= Current-source swapping in a resistance ratio measurement
{title2=$\sqrt{r_A r_B}$}
<Current-source swapping in a resistance ratio measurement> exchanges two excitation sources between the sensor and <reference resistor>, allowing an unequal-current contribution to be identified or canceled. The combining rule depends on the circuit topology. In the separate-branch model of <branch-current mismatch in a resistance ratio measurement>, the exact combination is a <geometric mean>.
Let $q=R_T/R_{\rm ref}>0$ and let the two positive source currents have a fixed <ratio> $k=I_A/I_B$. With source A on the sensor and B on the reference, the measured potential <ratio> is $r_A=qk$. After exchanging their assignments, $r_B=q/k$. Hence
$$
\widehat q=\sqrt{r_A r_B}=q.
$$
This assumes that the resistances, current outputs, potential-sensing gains and relevant thermal conditions are unchanged across the two acquisitions. It cancels the multiplicative source ratio, not an additive readout error or an inaccurate assigned reference resistance.
For an original example, let $q=0.2$ and $k=1.1$. The acquisitions give $r_A=0.22$ and $r_B=0.181818\ldots$. Their <geometric mean> is exactly $0.2$. Their <arithmetic mean> is $0.200909\ldots$, high by approximately $0.454545\%$. In general,
$$
\frac{(r_A+r_B)/2}{q}-1
=\frac{k+k^{-1}}2-1=\frac{(k-1)^2}{2k}.
$$
The arithmetic mean suppresses a small mismatch to second order here, but is not the exact inverse-pair cancellation.
The primary RTD guide uses source interchange in a \b[different] three-wire lead-cancellation circuit. Its two readings are averaged according to that circuit's equations; they must not be replaced automatically by the geometric-mean rule above.<A Basic Guide to RTD Measurements by Joseph Wu>[§1.4.2, printed p. 6, Figure 1-6 and equations (9)–(12), IDAC current chopping]{ref}
A <drift residual after current-source swapping> remains if the source ratios in the two acquisitions are $k_1$ and $k_2$. For unchanged $q$, the same calculation gives $\widehat q/q=\sqrt{k_1/k_2}$. A $0.2\%$ increase from $k_1$ to $k_2$ therefore makes the combined estimate low by $1-1/\sqrt{1.002}\approx0.0998502\%$. <Sensor self-heating in a ratiometric resistance measurement> can also change $q$ between source assignments.
A proposed apparatus check switches two characterized <current sources> between stable sensing <resistors>, waits for electrical settling, and compares both individual readings and their product. Reversing the order of the two acquisitions helps distinguish a repeatable source mismatch from drift during the sequence. This tests the stated circuit model without presuming that all source-swapping circuits use the same estimator.
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