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@ -352,12 +352,12 @@ Khac = -5e4 * ... % Gain
* Introduction :ignore:
This chapter presents a systematic approach to selecting and validating appropriate instrumentation for the nano active stabilization system (NASS), ensuring each component meets specific performance requirements.
This chapter presents an approach to selecting and validating appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements.
Figure ref:fig:detail_instrumentation_plant illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis.
The selection process follows a three-stage methodology.
First, dynamic error budgeting is performed in Section ref:sec:detail_instrumentation_dynamic_error_budgeting to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier).
This analysis employs the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
This analysis utilizes the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived.
Section ref:sec:detail_instrumentation_choice then presents the selection of appropriate components based on these noise specifications and additional requirements.
@ -439,7 +439,7 @@ The measured noise characteristics are then incorporated into the multi-body mod
<<sec:detail_instrumentation_dynamic_error_budgeting>>
** Introduction :ignore:
The primary goal of this analysis is to establish specifications for the maximum allowable noise levels in the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section ref:ssec:detail_instrumentation_cl_sensitivity).
This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
@ -584,36 +584,33 @@ exportFig('figs/detail_instrumentation_noise_sensitivities.pdf', 'width', 'wide'
#+end_src
#+name: fig:detail_instrumentation_noise_sensitivities
#+caption: Transfer function from noise sources to vertical motion errors
#+caption: Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.
#+RESULTS:
[[file:figs/detail_instrumentation_noise_sensitivities.png]]
** Estimation of maximum instrumentation noise
<<ssec:detail_instrumentation_max_noise_specs>>
From the previous analysis, the relationship between the noise of the instrumentation and its effect on the vertical error of the sample as a function of frequency has been established.
The next step involves determining specifications for each instrumentation component to ensure that the effect on the vertical error of the sample remains within acceptable limits.
The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of $100\,\text{nm}$, which corresponds to a maximum allowed vibration of $15\,\text{nm RMS}$.
Several assumptions regarding the noise characteristics have been made.
The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption.
Similarly, the noise corresponding to each strut is assumed to be uncorrelated.
This means that the power spectral densities (PSD) of the different noise sources can be summed.
Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated.
This means that the power spectral densities (PSD) of the different noise sources are summed.
The system symmetry has been utilized to simplify the analysis.
The system symmetry has been utilized to further simplify the analysis.
The effect of all struts on the vertical errors is identical, as verified from the extracted sensitivity curves.
Therefore, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of $\sqrt{6} \approx 2.5$ for RMS values.
In order to derive specifications in terms of noise spectral density for each instrumentation component, a white noise profile is assumed, which is typical for these components.
The noise specification is computed such that if all instrumentation components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications.
Based on this analysis, the obtained maximum noise levels are as follows: DAC maximum output noise ASD is established at $14\,\mu V/\sqrt{\text{Hz}}$, voltage amplifier maximum output voltage noise ASD at $280\,\mu V/\sqrt{\text{Hz}}$, and ADC maximum measurement noise ASD at $11\,\mu V/\sqrt{\text{Hz}}$.
In terms of RMS noise, these translate to less than $1\,\text{mV RMS}$ for the DAC, less than $20\,\text{mV RMS}$ for the voltage amplifier, and less than $0.8\,\text{mV RMS}$ for the ADC.
If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS. These specifications will guide the selection of appropriate instrumentation in Section ref:sec:detail_instrumentation_choice.
If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS.
#+begin_src matlab
% Maximum wanted effect of each noise source on the vertical error
@ -699,15 +696,12 @@ The amplifier should accept an analog input voltage, preferably in the range of
**** Small signal Bandwidth and Output Impedance
Two distinct bandwidth specifications are relevant for piezoelectric voltage amplifiers: large signal bandwidth and small signal bandwidth.
Large signal bandwidth relates to the output current capabilities of the amplifier and will be discussed in the next section.
Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system.
A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure ref:fig:detail_instrumentation_amp_output_impedance.
This model is valid for small signals and provides insight into the small signal bandwidth limitation [[cite:&fleming14_desig_model_contr_nanop_system, chap. 14]].
In this model, $R_o$ represents the output impedance of the amplifier.
When combined with the piezoelectric load (represented as a capacitance $C_p$), it forms a first order low pass filter described by equation ref:eq:detail_instrumentation_amp_output_impedance.
When combined with the piezoelectric load (represented as a capacitance $C_p$), it forms a first order low pass filter described by eqref:eq:detail_instrumentation_amp_output_impedance.
\begin{equation}\label{eq:detail_instrumentation_amp_output_impedance}
\frac{V_a}{V_i}(s) = \frac{1}{1 + \frac{s}{\omega_0}}, \quad \omega_0 = \frac{1}{R_o C_p}
@ -738,10 +732,10 @@ There are two limiting factors for large signal bandwidth that should be evaluat
1. Slew rate, which should exceed $2 \cdot V_{pp} \cdot f_r = 34\,V/ms$.
This requirement is typically easily met by commercial voltage amplifiers.
2. Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
To achieve high voltage at high frequency, the amplifier must provide substantial current.
To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
The maximum required current can be calculated as $I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A$.
Therefore, ideally, a voltage amplifier capable of providing $0.3\,A$ of current is needed.
Therefore, ideally, a voltage amplifier capable of providing $0.3\,A$ of current would be interesting for scanning applications.
#+begin_src matlab
%% Slew-rate specifications - Triangular scan
@ -759,7 +753,7 @@ max_current = 2*Vpp*f0*Cp % [A]
As established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the output noise of the voltage amplifier should be below $20\,\text{mV RMS}$.
It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure ref:fig:detail_instrumentation_amp_output_impedance.
Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used in the measurement (i.e., the low signal bandwidth considered) [[cite:&spengen20_high_voltag_amplif]].
Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement [[cite:&spengen20_high_voltag_amplif]].
For this application, the output noise must remain below $20\,\text{mV RMS}$ with a load of $8.8\,\mu F$ and a bandwidth exceeding $5\,\text{kHz}$.
@ -781,19 +775,19 @@ The PD200 from PiezoDrive was ultimately selected because it meets all the requi
#+caption: Specifications for the Voltage amplifier and considered commercial products
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :float t
| *Specification* | *PD200* | WMA-200 | LA75B | E-505 |
| | PiezoDrive | Falco | Cedrat | PI |
|--------------------------------------+-----------------------+-------------------------------------------+--------------+-----------|
| 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 $>15$ | 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) | (n/a) | (n/a) |
| Small Signal Bandwidth $> 5\,kHz$ | $6.4\,kHz$ | $300\,Hz$ | $30\,kHz$ | n/a |
| ($10\,\mu F$ load) | ($10\,\mu F$ load) | [fn:detail_instrumentation_1] | (unloaded) | (n/a) |
| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$[fn:detail_instrumentation_1] | n/a | n/a |
| *Specification* | *PD200* | WMA-200 | LA75B | E-505 |
| | PiezoDrive | Falco | Cedrat | PI |
|--------------------------------------+-----------------------+-------------------------------+--------------+-----------|
| 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 $>15$ | 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) | (n/a) | (n/a) |
| Small Signal Bandwidth $> 5\,kHz$ | $6.4\,kHz$ | $300\,Hz$ | $30\,kHz$ | n/a |
| ($10\,\mu F$ load) | ($10\,\mu F$ load) | [fn:detail_instrumentation_1] | (unloaded) | (n/a) |
| Output Impedance: $< 3.6\,\Omega$ | n/a | $50\,\Omega$ | n/a | n/a |
** ADC and DAC
**** Introduction :ignore:
@ -802,8 +796,7 @@ The proper selection of these components is critical for system performance.
**** Synchronicity and Jitter
For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements.
These factors significantly impact control performance, as highlighted in [[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]].
For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements [[cite:&abramovitch22_pract_method_real_world_contr_system;&abramovitch23_tutor_real_time_comput_issues_contr_system]].
Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals.
Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration.
@ -816,7 +809,7 @@ First, the /sampling frequency/ defines the interval between two sampled points
Then, the /bandwidth/ specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters.
Finally, /delay/ (or /latency/) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system.
Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and sampling frequency, but often at the cost of poor latency.
Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency.
Typically, the latency can reach 20 times the sampling period [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 8.4]].
Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor.
@ -824,14 +817,13 @@ For real-time control applications, SAR-ADCs (Successive Approximation ADCs) rem
**** ADC Noise
Based on the dynamic error budget established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$, equivalent to $0.8\,\text{mV RMS}$.
Based on the dynamic error budget established in Section ref:sec:detail_instrumentation_dynamic_error_budgeting, the measurement noise ASD should not exceed $11\,\mu V/\sqrt{\text{Hz}}$.
ADCs are subject to various noise sources.
Quantization noise, which results from the discrete nature of digital representation, is one of these sources.
To determine the minimum bit depth required to meet the noise specifications, the quantization noise must be analyzed.
To determine the minimum bit depth $n$ required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered.
Assuming an ideal ADC where quantization error is the only noise source, the quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the least significant bit.
Here, $\Delta V$ is the full range of the ADC in volts, $n$ is the bit depth, and $F_s$ is the sampling frequency in Hertz.
The quantization step size, denoted as $q = \Delta V/2^n$, represents the voltage equivalent of the least significant bit, with $\Delta V$ the full range of the ADC in volts, and $F_s$ the sampling frequency in Hertz.
