Power, Heat and Thermal Resistance

Suppose a 100 Ω resistor sits across a USB port's 5 V. It is a 1/4 W resistor from a beginner's kit. Lesson 01's check gives P = V² / R = 5² / 100 = 0.25 W, which is its power rating and no more. The check passes. Is the resistor cool enough to touch?

To answer, you need one more number for the part besides its rating: how many degrees it warms up for each watt.

From power to temperature

A resistor turns all the power in it into heat (lesson 01). Lesson 05's meter gives you both numbers you need. Across this resistor you would read 5 V, and through it 50 mA, so P = V · I = 5 V · 50 mA = 250 mW. The divider in lesson 03 does the same: it turns its current into heat all the time.

The heat leaves the part into the air around it. After you switch on, the resistor warms up until it sheds heat as fast as it makes it. Then its temperature stops climbing. That takes a while, and the part stays hot for a while after you switch off.

Four names describe where it settles:

• Ambient temperature (Ta) is the temperature of the air around the part. A room might be 25 °C. The air inside a car parked in the sun, or inside a closed box with a warm part in it, is much hotter.
• Temperature rise is how far above the ambient temperature the part settles.
• Thermal resistance, written θ (the Greek letter theta), is how many degrees a part's surface rises for each watt it turns into heat, for a stated mounting. Its unit is °C/W. Datasheets also write it in K/W (kelvin per watt). A step of one kelvin is the same size as a step of one degree Celsius, so the number is the same.
• Surface temperature is the temperature of the outside of the part, at the middle of its body, where the maker measures it.

The rise is the power times θ, and the surface sits that far above the air:

T = Ta + P · θ

The instrument writes this formula at the top, with P · θ, the rise, highlighted.

The resistors in this lesson are carbon-film resistors. In a carbon-film resistor, the resistance is a thin layer of carbon, the film, coated onto a small ceramic rod under the paint. Makers rarely print θ for these parts. Stackpole's datasheet for its CF carbon-film family gives a curve instead: the surface temperature rise against the power. At full rating the curve reads about 32 °C for the 1/4 W part and about 85 °C for the 1 W part. Divide by the power and you get θ: 32 / 0.25 = 128 °C/W and 85 / 1 = 85 °C/W. These θ values are typical, derived from the maker's temperature-rise curve. Another maker's curve, Vishay's, gives similar numbers.

The instrument makes three simplifications, so read its temperatures as "about":

• It draws a straight line from zero up to that point on the maker's curve. The real curve may bend a little below full power.
• It takes θ to be the same in any air, so the same power gives the same rise at 25 °C or at 85 °C.
• Its temperature is the surface at the middle of the body, in the maker's test mounting. The mounting is how the part is held: how long its legs are, the board it is soldered to, the parts around it and any box it sits in. On your board the number changes, because the rise depends on how much heat flows into the board. The film inside, where the heat is made, is at least as hot as the surface.

The instrument starts with the opening's resistor: 1/4 W under Resistor, 5 V in Supply, 100 Ω in R and 25 °C in Ambient temperature.

• Current: I = 5 V / 100 Ω = 50 mA
• Power: P = 5 V · 50 mA = 250 mW, all of the 1/4 W rating
• Temperature rise: 0.25 W · 128 °C/W = 32 °C
• Surface temperature: 25 °C + 32 °C = 57 °C

The instrument shows a Power of 250 mW with "50 mA through R", a Temperature rise of 32 °C with θ = 128 °C/W, and a Surface temperature of 57 °C with the note "can burn skin". Rating used reads 100 % "of 250 mW at 25 °C". That answers the opening: the resistor is inside its rating and too hot to touch.

Now pick 1 W under Resistor. The power is still 250 mW, but θ is now 85 °C/W:

• Temperature rise: 0.25 W · 85 °C/W = 21.25 °C, shown as 21 °C
• Surface temperature: 25 °C + 21.25 °C = 46.25 °C, shown as 46 °C

Rating used drops to 25 % "of 1 W at 25 °C", and the surface note says only "middle of the body". The 1 W part is bigger, 11.00 mm by 4.50 mm against 6.00 mm by 2.30 mm, so it has more surface to pass heat to the air. That is why it rises fewer degrees per watt, and why makers rate bigger resistors for more power.

Inside the rating, still too hot to touch

How hot is too hot depends on what you touch and for how long. CENELEC, the European body for electrical standards, publishes a guide to hot surfaces. It gives the burn threshold: the surface temperature at which skin starts to burn, for a given material and contact time. For bare metal, which burns fastest of the materials it lists:

• about 65–70 °C for 1 second, and about 55–59 °C for 10 seconds (read from the guide's curves);
• 51 °C for 1 minute.

For any material it gives 48 °C for 10 minutes and 43 °C for 8 hours. The instrument's "can burn skin" note appears from 51 °C, the 1-minute figure for metal. Below that it says where the temperature is measured and nothing more. The 1 W part's 46 °C is under the 1-minute figure, but it is not harmless: it is above the 8-hour figure.

The 1/4 W part at 57 °C is inside its rating and above 51 °C. A rating tells you how much heat the part can take without damage. It does not tell you whether the part is cool enough to touch.

