Reset dwell time below threshold in an RC pulse train ($t_{\mathrm{low}}$)

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 $V_b<V_{\mathrm{th}}<V_p$. 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:The below-threshold interval runs from the falling crossing to the next rising crossing. ThereforeIncluding $t_{\uparrow }$ 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.^[ref]

For an original exact example, take $V_0=1\mathrm{V}$, $\tau =1\mu \mathrm{s}$ and $T=D=\tau \log 2$. 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 $V_{\mathrm{th}}=0.5\mathrm{V}$, both crossing delays equal $\tau \log (4/3)$, so $t_{\mathrm{low}}=D\approx 0.693147\mu \mathrm{s}$. The reset margin is a positive $1/6\mathrm{V}$, yet a stipulated requirement of at least $0.8\mu \mathrm{s}$ low time is not met.

The strict crossing assumptions are essential. At $V_{\mathrm{th}}=V_b$, 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.