skin depth, 60 Hz, tubular conductor

[WhyDoYouAsk]

I'm trying to calculate the AC resistance at 60 Hz of a copper pipe with 3/4" I.D., 1/16" wall thickness, and 7/8" O.D., not near any other conductors.
Can anyone step me through this?
It must be several times the DC resistance but I'd like to know this AC value to at least two significant figures, if possible.
Thanks.

 

[roydm]

I dont think there is any diference at 60 Hz. If you are thinking of the skin effect doesn't that only apply to RF?
You don't say if the pipe is in a coil configuration so I assume it's straight.
Roy

 

[rbulsara]

Refer to this. It has a useful table.

http://www.acasolutions.com/resource_center/online_catalogs/pdfs/Substation-BusConductors.pdf

 

[bacon4life]

For a thin walled pipe, the 60 Hz skin effect will be very small. See

http://www.copperinfo.co.uk/busbars/pub22-copper-for-busbars/sec4.htm

for a graph.

Roydm, for large diameter conductors, the 60 Hz skin effect can be very significant. Thus the preference for hollow shapes for large bus conductors. At least one company also makes an almost hollow overhead conductor.

 

[dpc]

The Standard Handbook for Electrical Engineers (Fink and Beatty)has some additional information on skin effect in tubular conductors.

The ac resistance at 60 Hz will be somewhat higher than dc, but not anywhere near twice the dc resistance.

 

[waross]

My calculations show your pipe cross sectional area as between AWG 2/0 and AWG 3/0. Skin effect is usually slight with those sizes at 60 Hz. By going to a tubular configuration you have eliminated most of what started as a very small effect.
I suspect that errors in the figure used for the conductivity of the pipe will be greater than skin effect errors.

Bill
--------------------
"Why not the best?"
Jimmy Carter

 

[ScottyUK]

This might be a worthwhile read:

http://www.copperinfo.co.uk/busbars/pub22-copper-for-busbars/sec4.htm

The whole CDA document is an excellent reference.


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If we learn from our mistakes I'm getting a great education!

 

[WhyDoYouAsk]

Since a skin depth in copper is about 9 mm at 60Hz, I expected a much higher ratio of Rac/Rdc for the pipe in question.
The measurements and graphs in this thread show otherwise.
Thanks, all.

 

[davidbeach]

1/16" wall thickness is 1.587mm, why would you expect a significant difference in ac vs. dc resistance?

 

[WhyDoYouAsk]

I thought because most of the current @ 60Hz with this config. is trying to flow in empty space. To get the full benefit of the conductor size it should be a round conductor or greater than 18mm. . .?

 

[waross]

Skin effect doesn't quite work that way. There is a current gradient from the center of the conductor to the periphery. The closer you get to the surface, the higher the current density. In a hollow conductor, the area of lower current densities is eliminated.

Bill
--------------------
"Why not the best?"
Jimmy Carter

 

[marks1080]

Regarding your comment Bill:

The current densities for the hollow part of the conductor are obviously eliminated, but what happens to the actual current? Does the current that would have flowed in the hollow part just flow in the rest of the conductor? Does the current not flow at all (therefore raising the resistence of the overall conductor)? Or is it a combination of the two?

Thanks!
Mark

 

[dpc]

I think we're complicating this. Any conductor has a dc resistance (at a specific temperature). The ac resistance at a particular frequency will be higher than the dc resistance because the magnetic field in the conductor causes the current to be more concentrated at the extremes of the conductor with less current toward the center, so the current density is non-uniform resulting in higher net resistance.

This is true regardless of the shape of the conductor.

 

[WhyDoYouAsk]

The ACA solutions link has a lot of info in it.

With the help of a spreadsheet I have found that the (Rac/Rdc)/(Rindreact/Rdc) ratio rises more rapidly with wall thickness than a straight line relationship.

This is opposite to what you'd expect with a skin effect. That is, the thicker the conductor wall, the less resistance due to skin effect there should be.

I have to sleep on this one.

 

[davidbeach]

I'm not sure you understand the concept of skin effect. Skin effect is the current moving toward the skin of the conductor with increasing frequency. The thicker the conductor, the more resistance it would have compared to the dc resistance of the same conductor; skin effect increases resistance rather than reduces resistance.

 

[electricpete]

As David said, the resistance doesn't increase with thickening wall, but the ratio Rac/Rdc does.

You can solve skin effect resistance analytically using Bessel functions for simple solid conductors... I don't know of any closed form solutions for hollow conductors.

fwiw, here is a finite element solution using free F.E.M.M. software. Assuming copper conductivity of 58 MS/m, it predicts that the 60hz resistance of this geometry is 5.11282e-5 ohm/ft (slide 2) while the dc resistance is 5.11232e-5 ohm/ft (slide 3). I didn't go through the links, but I'd be interested if this agrees with what their tables show.

As you can see, there is not much difference but the 60hz is a tad higher. If it were a thicker wall, the crowding against the outer wall would be more noticeable and the difference in resistance would be more noticeable.

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[electricpete]

Some assumptions should be pointed out. It was assumed there was no effect on current distribution from adjacent conductors or ground plane. It was assumed solid copper conductor at 20C. Based on the assumed symmetry , the boundary condition can be used that vector magnetic potential A is constant at any circle concentric with the tube which has larger OD than the tube OD. I put my boundary condition at OD = 2". The output numbers for flux linkage, imaginary component of voltage (current angle is defined 0 by assumption) and impedance are not correct since we have not modeled flux out to inifinity. But the numbers for real component of voltage and real component of impedance and watts are correct to the best of my knowledge (within the assumptions). I verified this by re-running the scenario with a large simulation domain OD of 10". The results remained the same for real component of voltage and impedance and for watts.

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