 Missing comparator reset in an RC pulse train

Missing comparator reset in an RC pulse train occurs when the filtered output never spends a finite interval below the reset threshold between pulses. In the periodic first-order model, with $\alpha =e^{-T/\tau }$ and $\beta =e^{-D/\tau }$, the baseline is the minimum output and equalsFor a nonhysteretic ideal electronic comparator with $a=V_{\mathrm{th}}/V_0$, $V_b\ge V_{\mathrm{th}}$ leaves no finite low interval. At equality the waveform merely touches the threshold. This is different from a pulse whose peak never reaches threshold and therefore gives no finite high interval.

Solving the strict reset condition givesSeparate rising and falling transitions also require the peak to exceed the threshold. Both inequalities matter, so increasing the quiet interval is not automatically sufficient to produce an accepted output pulse from a weak input.

For an original three-regime example, take $\tau =1\mu \mathrm{s}$, $V_0=1\mathrm{V}$, $T=0.50\mu \mathrm{s}$ and $V_{\mathrm{th}}=0.40\mathrm{V}$. The reset boundary is $D\approx 0.46386\mu \mathrm{s}$. At $D=0.20\mu \mathrm{s}$, the periodic baseline/peak are about $0.63992/0.78160\mathrm{V}$: after startup the electronic comparator does not reset. At $D=1.0\mu \mathrm{s}$, they are $0.18632/0.50648\mathrm{V}$, allowing both transitions. At $D=10\mu \mathrm{s}$, the peak is only about $0.39348\mathrm{V}$, so no finite high interval is produced. The same input height and high duration can therefore give three different output patterns as repetition changes.

The threshold-reset margin for a periodic RC pulse train must also accommodate uncertainty and switching dynamics. With a Schmitt trigger, the lower switching threshold controls reset while the upper one controls acceptance. The capacitor-level inequalities alone do not specify propagation delay or a real electronic comparator's minimum switching dwell time.

A test records the source, capacitor output and electronic comparator output together while sweeping $D$. A steady high output with a pulsing input should be compared with the measured capacitor minimum before being attributed to a source failure. The RC decay model can be checked independently using the primary laboratory's natural-response method.^[ref]