#+end_export
* Introduction
The goal of this test bench is to characterize the Voltage amplifier [[https://www.piezodrive.com/drivers/pd200-60-watt-voltage-amplifier/][PD200]] from PiezoDrive.
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
| | |
| | *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
| | | |
| *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 $G_p(s)$ 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 output noise $n$.
This noise is described by its Power Spectral Density.
The objective is therefore to determine the transfer function $G_p(s)$ from the input voltage to the output voltage as well as the Power Spectral Density $S_n(\omega)$ of the amplifier output noise.
As both $G_p$ and $S_n$ depends on the load capacitance, they should be measured when loading the amplifier with a $10\,\mu F$ capacitor.
#+begin_src latex :file pd200-model-schematic.pdf
\begin{tikzpicture}
\node[block] (G) at (0,0){$G_p(s)$};
\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
** Introduction :ignore:
- Section [[sec:noise_setup]]
- Section [[sec:noise_model]]
- Section [[sec:noise_quantization]]
- Section [[sec:noise_preamp]]
- Section [[sec:noise_pd200]]
- Section [[sec:noise_dac]]
- Section [[sec:noise_full_measurement]]
- Section [[sec:noise_ssi2v]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no
addpath('./matlab/mat/');
addpath('./matlab/');
#+end_src
#+begin_src matlab :eval no
addpath('./mat/');
#+end_src
** 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.
** Model of the setup
<>
As shown in Figure [[fig:noise_meas_procedure]], there are 4 equipment involved in the measurement:
- a Digital to Analog Convert (DAC)
- the Voltage amplifier to be measured with a gain of 20 (PD200)
- a low noise voltage amplifier with a variable gain and integrated low pass filters and high pass filters
- an Analog to Digital Converter (ADC)
Each of these equipment has some noise:
- $q_{da}$: quantization noise of the DAC
- $n_{da}$: output noise of the DAC
- $n_p$: output noise of the PD200 (what we wish to characterize)
- $n_a$: input noise of the pre amplifier
- $q_{ad}$: quantization noise of the ADC
#+begin_src latex :file noise_meas_procedure.pdf
\begin{tikzpicture}
% DAC
\node[DAC] (DAC) at (0,0) {DAC};
\node[addb, right=0.4 of DAC] (addqda){};
\node[addb, right=0.4 of addqda] (addnda){};
% PD200
\node[block, right=1.2 of addnda] (Gp){$G_p(s)$};
\node[addb, right=0.4 of Gp] (addnp){};
% Pre Amp
\node[addb, right=1.2 of addnp] (addna){};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
% \draw[->] (const.east) -- node[sloped]{$/$} (DAC.west);
\draw[<-] (DAC.west) -- node[sloped]{$/$} ++(-1.0, 0);
\draw[->] (DAC.east) -- (addqda.west);
\draw[->] (addqda.east) -- (addnda.west);
\draw[->] (addnda.east) -- (Gp.west);
\draw[->] (Gp.east) -- (addnp.west);
\draw[->] (addnp.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addnda.north) -- ++(0, 0.6) node[below right](nda){$n_{da}$};
\draw[<-] (addqda.north) -- ++(0, 0.6) node[below right](qda){$q_{da}$};
\draw[<-] (addnp.north) -- ++(0, 0.6) node[below right](np){$n_{p}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (nda.north);
\coordinate[] (bot) at (Ga.south);
% DAC
\begin{scope}[on background layer]
\node[fit={(DAC.west|-bot) (addnda.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {DAC};
\end{scope}
% PD200
\begin{scope}[on background layer]
\node[fit={(Gp.west|-bot) (addnp.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {PD200};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_meas_procedure
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_meas_procedure.png]]
** Quantization Noise
<>
The quantization noise is something that can be predicted.
The Amplitude Spectral Density of the quantization noise of an ADC/DAC is equal to:
\begin{equation}
\Gamma_q(\omega) = \frac{q}{\sqrt{12 f_s}}
\end{equation}
with:
- $q = \frac{\Delta V}{2^n}$ the quantization in [V], which is the corresponding value in [V] of the least significant bit
- $\Delta V$ is the full range of the ADC in [V]
- $n$ is the number of bits
- $f_s$ is the sample frequency in [Hz]
#+begin_src matlab
adc = struct();
adc.Delta_V = 20; % [V]
adc.n = 16; % number of bits
adc.Fs = 20e3; % [Hz]
adc.Gamma_q = adc.Delta_V/2^adc.n/sqrt(12*adc.Fs); % [V/sqrt(Hz)]
#+end_src
The obtained Amplitude Spectral Density is src_matlab[:exports results :results value replace]{adc.Gamma_q} {{{results(=6.2294e-07=)}}} $V/\sqrt{Hz}$.
