From d3a9263f37b6d9556ff353042ca98ce29f0af199 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Wed, 8 May 2013 11:08:59 -0400 Subject: [PATCH] calibcant/theory.tex: Rework general derivations to remove G_1 (G_1p -> G_1) Values for the G constants are more easily calculated from the equipartition theorem than through chaining G definitions. I removed G_1 entirely, and renamed G_1p -> G_1. I also check my results against publishes sources. It would have been helpful if I had discovered burnham03 earlier :p. They have a general PSD formula, white and 1/f noise terms, and interference-based piezo calibration. Institutionalizing knowledge like this (vs. relying on vast literature searches) is a good reason to consolidate work around a small numer of open source tools. Sigh. --- src/calibcant/procedure.tex | 14 +-- src/calibcant/theory.tex | 215 +++++++++++++++++++++++++----------- 2 files changed, 155 insertions(+), 74 deletions(-) diff --git a/src/calibcant/procedure.tex b/src/calibcant/procedure.tex index 90aa66a..840db0a 100644 --- a/src/calibcant/procedure.tex +++ b/src/calibcant/procedure.tex @@ -143,12 +143,12 @@ vibrations are configurable (with \hFconfig, panel shows the $\PSD_f(V_p,f)$ with a fit following \cref{eq:psd-Vp-offset}. The constant offset $P_{0f}$, drawn as the horizontal line in the third panel, accounts for white noise - in the measurement circuit. The vertical line marks the peak - frequency $f_\text{max}$ (\cref{eq:peak-frequency}). Only data - in the blue region was used when computing the best fit. This - is the first vibration from the 2013-02-07T08-20-46 calibration, - yielding a fitted variance - $\avg{V_p(t)^2}=96.90\pm0.99\U{mV$^2$}$. + in the measurement circuit\citep{burnham03}. The vertical line + marks the peak frequency $f_\text{max}$ + (\cref{eq:peak-frequency}). Only data in the blue region was + used when computing the best fit. This is the first vibration + from the 2013-02-07T08-20-46 calibration, yielding a fitted + variance $\avg{V_p(t)^2}=96.90\pm0.99\U{mV$^2$}$. \label{fig:calibcant:vibration}} \end{center} \end{figure} @@ -177,7 +177,7 @@ attribute this to background white noise in the measurement circuit, and not due to cantilever oscillation. To avoid artificially inflating the estimated $\avg{V_p(t)^2}$, I created an alternative model for $\PSD_f(V_p,f)$ that adds a frequency-independent offset -$P_{0f}$. +$P_{0f}$\citep{burnham03}. \begin{equation} \PSD_f(V_p, f) = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} + P_{0f} \;. diff --git a/src/calibcant/theory.tex b/src/calibcant/theory.tex index 6f6f73e..abc8014 100644 --- a/src/calibcant/theory.tex +++ b/src/calibcant/theory.tex @@ -15,7 +15,7 @@ where $x$ is the displacement from equilibrium\index{$x$}, $F(t)$ is the external driving force\index{$F(t)$}. During the non-contact phase of calibration, $F(t)$ comes from random thermal noise. - +% \nomenclature{$\beta$}{Damped harmonic oscillator drag-acceleration coefficient $\beta \equiv \gamma/m$} \nomenclature{$\gamma$}{Damped harmonic oscillator drag coefficient @@ -101,8 +101,8 @@ For highly damped systems, the inertial term in \cref{eq:DHO} becomes insignificant ($m \rightarrow 0$). This model is commonly used for optically trapped