# Litz wire resistance

Discussion in 'Electronic Design' started by amdx, Mar 9, 2005.

1. ### amdxGuest

I'm looking for a formula that I can use to find the a.c. resistance of litz
wire at a frequency.
Also the a.c. resistance of a solid wire.

Specifically 12/36 litz
and #22 solid.

If possible please give me an example of it in use.
I have a book (Terman) that has two different formulas and I get the wrong
answer with both. (Ignorance on my part not the formula)

Mike

2. ### Winfield HillGuest

amdx wrote...
If the skin depth is more than the litz-wire constituent diameter,
the argument is that Rac = Rdc. I have found this postulate true
much of the time, but not all of the time. As for ac resistance
of ordinary (fat) wire, you have the skin-depth formulas. But in
real-world situations, with multiple-layer windings, the proximity
effect can increase the ac loss by up to 20x over simple skin-depth
calculations. See Snelling for a good discussion.

3. ### Terry GivenGuest

Fr = 1 + ((5*p^2-1)/45)*DELTA^4

p = no of effective layers

DELTA = layer thickness/skin depth = d/delta

d = copper wire OD*sqrt(pi/4) ie thickness of equivalent rectangular
conductor

This is only true at one frequency (ie that at which delta is
calculated). For arbitrary waveforms, you can do a Fourier transform to
get the coefficients of all the harmonics, and do lots of calculations.

Hurley et al published a cute way to extend this to any arbitrary
periodic waveform without knowledge of the fourier coefficients of the
waveform - "optimising the AX resistance of multilayer transformer
windings" IEEE trans. Power Electronics vol. 15 no. 2 March 2000, pp369-376.

multiply Dowells Fr (above) by:

[I'rms/(2*pi*Fo*Irms)]^2

Fo = fundamental frequency

Irms = rms value of I(t)

I'rms = rms value of dI(t)/dt

its pretty clear that for I(t) = Ipk*sin(2*pi*Fo*t) this scalar = 1

Cheers
Terry

4. ### The PhantomGuest

Charles Sullivan at Dartmouth and his students have done quite a bit
of work on this topic. Go have a look at their publications at:
http://engineering.dartmouth.edu/other/inductor/papers.shtml
The case of an isolated, solid, cylindrical wire is one of the few for
which an exact analytical solution exists. This result was known to
Maxwell.

Given: c, the conductivity of copper at room temp = 1/1724
f, the frequency in Hz
d, the diameter of the wire in inches
j, Sqrt(-1)
x, an auxiliary variable = 2.54 * Pi * Sqrt(2*j*f*c)

Then the ratio of the resistance of the wire at frequency f to the DC
resistance is given by:

Fr = Rac/Rdc = Re(x/2*(J0(x)/J1(x))

Re() means Real Part, and J0() is the Bessel function of the first
kind of order 0, J1() is the Bessel function of the first kind of
order 1.

For some reference points:
.1 inch dia wire @ 50 kHz, Rac/Rdc = 2.42058
.05 inch dia wire @ 50 kHz, Rac/Rdc = 1.33069
22 Ga wire @ 20 kHz, Rac/Rdc = 1.00467
22 Ga wire @ 50 kHz, Rac/Rdc = 1.02865
22 Ga wire @ 100 kHz, Rac/Rdc = 1.10733
22 Ga wire @ 300 kHz, Rac/Rdc = 1.59314

But, beware. As Win mentions, when you wind wire into a coil, the
proximity effect can make Rac/Rdc for the coil much larger than the
value for an isolated wire.

5. ### Fred BartoliGuest

Hi Win,

Do you remember in which instances this have been false? (maybe not true
litz wire but rather simply bundled wires)

I have a low noise preamp (2R equiv noise resistance, and next 0.2R/0.3R)
that have 1MHz BW. I'm planing to do the interconnect to the DUT with
specially made cable based on litz wire, but your comment worries me a
little.

Any thought?

BTW, did you receive the file I sent you a week ago?

