 Ratiometric measurement

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.