Temperature rise of a resistance

Suppose I have a resistance with a zero or very low temperature coefficient (its value stays constant with temperature variation).

At time zero the temperature is T0=20°C and a constant power P is applied (i.e. a voltage V=sqrt(P*R)). We know that at steady state the temperature rises from T0=20°C to Ts where Ts depends on many factors: mechanical charateristics of the resistance package and the capacity to dissipate electric heat with the air. Anyway at steady state an balance is reached and Ts is reached.

I'm interested in the function of T over time. I suppose it's an exponential function, but what is the time constant? And what's important for me: does this time constant depend on resistance value?

It the heat dissipation is a linear function of temperature - true of conductive heat loss, not true of convection or radiation - it will be exponential.

The time constant is given by the product of the heat capacity in joules per degree Celcius and the thermal resistance in joules per second per degree Celcius.

It won't depend on the resistance value. which doesn't have lot to do with heat capacity of resistor and the stucture on which it is mounted or the thermal resistance to ambient.

The temp curve will be kinda exponential looking, probably a bit flatter. The cooling mechanisms will be nonlinear, especially radiation.

Soldered to a PC board, you'll see multiple time constants all tangled. Heat flow is diffusive and messy.

The shape won't depend much on resistor value, given simlar power dissipation and physical structure and mounting.

An RTD is neat because it can act like a resistor but measure its own temperature.

Convection tends towards a square law - convection is faster when driven by larger temperature differences, and once it gets turbulent the boundary layer gets thinner as the convection gets faster.

Radiative emission depends on the fourth power of the absolute temperature of the radiator, which is even worse

Oh, really? What does that look like as a cooling curve?

The electrical resistance of the resistor doesn't come into it at all. Low temperature coefficient resistors tend to made as thin metal films, which are very thin indeed.

It doesn't act like a resistor. It is a resistor. RTD stands for resistive temperature detector.

It doesn't have a zero or all that low temperature coefficient of resistance.

It's a pity that John didn't bother to try to fit an exponential to any of those curves.

It will depend on the thermal capacity (specific heat) of the materials involved, and the thermal resistance of the structure to the surrounding environment (linearly dependent on surface area of the boundary). In electronics RC or RL.

There will be a thermal impedance between the power source and the environmental boundary that produces different spot temperatures, dependent on the homogeneity of the material volume.

Practically speaking, the quiescent condition is considered if your measurement doesn't change by a predetermined amount over three measurements made at predetermined intervals.

In a bang-bang controller situation, you're only concerned about deltaT between cycles.

If you know the surface area of the homogenous 'radiator' boundary, you can ballpark the surface temp rise above ambient as 1 degree C per milliwat per centimeter^2. +/-10% in free air.

Your heat source can be no cooler than that.

RL

The more interesting question (because you've specified that the temperature coefficient is very low for resistance) is what the two-terminal model suggests as a circuit perturbation. The resistor, in addition to having resistance, is a different material from the wiring, so each terminal of the resistor is a thermocouple, and the circuit will get a spurious voltage according to the terminals' temperature difference. There will also be heat DELIVERED to those terminals when DC current flows to the resistor (heat delivered to one terminal, and removed from the other terminal, by this unintended reversible heat engine).

The thermocouple effects may be very much influenced by the resistor value, because different resistor ranges might employ different materials, or different thermal coupling between the terminals.

Cooling is not linear on surface area. If you heat sink to an infinite plane, theta is not zero.

It's linear over practical ranges, on avaerage, in a defined environment.

If there's a range of conditions, you just pick the worst (and keep it out of direct sunlight, where permitted).

RL

But heat flow follows a diffusion equation, a thermal transient isn't characterized with those 'average' conditions; the problem is not an easy mathematical one to solve in a practical time-dependent case.

Automobile cooling, when the car gets parked, loses the pump; overpressure and coolant storage/return commonly happens, there's a tank to accomodate that. Turn off the electronic box, and the fan stops?

It's common to make a chassis out of 0.062" thick aluminum. And not unusual to heat sink a TO-220 transistor to that chassis.

So, what is theta of a TO-220 transistor bolted to such a chassis? Assume no insulator and an infinitely large chassis.

How about a SOT-89 soldered to an infinitely large 1 oz PCB plane?

Spreading thermal resistance soon makes a bigger sheet stop doing any good.

I'm talking boxes, not sheets.

I treat a U channel or flat plate as 2 dimensuional surfaces and use the standard surface rise equation, to get the average. There will be a delta T across the plate. At each spot point, the surface temperature will tell you how much power is being dissipated (per sq cm) from that location.

Fiddle with thickness, surface finish, color; its all the same. Force airflow, and it's a different story.

RL

The temperature rise of the heat source, your 'resistor' is determined by the thermal impedance to ambient.

As the physical size of the source reduces, it's thermal impedance to the surrounding dissipator rises. Early thermal modeling programs with point of source power elements showed ridiculous values at that one spot. Make it small enough and it'll approach the temperature of a star, without any effect on the surrounding larger area of conductively dissipative media.

RL

The top or bottom or side of a box is a sheet. If it's infinitely big, the transistor can't know if it's a box or a sheet.

So what's theta?

Sure. How many K/W? Theory is fun, until the transistor fails.

Infinite is theory, mw per cm^2 surface area is reality.

The aim is to reduce the thermal impedance to the practical physical boundary of, in most cases, the external ambient environment.

RL

So: given an infinite sheet of 0.062 thick aluminum, and a TO-220 bolted to it, what's the transistor theta?

And how big a sheet is close enough to infinite as makes no practical difference?

That is a very real situation and deserves real numbers.

Any guesses?

And a bonus question, how would theta change if we used a TO-247?

Infinity, by definition, is an unreal number.

I used to limit my calculations to hardware between matchbox and breadbox sizes - and have demonstrated them in physical hatrdware in more than a few instances when pointy heads wanted a chair and computer station occupied to 'solve the problem'.

I was surprised to see that iwas also true in a multi-moduled, multi-physical media hardware with vibrating surfaces that had basic conglomertates measuring 1.5' x 2' x4', and that also required stacking of units. Thankfully, I managed to avoid solar insolation.

That was a bitch to demonstrate, with thermocouples built-in and requiring many long-term (some hours) iterations with different power levels and stacking arangements.

After each of these past physical demonstrations, I had no further arguments, and the pointy heads went off to address other things that they didn't understand, elsewhere.

So, get real.

RL

You won't accept the concept of an infinite sheet? It's useful here.

No numbers. No guesses. OK.

You want guesses, I give you guesses. TO-220 about 1K/W (could be 0.8 could be 1.5)? TO-247 a wee bit lower, say 0.6 to 1.1 K/W ?

piglet

If the OP wants thermal impedance from his resistor to ambient, any ambient, he'd need to describe a physical interface, with dimensions.

If he wanted a thermal time constant, he'd need the volume and specific heat of the conductive media, and it's surface area contacting an ambient environment.

Infinite heatsinks don't change Rthjc of a semiconductor, just the temperature of it's mounting base.

RL