The quantization noise ranges between $\pm q/2$, and its probability density function is constant across this range (uniform distribution).
Since the integral of this probability density function $p(e)$ equals one, its value is $1/q$ for $-q/2 < e < q/2$, as illustrated in Figure ref:fig:detail_instrumentation_adc_quantization.
@ -851,11 +843,11 @@ Since the integral of this probability density function $p(e)$ equals one, its v
#+end_src
#+name: fig:detail_instrumentation_adc_quantization
#+caption: Probability density function $p(e)$ of the ADC error $e$
#+caption: Probability density function $p(e)$ of the ADC quantization error $e$
#+RESULTS:
[[file:figs/detail_instrumentation_adc_quantization.png]]
The variance (or time-average power) of the quantization noise is expressed by equation ref:eq:detail_instrumentation_quant_power:
The variance (or time-average power) of the quantization noise is expressed by 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}
@ -871,7 +863,7 @@ By Parseval's theorem, the power spectral density of the quantization noise $\Ph
\int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_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}
From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the minimum number of bits required to keep quantization noise below $\Gamma_{\text{max}}$ is calculated using equation ref:eq:detail_instrumentation_min_n.
From a specified noise amplitude spectral density $\Gamma_{\text{max}}$, the minimum number of bits required to keep quantization noise below $\Gamma_{\text{max}}$ is calculated using eqref:eq:detail_instrumentation_min_n.
\begin{equation}\label{eq:detail_instrumentation_min_n}
n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right)
@ -900,8 +892,8 @@ q_asd = sqrt(q_psd) % Quantization noise Amplitude Spectral Density [V/sqrt(Hz)]
**** DAC Output voltage noise
Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed $14\,\mu V/\sqrt{Hz}$, equivalent to $1\,\text{mV RMS}$.
This specification corresponds to a 13-bit $\pm 10\,V$ DAC, which is easily attainable with current technology.
Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed $14\,\mu V/\sqrt{\text{Hz}}$.
This specification corresponds to a $\pm 10\,V$ DAC with 13-bit resolution, which is easily attainable with current technology.
**** Choice of the ADC and DAC Board
@ -946,7 +938,7 @@ These include optical encoders (Figure ref:fig:detail_instrumentation_sensor_enc
#+end_figure
From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure ref:fig:detail_instrumentation_capacitive_implementation.
In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to relative rotations between the two ends of the APA, as shown in Figure ref:fig:detail_instrumentation_encoder_implementation.
In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure ref:fig:detail_instrumentation_encoder_implementation.
#+name: fig:detail_instrumentation_sensor_implementation
#+caption: Implementation of relative displacement sensor to measure the motion of the APA
@ -974,7 +966,7 @@ Based on this criterion, an optical encoder with digital output was selected, wh
The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table ref:tab:detail_instrumentation_sensor_specs, alongside alternative options that were considered.
#+name: tab:detail_instrumentation_sensor_specs
#+caption: Characteristics of the Vionic compared with the specifications
#+caption: Specifications for the relative displacement sensors and considered commercial products
#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xccc
#+attr_latex: :center t :booktabs t :float t
| *Specification* | *Renishaw Vionic* | LION CPL190 | Cedrat ECP500 |
@ -1322,8 +1314,6 @@ The measured voltage $n$ was then divided by 10000 to determine the equivalent n
In this configuration, the noise contribution from the ADC $q_{ad}$ is rendered negligible due to the high gain employed.
The resulting amplifier noise amplitude spectral density $\Gamma_{n_a}$ and the (negligible) contribution of the ADC noise are presented in Figure ref:fig:detail_instrumentation_femto_input_noise.
Additionally, verification was performed to ensure that the bandwidth of the instrumentation amplifier significantly exceeds 5kHz, thereby preventing any phase distortion within the frequency band of interest.
#+begin_src latex :file detail_instrumentation_femto_meas_setup.pdf
\begin{tikzpicture}
\node[block={0.6cm}{0.6cm}] (const) {$0$};
@ -1419,14 +1409,14 @@ exportFig('figs/detail_instrumentation_femto_input_noise.pdf', 'width', 'half',
** Digital to Analog Converters
**** Output Voltage Noise
To measure the output noise of the DAC, the setup schematically represented in Figure ref:fig:detail_instrumentation_dac_setup was utilized.
The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the quantization noise of the ADC.
The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC.
The Amplitude Spectral Density $\Gamma_{n_{da}}(\omega)$ of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement.
The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure ref:fig:detail_instrumentation_dac_output_noise.
The noise profile is predominantly white with an ASD of $0.6\,\mu V/\sqrt{\text{Hz}}$.
