= Reference-resistor error in ratiometric resistance measurement
<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 is
$$
\widehat R_T=rR_c,
\qquad\frac{\widehat R_T-R_T}{R_T}=\frac{R_c}{R_a}-1.
$$
Writing $R_a=R_c(1+e)$ 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 in
$$
-\frac{E}{1+E}\le\frac{\widehat R_T-R_T}{R_T}\le\frac{E}{1-E}.
$$
A 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 $149.880096\ldots\,\Omega$, 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.<A Basic Guide to RTD Measurements by Joseph Wu>[§2.1.3, printed pp. 10–11, reference accuracy and drift]{ref}
For small random errors, applying the <law of propagation of uncertainty> to $\widehat R_T=rR_c$ gives
$$
u^2(\widehat R_T)\simeq R_c^2u^2(r)+r^2u^2(R_c)+2rR_c\operatorname{Cov}(r,R_c).
$$
The <covariance> term must follow how the reference value and reading were obtained; it cannot simply be assumed zero when inputs are shared.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3, equation A-3]{ref}
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