Re: skin depth

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Malcolm Watts" <m.j.watts-at-massey.ac.nz> 

Hi Pete,

On 23 Sep 2003, at 7:56, Tesla list wrote:

 > Original poster: Peter Lawrence <Peter.Lawrence-at-Sun.COM>
 >
 >
 > I still find it very odd that apparently no one has done skin depth calcs
 > for cylindrical wire, only for hypothetical infinitely wide infinitely deep
 > flat plane conductors...
 >
 > Sounds like a good problem for a math-physics type, or even a
 > physical-simulation type.
 >
 > -Pete Lawrence.

The approximation I use for skin depth in copper wire is 66/SQRT(f)
mm.  and was arrived at after my studying about half a dozen texts
while doing research for a TC article. Resistivity is factored in and
is dependent on whether the copper is hard-drawn or annealed,
impurities etc. The factor in that formula is for annealed and it is
still only approximate.

      A post from someone else tried to ascribe attributes like "good"
to skin depth. My take on it is that skin effect just "is". To
understand the way it works, imagine three wires of widely differing
diameters, the first thin as, the second with 6x the diameter of the
first, the third considerably larger still. Suppose the first wire
diameter is comparable to 1 skin depth for that material at the
designated frequency of operation. Because the wire is so thin, skin
effect is almost non-existent and current flow is pretty much evenly
distributed across the cross-sectional area of the wire. Is that a
good thing? Not really - the resistance will be almost identical to
the DC resistance and it will be high because the cross-sectional
area is small.

      Take the second wire. Suppose it has a diameter = 6 skin depths.
1/e of the total current flowing will be flowing in a "skin" to a
depth of 1 skin depth (across the outer third of the diameter). The
current density at the centre of the wire is pretty low but not zero.
Skin effect is definitely at work. Bad? Not really - the cross-
sectional area of the wire in which current flows is considerably
greater than that for wire #1 and hence resistance is much lower for
a given length.

      Assume wire #3 has a diameter = 20 skin depths at the frequency
of interest. You will get almost no current flow at a distance of 5
or so skin depths from the surface and much beyond this, current will
actually be flowing in the opposite direction to the skin current.
Good, bad or ugly? Probably ugly in my book. Firstly, you are paying
for (expensive) conductor material which isn't actually being used
and simply adding to the weight. Secondly, the backward current flow
implies that the resistance at the frequency of interest might
actually be higher than a wire sized such that current flow is all
but zero at its centre. In a situation like this, pipe is a better
option by far. How thick should the pipe walls be? I'd personally go
for about 3 skin depths and if I wanted to decrease resistance still
further, go for either a larger diameter pipe while maintaining the
same wall thickness or if inductance was a consideration (normal :)
I'd probably flatten the pipe or go to ribbon to maintain the same
turn spacing.

       Unfortunately, it doesn't even end there. Conductors in  close
proximity influence the current densities in each other (proximity
effect) and again a degree of material wastage and resistance
increased is implied.

       One of the points about all this is that it is possible to a
reasonable degree to have some idea about how a coil design will
behave in real life while the design is still on the drawing board.
Another point is that given the behaviour of current flow in wires of
widely differing diameters, it is not surprising that most TC
resonators fall into a Q range of 150 - 300 or so.

Malcolm