%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); %% Path for functions, data and scripts addpath('./src/'); % Path for scripts addpath('./mat/'); % Path for data addpath('./STEPS/'); % Path for Simscape Model addpath('./subsystems/'); % Path for Subsystems Simulink files %% Colors for the figures colors = colororder; freqs = logspace(1,4,1000); % Frequency vector [Hz] %% Load computed requirements load('instrumentation_requirements.mat') % Small signal Bandwidth and Output Impedance % 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 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} % \end{equation} % #+name: fig:detail_instrumentation_amp_output_impedance % #+caption: Electrical model of a voltage amplifier with output impedance $R_0$ connected to a piezoelectric stack with capacitance $C_p$ % [[file:figs/detail_instrumentation_amp_output_impedance.png]] % Consequently, the small signal bandwidth depends on the load capacitance and decreases as the load capacitance increases. % For the APA300ML, the capacitive load of the two piezoelectric stacks corresponds to $C_p = 8.8\,\mu F$. % If a small signal bandwidth of $f_0 = \frac{\omega_0}{2\pi} = 5\,\text{kHz}$ is desired, the voltage amplifier output impedance should be less than $R_0 = 3.6\,\Omega$. Cp = 8.8e-6; % Capacitive load of the two piezoelectric actuators f0 = 5e3; % Wanted low signal bandwidth [Hz] Ro_max = 1/(2*pi*f0 * Cp); % Maximum wanted output impedance [Ohm] % Large signal Bandwidth % Large signal bandwidth relates to the maximum output capabilities of the amplifier in terms of amplitude as a function of frequency. % Since the primary function of the NASS is position stabilization rather than scanning, this specification is less critical than the small signal bandwidth. % However, considering potential scanning capabilities, a worst-case scenario of a constant velocity scan (triangular reference signal) with a repetition rate of $f_r = 100\,\text{Hz}$ using the full voltage range of the piezoelectric actuator ($V_{pp} = 170\,V$) is considered. % There are two limiting factors for large signal bandwidth that should be evaluated: % 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 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 would be interesting for scanning applications. %% Slew-rate specifications - Triangular scan Vpp = 170; % Full voltage scan [V] f0 = 100; % Repetition rate of the triangular scan [Hz] slew_rate = 1e-3*2*Vpp*f0 % Required slew rate [V/ms] %% Maximum Output Current - Triangular scan max_current = 2*Vpp*f0*Cp % [A] % #+name: fig:detail_instrumentation_adc_quantization % #+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 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} % To compute the power spectral density of the quantization noise, which is defined as the Fourier transform of the noise's autocorrelation function, it is assumed that noise samples are uncorrelated. % Under this assumption, the autocorrelation function approximates a delta function in the time domain. % Since the Fourier transform of a delta function equals one, the power spectral density becomes frequency-independent (white noise). % By Parseval's theorem, the power spectral density of the quantization noise $\Phi_q$ can be linked to the ADC sampling frequency and quantization step size eqref:eq:detail_instrumentation_psd_quant_noise. % \begin{equation}\label{eq:detail_instrumentation_psd_quant_noise} % \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 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) % \end{equation} % With a sampling frequency $F_s = 10\,\text{kHz}$, an input range $\Delta V = 20\,V$ and a maximum allowed ASD $\Gamma_{\text{max}} = 11\,\mu V/\sqrt{Hz}$, the minimum number of bits is $n_{\text{min}} = 12.4$, which is readily achievable with commercial ADCs. delta_V = 20; % +/-10 V Fs = 10e3; % Sampling Frequency [Hz] max_adc_asd = 11e-6; % V/sqrt(Hz) min_n = log2(delta_V/(sqrt(12*Fs)*max_adc_asd)) %% Estimate quantization noise of the ADC delta_V = 20; % +/-10 V n = 16; % number of bits Fs = 10e3; % [Hz] q = delta_V/2^n; % Quantization in [V] q_psd = q^2/12/Fs; % Quantization noise Power Spectral Density [V^2/Hz] q_asd = sqrt(q_psd) % Quantization noise Amplitude Spectral Density [V/sqrt(Hz)]