Remove unused sections
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@ -1915,135 +1915,6 @@ exportFig('figs/detail_instrumentation_pd200_tf.pdf', 'width', 'wide', 'height',
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#+RESULTS:
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[[file:figs/detail_instrumentation_pd200_tf.png]]
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**** Output Impedance :noexport:
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To confirms the model of the voltage amplifiers shown in Figure ref:fig:detail_instrumentation_amp_output_impedance (page pageref:fig:detail_instrumentation_amp_output_impedance), the output impedance $R_o$ of the PD200 was measured.
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A DAC with a constant output voltage (here 0.1V) is connected to the input of the PD200 amplifier.
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Then, the output voltage of the PD200 amplifier is measured in two conditions:
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- $V$ when the output is not connected to any load
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- $V_p$ when a load $R_{\text{load}} = 10\,\Omega$ is connected at the output of the amplifier
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The load and the output impedance form a voltage divider, and thus:
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\[ V_p = \frac{R_{\text{load}}}{R_{\text{load}} + R_o} V \]
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From the two values of voltage, the output impedance of the amplifier can be estimated from eqref:eq:detail_instrumentation_output_impedance_amp.
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\begin{equation}\label{eq:detail_instrumentation_output_impedance_amp}
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R_{o} = R_{\text{load}} \frac{V - V_p}{V}
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\end{equation}
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From this measurement, all the PD200 amplifiers are found to have an output impedance $R_{\text{out}} \approx 1\,\Omega$.
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With the capacitive load $C_p = 8.8\,\mu F$, the output resistor of the amplifier forms a low pass filter with a corner frequency equal to $f_0 = \frac{1}{R_{\text{out}} C_p}$.
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We get a corner frequency around $17\,\text{kHz}$ which is not far from the specified $7.4\,\text{kHz}$.
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#+begin_src matlab
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% Estimate output impedance of PD200 amplifiers
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R_load = 10; % Load [Ohm]
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% V Vp
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meas = [1.988, 1.794; % PD200 - 1
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1.990, 1.789; % PD200 - 2
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1.982, 1.795; % PD200 - 3
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1.984, 1.789; % PD200 - 4
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1.998, 1.810; % PD200 - 5
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1.984, 1.799];% PD200 - 6
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R_out = R_load * (meas(:,1) - meas(:,2))./meas(:,2); % Output impedance [Ohm]
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% Estimate corner frequency of the high pass filter
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Cp = 8.8e-6; % Capacitive load of the two piezoelectric actuators
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f0 = 1./(2*pi*R_out*Cp)
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#+end_src
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** Noise of the full setup with 16bits DAC :noexport:
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- [ ] Maybe not so useful
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Let's now measure the noise of the full setup in Figure ref:fig:noise_meas_procedure_bis and analyze the results.
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#+name: fig:noise_meas_procedure_bis
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#+caption: Sources of noise in the experimental setup
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#+RESULTS:
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[[file:figs/noise_meas_procedure.png]]
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#+begin_src matlab :exports none
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%% Full measurement chain from DAC to ADC
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pd200dac = {};
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for i = 1:7
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pd200dac(i) = {load(['mat/noise_PD200_' num2str(i) '_10uF_DAC.mat'], 't', 'Vout', 'notes')};
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end
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#+end_src
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#+begin_src matlab :exports none
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% Take into account the pre-amplifier gain
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for i = 1:7
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pd200dac{i}.Vout = pd200dac{i}.Vout/pd200dac{i}.notes.pre_amp.gain/20;
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pd200dac{i}.Vout = pd200dac{i}.Vout - mean(pd200dac{i}.Vout);
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end
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#+end_src
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#+begin_src matlab :exports none
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% Sampling time / frequency
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Ts = (pd200dac{1}.t(end) - pd200dac{1}.t(1))/(length(pd200dac{1}.t) - 1);
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Fs = 1/Ts;
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#+end_src
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The Amplitude Spectral Density of the measured noise is computed and the shown in Figure ref:fig:asd_noise_tot.
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We can very well see that to total measured noise is the sum of the DAC noise and the PD200 noise.
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#+begin_src matlab :exports none
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% Compute the PSD of the measured noise
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win = hanning(ceil(2/Ts));
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for i = 1:7
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[pxx, f] = pwelch(pd200dac{i}.Vout, win, [], [], Fs);
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pd200dac{i}.f = f;
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pd200dac{i}.pxx = pxx;
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end
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#+end_src
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#+begin_src matlab :exports none
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colors = get(gca,'colororder');
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figure;
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hold on;
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plot(egg.f, sqrt(egg.pxx)/20, 'DisplayName', '$\Gamma_{n_a}$');
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plot(pd200{1}.f, sqrt(pd200{1}.pxx), 'color', [colors(2, :), 0.5], 'DisplayName', '$\Gamma_{n_p}/|G_p|$');
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for i = 2:7
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plot(pd200{i}.f, sqrt(pd200{i}.pxx), 'color', [colors(2, :), 0.5], 'HandleVisibility', 'off');
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end
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set(gca,'ColorOrderIndex',3)
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plot(dac.f, sqrt(dac.pxx), 'DisplayName', '$|G_p| \cdot \Gamma_{n_{da}}$');
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plot([1 Fs/2], [adc.Gamma_q, adc.Gamma_q]./dac.notes.pre_amp.gain/20, 'k--', 'DisplayName', '$\Gamma_{q_{ad}}/|G_p G_a|$');
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plot(pd200dac{1}.f, sqrt(pd200dac{1}.pxx), 'color', [colors(4, :), 0.5], 'DisplayName', '$\Gamma_{tot}$');
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for i = 2:7
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plot(pd200dac{i}.f, sqrt(pd200dac{i}.pxx), 'color', [colors(4, :), 0.5], 'HandleVisibility', 'off');
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('ASD [$V/\sqrt{Hz}$]');
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legend('location', 'southeast');
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xlim([1, Fs/2]); ylim([1e-11, 1e-4]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/asd_noise_tot.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:asd_noise_tot
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#+caption: Amplitude Spectral Density of the measured noise and of the individual sources of noise
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#+RESULTS:
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[[file:figs/asd_noise_tot.png]]
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#+begin_important
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The input noise of the PD200 amplifier is limited by the output voltage noise of the DAC.
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Having a DAC with lower output voltage noise could lower the overall noise of the setup.
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SSI2V 20bits DACs are used in the next section to verify that.
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#+end_important
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** Linear Encoders
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To measure the noise $n$ of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured.
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@ -2103,7 +1974,7 @@ exportFig('figs/detail_instrumentation_vionic_asd.pdf', 'width', 'half', 'height
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#+attr_latex: :options [b]{0.48\linewidth}
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#+begin_minipage
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#+name: fig:detail_instrumentation_vionic_asd
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#+caption: Measured Amplitude Spectral Density of the encoder position noise
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#+caption: Measured Amplitude Spectral Density of the encoder noise
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#+attr_latex: :width 0.95\linewidth :float nil
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[[file:figs/detail_instrumentation_vionic_asd.png]]
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#+end_minipage
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Binary file not shown.
