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1 | 1 | \chapter{Definition and conversion of Q-factors} |
2 | 2 | \label{app:q_factors} |
3 | 3 |
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4 | | -In this appendix we consider a damped and driven harmonic oscillator described by the following equation of motion: |
| 4 | +The goal of this appendix is ot provide a clear definition of the Q-factor and to derive (using simulations) how the Q-factor can be obtained. In particular we will study a system described by the following differential equation: |
5 | 5 | \begin{equation} |
6 | | - \ddot{x} + \gamma \dot{x} + \omega_0^2 x = D(t) |
| 6 | + \ddot{x} - \gamma \dot{x} + \omega_0^2 x = D(t) |
| 7 | + \tag{damped h.o.} |
7 | 8 | \end{equation} |
8 | | -where $x$ is some amplitude in \unit{\meter}, $\gamma$ is the damping rate in \unit{\per\second}, $\omega_0 = 2\pi f_0$ is the resonance frequency in \unit{\radian\per\second} and $D(t)$ is the driving force. The Q-factor is a measure for the width (and height) of the resonance peak. |
| 9 | +where $x$ is the signal amplitude (we will use meters, but the results are independent of units), $\gamma$ is the damping rate in \unit{\per\second}, $\omega_0 = 2\pi f_0$ is the resonance frequency in \unit{\radian\per\second} and $D(t)$ is the driving force. We specifically consider the case where $0 < \gamma \leq 1$. For a more general treatment of Q-factors, including non-linear systems, see \citeauthor{wang_rigorous_2017}. |
9 | 10 |
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| 11 | +\section{Q-factor definition} |
10 | 12 | There are two common definitions of the Q-factor. The first is the \textit{bandwidth} definition, where the Q-factor is given by the ratio between the resonance frequency and the bandwidth of the resonance. |
11 | 13 | \begin{equation} |
12 | 14 | Q_\text{B} = \frac{f_0}{\Delta f} = \frac{\omega_0}{\Delta \omega} |
| 15 | + \tag{bandwidth definition} |
13 | 16 | \end{equation} |
14 | | -Given a resonance peak in a spectrum, the bandwidth is defined as the width of the peak at half the maximum amplitude (FWHM). The FWHM should be determined by fitting a Lorentzian (Cauchy distribution) to the peak in the power spectrum. |
| 17 | +where $\Delta \omega$ is the full width at half maximum (FWHM) of the resonance peak. The second definition is the \textit{energy} definition, where the Q-factor is given by the ratio between the energy stored in the oscillator and the energy dissipation (per cycle): |
15 | 18 | \begin{equation} |
16 | | - \mathcal{L}(f) \propto \frac{\Gamma}{(f - f_0)^2 + \Gamma^2} |
| 19 | + Q_\text{E} = \frac{\text{stored energy}}{\text{dissipated energy per cycle}} = f_0 \frac{\text{stored energy}}{\text{dissipation}} |
| 20 | + \tag{energy definition} |
17 | 21 | \end{equation} |
18 | | -In this case $\Gamma$ is the scale of the peak and the FWHM is given by $2\Gamma$. Additionally the FWHM is equal to the damping rate $\gamma$. When looking at a peak in a `normal' Fourier spectrum, meaning you look at the amplitude of the signal, the magnitude of the peak is instead described by: |
| 22 | + |
| 23 | +\section{FWHM} |
| 24 | +The first method of determining the Q-factor is by looking at the FWHM of the resonance peak. There are two ways of doing so, either by looking at the FWHM in the power spectrum or the FWHM in the Fourier spectrum. |
| 25 | + |
| 26 | +\subsection{Power spectrum} |
| 27 | +In the power spectrum (PSD), the peak is described by a Lorentzian distribution: |
19 | 28 | \begin{equation} |
20 | | - A(f) \propto \sqrt{\frac{1}{(f - f_0)^2 + \gamma^2f^2}} |
| 29 | + \mathcal{L}(f) \propto \frac{\frac{1}{2} \Gamma}{(f - f_0)^2 + \left(\frac{1}{2} \Gamma\right)^2} %\quad \text{or} \quad \mathcal{L}(\omega) \propto \frac{\frac{1}{2} \Gamma}{(\omega - \omega_0)^2 + \left(\frac{1}{2} \Gamma\right)^2} |
21 | 30 | \end{equation} |
22 | | -The FWHM in this case is given by $\sqrt{2}\gamma$. This means that the Q-factor of a `normal' Fourier spectrum deviates from the Q-factor of a power spectrum by a factor of $\sqrt{2}$! Since it is most common to look at the power spectrum, we will use the power spectrum definition of the Q-factor. |
23 | | - |
24 | | -Another way of defining the Q-factor is the \textit{energy} definition. The Q-factor is then given by the ratio between the energy stored in the oscillator and the energy dissipation. |