Minor $50\,\text{Hz}$ noise is present, along with some low frequency $1/f$ noise, but these are not expected to pose issues as they are well within specifications.
It should be noted that all DAC channels demonstrated similar performance, so only one channel's results are presented.
It should be noted that all DAC channels demonstrated similar performance, so only one channel measurement is presented.
#+begin_src latex :file detail_instrumentation_dac_setup.pdf
\begin{tikzpicture}
@ -1682,7 +1672,7 @@ From this, the Amplitude Spectral Density of the output voltage noise of the PD2
\end{equation}
The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure ref:fig:detail_instrumentation_pd200_noise.
Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or quantization noise.
Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise.
The measurements from all six amplifiers are displayed in this figure.
The noise spectrum of the PD200 amplifiers exhibits several sharp peaks.
@ -1794,7 +1784,7 @@ tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(pd200{1}.f, abs(pd200{1}.tf), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5, 'DisplayName', 'Measurement')
plot(pd200{1}.f, abs(squeeze(freqresp(Gp, pd200{1}.f, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', 'Model')
plot(pd200{1}.f, abs(squeeze(freqresp(Gp, pd200{1}.f, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$1^{st}$ order LPF')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude [V/V]'); set(gca, 'XTickLabel',[]);
@ -1822,14 +1812,14 @@ exportFig('figs/detail_instrumentation_pd200_tf.pdf', 'width', 'wide', 'height',
#+end_src
#+name: fig:detail_instrumentation_pd200_tf
#+caption: Identified dynamics from input voltage to output voltage
#+caption: Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier
#+RESULTS:
[[file:figs/detail_instrumentation_pd200_tf.png]]
** Linear Encoders
To measure the noise $n$ of the encoder, the head and ruler were rigidly fixed together to ensure that no actual motion would be detected.
Under these conditions, any measured signal $y_m$ would correspond solely to the encoder noise.
To measure the noise of the encoder, the head and ruler were rigidly fixed together to ensure that no actual motion would be detected.
Under these conditions, any measured signal would correspond solely to the encoder noise.
The measurement setup is shown in Figure ref:fig:detail_instrumentation_vionic_bench.
To minimize environmental disturbances, the entire bench was covered with a plastic bubble sheet during measurements.
@ -1894,7 +1884,6 @@ exportFig('figs/detail_instrumentation_vionic_asd.pdf', 'width', 'half', 'height
After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier.
The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure ref:fig:detail_instrumentation_cl_noise_budget.
The contribution from encoder noise was found to be negligible and is therefore not shown here.
The total motion induced by all noise sources combined is approximately $1.5\,\text{nm RMS}$, which remains well within the specified limit of $15\,\text{nm RMS}$.
This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application.

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@ -1,4 +1,4 @@
% Created 2025-03-17 Mon 21:28
% Created 2025-03-17 Mon 22:15
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -24,12 +24,12 @@
\clearpage
This chapter presents a systematic approach to selecting and validating appropriate instrumentation for the nano active stabilization system (NASS), ensuring each component meets specific performance requirements.
This chapter presents an approach to selecting and validating appropriate instrumentation for the Nano Active Stabilization System (NASS), ensuring each component meets specific performance requirements.
Figure \ref{fig:detail_instrumentation_plant} illustrates the control diagram with all relevant noise sources whose effects on sample position will be evaluated throughout this analysis.
The selection process follows a three-stage methodology.
First, dynamic error budgeting is performed in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting} to establish maximum acceptable noise specifications for each instrumentation component (ADC, DAC, and voltage amplifier).
This analysis employs the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
This analysis utilizes the multi-body model with a 2DoF APA model, focusing particularly on the vertical direction due to its more stringent requirements.
From the calculated transfer functions, maximum acceptable amplitude spectral densities for each noise source are derived.
Section \ref{sec:detail_instrumentation_choice} then presents the selection of appropriate components based on these noise specifications and additional requirements.
@ -46,7 +46,7 @@ The measured noise characteristics are then incorporated into the multi-body mod
\chapter{Dynamic Error Budgeting}
\label{sec:detail_instrumentation_dynamic_error_budgeting}
The primary goal of this analysis is to establish specifications for the maximum allowable noise levels in the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
The primary goal of this analysis is to establish specifications for the maximum allowable noise levels of the instrumentation used for the NASS (ADC, DAC, and voltage amplifier) that would result in acceptable vibration levels in the system.
The procedure involves determining the closed-loop transfer functions from various noise sources to positioning error (Section \ref{ssec:detail_instrumentation_cl_sensitivity}).
This analysis is conducted using the multi-body model with a 2-DoF Amplified Piezoelectric Actuator model that incorporates voltage inputs and outputs.
