Decay Time is Q Periods

Another well known rule of thumb is that the $ Q$ of a resonator is the number of ``periods'' under the exponential decay of its impulse response. More precisely, we will show that, for $ Q\gg 1/2$, the impulse response decays by the factor $ e^{-\pi}$ in $ Q$ cycles, which is about 96 percent decay, or -27 dB.

The impulse response corresponding to Eq.$ \,$(E.8) is found by inverting the Laplace transform of the transfer function $ H(s)$. Since it is only second order, the solution can be found in many tables of Laplace transforms. Alternatively, we can break it up into a sum of first-order terms which are invertible by inspection (possibly after rederiving the Laplace transform of an exponential decay, which is very simple). Thus we perform the partial fraction expansion of Eq.$ \,$(E.8) to obtain

$\displaystyle H(s) = \frac{g_1}{s-p_1} + \frac{g_2}{s-p_2}
$

where $ p_i$ are given by Eq.$ \,$(E.9), and some algebra gives
$\displaystyle g_1$ $\displaystyle =$ $\displaystyle -g\frac{p_1}{p_2-p_1}$ (E.12)
$\displaystyle g_2$ $\displaystyle =$ $\displaystyle g\frac{p_2}{p_2-p_1}$ (E.13)

as the respective residues of the poles $ p_i$.

The impulse response is thus

$\displaystyle h(t) = g_1 e^{p_1t} + g_2 e^{p_2t}.
$

Assuming a resonator, $ Q>1/2$, we have $ p_2 = \overline{p}_1$, where $ p_1=\sigma_p +j\omega_p = -\alpha + j\omega_d$ (using notation of the preceding section), and the impulse response reduces to

$\displaystyle h(t) = g_1\,e^{p_1 t} + \overline{g}_1\,e^{\overline{p}_1 t} = A\,e^{-\alpha t} \cos(\omega_p t + \phi)
$

where $ A$ and $ \phi$ are overall amplitude and phase constants, respectively.E.1

We have shown so far that the impulse response $ h(t)$ decays as $ e^{-\alpha t}$ with a sinusoidal radian frequency $ \omega_p=\omega_d$ under the exponential envelope. After Q periods at frequency $ \omega_p$, time has advanced to

$\displaystyle t_Q = Q\frac{2\pi}{\omega_p}
\approx \frac{2\pi Q}{\omega_0}
= \frac{\pi}{\alpha},
$

where we have used the definition Eq.$ \,$(E.7) $ Q\isdef \omega_0/(2\alpha)$. Thus, after $ Q$ periods, the amplitude envelope has decayed to

$\displaystyle e^{-\alpha t_Q} = e^{-\pi} \approx 0.043\dots
$

which is about 96 percent decay. The only approximation in this derivation was

$\displaystyle \omega_p = \sqrt{\omega_0^2 - \alpha^2} \approx \omega_0
$

which holds whenever $ \alpha\ll\omega_0$, or $ Q\gg 1/2$.


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Q as Energy Stored over Energy Dissipated
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Impulse Response