TCR-Matched Resistor Pairs: Ratio Drift Across Temperature

Distinguish a low coefficient from a matched coefficient

TCR describes fractional resistance change with temperature under a specified measurement convention. A pair's tracking behavior concerns the difference between its responses. Two coefficients of the same sign and nearly equal magnitude may produce better ratio stability than two smaller coefficients with opposite signs. Consequently, a purchasing requirement for low TCR should not be used as a substitute for a ratio-temperature requirement.

Define the ratio in a fixed direction, such as R2 divided by R1. Reversing numerator and denominator reverses the first-order sign of the drift. State the reference temperature and whether the requirement is an endpoint coefficient, a maximum deviation anywhere in the range, or both. These definitions become particularly important when comparing a material data sheet with measurements from a trimmed pair.

Calculate the common-temperature contribution

For a first-order resistance model, each element equals its reference resistance multiplied by one plus its coefficient times the temperature change. Dividing these expressions gives the exact ratio within that linear model. When the products of coefficient and temperature change are small, the ratio's fractional change is approximately the differential coefficient multiplied by the common temperature change.

As a design calculation, take coefficients of 40 and 55 parts per million per kelvin. Over a common rise of 60 kelvin, the first-order ratio change is 900 parts per million, or 0.09 percent. Those assumed values demonstrate sensitivity only; they do not specify a material or manufacturing capability. The useful conclusion is that a moderate difference between coefficients can consume a tight ratio allowance even when each individual value appears acceptable.

Keep individual temperatures when the pair is not isothermal

With unequal temperatures, the first-order ratio change becomes α2 times ΔT2 minus α1 times ΔT1. Even identical coefficients do not cancel if the temperature rises differ. For example, two assumed 50 ppm/K elements with a four-kelvin temperature difference acquire about 200 ppm of ratio change from that gradient alone. Improving coefficient matching will not remove this term.

A useful decomposition separates average temperature rise from the gradient between the two elements. The differential coefficient multiplies the average rise, while the average coefficient multiplies the gradient. This shows two distinct ways to improve the result: reduce differential material response or reduce thermal inequality. Geometry, spacing, heat-source proximity and attachment conditions influence the latter, so a material substitution alone may not solve the installed-circuit error.

Do not hide curvature inside one endpoint coefficient

An endpoint coefficient summarizes the slope between two chosen temperatures. Two resistance curves can share that slope while separating in the middle. A pair that meets a hot-end ratio requirement may therefore exceed its allowance at an intermediate temperature. Record enough points to observe curvature and any change in slope, especially when the required range spans distinct operating regimes.

Normalize each curve to its own measured resistance at the same reference temperature, then calculate the ratio at every paired temperature. Plot both individual normalized curves and their difference. If only the ratio plot is retained, large common movement becomes invisible; if only individual TCR values are retained, a narrow tracking requirement is difficult to judge. Both views are needed to explain the physical result.

Use a temperature sequence that reveals reversible and retained change

Begin with low-power readings at the reference condition. Move to selected temperatures with sufficient time for the substrate and fixture to equilibrate, not merely for the chamber air display to reach its setpoint. Return to the reference condition and repeat the starting measurement. A changed return value indicates retained movement or a measurement problem, not simply the reversible TCR being characterized.

Measure the two elements close together in time and alternate the reading order in a repeat sequence. Keep the same sensing locations and stimulus unless a deliberate comparison requires otherwise. Contact changes during thermal expansion can masquerade as resistor drift, so include a connection check and a stable measurement reference. Record actual temperature observations near the pair where practical rather than assigning every reading the chamber's nominal setting.

Separate coefficient mismatch from a thermal-layout problem

Compare a low-power temperature sweep with the powered assembly. The distinction between these two experiments is more informative than repeating one test at increasingly tight meter settings. Use the observations below to select the next controlled change, keeping the same physical pair whenever possible.

Make thermal symmetry a measurable design objective

Place the pair with attention to its entire heat path. Equal spacing on a drawing is not thermal symmetry if one resistor is near a terminal, clamp, heat spreader edge or active device. Compare the substrate support beneath each element and the conductor paths that carry heat away. Where power differs substantially, consider geometry or circuit arrangements that reduce temperature inequality without creating an unacceptable electrical compromise.

Do not assume a shared ceramic substrate guarantees uniform temperature. The useful question is whether the measured temperature difference and individual coefficients fit inside the ratio budget for each operating state. Include warm-up and load transitions when the system reads during them. A layout may have excellent equilibrium tracking while producing a temporary ratio excursion after one neighboring element is energized.

Specify the pair rather than a material slogan

The final requirement should identify the ratio direction, reference temperature, temperature range, allowed ratio deviation and powered conditions. Specify whether measurements occur before or after trimming, protective coating and assembly. If a material or process changes, recheck the paired curves; a previous tracking observation does not automatically transfer to a new termination combination or geometry.

For an RFQ, supply the actual error allocation rather than requesting the lowest possible TCR. An application dominated by a thermal gradient may benefit more from layout or power redistribution than from a tighter room-temperature match. Conversely, a nearly isothermal low-power network may justify a specific differential-temperature requirement. Keeping these cases separate makes the requested performance both technically meaningful and verifiable.