 Identifying flow-sensor gain and offset with timed transfers

Identifying flow-sensor gain and offset with timed transfers uses a known gas volume delivered over different durations to distinguish multiplicative response from accumulated zero error. In the model $Q_m=gQ+b$, a transfer of volume $V_0$ lasting $T_i$ producesThe observations form a linear regression with design matrix rows $(V_0,T_i)$. If $V_0\ne 0$, gain and offset are identifiable only when the durations are not all equal. Repeating one duration measures the same combination of $g$ and $b$ more precisely, without separating them.

For two noise-free transfers,Stipulated 3 L transfers lasting 1 and 6 s, measured as 3.020 and 3.120 L, give $\hat{b}=0.020\mathrm{L}{\mathrm{s}}^{-1}$ and $\hat{g}=1.000$. The different durations expose an offset that one transfer could misinterpret as a gain error.

Timing spread and gain-offset estimator covariance quantifies the design. Suppose $n$ observations have independent equal volume-error variance $s^2$, and define $S_{TT}={\sum }_i(T_i-\overline{T}{)}^2$. Ordinary least squares givesUnder the stipulated model, wider timing separation improves offset precision. For the two durations above and $s=0.010$ L, the offset's standard uncertainty is about $0.00283\mathrm{L}{\mathrm{s}}^{-1}$ and the gain's about 0.00406.

The reference syringe volume was treated as known in those formulas. If it is uncertain, repeated sensor tests share that reference uncertainty. Indeed the observations identify only the product $g{V}_0$ when both factors are unknown. Additional repetitions cannot establish an absolute gain without an independently characterized volume. The GUM treats fit uncertainties and common-reference correlations as distinct contributions.^[ref]

The primary spirometry statement's multiple-flow syringe checks test the practical response over different delivery rates.^[ref] Actual leaks, flow-dependent nonlinearities and uncertain timing require a richer model than a constant gain and offset.