 Ratiometric cancellation of RC timing gain errors

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 $0<\theta <H$,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 $\tau =0.8\mu \mathrm{s}$. The crossing is approximately $0.285340\mu \mathrm{s}$. 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 $G_{\theta }/G_H=1+ϵ$. Provided $0<a(1+ϵ)<1$,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.^[ref]