= Quiet interval required for an RC baseline timing budget
{title2=$D\ge-\tau\log(1-e^{-\delta/\tau})$}
The <quiet interval required for an RC baseline timing budget> controls how much a residual <capacitor> baseline may advance the next threshold crossing. In <residual baseline in RC pulse-width discrimination>, a positive baseline $V_b$ changes the relaxed crossing by
$$
\Delta t=\tau\log(1-V_b/V_0).
$$
For a permitted advance $\delta>0$, requiring $|\Delta t|\le\delta$ is equivalent to $V_b/V_0\le\epsilon$, where $\epsilon=1-e^{-\delta/\tau}$. To retain a fresh rising crossing, also require $\epsilon<a=V_{\rm th}/V_0$, equivalently $\delta< -\tau\log(1-a)$.
After a previous peak $V_p$, a zero-input low interval $D$ gives $V_b=V_pe^{-D/\tau}$. Hence a measured upper bound on that peak gives the sufficient interval
$$
D\ge\tau\max\left\{0,\log\frac{V_p}{V_0\epsilon}\right\}.
$$
Using only the model bound $V_p\le V_0$ recovers the conservative expression in the header, $D\ge-\tau\log(1-e^{-\delta/\tau})$. The peak bound and initial-state assumptions are part of the guarantee. The next pulse must still last long enough to produce the rising crossing; this timing bound alone does not guarantee acceptance.
For an original timing budget, let $\tau=2\,\mathrm{\mu s}$, $\delta=20\,\mathrm{ns}$ and $a=0.4$. Then $\epsilon\approx0.0099502<a$, and the conservative interval is $D\ge9.2203\,\mathrm{\mu s}$. If a validated peak bound is only $V_p\le0.60V_0$, it reduces to $8.1987\,\mathrm{\mu s}$. Simply calling either interval several time constants long would hide the actual timing specification.
For $\delta\ll\tau$, expansion gives $D\approx\tau\log(\tau/\delta)+\delta/2$ under the full-height peak bound. Tightening a small timing allowance by a factor of two therefore adds roughly $\tau\log2$ of quiet time. If the allowed advance is as large as the whole relaxed crossing time, a separate reset condition is still required; the logarithmic timing budget alone no longer ensures the initial output is below threshold.
A bench test measures the peak and the baseline while sweeping the low interval, keeping the threshold and pulse height fixed. MIT's RC laboratory supplies the natural-decay measurement used to check the underlying <time constant>.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2, repetitive excitation and RC decay]{ref} Probe <capacitance> and source resistance belong in that measured <time constant>.
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