Heat transfer mechanisms
The thermal chain of a transistor is solved exactly like a resistive divider: power plays the role of current, temperature that of voltage, and each interface that of a resistance. With that analogy, calculating a heat sink is a circuits exercise.
01Three ways to move heat
Heat always goes from the hot body to the cold one, and it arrives by three paths that act at the same time. In electronics all three are present in any piece of equipment, but they do not weigh the same.
| Mechanism | How it travels | Law | Where it dominates |
|---|---|---|---|
| Conduction | By contact, without the matter moving | \( \dot{Q} = kA\,\Delta T/L \) | From the chip to the package and from there to the heat sink |
| Convection | The fluid moves and carries the heat away | \( \dot{Q} = hA\,\Delta T \) | From the heat sink to the air |
| Radiation | Electromagnetic waves, no medium needed | \( \dot{Q} = \varepsilon\sigma A(T^4 - T_a^4) \) | Contributes little below 100 °C; in a vacuum it is the only thing left |
The third row has the fourth power, so radiation grows very fast with temperature. At 60 °C over a 25 °C ambient, polished aluminum (emissivity 0.05) radiates barely about 12 W per square meter; black anodized —emissivity close to 0.9— radiates almost twenty times more, and the heat sink gains ten or fifteen percent overall, because the fins shade one another. That is why heat sinks are black, and not for aesthetics.
02The analogy that solves everything
Fourier's law for a flat wall has exactly the form of Ohm's law, and that coincidence is no accident: both describe a flow pushed by a potential difference against a resistance.
| Thermal quantity | Unit | Equivalent to |
|---|---|---|
| Dissipated power \( \dot{Q} \) | W | Current |
| Temperature difference | K | Voltage |
| Thermal resistance \( R_{th} \) | K/W | Resistance |
| Thermal capacitance \( C_{th} \) | J/K | Capacitance |
The last row explains the transient regime: a heat sink with a lot of mass takes time to heat up, just as a large capacitor takes time to charge, and the time constant is \( R_{th}C_{th} \). That is why a transistor withstands power peaks far above what it could withstand in steady state.
03The junction-to-ambient chain
Heat is born at the semiconductor junction and has to reach the air. On the way it crosses three resistances in series, and the temperature drops across each one:
The complete thermal chain, with values from real datasheets. You move the power and the ambient temperature, choose the package, interface and heat sink, and the junction temperature appears along with how much power the assembly can handle.
Two apparently flat metal surfaces really touch over less than 1% of their area; the rest is air, which is an excellent insulator. The paste is not there because it conducts well —it conducts much worse than metal— but to displace that air. Hence more paste is worse: the layer has to be the minimum that fills the imperfections, and anything beyond that adds resistance.
04In the lab
Mount a power resistor on a heat sink with a thermocouple attached to the fin. Dissipate a known power, wait for steady state and calculate \( R_{sa} \) from the difference with respect to ambient. Compare with the heat sink datasheet.
Repeat the previous measurement with dry contact, with paste and with a silicone pad, always with the same tightening torque. Calculate \( R_{cs} \) in the three cases and rank them.
Record the heat sink temperature against time from the moment the power is applied until steady state. Fit an exponential, obtain the time constant and from it the thermal capacitance of the assembly.
05Common mistakes
- Using too much thermal paste. The useful layer is the minimum; the excess adds resistance.
- Using \( R_{ja} \) when there is a heat sink. That value assumes the package in open air and it is not valid for the real assembly.
- Calculating with the room's ambient. Inside the enclosure it is quite a bit hotter.
- Forgetting that mica insulates but gets in the way. It adds thermal resistance in exchange for electrical insulation.
- Designing for \( T_{j\max} \). That is the destruction limit, not the operating point: leave a margin.
- Always neglecting radiation. Below 100 °C it contributes little, but in a vacuum it is the only thing.
06Self-assessment
What role does power play in the thermal analogy?
That of current. The temperature difference plays voltage and the thermal resistance plays resistance.
Why are heat sinks black?
Because of emissivity: black anodizing radiates much better than polished aluminum, and that adds ten or fifteen percent.
What is thermal paste really for?
To displace the air trapped between two surfaces that only touch over a tiny fraction of their area.
How is the electronics of a satellite cooled?
Only by radiation: with no air there is no convection.
What sets the thermal time constant?
The product \( R_{th}C_{th} \), just as in an RC circuit. That is why a heat sink with mass withstands power peaks.
Which of the three resistances in the chain is chosen by buying?
\( R_{sa} \), the heat sink's. \( R_{jc} \) is set by the manufacturer of the semiconductor and \( R_{cs} \) depends on the mounting.
07Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). Conduction, convection and radiation with Fourier's law and Stefan–Boltzmann.
- Frank P. Incropera and David P. DeWitt. Fundamentals of Heat and Mass Transfer. 4th ed., Wiley, 1996 (Spanish edition, “Fundamentos de transferencia de calor”, Pearson, 1999). The reference of the discipline; the chapter on thermal resistances in series is exactly this unit.
- Gordon N. Ellison. Thermal Computations for Electronics. 2nd ed., CRC Press, 2020. Thermal design applied to electronic equipment, with real packages, heat sinks and enclosures.