Re: AC Resistance of wires - was 8 kHz Tesla Coil

["Tesla list" <tesla@xxxxxxxxxx>]

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

Hi Gerry,

Yes, Gary has written an excellent paper and did well to check out 
Termans work as well as Fraga, Predos, Chen, etc.. as to how they 
compare with low and high h/d coils at low and high frequency's. 
Actually, Fraga has a closed loop equation which includes both 
proximity and skin losses which would be ideal (page 6-19). It 
appears good for closewound coils with an h/d of >> 4. Because most 
coils are in that range (not all), I think it's worth a look.

Otherwise, yes, the table can be put to good use, or even better, 
Antonio's calculation is easily implemented:
Rac/Rdc = (wr/sd)2/(2wr/sd-1)
A 5% swing is good enough for the task.

BTW, T is temp (page 6-4, eq.7)

Gary makes some interesting observations throughout his work on these losses.
Here are a few which stood out to me:

"It appears that as coils get shorter and fatter, the interior 
current in the coil gets larger
and the effective resistance increases as compared with the 
predictions of Medhurst and Fraga." (page 6-28).

"This table (3) indicates that the proximity effect can easily double 
or triple the measured
input resistance over that predicted by Rac for a straight wire of 
the same length." (page 6-11).

"Figs. 3 and 4 also show another effect, a very interesting concept 
that is otherwise difficult
to explain. This concept is that there is little penalty in 
performance if one uses a smaller wire
in a coil. That is, the effect on spark length is not as strongly 
related to the wire resistance
as one would expect." (page 6-25).

Just some eye catching paragraphs.

Take care,
Bart

Tesla list wrote:
> Original poster: "Gerry  Reynolds" 
> <mailto:gerryreynolds@xxxxxxxxxxxxx><gerryreynolds@xxxxxxxxxxxxx>
>
> Hi Jim and Bart,
>
> The work that Dr Gary Johnson did for AC resistance seems to solve 
> the Rac/Rdc problem for round wires (no proximitry effects).  The 
> differential equation for the current density J(r) is:
>
> d^2 J/dr^2  +dJ/rdr +T^2 J =0  (not sure what T is)
>
> The solution is a Bessel function of the first kind zero order and 
> the solution does involve an infinite series. The current density is 
> complex and has real and imaginary parts that vary with radius from 
> the wire center.  He carves up the wire into cylindrical shells and 
> computes the average current density, cross sectional area, and 
> current for each shell (still a complex number).  He then computes 
> the power in each shell by multiplying the current by its complex 
> conjugate to get the real portion of I^2 for each shell.  From this, 
> the power in each shell is known.  He then sums up the shell powers 
> to get total power and divides by Rdc*|I|^2.  Now for the good 
> part.  He has created a table of Rac/Rdc for various ratios of 
> wire_radius(wr)/flat_plane_skin_depth(sd).  The following table 
> shows this for wr/sd up to 8.
>
> wr/sd      Rac/Rdc
> ------------------
>   1            1.020
>   2            1.263
>   3            1.763
>   4            2.261
>   5            2.743
>   6            3.221
>   7            3.693
>   8            4.154
>
> Bart,  what I'm thinking is since you compute the sd and know the 
> wr, you can just interpolate into the table and use the Rdc to 
> compute the Rac.
>
> Jim, how does Gary's table compare to the RDRE table???
>
> Gerry R.