= Ratiometric cancellation of RC timing gain errors
{title2=$t(GH,G\theta)=t(H,\theta)$}
<Ratiometric cancellation of RC timing gain errors> follows when the same positive gain multiplies every voltage level relevant to a threshold ratio. With zero initial baseline, fixed <time constant> $\tau$, high level $H>0$ and threshold $0<\theta<H$,
$$
t=\tau\log\frac{H}{H-\theta}=-\tau\log(1-a),\qquad a=\theta/H.
$$
Replacing $H,\theta$ by $GH,G\theta$ with $G>0$ leaves $a$ and the time unchanged. This is an exact invariance, not merely the first-order cancellation of two uncertainty terms. It is a <ratiometric measurement> property of the specified model.
For an original example, choose $H=2\,\mathrm V$, $\theta=0.6\,\mathrm V$ and $\tau=0.8\,\mathrm{\mu s}$. The crossing is approximately $0.285340\,\mathrm{\mu s}$. Gains $G=0.8$ and $G=1.2$ both give the same result when they scale high level and threshold together. A shared readout gain can cancel from the calculated time in the same way; whether a reference also changes the physical circuit must be stated separately.
The <mismatch remaining after common RC gain cancellation> depends on the relative gains, not their common magnitude. Let the high level use gain $G_H$ and the threshold gain $G_\theta$, with $G_\theta/G_H=1+\epsilon$. Provided $0<a(1+\epsilon)<1$,
$$
\Delta t=-\tau\log\left(1-\frac{a\epsilon}{1-a}\right)
\approx\frac{\tau a}{1-a}\epsilon.
$$
In the example, a one-percent relative mismatch delays the crossing by about $3.436\,\mathrm{ns}$, compared with $3.429\,\mathrm{ns}$ from the first-order expression. Thresholds nearer the high level amplify the residual.
A nonzero initial baseline must share the gain too: $t=\tau\log[(H-V_b)/(H-\theta)]$ is unchanged by multiplying all three levels by $G$. Scaling only the high level and threshold against a fixed nonzero baseline is another experiment. Likewise, additive errors follow the conditions in <common offset in RC timing>, not the two-level scaling rule.
Uncertainty in the <time constant>, independent threshold error, baseline memory, and failure to reach the threshold remain even when one common gain cancels. Their <covariance> structure must follow the complete measurement model, as in <correlated threshold and amplitude errors in RC timing> and NIST's propagation law.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.1 and A.3]{ref}
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