= Timing-skew bound for a ratiometric measurement
{title2=$|\log(r/r_0)|\le K|\delta t|$}
A <timing-skew bound for a ratiometric measurement> limits the resistance-ratio error caused by sampling the signal and reference at different times. It converts a bound on the fractional rate of change of the shared excitation into an acquisition-timing requirement. This is useful even when the two analog filters are exactly matched.
Use the model in <sampling skew after matched ratiometric filters>: the common filtered excitation is $F(t)>0$, the true resistance <ratio> is $r_0=R_T/R_{\rm ref}$, and the reference is sampled $\delta t$ after the signal. Then $r/r_0=F(t)/F(t+\delta t)$. Suppose $F$ is continuously differentiable and its <logarithmic derivative> obeys $|F'(s)/F(s)|\le K$ throughout the entire interval between the samples. The <fundamental theorem of calculus> gives
$$
\log\frac r{r_0}=-\int_t^{t+\delta t}\frac{F'(s)}{F(s)}\,ds,
\qquad \left|\log\frac r{r_0}\right|\le K|\delta t|.
$$
The same proof works for either sign of $\delta t$. Writing $\eta=K|\delta t|$, exponentiation yields the exact interval
$$
e^{-\eta}\le\frac r{r_0}\le e^{\eta},\qquad
\left|\frac r{r_0}-1\right|\le e^{\eta}-1.
$$
This is a deterministic bound under the stated slope assumption, rather than an expansion truncated at first order.
For an original timing budget, suppose $K=50\,\mathrm{s^{-1}}$ and require at most $0.1\%$ fractional resistance error from skew alone. It suffices to require
$$
|\delta t|\le\frac{\log(1.001)}{50\,\mathrm{s^{-1}}}
\approx19.9900\,\mathrm{\mu s}.
$$
The slightly smaller-than-$20\,\mathrm{\mu s}$ value accounts for the exponential upper bound. An excitation proportional to $e^{Ks}$ reaches one side of the logarithmic bound exactly, so the bound is attainable in this model.
A sufficient way to establish $K$ is to verify $F\ge F_{\min}>0$ and $|F'|\le M$, giving $K=M/F_{\min}$. A filter cutoff by itself supplies neither a positive excitation floor nor this fractional-slope bound. Near an excitation zero, a small absolute change can produce a large resistance-ratio error. The primary filter-design report establishes the need to match signal and reference dynamics and to allow filter settling; the inequality above adds an original unequal-time error budget.<RTD Ratiometric Measurements and Filtering Using the ADS1148 and ADS1248 Family of Devices by Luis Chioye>[Abstract, §2 and §3.4, printed p. 11, matching and initial settling]{ref}
A proposed bench check uses a stable sensor substitute and <reference resistor>, records the filtered excitation with an <oscilloscope>, and deliberately changes the acquisition delay. Compare each observed <ratio> with the interval predicted from a verified excitation floor and slope bound. This isolates timing effects from a fixed <reference-resistor error in ratiometric resistance measurement>.
Back to article page
