= Threshold timing sensitivity of an RC step response
{title2=$\partial t_\uparrow/\partial V_{\rm th}=\tau/(V_0-V_{\rm th})$}
<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 $0<V_{\rm th}<V_0$, the rising crossing is
$$
t_\uparrow=\tau\log\frac{V_0}{V_0-V_{\rm th}}.
$$
Holding the other inputs fixed gives
$$
\frac{\partial t_\uparrow}{\partial V_{\rm th}}=\frac{\tau}{V_0-V_{\rm th}},\qquad
\frac{\partial t_\uparrow}{\partial V_0}=-\frac{\tau V_{\rm th}}{V_0(V_0-V_{\rm th})},\qquad
\frac{\partial t_\uparrow}{\partial\tau}=\frac{t_\uparrow}{\tau}.
$$
A threshold approaching the step height encounters a flatter part of the exponential response, making the first sensitivity large.
For an original timing estimate, choose $V_0=2\,\mathrm V$, $\tau=0.5\,\mathrm{\mu s}$, and effective small threshold-noise <standard deviation> $4\,\mathrm{mV}$. At $V_{\rm th}=0.4\,\mathrm V$, 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, $dV_{\rm th}=a\,dV_0$ and $V_{\rm th}=aV_0$, 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
The local <noise> rule also follows from dividing a small voltage perturbation by the crossing slope $(V_0-V_{\rm th})/\tau$. 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.
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