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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.

Joseph Wu is the named author of Texas Instruments' 2018 guide to resistance thermometer measurements, revised in 2023. The guide connects sensor wiring, excitation electric current, reference resistors and analog-to-digital converters rather than treating the converter as a complete thermometer by itself.^[ref]

One concrete design distinction is between two-wire resistance measurement and four-terminal sensing. In the two-wire arrangement, the same leads carry excitation electric current and contribute to the measured electric potential difference. Their electrical resistance is therefore counted with the sensor. Separate sense leads with high input impedance can exclude that series drop, although input leakage and other nonidealities still require attention.^[ref]

For an original ideal example, let a sensor have electrical resistance $R_T=100\Omega$, with two $1\Omega$ leads carrying $I=1\mathrm{mA}$. Two-wire resistance measurement givesThe $2\Omega$ lead error remains even when a shared reference resistor cancels excitation variations in the ratiometric measurement. Ideal four-terminal sensing instead measures $I{R}_T=100\mathrm{mV}$ directly across the sensor. This illustrates the platinum resistance thermometer lead resistance error mechanism without assuming a particular resistance-to-temperature calibration.

A proposed bench comparison replaces the sensor with a stable calibrated resistor and adds known series resistors to its current leads. Comparing two-wire and four-terminal sensing readings then tests lead-drop rejection without changing sensor temperature. The calculation distinguishes the wiring error from sensor self-heating in a ratiometric resistance measurement, which changes the sensor's actual electrical resistance.

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 $R_{\mathrm{ref}}$. Their electric potential differences are $V_T=I{R}_T$ and $V_{\mathrm{ref}}=I{R}_{\mathrm{ref}}$. An ideal normalized analog-to-digital converter reading is thereforeThe 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.^[ref]

For an original ideal example, take $R_T=100\Omega$, $R_{\mathrm{ref}}=2000\Omega$ and $I=1\mathrm{mA}$. Then $V_T=0.100\mathrm{V}$, $V_{\mathrm{ref}}=2.00\mathrm{V}$ 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 $105\Omega$.

The reference-resistor error in ratiometric resistance measurement remains even with perfect shared excitation. If the actual reference resistor is $R_{\mathrm{ref},\mathrm{nom}}(1+e)$ but the calculation uses its nominal value, thenThus 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.^[ref]

Nor does the ratio remove sensor self-heating in a ratiometric resistance measurement. Joule heating dissipates $P=I^2{R}_T$, 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.^[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 $r=V_T/V_{\mathrm{ref}}$ obeysThe 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.^[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 $R_T(h\ast I)$ and $R_{\mathrm{ref}}(h\ast I)$, so their ratio is still $R_T/R_{\mathrm{ref}}$ wherever defined. Here $\ast$ denotes convolution. Different filters or different sampling times can leave an excitation-dependent error.^[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.

A reference resistor is a resistor whose assigned electrical resistance supplies the comparison value in a resistance-measurement circuit. Its value may come from measurement calibration rather than its nominal marking. The useful reference is its value under the stated temperature, loading and connection conditions, together with the associated measurement uncertainty.

In a shared-excitation resistance ratio measurement, the sensor and reference resistor carry the same electric current. Measuring their electric potential differences gives $r=V_T/V_{\mathrm{ref}}=R_T/R_{\mathrm{ref}}$, so $R_T=r{R}_{\mathrm{ref}}$. A reference value error therefore sets a scale error even when excitation cancels perfectly. Reference-resistor error in ratiometric resistance measurement derives the exact relation and distinguishes a correction from its remaining measurement uncertainty.^[ref]

The connections define which resistance is compared. In four-terminal sensing, two leads carry the electric current while a separate high-impedance pair senses the electric potential difference at the reference terminals. This avoids including the current-lead drops in the assigned reference value, within the sensing instrument's limitations.

For an original ideal example, a $100\Omega$ reference resistor carries $1\mathrm{mA}$ through two current leads, each of electrical resistance $0.20\Omega$. The reference element has a potential difference of $0.1000\mathrm{V}$, while a reading outside both leads is $0.1004\mathrm{V}$. If this latter reading is used as the denominator while the calculation assumes $100\Omega$, a true $50\Omega$ sensor is inferred asSensing directly at the reference terminals removes this idealized lead contribution. It does not correct a wrong reference value. The circuit guide explicitly shows a Kelvin connection at the reference terminals for this purpose.^[ref]

A small temperature change shifts the value approximately by $\Delta R_{\mathrm{ref}}/R_{\mathrm{ref}}\simeq {\alpha }_R\Delta T$, with temperature coefficient of resistance ${\alpha }_R$. Excitation also dissipates electric power $I^2{R}_{\mathrm{ref}}$; a reference with a different temperature or load from its measurement calibration may require a correction or additional measurement uncertainty. A proposed check compares the reference before and after the measurement sequence under the same connection and loading conditions. Repeating the comparison at a lower electric current tests for a heating contribution. Stability of the observed ratio alone cannot certify the reference's assigned value.

