre_amp.gain;
#+end_src

#+begin_src matlab
figure;
hold on;
plot(pd200.t, 1e3*pd200.Vout)
plot(la75.t, 1e3*la75.Vout)
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
% ylim([-3, 3]);
#+end_src

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

% Hanning window
win = hanning(ceil(0.5/Ts));

[pxx, f] = pwelch(pd200.Vout, win, [], [], Fs);
pd200.pxx = pxx;
pd200.f = f;

[pxx, f] = pwelch(la75.Vout, win, [], [], Fs);
la75.pxx = pxx;
la75.f = f;
#+end_src

#+begin_src matlab :exports none
colors = get(gca,'colororder');

figure;
hold on;
plot(pd200.f, sqrt(pd200.pxx), 'DisplayName', 'PD200');
plot(la75.f,  sqrt(la75.pxx),  'DisplayName', 'LA75');
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

* Conclusion
<<sec: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]   |        -        |