TCR Uncertainty When Temperature and Resistance Errors Are Correlated

Measurement purpose

Evaluate the uncertainty of an endpoint TCR coefficient while preserving supported correlations among electrical and thermal inputs.

Specimens and conditions

Paired physical state

Use resistance and temperature observations from the same identified specimen and specified thermal sequence.

Coefficient definition

State endpoint order, reference resistance and whether the result is a chord coefficient rather than a local slope.

Equipment and records required

• Electrical uncertainty evidence: Identify which gain, offset, range, contact and repeatability contributions are shared between readings.
• Thermal uncertainty evidence: Separate common calibration components from endpoint-specific gradients, attachment errors and instability.

Method sequence

1. Define the model

   Express the coefficient as a function of measured quantities and physical correction inputs.

   Record: Equation, units and covariance assumptions.

2. Calculate sensitivities

   Evaluate derivatives at the measured values and verify signs and dimensions.

   Record: Contribution worksheet in ppm/K.

3. Test assumptions

   Compare direct shared-error calculations with the linearized budget and investigate large nonlinear effects.

   Record: Sensitivity checks, correlation evidence and reporting scope.

Decision and uncertainty

Accept the budget only when shared terms have a defensible physical basis and are neither omitted nor counted twice.

First-order propagation can be inadequate when the temperature interval is poorly determined, the denominator approaches zero, or the measurement model is strongly nonlinear.

The metrology reviewer approves correlation and distribution assumptions; the coefficient requirement remains controlled by the agreed test definition.

Traceable outputs

Start with the coefficient actually reported

For a coefficient normalized to the first endpoint resistance, write alpha = (R2 − R1)/(R1 × d), where d = T2 − T1. The fractional result has units per kelvin; multiply by one million to report ppm/K. A different reference resistance produces different sensitivities.

Treat R1 as both a measured endpoint and the denominator. It is not an independent third reading unless the method truly measures another reference value. Creating separate unrelated uncertainty entries for the same R1 in the difference and denominator loses the dependency built into the equation.

Retain signed sensitivity coefficients

The resistance sensitivities are cR2 = 1/(R1d) and cR1 = −R2/(R1²d). The temperature sensitivities are cT2 = −alpha/d and cT1 = alpha/d. Multiply these coefficients by input standard uncertainties using consistent units before comparing contributions.

The signs matter in covariance terms even though they disappear in the squared independent terms. A positive correlation between inputs with opposite sensitivities can reduce uncertainty. Removing signs before combination would turn a legitimate cancellation into an added error and produce an incorrect budget.

Calculate a transparent independent-input example

Assume R1 = 1,000 ohms, R2 = 1,002 ohms and d = 50 kelvin. The coefficient is 40 ppm/K. If each endpoint resistance has independent standard uncertainty 0.01 ohm, the electrical contribution is approximately 0.2831 ppm/K after combining sensitivities of 20 and −20.04 ppm/K per ohm.

If each endpoint temperature has independent standard uncertainty 0.1 kelvin, the uncertainty of the difference is about 0.1414 kelvin. With a coefficient sensitivity magnitude of 0.8 ppm/K per kelvin, the thermal contribution is about 0.1131 ppm/K. These assumed components illustrate propagation, not an achieved measurement capability.

Distinguish common temperature offset from interval error

With the same 0.1-kelvin endpoint uncertainties and correlation 0.9, the standard uncertainty of their difference becomes approximately 0.04472 kelvin, giving about 0.03578 ppm/K in the example. This follows from u²(d) = u²(T1) + u²(T2) − 2cov(T1,T2).

An exactly common additive thermometer offset cancels from the numerical temperature difference. It does not necessarily establish that the physical interval lies at the required absolute temperatures. On a curved resistance-temperature characteristic, shifting both true endpoints can change the chord being measured. Keep that interval-placement question distinct from subtraction arithmetic.

Check resistance gain and resistance offset separately

If both resistance readings are multiplied by exactly the same gain factor, that factor cancels from (R2 − R1)/R1. This exact algebra is more informative than assigning two independent copies of one gain specification. Range changes or temperature-dependent gain can break the shared-factor assumption.

A common additive resistance offset behaves differently. It cancels from R2 − R1 but remains in R1. In the numerical example, the summed sensitivity to a common offset is −0.04 ppm/K per ohm. A 0.01-ohm standard common offset therefore contributes 0.0004 ppm/K, not zero. Unshared contact errors may be much larger and require separate entries.

Classify each contribution before combining it

Use instrument documentation and method evidence to decide how a term is shared. A convenient correlation coefficient chosen to obtain a smaller final number has no metrological basis.

Choose one representation of shared errors

One approach uses endpoint uncertainties plus covariance. Another introduces a common gain or offset variable directly into the measurement equation and assigns residual errors separately. Either can work. Combining both approaches for the same physical component counts it twice.

For each component, document its origin, probability interpretation and shared scope. A manufacturer's bounded accuracy term is not automatically a standard uncertainty. Nor does using the same instrument guarantee perfect correlation for every error it produces. Noise, range switching, environmental changes and contacts can remain endpoint-specific.

Verify the budget with controlled perturbations

Perturb one model input at a time and compare the resulting coefficient change with the derivative prediction. Apply a common gain to both resistance endpoints and confirm exact cancellation. Apply a common offset and confirm the smaller nonzero effect. These tests expose sign, unit and spreadsheet-reference errors.

Report the coefficient definition, raw values, standard uncertainty and any justified expanded uncertainty with its coverage convention. Retain the correlation rationale and temperature assignment. If a narrow interval makes the uncertainty unacceptably large, changing the test interval is a technical decision that must remain consistent with the required TCR specification, not an undisclosed numerical adjustment.