 Common offset in RC timing

A common offset in RC timing cancels only when every level used in the crossing model shares the same additive change. For high level $H$, threshold $\theta$, and initial baseline $V_b$, assume $V_b<\theta <H$ and fixed time constant $\tau$. The crossing time isAdding $B$ to all three levels leaves both differences unchanged, so the invariance in the header is exact. It describes a shared readout zero, or an equivalently translated step with a translated initial condition, rather than every possible physical offset experiment.

The three sensitivities make the cancellation visible locally:A common random offset has covariance matrix $u_B^{2}11^{\mathsf{T}}$ for these three inputs and contributes zero to the propagated timing variance. NIST's primary law retains exactly such covariance terms; setting them to zero would represent another measurement model.^[ref]

For an original nonzero-baseline example, take $H=2\mathrm{V}$, $\theta =0.8\mathrm{V}$, $V_b=0.2\mathrm{V}$ and $\tau =0.6\mu \mathrm{s}$. The crossing is about $0.243279\mu \mathrm{s}$. Adding $20\mathrm{mV}$ to all three levels leaves it exactly unchanged. Adding the same offset only to $H$ and $\theta$ while keeping the baseline fixed instead increases it by about $6.630\mathrm{ns}$. The two-level sensitivity is $c_H+c_{\theta }=\tau /(H-V_b)$, predicting $6.667\mathrm{ns}$ to first order.

A calibration test must state whether the initial capacitor output, electronic comparator reference, input level and readout share the offset. Independent threshold errors, uncertainty of the time constant, and baseline memory remain even when this one common component cancels. The full distinction is developed in correlated threshold and amplitude errors in RC timing; a common gain is a separate invariance treated in ratiometric cancellation of RC timing gain errors.