Heat sink calculator

Keeping 10 W under a 110 °C junction at 40 °C ambient allows 7.0 K/W in total. Subtract the package's 1.5 K/W junction to case and 0.33 K/W for the pad. The heat sink must then be 5.17 K/W or better, from TI SPRA953's Equation 5. It holds for tabbed packages in steady state, not leaded plastic parts that cool through the board.

The thermal resistance chain from junction to ambient, solved both ways: the sink-to-ambient rating a device's power and junction limit allow, and the junction temperature a chosen heat sink actually gives, with the thermal interface estimated from the pad's thickness, conductivity and area rather than forgotten.

Fig 1 — the thermal chain at 10.0 W: ambient 40 °C, sink 80.0 °C, case 83.3 °C, junction 98.3 °C against a 110 °C limit.

Power dissipated (W)

Ambient temperature (°C)

Junction temperature limit (°C)

Junction-to-case RθJC (K/W)

Interface thickness (mm)

Interface conductivity (W/m·K)

Interface contact area (mm²)

Heat sink rating RθSA (0 to solve only) (K/W)

System RθJA for Eq 6 (0 to skip) (K/W)

Interface RθCS = T / (k·A), SPRA953 Eq 7: 0.333 K/W · 3.3 °C across it at 10 W
Whole chain the 110 °C limit allows at 10 W: 7.00 K/W
Heat sink rating required, RθSA: 5.17 K/W or better
With a 4 K/W sink: sink · case · junction: 80.0 °C · 83.3 °C · 98.3 °C
Margin to the limit: 11.7 °C

How this is calculated

Standard: TI SPRA953 — Semiconductor and IC Package Thermal Metrics (Rev D)

$T_J = T_A + P\,(R_{\theta JC} + R_{\theta CS} + R_{\theta SA})$: SPRA953 Eq 5, "the proper application of RθJC for those instances when a high-efficiency heat sink is applied to the top surface of a device for which RθJC is small compared to RθJA". Solved for RθSA it gives the sink the limit allows.
$R_{\theta CS} = \frac{T}{k\,A}$: Eq 7: interface thickness over conductivity times contact area. The note calls it "merely an estimate, because the thermal interfacial resistance that can be developed between any two surfaces is neglected", and prefers a measurement.
$T_J = T_A + P\,\frac{R_{\theta JA}\,(R_{\theta JC} + R_{\theta CS} + R_{\theta SA})}{R_{\theta JA} + R_{\theta JC} + R_{\theta CS} + R_{\theta SA}}$: Eq 6, "more accurate than Equation 5 for any combination of RθJA, RθJC, or Rθ(SA) if RθJA is known for the system configuration": the board path in parallel with the sink path.

Assumptions

A package whose heat leaves mainly through the sinked surface: a tab, slug or exposed pad. SPRA953 states that for a leaded plastic package most of the heat leaves through the board, and that using RθJC there is "traditional, but invalid".
Steady state. The chain has no thermal capacitance; pulses shorter than the package's thermal time constant see a lower impedance.
The heat sink rating applies at the orientation, airflow and rise it was characterised at, with the ambient measured where the sink's datasheet measured it.
RθJA for Eq 6 is the value in this system, measured or modelled; the datasheet figure is for a JEDEC test board and is not it.
One device on the sink. Several share the sink's rating between their total power.

What sets the heat sink a device needs

Heat leaves a junction the way current leaves a source: through a chain of resistances, each dropping temperature in proportion to the power that flows. SPRA953 writes the chain for a device with a heat sink as its Equation 5, T_J = T_A + P·(RθJC + RθCS + RθSA): junction to case through the package, case to sink through the thermal interface, sink to ambient through the fins. Everything the calculator does is that one line, solved for whichever term is missing. Given the power, the ambient and the junction limit, the whole chain may be at most (T_J,max − T_A)/P kelvin per watt; subtract what the package and the interface take and what is left is the rating the heat sink must beat.

The note is careful about when the equation applies. RθJC "was originally devised to allow estimation of the thermal performance of a package when a heat sink was attached", and it is measured with the case pressed against a cold plate. Equation 5 is therefore "the proper application of RθJC for those instances when a high-efficiency heat sink is applied to the top surface of a device for which RθJC is small compared to RθJA", which is to say a power package with a tab or a slug: a TO-220, a D²PAK, a TO-247. For a plastic package with pins and no tab, most of the heat leaves through the board, and the note is explicit that using RθJC to estimate the junction from the case temperature there is "traditional, but invalid".

The interface is the term people leave out. SPRA953's Equation 7 estimates it as RθCS = T/(k·A), the thickness of the pad or grease over its conductivity times the area it covers, and calls that "merely an estimate, because the thermal interfacial resistance that can be developed between any two surfaces is neglected." At 10 W, a 0.1 mm pad of 3 W/m·K over a square centimetre is 0.33 K/W and 3.3 °C; the same pad on a 0.25 cm² tab is 13 °C, which is more than many heat sinks are worth. The calculator takes the three figures and shows the loss across the joint on its own.

