 Threshold timing sensitivity of an RC step response

Threshold timing sensitivity of an RC step response measures how a crossing time changes when the absolute threshold or signal height changes. For a relaxed step response $v(t)=V_0(1-e^{-t/\tau })$, with $0<V_{\mathrm{th}}<V_0$, the rising crossing isHolding the other inputs fixed givesA threshold approaching the step height encounters a flatter part of the exponential response, making the first sensitivity large.

For an original timing estimate, choose $V_0=2\mathrm{V}$, $\tau =0.5\mu \mathrm{s}$, and effective small threshold-noise standard deviation $4\mathrm{mV}$. At $V_{\mathrm{th}}=0.4\mathrm{V}$, local linear propagation gives timing standard deviation about $1.25\mathrm{ns}$. At $1.9\mathrm{V}$, it gives about $20\mathrm{ns}$. The latter threshold also needs a longer input pulse to be reached at all, as shown in threshold-crossing time of a finite rectangular pulse through an RC load.

The two voltage sensitivities can cancel under a different measurement calibration model. If the threshold is set to a fixed fraction $a$ of the same varying height, $d{V}_{\mathrm{th}}=ad{V}_0$ and $V_{\mathrm{th}}=a{V}_0$, their contributions sum to zero. The crossing becomes $-\tau \log (1-a)$, independent of amplitude in this ideal model. Such correlated threshold and amplitude errors in RC timing must retain covariance rather than add two independent voltage uncertainties.^[ref]

The local noise rule also follows from dividing a small voltage perturbation by the crossing slope $(V_0-V_{\mathrm{th}})/\tau$. It assumes a well-defined nearby crossing and sufficiently small perturbations; it does not describe multiple noise crossings, comparator switching dynamics, or a pulse ending before the threshold. A bench test can compare controlled threshold shifts with oscilloscope crossing times while holding the actual step height and initial state fixed.