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Two laboratories can report different temperature coefficients for the same resistance curve because their reference temperatures, intervals or normalization conventions differ. Comparing the numbers before reconciling those definitions can create a false material discrepancy. Preserve the measured resistance–temperature points and state precisely which slope each coefficient represents. Re-expressing the curve changes its coefficients, not the physical resistor.
Key design decisions
- Name both the temperature interval and the resistance used as the denominator.
- Distinguish an endpoint coefficient from a local slope and a maximum deviation requirement.
- Rebase every polynomial coefficient consistently when the reference temperature changes.
- Keep measured return-to-reference changes separate from reversible temperature behavior.
1. Identify the quantity behind the TCR number
An interval coefficient divides the resistance difference between two temperatures by a stated reference resistance and by the temperature difference. The reference resistance may be the value at one endpoint or at another agreed temperature. The units ppm/K and ppm/°C are equivalent for temperature differences, but the denominator and interval must still match.
A local coefficient can mean the slope divided by the resistance at that same temperature. Another report may give the tangent slope of a curve normalized to one fixed reference resistance. These local definitions differ away from the normalization point. Write the equation used rather than assuming that the abbreviation TCR uniquely identifies it.
2. Start the reconciliation with resistance in ohms
Ask for the actual sample temperatures and resistance readings, including specimen identity and the measurement state. Retaining only a ppm/K value loses information needed to distinguish a changed interval from a changed resistor. If the data are already normalized, obtain the numerical reference resistance and the temperature associated with it.
Recalculate both laboratories' reported coefficients from the same raw curve using each declared convention. Agreement with both calculations can resolve an apparent discrepancy without changing the product. If the raw curves themselves disagree at matched temperatures, the issue is no longer simply reporting convention and should return to measurement conditions and specimen history.
3. Declare a curved model only over the supported interval
A quadratic representation can be convenient when the observed resistance curve contains modest smooth curvature. Write it relative to an explicit temperature origin, with coefficients in fractional units. The linear coefficient has units K⁻¹ and the quadratic coefficient K⁻². A coefficient quoted in ppm/K² must be multiplied by one millionth before use in the fractional equation.
The representation is a fitted approximation, not a universal law for fired thick-film resistors. Three distinct temperature points determine a quadratic exactly but leave no independent point with which to assess its shape. Use additional justified observations and examine the residuals before relying on the curve between points or at the ends of its measured range.
R(T) = R0[1 + a(T − T0) + b(T − T0)²]
- R0 is resistance at the original reference temperature T0, in ohms.
- a and b are the coefficients of the stated quadratic, in K⁻¹ and K⁻² respectively.
- Temperature differences may be expressed in kelvin or degrees Celsius with consistent numerical increments.
A reversible single-valued curve over the tested temperature interval, at the defined electrical excitation and specimen state; model adequacy must be checked against observations.
4. Calculate several legitimate coefficients from one curve
Assume R25 is 1,000 Ω, a is 20 ppm/K and b is 0.2 ppm/K². At 75 °C, the 50 K increment gives R75 = 1,001.5 Ω. The 25-to-75 °C endpoint coefficient normalized by R25 is therefore 30 ppm/K. The tangent coefficient at 25 °C is 20 ppm/K.
At 75 °C, the derivative is 0.040 Ω/K. Dividing that by the original 1,000 Ω gives a fixed-reference tangent slope of 40 ppm/K, whereas dividing by the local 1,001.5 Ω gives approximately 39.940 ppm/K. All four values describe the same assumed curve under different definitions. None should replace another without retaining its interval and denominator.
5. Rebase the complete curve rather than moving its label
Let the new origin T1 be T0 plus h. The resistance at the new origin is R1 = R0(1 + ah + bh²). When the same curve is expressed as R1[1 + a1(T − T1) + b1(T − T1)²], its new coefficients are a1 = (a + 2bh)/(1 + ah + bh²) and b1 = b/(1 + ah + bh²).
For the preceding example rebased from 25 °C to 75 °C, R1 is 1,001.5 Ω, a1 is approximately 39.940 ppm/K and b1 approximately 0.199700 ppm/K². Merely changing the temperature label while leaving a and b unchanged would predict a different curve. Check the transformation by calculating a temperature other than either reference using both representations; the resistance must agree to the retained numerical precision.
6. Check an interior extremum when the endpoint slope is small
Consider a different assumed curve with R25 = 1,000 Ω, a = −20 ppm/K and b = 0.4 ppm/K². Its values at 25 °C and 75 °C are both 1,000 Ω, giving a zero endpoint coefficient over that interval. At 50 °C, however, the model gives 999.75 Ω, a deviation of −250 ppm from R25.
The stationary point occurs where a + 2b(T − T0) equals zero, provided b is nonzero and the point lies inside the validated interval. Evaluate that point alongside the interval endpoints when the requirement limits the maximum resistance deviation. A zero chord is not the same requirement as a flat curve, and a local zero slope does not imply zero change over an extended interval.
7. Select the correction that matches the disagreement
Use the raw curve to determine whether the disagreement concerns reporting, model adequacy or physical repeatability. Changing a coefficient definition is appropriate only for the first of these cases.
| Disagreement | Required reconciliation | What must remain separate |
|---|---|---|
| Same resistance points but different interval coefficients | Match endpoints and normalization resistance | Material change versus arithmetic convention |
| Same fitted curve but different linear coefficients | Rebase origin, reference resistance and quadratic term together | Coefficient change versus physical curve change |
| Endpoint coefficient is near zero but intermediate values move | Check measured interior points and a justified stationary point | Chord requirement versus maximum deviation |
| Heating and cooling differ at the same temperature | Preserve separate branches and investigate state dependence | Reversible curve versus hysteresis |
| Return-to-reference value has shifted | Retain the sequence and the changed baseline | Temperature dependence versus retained drift |
| Quadratic fit has structured residuals | Review model and measurement conditions within the tested range | Convenient polynomial versus validated representation |
8. Deliver a coefficient record that survives a comparison
Report the sample, process stage, actual temperatures, resistance values, electrical stimulus and sequence with the coefficient definition. Include the numerical normalization resistance, temperature interval and sign convention. For a fitted model, retain the original data, units, reference origin, residuals and usable interval as well as the coefficients.
Rebasing does not remove temperature uncertainty, self-heating, hysteresis or retained drift. If a reference measurement after the cycle differs from the starting value, state which observation was used for normalization and investigate the time history before merging all readings into one reversible curve. Keeping these distinctions explicit lets customers compare reports fairly without concealing an actual material or measurement discrepancy.
Send the complete TCR reporting definition
Provide enough raw temperature data to separate a convention mismatch from a real resistance-curve change.
- Individual specimen identity, material stack and measured processing stage
- Actual resistance–temperature points with heating, cooling and return sequence
- Reported interval, normalization resistance and exact coefficient equation
- Electrical excitation, thermal equilibration method and measurement uncertainty
- Polynomial coefficients with units, reference temperature, residuals and fitted range
- Whether acceptance limits endpoint slope, local coefficient or maximum resistance deviation
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