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An observed ratio change can be consistent with several combinations of temperature-coefficient mismatch and unequal element temperature. The diagnostic task is to determine which combination explains the installed pair without fitting an arbitrary gradient to every discrepancy. Expressing temperature as a mean rise and a signed difference gives a useful contribution ledger. Retaining the ratio denominator also shows when a first-order estimate is too coarse for the available error budget.
Key design decisions
- Normalize the observed ratio to the same reference used to characterize each resistor's temperature response.
- Calculate the common-temperature mismatch term and the gradient term separately before judging their sum.
- Infer a temperature difference only when independent coefficient and mean-temperature information make the inverse calculation sufficiently sensitive and credible.
Keep the measured ratio and reference pair in one direction
Label the denominator resistor R1 and numerator resistor R2 on the schematic, fixture and data columns. Let Q0 be their measured ratio at the selected reference temperature and defined low-power condition. The diagnostic observation is e = Q/Q0 minus one. This removes initial ratio offset from the thermal-change calculation while preserving the possibility that the absolute values later move for other reasons.
Use individual low-power resistance-versus-temperature data obtained after the relevant trim and protective processes. A catalog tracking label does not establish the two coefficients needed for a powered installation. If the curves are nonlinear, preserve their measured functions and solve with those functions over the actual interval. The linear treatment below is a deliberately local model. It should not be extended through a temperature range where the slope or return-to-reference value changes materially.
Assign separate signs to mean heating and the element difference
Define M as the average of the two element temperature rises from the reference, and G as the temperature of element two minus that of element one. Then the individual rises are M minus G/2 and M plus G/2. Also define the mean coefficient alpha-bar and the coefficient difference delta-alpha as alpha2 minus alpha1. Positive G means the numerator is warmer, regardless of where it sits physically.
Under the linear resistance model, the numerator of the fractional ratio change splits exactly into delta-alpha times M plus alpha-bar times G. The first term describes mismatch under shared heating. The second describes unequal temperature acting through the average coefficient. Keeping the signs matters: the two contributions can cancel at one operating state, creating an apparently stable ratio that moves sharply when the load or mounting changes.
Retain the denominator when converting contributions into ratio error
The contribution sum is not itself the exact fractional ratio change. It must be divided by the normalized denominator resistance. This denominator follows the temperature of element one and makes reversing the ratio slightly different from merely reversing the sign. For broad temperature changes or tight error allocations, calculate the exact expression within the chosen linear model before deciding whether the approximation is adequate.
A small denominator correction does not prove the underlying thermal model is accurate. Uncertain element temperatures, curve curvature, voltage dependence and retained material change can exceed the arithmetic correction. Compare the approximation error with those other uncertainties rather than adding unnecessary numerical precision. The equation describes ratio drift only; translating it into amplifier gain or divider-output error requires the circuit's transfer sensitivity.
e = [δα·M + ᾱ·G] / [1 + α1(M − G/2)]
- e = (R2/R1)/Q0 − 1 is dimensionless fractional ratio change.
- δα = α2 − α1 and ᾱ = (α1 + α2)/2; coefficients are fractional change per kelvin.
- M is mean temperature rise in kelvin and G is the signed element-temperature difference.
Each resistance is locally linear in its own effective temperature, coefficients refer to the same reference state, and nonthermal electrical or retained changes are negligible or separately corrected.
Calculate a pair whose two thermal terms reinforce
Assume a hypothetical pair has alpha1 = 40 ppm/K and alpha2 = 60 ppm/K. Its mean temperature rise is 50 kelvin and the numerator is four kelvin warmer. The element rises are therefore 48 and 52 kelvin. Differential TCR contributes 20 × 50 = 1,000 ppm, and the gradient contributes 50 × 4 = 200 ppm. Their unnormalized sum is 1,200 ppm.
The denominator is 1 plus 40 × 10 to the power minus six times 48, or 1.00192. Dividing 0.0012 by 1.00192 gives approximately 0.00119770: 1,197.70 ppm, or 0.119770 percent ratio drift. The first-order result is about 2.30 ppm higher. If the gradient were minus four kelvin with the same mean rise, the gradient term would oppose the mismatch contribution. Agreement at one gradient direction would consequently not establish good coefficient tracking.