@ -1,4 +1,4 @@
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% Created 2025-03-15 Sat 10:44
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% Created 2025-03-17 Mon 17:41
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% Intended LaTeX compiler: pdflatex
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\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
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@ -43,31 +43,29 @@ The received instrumentation are characterized in Section \ref{sec:detail_instru
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\end{figure}
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\chapter{Dynamic Error Budgeting}
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\label{sec:org5318a4e}
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\label{sec:detail_instrumentation_dynamic_error_budgeting}
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\textbf{Goal}:
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\begin{itemize}
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\item Write specifications regarding the maximum noise of instrumentation (ADC, DAC and voltage amplifier)
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\item Obtain specifications regarding the maximum noise of instrumentation (ADC, DAC and voltage amplifier)
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such that it induces acceptable vibrations levels
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\end{itemize}
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\textbf{Procedure}:
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\begin{itemize}
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\item Get closed-loop transfer functions from disturbance sources (noise of ADC, DAC and amplifier noise) to positioning error
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This is done using the multi-body model, with 2DoF APA model (having voltage input and outputs)
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\item Get closed-loop transfer functions from noise sources (noise of ADC, DAC and amplifier noise) to positioning error.
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This is done using the multi-body model, with 2DoF APA model (having voltage input and outputs).
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\item Focus is made on the vertical direction, as it is the direction with the most stringent requirements.
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If horizontal directions are considered, requirements are just less stringent than for the vertical direction.
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\item Deduce the maximum acceptable ASD of the noise sources
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\item Deduce the maximum acceptable ASD of the noise sources.
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\end{itemize}
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As the voltage amplifier gain will impact how the DAC noise will be amplified, some assumption are made:
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As the voltage amplifier gain will impact how the DAC noise is amplified, some assumption are made:
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\begin{itemize}
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\item we want to apply -20 to 150V to the stacks
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\item Typical ADC are +/-10V
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\item Assumption of voltage amplifier with gain 20
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\end{itemize}
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\section{Closed-Loop Sensitivity to Instrumentation Disturbances}
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\label{sec:org0921bec}
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The following noise sources are considered (Figure \ref{fig:detail_instrumentation_plant}):
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\begin{itemize}
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@ -89,20 +87,20 @@ The lateral error was also considered, but the specifications are less stringent
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\end{figure}
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\section{Estimation of maximum instrumentation noise}
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\label{sec:orgd3ab02f}
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From previous analysis, we know how the noise of the instrumentation will affect the vertical error of the sample.
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From previous analysis, we know how the noise of the instrumentation will affect the vertical error of the sample as a function of frequency.
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Now, we want to determine specifications for each instrumentation such that the effect on the vertical error of the sample is within specifications.
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Most stringent requirement:
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\begin{itemize}
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\item vertical vibrations less than the smallest expected beam size of 100nm
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\item Vertical vibrations less than the smallest expected beam size of 100nm
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\item This corresponds to a maximum allowed vibration of 15nm RMS
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\end{itemize}
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Assumption on the noise:
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\begin{itemize}
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\item uncorrelated, which is reasonable.
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\item DAC, DAC and amplifier noise are uncorrelated, which is reasonable.
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Noise corresponding each strut are each uncorrelated.
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This means that the PSD of the different noise sources adds up.
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\end{itemize}
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@ -112,10 +110,7 @@ Use of system symmetry to simplify the analysis:
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Therefore only one strut can be considered for this analysis, and the total effect of the six struts is just six times the effect of one strut (in terms of power, but in terms of RMS value it's only sqrt(6)=2.5)
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\end{itemize}
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In order to have specifications for the noise of the instrumentation, assumptions:
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\begin{itemize}
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\item flat noise, which is quite typical
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\end{itemize}
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In order to have specifications in terms of noise spectral density of each instrumentation, a white noise is assumed, which is quite typical.
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The noise specification is computed such that if all the instrumentation have this maximum noise, the specification in terms of vertical error is still respected.
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This is a pessimistic choice, but it gives a rough idea of the specifications.
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@ -149,30 +144,24 @@ RMS Noise & \(0.8\,mV\,\text{RMS}\) & \(1\,mV\,\text{RMS}\) & \(20\,mV\,\text{RM
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If the Amplitude Spectral Density of the noise of the ADC, DAC and voltage amplifiers are all below the specified maximum noises, then the induced vertical error will be below 15nmRMS.
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Such specification will guide the choice of instrumentation in Section \ref{sec:detail_instrumentation_choice}.
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\chapter{Choice of Instrumentation}
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\label{sec:orge67db89}
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\label{sec:detail_instrumentation_choice}
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In previous section: noise characteristics.
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In this section, other characteristics (range, bandwidth, etc\ldots{})
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ADC, DAC, Voltage amplifier, Encoder
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\textbf{Model of each instrument} (transfer function + noise source).
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In this section, also tell which instrumentation has been bought, and different options.
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Based on:
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\begin{itemize}
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\item[{$\square$}] block diagram of the model of the amplifier
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\item noise specifications extracted from Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}
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\item other specifications (input/output range, bandwidth, etc\ldots{})
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\end{itemize}
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The most adequate ADC, DAC, Voltage amplifier, and relative positioning sensor are found.
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Different options that were considered are presented, and the choice of instrumentation is explained.
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\section{Piezoelectric Voltage Amplifier}
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\label{sec:org584716c}
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There are several characteristics of the piezoelectric voltage amplifiers that should be considered.
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To be able to use the full stroke of the piezoelectric actuator, the voltage output should be between -20 and 150V.
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It should accept an analog input voltage, preferably between -10 and 10V.
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It should accept an analog input voltage, preferably between -10 and 10V, as it is quite typical for DAC.
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\paragraph{Small signal Bandwidth and Output Impedance}
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\label{sec:orgf100305}
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There are two bandwidth that should be considered for a piezoelectric voltage amplifier: large signal bandwidth and small signal bandwidth.
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Large signal bandwidth are linked to the output current capacities of the amplifier and will be discussed next.
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@ -180,7 +169,7 @@ Large signal bandwidth are linked to the output current capacities of the amplif
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Small signal bandwidth of the voltage amplifier is very important for feedback applications as it can limit the bandwidth of the complete feedback system.
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A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure \ref{fig:detail_instrumentation_amp_output_impedance}.
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This model is only valid for small signals, but it well models the small signal bandwidth limitation \cite[, chap. 14]{fleming14_desig_model_contr_nanop_system}.
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This model is only valid for small signals, but it is useful to understand the small signal bandwidth limitation \cite[, chap. 14]{fleming14_desig_model_contr_nanop_system}.
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\(R_o\) corresponds to the output impedance of the amplifier.
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With the piezoelectric load that corresponds to a capacitance \(C_p\), it forms a first order low pass filter \eqref{eq:detail_instrumentation_amp_output_impedance}.
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@ -194,98 +183,93 @@ With the piezoelectric load that corresponds to a capacitance \(C_p\), it forms
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\caption{\label{fig:detail_instrumentation_amp_output_impedance}Electrical model of a voltage amplifier with output impedance \(R_0\) connected to a piezoelectric stack with capacitance \(C_p\)}
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\end{figure}
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Therefore, the small signal bandwidth is load dependent.
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As the capacitance load of the two piezoelectric stacks correspond to a capacitance \(C_p = 8.8\,\mu F\).
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If a small signal bandwidth of \(f_0 = \frac{\omega_0}{2\pi} = 5\,kHz\) is wanted, it corresponds to a maximum output impedance of \(R_0 = 3.6\,\Omega\).
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Therefore, the small signal bandwidth is load dependent (it decreases as the load capacitance increases).
