Re: 8 kHz Tesla Coil

["Tesla list" <tesla@xxxxxxxxxx>]

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>