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Re: 8 kHz Tesla Coil

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


Hi Jim,

I just reread your posting and I'm thinking you are using the skin depth of a flat plane conductor in the equations I quoted. I think those equations are using a skin depth for a round wire instead of a flat plane in which the equation for skin depth you quoted would not apply. I dont know if such an equation exist in reality but the site I'm quoting from gives a SD for round conductors that seems to give a little larger value than what Bart's JAVATC program calculates. Several people in this group said that the SD of a round conductor was different than for a flat plane.

Maybe the real question is: if there was a round conductor skin depth equation, would "what I quoted" be true. If you calculate the resistance of the flat plane conductor that is only one skin depth thick using the flat plane skin depth value, you get the AC resistance. So, is it possible to do the same with a round conductor using a round conductor skin depth??? Some web sites seems to think so. See comments below.

Gerry R.


Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>

Rac/Rdc = wire_cross_sectional_area / (wire_cross_sectional_area - area_internal_to_the_skin_depth)

NO... This is NOT true. It will give you the wrong answer, except in certain limiting cases (i.e. diameter >> skin depth)

It's a nice conceptual model, but mathematically incorrect. It's even "approximately" right, but it's incorrect to cast it as an equation, because for situations of great interest to coilers, it's wrong.


So, for skin depth (sd) greater or equal to the radius (r) of the wire, Rac = Rdc. 

True???


For sd less than r:

Rac/Rdc = pi*r^2 / (pi*r^2 - pi*[r-sd]^2) which reduces to:

Rac/Rdc = r^2 / (r^2 - [r-sd]^2)

Wrong... only true for r >> sd (e.g. R > 5-10 times sd)

The equation doesn't use the flat plane sd here. It uses a round conductor skin depth.



Skin depth (as a number) only applies to an infinite flat plate, and is defined as sqrt(lambda/(pi*sigma*mu*c))

Or, more familiarily = sqrt(1/(f*pi*sigma*mu))
mu = mu0 * murelative
murelative = 1 for nonmagnetic materials
mu0 = 4e-7 *pi
sigma is the conductivity


Certainly, the equation you quote is for a flat plane conductor.


Here's an example:
Assume Copper (sigma = 5.8E7 S/m), and 100 kHz...
sd is about 0.21 mm

r = .5 mm  (i.e. a 1mm diameter wire)

By your equation {Rac/Rdc = r^2 / (r^2 - [r-sd]^2)} one gets Rac/Rdc = 1.51

However, if one uses the tables in something like Reference Data for Radio Engineers, one gets a correction factor Rac/Rdc = 1.2 or 1.3

Using the sd for a flat plane. But what about the sd for a round conductor??? If the sd for a round conductor was indeed larger, the ratio would be smaller as suggested by the RDRE


it's true that for secondary wire that's say, AWG 24 (20 mil diam = about .25 mm radius), you're getting close to where the skin depth is comparable to the radius, so Rac is approximately equal to Rdc (within 5%)

Good, so for a wire that is 2SD's in diameter, one can use the Rdc to calculate the Q (within 5%)


It's also true that for large primary conductors, like 1/4" OD tubing (i.e. r = 3mm), the model of a thin tube of diameter D and thickness sd is reasonably accurate. (at least, the error is <5%)

This fits my understanding since a large diameter tube (relative to SD) would approach the physics of a flat plane. This might suggest that SD(round_conductor)/SD(flat_plane) could be a function of r/sd where the sd ratio would approach 1 as r/sd becomes large.


However, those 1-2 mm diameters (approx AWG18 - AWG12) where both approximations are most inaccurate are also those sizes likely to be contemplated by high power coil builders who are most concerned about efficiency.

Not sure what the final answer is other than using the RDRE tables (assuming these are based on measurements and should be considered accurate). Right now I think the only info provided by any of the TC programs is Rdc. Maybe an estimate would be better than nothing. Maybe the RDRE table info could be incorporated into the programs if an equation can not be found.