The Electrical Resistance Temperature Relationship: Thermal Runaway and Heatsink Math

The electrical resistance temperature relationship dictates that as conductors and semiconductors heat up, their electrical resistance changes—usually increasing. This positive feedback loop is the root cause of thermal runaway in power electronics. When current flows through a material, it generates heat ($P = I^2R$). If the material has a Positive Temperature Coefficient (PTC), that heat raises its resistance, which in turn generates even more heat for the same current. Without a properly calculated thermal path to ambient air, this cycle ends in melted solder, cracked silicon, or a fire.

To design reliable circuits, you cannot just look at a component's maximum current rating; you must model the thermal path from the silicon junction to the room air. Here is how to translate the physics of resistance into actionable thermal management math.

Material TCR and the Thermal Runaway Threshold

The Temperature Coefficient of Resistance (TCR) quantifies how much a material's resistance changes per degree Celsius of temperature rise. It is usually expressed in parts per million per degree Celsius (ppm/°C) or as a percentage. Understanding the TCR of your specific materials tells you whether a component will self-stabilize or spiral into thermal runaway.

For example, standard silicon power MOSFETs exhibit a strong positive TCR in their channel resistance ($R_{DS(on)}$). As the junction heats up, $R_{DS(on)}$ can double or triple compared to its 25°C datasheet value. If your heatsink is undersized, the increased $R_{DS(on)}$ dissipates more wattage, driving the temperature higher until the 175°C silicon limit is breached. Conversely, materials like Manganin are engineered for near-zero TCR to serve as stable current shunts, while NTC thermistors intentionally use a negative TCR for temperature sensing.

Thermal Path Math: Sizing the Heatsink

To prevent the electrical resistance temperature relationship from destroying your silicon, you must calculate the required thermal resistance of your heatsink. Thermal resistance ($R_{\theta}$) is measured in °C/W and acts exactly like electrical resistance in Ohm's Law, where temperature is voltage and power dissipation is current.

The governing equation for the total thermal path from the semiconductor junction to the ambient air is:

$T_J = T_A + P_D \times (R_{\theta JC} + R_{\theta CS} + R_{\theta SA})$

• $T_J$: Junction Temperature (°C)
• $T_A$: Ambient Air Temperature (°C)
• $P_D$: Power Dissipated (Watts)
• $R_{\theta JC}$: Junction-to-Case thermal resistance (from datasheet)
• $R_{\theta CS}$: Case-to-Sink thermal resistance (thermal paste/pad)
• $R_{\theta SA}$: Sink-to-Ambient thermal resistance (the heatsink itself)

Worked Example: Sizing a Heatsink for an IRF3205

Let's say you are switching a 25A load using an IRF3205 power MOSFET in a TO-220 package. Your calculated conduction and switching losses total $P_D = 25W$. The enclosure ambient temperature ($T_A$) reaches 45°C on a summer day.

While the IRF3205 datasheet lists an absolute maximum $T_J$ of 175°C, operating silicon at 175°C guarantees a short lifespan due to accelerated electromigration. A standard reliability rule of thumb is to keep $T_J$ below 125°C.

1. Calculate maximum allowed total thermal resistance ($R_{\theta JA}$):
   $R_{\theta JA(max)} = (T_J - T_A) / P_D = (125 - 45) / 25 = 3.2 °C/W$
2. Subtract known resistances:
   The IRF3205 datasheet specifies $R_{\theta JC} = 0.75 °C/W$.
   We will use a Bergquist Sil-Pad 400VOS thermal interface material, which yields $R_{\theta CS} \approx 0.50 °C/W$ for a TO-220 footprint.
3. Solve for the required heatsink ($R_{\theta SA}$):
   $R_{\theta SA} = 3.2 - 0.75 - 0.50 = 1.95 °C/W$

You need a heatsink with a thermal resistance of 1.95 °C/W or lower. Browsing DigiKey's heatsink selection guide and filtering for TO-220 packages with natural convection, the Aavid (Boyd) 528002B02500G extruded aluminum heatsink is rated at approximately 1.8 °C/W in still air. This provides a 0.15 °C/W safety margin, keeping your junction safely under 125°C.

Derating Curves, Airflow, and Enclosure Reality

Datasheets include a "Power Derating Curve" graph that plots maximum allowable power dissipation against ambient temperature. Many hobbyists misread this chart. The slope of the derating line is exactly equal to $-1 / R_{\theta JA}$. If the curve shows that power must be derated to zero watts at 175°C ambient, and the max power at 25°C is 100W, the implied $R_{\theta JA}$ is $(175 - 25) / 100 = 1.5 °C/W$. This specific $R_{\theta JA}$ usually assumes the part is mounted to an infinite heatsink or a massive copper plane, not sitting in free air.

If you rely on the datasheet's "free air" $R_{\theta JA}$ (often 40 to 62 °C/W for a TO-220), your 25W dissipation would result in a $1000°C+$ temperature rise, instantly vaporizing the part. You must alter the physical environment to buy thermal headroom.

As noted in All About Circuits' thermal design tutorials, placing a heatsink inside a sealed plastic enclosure is a common mistake. The air inside the box heats up, raising the local $T_A$ and rendering the heatsink's external rating useless. If the enclosure must be sealed, the heatsink must be mounted to the outside of the chassis, or the chassis itself must act as the heatsink.

Failure Signatures: How Hot is Too Hot?

When the electrical resistance temperature relationship drives a component past its thermal limits, the failure is rarely instant unless you hit the silicon melting point. Instead, thermal stress causes progressive degradation that manifests in specific, recognizable signatures on the bench.

1. Silicon Electromigration and Threshold Shift

Operating a MOSFET consistently above 125°C accelerates electromigration—the physical movement of metal atoms in the gate and interconnect layers due to high current density and heat. The signature failure is a gradual shift in the gate threshold voltage ($V_{GS(th)}$). The part will begin to turn on partially at lower gate voltages, leading to shoot-through in half-bridge circuits and eventual catastrophic shorting.

2. Solder Joint Creep Fatigue

Lead-free SAC305 solder has a melting point of roughly 217°C (490 K). In materials science, "creep" (slow plastic deformation under stress) begins at about 0.5 times the absolute melting temperature. For SAC305, that is roughly 108°C. If your PCB or component leads operate above 100°C continuously, the thermal expansion mismatch between the silicon die, the copper leadframe, and the FR4 board will cause the solder joints to micro-crack over thousands of thermal cycles. The signature is an intermittent open circuit that works when cold but fails when hot.

3. FR4 Glass Transition (Tg) Delamination

Standard FR4 PCB material has a Glass Transition Temperature ($T_g$) of 130°C to 140°C. Above this temperature, the epoxy resin softens and loses its mechanical rigidity. If a copper trace is carrying high current and self-heating due to its PTC resistance, and the local board temp crosses $T_g$, the Z-axis expansion coefficient spikes by 300%. This rapidly breaks plated through-holes (vias) and causes the copper trace to delaminate and lift off the board. The visual signature is a browned, blistered solder mask and a lifted trace.

By respecting the electrical resistance temperature relationship and doing the $R_{\theta}$ math upfront, you ensure your components stay in the safe zone where resistance is stable, solder remains rigid, and silicon lives for decades.