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The capacitance load of the two piezoelectric stacks in the APA300ML correspond to a capacitance \(C_p = 8.8\,\mu F\).
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If a small signal bandwidth of \(f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}\) is wanted, the voltage amplifier output impedance should be smaller than \(R_0 = 3.6\,\Omega\).
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\paragraph{Large signal Bandwidth}
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\label{sec:orga060611}
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Large signal bandwidth are linked to the maximum output capabilities of the amplifiers in terms of amplitude as a function of frequency \cite{spengen16_high_voltag_amplif}.
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As the primary objective of the NASS is to stabilize the position and not to perform scans, this specification is not as important as the small signal bandwidth.
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However, let's take into account scanning capabilities.
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In the worst case: constant velocity scan (i.e. triangular reference signal) with a repetition rate of \(f_r = 100\,\text{Hz}\) and using full voltage capabilities of the piezoelectric actuator \(V_{pp} = 170\,V\).
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However, let's take into account scanning capabilities, and consider the worst case scenario of a constant velocity scan (i.e. triangular reference signal) with a repetition rate of \(f_r = 100\,\text{Hz}\) and using full voltage capabilities of the piezoelectric actuator \(V_{pp} = 170\,V\).
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There are two things to consider:
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\begin{itemize}
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\item Slew rate that should be above \(2 \cdot V_{pp} \cdot f_r = 34\,V/ms\)
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This specification is easily achieved
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This specification is easily achieved by commercial voltage amplifiers.
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\item Current output capabilities: as the capacitance impedance decreases the the inverse of the frequency, it can reach very low values at high frequency
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In order to reach high voltage at high frequency, the required current that the voltage amplifier needs to provide may reach very large values.
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\(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A\)
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\end{itemize}
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So ideally, a voltage amplifier capable of providing \(0.3\,A\) of current is wanted.
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\paragraph{Output voltage noise}
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\label{sec:orga1946e3}
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As discussed in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the output noise of the voltage amplifier should be smaller than \(20\,mV\,\text{RMS}\).
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As explained in \cite{spengen20_high_voltag_amplif}, the load capacitance of the piezoelectric stack filters the output noise of the amplifier (low pass filter of Figure \ref{fig:detail_instrumentation_amp_output_impedance}).
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Therefore, when comparing noise of different voltage amplifiers, it should be noted what capacitance of the load is considered (i.e. the low signal bandwidth considered).
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Therefore, when comparing noise indicated in the datasheet of different voltage amplifiers, it is important to check what is the considered capacitance of the load (i.e. the low signal bandwidth considered).
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Here, the output noise should be smaller than 20mVRMS for a load of 8.8uF.
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Here, the output noise should be smaller than 20mVRMS for a load of 8.8uF and a bandwidth larger than 5kHz.
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\paragraph{Choice of voltage amplifier}
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\label{sec:org3c7c558}
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The specifications as well as the amplifier characteristics as shown in the datasheet are summarized in Table \ref{tab:pd200_characteristics}.
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The most important characteristics are the (small signal) bandwidth > 5 [kHz] and the output voltage noise (< 20 [mV RMS]).
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The specifications are summarized in Table \ref{tab:detail_instrumentation_amp_choice}.
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The most important characteristics being the (small signal) bandwidth > 5 [kHz] and the output voltage noise (< 20 [mV RMS]).
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Several voltage amplifiers were considered, with datasheet information summarized in Table \ref{tab:detail_instrumentation_amp_choice}.
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\begin{itemize}
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\item Issue for the selection: manufacturers are not specifying the output noise as a function of frequency (i.e. the ASD of the noise), but only the RMS value (i.e. the integrated value over all frequency).
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It does not take into account the frequency dependency of the noise, that is very important to perform error budgets
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Also, the load the estimate the bandwidth and noise is often not mentioned (or no load for the bandwidth, and high capacitive load for the noise).
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\item Explain why the PD200 was selected:
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Also, the load used to estimate the bandwidth and noise is often not mentioned.
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Most of the time, the bandwidth is indicated with very little load and the noise with high load.
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It renders the comparison between different models more complex.
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\item The chosen model is the PD200 from PiezoDrive.
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\begin{itemize}
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\item fulfill the specification
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\item clear documentation, especially about noise and bandwidth (Figure \ref{fig:detail_instrumentation_pd200_specs})
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\item It fulfill the specification
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\item It has clear documentation, especially about noise and bandwidth
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\end{itemize}
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\end{itemize}
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\begin{table}[htbp]
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\caption{\label{tab:pd200_characteristics}Characteristics of the PD200 compared with the specifications}
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\caption{\label{tab:detail_instrumentation_amp_choice}Characteristics of the PD200 compared with the specifications}
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\centering
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\begin{tabularx}{\linewidth}{Xcccc}
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\toprule
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\textbf{Specification} & \textbf{PD200} & WMA-200 & LA75B & E-505\\
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& PiezoDrive & Falco & Cedrat & PI\\
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\midrule
|
||||
Input Voltage Range: \(\pm 10\,V\) & \(\pm 10\,V\) & \(\pm8.75\,V\) & \(-1/7.5\,V\) & \\
|
||||
Output Voltage Range: \(-20/150\,V\) & \(-50/150\,V\) & \(\pm 175\,V\) & \(-20/150\,V\) & -30/130\\
|
||||
Gain & 20 & 20 & 20 & 10\\
|
||||
Output Current \(> 50\,mA\) & \(900\,mA\) & \(150\,mA\) & \(360\,mA\) & \(215\,mA\)\\
|
||||
Gain \(>15\) & 20 & 20 & 20 & 10\\
|
||||
Output Current \(> 300\,mA\) & \(900\,mA\) & \(150\,mA\) & \(360\,mA\) & \(215\,mA\)\\
|
||||
Slew Rate \(> 34\,V/ms\) & \(150\,V/\mu s\) & \(80\,V/\mu s\) & n/a & n/a\\
|
||||
Output noise \(< 20\,mV\ \text{RMS}\) & \(0.7\,mV\,\text{RMS}\) & \(0.05\,mV\) & \(3.4\,mV\) & \(0.6\,mV\)\\
|
||||
(10uF load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & & \\
|
||||
Small Signal Bandwidth \(> 5\,kHz\) & \(6.4\,kHz\) & \(300\,Hz\) & \(30\,kHz\) & n/a\\
|
||||
(\(10\,\mu F\) load) & (\(10\,\mu F\) load) & \footnotemark & (unloaded) & \\
|
||||
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\)\textsuperscript{\ref{org28ed3ea}} & n/a & n/a\\
|
||||
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\)\textsuperscript{\ref{orgcbb55af}} & n/a & n/a\\
|
||||
\bottomrule
|
||||
\end{tabularx}
|
||||
\end{table}\footnotetext[1]{\label{org28ed3ea}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
|
||||
\end{table}\footnotetext[1]{\label{orgcbb55af}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
|
||||
|
||||
|
||||
\section{ADC and DAC}
|
||||
\label{sec:org80ca7dd}
|
||||
Analog to digital converters and digital to analog converters are very important to convert signals from the RT controller that only uses digital numbers to the physical plant, which is of course affected by analog signals.
|
||||
|
||||
\paragraph{Synchronicity and Jitter}
|
||||
\label{sec:orge597a56}
|
||||
|
||||
For control systems, it is very important that the inputs and outputs are sampled synchronously with the controller and with low jitter \cite{abramovitch22_pract_method_real_world_contr_system,abramovitch23_tutor_real_time_comput_issues_contr_system}.
