= 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> $R_{\rm ref}$. Their <electric potential differences> are $V_T=IR_T$ and $V_{\rm ref}=IR_{\rm ref}$. An ideal normalized <analog-to-digital converter> reading is therefore
$$
r=\frac{V_T}{V_{\rm ref}}=\frac{R_T}{R_{\rm ref}},\qquad R_T=rR_{\rm ref}.
$$
The <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>.<A Basic Guide to RTD Measurements by Joseph Wu>[§2.1.3, pp. 10–11, shared-current resistance measurement]{ref}
For an original ideal example, take $R_T=100\,\Omega$, $R_{\rm ref}=2000\,\Omega$ and $I=1\,\mathrm{mA}$. Then $V_T=0.100\,\mathrm V$, $V_{\rm ref}=2.00\,\mathrm V$ 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 $105\,\Omega$.
\Image[/ratiometric-resistance-excitation.png]
{title=Shared excitation cancels in a resistance ratio; a fixed independent reference retains the excitation error. Original ideal calculation, not measured data.}
The <reference-resistor error in ratiometric resistance measurement> remains even with perfect shared excitation. If the actual <reference resistor> is $R_{\rm ref,nom}(1+e)$ but the calculation uses its nominal value, then
$$
\widehat R_T=rR_{\rm ref,nom}=\frac{R_T}{1+e}.
$$
Thus 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.<A Basic Guide to RTD Measurements by Joseph Wu>[§2.1.3, p. 10, reference-resistor requirements]{ref}
Nor does the <ratio> remove <sensor self-heating in a ratiometric resistance measurement>. <Joule heating> dissipates $P=I^2R_T$, 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.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.1, equation (4.1.1), self-heating and excitation-current dependence]{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 $r=V_T/V_{\rm ref}$ obeys
$$
u_r^2\simeq\frac{u_T^2}{V_{\rm ref}^2}+\frac{V_T^2u_{\rm ref}^2}{V_{\rm ref}^4}-\frac{2V_T\operatorname{Cov}(V_T,V_{\rm ref})}{V_{\rm ref}^3}.
$$
The 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.1–A.3, complete measurement model and covariance terms]{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 $R_T(h*I)$ and $R_{\rm ref}(h*I)$, so their <ratio> is still $R_T/R_{\rm ref}$ wherever defined. Here $*$ denotes <convolution>. Different filters or different sampling times can leave an excitation-dependent error.<RTD Ratiometric Measurements and Filtering Using the ADS1148 and ADS1248 Family of Devices by Luis Chioye>[First-page abstract and low-pass filter design discussion, matching input and reference filters]{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.
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