@ -67,35 +67,32 @@ The transfer functions from these three noise sources (for one strut) to the ver
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_noise_sensitivities.png}
\caption{\label{fig:detail_instrumentation_noise_sensitivities}Transfer function from noise sources to vertical motion errors}
\caption{\label{fig:detail_instrumentation_noise_sensitivities}Transfer function from noise sources to vertical motion errors, in closed-loop with the implemented HAC-LAC strategy.}
\end{figure}
\section{Estimation of maximum instrumentation noise}
\label{ssec:detail_instrumentation_max_noise_specs}
From the previous analysis, the relationship between the noise of the instrumentation and its effect on the vertical error of the sample as a function of frequency has been established.
The next step involves determining specifications for each instrumentation component to ensure that the effect on the vertical error of the sample remains within acceptable limits.
The most stringent requirement for the system is maintaining vertical vibrations below the smallest expected beam size of \(100\,\text{nm}\), which corresponds to a maximum allowed vibration of \(15\,\text{nm RMS}\).
Several assumptions regarding the noise characteristics have been made.
The DAC, ADC, and amplifier noise are considered uncorrelated, which is a reasonable assumption.
Similarly, the noise corresponding to each strut is assumed to be uncorrelated.
This means that the power spectral densities (PSD) of the different noise sources can be summed.
Similarly, the noise sources corresponding to each strut are also assumed to be uncorrelated.
This means that the power spectral densities (PSD) of the different noise sources are summed.
The system symmetry has been utilized to simplify the analysis.
The system symmetry has been utilized to further simplify the analysis.
The effect of all struts on the vertical errors is identical, as verified from the extracted sensitivity curves.
Therefore, only one strut is considered for this analysis, and the total effect of the six struts is calculated as six times the effect of one strut in terms of power, which translates to a factor of \(\sqrt{6} \approx 2.5\) for RMS values.
In order to derive specifications in terms of noise spectral density for each instrumentation component, a white noise profile is assumed, which is typical for these components.
The noise specification is computed such that if all instrumentation components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
The noise specification is computed such that if all components operate at their maximum allowable noise levels, the specification for vertical error will still be met.
While this represents a pessimistic approach, it provides a reasonable estimate of the required specifications.
Based on this analysis, the obtained maximum noise levels are as follows: DAC maximum output noise ASD is established at \(14\,\mu V/\sqrt{\text{Hz}}\), voltage amplifier maximum output voltage noise ASD at \(280\,\mu V/\sqrt{\text{Hz}}\), and ADC maximum measurement noise ASD at \(11\,\mu V/\sqrt{\text{Hz}}\).
In terms of RMS noise, these translate to less than \(1\,\text{mV RMS}\) for the DAC, less than \(20\,\text{mV RMS}\) for the voltage amplifier, and less than \(0.8\,\text{mV RMS}\) for the ADC.
If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS. These specifications will guide the selection of appropriate instrumentation in Section \ref{sec:detail_instrumentation_choice}.
If the Amplitude Spectral Density of the noise of the ADC, DAC, and voltage amplifiers all remain below these specified maximum levels, then the induced vertical error will be maintained below 15nm RMS.
\chapter{Choice of Instrumentation}
\label{sec:detail_instrumentation_choice}
@ -109,15 +106,12 @@ The amplifier should accept an analog input voltage, preferably in the range of
\paragraph{Small signal Bandwidth and Output Impedance}
Two distinct bandwidth specifications are relevant for piezoelectric voltage amplifiers: large signal bandwidth and small signal bandwidth.
Large signal bandwidth relates to the output current capabilities of the amplifier and will be discussed in the next section.
Small signal bandwidth is particularly important for feedback applications as it can limit the overall bandwidth of the complete feedback system.
A simplified electrical model of a voltage amplifier connected to a piezoelectric stack is shown in Figure \ref{fig:detail_instrumentation_amp_output_impedance}.
This model is valid for small signals and provides insight into the small signal bandwidth limitation \cite[, chap. 14]{fleming14_desig_model_contr_nanop_system}.
In this model, \(R_o\) represents the output impedance of the amplifier.
When combined with the piezoelectric load (represented as a capacitance \(C_p\)), it forms a first order low pass filter described by equation \ref{eq:detail_instrumentation_amp_output_impedance}.
When combined with the piezoelectric load (represented as a capacitance \(C_p\)), it forms a first order low pass filter described by \eqref{eq:detail_instrumentation_amp_output_impedance}.
\begin{equation}\label{eq:detail_instrumentation_amp_output_impedance}
\frac{V_a}{V_i}(s) = \frac{1}{1 + \frac{s}{\omega_0}}, \quad \omega_0 = \frac{1}{R_o C_p}
@ -145,18 +139,18 @@ There are two limiting factors for large signal bandwidth that should be evaluat
\item Slew rate, which should exceed \(2 \cdot V_{pp} \cdot f_r = 34\,V/ms\).
