= Reset dwell time below threshold in an RC pulse train
{title2=$t_{\rm low}$}
The <reset dwell time below threshold in an RC pulse train> is the continuous time for which the filtered <electric potential difference> lies below an <electronic comparator>'s threshold between successive accepted pulses. A positive <threshold-reset margin for a periodic RC pulse train> ensures a crossing in the ideal model, but does not ensure a long enough low interval for the intended electronics.
Consider the periodic <RC low-pass filter> response to a high level $V_0>0$ lasting $T$, followed by zero input for a quiet time $D$. Let $\tau>0$ be the <time constant>, $V_b$ the periodic baseline at the start of the high interval and $V_p$ the peak at its end. Assume a nonhysteretic ideal <electronic comparator> and $V_b<V_{\rm th}<V_p$. Solving the two exponential segments gives the rising crossing time $t_\uparrow$, measured from the high interval's start, and the falling crossing delay $s_\downarrow$, measured from its end:
$$
t_\uparrow=\tau\log\frac{V_0-V_b}{V_0-V_{\rm th}},\qquad
s_\downarrow=\tau\log\frac{V_p}{V_{\rm th}}.
$$
The below-threshold interval runs from the falling crossing to the next rising crossing. Therefore
$$
t_{\rm low}=D-s_\downarrow+t_\uparrow.
$$
Including $t_\uparrow$ matters: the low interval usually extends into the next input high interval. The natural <exponential decay> underlying these segments is directly testable with the switched-RC apparatus in the MIT laboratory.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed p. 1, natural-response and time-constant experiment]{ref}
For an original exact example, take $V_0=1\,\mathrm V$, $\tau=1\,\mathrm{\mu s}$ and $T=D=\tau\log2$. The periodic solution in <missing comparator reset in an RC pulse train> gives $V_b=1/3\,\mathrm V$ and $V_p=2/3\,\mathrm V$. At $V_{\rm th}=0.5\,\mathrm V$, both crossing delays equal $\tau\log(4/3)$, so $t_{\rm low}=D\approx0.693147\,\mathrm{\mu s}$. The reset margin is a positive $1/6\,\mathrm V$, yet a stipulated requirement of at least $0.8\,\mathrm{\mu s}$ low time is not met.
The strict crossing assumptions are essential. At $V_{\rm th}=V_b$, the waveform only touches the threshold and the low interval vanishes. At or above the peak there is no finite high interval to separate. A <Schmitt trigger> uses distinct switching thresholds, so its output-state interval requires the corresponding falling and rising crossings rather than substituting one threshold indiscriminately.
A proposed measurement records the <pulse generator>, <capacitor> and <electronic comparator> outputs on an <oscilloscope>. Measure the low interval between actual output edges while also locating both capacitor-level crossings. Their difference can expose switching delays; the capacitor formula alone does not specify a real device's minimum pulse duration.
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