Re: 8 kHz Tesla Coil

["Tesla list" <tesla@xxxxxxxxxx>]

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)

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)

For proximitry effects for a close wound coil, divide the wire into 
quadrants like below:

...(X)(X)(X)...

If you assume that the proximitry of left and right adjacent wires 
forces the AC current out of the left and right quadrants and into 
the upper and lower quadrants, then the effective area is further cut 
in half and Rac_with_proximitry will be twice the Rac as calculated 
from above.  Note this is just a estimate but could be compared to 
the tabular data in that reference book for radio engineers that I 
can't remember the name of.

Also calculated Q values based on Rac_with_proximitry using the 
following formula:

Q = sqrt (L/C)  /  R

could be compared to measured Q to determine the accuracy of the 
estimate. Preferably, this should be done with coils having different 
fo's and wire diameters.  I will measure the Q of my coil when I get a chance.

Gerry R.

(see comments below)

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

> I've searched high and low for the reference to the equation, but I 
> can't find it. It is something I picked up from Googling and didn't 
> save the document (dummy me). You are correct, Rdc added to Rac 
> makes no sense. I'm sure I wrote it down as it was shown. I suspect 
> this may have come out of a text book, but I think it's use was not 
> interpreted as intended. I'm certain this was a skin effect approximation.
>
> I've found another approximation calculator (excel throw-together) 
> for Rac and skin depth in cylindrical conductors. It gave 37 ohms on 
> your coil. If you remember, I showed 98 after adding the 61 Rdc (per 
> the equation), which is a difference of 37 ohms! It appears the 
> equation was using Rdc as part of it's approximation, but Rdc should 
> have been removed to identify Rac, such as:
>
> Rac = Rdc(1+(r/(2sD))-Rdc

this equation reduces to Rac/Rdc = r/2sD  and I dont believe it 
unless Rac is defined as the incremental increase of resistance above 
Rdc.  Also the equation doesn't seem to take conduction areas into account.


> where:
>
> Rdc = DC resistance of winding
> r = radius of wire diameter in inches
> sD = skin depth in inches
>
> For reference, another Rac approximation I found from Michael Mirmak 
> of Intel Corp (was for pcb traces originally using trace heights and 
> widths) I modified for a round conductor: It follows the excel 
> calculator for high and low frequency's.
>
> Rac = L(((3.318*10^-7)*F^0.5)/(4*d))
> (should be ready for excel - just insert the values)
>
> where:
> L = Length of winding in inches
> F = Frequency in Hz
> d = wire diameter in inches

I think in all these equations, it is important to understand how 
they define Rac as it seems there isn't always consistency.



> BTW, here's the excel file mentioned:
> http://home.swipnet.se/2ingandlin/Skin_depth_calc.xls
>
> Of course, we don't know how well any of these approximations work 
> for our coils. It would be interesting to measure and find the 
> proximity losses. Empirically, we could likely come up with 
> something similar, which of course gets better with time. So far, no 
> one is doing this. When I get my coils set back up, I'll have to 
> perform some Q measurements.