Now give the 1 W part its own full rating. Keep 1 W under Resistor and type 10 in Supply:

• Power: P = 10² / 100 = 1 W, with I = 10 V / 100 Ω = 100 mA
• Temperature rise: 1 W · 85 °C/W = 85 °C
• Surface temperature: 25 °C + 85 °C = 110 °C

The instrument shows 1 W "100 mA through R", a rise of 85 °C, a surface of 110 °C "can burn skin", and Rating used 100 % "of 1 W at 25 °C". Compare that with the 1/4 W part's 57 °C at its rating. Per watt the 1 W part runs cooler (85 °C/W against 128 °C/W), but at its rating it takes four times the power, so it runs hotter. 110 °C is far past the 1-second figure for metal.

The habit that helps is headroom: the part of the rating you leave unused. This course uses a rule of thumb: plan for about half the rating. The rule is a course convention, not a figure from a datasheet. That is why Rating used changes colour above 50 %, and again above 100 %. Pick 1/4 W under Resistor and type 3.5 in Supply:

• Power: P = 3.5² / 100 = 122.5 mW, shown as 123 mW
• Rating used: 49 %, under the 50 % line
• Temperature rise: 0.1225 W · 128 °C/W = 15.68 °C, shown as 16 °C
• Surface temperature: 41 °C

Headroom also keeps the part cooler: 41 °C instead of 57 °C. It leaves room for the next problem too: hot air.

Hot air shrinks the rating

A resistor's power rating holds only while the ambient temperature is 70 °C or below. Above that, the rating falls in a straight line, down to zero at 155 °C. 155 °C is also the top of the part's operating temperature range. This fall is called derating, and the graph of it in a datasheet is the derating curve. Stackpole and Yageo print the same curve for their carbon-film parts. So lesson 01's 1/4 W really means 1/4 W at 70 °C or below.

Room air is far from 70 °C. Air inside a closed box is not: every warm part in the box heats the air around the others. Type 5 in Supply and 85 in Ambient temperature, with the 1/4 W part still selected:

• Derating: (155 − 85) / 85 = 0.8235 of the rating
• Rating now: 0.25 W · 0.8235 = 205.9 mW, shown as 206 mW
• Rating used: 250 mW / 205.9 mW = 121.4 %
• Temperature rise: still 32 °C; surface temperature 85 °C + 32 °C = 117 °C

Rating used reads 121.4 % "of 206 mW at 85 °C", and the resistor in the drawing turns the hot colour. The power and the rise did not change. The ambient temperature moved two things: the starting point of the surface temperature, and the rating itself.

The 1 W part can take more. Pick 1 W and type 9 in Supply. The power is 9² / 100 = 810 mW. The rating at 85 °C is 1 W · 0.8235 = 823.5 mW, shown as 824 mW, so Rating used reads 98.4 %. The rise is 0.81 W · 85 °C/W = 68.85 °C, shown as 69 °C, and the surface is 85 °C + 68.85 °C = 153.85 °C, shown as 154 °C.

That is close to 155 °C. Type 10 in Supply and 70 in Ambient temperature. The part takes 1 W, its full rating, and rises 85 °C to a surface of 70 + 85 = 155 °C, with Rating used at 100 %. Now type 85 in Ambient temperature: Rating used goes to 121.4 % and the surface reads 170 °C, "above the part's 155 °C limit". For the 1 W part the numbers line up: at any ambient temperature above 70 °C, its derated rating is the power that brings the surface estimate to 155 °C.

The 1/4 W part's rating leaves more room by this estimate. At its rating in 70 °C air its surface is about 70 + 32 = 102 °C (pick 1/4 W, type 5 in Supply and 70 in Ambient temperature). By the estimate it would take (155 − 70) / 128 = 664 mW to reach 155 °C. Follow the rating anyway. The surface is not the hottest point, because the film inside makes the heat. And your board is not the maker's test mounting, so your part may run hotter than the curve says.

Past the rating

Suppose a 220 Ω, 1/4 W resistor sat across 12 V, about what a car's electrical system runs at. Try it in the instrument, not on the bench. Pick 1/4 W, then type 12 in Supply, 220 in R and 25 in Ambient temperature:

• Power: P = 12² / 220 = 654.5 mW, shown as 655 mW, with I = 12 V / 220 Ω = 54.5 mA
• Rating used: 0.6545 W / 0.25 W = 261.8 %

Rating used reads 261.8 % "of 250 mW at 25 °C". Temperature rise and Surface temperature show —, with the note "past its rating: no maker data". The maker's curve stops at the rating, so the instrument gives no temperature rather than a guess.

The fix here is the 1 W part. Pick 1 W: Rating used drops to 65.5 %, the rise is 0.6545 W · 85 °C/W = 55.64 °C, shown as 56 °C, and the surface is 81 °C, "can burn skin". The part is now inside its rating, but it still runs hot. Mount it where nothing touches it.

Lesson 01's last example put 22 V across a 220 Ω, 1/4 W resistor: 22² / 220 = 2.2 W. Pick 1/4 W and type 22 in Supply. The instrument shows 2.2 W and Rating used at 880 %. Even the 1 W part is at 220 %. Neither part can take it. The answer is a larger resistance, a lower voltage, or a part rated for much more power.

Lesson 02 showed that two equal 1/4 W resistors in parallel can share 1/2 W. That still holds for the power. But two warm parts side by side heat the air around each other, so each runs hotter than it would alone. Leave space between hot parts.

Where θ comes back

The same T = Ta + P · θ gives the temperature of a regulator (module 05) and of a transistor, a part that switches or controls current (module 03). Their datasheets give θ for the part. The copper tracks on a circuit board heat up the same way when current flows through them (module 11).