beads\citep{bechhoefer02}. Because it is simpler and solutions are more easily -available\citep{bechhoefer02,grossman05}, it will serve to outline the -general approach before we dive into the general case. +available\citep{bechhoefer02,burnham03,grossman05}, it will serve to +outline the general approach before we dive into the general case. Fourier transforming \cref{eq:DHO} with $m=0$ and applying \cref{eq:four-deriv} we have @@ -127,9 +127,12 @@ Because thermal noise is white (not autocorrelated + Wiener-Khinchin Theorem), we can write the one sided thermal power spectral density per unit time as \begin{equation} - \PSD(F, \omega) = G_0 + G_0 \equiv \PSD(F, \omega) = \normLimT 2 \magSq{F(\omega)} \;. \label{eq:GOdef} % label O != zero \end{equation} +% +\nomenclature{$G_0$}{The power spectrum of the thermal noise in + angular frequency space (\cref{eq:GOdef}).} Plugging \cref{eq:GOdef} into \cref{eq:ODHO-psd-F} we have \begin{equation} @@ -146,24 +149,35 @@ time yields \iOInf{\omega}{\PSD(x, \omega)} = \iOInf{\omega}{\frac{G_0}{\kappa^2 + \gamma^2\omega^2}} = \frac{G_0}{\gamma}\iOInf{z}{\frac{1}{\kappa^2 + z^2}} - = \frac{G_0 \pi}{2 \gamma \kappa} \;, + = \frac{\pi G_0}{2 \gamma \kappa} \;, \label{eq:ODHO-psd-int} \end{align} -where the integral is solved in \cref{sec:integrals:highly-damped}. +where we made the simplifying replacement $z\equiv\gamma\omega$, so +$\dd \omega = \dd z/\gamma$. The integral is solved in +\cref{sec:integrals:highly-damped}. Plugging into our corollary to Parseval's theorem (\cref{eq:parseval-var}), \begin{equation} - \avg{x(t)^2} = \frac{G_0 \pi}{2 \gamma \kappa} \;. \label{eq:ODHO-var} + \avg{x(t)^2} = \frac{\pi G_0}{2 \gamma \kappa} \;. \label{eq:ODHO-var} \end{equation} Plugging \cref{eq:ODHO-var} into \cref{eq:equipart} we have \begin{align} - \kappa \frac{G_0 \pi}{2 \gamma \kappa} &= k_BT \\ + \kappa \frac{\pi G_0}{2 \gamma \kappa} &= k_BT \\ G_0 &= \frac{2 \gamma k_BT}{\pi} \;. \label{eq:ODHO-GO} \end{align} Combining \cref{eq:ODHO-psd-GO,eq:ODHO-GO}, we expect $x(t)$ to have a -power spectral density per unit time given by +power spectral density per unit time given by\footnote{% + \cref{eq:ODHO-psd} is Eq.~(A12) from \citet{bechhoefer02} (who's + $\tau_0\equiv\gamma/\kappa$), except that they're missing a factor + of $1/\pi$. + \cref{eq:ODHO-psd} is also Eq.~(8) from \citet{burnham03}, where + their damping coefficient $b$ is equivalent to our $\gamma$, their + frequency $\nu$ is equivalent to our $f=\omega/2\pi$, and their roll + off frequency $\nu_R\equiv k/2\pi b$ is equivalent to our + $\kappa/2\pi\gamma$. +} \begin{equation} \PSD(x, \omega) = \frac{2}{\pi} \cdot @@ -208,41 +222,50 @@ We compute the \PSD\ by plugging \cref{eq:DHO-xmag} into \index{PSD@\PSD} Plugging \cref{eq:GOdef} into \cref{eq:DHO-psd-F} we have -\begin{align} - \PSD(x, \omega) &= \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;, - \label{eq:model-psd} \\ - &= \frac{G_1}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2} \;, - \label{eq:model-psd-Gone} -\end{align} -where $G_1\equiv G_0/m^2$ consolidates the unknown fitting parameters -without loss of generality. +\begin{equation} + \PSD(x, \omega) = \frac{G_0/m^2}{(\omega_0^2-\omega^2)^2 +\beta^2\omega^2}\;, + \label{eq:model-psd} \;. +\end{equation} Integrating