This is a theoretical question about the maximum current carrying capacity of a copper conductor.
I discussed this question with a collegue but we couldn't come to a conclusion. Background was that researching Power Mosfets I've seen several Mosfets in a TO-247 package that have several 100A continuous (!) drain current ratings and pulse current ratings of even more than 1000A.

Now the question is not about the FET characteristics and what's the point in having for example a drain current limit rating (silicon limited) of 350A if the drain current limit rating (wire bond limited) is only 195A like in the datasheet I attached.

The question is if there is a limit in current density, means in the amount of electrons that can pass through a certain area of a conductor. I understand that the practical limit is given by the temperature rise that the current causes which will eventually melt the conductor ( like in a fuse). The question is theoretical: If I had the possibility to maintain the temperature under the melting point of the conductor, is there any other parameter that limits maximum current density? Or could it be infinite?

 

It's an excellent question and I personally don't know the answer (gotta get that out of the way at the beginning). But I'll make a guess. I'm not going to bother dragging old textbooks out, so I'm winging it.

I'd say there's an upper limit in real conductors because you wouldn't be able to remove the internally-generated Joule heating fast enough at some point. The mechanism of heat removal would be a colder solid or fluid on the exterior of the conductor (using conduction and/or convection); thus, heat generated internal to the conductor would have to flow to the surface of the conductor and this will be limited by the phonon velocities. Even if that weren't true, there's a practical limit to the amount of heat that can be removed through any of the transport forms, so there still will be a point where the conductor melts.

I know that doesn't answer your question because you said to assume all the heat generated could be removed. AFAIK, there's no other limiting phenomenon (e.g., you were probably thinking of the analogous critical magnetic field in a superconductor caused by the current the conductor is carrying).

There's only a finite number of charge carriers in a real conductor and there might be some upper physical limit to the allowed drift velocity; if so, this would also imply a limit.

I'd suggest you'd want to get in touch with folks who do work at high current densities; one such place is there are folks at national labs who do work on exploding wires (well, they were doing it 35 years ago when I talked to them).

 

Exploding wires I heard about that to generate an EMP, interesting subject too...

Good point your explanation of internal heating, but I was assuming that heating was not an issue.

I was thinking there might be a limit because of the quantity of free electrons that "fit" into a certain cross section (their speed is limited too). I don't know if such a limit exists, but if it does then it would be extremely high, because lightning rods for example can handle several hundreds of kA and they obviously don't melt.

 

Well, I still think it's an interesting question and I don't know the answer. It would be hard to experimentally measure it on common conductor sizes because the currents needed are rather large -- and there's no way to get around the Joule heating.

Just last week I was wondering if and how the resistance of a copper conductor changes as a function of current through it. Thanks to Ohm, we all know its current to voltage ratio is constant for small currents; however, what happens to the material when it is significantly heated by the power dissipated? Can there be things like grain growth or fracturing when the self-heating is large so that the conductor's behavior back at room temperature is changed permanently?

I made some quicky measurements by taking apart a piece of 16 gauge zip cord to get the individual conductors. As I recall, they were about 10 mils in diameter, meaning they were 30 AWG wire. I clipped some alligator clips to a couple cm of this 30 ga. wire for a current connection, then measured the voltage drop across the wire with two IC clips clipped to the wire. That tiny wire was able to handle 9 A of current through it without failing! That was the limit of the current my power supply could supply, so I wasn't able to answer the question. And, to add insult to injury, I can't remember the name or location of the file where I typed in the results, so I can't show the plotted results. As I recall, the current vs. voltage curve is a power function with the exponent less than 1; very similar to what you get if you measure the i-V curve of a light bulb.

BTW, that 9 A meant that the current density was 177 A/mm2; typical NEC current density values are in the range of 4 to 6 A/mm2. I did release a bit of magic smoke because I could smell it and the surface of the copper wire turned black, so there was definitely some heating going on.

 

Given that power plants operate in the Megawatt range I'm going to guess and say that we aren't really that close to a practical limit. I remember my instructor explaining how a connector or wire literally blew up because it had developed a couple of milliohms of resistance in a high current area, but that was wattage/heat issue.

 

The "Standard Handbook for Electrical Engineers" [Edited Fink & Carrol] devotes a section to Fusible Metals & Alloys providing some formulas and graphical data including a nomograph of fusing time vs current for various copper wire gauges. Fuse design & properties have been the subject of active research for many years. I recollect a research group doing such work at Sydney University in the 1980's (?) where high speed photography was used to study fuse behavior. Work by H W Baxter in the last century in this area is considered seminal.

