Re: skin depth in round conductors Re: 8 kHz Tesla Coil

["Tesla list" <tesla@xxxxxxxxxx>]

Original poster: "Barton B. Anderson" <bartb@xxxxxxxxxxxxxxxx>

Hi Jim, All,

In every reference I've been reading regarding skin depth, I can find 
nothing stating round conductors and sheet conductors have a 
difference in depth penetration due to frequency, and it just doesn't 
make sense that they would (at least, I'm not getting it). The only 
difference I can find is that for round conductors, the math gets 
messy to define exactly when the abrupt change occurs and tails off 
toward zero.

Skin depth is defined as the distance from the surface of a conductor 
where the current density is 1/e times the surface current density. 
This is nothing more than a density ratio used to describe the 
effective conducting area. Once the conductor is small enough that 
the conductor no longer has a ratio less than the wire diameter, 
there can be no losses associated with skin depth. This does not mean 
there are no losses associated with AC currents. But, to apply those 
losses to skin depth is no longer appropriate. Proximity losses, absolutely!

Skin depth occurs because a changing flux induces a voltage loop or 
eddy current which is coincident with the voltage. This eddy 
reinforces the main current at the surface and opposes the current in 
the center of the conductor. The result is that as frequency rises, 
current density increases at the surface and tails off exponentially 
toward zero at the center because of these frequency dependent eddy currents.

It should be noted that the current is not uniform around the wire. 
The current density will occur adjacent to magnetic fields. In an 
iron core transformer, the current flow through a conductor is drawn 
to the high permeability core side of the winding. In an air core 
transformer, it's toward adjacent wires. In a helical single layer 
coil, toward the 2 adjacent wires. In transformer design, the 
designer decrease the layers to minimize losses which is why novel 
winding techniques are thought of because of the huge losses with 
adjacent currents (high frequency or low).

For an air core transformer at a high frequency where depth 
penetration crowds toward the surface, the self inductance of the 
wire must also change due to the changing flux and portion of the 
conductor which is effectively conducting current, and which is also 
crowded toward adjacent wires. Thus, as frequency increases, the coil 
inductance must also move away from the DC inductance. We certainly 
know this to be true in our measurements.

But at low frequency, where sD is no longer valid, Rac must be due to 
proximity losses in the conductors and to dielectric heating. Out of 
all the study which has been done on RF from pcb traces operating in 
the GHz to antenna theory, I just can't bite on why the conductor 
must be large in size. In everything I'm reading, it is actually 
typical to reduce losses by reducing the wire size until the skin 
depth is no longer affecting the conductor and then to add multiple 
conductors for the "same" current flow to minimize proximity losses 
and power losses. I think this is where the 5 x recommendation is 
appropriate, I do not believe it was meant to show the conductor 
itself should be 5 x the penetration depth. That should actually 
create a problem because it causes the sD induced eddy's I discussed earlier.

Sorry for the long post, but it's one of the area's I think I 
understand, then I don't, then I do, then I don't, ... Are there any 
references anyone can point me to for the 5 x sd recommendation for a 
"single" conductor wire size or similar recommendation?

Take care,
Bart



Tesla list wrote:

> Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>
>
> At 07:43 AM 9/22/2005, Tesla list wrote:
>
> > Original poster: "Gerry  Reynolds" <gerryreynolds@xxxxxxxxxxxxx>
> >
> > Hi Jim,
> >
> > Lets say that the skin depth is 10 mils as calculated from the flat 
> > plane formula.  Are you saying the skin depth in a round conductor 
> > (at the same frequency) is smaller than the 10 mils???
>
>
> Yes, that's exactly what it is.. (bearing in mind that skin depth is 
> a mathematical fiction to use to calculate Rac from geometry and the 
> resitivity of the material)
>
>
> > If so, by how much approximately??
>
>
> Depends on the diameter of the conductor.  There's a table in the 
> "Reference data for radio engineers" book, as well as others.
>
> >  Also, does the current fall exponentially???
>
>
> Yes, in a flat plate.