|
||||
|
||||
Therefore, the ADC and DAC:
|
||||
\begin{itemize}
|
||||
\item Needs to be well interfaced with the Speedgoat
|
||||
\item Sampling frequency of 10kHz
|
||||
\item Synchronous with the Speedgoat
|
||||
\end{itemize}
|
||||
Therefore, the ADC and DAC needs to be well interfaced with the Speedgoat, and triggered synchronously with the computation of the control signals.
|
||||
For that reason, it was decided to first look for ADC and DAC sold by Speedgoat (the RT controller used).
|
||||
|
||||
\paragraph{Sampling Frequency, Bandwidth and delays}
|
||||
\label{sec:org41d5214}
|
||||
|
||||
Several requirements that may appear the same but are different:
|
||||
Several requirements may appear the same but are very different in nature:
|
||||
\begin{itemize}
|
||||
\item Sampling frequency: defines the interval between two sampled points, also determines the Nyquist frequency
|
||||
\item Bandwidth: defines the maximum frequency of a measured signal (typically specified as the -3dB point), usually limited by implemented anti-aliasing filters
|
||||
@ -295,35 +279,24 @@ Several requirements that may appear the same but are different:
|
||||
Sigma-Delta ADC can have extremely good noise characteristics, high bandwidth and sampling frequency but very poor latency.
|
||||
Typically, the latency can reach 20 times the sampling period \cite[, chapt. 8.4]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
|
||||
|
||||
Therefore, Sigma-Delta ADC are very well used for signal acquisition, but has limited use for real-time control where latency is critical.
|
||||
Therefore, Sigma-Delta ADC are very much used for signal acquisition applications, but has limited use for real-time control where latency is critical.
|
||||
Therefore, for real time control applications, SAR-ADC (Successive approximation ADCs) is still the mostly applied type because of its single sample latency.
|
||||
|
||||
\paragraph{ADC Noise}
|
||||
\label{sec:org323f222}
|
||||
|
||||
From the dynamical error budget in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}
|
||||
Measurement noise ASD should be bellow 11uV/sqrt(Hz), 0.8mV RMS
|
||||
|
||||
Linearity is not a concerned as it is not used for positioning but only for active damping
|
||||
So the accuracy of the measurement is not a concern
|
||||
|
||||
ADC are affected by various noise sources
|
||||
One of them is quantization noise, and is linked to the fact that input/output values can only take a finite number of values.
|
||||
Let's first find the number of bits such that the quantization noise is fulfilling the requirements.
|
||||
|
||||
Let's first suppose that the ADC is ideal and the only noise comes from the quantization error.
|
||||
Interestingly, the noise amplitude is uniformly distributed.
|
||||
|
||||
Let's note:
|
||||
\begin{itemize}
|
||||
\item \(q = \frac{\Delta V}{2^n}\) the quantization in [V], which is the corresponding value in [V] of the least significant bit
|
||||
\item \(\Delta V\) is the full range of the ADC in [V]
|
||||
\item \(n\) is the number of ADC's bits
|
||||
\item \(f_s\) is the sample frequency in [Hz]
|
||||
\end{itemize}
|
||||
Let's note \(q = \frac{\Delta V}{2^n}\) 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 ADC's bits and \(F_s\) is the sample frequency in [Hz].
|
||||
|
||||
The quantization noise can take a value between \(\pm q/2\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
|
||||
Since the integral of the probability density function is equal to one, its value will be \(1/q\) for \(-q/2 < e < q/2\) (Fig. \ref{fig:detail_instrumentation_adc_quantization}).
|
||||
Since the integral of the probability density function \(p(e)\) is equal to one, its value is \(1/q\) for \(-q/2 < e < q/2\) as illustrated in Figure \ref{fig:detail_instrumentation_adc_quantization}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -331,48 +304,44 @@ Since the integral of the probability density function is equal to one, its valu
|
||||
\caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC error \(e\)}
|
||||
\end{figure}
|
||||
|
||||
Now, we can calculate the time average power of the quantization noise as
|
||||
\begin{equation}
|
||||
The variance (or time average power) of the quantization noise is \eqref{eq:detail_instrumentation_quant_power}.
|
||||
|
||||
\begin{equation}\label{eq:detail_instrumentation_quant_power}
|
||||
P_q = \int_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
|
||||
\end{equation}
|
||||
|
||||
The other important parameter of a noise source is the power spectral density (PSD), which indicates how the noise power spreads in different frequency bands.
|
||||
To find the power spectral density, we need to calculate the Fourier transform of the autocorrelation function of the noise.
|
||||
|
||||
Assuming that the noise samples are not correlated with one another, we can approximate the autocorrelation function with a delta function in the time domain.
|
||||
Now, the goal is to compute the power spectral density of the quantization noise, which is by definition the Fourier transform of the autocorrelation function of the quantization noise.
|
||||
Assuming that the noise samples are not correlated with one another, the autocorrelation function can be approximated with a delta function in the time domain.
|
||||
Since the Fourier transform of a delta function is equal to one, the power spectral density will be frequency independent (i.e. white noise).
|
||||
Therefore, the quantization noise is white noise with total power equal to \(P_q = \frac{q^2}{12}\).
|
||||
Therefore, thanks the the Perseval's theorem, the quantization noise is a white noise with total power equal to \(P_q = \frac{q^2}{12}\).
|
||||
|
||||
Thus, the two-sided PSD (from \(\frac{-f_s}{2}\) to \(\frac{f_s}{2}\)), we should divide the noise power \(P_q\) by \(f_s\):
|
||||
Thus, the two-sided PSD (from \(\frac{-F_s}{2}\) to \(\frac{F_s}{2}\)), we should divide the noise power \(P_q\) by \(F_s\):
|
||||
\begin{equation}
|
||||
\int_{-f_s/2}^{f_s/2} \Gamma(f) d f = f_s \Gamma = \frac{q^2}{12}
|
||||
P_q = \int_{-F_s/2}^{F_s/2} \Gamma(f) d f = F_s \Gamma = \frac{q^2}{12}
|
||||
\end{equation}
|
||||
|
||||
Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\Gamma &= \frac{q^2}{12 f_s} \\
|
||||
&= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f_s} \quad \text{in} \quad \left[ \frac{V^2}{Hz} \right]
|
||||
\end{aligned}
|
||||
Finally, the Power Spectral Density of the quantization noise of an ADC is equal to \eqref{eq:detail_instrumentation_psd_quant_noise}.
|
||||
|
||||
\begin{equation}\label{eq:detail_instrumentation_psd_quant_noise}
|
||||
\Gamma_q = \frac{q^2}{12 F_s} = \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 F_s} \quad \text{in } \left[ \frac{V^2}{\text{Hz}} \right]
|
||||
\end{equation}
|
||||
|
||||
The minimum number of bits so that the quantization noise is above some define values can be computed using:
|
||||
\begin{equation}
|
||||
n_{\text{min}} = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 Fs} \Phi_{\text{max}}} \right)
|
||||
From a defined noise amplitude spectral density \(\Phi_{\text{max}}\), the minimum number of bits so that the quantization noise is below \(\Phi_{\text{max}}\) can be computed from \eqref{eq:detail_instrumentation_min_n}.