over positive $\omega$ to find the total power per unit time yields -\begin{align} +\begin{equation} \iOInf{\omega}{\PSD(x, \omega)} - &= \frac{G_0}{2m^2} - \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}} - = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2} - = \frac{G_0 \pi}{2m^2\beta\omega_0^2} - = \frac{G_0 \pi}{2m^2\beta \frac{\kappa}{m}} \\ - &= \frac{G_0 \pi}{2m \beta \kappa} \;, + = \frac{G_0}{2m^2} + \iInfInf{\omega}{\frac{1}{(\omega_0^2-\omega^2)^2 + \beta^2\omega^2}} + = \frac{G_0}{2m^2} \cdot \frac{\pi}{\beta\omega_0^2} + = \frac{\pi G_0}{2m^2 \beta \omega_0^2} \label{eq:DHO-psd-int} -\end{align} -where the integration is solved in \cref{sec:integrals:general}. +\end{equation} +where the integration is solved in \cref{sec:integrals:general}\footnote{ + Comparing \cref{eq:ODHO-psd-int,eq:DHO-psd-int}, we see + \begin{equation} + \frac{\pi G_0}{2m^2 \beta \omega_0^2} + = \frac{\pi G_0}{2m^2 \frac{\gamma}{m} \frac{k}{m}} + = \frac{\pi G_0}{2 \gamma \kappa} \;. + \end{equation} + This is not a coincidence. Both spectra satisfy the equipartion + theorem, so + \begin{equation} + \iOInf{\omega}{\PSD(x, \omega)} = \avg{x(t)^2} = \frac{k_BT}{\kappa} \;, + \end{equation} + which is the same for both cases. +}. By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have -\begin{align} +\begin{equation} \avg{x(t)^2} - &= \frac{G_0 \pi}{2m^2\beta\omega_0^2} \;, \label{eq:DHO-var} \\ - &= \frac{G_1 \pi}{\beta\omega_0^2} \;, \label{eq:DHO-var-Gone} -\end{align} + = \frac{\pi G_0}{2m^2 \beta \omega_0^2} \;. \label{eq:DHO-var} +\end{equation} Plugging \cref{eq:DHO-var} into the equipartition theorem -(\cref{eq:equipart}) we have +(\cref{eq:equipart}) we can reproduce \cref{eq:ODHO-GO}. \begin{align} - \kappa \frac{G_0 \pi}{2m \beta \kappa} &= k_BT \, \\ - G_0 &= \frac{2}{\pi} k_BT m \beta \;, \label{eq:GO} \\ - G_1 &\equiv \frac{G_0}{m^2} = \frac{2}{\pi m} k_BT \beta \;. \label{eq:Gone} + \kappa \frac{\pi G_0}{2m^2 \beta \omega_0^2} &= k_BT \, \\ + G_0 &= \frac{2m^2 \beta \omega_0^2 k_BT}{\pi \kappa} + = \frac{2m^2 \beta \frac{\kappa}{m} k_BT}{\pi \kappa} + = \frac{2m \beta k_BT}{\pi} + = \frac{2m \frac{\gamma}{m} k_BT}{\pi} + = \frac{2 \gamma k_BT}{\pi} \;. \label{eq:GO} \end{align} Combining \cref{eq:model-psd,eq:GO}, we expect $x(t)$ to have a power @@ -258,10 +281,10 @@ As expected, we can recover the extremely overdamped form \cref{eq:ODHO-psd} from the general form \cref{eq:DHO-psd}. Plugging in for $\beta\equiv\gamma/m$ and $\omega_0\equiv\sqrt{\kappa/m}$, \begin{align} - \lim_{m\rightarrow 0} \PSD(x, \omega) - &= \lim_{m\rightarrow 0} \frac{2 k_BT \gamma} + \limX{m}{0} \PSD(x, \omega) + &= \limX{m}{0} \frac{2 k_BT \gamma} { \pi m^2 \p[{\p({\frac{\kappa}{m}-\omega^2})^2 + \frac{\gamma^2}{m^2}\omega^2}] } - = \lim_{m\rightarrow 0} \frac{2 k_BT \gamma} + = \limX{m}{0} \frac{2 k_BT \gamma} { \pi \p[{(\kappa-m\omega^2)^2 + \gamma^2\omega^2}] } \\ &= \frac{2}{\pi} \cdot @@ -280,38 +303,56 @@ into \cref{eq:DHO}, + \omega_0^2 \frac{V_p}{\sigma_p} &= F(t) \\ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p - &= \sigma_p\frac{F(t)}{m} \\ + &= \sigma_p\frac{F(t)}{m} \label{eq:DHO-ddt-Vp} \\ \ddt{V_p} + \beta\dt{V_p} + \omega_0^2 V_p - &= \frac{F(t)}{m_p} \;, + &= \frac{F_p(t)}{m} \;, \end{align} -where $m_p\equiv m/\sigma_p$. This has the same form as +where $F_p(t)\equiv \sigma_p F(t)$. This has the same form as \cref{eq:DHO}, which can be rearranged to: \begin{align} \ddt{x} + \frac{\gamma}{m} \dt{x} + \frac{\kappa}{m} x &= \frac{F(t)}{m} \\ \ddt{x} + \beta \dt{x} + \omega_0^2 x &= \frac{F(t)}{m} \;, \end{align} so the \PSD\ of $V_p(t)$ will be the same as the \PSD\ of $x(t)$, -after the replacements $x\rightarrow V_p(t)$ and $m\rightarrow m_p$. -Making these replacements in \cref{eq:model-psd-Gone,eq:DHO-var-Gone}, -we have +after the replacements $x\rightarrow V_p(t)$, $F\rightarrow F_p$, and +(because of \cref{eq:GOdef}) $G_0\rightarrow\sigma_p^2G_0$. Making +these replacements in \cref{eq:model-psd,eq:DHO-var}, we have \begin{align} - \PSD(V_p, \omega) &= \frac{G_{1p}} + \PSD(V_p, \omega) &= \frac{\sigma_p^2 G_0/m^2} { (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } \\ - \avg{V_p(t)^2} &= \frac{\pi G_{1p}}{2\beta\omega_0^2} - = \frac{\pi \sigma_p^2 G_{1}}{2\beta\omega_0^2} - = \sigma_p^2 \avg{x(t)^2} \;, + \avg{V_p(t)^2} &= \frac{\pi \sigma_p^2 G_0}{2 m^2 \beta \omega_0^2} + = \sigma_p^2 \avg{x(t)^2} \;. +\end{align} +The scaling parameters cannot be independently fit though, so lets +condense the power spectrum of the right hand side of +\cref{eq:DHO-ddt-Vp} into a single +\begin{equation} + G_1 \equiv \frac{\sigma_p^2 G_0}{m^2} \;. \label{eq:Gone-def} +\end{equation} +This gives +\begin{align} + \PSD(V_p, \omega) + &= \frac{G_1}{ (\omega_0^2-\omega^2)^2 + \beta^2\omega^2 } + \label{eq:psd-Vp-Gone} \\ + \avg{V_p(t)^2} &= \frac{\pi G_1}{2 \beta \omega_0^2} + = \sigma_p^2 \avg{x(t)^2} \;. + \label{eq:avg-Vp-Gone} \end{align} -where $G_{1p}\equiv G_0/m_p^2=\sigma_p^2 G_1$. +% +\nomenclature{$G_1$}{The scaled power spectrum of the thermal noise in + angular frequency space (\cref{eq:Gone-def}).} + Plugging into the equipartition theorem (\cref{eq:equipart_k}) yields \begin{align} \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} - = \frac{2 \beta\omega_0^2 \sigma_p^2 k_BT}{\pi G_{1p}} \;. + = \frac{2 \beta \omega_0^2 \sigma_p^2 k_BT}{\pi G_1} \;. \end{align} - -From \cref{eq:Gone}, we find the expected value of $G_{1p}$ to be +Shifting this around, we can find the expected value of $G_1$. \begin{equation} - G_{1p} \equiv \sigma_p^2 G_1 = \frac{2}{\pi m} \sigma_p^2 k_BT \beta \;. - \label{eq:Gone-p} + G_1 = \frac{2 \beta \omega_0^2 \sigma_p^2 k_BT}{\pi \kappa} + = \frac{2 \beta \frac{\kappa}{m} \sigma_p^2 k_BT}{\pi \kappa} + = \frac{2 \beta \sigma_p^2 k_BT}{\pi m} + \label{eq:Gone} \end{equation} \subsection{Fitting deflection voltage in frequency space} @@ -347,35 +388,75 @@ The variance of the function $x(t)$ is then given by plugging into = \iOInf{f}{\frac{1}{2\pi}\PSD_f(x,f)2\pi\cdot} = \iOInf{f}{\PSD_f(x,f)} \;. \end{align} -We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit}. +We can now extract \cref{eq:psd-Vp,eq:Vp-from-freq-fit} from +\cref{eq:psd-Vp-Gone,eq:avg-Vp-Gone}. \begin{align} \begin{split} - \PSD_f(V_p, f) &= 2\pi\PSD(V_p,\omega) - = \frac{2\pi G_{1p}}{(4\pi f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2} - = \frac{2\pi G_{1p}}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\ - &= \frac{G_{1p}/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}} + \PSD_f(V_p, f) &= 2\pi\PSD(V_p, \omega) + = \frac{2\pi G_1}{(4\pi^2f_0^2-4\pi^2f^2)^2 + \beta^2 4\pi^2f^2} + = \frac{2\pi G_1}{16\pi^4(f_0^2-f^2)^2 + \beta^2 4\pi^2f^2} \\ + &= \frac{G_1/8\pi^3}{(f_0^2-f^2)^2 + \frac{\beta^2 f^2}{4\pi^2}} = \frac{G_{1f}}{(f_0^2-f^2)^2 + \beta_f^2 f^2} \end{split} \\ - \avg{V_p(t)^2} &= \frac{\pi G_{1f}}{2\beta_f f_0^2} \;. -% = \frac{\pi G_{1p} / (2\pi)^3}{2\beta/(2\pi) \omega_0^2/(2\pi)^2} -% = \frac{\pi G_{1p}}{2\beta\omega_0^2} = \avg{V_p(t)^2} % check! + \avg{V_p(t)^2} + &= \frac{\pi \frac{G_1}{(2\pi)^3}} + {2 \frac{\beta}{2\pi} \p({\frac{\omega_0}{2\pi}})^2} + = \frac{\pi G_{1f}}{2 \beta_f f_0^2} \;. \end{align} where $f_0\equiv\omega_0/2\pi$, $\beta_f\equiv\beta/2\pi$, and -$G_{1f}\equiv G_{1p}/8\pi^3$. Finally, we can generate -\cref{eq:kappa}. +$G_{1f}\equiv G_1/8\pi^3$. Finally, we can generate +\cref{eq:kappa} from \cref{eq:equipart_k,eq:x-from-Vp}. \begin{align} \kappa &= \frac{\sigma_p^2 k_BT}{\avg{V_p(t)^2}} = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi G_{1f}} \;. \end{align} -From \cref{eq:Gone-p}, we expect $G_{1f}$ to be +Shifting this around, we can find the expected value of $G_{1f}$. \begin{equation} - G_{1f} = \frac{G_{1p}}{8\pi^3} - = \frac{\frac{2}{\pi m} \sigma_p^2 k_BT \beta}{8\pi^3} - = \frac{\sigma_p^2 k_BT \beta}{4\pi^4 m} \;. + G_1 = \frac{2 \beta_f f_0^2 \sigma_p^2 k_BT}{\pi \kappa} + = \frac{2 \beta_f \frac{\kappa}{4\pi^2 m} \sigma_p^2 k_BT}{\pi \kappa} + = \frac{\beta_f \sigma_p^2 k_BT}{2\pi^3 m} \label{eq:Gone-f} \end{equation} +Plugging \cref{eq:Gone-f} into \cref{eq:psd-Vp}, we have +\begin{equation} + \PSD_f(V_p, f) = \frac{\sigma_p^2 k_BT \beta_f}{2\pi^3 m} \cdot + \frac{1}{(f_0^2-f^2)^2 + \beta_f^2 f^2} +\end{equation} +From which we can recover \citet{burnham03}'s Eq.~(6). +\begin{align} + \PSD_f(x, f) &= \frac{\PSD_f(V_p, f)}{\sigma_p^2} + = \frac{k_BT \colA{\beta_f}}{2\pi^3 m} \cdot + \frac{1}{(f_0^2-f^2)^2 + \colA{\beta_f^2} f^2} \\ + &= \frac{k_BT \colAB{f_0}}{2\pi^3 m \colA{Q}} \cdot + \frac{1}{\colB{(f_0^2}-f^2)^2 + \frac{\colAB{f_0^2}f^2}{\colA{Q^2}}} + = \frac{k_BT}{2\pi^3 m Q \colAB{f_0^3}} \cdot + \frac{1}{(\colB{1}-\frac{f^2}{\colB{f_0^2}})^2 + + \frac{f^2}{\colB{f_0^2}Q^2}} \\ + &= \frac{k_BT}{\colB{2\pi^3} m Q \colAB{\p({\frac{\omega_0}{2\pi}})^3}} + \cdot + \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} + = \frac{\colB{4}k_BT}{m Q \colB{\omega_0}\colAB{\omega_0^2}} \cdot + \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} \\ + &= \frac{4k_BT}{\colB{m} Q \omega_0\frac{\colA{\kappa}}{\colAB{m}}} \cdot + \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} + = \frac{4 k_BT}{\omega_0 Q \kappa} + \frac{1}{(1-\frac{f^2}{f_0^2})^2 + \frac{f^2}{f_0^2Q^2}} \;, + \label{eq:psd-f-x} +\end{align} +where $Q$ is the quality factor\citep{burnham03} +\begin{equation} + Q \equiv \frac{\sqrt{\kappa m}}{\gamma} + = \sqrt{\frac{\kappa}{m}}\frac{m}{\gamma} + = \frac{\omega_0}{\beta} + = \frac{2\pi f_0}{2\pi\beta_f} + = \frac{f_0}{\beta_f} \;. + \label{eq:Q} +\end{equation} +% +\nomenclature{$Q$}{Quality factor of a damped harmonic oscillator. + $Q\equiv \frac{\sqrt{\kappa m}}{\gamma}$ (\cref{eq:Q}).} % TODO: re-integrate the following -- 2.26.2