Time above threshold for an RC-filtered rectangular pulse ($W=t_{\downarrow }-t_{\uparrow }$)

The time above threshold for an RC-filtered rectangular pulse is the separation of its rising and falling threshold crossings, rather than the applied pulse duration. In the relaxed first-order model, put $a=V_{\mathrm{th}}/V_0$ with $0<a<1$ and $T>T_{\min }=-\tau \log (1-a)$. The parent crossing calculation givesThis excludes comparator propagation dynamics and uses the same threshold on both edges. A Schmitt trigger with unequal switching thresholds needs a different formula.

The positive-width relation can be inverted exactly:Its derivative is $dW/dT=1/(1-e^{-T/\tau })>1$. Thus a measured positive $W$ determines a unique input duration under the stated model. A zero-width observation does not: all rejected durations $T\le T_{\min }$ have that same outcome.

For an original example, take $a=0.25$, $\tau =1\mu \mathrm{s}$ and $T=2\mu \mathrm{s}$. The output remains above threshold for about $2.9532\mu \mathrm{s}$, longer than the input pulse because of its decaying tail. Substitution into the inverse relation recovers $2\mu \mathrm{s}$.

For long pulses, $W-T$ tends to $\tau \log [(1-a)/a]$. It is positive below a half-height threshold and negative above one. At exactly half height, $W<T$ for every finite accepted pulse, with the difference tending to zero for long pulses. Consequently a measured above-threshold width cannot be equated to the input width without the threshold and filter model.

A pulse-generator/oscilloscope comparison can record both widths while varying threshold. The measured RC decay should be checked independently, using the natural-response method in the primary MIT laboratory.^[ref] Initial residual charge, hysteresis, and different rising/falling comparator delays alter the observation; the algebra above only inverts the ideal capacitor waveform.