|
||||
|
||||
\begin{equation}\label{eq:detail_instrumentation_min_n}
|
||||
n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 Fs} \Phi_{\text{max}}} \right)
|
||||
\end{equation}
|
||||
|
||||
With a sampling frequency \(F_s = 10\,kHz\), a full range of \(\Delta V = 20\,V\) and a maximum allowed ASD \(\Phi_{\text{max}} = 11\,\mu V/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is easily satisfied.
|
||||
With a sampling frequency \(F_s = 10\,kHz\), a full range of \(\Delta V = 20\,V\) and a maximum allowed ASD \(\Phi_{\text{max}} = 11\,\mu V/\sqrt{Hz}\), the minimum number of bits is \(n_{\text{min}} = 12.4\), which is easily satisfied by commercial ADCs.
|
||||
|
||||
\paragraph{DAC Output voltage noise}
|
||||
\label{sec:org30caf00}
|
||||
|
||||
Similarly, the DAC output voltage noise ASD should be below \(14\,\mu V/\sqrt{Hz}\), 1mV RMS.
|
||||
This corresponds to a 13bits +/-10V DAC.
|
||||
Similarly to ADC, the DAC output voltage noise ASD should be below \(14\,\mu V/\sqrt{Hz}\), 1mV RMS.
|
||||
This corresponds to a 13bits +/-10V DAC, which is easily satisfied.
|
||||
|
||||
\paragraph{Choice of the ADC and DAC Board}
|
||||
\label{sec:org1df7a52}
|
||||
|
||||
Based on the above analysis, the choice of ADC and DAC is quite simple.
|
||||
Based on the above analysis, the choice of ADC and DAC is quite straightforward.
|
||||
|
||||
Integrated in Speedgoat for best synchronicity.
|
||||
Chosen model: IO131:
|
||||
@ -396,7 +365,6 @@ Noise is not specified, but as it has 16 bits resolution, it should be well belo
|
||||
It will be experimentally measured in Section \ref{sec:detail_instrumentation_characterization}.
|
||||
|
||||
\section{Relative Displacement Sensors}
|
||||
\label{sec:org2dd10f4}
|
||||
|
||||
Specifications:
|
||||
\begin{itemize}
|
||||
@ -471,6 +439,7 @@ The specifications are summarized in Table \ref{tab:detail_instrumentation_senso
|
||||
\toprule
|
||||
\textbf{Specification} & \textbf{Renishaw Vionic} & LION CPL190 & Cedrat ECP500\\
|
||||
\midrule
|
||||
Technology & Digital Encoder & Capacitive & Eddy Current\\
|
||||
Bandwidth \(> 5\,\text{kHz}\) & \(> 500\,\text{kHz}\) & 10kHz & 20kHz\\
|
||||
Noise \(< 6\,nm\,\text{RMS}\) & 1.6 nm rms & 4 nm rms & 15 nm rms\\
|
||||
Range \(> 100\,\mu m\) & Ruler length & 250 um & 500um\\
|
||||
@ -481,34 +450,27 @@ Digital Output & \(\times\) & & \\
|
||||
\end{table}
|
||||
|
||||
\chapter{Characterization of Instrumentation}
|
||||
\label{sec:org3c1dd2e}
|
||||
\label{sec:detail_instrumentation_characterization}
|
||||
All the instrumentation was then procured and tested individually to verify whether is fulfils the specifications or not.
|
||||
|
||||
\begin{itemize}
|
||||
\item[{$\square$}] Code couleur:
|
||||
\begin{center}
|
||||
\begin{tabular}{ll}
|
||||
Yellow & ADC\\
|
||||
Rouge & PD200\\
|
||||
Vert & Femto\\
|
||||
blue & DAC\\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\item[{$\square$}] Make sure that at some point I talk about twisted pairs etc.. Maybe use the nice schematic?
|
||||
\end{itemize}
|
||||
All the instrumentation was then procured and tested individually to verify whether it fulfils the specifications or not.
|
||||
\section{Analog to Digital Converters}
|
||||
\label{sec:orga04d2a2}
|
||||
Internally uses the AD7609 ADC from Analog Devices.
|
||||
200kSPS, 16 bits, +/-10V
|
||||
The ADC of the IO318 cards:
|
||||
\begin{itemize}
|
||||
\item have differential inputs
|
||||
\item internally uses the AD7609 ADC from Analog Devices.
|
||||
\item capable of 200kSPS, 16 bits, +/-10V
|
||||
\end{itemize}
|
||||
|
||||
\paragraph{Measured Noise}
|
||||
\label{sec:org485cf0d}
|
||||
The ADC noise of the IO131 was simply measured by short-circuiting its input with a 50 Ohm resistor.
|
||||
Results are shown in Figure \ref{fig:detail_instrumentation_adc_noise_measured}.
|
||||
The ADC noise is a white noise with an amplitude spectral density of \(5.6\,\mu V/\sqrt{Hz}\).
|
||||
|
||||
RMS value of 0.4mV.
|
||||
The measurement of the ADC noise was done by short-circuiting its input with a 50 Ohm resistor and recording the digital value at 10kHz.
|
||||
The amplitude spectral density of the recorded values are computed and shown in Figure \ref{fig:detail_instrumentation_adc_noise_measured}.
|
||||
The ADC noise is a white noise with an amplitude spectral density of \(5.6\,\mu V/\sqrt{Hz}\) (RMS value of 0.4mV), which fulfills the specifications.
|
||||
All ADC channels are measuring the same, so the noise of only one channel is here shown.
|
||||
|
||||
If required, it is possible to apply some oversampling to lower the obtained noise as explained in \cite{lab13_improv_adc}.
|
||||
To have additional \(w\) bits of resolution, the oversampling frequency \(f_{os}\) should be \(f_{os} = 4^w \cdot f_s\).
|
||||
As the ADC can work at 200kSPS, and the real time controller only runs at 10kSPS, an oversampling factor of 16 can be used to have approximately two more bits of resolution (i.e. reducing the noise by a factor 4).
|
||||
This works because the noise can be approximated by a white noise and the amplitude is larger than 1 LSB (0.3 mV) \cite{hauser91_princ_overs_conver}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -516,39 +478,35 @@ RMS value of 0.4mV.
|
||||
\caption{\label{fig:detail_instrumentation_adc_noise_measured}Measured ADC noise (IO318)}
|
||||
\end{figure}
|
||||
|
||||
If required, it is possible to apply some oversampling to lower the obtained noise \cite{lab13_improv_adc}.
|
||||
To have additional \(w\) bits of resolution, the oversampling frequency \(f_{os}\) should be:
|
||||
\begin{equation}
|
||||
f_{os} = 4^w \cdot f_s
|
||||
\end{equation}
|
||||
|
||||
As the ADC can work at 200kSPS, and we only need 10kSPS, we can have an oversampling factor of 16 and have two more bits of resolution (i.e. reducing the noise by a factor 4).
|
||||
This works because the noise can be approximated by a white noise and the amplitude is larger than 1 LSB (0.3 mV) \cite{hauser91_princ_overs_conver}.
|
||||
|
||||
|
||||
\paragraph{Reading of piezoelectric force sensor}
|
||||
\label{sec:org05565b5}
|
||||
|
||||
There are few other things to consider when measuring the voltage generated by a piezoelectric stack.
|
||||
|
||||
The setup is shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc_setup} where two stacks are used as actuator (in parallel) and one stack is used as sensor.
