= Range-overlap calibration
\b[Range-overlap calibration] establishes or checks the relationship between instrument settings using inputs that both can measure validly. In <gain ranging>, it tests whether changing sensitivity preserves the inferred <measurand>. It can determine relative scale and offset between ranges, but agreement between ranges alone does not establish their absolute accuracy or independent <metrological traceability>.
For an original two-range <measurement model>, let the same stable <electric current> $I$ produce
$$
V_L=G_LI+b_L,\qquad V_H=G_HI+b_H.
$$
Eliminating $I$ gives
$$
V_H=aV_L+c,\qquad a=G_H/G_L,\qquad c=b_H-ab_L.
$$
An overlap experiment therefore needs enough distinct, stable input levels to distinguish relative gain from offset. One common input gives one equation for $a$ and $c$ and cannot identify both without additional information.
For example, suppose $G_L=0.10\,\mathrm{V/nA}$, $b_L=0.01\,\mathrm V$, $G_H=0.50\,\mathrm{V/nA}$ and $b_H=0.02\,\mathrm V$. Currents of $1$, $2$ and $3\,\mathrm{nA}$ give output pairs $(0.11,0.52)$, $(0.21,1.02)$ and $(0.31,1.52)\,\mathrm V$. They satisfy $V_H=5V_L-0.03\,\mathrm V$. Dividing raw outputs by their nominal gains without removing the offsets would not give consistent current estimates.
A practical test can drive a switchable <transimpedance amplifier> from a stable current source, record multiple overlap levels, and alternate the setting order to reveal <instrumental drift>. Avoid <amplifier clipping>, allow <settling time>, and retain the bandwidth and temperature conditions. The manufacturer tables show why range-dependent noise and gain drift must be included rather than assumed equal.<Datasheet DLPCA-200 by FEMTO>[p. 2, gain accuracy, gain drift, input limits and gain-dependent performance]{ref}
At each level, compare corrected estimates $I_H$ and $I_L$ through their difference $d=I_H-I_L$. The <measurement uncertainty> of that difference depends on <covariance>:
$$
u^2(d)=u^2(I_H)+u^2(I_L)-2\operatorname{cov}(I_H,I_L).
$$
A common reference can move both estimates together and disappear from this difference. Thus good overlap agreement checks a relative relationship; it cannot by itself exclude a shared calibration error. The general uncertainty-combination law explicitly retains correlated contributions.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated input quantities]{ref}
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