This requirement is typically easily met by commercial voltage amplifiers.
\item Current output capabilities: as the capacitive impedance decreases inversely with frequency, it can reach very low values at high frequencies.
To achieve high voltage at high frequency, the amplifier must provide substantial current.
To achieve high voltage at high frequency, the amplifier must therefore provide substantial current.
The maximum required current can be calculated as \(I_{\text{max}} = 2 \cdot V_{pp} \cdot f \cdot C_p = 0.3\,A\).
\end{enumerate}
Therefore, ideally, a voltage amplifier capable of providing \(0.3\,A\) of current is needed.
Therefore, ideally, a voltage amplifier capable of providing \(0.3\,A\) of current would be interesting for scanning applications.
\paragraph{Output voltage noise}
As established in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the output noise of the voltage amplifier should be below \(20\,\text{mV RMS}\).
It should be noted that the load capacitance of the piezoelectric stack filters the output noise of the amplifier, as illustrated by the low pass filter in Figure \ref{fig:detail_instrumentation_amp_output_impedance}.
Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used in the measurement (i.e., the low signal bandwidth considered) \cite{spengen20_high_voltag_amplif}.
Therefore, when comparing noise specifications from different voltage amplifier datasheets, it is essential to verify the capacitance of the load used during the measurement \cite{spengen20_high_voltag_amplif}.
For this application, the output noise must remain below \(20\,\text{mV RMS}\) with a load of \(8.8\,\mu F\) and a bandwidth exceeding \(5\,\text{kHz}\).
@ -191,10 +185,10 @@ Output noise \(< 20\,mV\ \text{RMS}\) & \(0.7\,mV\,\text{RMS}\) & \(0.05\,mV\)
(10uF load) & (\(10\,\mu F\) load) & (\(10\,\mu F\) load) & (n/a) & (n/a)\\
Small Signal Bandwidth \(> 5\,kHz\) & \(6.4\,kHz\) & \(300\,Hz\) & \(30\,kHz\) & n/a\\
(\(10\,\mu F\) load) & (\(10\,\mu F\) load) & \footnotemark & (unloaded) & (n/a)\\
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\)\textsuperscript{\ref{orgd8b72a7}} & n/a & n/a\\
Output Impedance: \(< 3.6\,\Omega\) & n/a & \(50\,\Omega\) & n/a & n/a\\
\bottomrule
\end{tabularx}
\end{table}\footnotetext[1]{\label{orgd8b72a7}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
\end{table}\footnotetext[1]{\label{orge65a0e3}The manufacturer proposed to remove the \(50\,\Omega\) output resistor to improve to small signal bandwidth above \(10\,kHz\)}
\section{ADC and DAC}
Analog-to-digital converters and digital-to-analog converters play key roles in the system, serving as the interface between the digital RT controller and the analog physical plant.
@ -202,8 +196,7 @@ The proper selection of these components is critical for system performance.
\paragraph{Synchronicity and Jitter}
For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements.
These factors significantly impact control performance, as highlighted in \cite{abramovitch22_pract_method_real_world_contr_system,abramovitch23_tutor_real_time_comput_issues_contr_system}.
For control systems, synchronous sampling of inputs and outputs of the real-time controller and minimal jitter are essential requirements \cite{abramovitch22_pract_method_real_world_contr_system,abramovitch23_tutor_real_time_comput_issues_contr_system}.
Therefore, the ADC and DAC must be well interfaced with the Speedgoat real-time controller and triggered synchronously with the computation of the control signals.
Based on this requirement, priority was given to ADC and DAC components specifically marketed by Speedgoat to ensure optimal integration.
@ -216,7 +209,7 @@ First, the \emph{sampling frequency} defines the interval between two sampled po
Then, the \emph{bandwidth} specifies the maximum frequency of a measured signal (typically defined as the -3dB point) and is often limited by implemented anti-aliasing filters.
Finally, \emph{delay} (or \emph{latency}) refers to the time interval between the analog signal at the input of the ADC and the digital information transferred to the control system.
Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and sampling frequency, but often at the cost of poor latency.
Sigma-Delta ADCs can provide excellent noise characteristics, high bandwidth, and high sampling frequency, but often at the cost of poor latency.
Typically, the latency can reach 20 times the sampling period \cite[, chapt. 8.4]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
Consequently, while Sigma-Delta ADCs are widely used for signal acquisition applications, they have limited utility in real-time control scenarios where latency is a critical factor.