** Pre Amplifier noise measurement
<>
First, we wish to measure the noise of the pre-amplifier.
To do so, the input of the pre-amplifier is shunted such that there is 0V at its inputs.
Then, the gain of the amplifier is increase until the measured signal on the ADC is much larger than the quantization noise.
The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal $n$ is computed.
Finally, the Amplitude Spectral Density of $n_a$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_a}(\omega) \approx \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
This is true if the quantization noise $\Gamma_{q_{ad}}$ is negligible.
#+begin_src latex :file noise_measure_setup_preamp.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% Pre Amp
\node[addb, right=0.6 of const] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_preamp
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_preamp.png]]
#+begin_src matlab :exports none
% Load Data
preamp = load('mat/noise_preamp_5113.mat', 't', 'Vn', 'notes');
#+end_src
The gain of the low noise amplifier is set to src_matlab[:exports results :results value replace]{ans = preamp.notes.pre_amp.gain} {{{results(=50000=)}}}.
#+begin_src matlab :exports none
% Compute the equivalent voltage at the input of the amplifier
preamp.Vn = preamp.Vn/preamp.notes.pre_amp.gain;
preamp.Vn = preamp.Vn - mean(preamp.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (preamp.t(end) - preamp.t(1))/(length(preamp.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab
% Hanning window
win = hanning(ceil(0.5/Ts));
% Power Spectral Density
[pxx, f] = pwelch(preamp.Vn, win, [], [], Fs);
% Save the results inside the struct
preamp.pxx = pxx;
preamp.f = f;
#+end_src
The obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier is shown in Figure [[fig:asd_preamp]].
The obtained noise amplitude is very closed to the one specified in the documentation of $4nV/\sqrt{Hz}$ at 1kHZ.
#+begin_src matlab :exports none
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./preamp.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'northeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_preamp.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:asd_preamp
#+caption: Obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier
#+RESULTS:
[[file:figs/asd_preamp.png]]
** PD200 noise measurement
<>
The input of the PD200 amplifier is shunted such that there is 0V between its inputs.
Then the gain of the pre-amplifier is increased in order to measure a signal much larger than the quantization noise of the ADC.
We compute the Amplitude Spectral Density of the measured signal $\Gamma_n(\omega)$.
The Amplitude Spectral Density of $n_p$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_p}(\omega) = \frac{\Gamma_n(\omega)}{|G_a(\omega)|}
\end{equation}
And we verify that this is indeed the noise of the PD200 and not the noise of the pre-amplifier by checking that:
\begin{equation}
\Gamma_{n_p} \ll \Gamma_{n_a}
\end{equation}
#+begin_src latex :file noise_measure_setup_pd200.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% PD200
\node[block, right=0.6 of const] (Gp){$G_p(s)$};
\node[addb, right=0.4 of Gp] (addnp){};
% Pre Amp
\node[addb, right=1.2 of addnp] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- (Gp.west);
\draw[->] (Gp.east) -- (addnp.west);
\draw[->] (addnp.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0) node[above left]{$n$};
\draw[<-] (addnp.north) -- ++(0, 0.6) node[below right](np){$n_{p}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% PD200
\begin{scope}[on background layer]
\node[fit={(Gp.west|-bot) (addnp.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {PD200};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_pd200
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_pd200.png]]
#+begin_src matlab :exports none
%% Load all the measurements
pd200w = {};
for i = 1:7
pd200w(i) = {load(['mat/noise_PD200_' num2str(i) '_3uF_warmup.mat'], 't', 'Vn', 'notes')};
end
#+end_src
#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
pd200w{i}.Vn = pd200w{i}.Vn/pd200w{i}.notes.pre_amp.gain;
end
#+end_src
The measured low frequency noise $n_p$ of one of the amplifiers is shown in Figure [[fig:pd200_noise_time_lpf]].
It is very similar to the one specified in the datasheet in Figure [[fig:pd200_expected_noise]].
#+begin_src matlab :exports none
% Compute the low frequency noise
G_lpf = 1/(1 + s/2/pi/20);
t_max = 40;
figure;
hold on;
plot(pd200w{1}.t(1:t_max/Ts), lsim(G_lpf, 1e3*pd200w{1}.Vn(1:t_max/Ts), pd200w{1}.t(1:t_max/Ts)))
hold off;
xlabel('Time [s]');
ylabel('Voltage [mV]');
ylim([-3, 3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_noise_time_lpf.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:pd200_noise_time_lpf
#+caption: Measured low frequency noise of the PD200 from 0.01Hz to 20Hz
#+RESULTS:
[[file:figs/pd200_noise_time_lpf.png]]
The obtained RMS and peak to peak values of the measured noises are shown in Table [[tab:rms_pkp_noise]].