 

The "Standard Handbook for Electrical Engineers" [Edited Fink & Carrol] devotes a section to Fusible Metals & Alloys providing some formulas and graphical data including a nomograph of fusing time vs current for various copper wire gauges.

Interesting t_n_k -- but I don't have a copy of that book and the library is miles away. If you feel it's possible, would you mind posting a fair use image of the nomograph? I'd be interested in filing the information away, as that's something that occasionally crops up in my twisted thinking...

 

Interesting t_n_k -- but I don't have a copy of that book and the library is miles away. If you feel it's possible, would you mind posting a fair use image of the nomograph? I'd be interested in filing the information away, as that's something that occasionally crops up in my twisted thinking...

see the attached extract from the book. You can also find the whole book on the internet.

I think I'm gonna ask the same question on a physics forum, and will post the answer here, if there is any.

 

Nomograph attached

 

Thanks for the nomograph, t_n_k. My curiosity just got the better of me and I took a strand of wire out from an 8 inch piece of 16 gauge zip cord. A micrometer shows the wire diameter to be 5.7 mils; the closest AWG is 36, not 30 like my poor memory remembered above. The nomograph doesn't appear to be terribly accurate; I extrapolated with a pencil to get the fusing time for 3 A of current through this chunk of wire and it comes out to about 1.5 s. However, I ran that current though the wire for more than a minute and nothing happened other than a bit of discoloring. While my extrapolation might be beyond what the author recommends for the nomograph, I'd certainly use it for just order of magnitude estimates unless I could measure a few data points on it. Since I don't work in industry anymore, I don't have access to the hefty power supplies needed for such work...

 

If you apply Onderdonk's formula

\(I=A \sqrt{\frac{log(\frac{T_m-T_a}{234+T_a}+1)}{33S}}\)

then your estimate seems correct.

I think CSA 'A' for 36AWG is 25 circular mils (the anticipated unit for area in the formula).

Tm for copper is 1083 C. With Ta=20 C and S (melt time) =1.5 sec then a current of 3A should about do it.

Perhaps it has a lot to do with how effectively or not the heat is conducted away from the wire. Presumably these relationships were derived for carefully controlled conditions.

 

Well, one ugly fact trumps a beautiful formula... My carefully controlled conditions were a chunk of wire 195 mm long held in a semicircle in the air by two IC clips and supplied by a constant current power supply. I think I'll try again with a different supply and use a scope to monitor the voltage across the wire to see if I can detect the fusing point. I've got too many scars from too many decades to blindly trust a formula like this without one or two corroborating data points...

 

All power to informed skepticism. Others must of necessity be more gullible.

Obviously some folk ascribe credence to the work done by researchers such as Preece & Onderdonk.

http://www.cirris.com/testing/resistance/fuse.html

Just Googling "Onderdonks formula fusing" gave many such examples.

As I said, the test conditions are probably key to the the use of such formulae & nomographs etc. Were the tests done under adiabatic conditions for instance? One would have to go back to the original research reports & papers.

You are correct to be skeptical. As you note, there are probably many instances where equations are pressed into service with little regard to the underlying limitations or assumptions.

 

One would have to go back to the original research reports & papers.

Agreed. The nomograph doesn't appear useful unless one has the supporting usage assumptions. Was there any information on its use in the reference you got it from? If not, anti-kudos to the author(s)/editors at McGraw-Hill.

Actually, I wasn't being overly skeptical -- I could just tell that the nomograph was likely based on empirical data and I wanted to verify one point for myself. Alas, a short (10 minute) googling for the equation didn't turn up any useful references to the original work. Maybe someone with access to a university library could be helpful and dig up the original paper.

 

A commonly used relationship equivalent to Onderdonk's formula is the Insulated Cable Engineers Association [ICEA] version - as shown in the attached png file.

My understanding is that this formula applies to insulated cables and therefore assumes an adiabatic model estimate of the conductor fusing current.

 

This paper specifically mentions the formula of my previous post.

 

A practical answer would be: Amperes/ Circular Mil./ Unit length.
Conditions: STP,(standard temperature and pressure) or other.
Environment: Still air or forced air cooling ( Volume /Time) or other.
Material: Cu., Au., Ag or other.

Cheers, DPW [ Everything has limitations...and I hate limitations.]

 

Do any of the formulas take skin effect into account? While it may not be much, I'm sure there is some.