Coupled Resistors: Joint Electrothermal Stability Beyond Single-Element Tests

Define perturbations around an established operating point

Let x1 and x2 denote small temperature changes from a specified equilibrium of the two resistor regions. Each region has thermal capacitance Ci, a conductance gi to a fixed-temperature surrounding boundary, and reciprocal inter-region conductance k. The conductances are positive and locally constant. This two-node description requires each represented region to be sufficiently uniform for a single temperature to be useful.

Define Si as the derivative of electrical power Pi with respect to its own temperature at the selected electrical operating state. In this model the electrical channels are independent, so changing one resistor's temperature does not directly change the other channel's power. Shared current paths, active control delays or supply interaction require additional electrical cross-terms and are outside the following two-slope calculation.

Include the electrical slope in each heat balance

For node one, incremental electrical heating is S1 x1, heat flow to its fixed boundary is g1 x1, and heat exchange toward node two is k times x1 minus x2. The second node has the corresponding reverse exchange. Thus C1 times the derivative of x1 equals minus a1 x1 plus k x2, with a1 defined as g1 plus k minus S1. Define a2 in the same way.

The inter-node terms cancel when the two energy balances are added. The coupling transfers heat but does not remove it from the combined system. A positive Si means rising temperature adds electrical power locally; a negative slope provides a restoring contribution. Determine the slope from the actual source and resistance law, not from TCR sign alone.

Require the determinant as well as the diagonal terms

For positive a1 and a2, the system matrix has a negative trace. Its determinant is a1 a2 minus k squared, divided by C1 C2. Both perturbation modes decay only when a1 a2 is greater than k squared. Positive diagonal restoring coefficients alone do not satisfy this condition because the off-diagonal thermal feedback can overcome their product.

The same result follows by substituting a trial exponential into both equations and requiring negative modal growth rates. Equality produces a zero linear mode, for which this model does not establish return to equilibrium. If the determinant is negative, one rate is positive and the equilibrium is locally unstable. Thermal capacitance affects the rates, but cannot change the sign of that determinant when both capacitances remain positive.

Compare single-enabled and both-enabled slope states

Assume g1 and g2 are each 0.020 W/K, k is 0.030 W/K, and both capacitances are 0.100 J/K. For an enabled channel, take its local power slope as 0.030 W/K; for a disabled channel, take that slope as zero. These are illustrative local coefficients. Each operating state still needs its own physically established equilibrium and coefficient validation before this comparison is used for an actual circuit.

With only channel one enabled, a1 is 0.020 W/K and a2 is 0.050 W/K. Their product is 0.001000, exceeding k squared of 0.000900 in matching squared-conductance units. The two calculated rates are approximately minus 0.01459 and minus 0.68541 per second. Both modes decay. By symmetry, the other single-enabled state also passes this local test.

With both channels enabled, a1 and a2 each become 0.020 W/K. Their product falls to 0.000400, below k squared. The rates are now plus 0.100 and minus 0.500 per second. The same passive thermal coupling is present, but the changed electrical slopes create a growing mode that the single-enabled tests did not exercise.

Explain how a stable ratio can hide a growing temperature

For identical nodes with equal enabled slopes, define common temperature perturbation as half the sum of x1 and x2, and differential perturbation as half their difference. The common-mode rate is S minus g, divided by C. The differential-mode rate is S minus g minus twice k, divided by C. Coupling suppresses the difference but disappears from the common-mode equation.

In the both-enabled example, a small common perturbation grows by a factor of e in ten seconds, while a pure differential perturbation decays by the same factor in two seconds. Matched resistor temperature characteristics can consequently preserve their ratio while their common temperature departs from equilibrium. This is why a ratio-only observation is insufficient for this specific stability question.

Change the path that controls the unstable mode

Increasing coupling alone cannot stabilize the common mode of the identical-node model because k does not appear in its rate. Increasing each conductance to the fixed boundary from 0.020 to 0.040 W/K, with the other example coefficients unchanged, changes the common rate to minus 0.100 per second and the differential rate to minus 0.700 per second. Both then decay locally.

Alternatively, a revised electrical operating condition can reduce the incremental power slope, but the complete source behavior must support that change. Adding thermal mass merely slows the original positive rate; it does not make it negative. Distinguish these decisions from geometry changes intended only to reduce ratio mismatch, since stronger matching between temperatures and stable absolute temperature are different objectives.

Validate the local model without driving an uncontrolled excursion

Estimate thermal coefficients and electrical slopes within a bounded permitted region, with independent temperature observation and protection appropriate to the assembly. Use small perturbations around controlled operating states to compare common and differential recovery. Retain actual element powers with the temperature records so changing excitation is not mistaken for passive heat exchange.

Do not extrapolate a positive linear growth rate to an unlimited temperature or a predicted failure time. Resistance curvature, source limiting, heat-transfer changes and protection may intervene. Conversely, those later effects do not establish that the requested operating point is stable. The relevant acceptance result is supported return behavior within the required envelope, not survival of a large uncontrolled excursion.

Specify joint operation as its own acceptance state

Provide the resistor layout, independent channel circuits, intended simultaneous operation and temperature-dependent electrical behavior. Identify which measured quantity represents each thermal node and how nearby powered devices enter the balance. Where the two-node reduction is inadequate, retain more nodes or a distributed model rather than forcing a fitted ratio trace into this structure.

The engineering record should contain a separate joint-operation stability conclusion alongside ratio-drift and single-channel results. Record the coefficient ranges that control the determinant, not just one nominal answer. A modest positive nominal margin can be lost if heat removal weakens or the power slope increases, so repeat the calculation across the supported local parameter range before approving the shared-substrate operating state.