#+begin_src matlab :exports none
%% Compute the RMS and Peak to Peak noise for the low frequency noise
Vn_rms = zeros(7,1); % RMS value [uV rms]
Vn_pkp = zeros(7,1); % Peak to Peak Value in 20Hz bandwidth [mV]
for i = 1:7
Vn_rms(i) = 1e6*rms(pd200w{i}.Vn);
Vn_lpf = lsim(1/(1 + s/2/pi/20), pd200w{i}.Vn, pd200w{i}.t);
Vn_pkp(i) = 1e3*(max(Vn_lpf)-min(Vn_lpf));
end
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([[714; Vn_rms], [4.3; Vn_pkp]], {'Specification [10uF]', 'PD200_1', 'PD200_2', 'PD200_3', 'PD200_4', 'PD200_5', 'PD200_6', 'PD200_7'}, {'*RMS [uV]*', '*Peak to Peak [mV]*'}, ' %.1f ');
#+end_src
#+name: tab:rms_pkp_noise
#+caption: RMS and Peak to Peak measured low frequency noise (0.01Hz to 20Hz)
#+attr_latex: :environment tabularx :width \linewidth :align lXX
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
| | *RMS [uV]* | *Peak to Peak [mV]* |
|----------------------+------------+---------------------|
| Specification [10uF] | 714.0 | 4.3 |
| PD200_1 | 565.1 | 3.7 |
| PD200_2 | 767.6 | 3.5 |
| PD200_3 | 479.9 | 3.0 |
| PD200_4 | 615.7 | 3.5 |
| PD200_5 | 651.0 | 2.4 |
| PD200_6 | 473.2 | 2.7 |
| PD200_7 | 423.1 | 2.3 |
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200w{1}.t(end) - pd200w{1}.t(1))/(length(pd200w{1}.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab :exports none
win = hanning(ceil(0.5/Ts));
for i = 1:7
[pxx, f] = pwelch(pd200w{i}.Vn, win, [], [], Fs);
pd200w{i}.f = f;
pd200w{i}.pxx = pxx;
end
#+end_src
The Amplitude Spectral Density of the measured noise is now computed and shown in Figure [[fig:asd_noise_3uF_warmup]].
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot(pd200w{1}.f, sqrt(pd200w{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./pd200w{1}.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_3uF_warmup.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_3uF_warmup
#+caption: Amplitude Spectral Density of the measured noise
#+RESULTS:
[[file:figs/asd_noise_3uF_warmup.png]]
** DAC noise measurement
<>
In order not to have any quantization noise, we impose the DAC to output a zero voltage.
The gain of the low noise amplifier is adjusted to
The Amplitude Spectral Density $\Gamma_n(\omega)$ of the measured signal is computed.
The Amplitude Spectral Density of $n_{da}$ can be computed taking into account the gain of the pre-amplifier:
\begin{equation}
\Gamma_{n_{da}}(\omega) = \frac{\Gamma_m(\omega)}{|G_a(\omega)|}
\end{equation}
And it is verify that the Amplitude Spectral Density of $n_{da}$ is much larger than the one of $n_a$:
\begin{equation}
\Gamma_{n_{da}} \gg \Gamma_{n_a}
\end{equation}
#+begin_src latex :file noise_measure_setup_dac.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
% DAC
\node[DAC, right=0.6 of const] (DAC) {DAC};
\node[addb, right=0.4 of DAC] (addnda){};
% Pre Amp
\node[addb, right=1.2 of addnda] (addna) {};
\node[block, right=0.4 of addna] (Ga) {$G_a(s)$};
% ADC
\node[addb, right=1.2 of Ga] (addqad){};
\node[ADC, right=0.4 of addqad] (ADC) {ADC};
\draw[->] (const.east) -- node[sloped]{$/$} (DAC.west);
\draw[->] (DAC.east) -- (addnda.west);
\draw[->] (addnda.east) -- (addna.west);
\draw[->] (addna.east) -- (Ga.west);
\draw[->] (Ga.east) -- (addqad.west);
\draw[->] (addqad.east) -- (ADC.west);
\draw[->] (ADC.east) -- node[sloped]{$/$} ++(1.0, 0);
\draw[<-] (addnda.north) -- ++(0, 0.6) node[below right](nda){$n_{da}$};
\draw[<-] (addna.north) -- ++(0, 0.6) node[below right](na){$n_{a}$};
\draw[<-] (addqad.north) -- ++(0, 0.6) node[below right](qad){$q_{ad}$};
\coordinate[] (top) at (na.north);
\coordinate[] (bot) at (Ga.south);
% DAC
\begin{scope}[on background layer]
\node[fit={(DAC.west|-bot) (addnda.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {DAC};
