Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>
At 06:26 AM 10/1/2005, Tesla list wrote:
Original poster: "Gerry Reynolds" <gerryreynolds@xxxxxxxxxxxxx>
Hi Jim,
OK, so what is the definition of skin depth for round conductors
with no proximitry effects or is there one???
Skin depth is a "fiction" used to allow calculation of Rac using an
"equivalent" cross sectional area with uniform current density. I
suppose one could define a skin depth for a round conductor, in the
sense of it would be the "wall thickness" of an idealized tube with
a cross sectional area that would give you the right Rac. However,
you couldn't calculate that skin depth with a simple equation.
In the infinite flat plate case, it turns out that it's simple to
derive the relationship because the fall off in current density is
exponential. You then get the fairly simple relationship for skin
depth (squareroot of frequency, mu, and sigma).
In finite thickness conductors, there IS an equation, but it's non
trivial, mostly because the integration has to deal with the fields
from the "other side of the conductor". For a infinite extent thin
layer, it's reasonably straightforward, but for the round conductor
case it gets fairly tricky (as in, it requires better calculus
skills than I have to derive it from scratch).
The analysis and derivation HAS been done, but I don't happen to
have it handy. I'd be willing to bet it involves Bessel functions
(a safe bet when there's circular symmetry involved). The
computations would be moderately complex (probably involving a
series expansion).
Until recently, such things would be done by table lookups and
graphs, since they're compatible with the precision required in most
cases, and calculating the series by hand with a slide rule or a
calculator would be tedious.
Since, in real applications you'd also have other factors to worry
about (mostly, the fact that the wire probably isn't straight, nor
is it likely the wire is a long way from all other conductors) the
usefulness of a "skin effect only" equation is limited.
Today, people who have to do this sort of thing as part of a job and
need a number more accurate than 5% (which you can get from the
tables) would use one of the finite element EM modeling programs out
there. That would take care of all the factors at once.
Analytic expressions would most likely be useful either as an
academic exercise (e.g. a topic for a master's thesis) or to
validate a modeling code for "known cases".
So.. calculating skin depth (for infinite flat plate) is mostly a
preliminary step to determine whether you are safe in approximating
uniform current density over the whole conductor (sd>>diameter),
approximating as a thinwalled tube (sd<<diameter), or whether you
have a bunch of work in front of you to figure it out exactly. It
might well be that the two bounding cases are close enough in value
(e.g. <5%) that it doesn't make any difference, because you know the
true value will be in between, and there will be other things in
your system that make more difference.
Calculating skin depth is also handy when you are doing something
like estimating required wall thickness for a shielding can. You
can calculate skin depth, and then figure that if the wall is
greater than 10 times that, not much field will be getting out.
Gerry R.
Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>