|
||||
The voltage amplifier used has a gain of 20 [V/V].
|
||||
To further verify that the ADC can effectively measure the voltage generated by a piezoelectric stack without issue, a test with the APA95ML was performed.
|
||||
The setup is shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc_setup} where two stacks are used as actuator (in parallel) and one stack is used as a sensor.
|
||||
The voltage amplifier used has a gain of 20.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/detail_instrumentation_force_sensor_adc_setup.png}
|
||||
\caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup}
|
||||
\caption{\label{fig:detail_instrumentation_force_sensor_adc_setup}Schematic of the setup to validate the use of the ADC for reading the force sensor volage}
|
||||
\end{figure}
|
||||
|
||||
Steps signals was generated using the DAC with an amplitude of 1V, and the ADC signal was recorded.
|
||||
The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure \ref{fig:detail_instrumentation_step_response_force_sensor}.
|
||||
The measured voltage shows an exponential decay which indicates that the charge across the capacitor formed by the stack is discharging into a resistor.
|
||||
This corresponds to an RC circuit with a time constant \(\tau = R_i C_p\).
|
||||
Therefore, first order high filter, with corner frequency \(1/\tau\).
|
||||
|
||||
The exponential curves are fitted and a value of \(\tau = 6.5\,s\) and an offset voltage of \(2.26\,V\) are found.
|
||||
Two things can be observed:
|
||||
\begin{itemize}
|
||||
\item an offset voltage of \(2.26\,V\)
|
||||
\item the measured voltage shows an exponential decay response to the step input
|
||||
\end{itemize}
|
||||
|
||||
With the capacitance being \(C_p = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows \(R_i = \frac{\tau}{C_p} = 1.5\,M\Omega\).
|
||||
This can be understood by the electrical schematic shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc} were the ADC has some input impedance \(R_i\) and input bias current \(i_n\).
|
||||
|
||||
|
||||
The input impedance \(R_i\) of the ADC with the capacitance \(C_p\) of the piezoelectric stack sensor forms an RC circuit with a time constant \(\tau = R_i C_p\).
|
||||
The charge generated by the piezoelectric effects across its capacitance is discharging into the input resistor of the ADC.
|
||||
Therefore, the transfer function from the generated voltage \(V_p\) to the measured voltage \(V_{\text{ADC}}\) is a first order high filter, with time constant \(\tau\).
|
||||
The an exponential curve was fitted to the experimental data and a time constant \(\tau = 6.5\,s\) was obtained.
|
||||
With the capacitance of the piezoelectric sensor stack being \(C_p = 4.4 \mu F\), the internal impedance of the Speedgoat ADC can be computed as follows \(R_i = \frac{\tau}{C_p} = 1.5\,M\Omega\).
|
||||
It is close to the specified value of \(1\,M\Omega\) found in the datasheet
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@ -562,37 +520,32 @@ It is close to the specified value of \(1\,M\Omega\) found in the datasheet
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_step_response_force_sensor.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Signals}
|
||||
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor}Measured Signals}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}). Measured voltage \(V_s\) while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).}
|
||||
\caption{\label{fig:detail_instrumentation_force_sensor}Electrical schematic of the ADC measuring the piezoelectric force sensor (\subref{fig:detail_instrumentation_force_sensor_adc}), adapted from \cite{reza06_piezoel_trans_vibrat_contr_dampin}. Measured voltage \(V_s\) while step voltages are generated for the actuator stacks (\subref{fig:detail_instrumentation_step_response_force_sensor}).}
|
||||
\end{figure}
|
||||
|
||||
As shown in Figure \ref{fig:detail_instrumentation_step_response_force_sensor}, the voltage across the Piezoelectric sensor stack shows a constant voltage offset.
|
||||
This can be explained by looking at the electrical model shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc} (taken from \cite{reza06_piezoel_trans_vibrat_contr_dampin}).
|
||||
The constant voltage offset can be explained by the input bias current \(i_n\) of the ADC, represented in Figure \ref{fig:detail_instrumentation_force_sensor_adc}
|
||||
At DC, the impedance of the piezoelectric stack is much larger that the input impedance of the ADC, and therefore the input bias current \(i_n\) passing through the internal resistance \(R_i\) produces a constant voltage offset \(V_{\text{off}} = R_i i_n\).
|
||||
The input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\,\mu A\).
|
||||
|
||||
The differential amplifier in the Speedgoat has some input bias current \(i_n\) that induces a voltage offset \(V_{\text{off}}\) across its own internal resistance \(R_i\).
|
||||
Note that the impedance of the piezoelectric stack is much larger that that at DC.
|
||||
|
||||
Therefore, the input bias current \(i_n\) is estimated from \(i_n = V_{\text{off}}/R_i = 1.5\,\mu A\).
|
||||
|
||||
In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor is added in parallel to the force sensor.
|
||||
In order to reduce the input voltage offset and to increase the corner frequency of the high pass filter, a resistor \(R_p\) is added in parallel to the force sensor, as illustrated in Figure \ref{fig:detail_instrumentation_force_sensor_adc_R}
|
||||
|
||||
It has two effects:
|
||||
\begin{itemize}
|
||||
\item Reduction of input voltage offset:
|
||||
\[ V_{off} = \frac{R_a R_{in}}{R_a + R_{in}} i_n \]
|
||||
\[ V_{\text{off}} = \frac{R_p R_{in}}{R_p + R_{in}} i_n \]
|
||||
\item Increase the high pass corner frequency \(f_c\)
|
||||
\[ C_p \frac{R_{in}R_a}{R_{in} + R_a} = \tau_c = \frac{1}{f_c} \]
|
||||
\[ R_a = \frac{R_i}{f_c C_p R_i - 1} \]
|
||||
\[ \frac{R_i R_p}{R_i + R_p} C_p = \tau_c = \frac{1}{2 \pi f_c} \]
|
||||
\[ R_p = \frac{R_i}{2 \pi f_c C_p R_i - 1} \]
|
||||
\end{itemize}
|
||||
|
||||
|
||||
The resistor is chosen such that the high pass corner frequency is equal to 3Hz.
|
||||
This corresponds to a resistor of \(R_p = 80\,k\Omega\).
|
||||
|
||||
The resistor is chosen such that the high pass corner frequency is equal to \(0.5\,\text{Hz}\).
|
||||
This corresponds to a resistor of \(R_p = 76\,k\Omega\).
|
||||
With this parallel resistance value, the voltage offset would be \(V_{\text{off}} = 0.11\,V\), which is much more acceptable.
|
||||
|
||||
To validate this, a resistor \(R_p \approx 82\,k\Omega\) is then added in parallel with the force sensor as shown in Figure \ref{fig:detail_instrumentation_force_sensor_adc_R}.
|
||||
After the resistor is added, the same steps response is performed.
|
||||
After the resistor is added, the same steps response were performed (Figure \ref{fig:detail_instrumentation_step_response_force_sensor_R}).
|
||||
And indeed, we obtain a much smaller offset voltage (\(V_{\text{off}} = 0.15\,V\)) and a much faster time constant (\(\tau = 0.45\,s\)).
|
||||
This validates the model of the ADC and the effectiveness of the added resistor.
|
||||
|
||||
@ -607,35 +560,23 @@ This validates the model of the ADC and the effectiveness of the added resistor.
