Sampling skew after matched ratiometric filters ($\delta t$)

Sampling skew after matched ratiometric filters is the timing difference between the signal and reference samples in a ratiometric measurement, after their analog paths have the same response. Shared wiring and matched filters cancel excitation at the same effective time; they do not make two different times interchangeable.

For constant sensor and reference resistances, let the common filtered excitation be $F(t)=(h\ast I)(t)$, where $h$ is the common impulse response and $\ast$ denotes convolution. With consistent initial conditions, the two filtered electric potential differences are $R_{T}F(t)$ and $R_{\mathrm{ref}}F(t)$. If the sensor is sampled at $t$ but the reference at $t+\delta t$, the measured ratio isBoth denominator factors must be nonzero. The sampling instants here mean the effective acquisition times; conversion-result arrival times need not be those instants. Ratiometric cancellation with matched signal and reference filters treats the equal-time cancellation, supported by the primary circuit report's matching requirement.^[ref]

For positive differentiable $F$ and small skew, expanding the logarithm givesA later reference sample during a rising excitation therefore makes the inferred resistance low. The relevant rate is the fractional slope $F^{\prime }/F$, not the absolute slope alone. A timing-skew bound for a ratiometric measurement can express a worst-case requirement using a bound on this logarithmic slope.

For an original local example, suppose $F(t)=F_0{e}^{\kappa t}$ over the acquisition window, with $\kappa =20{\mathrm{s}}^{-1}$ and $\delta t=100\mu \mathrm{s}$. Then the exact scale error is $e^{-0.002}-1\approx -0.199800\%$. The small-skew estimate is $-0.200000\%$. Doubling the common amplitude $F_0$ leaves both errors unchanged; reversing the sample order changes their sign to first order. This is a stipulated short-window excitation model, not an instrument-performance claim.

A focused bench check replaces the sensor with a stable resistor, imposes a controlled excitation change, and compares simultaneous acquisition with deliberately delayed reference sampling. Recording the excitation on an oscilloscope supplies its fractional slope. Repeating on a settled plateau separates timing error from reference-resistor error in ratiometric resistance measurement and other steady scale errors.

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 $r_0=R_T/R_{\mathrm{ref}}$, 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 $|F^{\prime }(s)/F(s)|\le K$ throughout the entire interval between the samples. The fundamental theorem of calculus givesThe same proof works for either sign of $\delta t$. Writing $\eta =K|\delta t|$, exponentiation yields the exact intervalThis 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 requireThe slightly smaller-than-$20\mu \mathrm{s}$ 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 $|F^{\prime }|\le M$, 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.^[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.