\end{scope}
% 5113
\begin{scope}[on background layer]
\node[fit={(addna.west|-bot) (Ga.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Pre Amp};
\end{scope}
% ADC
\begin{scope}[on background layer]
\node[fit={(addqad.west|-bot) (ADC.east|-top)}, inner sep=10pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {ADC};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:noise_measure_setup_dac
#+caption: Sources of noise in the experimental setup
#+RESULTS:
[[file:figs/noise_measure_setup_dac.png]]
#+begin_src matlab :exports none
dac = load('mat/noise_preamp_5113_dac.mat', 't', 'Vn', 'notes');
#+end_src
#+begin_src matlab :exports none
dac.Vn = dac.Vn/dac.notes.pre_amp.gain;
dac.Vn = dac.Vn - mean(dac.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (dac.t(end) - dac.t(1))/(length(dac.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab :exports none
win = hanning(ceil(0.5/Ts));
[pxx, f] = pwelch(dac.Vn, win, [], [], Fs);
dac.pxx = pxx;
dac.f = f;
#+end_src
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
set(gca,'ColorOrderIndex',3)
plot(dac.f, sqrt(dac.pxx), 'DisplayName', '$\Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./dac.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_dac.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_dac
#+caption:
#+RESULTS:
[[file:figs/asd_noise_dac.png]]
** Total noise measurement
<>
Let's now analyze the measurement of the setup in Figure [[fig:noise_meas_procedure]].
#+begin_src matlab :exports none
%% Load all the measurements
pd200dac = {};
for i = 1:7
pd200dac(i) = {load(['mat/noise_PD200_' num2str(i) '_3uF_DAC.mat'], 't', 'Vn', 'notes')};
end
#+end_src
#+begin_src matlab :exports none
%% Take into account the pre-amplifier gain
for i = 1:7
pd200dac{i}.Vn = pd200dac{i}.Vn/pd200dac{i}.notes.pre_amp.gain;
pd200dac{i}.Vn = pd200dac{i}.Vn - mean(pd200dac{i}.Vn);
end
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (pd200dac{1}.t(end) - pd200dac{1}.t(1))/(length(pd200dac{1}.t) - 1);
Fs = 1/Ts;
#+end_src
The PSD of the measured noise is computed and the ASD is shown in Figure [[fig:asd_noise_tot]].
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
for i = 1:7
[pxx, f] = pwelch(pd200dac{i}.Vn, win, [], [], Fs);
pd200dac{i}.f = f;
pd200dac{i}.pxx = pxx;
end
#+end_src
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
plot(pd200w{2}.f, sqrt(pd200w{2}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}$');
for i = 2:7
plot(pd200w{i}.f, sqrt(pd200w{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',3)
plot(dac.f, 20*sqrt(dac.pxx), 'DisplayName', '$|G_p| \cdot \Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./dac.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
plot(pd200dac{2}.f, sqrt(pd200dac{2}.pxx), 'color', [colors(4, :), 0.5], 'DisplayName', '$\Gamma_{tot}$');
for i = 2:7
plot(pd200dac{i}.f, sqrt(pd200dac{i}.pxx), 'color', [colors(4, :), 0.5], 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_noise_tot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_noise_tot
#+caption: Amplitude Spectral Density of the measured noise and of the individual sources of noise
#+RESULTS:
[[file:figs/asd_noise_tot.png]]
#+begin_important
The output noise of the PD200 amplifier is limited by the noise of the DAC.
Having a DAC with lower noise could lower the output noise of the PD200.
SSI2V DACs will be used to verify that.
#+end_important
** 20bits DAC noise measurement
<>
Let's now measure the noise of another DAC called the "SSI2V" ([[file:doc/\[SSI2V\]Datasheet.pdf][doc]]).
It is a 20bits DAC with an output of +/-10.48 V and a very low noise.
The measurement setup is the same as the one in Figure [[fig:noise_measure_setup_dac]].
