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December 1998

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Tue, 15 Dec 1998 21:16:34 -0800
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> Hi,
> I have a little physics problem maybe someone can help me with. I'm
> looking for an equation to solve for the temperature rise in a trace
> with a given amperage over a given period of time. Here's the problem:
> A copper (inner layer) trace will have 6 amps on it (under short
> circuit conditions) for a maximum of 128 microseconds. For a given
> trace width and length, I can calculate the DC resistance. What I
> can't calculate is the temperature rise on that trace over that
> time. Obviously, I need to know this to ensure I don't approach
> the fusing temperature of the copper.
> I'll turn this into a science project if I have to, but I'm
> wondering if anyone has this equation at their fingertips.
> Thanks in advance if you do.

A while back, there was quite a discussion about fusing
currents in traces here on TechNet. A gentleman by the
name of Ralph Hersey who was a major contributor to this
group who has since retired (but I'm sure is lurking around)
had a lot of information on just this type of subject.

Ralph, are you there?

As far as rise in temp, I've got just some random thoughts.
First I don't think you'll get a very accurate equation
if derived.  Too emprical. Second, it's got to be related
to the specific emissitivity of Cu.  Surpass it and there
will be enough power to change by a single phase to liquid
(melt the Cu) or double phase into a gas (explode).
I'm way over simplifying it all.

Soooo, I'd start by working backwards from ye old
Stefen-Boltzmann law ...   < rumaging thru some old books... >
See, Ralph could do this stuff off the top of his head.

The rate at which an object of surface area A and absolute temp
T emits radiation (watts P) with e = emissivity (0 for perfect
reflector, 1 for black body) and sigma = 5.67*10^-8 W/m^2 is

 R = P/A = e*sigma*T^4

Radiating power is related to the fourth power of temp.
So, temp must be related to the fourth root of power and
inversely to area.

So

    T = [ P/(A*e*sigma)]^0.25

or very roughly  (e = 1, black body assumption)

    T = [ P/(A*5.67*10^-8) ]^0.25

or e = 1/2 (halfway between perfect reflector and black body)

   T = [ 2*P/(A*5.67*10^-8) ]^0.25

or e = 1/3 ...  etc ...  A spreadsheet would do real fine here.

Then, test it against some not so destructive steady state
currents to get some data empirically.

Remember, there's units of Watts per square meter in there.
Adjust accordingly.  Mileage may very ... by A LOT.

Then, depending upon how it all goes, write a paper on it
and submit to the good people at the IPC for presentation.
I think it would really help all of us.  And give you that
15 minutes of infamy.

Regards,  Doug

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