Shared-excitation resistance ratio measurement

A shared-excitation resistance ratio measurement compares two electrical resistances by passing the same electric current through them and comparing their electric potential differences. It is the circuit realization of a ratiometric measurement: the unknown excitation becomes a common multiplicative input.

For a sensor $R_T$ in series with a reference resistor $R_{\mathrm{ref}}$, ideal high-input-impedance sensing givesThe electric current must be nonzero and shared at the time relevant to each sample. One implementation feeds $V_T$ to an analog-to-digital converter input and $V_{\mathrm{ref}}$ to its external reference pins; the conversion is then scaled by their ratio. The primary circuit guide shows this wiring and includes the programmed gain when translating a digital code to resistance.^[ref]

For an original check, choose $R_T=180\Omega$ and $R_{\mathrm{ref}}=1200\Omega$. Excitation at $0.4\mathrm{mA}$ gives $0.072\mathrm{V}$ and $0.480\mathrm{V}$; excitation at $1.2\mathrm{mA}$ gives $0.216\mathrm{V}$ and $1.440\mathrm{V}$. Both give $r=0.15$ and recover $180\Omega$. This calculation assumes unchanged resistances; the larger excitation dissipates nine times the sensor electric power, making sensor self-heating in a ratiometric resistance measurement an independent test.

A common supply is insufficient if different branch currents pass through the two resistances. With currents $I_T,I_{\mathrm{ref}}$,This is branch-current mismatch in a resistance ratio measurement. Sequential sampling of a changing excitation has the same problem unless the two paths retain the same effective excitation history. Ratiometric cancellation with matched signal and reference filters specifies the dynamic condition.

Wiring matters even with an ideal shared current source. A two-wire sensor reading includes lead electrical resistance; four-terminal sensing separates the current leads from high-impedance potential-sensing leads. The reference itself needs the same care if lead drops would otherwise enter its measured denominator.^[ref]

A bench check can replace the sensor with a stable known resistor, sweep excitation, and compare each measured potential with their ratio. A constant ratio accompanied by changing individual potentials tests shared excitation. It does not establish reference accuracy, eliminate sensor heating, or prove operation outside the converter's input and reference ranges.

Branch-current mismatch in a resistance ratio measurement is a residual scale error when the sensor and reference resistor carry unequal electric currents. A common supply or common nominal current setting does not establish the shared excitation required by a shared-excitation resistance ratio measurement.

Suppose separate branches carry positive currents $I_T$ and $I_{\mathrm{ref}}$. Ideal potential sensing givesThus the fractional resistance error is exactly $I_T/I_{\mathrm{ref}}-1$. If $I_T=I_0(1+{ϵ}_T)$ and $I_{\mathrm{ref}}=I_0(1+{ϵ}_{\mathrm{ref}})$, the common scale $I_0$ cancels, whileIn an unbranched series circuit the same electric current passes through both components; this particular error instead concerns separate paths, leakage, or unequal effective sampling times.

For an original example, use $R_T=200\Omega$, $R_{\mathrm{ref}}=2000\Omega$, $I_T=1.002\mathrm{mA}$ and $I_{\mathrm{ref}}=0.998\mathrm{mA}$. Their potentials are $0.2004\mathrm{V}$ and $1.996\mathrm{V}$. The inferred resistance is $200.801603\dots \Omega$, high by approximately $0.400802\%$. Each current is only $0.2\%$ from its nominal setting, but the opposed errors add to first order.

A useful ideal comparison is current-source swapping in a resistance ratio measurement. In two otherwise unchanged acquisitions, assign source A to the sensor and source B to the reference, then exchange them. If $q=R_T/R_{\mathrm{ref}}$, the two ratios obeyThis exact geometric mean cancellation assumes unchanged source outputs and resistances between acquisitions; switching transients and sensor self-heating in a ratiometric resistance measurement can violate that assumption. The primary circuit guide independently treats current-source mismatch and source interchange in its three-wire lead-cancellation circuit; that topology's correction must follow its own equations rather than be replaced by the two-branch model above.^[ref]

A proposed test measures both branch currents through characterized sensing resistors, then changes their assignments while holding the sensor environment fixed. Agreement after swapping tests the mismatch model; agreement from merely changing the common nominal current tests a different cancellation.

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.