#+begin_src matlab :exports none
ssi2v = load('mat/noise_preamp_5113_SSI2V.mat', 't', 'Vn', 'notes');
#+end_src
#+begin_src matlab :exports none
ssi2v.Vn = ssi2v.Vn/ssi2v.notes.pre_amp.gain;
ssi2v.Vn = ssi2v.Vn - mean(ssi2v.Vn);
#+end_src
#+begin_src matlab :exports none
% Sampling time / frequency
Ts = (ssi2v.t(end) - ssi2v.t(1))/(length(ssi2v.t) - 1);
Fs = 1/Ts;
#+end_src
#+begin_src matlab
win = hanning(ceil(0.5/Ts));
[pxx, f] = pwelch(ssi2v.Vn, win, [], [], Fs);
ssi2v.pxx = pxx;
ssi2v.f = f;
#+end_src
The obtained noise of the SSI2V DAC is shown in Figure [[fig:asd_ssi2v_noise]] and compared with the noise of the 16bits DAC.
It is shown to be much smaller (~1 order of magnitude).
#+begin_src matlab :exports none
colors = get(gca,'colororder');
figure;
hold on;
plot(preamp.f, sqrt(preamp.pxx), 'DisplayName', '$\Gamma_{n_a}$');
set(gca,'ColorOrderIndex',3)
plot(dac.f, sqrt(dac.pxx), 'DisplayName', '$\Gamma_{n_{da}}$');
plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./ssi2v.notes.pre_amp.gain, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_a|$');
set(gca,'ColorOrderIndex',5)
plot(ssi2v.f, sqrt(ssi2v.pxx), 'DisplayName', '$\Gamma_{n_{SSI2V}}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
legend('location', 'southeast');
xlim([1, Fs/2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/asd_ssi2v_noise.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:asd_ssi2v_noise
#+caption: Amplitude Spectral Density of the SSI2V DAC's noise
#+RESULTS:
[[file:figs/asd_ssi2v_noise.png]]
#+begin_important
Using the SSI2V as the DAC with the PD200 should give much better noise output than using the 16bits DAC.
The limiting factor should then be the noise of the PD200 itself.
#+end_important
** Tests :noexport:
*** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+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
* 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]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+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
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:
\[ \omega_{\text{max}} = \frac{1}{20 C V_{in}} I_{out,\text{max}} \]
$\omega_max$ 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]]
** Obtained Transfer Functions
Several identifications using sweep sin were performed with input voltage amplitude ranging from 0.1V to 4V.
#+begin_src matlab :exports none
%% Load all the measurements
Vin_ampl = {'0_1', '0_5', '1', '2', '4'};
pd200 = {};
for i = 1:length(Vin_ampl)
pd200(i) = {load(['tf_pd200_7_' Vin_ampl{i} 'V.mat'], 't', 'Vin', 'Vout', 'notes')};
end
#+end_src
#+begin_src matlab :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
% Hannning Windows
win = hanning(ceil(0.5*Fs));
% Compute all the transfer functions
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 frequency response functions are shown in Figure [[fig:pd200_tf_voltage]].
As the input voltage increases, the voltage drop is increasing.
#+begin_src matlab :exports none
figure;
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
for i = 1:length(pd200)
plot(pd200{i}.f, abs(pd200{i}.tf), 'DisplayName', sprintf('$V_{in} = %.1f [V]$', pd200{i}.notes.pd200.Vin))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([19, 21]);
legend('location', 'northeast');
ax2 = nexttile;
hold on;
for i = 1:length(pd200)
plot(pd200{i}.f, 180/pi*angle(pd200{i}.tf))
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:5:360);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
ylim([-15, 5]);
linkaxes([ax1,ax2],'x');
xlim([10, 5e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/pd200_tf_voltage.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:pd200_tf_voltage
#+caption: Transfer function for the PD200 amplitude between $V_{in}$ and $V_{out}$ for multiple voltage amplitudes
#+RESULTS:
[[file:figs/pd200_tf_voltage.png]]
The small signal transfer function of the amplifier can be approximated by a first order low pass filter.
#+begin_src matlab
Gp = 19.95/(1 + s/2/pi/35e3);
#+end_src
The comparison from the model and measurements are shown in Figure [[fig:tf_pd200_model]].
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
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
plot(freqs, abs(squeeze(freqresp(Gp, freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([19, 21]);
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
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp, freqs, 'Hz'))), 'k--', 'DisplayName', '$G_p(j\omega)$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-5:1:5);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
ylim([-3, 1]);
legend('location', 'southwest');
linkaxes([ax1,ax2],'x');
xlim([10, 1e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/tf_pd200_model.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:tf_pd200_model
#+caption: Comparison of the model transfer function and the measured frequency response function
#+RESULTS:
[[file:figs/tf_pd200_model.png]]
* 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
| | | | |
| *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] | - |