6. ### Winfield HillGuest

Fred Bartoli wrote...
No, I generally use many-stranded litz wire from one of my custom-
made rolls. The cases I have seen where Rac for litz exceeds Rdc
are generally situations where the magnetic flux becomes highly
concentrated in one portion of the windings. E.g., in a toroid,
or in a many-layered coil situation where there would be a high
Rac proximity effect increase for ordinary thick wire.
I think you'll be OK. But don't know if litz is necessary, because
in a bridge, etc., fixture and wiring Rac can be nulled out.
Yes, now, where did I put that?

7. ### Fred BartoliGuest

Oh, that makes sense now. I guess you probably were in a situation where the
length of the "high concentration zone" was lower than the litz
"interwowing" step length (sorry I miss the word for this). This would
prevent the litz wire to do its work on that portion. Now, it you had
several layers, or several passages in the same zone, I think that would
have mitigated the effects on the average, unless you had a cumulative
effect due to bad luck (litz interwowing length about the wire length
between 2 passages).

Now that I understand why, I think too it'll be OK.
And yes, I'll need this kind of link. The preamp is not for a bridge but for
qualifying very low noise levels out of a low noise supply and after an RC
LPF (200nVrms over 1MHz BW).

I can send it again along with a pretty girl pic to make sure you'll
remember this time

8. ### John WoodgateGuest

I read in sci.electronics.design that Winfield Hill <[email protected]_rowland-
Is it still in the cake? (;-)

9. ### GenomeGuest

There is this nice librarian at the Central Application, Philips Product
Division Electronic Components and Materials. Eindhoven, The
Netherlands.....

Who, if you can find an appropriate e-mail and make the request will snail
mail you a copy of E.A.B No.3 parts 1 and 2.

That's Electronic Applications Bulletin. Vol 35, No.3 May 1978.

It's by J.Jongsma.

Bugger it..... I shall scan the bastard and post it in ABSE.... that'll be

I'll leave you know when it's up.

If you don't have access to the above and want me to e-mail the thing then
my e-mail addy as shown is valid.

DNA

10. ### amdxGuest

"Fred Bartoli"
Sounds like I'd like that file also!
Mike

11. ### amdxGuest

I'll be looking for it.
Thanks, Mike

12. ### GenomeGuest

Posted, you'll have to fiddle the page order. It's 18 jpgs.

DNA

13. ### The PhantomGuest

The Jongsma paper is good, but somewhat dated. He uses the Dowell method (see his
references) with modifications by Snelling.

Be aware that recent research:
http://thayer.dartmouth.edu/other/inductor/papers/newcalc.pdf
has shown that the Dowell method and Ferreira's method can have substantial errors.

Sullivan has published a further simplified treatment in the very recent paper:
"Simplified High-Accuracy Calculation of Eddy-Current Loss in Round-Wire Windings", IEEE
2004 PESC (Power Electronics Specialists Conference), Page 873.

14. ### leggGuest

I've been trying to get a look at this article by Jongsma, since '82.

Did they, perhaps, also copy you Publication#207 from '86?

RL

15. ### GenomeGuest

Unfortunately not, I didn't request it.

I've scanned the other one and posted it to the binaries newsgroup.

Like I said I e-mailed someone, it might have been the webmaster at Philips
website. He passed the request on and the librarian offered to snail mail me
photocopies.

DNA

16. ### Winfield HillGuest

Genome wrote...
You put up pages 162 and 163, and page 211. What about the rest of it?

17. ### GenomeGuest

Like it's my fucking problem that e-mail and newsgroup postings are shite.

You're the old bloke, why haven't you sorted it out?

http://genome.dna.tripod.com/jongsma/jongsma.html

DNA

18. ### Winfield HillGuest

Genome wrote...
Hey, good job, DNA!!! I combined the .gifs into two .pdf files,

Wonderful!

DNA

20. ### Terry GivenGuest

Hi Win,

thanks for that. Just one question - what do I do with the pile of ASCII
stuff I see in SED?

Cheers
Terry