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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]

Sampling skew after matched ratiometric filters is the timing difference between the signal and reference samples in a ratiometric measurement, after their analog paths have the same response. Shared wiring and matched filters cancel excitation at the same effective time; they do not make two different times interchangeable.

For constant sensor and reference resistances, let the common filtered excitation be $F(t)=(h\ast I)(t)$, where $h$ is the common impulse response and $\ast$ denotes convolution. With consistent initial conditions, the two filtered electric potential differences are $R_{T}F(t)$ and $R_{\mathrm{ref}}F(t)$. If the sensor is sampled at $t$ but the reference at $t+\delta t$, the measured ratio isBoth denominator factors must be nonzero. The sampling instants here mean the effective acquisition times; conversion-result arrival times need not be those instants. Ratiometric cancellation with matched signal and reference filters treats the equal-time cancellation, supported by the primary circuit report's matching requirement.^[ref]

For positive differentiable $F$ and small skew, expanding the logarithm givesA later reference sample during a rising excitation therefore makes the inferred resistance low. The relevant rate is the fractional slope $F^{\prime }/F$, not the absolute slope alone. A timing-skew bound for a ratiometric measurement can express a worst-case requirement using a bound on this logarithmic slope.

For an original local example, suppose $F(t)=F_0{e}^{\kappa t}$ over the acquisition window, with $\kappa =20{\mathrm{s}}^{-1}$ and $\delta t=100\mu \mathrm{s}$. Then the exact scale error is $e^{-0.002}-1\approx -0.199800\%$. The small-skew estimate is $-0.200000\%$. Doubling the common amplitude $F_0$ leaves both errors unchanged; reversing the sample order changes their sign to first order. This is a stipulated short-window excitation model, not an instrument-performance claim.

A focused bench check replaces the sensor with a stable resistor, imposes a controlled excitation change, and compares simultaneous acquisition with deliberately delayed reference sampling. Recording the excitation on an oscilloscope supplies its fractional slope. Repeating on a settled plateau separates timing error from reference-resistor error in ratiometric resistance measurement and other steady scale errors.

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 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.