= Sensor self-heating in a ratiometric resistance measurement
{title2=$P=I^2R$}
<Sensor self-heating in a ratiometric resistance measurement> occurs when the sensing <electric current> changes the sensor's own <temperature> through <Joule heating>. A <ratiometric measurement> can remove the direct dependence of the electrical reading on that <electric current> while still reading the heated sensor correctly. For a <resistance thermometer>, that means the indicated <temperature> can differ from the unperturbed surroundings.
With actual sensor <electrical resistance> $R_T(I)$ and a stable <reference resistor> $R_{\rm ref}$ carrying the same <electric current>,
$$
r(I)=\frac{IR_T(I)}{IR_{\rm ref}}=\frac{R_T(I)}{R_{\rm ref}},\qquad P=I^2R_T(I).
$$
The cancellation leaves the dependence of $R_T$ on $I$ intact. In a small, steady perturbation, a model with <thermal resistance> $R_{\rm th}$ gives $\Delta T\simeq R_{\rm th}I^2R_T(0)$ and
$$
\Delta R_T\simeq R_T(0)\alpha_R\Delta T,
$$
where $\alpha_R$ is the <temperature coefficient of resistance>. These approximations require small heating and thermal conditions that remain comparable between readings. The BIPM guide distinguishes heat flow inside the probe from heat flow into its surroundings; changing the surrounding medium can change the effect.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.1, printed pp. 24–25, internal and external heating and medium dependence]{ref}
A practical diagnostic is a two-current <extrapolation>. Model $R_T(I)\simeq R_0+cI^2$ over the tested range. Measurements $R_1,R_2$ at distinct nonzero current magnitudes $I_1,I_2$ imply
$$
R_0\simeq\frac{I_2^2R_1-I_1^2R_2}{I_2^2-I_1^2}.
$$
$R_0$ is an inferred zero-power <electrical resistance>, not an electrical reading taken with zero excitation. The guide describes this squared-current method and requires a stable surrounding <temperature> and settled readings.<Guide on Secondary Thermometry: Industrial Platinum Resistance Thermometers by Jonathan Pearce and others>[§4.1, printed p. 24, equation (4.1.1)]{ref}
For an original numerical example, suppose a $1000\,\Omega$ <reference resistor> gives ratios $0.120006$ at $0.5\,\mathrm{mA}$ and $0.120024$ at $1.0\,\mathrm{mA}$. The inferred resistances are $120.006\,\Omega$ and $120.024\,\Omega$. Since $I_2^2=4I_1^2$, the estimate is $(4R_1-R_2)/3=120.000\,\Omega$. Both ratios canceled their excitation exactly, yet they disagree because the sensor changed.
A bath experiment should alternate the two excitation settings, allow thermal settling and monitor the bath <temperature>. Repeated low/high/low readings help reveal drift rather than attributing every difference to heating. <Resistance thermometer self-heating> gives the wider physical setting; <reference-resistor error in ratiometric resistance measurement> and mismatched filters are separate contributions.
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