Reference-resistor error in ratiometric resistance measurement is the error left when an instrument converts a measured resistance ratio using an inaccurate value for its reference resistor. Shared excitation removes one nuisance input; it does not establish the denominator's true electrical resistance.

Let $R_a$ be the actual reference resistor value and $R_c$ the value used in calculation. For an ideal simultaneous reading $r=R_T/R_a$, the inferred sensor resistance isWriting $R_a=R_c(1+e)$ gives the exact fractional error $-e/(1+e)$. The familiar $-e$ is its first-order approximation. If a deterministic bound is $|e|\le E<1$, the inferred fractional error lies inA symmetric reference tolerance therefore produces a slightly asymmetric resistance-error interval.

For an original example, suppose the instrument uses $R_c=2000\Omega$ while the actual reference resistor is $2001.6\Omega$. A true $150\Omega$ sensor is reported as $149.880096\dots \Omega$, an error near $-0.07994\%$. Multiplying the shared excitation by any common positive number leaves this error unchanged in the ideal model. Using a measured and corrected $R_c$ instead of its nominal marking can remove the known part, while the corrected value retains its own measurement uncertainty.

Temperature variation can produce the remaining error after measurement calibration. Locally, $e(T)\simeq e(T_0)+{\alpha }_R(T-T_0)$, where ${\alpha }_R$ is the temperature coefficient of resistance. For example, a $20$-parts-per-million-per-kelvin coefficient over $15\mathrm{K}$ contributes roughly $300$ parts per million; it survives excitation cancellation. The primary circuit guide explicitly requires reference accuracy and low drift, and separates these requirements from shared-current cancellation.^[ref]

For small random errors, applying the law of propagation of uncertainty to ${\hat{R}}_T=r{R}_c$ givesThe covariance term must follow how the reference value and reading were obtained; it cannot simply be assumed zero when inputs are shared.^[ref]

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 $R_{\mathrm{ref}}$ carrying the same electric current,The cancellation leaves the dependence of $R_T$ on $I$ intact. In a small, steady perturbation, a model with thermal resistance $R_{\mathrm{th}}$ gives $\Delta T\simeq R_{\mathrm{th}}{I}^2{R}_T(0)$ andwhere ${\alpha }_R$ 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.^[ref]

A practical diagnostic is a two-current extrapolation. Model $R_T(I)\simeq R_0+c{I}^2$ over the tested range. Measurements $R_1,R_2$ at distinct nonzero current magnitudes $I_1,I_2$ imply$R_0$ 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.^[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.

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.

A timing-skew bound for a ratiometric measurement limits the resistance-ratio error caused by sampling the signal and reference at different times. It converts a bound on the fractional rate of change of the shared excitation into an acquisition-timing requirement. This is useful even when the two analog filters are exactly matched.

Use the model in sampling skew after matched ratiometric filters: the common filtered excitation is $F(t)>0$, the true resistance ratio is $r_0=R_T/R_{\mathrm{ref}}$, and the reference is sampled $\delta t$ after the signal. Then $r/r_0=F(t)/F(t+\delta t)$. Suppose $F$ is continuously differentiable and its logarithmic derivative obeys $|F^{\prime }(s)/F(s)|\le K$ throughout the entire interval between the samples. The fundamental theorem of calculus givesThe same proof works for either sign of $\delta t$. Writing $\eta =K|\delta t|$, exponentiation yields the exact intervalThis is a deterministic bound under the stated slope assumption, rather than an expansion truncated at first order.

For an original timing budget, suppose $K=50{\mathrm{s}}^{-1}$ and require at most $0.1\%$ fractional resistance error from skew alone. It suffices to requireThe slightly smaller-than-$20\mu \mathrm{s}$ value accounts for the exponential upper bound. An excitation proportional to $e^{Ks}$ reaches one side of the logarithmic bound exactly, so the bound is attainable in this model.

A sufficient way to establish $K$ is to verify $F\ge F_{\min }>0$ and $|F^{\prime }|\le M$, giving $K=M/F_{\min }$. A filter cutoff by itself supplies neither a positive excitation floor nor this fractional-slope bound. Near an excitation zero, a small absolute change can produce a large resistance-ratio error. The primary filter-design report establishes the need to match signal and reference dynamics and to allow filter settling; the inequality above adds an original unequal-time error budget.^[ref]

A proposed bench check uses a stable sensor substitute and reference resistor, records the filtered excitation with an oscilloscope, and deliberately changes the acquisition delay. Compare each observed ratio with the interval predicted from a verified excitation floor and slope bound. This isolates timing effects from a fixed reference-resistor error in ratiometric resistance measurement.