Current-source swapping in a resistance ratio measurement ($\sqrt{r_A{r}_B}$)

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 $q=R_T/R_{\mathrm{ref}}>0$ 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, $r_B=q/k$. HenceThis 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 $r_B=0.181818\dots$. Their geometric mean is exactly $0.2$. Their arithmetic mean is $0.200909\dots$, high by approximately $0.454545\%$. In general,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 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.^[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 $\hat{q}/q=\sqrt{k_1/k_2}$. A $0.2\%$ increase from $k_1$ to $k_2$ therefore makes the combined estimate low by $1-1/\sqrt{1.002}\approx 0.0998502\%$. 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.

A drift residual after current-source swapping remains when the current ratio changes between the two source assignments. The geometric mean cancellation in current-source swapping in a resistance ratio measurement is exact for a fixed multiplicative mismatch; it does not make a time-varying mismatch constant.

Let the true resistance ratio $q$ remain fixed. At the two acquisition times $t_1,t_2$, write $k_i=I_A(t_i)/I_B(t_i)>0$. With opposite source assignments,For $g(t)=\log k(t)$, the logarithmic error is $[g(t_1)-g(t_2)]/2$. A full-interval bound $|g^{\prime }(t)|\le \gamma$ therefore givesFor an original example, a logarithmic source-ratio drift of $0.01{\mathrm{s}}^{-1}$ over a $0.2\mathrm{s}$ acquisition separation produces $\hat{q}/q=e^{-0.001}\approx 0.9990005$. This is a $0.0999500\%$ underestimate for that drift direction. Shortening the separation reduces this particular residual, provided both readings still follow electrical settling.

A useful original logarithmic variant of an ABBA measurement sequence uses source-assignment signs $s=(+1,-1,-1,+1)$ at equally spaced times $0,\Delta ,2\Delta ,3\Delta$. The four readings obey $\log r_i=\log q+s_{i}g(t_i)$. Combining them as $\hat{q}=(r_1{r}_2{r}_3{r}_4{)}^{1/4}$ givesBoth the constant and linear logarithmic drift terms cancel because the signed time sums vanish. Unequal timing, nonlinear drift and a changing sensor resistance can leave residuals. If $q$ itself varies, the combination includes its geometric average over the acquisitions rather than automatically recovering a single-time value.

The primary RTD guide establishes source interchange in its three-wire circuit; its topology-specific arithmetic averaging is distinct from the separate-branch logarithmic model above.^[ref] A proposed bench test imposes a slow controlled source-ratio variation, timestamps the readings, and compares a two-reading swap with the balanced four-reading sequence while using stable sensing resistors.