= RC pulse-width discrimination
{c}
<RC pulse-width discrimination> converts an input-duration test into a threshold observation on an <RC circuit>. For a relaxed first-order load with <time constant> $\tau>0$, a positive rectangular input of height $V_0$ and duration $T$ reaches peak output $V_0(1-e^{-T/\tau})$. An ideal <electronic comparator> has a nonzero interval above fixed threshold $0<V_{\rm th}<V_0$ precisely when
$$
T>\tau\log\frac{V_0}{V_0-V_{\rm th}}.
$$
Equality only touches the threshold at the peak. The parent <threshold-crossing time of a finite rectangular pulse through an RC load> derives the full response and the rejected-pulse boundary.
This is a duration discriminator only while the input height is controlled. At a fixed duration, the same condition is
$$
V_0>\frac{V_{\rm th}}{1-e^{-T/\tau}}.
$$
A larger input can pass the threshold even when a smaller pulse of exactly the same duration fails. The minimum accepted duration decreases with height, with derivative $-\tau V_{\rm th}/[V_0(V_0-V_{\rm th})]$.
For an original example, choose $\tau=2\,\mathrm{\mu s}$, $V_{\rm th}=0.40\,\mathrm V$, and $T=1\,\mathrm{\mu s}$. The minimum height for a nonzero comparator interval is about $1.0166\,\mathrm V$. A $0.8\,\mathrm V$ pulse peaks at about $0.3148\,\mathrm V$ and is rejected; a $1.2\,\mathrm V$ pulse peaks at about $0.4722\,\mathrm V$ and is accepted. Treating this observation as a width-only decision would conflate two different inputs.
A pulse-generator experiment therefore sweeps width at a recorded height and threshold, measures the actual capacitor waveform on an <oscilloscope>, and allows the initial state to relax. A <residual baseline in RC pulse-width discrimination> changes the next pulse's peak and invalidates the relaxed-state formula. Probe and comparator <input capacitance> also belong in the effective <time constant>. The natural-decay apparatus in MIT's RC laboratory supplies a direct component-level check of that <time constant>.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2, square-wave decay and sinusoidal measurements]{ref} Comparator propagation delay and hysteresis are additional dynamics, not part of the ideal RC acceptance inequality.
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