Use an observed ratio to estimate a gradient only with independent inputs
Rearranging the equation gives G = [e(1 + alpha1 M) minus delta-alpha M] divided by [alpha-bar plus e alpha1/2]. With the preceding assumed coefficients, mean rise and exact calculated error, it recovers four kelvin. In an experiment, this calculation is useful only when e is measured independently and the coefficient and mean-temperature inputs are established elsewhere. Reconstructing a ratio from temperatures inferred from that same ratio is a consistency identity, not validation.
The denominator of this inverse expression reveals an important limitation. When the mean coefficient is near zero, a considerable gradient can produce little ratio response. Small electrical errors then imply very large changes in inferred temperature. Treat such a result as poorly identifiable rather than concluding the substrate is extremely hot. The inferred temperature is also an effective electrical average for each resistor, not a measurement of its hottest point or a thermal safety limit.
Decide whether the inferred difference can resolve the engineering question
Evaluate uncertainty through the complete inverse expression using sensitivities or a numerical propagation with supported input distributions. Coefficients, mean temperature and ratio measurement can be correlated. A common thermometer error may affect both fitted coefficients; sequential resistance readings may share instrument drift. Keep those relationships rather than assigning independent uncertainties to convenient table columns.
A simple sensitivity check already exposes weak experiments. With mean TCR near 50 ppm/K, an assumed ratio uncertainty of 30 ppm corresponds to roughly 0.6 kelvin of gradient uncertainty before other terms. A one-ppm/K uncertainty in differential TCR multiplied by a 50-kelvin mean rise produces 50 ppm, equivalent to about one kelvin of gradient. These illustrative contributions show why extra ratio-display digits may contribute little when coefficient characterization dominates.
| Input or observation | Diagnostic role | Required independent support |
|---|---|---|
| Measured normalized ratio change | Defines the change being explained | Stable reference ratio, measurement nodes and acquisition timing |
| Differential TCR | Predicts response to the mean rise | Paired low-power temperature curves after final relevant processing |
| Mean TCR | Sets sensitivity to a gradient | Individual coefficient estimates with interval and uncertainty |
| Mean element-temperature rise | Separates shared heating from unequal heating | Temperature observation or validated thermal characterization |
| Return-to-reference ratio | Tests whether a reversible thermal interpretation remains valid | Repeated reference readings after controlled recovery |
Challenge the attribution with an independently changed thermal condition
First compare a low-power, approximately uniform-temperature observation with the predicted mismatch contribution. Then evaluate the installed power state using an independent indication of mean heating and, where practical, local temperature near each element. Change a thermal boundary that is expected to alter G while keeping the electrical topology and mean condition as comparable as possible. A successful model should predict the direction and approximate amount of that change without refitting the coefficients.
Use more than one mean-temperature level and gradient direction if the fixture permits those comparisons safely. Repeat the ratio after returning to the original condition. Record power and measurement timing so a change caused by receiver loading or warm-up is not attributed solely to gradient. Compare prediction residuals with the uncertainty ledger. A residual that exceeds it calls for another physical term or a better measurement, not an undocumented adjustment of the TCR values.
Recognize signatures that invalidate the thermal explanation
An error that changes immediately with voltage before appreciable heating suggests loading, voltage dependence or acquisition effects. A ratio that remains displaced after the original temperature is restored suggests retained change, contact movement or an unstable reference. A fitted gradient that changes dramatically when the ratio order is reversed indicates a calculation or normalization problem, because the inferred physical temperature difference should remain consistent after the sign conventions are transformed.
If a thermal boundary reversal changes the observed error in the expected direction but by an incorrect amount, inspect the coefficient interval, mean-temperature estimate and spatial averaging before redesigning the resistor pair. If uncertainty makes several causes plausible, report that ambiguity explicitly and identify the next measurement that can separate them. The quotation discussion should carry a contribution ledger, measured residuals and an actionable thermal or material question, rather than simply asking for a lower TCR number.
Provide the thermal-ratio diagnosis inputs
Independent electrical and temperature observations make the mismatch and gradient terms separable.
- Pair schematic, ratio direction, reference values and permitted thermal ratio drift under the actual receiver loading.
- Individual post-process resistance-temperature curves, coefficient intervals and uncertainty including return-to-reference data.
- Measured operating ratio, element power states, mean-temperature evidence and available local-temperature observations with timing.
- Mounting and neighboring heat sources, proposed thermal comparison, measured prediction residuals and the required diagnostic resolution.
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