| 31 | +where $\Gamma$ is the FWHM of the peak. Coincidentally, the FWHM is equal to the damping rate $\gamma$. The Q-factor in terms of the FWHM in the power spectrum is then given by: |
25 | 32 | \begin{equation} |
26 | | - Q_\text{E} = 2 \pi \frac{\text{stored energy}}{\text{dissipated energy per cycle}} |
| 33 | + Q_\text{B} = \frac{f_0}{\Gamma} = \frac{f_0}{\gamma} |
27 | 34 | \end{equation} |
28 | | -Or equivalently: |
| 35 | +There is however a caveat. When looking at a peak in a `normal' Fourier spectrum, meaning you look at the amplitude of the signal, the magnitude of the peak is instead described by (this can be found in most textbooks on classical mechanics, such as Chapter~3.6 in \citeauthor{fowles_analytical_2005}): |
29 | 36 | \begin{equation} |
30 | | - Q_\text{E} = 2 \pi f_0 \frac{\text{stored energy}}{\text{dissipation}} |
| 37 | + A(f) \propto \sqrt{\frac{1}{(f^2 - f_0^2)^2 + \gamma^2f^2}} |
31 | 38 | \end{equation} |
32 | | -The energy of a damped harmonic oscillator decays exponentially with time where the decay time is given by $1/\gamma$. Thus the Q-factor in terms of $\gamma$ is given by: |
| 39 | +The FWHM in this case is approximated by $\sqrt{2}\gamma$. This means that the Q-factor of a `normal' Fourier spectrum deviates from the Q-factor of a power spectrum by a factor of $\sqrt{2}$! Since it is most common to look at the power spectrum, we will use the power spectrum definition of the Q-factor. |
| 40 | + |
| 41 | +\section{Decay time} |
| 42 | +Another simple way of measuring the Q-factor of a resonator is by looking at the decay time of the signal. There are two ways of doing so, either by looking at the decay in amplitude or the decay in energy. They are related to $\gamma$ as $\gamma = 1 / \tau_\text{energy} = 2 / \tau_\text{amplitude}$. To derive an expression for the Q-factor in terms of the decay time, we use that the energy dissipation is given by $\dot{E} = -\tau_\text{energy}E$. Substituting this into the energy definition of the Q-factor gives: |
33 | 43 | \begin{equation} |
34 | | - Q_\text{E} = 2 \pi f_0 \frac{E(t)}{\tau E(t)} = 2 \pi f_0 \tau_\text{energy} = \frac{2\pi f_0}{\gamma} |
| 44 | + Q_\text{E} = f_0 \frac{E(t)}{\tau_\text{energy} E(t)} = f_0 \tau_\text{energy} = \frac{1}{2} f_0 \tau_\text{amplitude} = \frac{f_0}{\gamma} |
35 | 45 | \end{equation} |
36 | | -Since we often observe a decaying amplitude in the time domain, to convert this decay time to the energy domain we can use $\tau_\text{energy} = \frac{1}{2} \tau_\text{amplitude}$. To convert between any Q-factor we assert that they must result in the same damping rate $\gamma$. As this is the physical quantity that underlies the system. As a reference we summarize the conversion factors in the table below. |
| 46 | +This also means that the value of $Q_\text{B}$ is the same as the value of $Q_\text{E}$! |
| 47 | + |
| 48 | +\section{Summary} |
| 49 | +In summary, calculating a Q-factor can be done in three ways: by looking at the decay time, the FWHM in the power spectrum or the FWHM in the Fourier spectrum. The table below serves as a tool to convert between the different definitions of the Q-factor by using $Q = \frac{f_0}{\gamma}$. |
37 | 50 |
|
38 | 51 | \begin{table}[h] |
39 | 52 | \centering |
40 | 53 | \begin{tabular}{ccc} |
41 | 54 | \toprule |
42 | | - \textbf{Observation} & $\bm{Q_B}$ & $\bm{Q_E}$ \\ |
| 55 | + \textbf{Observation} & $\bm{\gamma}$ & $\bm{Q}$ \\ |
43 | 56 | \midrule |
44 | | - FWHM in PSD ($\Delta f$) & $\frac{f_0}{\Delta f}$ & $2\pi \frac{f_0}{\Delta f}$ \\ |
45 | | - FWHM in FFT ($\Delta f$) & $\sqrt{2} \frac{f_0}{\Delta f}$ & $2\pi \sqrt{2} \frac{f_0}{\Delta f}$ \\ |
46 | | - Decay in energy ($\tau_\text{energy}$) & $f_0 \tau_\text{energy}$ & $2 \pi f_0 \tau_\text{energy}$ \\ |
47 | | - Decay in amplitude ($\tau_\text{amplitude}$) & $\frac{1}{2} f_0 \tau_\text{amplitude}$ & $\pi f_0 \tau_\text{amplitude}$ \\ |
| 57 | + FWHM in power spectrum ($\Delta f = \Gamma$) & $\Delta f$ & $f_0 / \Delta f$ \\ |
| 58 | + FWHM in Fourier spectrum ($\Delta f$) & $\Delta f / \sqrt{2}$ & $\sqrt{2} f_0 / \Delta f$ \\ |
| 59 | + Energy decay ($\tau_\text{energy}$) & $1 / \tau_\text{energy}$ & $f_0 \tau_\text{energy}$ \\ |
| 60 | + Amplitude decay ($\tau_\text{amplitude}$) & $2 / \tau_\text{amplitude}$ & $f_0 \tau_\text{amplitude}$ \\ |
48 | 61 | \bottomrule |
49 | 62 | \end{tabular} |
50 | 63 | \end{table} |
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