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_step_response_force_sensor_R.png}
|
||||
\end{center}
|
||||
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor_R}Signals}
|
||||
\subcaption{\label{fig:detail_instrumentation_step_response_force_sensor_R}Measured Signals}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:detail_instrumentation_force_sensor_R}Effect of an added resistor \(R_p\) in parallel to the force sensor. The electrical schematic is shown in (\subref{fig:detail_instrumentation_force_sensor_adc_R}) and the measured signals in (\subref{fig:detail_instrumentation_step_response_force_sensor_R}).}
|
||||
\end{figure}
|
||||
|
||||
\section{Instrumentation Amplifier}
|
||||
\label{sec:org4ed7494}
|
||||
|
||||
Because the ADC noise may be too large to measure noise of other instruments (anything below \(5.6\,\mu V/\sqrt{Hz}\) cannot be distinguish from the noise of the ADC itself), a low noise instrumentation amplifier can be used.
|
||||
Here, a Femto DLPVA-101-B-S amplifier, with gains from 20dB up to 80dB, was used.
|
||||
|
||||
Different instrumentation amplifiers were used:
|
||||
\begin{itemize}
|
||||
\item EG\&G 5113, 4nV/sqrt(Hz), gain up to 100000 (100dB)
|
||||
\item Femto DLPVA-101-B-S 2nV/sqrt(Hz), gain from 20 to 80dB
|
||||
\item Koheron AMP200, 2.4nV/sqrt(Hz), gain up to 100
|
||||
\end{itemize}
|
||||
But first, the input\footnote{For variable gain amplifiers, it is usual to refer to the input noise rather than the output noise, as the input referred noise is almost independent on the chosen gain.} noise of the amplifier was characterized.
|
||||
To do so, its input was short circuited with a \(50\,\Omega\) resistor, and the output voltage was measured by the ADC (Figure \ref{fig:detail_instrumentation_femto_meas_setup}).
|
||||
The maximum amplifier gain of 80dB (i.e. 10000) was used.
|
||||
|
||||
Here, the Femto amplifier is used.
|
||||
|
||||
But first, the input noise of the amplifier is characterized.
|
||||
\begin{itemize}
|
||||
\item Footnote: for variable gain amplifiers, it is more convenient to refer to the input noise rather than the output noise.
|
||||
\end{itemize}
|
||||
|
||||
Its input is short circuited, and the output voltage is measured by the ADC.
|
||||
The maximum amplifier gain of 80dB (i.e. 10000) is used.
|
||||
|
||||
The measured voltage is then divided by 10000 to obtain the equivalent noise at the input of the voltage amplifier.
|
||||
In that case, the noise of the ADC is negligible, thanks to the high gain used.
|
||||
The measured voltage \(n\) was then divided by 10000 to obtain the equivalent noise at the input of the voltage amplifier \(n_a\).
|
||||
In that case, the noise of the ADC \(q_{ad}\) is negligible, thanks to the high gain used.
|
||||
The obtained amplifier noise ASD \(\Gamma_{n_a}\) and the (negligible) contribution of the ADC noise are shown in Figure \ref{fig:detail_instrumentation_femto_input_noise}.
|
||||
|
||||
It was also verified that the bandwidth of the instrumentation amplifier is much larger than 5kHz such that not phase drop are added by the use of the amplifier in the frequency band of interest.
|
||||
|
||||
@ -654,22 +595,11 @@ It was also verified that the bandwidth of the instrumentation amplifier is much
|
||||
\end{minipage}
|
||||
|
||||
\section{Digital to Analog Converters}
|
||||
\label{sec:org7adcab5}
|
||||
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.
|
||||
\paragraph{Output Voltage Noise}
|
||||
In order to measure the output noise of the DAC, the measurement setup schematically represented in Figure \ref{fig:detail_instrumentation_dac_setup} was used.
|
||||
The DAC was instructed to output constant voltage (here zero), and 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 measurement setup is schematically represented in Figure \ref{fig:detail_instrumentation_dac_setup}.
|
||||
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}
|
||||
The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed and it was verified the contribution of the ADC noise and amplifier noise are negligible.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -680,10 +610,13 @@ And it is verified that the Amplitude Spectral Density of \(n_{da}\) is much lar
|
||||
The obtained Amplitude Spectral Density of the DAC's output voltage is shown in Figure \ref{fig:detail_instrumentation_dac_output_noise}.
|
||||
It is almost white noise with an ASD of 0.6uV/sqrt(Hz).
|
||||
There is a little bit of 50Hz, and some low frequency noise (thermal noise?) which are not foreseen to be an issue as it will be inside the bandwidth.
|
||||
Note that all channels are measuring the same, so only one channel is shown here.
|
||||
|
||||
DAC is directly wired to the ADC.
|
||||
The transfer function from DAC to ADC is computed.
|
||||
It corresponds to 1 sample delay (Figure \ref{fig:detail_instrumentation_dac_adc_tf}).
|
||||
\paragraph{Delay from ADC to DAC}
|
||||
In order to measure the transfer function from DAC to ADC and verify that the bandwidth of both instrument is high enough, the DAC output was directly wired to the ADC input.
|
||||
A white noise signal was generated by the DAC, and the ADC signal was recorded.
|
||||
The obtained frequency response function from the digital DAC signal to the digital ADC signal is shown in (Figure \ref{fig:detail_instrumentation_dac_adc_tf}).
|
||||
It corresponds to 1 sample delay, which is corresponding to the specifications.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}{0.48\textwidth}
|
||||
@ -702,16 +635,11 @@ It corresponds to 1 sample delay (Figure \ref{fig:detail_instrumentation_dac_adc
|
||||
\end{figure}
|
||||
|
||||
\section{Piezoelectric Voltage Amplifier}
|
||||
\label{sec:orgd93524f}
|
||||
|
||||
|
||||
\paragraph{Output Voltage Noise}
|
||||
\label{sec:orgc975b4b}
|
||||
The measurement setup is shown in Figure \ref{fig:detail_instrumentation_pd200_setup}.
|
||||
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.
|
||||
|
||||
Two piezoelectric stacks of the APA95ML are connected to the PD200 output to have appropriate load.
|
||||
The input of the PD200 amplifier is shunted with a \(50\,\Omega\) resistor such that only the noise of the amplifier itself is measured.
|
||||
The gain of the pre-amplifier was increased in order to measure a signal much larger than the noise of the ADC.
|
||||
Two piezoelectric stacks of the APA95ML were connected to the PD200 output to provide an appropriate load.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -720,22 +648,17 @@ Two piezoelectric stacks of the APA95ML are connected to the PD200 output to hav
|
||||
\end{figure}
|
||||
|
||||
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:
|
||||
The Amplitude Spectral Density of the output voltage noise of the PD200 amplifier \(n_p\) is then computed taking into account the gain of the pre-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}
|
||||
The Amplitude Spectral Density of the measured output noise of the PD200 is computed and shown in Figure \ref{fig:detail_instrumentation_pd200_noise}.
|
||||
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 (i.e. what is measured is indeed the PD200 noise).
|
||||
Here, the measured noise of the six received amplifiers are all shown.
|
||||
|
||||
The Amplitude Spectral Density of the measured \textbf{input} noise is computed and shown in Figure \ref{fig:detail_instrumentation_pd200_noise}.
|
||||
|
||||
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.
|
||||
|
||||
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.
|
||||
The Amplitude Spectral Density of the output voltage noise of the PD200 amplifiers present sharp peaks.
