Quiet interval required for an RC baseline timing budget ($D\ge -\tau \log (1-e^{-\delta /\tau })$)

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 byFor a permitted advance $\delta >0$, requiring $|\Delta t|\le \delta$ is equivalent to $V_b/V_0\le ϵ$, where $ϵ=1-e^{-\delta /\tau }$. To retain a fresh rising crossing, also require $ϵ<a=V_{\mathrm{th}}/V_0$, equivalently $\delta <-\tau \log (1-a)$.

After a previous peak $V_p$, a zero-input low interval $D$ gives $V_b=V_p{e}^{-D/\tau }$. Hence a measured upper bound on that peak gives the sufficient intervalUsing 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 $\tau =2\mu \mathrm{s}$, $\delta =20\mathrm{ns}$ and $a=0.4$. Then $ϵ\approx 0.0099502<a$, and the conservative interval is $D\ge 9.2203\mu \mathrm{s}$. If a validated peak bound is only $V_p\le 0.60V_0$, it reduces to $8.1987\mu \mathrm{s}$. 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 $\tau \log 2$ 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.^[ref] Probe capacitance and source resistance belong in that measured time constant.