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Luis Chioye writes about precision analog-to-digital converters and their measurement circuits. The author biography in a 2013 Texas Instruments reference design identifies him as an applications engineer in its Precision Analog group, with degrees in electrical engineering from the University of Arizona and Walden University/NTU School of Engineering and Applied Science.^[ref]

His work includes ratiometric measurement with resistance thermometers: sharing an excitation electric current can cancel its variation, but the input and reference filters must preserve that shared variation consistently.^[ref] This is the engineering distinction captured by ratiometric cancellation with matched signal and reference filters.

With Mike Beckman, Chioye also published a thermocouple reference design. Its reference-junction temperature sweep in thermocouple testing keeps the measurement junction in a stable bath while changing the temperature of the connector and converter board; the acquisition computer stays outside the temperature-controlled chamber. This isolates reference-side effects from changing the temperature being measured.^[ref]

An original ideal calculation explains the test. Let $E(T)$ be the thermocouple electromotive force response from the Seebeck effect relative to $0{}^{\circ }\mathrm{C}$. With isothermal connector junctions at $T_c$, the measured electric potential difference is $V=E(T_h)-E(T_c)$. Cold-junction compensation reconstructs $E(T_h)=V+E(T_c)$ before applying the inverse function of the response. In the stipulated linear model $E(T)=ST$ with $S=40\mu V/K$, a fixed $T_h=100{}^{\circ }\mathrm{C}$ gives $V=3.2\mathrm{mV}$ at $T_c=20{}^{\circ }\mathrm{C}$ and $2.8\mathrm{mV}$ at $30{}^{\circ }\mathrm{C}$. Correct compensation gives the same $100{}^{\circ }\mathrm{C}$ in both cases. These are constructed values, not measured performance or a calibration table for a particular thermocouple.

Cold-junction thermal coupling matters because the temperature sensor must represent the connector temperature. A precise converter reading cannot by itself remove an unmeasured temperature difference between those locations.^[ref]

Ratiometric cancellation with matched signal and reference filters preserves a resistance ratio despite changing excitation when both paths apply the same dynamics to the shared electric current. Matching only the steady gain is insufficient during a transient.

Let constant sensor and reference resistances be $R_T,R_{\mathrm{ref}}$, and let both paths be linear time-invariant systems with impulse responses $h_T,h_{\mathrm{ref}}$. With consistent initial conditions, their filtered electric potential differences arewhere $\ast$ is convolution. If $h_T=h_{\mathrm{ref}}$, simultaneous sampling gives $y_T/y_{\mathrm{ref}}=R_T/R_{\mathrm{ref}}$ whenever the denominator is nonzero. Arbitrary unequal stored initial states need not cancel. Nor can a time-varying $R_T$ generally be pulled outside the convolution.

For unit steady gains, write $I(t)=I_0(1+\eta (t))$ after settling at $I_0$. Define $q_T=h_T\ast \eta$ and $q_{\mathrm{ref}}=h_{\mathrm{ref}}\ast \eta$. ThenThis displays exactly which excitation component survives. For small fluctuations the fractional residual is $q_T-q_{\mathrm{ref}}$, so both amplitude and timing differences between the transfer functions matter.

As an original transient example, step the excitation up by ten percent from a settled state. Two first-order low-pass filters with time constants ${\tau }_T=1\mathrm{ms}$ and ${\tau }_{\mathrm{ref}}=2\mathrm{ms}$ give, for $t\ge 0$,At $t=1\mathrm{ms}$ the reading is high by approximately $2.2962\%$, although both filters have the same final gain and the error tends to zero. Setting both time constants to $1\mathrm{ms}$ cancels this particular excitation step at every time. Sampling skew after matched ratiometric filters can still break simultaneous cancellation even with identical hardware.

The primary ADS1248 experiment used a resistance decade box, a $1\mathrm{mA}$ excitation and a $2\mathrm{k\Omega }$ reference. Reference-only filtering produced signal-dependent noise; matching the signal and reference paths substantially reduced that dependence. The report provides the circuit and noise-versus-input plots, rather than merely recommending a common nominal cutoff.^[ref]

A ratiometric measurement infers a quantity from a ratio of two measured signals chosen to share an unwanted multiplicative variation. The useful question is which inputs actually share that variation: a reference resistor error or a change in the sensor itself generally survives the cancellation.

In a shared-excitation resistance ratio measurement, a current source passes the same electric current $I$ through a sensor of electrical resistance $R_T$ and a reference resistor $R_{\mathrm{ref}}$. Their electric potential differences are $V_T=I{R}_T$ and $V_{\mathrm{ref}}=I{R}_{\mathrm{ref}}$. An ideal normalized analog-to-digital converter reading is thereforeThe current source need not have a perfectly constant output for this cancellation, provided both measurements sample the same electric current, the denominator is nonzero, and the converter stays within its operating ranges. This is a standard arrangement for a resistance thermometer.^[ref]

For an original ideal example, take $R_T=100\Omega$, $R_{\mathrm{ref}}=2000\Omega$ and $I=1\mathrm{mA}$. Then $V_T=0.100\mathrm{V}$, $V_{\mathrm{ref}}=2.00\mathrm{V}$ and $r=0.05$. Increasing $I$ by $5\%$ gives $0.105\mathrm{V}$ and $2.10\mathrm{V}$: the ratio remains $0.05$. If the converter instead uses an independent fixed $2\mathrm{V}$ reference and the calculation assumes the original $1\mathrm{mA}$, the inferred electrical resistance becomes $105\Omega$.

The reference-resistor error in ratiometric resistance measurement remains even with perfect shared excitation. If the actual reference resistor is $R_{\mathrm{ref},\mathrm{nom}}(1+e)$ but the calculation uses its nominal value, thenThus a $+0.1\%$ reference error makes the nominal estimate of a true $100\Omega$ sensor about $99.9001\Omega$. Reference accuracy and its variation with temperature still matter.^[ref]

Nor does the ratio remove sensor self-heating in a ratiometric resistance measurement. Joule heating dissipates $P=I^2{R}_T$, which is $0.1\mathrm{mW}$ in the example. A temperature rise changes the actual electrical resistance of a resistance thermometer; the ratio then measures that changed resistance. Reducing the electric current or comparing several currents can reveal this physical effect.^[ref]

The law of propagation of uncertainty makes the role of covariance explicit. For small perturbations of nonzero mean signals, the first-order measurement uncertainty of $r=V_T/V_{\mathrm{ref}}$ obeysThe contribution from a shared excitation fluctuation cancels through the covariance term. Treating the two signals as independent random variables would discard that cancellation. Other errors need their own inputs in the complete model.^[ref]

For time-varying excitation, ratiometric cancellation with matched signal and reference filters requires matching dynamics as well as shared wiring. With constant resistances and identical linear time-invariant systems of impulse response $h$, the filtered signals are $R_T(h\ast I)$ and $R_{\mathrm{ref}}(h\ast I)$, so their ratio is still $R_T/R_{\mathrm{ref}}$ wherever defined. Here $\ast$ denotes convolution. Different filters or different sampling times can leave an excitation-dependent error.^[ref]

A simple proposed bench test first replaces the sensor by a stable calibrated resistor, sweeps the electric current within the instrument limits, and records both electric potential differences and their ratio. That separates electrical cancellation from changing sensor temperature. Repeating with a resistance thermometer in a stable bath then tests Joule heating; changing the reference resistance or one filter tests the predicted failure modes. These are proposed checks of the model, rather than reported experimental results.

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.