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Re: AC Resistance of wires - was 8 kHz Tesla Coil
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- Subject: Re: AC Resistance of wires - was 8 kHz Tesla Coil
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- Date: Tue, 04 Oct 2005 05:25:58 -0600
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Original poster: "Gerry Reynolds" <gerryreynolds@xxxxxxxxxxxxx>
Hi Dr R,
wr is the wire radius and sd is the skin depth
----- Original Message ----- From: "Tesla list" <tesla@xxxxxxxxxx>
Sent: Monday, October 03, 2005 1:30 PM
Subject: Re: AC Resistance of wires - was 8 kHz Tesla Coil
Original poster: "D.C. Cox" <resonance@xxxxxxxxxx>
Is "wr" the DC wire resistance?
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.
Tesla list wrote:
Original poster: "Gerry Reynolds"
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.
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???