Tony said:So another way to look at is "effective impedance" from the ratio of utilization of AC/DC currents from geometry of skin effects, but not a change in sacred terms like Resistance or Conductance.
Thanks, "The Electrician"......your point above really adds weight to what I say in the top post, -that indeed the expensive_to_manufacture multi strand wire for the flyback smps seems like complete overkill...because as you show, the ratio of RAC/RDC is close to unity for a 26 gauge wire.If we take the radius of a 26 ga wire as .202 mm, the exact value of Rac/Rdc at 100 kHz is 1.0179. The expression g=1361/(6641*pi*sqrt(f)*R) gives 1.0212, quite close to the exact value.
FB said:don't see how rac=rdc for any given strand of wire...I mean, as we said, rdc always less than rac for any given wire.
These are not 100 percent accurate but they are very close to the NBS numbers. For example, the NBS number for #26 wire at 100kHz is close to K=1.0165 while this formula gives a result of K=1.013 which is close enough for now. At 300kHz the NBS shows about 1.135 while this formula gives a result of 1.11, which is still close enough given the other variables that we cant really account for anyway.
I might work on getting the formula down to 0.1 percent accuracy, but then again it doesnt matter much due to the other errors that will be present anyway.
Hi, Tony, Electrician, Kiss,
Very interesting replies, but i do differ on some points here.
I thought i made it clear that i dont mind using other terms, like conductance instead of conductivity. But we have to understand why we would want to use use a term before we can knock using it.
Why do we use 'resistivity' in copper wires at DC? It's because it is a property of the MATERIAL itself. It fits into the equation in such a way that it allows us to use it for different physical geometries and it is thus convenient, and the only time we have to change it for DC is when we change material. If we look at the skin effect, we see that copper has a different property when it is exposed to AC instead of DC. We dont have to consider anything but the material specs just as we do with DC, and it does not involve a phase change so the term impedance seems too inappropriate.
Sound too strange? I suppose it does, but that's nothing compared to new materials that will come out in the future which have non anisotropic conduction properties of who's resistivity will have to be defined in two or more different directions even for DC. I am guessing you will hate that stuff
But if that sounds too futuristic for you (we know those things take forever to become available sometimes) then simply look up some silver plated copper wire. A quote from a paper by an IREE member:
START QUOTE
Use of Silver Plating to Reduce Losses
Pure silver has a d.c. conductivity only 5 percent higher than that of copper and at
radio frequencies where the relative conductivity is proportional to the
square root of the d.c. values
END QUOTE
Note two things here:
1. The use of the word, "conductivity".
2. The use of the phrase, "d.c. conductivity".
They aren't talking about copper; they're talking about conductor-insulator composite. I've never said that there can't be some materials whose resistivity and conductivity change with applied frequency, but copper isn't one of them until the frequency reaches optical range. We aren't talking about frequencies that high in this thread. I don't know what the "percolation threshold" is, but I do know that the resistivity (and conductivity) of copper doesn't change at AC frequencies until the frequency reaches the optical range.Here's another quote:
Experimental study of the three-dimensional ac conductivity and dielectric constant of a conductor-insulator composite near the percolation threshold
Yi Song, Tae Won Noh, Sung-Ik Lee, and James R. Gaines
Phys. Rev. B 33, 904 – Published 15 January 1986
So before this and still yet i have no trouble using the word "conductivity" or "resistivity" to help explain this phenomenon. I really think it is silly to argue against such a thing as it just means opening your mind a little more about things that sound unfamiliar.
Also, this website gives RAC/RDC = 1 for 25 gauge copper wire.....I don't see how rac=rdc for any given strand of wire...I mean, as we said, rdc always less than rac for any given wire.
https://daycounter.com/Calculators/SkinEffect/Skin-Effect-Calculator.phtml
Hello again,
Here is an update on the previous simplified formula.
First, as before, we have:
x=2*pi*r*sqrt(2*f/172400)
r in millimeters and f in Hertz as before, and
the 172400 is just the common resistivity for copper adjusted from abohm centimeters and scaling the radius to millimeters.
Then this simple witch is a pretty incredible fit for 1<=x<=3 which is:
K=(2*log(10)*x^4+501)/(2*x^4+501)
and that is an incredible fit already, but using a simple optimization process we can get the error down to about one ten thousandth of one percent (a factor of 1e-6) using just six significant figures for the constants:
K=(2.30242*x^4+250.672)/(x^4+250.668)
(this comes about using x=1,2, and 3 for test points only)
and this is valid for 1<=x<=3, and for x=0 to x=1 K is almost zero anyway, and for x>3 we have an almost straight line at least up to x=100, so that is easy to calculate too.
Here's an even simpler version not too bad which is self complete:
K=(2.3*f^2*r^4+1.2e9)/(f^2*r^4+1.2e9)
again for 1<=x<=3, and again f in Hertz and r in millimeters.
I had a feeling this problem was well suited to a witch, i just at first didnt know which witch (pun intended)
Execellent job finding a simple but good approximation for the case where the skin depth is near the radius.
Here's a plot showing the exact Rac/Rdc compared to your approximation (exact in blue, your approximation in red):
Here's the error in your approximation:
Here's the error in your improved approximation. I see that the error is closer to zero at your test points 1, 2 and 3:
Here's the error in your self complete version for 26 gauge wire up to 500 kHz:
Any of these approximations you've derived is MUCH better than anyone would ever need for practical work. Excellent work, MrAl!
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