@ -224,14 +217,13 @@ For real-time control applications, SAR-ADCs (Successive Approximation ADCs) rem
\paragraph{ADC Noise}
Based on the dynamic error budget established in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise ASD should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\), equivalent to \(0.8\,\text{mV RMS}\).
Based on the dynamic error budget established in Section \ref{sec:detail_instrumentation_dynamic_error_budgeting}, the measurement noise ASD should not exceed \(11\,\mu V/\sqrt{\text{Hz}}\).
ADCs are subject to various noise sources.
Quantization noise, which results from the discrete nature of digital representation, is one of these sources.
To determine the minimum bit depth required to meet the noise specifications, the quantization noise must be analyzed.
To determine the minimum bit depth \(n\) required to meet the noise specifications, an ideal ADC where quantization error is the only noise source is considered.
Assuming an ideal ADC where quantization error is the only noise source, the quantization step size, denoted as \(q = \Delta V/2^n\), represents the voltage equivalent of the least significant bit.
Here, \(\Delta V\) is the full range of the ADC in volts, \(n\) is the bit depth, and \(F_s\) is the sampling frequency in Hertz.
The quantization step size, denoted as \(q = \Delta V/2^n\), represents the voltage equivalent of the least significant bit, with \(\Delta V\) the full range of the ADC in volts, and \(F_s\) the sampling frequency in Hertz.
The quantization noise ranges between \(\pm q/2\), and its probability density function is constant across this range (uniform distribution).
Since the integral of this probability density function \(p(e)\) equals one, its value is \(1/q\) for \(-q/2 < e < q/2\), as illustrated in Figure \ref{fig:detail_instrumentation_adc_quantization}.
@ -239,10 +231,10 @@ Since the integral of this probability density function \(p(e)\) equals one, its
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_adc_quantization.png}
\caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC error \(e\)}
\caption{\label{fig:detail_instrumentation_adc_quantization}Probability density function \(p(e)\) of the ADC quantization error \(e\)}
\end{figure}
The variance (or time-average power) of the quantization noise is expressed by equation \ref{eq:detail_instrumentation_quant_power}:
The variance (or time-average power) of the quantization noise is expressed by \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}
@ -258,7 +250,7 @@ By Parseval's theorem, the power spectral density of the quantization noise \(\P
\int_{-F_s/2}^{F_s/2} \Phi_q(f) d f = \int_{-q/2}^{q/2} e^2 p(e) de \quad \Longrightarrow \quad \Phi_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}
From a specified noise amplitude spectral density \(\Gamma_{\text{max}}\), the minimum number of bits required to keep quantization noise below \(\Gamma_{\text{max}}\) is calculated using equation \ref{eq:detail_instrumentation_min_n}.
From a specified noise amplitude spectral density \(\Gamma_{\text{max}}\), the minimum number of bits required to keep quantization noise below \(\Gamma_{\text{max}}\) is calculated using \eqref{eq:detail_instrumentation_min_n}.
\begin{equation}\label{eq:detail_instrumentation_min_n}
n = \text{log}_2 \left( \frac{\Delta V}{\sqrt{12 F_s} \cdot \Gamma_{\text{max}}} \right)
@ -268,8 +260,8 @@ With a sampling frequency \(F_s = 10\,\text{kHz}\), an input range \(\Delta V =
\paragraph{DAC Output voltage noise}
Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed \(14\,\mu V/\sqrt{Hz}\), equivalent to \(1\,\text{mV RMS}\).
This specification corresponds to a 13-bit \(\pm 10\,V\) DAC, which is easily attainable with current technology.
Similar to the ADC requirements, the DAC output voltage noise ASD should not exceed \(14\,\mu V/\sqrt{\text{Hz}}\).
This specification corresponds to a \(\pm 10\,V\) DAC with 13-bit resolution, which is easily attainable with current technology.
\paragraph{Choice of the ADC and DAC Board}
@ -312,7 +304,7 @@ These include optical encoders (Figure \ref{fig:detail_instrumentation_sensor_en
\end{figure}
From an implementation perspective, capacitive and eddy current sensors offer a slight advantage as they can be quite compact and can measure in line with the APA, as illustrated in Figure \ref{fig:detail_instrumentation_capacitive_implementation}.
In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to relative rotations between the two ends of the APA, as shown in Figure \ref{fig:detail_instrumentation_encoder_implementation}.
