Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>
At 07:28 AM 9/30/2005, Tesla list wrote:
Original poster: "Gerry Reynolds" <gerryreynolds@xxxxxxxxxxxxx>
Hi Bart,
I'm not sure what is being said about it giving 37 ohms for my
coil, but if Rac is defined as the effective resistance when runing
AC current thru it, the AC resistance can never be less than the DC
resistance. Skin depth is defined (for the benefit of others) as
the depth of the conductor from which if you throw away the
interior conductor and keep the exterior conductor and calculate
the DC resistance from the resulting area, you will have the Rac of
the wire for the frequency of the AC current (no proximity effects
included yet). In other words:
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. 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)
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
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
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%)
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%)
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.