AC Resistance of wires - was 8 kHz Tesla Coil

["Tesla list" <tesla@xxxxxxxxxx>]

Original poster: "Gerry  Reynolds" <gerryreynolds@xxxxxxxxxxxxx>

Hi Jim,

I'm going thru the document written by Dr Gary Johnson where he goes 
into the Bessel functions and developes a table of Rac/Rdc.  He also 
goes into proximitry effects.  I skimmed thru that part and it looks 
like adjustments are made on Rac.  So first step is to get Rac with 
no proximitry effects. He also presents an approximation 
table/equation that appears to be within 1% of the Bessel function 
solutions.  This might be a better approach since his Bessel based 
table only goes up to a r/sd of 8 and I dont think we want to compute 
the bessel function series for larger relative wire sizes.  It looks 
like the functions have been normalized and we dont need to "have a 
table for each awg".

The approximations presented were developed by Fredrick Terman and 
published in Radio Engineers Handbook, McGraw-Hill 1943.  It uses an 
index value of X that is defined as:

(general equation - in meters)

X = pi * d sqrt(2f/(rho*10^7))

where d is the conductor diameter in meters
          f is the frequency in Hz
         rho is the resistivity in ohm meters (for copper rho = 
1.724x10^-8 ohm meters)

(for Copper using mils for wire diameter)

X = 0.271 * dm sqrt(fMHz)

where dm is wire diameter in mils
         fMHz is frequency in MHz

for X between 0 and 3.0 he uses a table

X   Rac/Rdc
-------------
0.0    1.0000
0.5    1.0003
0.6    1.0007
0.7    1.0012
0.8    1.0021
0.9    1.0034
1.0    1.005
1.1    1.008
1.2    1.011
1.3    1.015
1.4    1.020
1.5    1.026
1.6    1.033
1.7    1.042
1.8    1.052
1.9    1.064
2.0    1.078
2.2    1.111
2.4    1.152
2.6    1.201
2.8    1.256
3.0    1.318

for X > 3.0,  Rac/Rdc becomes a linear function of X:

Rac/Rdc = 0.3535X + 0.264

This approach should be easily incorporated into a program like 
JAVATC. I'll try to report on proximitry effects sometime this week.

Gerry R




> Original poster: Jim Lux <jimlux@xxxxxxxxxxxxx>
>
> At 09:28 PM 10/1/2005, you wrote:
> > Original poster: "Gerry  Reynolds" <gerryreynolds@xxxxxxxxxxxxx>
> >
> > Hi Jim,
> >
> > I went thru all my text books and did find a section on skin depth 
> > for round conductors.  Funny that you mentioned it, it does involve 
> > Bessel functions. I will study up on it and report what I can learn 
> > from it.  I'm hoping that we can find an easy way to come close to 
> > finding the Rac without proximitry effects first then we can tackle 
> > those effects later. Another approach might be to have a RDRE table 
> > for each wire guage that would give the Rac/Rdc vs frequency.  This 
> > should be easy to incorporate into a program and easy to 
> > interpolate between frequency points.
>
> That would work..
> BUT, I think there are enough other factors (proximity effect) that 
> a "better" overall solution would be useful.
>
> One might be able to build up some sort of useful table with wire 
> diameter, frequency, and spacing as the independent variables.  Or, 
> use the long form equations or a FEM program to calculate the 
> numbers, then find a simple interpolating function that works nicely.
>
> Or, just grind out the explicit equations for the whole thing.  Once 
> it's coded, everyone can use it.
>
> Probably the real question is to settle on what level of precision 
> you think you might need.  In general, I'd think that trying to get 
> better than 1% is not worth it, and at that level, a simple 
> approximation might work just fine.
>
> > Gerry R.