A.C. resistance of non-cylindrical conductors

[cjs94]

Hi,

Can someone please explain how to calculate the A.C. resistance of non-cylindrical conductors? All the references I can find online assume round wires and I'm not sure the same equations would apply for other geometries?

Thanks,
Chris

 

[Jeff Rosenbury]

Pure resistance depends on the cross sectional area of the material (as well as the material itself of course). But for AC signals this can become complicated. They have an impedance that's based on a combination of inductance, capacitance, and resistance.

Do you just want the resistive component of the characteristic impedance? If that's the the case, you need to figure something called the skin depth. AC signals tend to stay near the surface of the conductor because magnetically induced counter-currents prevent current in the middle of the conductor. Here's a calculator of the skin depth.

The skin depth is the depth which the current is reduced 1 neper (8.65 dB in more normal units). With this knowledge in hand, use the cross sectional geometry to integrate the total conductance.

One possible problem that's listed in the link, but might be missed is that the skin depth depends on the dielectric. If part of your conductor has a high dielectric, most of the current will want to flow there (this is common in stripline, etc.). (It depends on the permeability as well, but that's usually a relative 1 for conductors.)

OTOH, if you need the characteristic impedance of a waveguide of some sort, let us know. That is a much more complicated problem.

 

[cjs94]

I need to find the resistive component due to skin depth. I know the equation for skin depth and have been given the formula which relates AC to DC resistance for round wires: $$R_{ac} = 0.25(1+\frac{D}{\delta})R_{dc}$$

I need to calculate the AC resistance for a square and rectangular cross-section conductor (this is an assignment question), but I don't know how that formula has been derived and so can't tell if it is valid for non-round conductors.

I (rather naively) assumed I could simply calculate the new cross-sectional area due to skin depth and use that to calculate the resistance with the standard equation: $$R = \frac{l}{\sigma A}$$

But that doesn't agree with the first formula, so obviously I'm wrong.

 

[Jeff Rosenbury]

Your first equation is a special case and of limited use here. Your second equation is correct as long as you realize sigma (the conductivity) is a function which changes with the depth in the material rather than the simple DC constant due to the material.

BTW, please post homework questions in the homework forum using the homework template.

There are likely other ways of solving this problem which might be in your textbook.

 

[stedwards]

There's a formula here for rectangular cross section pcb traces, where the dielectric is not considered.

 

[cjs94]

Your first equation is a special case and of limited use here. Your second equation is correct as long as you realize sigma (the conductivity) is a function which changes with the depth in the material rather than the simple DC constant due to the material.

Thanks, I hadn't appreciated that sigma was not constant for a material. I'm looking into that now, but while I can see its dependence on frequency I can't see how it is affected by the geometry. Do you have any references which show how the first equation is derived and why it is a special case?

BTW, please post homework questions in the homework forum using the homework template.

Noted, but I didn't think this question was a good fit for that forum - I'm not asking for specific help on a question but rather trying to improve my knowledge of the theory involved. Perhaps I could have phrased my questions in a better way.

There are likely other ways of solving this problem which might be in your textbook.

Sadly not. This is a course on EMC and doesn't delve into that kind of theory in great detail.

 

[meBigGuy]

Not a simple subject
"Skin-effect in the case of a conductor of an arbitrary cross section"
http://link.springer.com/article/10.1007/BF02107396