|
||||
The reason for all these peaks is not clear, but as their amplitude are bellow the specifications, it should not pose any issue.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -744,7 +667,6 @@ It is not clear yet what causes such peaks and if these peaks have high influenc
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Small Signal Bandwidth}
|
||||
\label{sec:orgff5c292}
|
||||
|
||||
Here the small signal dynamics of all the PD200 amplifiers are identified.
|
||||
|
||||
@ -754,10 +676,10 @@ The output voltage of the PD200 amplifier is measured thanks to the monitor volt
|
||||
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.
|
||||
|
||||
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.
|
||||
All six received amplifiers are measuring the same regarding their transfer functions.
|
||||
The amplitude is constant over a wide frequency band and the phase drop is limited to less than 1 degree up to 500Hz, which is well within the specifications.
|
||||
|
||||
The identified dynamics in Figure \ref{fig:detail_instrumentation_pd200_tf} can very well be modeled this dynamics with a first order low pass filter (even a constant could work fine).
|
||||
The identified dynamics in Figure \ref{fig:detail_instrumentation_pd200_tf} can very well be modeled with a first order low pass filter or even a simple constant.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -765,75 +687,39 @@ The identified dynamics in Figure \ref{fig:detail_instrumentation_pd200_tf} can
|
||||
\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage}
|
||||
\end{figure}
|
||||
|
||||
\paragraph{Output Impedance}
|
||||
\label{sec:orgb2297c8}
|
||||
|
||||
The goal of this experimental setup is to estimate the output impedance \(R_\text{out}\) of the PD200 voltage amplifiers.
|
||||
A DAC with a constant output voltage (here 0.1V) is connected to the input of the PD200 amplifier.
|
||||
|
||||
Then, the output voltage of the PD200 amplifier is measured in two conditions:
|
||||
\begin{itemize}
|
||||
\item \(V\) when the output is not connected to any load
|
||||
\item \(V_p\) when a load \(R_{\text{load}} = 10\,\Omega\) is connected at the output of the amplifier
|
||||
\end{itemize}
|
||||
|
||||
The load and the output impedance form a voltage divider, and thus:
|
||||
\[ V^\prime = \frac{R}{R + R_\text{out}} V \]
|
||||
|
||||
From the two values of voltage, the output impedance of the amplifier can be estimated:
|
||||
\begin{equation}
|
||||
R_{\text{out}} = R_{\text{load}} \frac{V - V_p}{V}
|
||||
\end{equation}
|
||||
|
||||
From this measurement, all the PD200 amplifiers are found to have an output impedance \(R_{\text{out}} \approx 1\,\Omega\).
|
||||
|
||||
With the capacitive load \(C_p = 8.8\,\mu F\), the output resistor of the amplifier forms a low pass filter with a corner frequency equal to \(f_0 = \frac{1}{R_{\text{out}} C_p}\).
|
||||
We get a corner frequency around \(10\,\text{kHz}\) which is not far from the specified \(7.4\,\text{kHz}\).
|
||||
|
||||
\paragraph{Conclusion}
|
||||
\label{sec:org1118cdb}
|
||||
|
||||
\begin{table}[htbp]
|
||||
\caption{\label{tab:table_name}Measured characteristics, Manual characterstics and specified ones}
|
||||
\centering
|
||||
\begin{tabularx}{\linewidth}{Xcc}
|
||||
\toprule
|
||||
\textbf{Characteristics} & \textbf{Specification} & \textbf{Measurement}\\
|
||||
\midrule
|
||||
Input Voltage Range & +/- 10 [V] & -\\
|
||||
Output Voltage Range & -20/150 [V] & -\\
|
||||
Gain & 20 & 20\\
|
||||
Maximum RMS current & > 50 [mA] & \\
|
||||
Maximum Pulse current & - & \\
|
||||
Slew Rate & - & \\
|
||||
Noise (10uF load) & < 2 [mV rms] & \\
|
||||
Small Signal Bandwidth (10uF load) & > 5 [kHz] & \\
|
||||
Large Signal Bandwidth (150V, 10uF) & - & \\
|
||||
\bottomrule
|
||||
\end{tabularx}
|
||||
\end{table}
|
||||
|
||||
\section{Linear Encoders}
|
||||
\label{sec:orgddea30b}
|
||||
|
||||
To measure the noise \(n\) of the encoder, one can rigidly fix the head and the ruler together such that no motion should be measured.
|
||||
Then, the measured signal \(y_m\) corresponds to the noise \(n\).
|
||||
|
||||
The measurement bench is shown in Figures \ref{fig:meas_bench_top_view} and \ref{fig:meas_bench_side_view}.
|
||||
The measurement bench is shown in Figure \ref{fig:detail_instrumentation_vionic_bench}.
|
||||
Note that the bench is then covered with a ``plastic bubble sheet'' in order to keep disturbances as small as possible.
|
||||
Then, and for all the six encoders, the measured motion during 100s with a sampling frequency of 20kHz.
|
||||
|
||||
Then, and for all the 7 encoders, we record the measured motion during 100s with a sampling frequency of 20kHz.
|
||||
The obtained amplitude spectral density of the measured displacement (i.e. measurement noise) is shown in Figure \ref{fig:detail_instrumentation_vionic_asd}.
|
||||
It corresponds to a white noise, with an amplitude \(\approx 1\,nm\,\text{RMS}\).
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:orgc63ce3f}
|
||||
\begin{minipage}[b]{0.48\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_vionic_bench.jpg}
|
||||
\captionof{figure}{\label{fig:detail_instrumentation_vionic_bench}Test bench used to measured the encoder noise}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}[b]{0.48\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_instrumentation_vionic_asd.png}
|
||||
\captionof{figure}{\label{fig:detail_instrumentation_vionic_asd}Measured Amplitude Spectral Density of the encoder noise}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\begin{itemize}
|
||||
\item[{$\square$}] Compare with measurement noise? or effect of measurement noise => higher so it's OK?
|
||||
\item[{$\square$}] Or compare with specifications? (but we don't have specifications for ASD, only RMS)
|
||||
\item[{$\square$}] Or maybe put the ASD of the measured vibration in simulation (or noise budget based on disturbances) => show that we are bellow disturbances so it should not be limited
|
||||
\end{itemize}
|
||||
\section{Noise budgeting from measured instrumentation noise}
|
||||
|
||||
From all the measured noises, compute the obtained PSD error in Y and Z (show PSD of individual + Sum + Cumulative?)
|
||||
Once all the instrumentation noise were characterized, the effect of the instrumentation noise on the sample's vibration can be assessed using the multi-body model.
|
||||
|
||||
The obtained vertical motion induced by the ADC noise, DAC noise and voltage amplifier noise is displayed in Figure \ref{fig:detail_instrumentation_cl_noise_budget} (the effect of encoder noise is negligible).
|
||||
|
||||
The total motion induced by all the noise sources is around \(1.5\,nm\) which is well within the specifications.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -842,7 +728,6 @@ From all the measured noises, compute the obtained PSD error in Y and Z (show PS
|
||||
\end{figure}
|
||||
|
||||
\chapter*{Conclusion}
|
||||
\label{sec:org9929878}
|
||||
\label{sec:detail_instrumentation_conclusion}
|
||||
|
||||
\begin{itemize}
|
||||
|
||||
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Reference in New Issue
Block a user