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



Original poster: "D.C. Cox" <resonance@xxxxxxxxxx>




Is "wr" the DC wire resistance?

Dr. Resonance



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.