When the package's junction-to-ambient in the actual system is known, Equation 6 does better than Equation 5 "for any combination of RθJA, RθJC, or Rθ(SA)": it puts the board path in parallel with the sink path, so the junction sits a little lower than the chain alone predicts and the calculator reports how much of the heat each path carries. The datasheet RθJA is not that number; it belongs to a JEDEC test board, which the θJA article takes apart, and the field is for a measured or modelled figure.

Heat sink chart: the rating the limit allows

The sink-to-ambient rating Equation 5 allows for the powers and temperature rises a design actually has, computed by the calculator above with RθJC = 1.5 K/W and a 0.3 K/W interface. A dash means the package and the joint alone already use the whole budget, and no heat sink helps; a rating in single digits is a small extruded sink in free air, and one below 1 K/W wants a fan.

Power	30 °C rise, junction over ambient	50 °C rise, junction over ambient	70 °C rise, junction over ambient
1 W	28.2 K/W	48.2 K/W	68.2 K/W
2 W	13.2 K/W	23.2 K/W	33.2 K/W
5 W	4.2 K/W	8.2 K/W	12.2 K/W
10 W	1.2 K/W	3.2 K/W	5.2 K/W
20 W	—	0.7 K/W	1.7 K/W
50 W	—	—	—

Read the 50 W row against the 5 W row: ten times the power leaves a tenth of the budget for the whole chain, and the fixed 1.8 K/W of package and joint eats most of it. Past a certain power the heat sink is no longer the problem; the package is, and the answer is a lower RθJC part or a second device to share the load.

Worked example: 10 W in a TO-220 with a 4 K/W sink

The defaults: 10 W dissipated, 40 °C ambient, a 110 °C design limit on the junction, RθJC of 1.5 K/W, a 0.1 mm pad of 3 W/m·K over 100 mm², and a 4 K/W heat sink in hand.

RθCS        = 0.1 mm / (3 W/m·K × 100 mm²)          = 0.333 K/W          (Eq 7)
chain budget = (110 − 40) / 10 W                     = 7.0 K/W
RθSA needed  = 7.0 − 1.5 − 0.333                     = 5.17 K/W or better  (Eq 5 solved)

with 4 K/W:  sink   = 40 + 10 × 4                    = 80.0 °C
             case   = 80 + 10 × 0.333                = 83.3 °C
             junction = 83.3 + 10 × 1.5              = 98.3 °C   →  11.7 °C of margin   (Eq 5)
with RθJA = 40 K/W known:  5.83 ∥ 40 = 5.09 K/W      →  90.9 °C, 87 % through the sink  (Eq 6)

The sink passes with room to spare, and the split of the 58 °C rise is the useful part: 40 °C of it is the heat sink, 15 °C the package and 3 °C the joint. Halving the sink's rating would take 20 °C off the junction; halving the pad thickness, 1.7 °C. That ordering is normal for a tabbed package on a small sink, and reverses only when the sink is large and the joint is bad.

Where the thermal chain stops being valid

Packages that cool through the board. SPRA953 puts it plainly: in a JEDEC still-air measurement "almost 70%–95% of the power generated by the chip is dissipated from the test board, not from the surfaces of the package." For a leaded plastic package with no tab, RθJC describes a path most of the heat does not take, and a top-side heat sink does less than its rating suggests. Equation 6 with a real RθJA is the honest version; theLDO thermal calculator and thevia calculator are the tools for a package that sinks into copper instead.

The heat sink's rating is conditional. A datasheet RθSA is for a stated orientation in free air, or a stated airflow, at a stated temperature rise; the same extrusion mounted horizontally, or boxed, or at a smaller rise, is worse. The note reminds that the ambient in Equation 5 is "at the location used for characterizing Rθ(SA), usually some distance away from the heat sink", and inside an enclosure that air is warmer than the room.

The interface estimate neglects contact resistance.Two machined faces pressed together touch at their high points; the grease or pad fills the rest, and Equation 7 counts only the fill. Low clamping force, a warped tab or a dry joint can double the real figure, which is why SPRA953 says the best method is to measure it.

Steady state only. The chain has no capacitance in it. A pulse shorter than the package's thermal time constant sees a lower resistance, which is what transient thermal impedance curves are for; a pulse longer than the heat sink's sees the full chain.

Several devices on one sink. The sink's rating is for the total power it carries; two 10 W parts on a 4 K/W sink raise it 80 °C, not 40, and each device then sees the other's heat in its ambient.

Common heat sink mistakes

Sizing the sink from the junction limit and forgetting the package and the joint. The sink gets what is left after RθJC and RθCS; on a hot part at high power that can be less than nothing.
Using the datasheet RθJA as the "system" figure for Equation 6. That number belongs to a JEDEC board with a defined copper area; the note's whole point is that it is not the board in the product.
Trusting a dry joint. Bare metal on bare metal leaves air in the gaps, and air is a thermal insulator; the pad or grease is not optional, and its thickness is the third term of the chain.
Reading RθJC off a leaded plastic package and attaching a top-side sink. Most of the heat still leaves through the leads into the board, and the sink cools the plastic more than the die.
Designing to the absolute maximum junction temperature. It is a survival limit, not an operating point; the 15 to 25 °C of margin is what buys the lifetime.