In contrast, optical encoders are bigger and they must be offset from the strut's action line, which introduces potential measurement errors (Abbe errors) due to potential relative rotations between the two ends of the APA, as shown in Figure \ref{fig:detail_instrumentation_encoder_implementation}.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -338,7 +330,7 @@ Based on this criterion, an optical encoder with digital output was selected, wh
The specifications of the considered relative motion sensor, the Renishaw Vionic, are summarized in Table \ref{tab:detail_instrumentation_sensor_specs}, alongside alternative options that were considered.
\begin{table}[htbp]
\caption{\label{tab:detail_instrumentation_sensor_specs}Characteristics of the Vionic compared with the specifications}
\caption{\label{tab:detail_instrumentation_sensor_specs}Specifications for the relative displacement sensors and considered commercial products}
\centering
\begin{tabularx}{0.8\linewidth}{Xccc}
\toprule
@ -458,8 +450,6 @@ The measured voltage \(n\) was then divided by 10000 to determine the equivalent
In this configuration, the noise contribution from the ADC \(q_{ad}\) is rendered negligible due to the high gain employed.
The resulting amplifier noise amplitude spectral density \(\Gamma_{n_a}\) and the (negligible) contribution of the ADC noise are presented in Figure \ref{fig:detail_instrumentation_femto_input_noise}.
Additionally, verification was performed to ensure that the bandwidth of the instrumentation amplifier significantly exceeds 5kHz, thereby preventing any phase distortion within the frequency band of interest.
\begin{minipage}[b]{0.48\linewidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_instrumentation_femto_meas_setup.png}
@ -477,14 +467,14 @@ Additionally, verification was performed to ensure that the bandwidth of the ins
\section{Digital to Analog Converters}
\paragraph{Output Voltage Noise}
To measure the output noise of the DAC, the setup schematically represented in Figure \ref{fig:detail_instrumentation_dac_setup} was utilized.
The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the quantization noise of the ADC.
The DAC was configured to output a constant voltage (zero in this case), and the gain of the pre-amplifier was adjusted such that the measured amplified noise was significantly larger than the noise of the ADC.
The Amplitude Spectral Density \(\Gamma_{n_{da}}(\omega)\) of the measured signal was computed, and verification was performed to confirm that the contributions of ADC noise and amplifier noise were negligible in the measurement.
The resulting Amplitude Spectral Density of the DAC's output voltage is displayed in Figure \ref{fig:detail_instrumentation_dac_output_noise}.
The noise profile is predominantly white with an ASD of \(0.6\,\mu V/\sqrt{\text{Hz}}\).
Minor \(50\,\text{Hz}\) noise is present, along with some low frequency \(1/f\) noise, but these are not expected to pose issues as they are well within specifications.
It should be noted that all DAC channels demonstrated similar performance, so only one channel's results are presented.
It should be noted that all DAC channels demonstrated similar performance, so only one channel measurement is presented.
\begin{figure}[htbp]
\centering
@ -536,7 +526,7 @@ From this, the Amplitude Spectral Density of the output voltage noise of the PD2
\end{equation}
The computed Amplitude Spectral Density of the PD200 output noise is presented in Figure \ref{fig:detail_instrumentation_pd200_noise}.
Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or quantization noise.
Verification was performed to confirm that the measured noise was predominantly from the PD200, with negligible contributions from the pre-amplifier noise or ADC noise.
The measurements from all six amplifiers are displayed in this figure.
The noise spectrum of the PD200 amplifiers exhibits several sharp peaks.
@ -563,13 +553,13 @@ The identified dynamics shown in Figure \ref{fig:detail_instrumentation_pd200_tf
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_instrumentation_pd200_tf.png}
\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage}
\caption{\label{fig:detail_instrumentation_pd200_tf}Identified dynamics from input voltage to output voltage of the PD200 voltage amplifier}
\end{figure}
\section{Linear Encoders}
To measure the noise \(n\) of the encoder, the head and ruler were rigidly fixed together to ensure that no actual motion would be detected.
Under these conditions, any measured signal \(y_m\) would correspond solely to the encoder noise.
To measure the noise of the encoder, the head and ruler were rigidly fixed together to ensure that no actual motion would be detected.
Under these conditions, any measured signal would correspond solely to the encoder noise.
The measurement setup is shown in Figure \ref{fig:detail_instrumentation_vionic_bench}.
To minimize environmental disturbances, the entire bench was covered with a plastic bubble sheet during measurements.
@ -596,7 +586,6 @@ The noise profile exhibits characteristics of white noise with an amplitude of a
After characterizing all instrumentation components individually, their combined effect on the sample's vibration was assessed using the multi-body model developed earlier.
The vertical motion induced by the noise sources, specifically the ADC noise, DAC noise, and voltage amplifier noise, is presented in Figure \ref{fig:detail_instrumentation_cl_noise_budget}.
The contribution from encoder noise was found to be negligible and is therefore not shown here.
The total motion induced by all noise sources combined is approximately \(1.5\,\text{nm RMS}\), which remains well within the specified limit of \(15\,\text{nm RMS}\).
This confirms that the selected instrumentation, with its measured noise characteristics, is suitable for the intended application.