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Re: 8 kHz Tesla Coil



Original poster: "Gerry  Reynolds" <gerryreynolds@xxxxxxxxxxxxx>

Hi Jim,

I went thru all my text books and did find a section on skin depth for round conductors. Funny that you mentioned it, it does involve Bessel functions. I will study up on it and report what I can learn from it. I'm hoping that we can find an easy way to come close to finding the Rac without proximitry effects first then we can tackle those effects later. Another approach might be to have a RDRE table for each wire guage that would give the Rac/Rdc vs frequency. This should be easy to incorporate into a program and easy to interpolate between frequency points.

Gerry R.

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>